International Macroeconomics a Concise Course for Advanced Undergraduates

International Macroeconomics A Concise Course for Advanced Undergraduates

Gregor W. Smith

April 2021 Preface

Welcome! This book studies issues in international macroeconomics (also known as international finance or open economy macroeconomics) and examines the interaction of national economies through international financial markets. We try to understand and explain things like capital flows (international borrowing and lending), sovereign debt and default, and how goods prices are related across countries. On the financial side we shall study floating nominal exchange rates (spot and forward), speculative attacks, and interest- rate differentials. We also learn about the history of the international monetary system and policy issues such as exchange-rate management, currency unions, and speculative currency crises. The emphasis is on predictions and empirical evidence. Overall the book aims to help you learn some new and thought-provoking things about the world, while putting to work your toolkit from previous economics courses. This course introduces these central topics in international macroeconomics for advanced undergraduate students. For the book to be useful, you should have courses in intermediate microeconomics and macroeconomics and an introductory statistics course. We shall make some of our discoveries using basic econometrics (in the form of linear regressions) and so we’ll rely on your knowing how to run a regression in any statistical software package you like. I should warn you about some omissions, first in content then in form. First, the emphasis is mostly on developed economies with open capital markets. There is not as much focus on EMEs (emerging market economies) as yet. Also, we do not assemble all the elements into a general-equilibrium model with multiple shocks. That interesting task is left to graduate-level courses. Second, the book does not yet include pictures that help us understand the theory. Sorry. This is a low-budget operation. We’ll fill those in as we go along. It also does not include time-series plots of data even though we discuss those at various points. That omission is by design. Books in this field rapidly become out-of-date, so we shall learn how to produce our own, up-to-date plots at several points and in the exercises. The table of contents is up next. Each chapter ends with some suggestions for further reading and references cited in the chapter. And then most chapters really end with a set of practice questions. Answers to the questions are collected at the end of the book. The book is based on a 12-week course (ECON 426) at Queen’s University, Canada. I am grateful to the cohorts of students in that course who have offered me feedback, whether in writing or in the form of a raised eyebrow or a gently rolled eye, that has helped me to improve the course. Now a warning, dear reader: The material in this book is copyrighted and is for the sole use of students registered in Economics 426. The book may be downloaded for a registered student’s personal use, but shall not be distributed or disseminated to anyone other than students registered in Economics 426. Failure to abide by these conditions is a breach of copyright, and may also constitute a breach of academic integrity under the University Senate’s Academic Integrity Policy Statement. Table of Contents

Page

1. Mysteries, Trilemma, and Tools 1 1.1 Four Mysteries of International Macroeconomics 1 1.2 The Open-Economy Trilemma 9 1.3 Ten Mathematical and Statistical Tools 13

Part A: Capital Flows

2. Basics of International Capital Flows 18 2.1 Accounting 18 2.2 Consumption Plans 24 2.3 Recipe 25 2.4 Results 27 2.5 Fiscal Policy 29 2.6 Large Economies 31 2.7 Global Imbalances 32

3. Sovereign Debt 38 3.1 Sanctions 38 3.2 Reputation 44 3.3 Debt Overhang and Debt Relief 47

4. Measuring Capital Market Integration 56 4.1 Savings-Investment Correlations 56 4.2 Home Bias 57 4.3 Consumption Correlations in Seven Steps 63

Part B: Exchange Rates and Asset Prices

5. Real Exchange Rates 72 5.1 Predictability and Persistence 72 5.2 PPP 77 5.3 Decomposing the Real Exchange Rate 83 5.4 China’s Exchange-Rate Policy 89 6. Floating Nominal Exchange Rates 94 6.1 ISO 4217, Data, Volume, and Floating Incidence 94 6.2 CIP and UIP 94 6.3 Monetary Model 97 6.4 Evidence on Floating Exchange Rates 102 6.5 Bubbles 105 6.6 Currency Transaction Tax 107 6.7 Testing UIP or Unbiasedness 108

7. Asset Pricing 115 7.1 Deriving the Euler Equation 116 7.2 Asset pricing using the Euler Equation 116 7.3 A Discount Bond 118 7.4 Equities and Risk Premia 120 7.5 Bond Default Risk 123 7.6 Nominal Bonds 125 7.7 The Term Structure of Interest Rates 125 7.8 Forecasting with the Term Structure 128 7.9 Risk Premia in the Foreign Exchange Market 130

Part C: Monetary Policy

8. Currrency Wars 136 8.1 Case Studies 136 8.2 Intervention 139

9. Fixed Exchange Rates 143 9.1 Operating a Fixed Exchange Rate System 143 9.2 Speculative Attacks 149 9.3 Exchange-Rate Crises in EMEs 155

10. Currency Unions and Exchange-Rate Regime Choice 160 10.1 Arguments for Exchange-Rate Stability 160 10.2 Arguments for Floating 163 10.3 Varieties of Exchange-Rate Stability 166 10.4 Actual Regime Choices 169

Answers to Exercises 173 Chapter 1. Mysteries, Trilemma, and Tools

We’ll do three things in this introductory chapter. First, we’ll look at four mysteries or puzzles, to introduce the field and to help you decide whether you want to read on. Second, we’ll learn about the open-economy trilemma, a feature of macroeconomic policy that helps us understand the international financial arrangements followed in different countries and over the span of history. Third, we’ll list some mathematical and statistical tools that will be used later on so you can have an early view of anything unfamiliar and (I hope) be reassured that nothing too daunting lies ahead.

1.1 Four Mysteries of International Macroeconomics

1.1.1 Why Doesn’t Capital Flow from Rich to Poor Countries?

Before we begin we need to take a moment to discuss the word capital. In economics it means things like machinery, equipment, and buildings: the K that enters the production function. This concept is sometimes broadened to include human capital, meaning accumulated skills or education. (In other social sciences people sometimes use the word diﬀerently, to refer to a social class.)

But, confusingly, capital can also mean ﬁnancial assets. It is this second meaning we usually study when we talk about capital ﬂowing betwen countries. Imagine a loan from one country to another. (Of course that could be used to buy or rent physical capital).

The same ambiguity arises with the word investment. The ﬁrst meaning refers to the change in the physical capital stock, I, that also shows up in the national accounts. The second meaning, which we focus on in this book, refers to the ﬂow of investment capital between countries. That occurs when citizens of one country buy assets located in another country.

On to our ﬁrst mystery. Hispaniola is an island in the West Indies that is shared by two countries: Haiti and the Dominician Republic. Take a moment and look up the population and GDP per capita of each country.

1 I would guess that you ﬁnd the populations are roughly similar but that GDP per capita on the east side of the island is roughly ten times larger than on the west side. It is very important to pause and think about why that is so. But for now let us consider the implications for capital ﬂows.

According to growth theory, rich countries have a high ratio of physical capital to labour (K/L) and poor countries have a low ratio. And thus the marginal product of capital (the slope of the intensive-form production function) is higher in poor countries than in rich countries which means the real interest rate is higher there too.

The basic growth model from intermediate macroeconomics thus predicts that financial capital should flow from a rich country to a poor one, to take advantage of the higher returns there. (The same model predicts labour should flow in the opposite direction, whether temporarily or permanently. We could also study the evidence for that occurring.) Here is the first mystery: In practice these capital flows are very small. For example, private capital inflows to Haiti are very small, and its GDP per capita does not seem to be converging towards the value in the Dominican Republic.

International capital ﬂows are enormous, as we’ll see in chapter 2. But they almost all are between relatively rich countries. In fact, there even are examples of large capital ﬂows from middle-income countries (like China) to rich countries (like the US).

Implictly here we’re deﬁning a mystery or puzzle as a discrepancy between an economic theory and the evidence. So I suppose this is just a polite way of saying the theory is missing something. When you think about what that thing is, I suspect two things come to mind.

First, perhaps returns (or the real interest rate) really are not higher in Haiti than they are in the Dominican Republic. In turn, that could be because Haiti lacks a third factor that matters in the production function, like good roads or reliable electricity. If you put this third factor in the production function and make it scarce (rather than relatively abundant, like labour) you can see that that might deliver a low marginal product of capital and real interest rate.

Second, perhaps there are aspects of the legal system or property rights or policy

2 barriers that explain the absence of capital flows. The lack of capital flows to North Korea is no mystery. One possible barrier is the presence of sovereign risk. That is a legal characteristic of loans to sovereign governments. In general, they cannot be pursued in the courts for repayment. So, that risk of course might explain why there are few loans to EME governments that have a history of defaulting. Argentina unfortunately is one example of such a country. You can see that banks would be unlikely to lend to sovereign governments if they thought they would not be repaid. And this is unfortunate since such loans might help finance important development projects. So this feature too might resolve the mystery, but it leads to an opposite one: Why are their any capital flows (lending) to sovereign governments? In chapter 3 we’ll study this issue in detail.

1.1.2 Why are Real Exchange Rates so Persistent?

The title of the second mystery might not sound gripping, so I’ll rephrase it. Why are prices persistently diﬀerent in neighbouring countries even when they trade a lot?

To see the evidence, we need to deﬁne the nominal and real exchange rates and intro-

duce some notation. The nominal exchange rate, labelled St is the price of a unit of foreign exchange (such as a USD) in period t. Please note that in this ﬁeld we usually measure that in units of the domestic currency. That means that an increase is a depreciation. In other words, it now costs more to buy a USD. And a decrease is an appreciation of the domestic currency. This takes a bit of practice because news sources often report the exchange rate in units of foreign currency, like USD. But it makes sense, for example, in

an ongoing inﬂation where all prices tend to rise and St does too.

While the nominal exchange rate is the relative price of two currencies, the real exchange rate is the relative price of two baskets of goods. It is deﬁned like this:

∗ StPt Qt = . (1.1) Pt

On the denominator is Pt, the price of a basket of domestic goods, perhaps measured by ∗ the CPI. On the numerator we take the foreign price Pt which is in USD, say, and convert it to units of domestic currency by multiplying by St which is domestic currency/foreign

3 currency. As a result, the ratio Qt is a pure number that tells us the relative price. If

Qt > 1 then things are more expensive in the US and if Qt < 1 then they are more expensive at home.

For more than a century economists have hypothesized that there is a tendency for

Qt to move towards 1. For CPIs this hypothesis is called purchasing power parity (PPP) while for the prices of individual goods it is called the la of one price (LOP). The idea is that demand will shift to the low-price location, supply will shift to the high-price location, and so prices will tend to be equalized when quoted in the same currency.

In the exercises, we ask you to collect some price indexes and an exchange rate, construct the real exchange rate, and create a time-series plot. You will see that the resulting series is not necessarily centred at 1. That is simply because the CPIs do not have the same base year, and so is not a rejection of PPP. But, you’ll also see that Qt is highly persistent, meaning that it tends to wander in large swings over time rather than being stable. That means that prices can be persistently higher in one country than another, even when the countires share a border and have a lot of trade. One also tends to observe that St and Qt tend to move together: The nominal and real exchange rates are highly correlated.

You might pause to think about any resolution of this puzzle that comes to mind. In chapter 5 we’ll look at the evidence and potential resolutions in detail.

1.1.3 Why are Changes in Exchange Rates so Unpredictable?

Before we look at the third mystery we need a warning about terminology. Exchange rates fall or rise. Currencies appreciate or depreciate. Thus one never says “The exchange rate appreciated.” That would be like saying “The temperature became hotter.”

On to the mystery. Use lower-case letters to denote natural logarithms, so st ≡ ln St. The rate of depreciation is:

St − St−1 ∆St ≡ ≈ ∆st. (1.2) St−1 St−1

4 To see the approximation at the end of equation (1.2) notice that it is exactly true for the inﬁnitesimal changes studied in calculus:

1 d(lnS)/dt = dS/dt. (1.3) S

Thus (1.2) is an approximation that will hold for small changes. The change in the log is approximately the growth rate. Usually we multiply these by 100 to report them in percentage terms.

As you might imagine, there is great interest in trying to predict exchange-rate

changes. Knowing something about future values of St would be very helpful to importers, to exporters, and to speculators. As we’ll see in chapter 7, the foreign exchange market is the largest and most liquid ﬁnancial market in the world.

The mystery, or at least very interesting fact, is that changes in st (appreciations or depreciations) are unpredictable, meaning that they are uncorrelated with anything known at the beginning of the period. To see how to report on or check this evidence for yourself, imagine collecting St and forming st. Then lag st to produce st−1, for example by copying a column in a spreadsheet and then pasting it shifted down one row. Finally, run this regression:

st = α + βst−1 + t. (1.4)

This is called a ﬁrst-order autoregression because you are regressing st on one lag of itself.

Remember t is an error term that allows for the fact this won’t ﬁt perfectly. It is not something you include in the regression as a variable.

Forecasting just involves running a regression and using ﬁtted values. So if we run this regression in past data then our prediction or forecast for next period will just be:

ˆ sˆt+1 =α ˆ + βst. (1.5)

I predict you will ﬁnd thatα ˆ is very close to 0 and βˆ is very close to 1. (Check these by constructing t-tests for those two hypotheses if you wish.) Equivalently, the scatter plot with st−1 (from last month or quarter) on the x-axis and st on the vertical axis will lie along the 45◦ line.

5 We can rearrange (1.4) by subtracting st−1 from each side:

∆st = α + (β − 1)st−1 + t. (1.6)

One could also estimate α and β with this second regression.

Now if β = 1 and α = 0 these two statistical models become

st = st−1 + t (1.7)

which is sometimes called a random walk and

∆st = t. (1.8)

Equation (1.7) says that st can be predicted with its own past value, st−1. Notice that you do not need to know any coeﬃcients. So if someone asks you to forecast the exchange rate you simply ask them the current value then say “What a coincidence! That is my forecast.”

Equation (1.8) says that the change in the log exchange rate (or rate of appreciation)

cannot be predicted though. There is no correlation between ∆st and st−1. So knowing the current value tells us nothing about the subsequent direction or scale of change. That is another reason this is called a random walk. Increases and decreases are equally likely.

I’ve suggested that β = 1 (or at least that there is little evidence against that null hypothesis, measured with a p-value). But let us imagine that β < 1 and see what that would imply. Suppose for example that β = 0.9, so

st = α + 0.9st−1 + t, (1.9)

and, subtracting st−1 from each side again:

∆st = α − 0.1st−1 + t. (1.10)

In this case the change would be predictable, because it is correlated with the lagged level. This statistical model or regression is sometimes said to display mean reversion meaning

6 that if the exchange rate has been high it tends to fall and if it has been low it tends to rise.

You can see that pattern appears in the negative coeﬃcient on st−1 in equation (1.10). Low values will usually be followed by increases and high ones by decreases. Mean reversion appears in lots of random variables—like one’s body temperature—but not usually in exchange rates.

We’ve been using a linear regression to figure out how to make predictions or forecasts or, in the case of ∆st, show that something is unpredictable by showing a regression coefficient (here βˆ − 1) is not statistically different from zero. But what about trying to forecast an appreciation or depreciation with other regressors? Imagine running a regression like this:

∆st = α + βst−1 + γxt−1 + t. (1.11)

Here x is some other variable and note we use the lagged value. A further lag like st−2 would also be a candidate one could investigate. If you foundγ ˆ was signiﬁcantly diﬀerent from zero then that would imply some predictability.

Many, many studies have examined regressions like (1.11). Only rarely does one find a significant regression coefficientγ ˆ. This regression is easy to run and predicting exchange- rate changes would be very profitable, so it is tempting to do a search of various databases

until one ﬁnds one or more variables xt that are statistically signiﬁcant.

But let me warn you of a statistical pitfall here, in the form of type I error or the probability of a false positive result. Recall from statistics that something like a t-statistic is a random variable. Under the null hypothesis that γ = 0, the t-statistic to test that hypothesis (which is justγ ˆ minus zero and divided by its standard error) can lie above or below a critical value like 1.96 5% of the time simply due to randomness. This same syndrome arises when one searches past history for correlations, or signiﬁcant, large values γˆ. Using a test with probability of type I error of 5% (also called test size), if you study

20 candidates for xt you will ﬁnd 1 (5% of 20) of them is statistically signiﬁcant. This search is sometimes called data-mining, a term used here in a pejorative sense. Analysts sometimes report these patterns but then they turn out not to be present once one tries to use them in later predictions.

7 I don’t know the answer to the question of why ﬂoating nominal exchange rates seem

to follow random walks. We’ll look at models of st in chapter 6 and see how economists have tried to explain them. Meanwhile, perhaps the real mystery is why so many commentators make predictions for these changes. You can safely disregard those.

1.1.4 Why do High-Interest-Rate Currencies often Appreciate?

We often refer to currencies with their ISO 4217 codes: EUR, USD, JPY, GBP, CAD, and so on. For the currencies I’ve just listed the random walk model is very difficult to beat. But there are some currencies where there seems to be a predictable rate of depreciation, due to ongoing, looser monetary policy. The exchange rate is the relative price of two monies so it makes sense that if one is being printed much more rapidly that it will tend to fall in value. Two examples are the South Afircan rand (ZAR) and the Turkish lira (TRY). Please take a moment and look at the Economic & financial indicators near the back of any issue of The Economist. You will see that these currencies tend to depreciate against the USD.

From the same tables you will see that South Africa and Turkey tend to have higher interest rates. To see that this makes sense, suppose that in the US the return on holding ∗ a domestic bond is it. Use a to denote foreign variables. The return on holding a foreign bond is:

∗ it + ∆st+1. (1.12)

An investor parking money in the foreign currency for a year will earn a higher interest rate but also will experience a depreciation. If we are vieweing this from the perspective of an investor in the US, then they will look at the high-frequency version tables in The ∗ Economist and see that (a) it > it, and (b) ∆st+1 is negative. In other words, the USD will appreciate against the ZAR or TRY in a typical time span.

Now a widely-studied hypothesis says that these two eﬀects oﬀset each other on average:

∗ it = it + Et∆st+1, (1.13)

8 where Et denotes the expectation or forecast at time t. This is called uncovered interest rate parity (UIP) and we’ll study it in chapter 6. The idea behind the hypothesis is that if this were not true then funds would ﬂow to make the two sides equal. Or, interest rates in accounts denominated in TRY are higher to compensate investors for the expected depreciation of that currency.

Now for the mystery. Even though I’ve said that UIP (1.13) explains the interest- differential for Turkey or South Africa, it does a surprisingly bad job for countries without high monetary growth and high inflation. A classic example involves Australia and Japan where, for many years, the Australian interest rate was well above the Japanese one on a similar deposit. According to UIP, the AUD should have steadily depreciated against the JPY. In fact, in many years it appreciated. That yielded investors a bonus. They borrowed in JPY and invested in AUD (an action called a carry trade) and were rewarded with both a higher interest rate and an appreciation. The mystery is how such profits are possible when these markets are so liquid.

1.2 The Open-Economy Trilemma

Having looked at four examples of empirical questions we study, we next turn to some important context for policy. One of our goals in this course is to understand how international financial arrangements operate in different countries and at different time periods. Ultimately, we would like to study the normative question of what arrangements might be best, and whether those are the same for diferent countries. To understand these policy choices, imagine looking at three aspects:

(a) a ﬁxed exchange rate;

(b) international capital mobility;

Why might you view each of these as a good thing? Often, economists have argued that a fixed or stable exchange rate (or even a common currency) promotes trade by eliminating the uncertainty from exchange-rate fluctuations. And they’ve argued capital mobility might be good because it allows funds to flow to where returns are highest (so

9 that it helps borrowers in poor countries and savers in rich ones) or allows savers to diversify their portfolios. And, ﬁnally, they’ve argued that monetary policy autonomy is good because it allows the cntral bank to use the tools of domestic monetary policy to combat local recessions.

The open-economy trilemma points out that a country can choose only two of these three features. To see why, imagine that you have two of the features and want to add the third one. Suppose that you have a fixed exchange rate and monetary policy autonomy. A recession begins in your country only, and so you wish to put your autonomy to work and use monetary policy to lower interest rates. What happens? With an open capital market, capital leaves your currency in search of higher interest rates in the rest of the world. As a result, your currency depreciates and so is obviously no longer fixed. The conclusion is that you need capital controls (limits on capital mbility) if you have a fixed exchange rate and monetary policy autonomy.

By the same reasoning, if you have monetary policy autonomy and an open capital market then you will need to experience currency ﬂuctuations. And, if you have a ﬁxed exchange rate and an open capital market then there will be no scope for an independent, domestic monetary policy.

The ideal choice depends on empirical evidence we shall study in this course. Let us look at three examples. First, if an analyst found that exchange-rate stability had a large effect on trade volume (and thought that in turn promoted efficiency or productivity) then presumably they would recommend a fixed exchange rate. In chapter 10 we’ll see how to study and assess some of the evidence on this effect. Second, if you calculated that the gains from being able to borrow or lend internationally—for example through diversification—were very large then you might recommend open capital markets. We’ll see how to assess those diversification gains in chapter 4. And finally, the benefits of monetary policy autonomy depend on (a) an alert and capable central bank and (b) a significant local component to recessions. A country without an experienced monetary authority or where the business cycle was strongly correlated with that in the US, say, might decide to fix its exchange rate with the USD or even adopt the USD as its currency

10 (dollarize) rather than having a ﬂoating exchange rate or limiting capital ﬂows.

In practice the choice of system no doubt depends on past history and on many political considerations, so trying to rationalize what countries actually choose may be a diﬃcult task. But at least we can use the trilemma to understand the constraints imposed by their diﬀerent choices, in the cross-section of contempoary countries.

We also can use the trilemma to better understand history. Building up your institutional and historical knowledge in international ﬁnance is interesting and rewarding, but it also can be a bit daunting. The trilemma serves as a useful way to organize information about both past and current policy regimes. Let us look at some examples next.

1. The classical gold standard, from 1870 to 1914, is our first example. In this period, many countries fixed the value of their currency in terms of gold, which means that they were also fixed relative to one another. And this was an era of very open international capital flows. For example, there were large outflows from northern Europe to finance EMEs like Argentina, Canada, and Russia. From the logic of the trilemma then, you can tell that there were not really independent monetary policies during this period.

2. During the 1914–1918 War most countries left the gold standard and also had capital controls. Then during the interwar period, from 1918 to 1939 a number of countries experimented with ﬂoating exchange rates. One reason was that in the early 1930s a series of countries allowed their currencies to depreciate as they followed expansionary monetary policy to try to combat the Great Depression. According to the trilemma, then, they could have had open capital markets, though in practice these markets were not as open as they had been in the late 19th century.

3. Our next example is the Bretton Woods system, in eﬀect from 1950 to 1970. The name comes from a town in New Hampshire where delegates negotiated this system in 1944. (The head of the Canadian delegation was W. A. Mackintosh, later Principal of Queen’s Univeristy.) This system was designed to restore international trade partly through exchange-rate stability. And it allowed for some independent monetary policies, as memories of the need for such tools were prominent after the 1930s. Thus you can tell

11 that capital markets were not open between countries. They used a variety of taxes and legal limits to control capital ﬂows and hence maintain ﬁxed exchange rates, though there were periodic revaluations or devaluations. These controls lasted a long time: up to the late 1970s in the UK and the early 1980s in France.

4. As capital markets became more open in the 1960s and then early 1970s the Bretton Woods system broke down because maintaining fixed exchange rates required that countries followed similar monetary policies. In fact they were not prepared to surrender monetary policy autonomy. For example, the US followed a more expansionary policy than West Germany did, and so the USD tended to depreciate against the deutsche mark, violating the Bretton Woods settings. Thus many countries switched to systems with open capital markets and floating exchange rates. That same system remains in effect today for many wealthy countries like the US, Japan, the UK, Canada, Sweden, Korea, and Switzerland.

Residents of these countries probably take this outcome for granted, so it is worth remembering that many countries in the world still have controls on international capital ﬂows. When there are restrictions on foreign exchange transactions to buy and sell assets we say that the relevant currency is not convertible on capital account. Sometimes there are limits on foreign exchange transactions for goods and services too, in which case the currency isn’t converible on current account either.

5. In general, policymakers tend to be concerned about big fluctuations in the exchange rate. (And you might pause to ask yourself why.) And, after the widespread float in the early 1970s, exchange rates were more volatile than most people expected. They still are very difficult to explain (as we’ll see in chapter 6) and the changes are virtually impossible to predict, as we saw in section 1.1.3.

Thus, there also have been new cases of ﬁxed exchange rates and open capital markets, just as in the gold standard period. The most well-known example is the euro area, where a common currency is designed to promote trade among members, capital markets are very open, and thus members do not follow independent monetary policies but instead use the same currency.

6. Our ﬁnal example focuses on the US and China today. We know that these two

12 large economies have independent monetary policies. And they do not have open capital markets. In particular, there are restrictions on both inflows and outflows in China. Thus we know from the trilemma that there is scope to stabilize the exchange rate. And the monetary authority in China (the PBOC) is often described as doing this, unofficially. For example, it has at times kept the value of the CNY low so as to promote exports and at other times has acted to stabilize the exchange rate against a basket of other currencies. Some commentators have called this syetm Bretton Woods II.

1.3 Ten Mathematical and Statistical Tools

Before we conclude this chapter let us brieﬂy list some of the tools we’ll need for the rest of this book.

1. A calculator. You’ll need to be able to do calculations on your phone or calculator (for tests perhaps) both to ﬁgure out predictions like the UIP formula above and to solve more complicated economic models, starting in chapter 2.

2. The log approximation. For small values of x you will ﬁnd that ln(1+x) ∼ x. This arises simply because the ln function crosses the x-axis at 1 and there has a slope near 45◦. You can check this approximation using your calculator.

3. Rules for growth rates. We’ve already seen that for inﬁnitesimal changes

dx/dt/x = d ln x/dt (1.14) or in discrete time

∆xt/xt−1 ∼ ∆ ln xt. (1.15)

Again you can check that approximation with your calculator or in a spreadsheet if you wish. We also need the rules for growth rates of products and ratios, which you can derive using the product and quotient rules of calculus. For products, d(xy)/dt dx/dt dy/dt = + . (1.16) xy x y i.e. the growth rate of a product is the sum of the growth rates. For ratios, d(x/y)/dt dx/dt dy/dt = − . (1.17) (x/y) x y

13 i.e. the growth rate of a ratio is the growth rate of the numerator minus the growth rate of the denominator. These two rules are approximate for discrete growth rates.

4. Geometric series. Several times we’ll need to rely on the formula for an inﬁnite, geometric series. Suppose 0 < λ < 1, then:

1 1 + λ + λ2 + ... = . (1.18) 1 − λ

Suppose instead that we begin part way along this series:

λ2 λ2 + λ3 + λ4 + ... = λ2(1 + λ + λ2 + ...) = . (1.19) 1 − λ

That means that we also can ﬁnd an expression for the ﬁrst few terms if the series is truncated:

1 1 + λ = − λ2(1 + λ + λ2 + ...) = (1 − λ2 + λ + λ2 + ...). (1.20) 1 − λ

5. Moments of functions of random variables. Your introductory statistics textbook has a section on this topic, but here are some reminders. First, suppose that x and y are random variables and y = a + bx. Then for means or expectations: E(y) = a + bE(x). For 2 2 2 variances: σy = b σx. That means that in standard deviations σy = bσx.

Next suppose that we study a third random variable w which is a combination of x and y: w = cx + dy (1.21) where c and d are numbers. Then E(w) = cE(x) + dE(y), which is easy to remember, while 2 2 2 2 2 σw = c σx + d σy + 2cdσxy, (1.22) which is not quite as easy to remember. Here σxy is the covariance, which we can also write as ρxyσxσy where ρxy is the correlation. This formula is vital when we study portfolio decisions in chapter 4.

Finally, on a few occasions we’ll see examples where y = f(x) and f is not a linear function like the one above. In general we cannot ﬁgure out the mean of y without

14 knowing the complete distribution function for x. But we can know a result called Jensen’s Inequality. That says that:

E[(f(x)] > (<)f[E(x)] if f 00(x) > (<)0. (1.23)

For example, suppose that f(x) = 1/x. Here f 00(x) = 2x−3 > 0 so E(1/x) > 1/E(x). This reminds us that we cannot simply reverse the order of the f and E operators for a nonlinear function.

6. The covariance decomposition. We’ll need one more statistical result when we study asset-pricing. Remember the deﬁnition of a covariance:

cov(x, y) = E[x − E(x)][y − E(y)]

= E[xy − yE(x) − xE(y) + E(x)E(y)] (1.24) = E(xy) − E(y)E(x) − E(x)E(y) + E(x)E(y)

= E(xy) − E(x)E(y).

Rearranging this formula tells us about the expected value of a product of random variables:

E(xy) = E(x)E(y) + cov(x, y). (1.25)

7. Solving a diﬀerence equation backwards or forwards by repeated substitution. We’ll see how to do this in chapters 5 and 6. It’s simple.

8. Guess-and-verify solutions to pricing problems. We’ll see how to do this in chapter 6. It’s simple too and requires only a bit of patience.

9. Scatter plot. In section 1.1 we discussed running a linear regression in Excel or Stata or some other package. When we have only one regressor we can see what the regression line looks like using a scatter plot in Excel, so we’ll sometimes need to report our ﬁndings that way.

10. Time-series plot. When we track variables over time (for example the log real exchange rate qt) we’ll need to produce a time-series plot in Excel, with the values of the variable on the vertical axis and time units labelled on the horizontal axis. Producing

15 these plots is always a good idea even if you do not use them in your reporting, simply as a way to check for missing data or erroneous data transformations, before calculating statistics. Colleagues, plot your data!

Further Reading

For more detailed background I can recommend the macroeconomics half of Paul Krug- man, Marc Melitz, and Maurice Obstfeld’s International Economics, or Robert Feenstra and Alan Taylor’s International Economics. Another great book is International Macroe- conomics by Stephanie Schmitt-Grohé,Martín Uribe, and Michael Woodford. These three books are very different but all excellent, so if you can read some of any of them that would be helpful.

If you want to know the history of Canada’s international monetary arrangements, please download a free copy of James Powell’s (2005) A History of the Canadian Dollar (Ottawa: Bank of Canada).

16 Part A: Capital Flows

17 Chapter 2. Basics of International Capital Flows

Let us begin by asking why we spend time studying international capital ﬂows. They matter for:

(a) growth in borrowing countries (e.g. in the history of Canada, Latin America, Australia, Sweden)

(b) consumption growth in lending countries (e.g. Norway, Singapore, Switzerland)

(d) the possibility of sudden reversals.

Economic & ﬁnancial indicators at the back of each issue of The Economist show enormous variation over countries in the direction and scale of these ﬂows. And there is great persistence over time. Countries that lend internationally tend to do so year after year. For example, for the last 20 years the US has borrowed several hundred billion dollars each year. Why? It is not easy to see a pattern in which countries lend and which borrow. Why do China, Norway, and Russia lend while India, Australia, and Poland borrow?

We measure capital ﬂows with the current account balance, which we’ll denote CA. It is tempting to think that a current account deﬁcit is somehow imprudent. As we’ll see that is not necessarily the case. And we’ll also see that we can avoid some mistakes in policy analysis simply by remembering national accounting. Here is an example, overheard by a guest commentator on a TV network which shall remain CNN.

Free trade between the US and Mexico is bad for the US. Wages are much lower in Mexico than in the US. That fact means that (a) Mexico will export to the US much more than it imports from the US, and (b) US companies will invest in Mexico, not in the US, to take advantage of the low wages there, so capital will leave the US.

Do you see anything wrong with this statement?

2.1 Accounting

2.1.1. The Budget Diﬀerence Equation

18 To form a model of capital ﬂows (or the current account) let us begin with a two- period model with no uncertainty. The periods are labelled 1 and 2, and past debts are inherited from time 0. To study budgets, begin with the private sector. It has income Y and spends C on consumption and I on investment in physical capital after paying taxes

T . Its stock of assets is labelled Ap (where the A stands for assets, just as a mnemonic) and evolves like this:

Ap1 = (1 + r)Ap0 + Y1 − T1 − I1 − C1. (2.1)

Meanwhile, government debt or assets evolves like this:

Ag1 = (1 + r)Ag0 + T1 − G1. (2.2)

For the record, T − G is called the primary budget surplus or deﬁcit while T − G + rA is usually just called the budget surplus or deﬁcit. The primary version does not include interest payments.

For a country, assets or debt are given by A = Ap +Ag so adding (2.1) and (2.2) gives:

A1 = (1 + r)A0 + Y1 − C1 − G1 − I1. (2.3) or

∆A1 = rA0 + NX. (2.4) where NX = Y − C − I from national accounting. We sometimes uses TB (trade balance) as a synonym for NX (net exports).

Remember the difference between GDP and GNP? GDP plus net factor payments gives GNP: Y1 + rA0. Net factor payments includes things like wages paid to visiting opera singers or hockey players but the largest component consists of interest and profits that flow across borders, here labelled rA0. National saving S is given by the income earned by residents (that’s GNP) minus their spending on consumption and investment.

S1 = Sp,1 + Sg,1 = rA0 + Y1 − C1 − G1. Thus

∆A1 = rA0 + NX = S1 − I1 = CA1 = −KA1. (2.5)

19 Thus the change in net foreign assets equals saving minus investment which equals the current account balance which also equals the negative of the capital account balance KA. (Remember that a country with a current account surplus is exporting capital. Capital flows out which implies a capital account deficit.) So the current account is (a) savings minus investment; (b) the trade balance plus investment income; (c) the change in NFA (minus the capital account). This important identity tells us that we can predict or explain capital flows (changes in net foreign assets) by studying things that influence savings and invesment.

Detour: Debt-to-Income Ratios. While we’re working with this equation let us detour to look at the government version. It applies for any two adjacent time periods, say from t − 1 to t instead of 0 to 1. So, here it is again:

Agt = (1 + r)Agt−1 + Tt − Gt (2.6)

Suppose that the real interest rate is given by r = i − π where π is the inﬂation rate.

Suppose that output Yt grows at rate g. Challenge: Try to show that: A A T − G gt ' (1 + i − π − g) gt−1 + t t . (2.7) Yt Yt−1 Yt

This equation describes the evolution of the debt-to-income ratio for the government. That ratio is a standard way of reporting on the size of public debt. Using equation (2.7) lets us see how it changes from year to year. Imagine that you know this year’s primary surplus, debt, and GDP. If you forecast g, i, and π over the coming year you then can forecast the debt-to-income ratio. And you can simply enter this formula in a spreadsheet to project the ratio into the future.

This tool is helpful to understand the European debt crisis of 2010. Countries like

Greece and Portugal had Ag negative (government debt) and large as a ratio to Y . And they were growing slowly. Projections showed the absolute value of the ratio rising into the future, which led ﬁnancial markets to consider the possibility of defaults. The idea is that if the debt becomes very large as a share of GDP then paying the interest will be too costly and the government will default (or reschedule) its debt.

20 Notice that our example (2.7) assumes that i is ﬁxed. What would you expect to happen to the interest rate at which a government can borrow as its debt-to-income ratio rises? How would that feedback aﬀect the path of the ratio?

Looking back at the history for Greece, it essentially did default in that much of its debt payments were renegotiated or rescheduled. And it received a large bailout from the IMF and the EU. But it continues to face a high debt-to-income ratio that constrains decisions about G and T .[End of the Detour.]

Back to our theory. Assume r is given (though it can change). This is a small, open economy which means that events within it do not aﬀect the world real interest rate. Remember that t = 0, 1, 2. The idea is to make this theory as simple as possible by limiting

the number of time periods we need to keep track of. That means A2 = 0: There will not be any outstanding loans at the end of the last time period.

Using equation (2.5):

∆A2 = 0 − A1 = CA2, (2.8)

but

A1 = A0 + CA1, (2.9)

A0 = −CA1 − CA2. (2.10)

A country that starts with debt (A0 < 0) must run current-account surpluses (in total

in the future) while a country that starts with assets (A0 > 0) can run current-account deﬁcits. Equation (2.10) will be a useful aid in calculation. If we know A0 and CA1 we

can use it to ﬁnd CA2.

Again start with the basic equation:

A2 = (1 + r)A1 + TB2 (2.11) 2 = (1 + r) A0 + (1 + r)TB1 + TB2.

We get this result by lagging the whole equation and substituting for A1. That is sometimes called solving a diﬀerence equation backwards (thing 7 in section 1.3). But remember that

21 A2 = 0, so rearranging gives us: TB TB A = − 1 − 2 (2.12) 0 (1 + r) (1 + r)2 which is called a present-value version of the budget constraint.

This accounting will serve us well in the theory coming up later in this chapter. But before leaving the topic of accounting, we need to look at three important upgrades to this basic version.

2.1.2 Beyond the Basics: Large Gross Positions

A very important trend over the past several decades is that gross ﬂows and stocks are much larger than net ones, and have grown much more rapidly. Assets and liabilities can diﬀer by (a) type of investment and (b) currency. A country can simultaneously lend and borrow and thus have both assets and liabilities. CA measures its net lending or borrowing and B records its net foreign assets (NFA).

Example 1: Suppose someone has assets of zero and liabilities of 100. Thus A0 = −100.

Suppose r = 0.03 and NX = −5. Then ∆A1 = −3 − 5 = −8 and so A1 = −108.

Example 2: Imagine neighbours who tell you they have substantial debts yet net investment income. How is this possible?

To see this example in detail, imagine assets of 100 (as foreign direct investment, FDI) and liabilities of 200 (as bonds). So again A0 = −100 just as in Example 1. But suppose that on assets r = 0.10 and on bonds r = 0.03 again. What is net investment income

(NFP)? If again NX = −5 what is ∆A1?

What is the name of these neighbours?

In recent years the size and sustainability of the US current account deﬁcit has been much-debated by economists. We’ll discuss this issue in section 2.7. But there is an aspect of measurement that also is worth thinking about carefully.

Here is the issue. The US has been running large current account deﬁcits and so

has become a net debtor. But its annual net factor payments—the term rA0 above—is

positive. How can A0 be negative if rA0 is positive?

22 Persistent diﬀerences in rates of return on assets and liabilities may explain the facts for the US. That is the feature built into Example 2. That means that the US acts as a giant hedge fund, borrowing from the rest of the world at a low interest rate and investing in high-yielding projects overseas.

However economists Ricardo Hausmann and Frederico Sturznegger have argued that the stock of debt must be mismeasured and that A0 must be positive. They call the missing foreign assets ‘dark matter’, for we can observe it only because of the income ﬂow.

Tax havens are one source of mis-measurement. For example, US investors may hold bonds issued in the Cayman Islands by Vale Overseas Ltd, a subsidiary of Brazil’s Vale SA. Official data lists the Cayman Islands as the destination of the capital flow, whereas the ultimate destination is in Brazil. Similarly, large firms in China have a corporate structure called a variable interest entity (VIE) . One effect of these structures is that foreign investors own shares in a tax-haven-based shell company, even though the ultimate equity position is in China. In these cases US assets in Brazil and China will be under- estimated.

Finally, we’ve seen that foreign assets and liabilities can diﬀer in their interest rates. They also diﬀer in currency.

Example 3: Suppose (realistically) that Canada has liabilities in CAD, but assets denominated in foreign currency. What happens to NFA when the CAD depreciates? You can see that a depreciation reduces the value of liabilities relative to assets, so the value of NFA rises, which may act as a stabilizer in a ﬁnancial crisis. Note that some EMEs have the opposite mismatch: Their liabilities are largely in foreign currency (USD). For them, a depreciation can sharply reduce national wealth.

Overall this valuation eﬀect, with large gross positions, can lead to a large change in NFA in addition to the change caused by the CA. Thus we can upgrade our accounting to:

∆A = CA + valuation eﬀects. (2.13)

These valuation eﬀects can be due to ﬂuctuations in exchange rates or stock markets or real estate markets.

23 In the longer term though, changes in NFA are driven mainly by CA, so next we return to constructing a theory to try to explain that important measure.

2.2 Consumption Plans

The CA depends on savings decisions so let us try to model those. Consider two wats to do this:

Consumption-smoothing: C1 = C2. This simply asserts that households smooth consumption over time.

Utility maximization: Maximize

U(C1) + βU(C2). (2.14)

Here U is a utility function and β is called a discount factor. It allows for the possibility that future utility might be down-weighted relative to current utility. So β ∈ (0, 1].

Either way, for simplicity ﬁrst suppose that there is no investment (so this is an ‘endowment economy’) and no government spending. Then the present-value budget constraint (2.12) becomes: Y C (1 + r)A + Y + 2 = C + 2 . (2.15) 0 1 1 + r 1 1 + r

Please take a moment and maximize (2.14) subject to (2.15), perhaps by forming a Lagrangean. You will see that the ﬁrst-order conditions combine to give:

0 0 U (C1) = β(1 + r)U (C2). (2.16)

This very important result is called the Euler equation, named for the Swiss mathematician Leonhard Euler (1707–1783) and pronounced ‘Oiler’. It basically says that the slope of the

budget line is equal to the slope of the indiﬀerence curve between C1 and C2.

Let us see what the Euler equation looks like for several choices of the utility function U.

Example 1: First suppose that utility is quadratic: b U(C) = C − C2. (2.17) 2

24 Notice that we can get U 0 > 0 and U 00 < 0 so this indeed looks like a concave utility function. The Euler equation is:

1 − bC1 = β(1 + r)(1 − bC2) (2.18)

Finally, if by chance β = 1/(1 + r then C1 = C2 which is consumption smoothing. So we’ve shown that there is a version of the utility maximization problem that delivers that result.

Example 2: Suppose that U(C) = ln(C) (log utility) so:

1 1 = β(1 + r) . (2.19) C1 C2

Example 3: Suppose that we have U(C) = C1−α/(1 − α). This is sometimes called power utility or CRRA utility, because α > 0 is the coeﬃcient of relative risk aversion. The Euler equation is:

−α −α C1 = β(1 + r)C2 . (2.20)

Notice that Example 2 is a special case of Example 3 with α = 1.

2.3 Recipe

In this section we pause and outline the recipe for studying the two-period model of the current account. You’ll ﬁnd that there are a lot of labelled variables. The good news is that most of them are exogenous, which means that the model is easy to solve and that it allows us to study many diﬀerent scenarios. So here is the recipe we’ll follow ...

Step 1: Budget

A1 = (1 + r)A0 + Y1 − C1 (2.21) and

A2 = (1 + r)A1 + Y2 − C2, (2.22)

25 where for simplicity we begin with an endowment economy with no investment or govern-

ment. Remember that A2 = 0 so combining these equations to remove A1 gives us:

Y C A (1 + r) + Y + 2 = C + 2 . (2.23) 0 1 1 + r 1 1 + r

We inherit A0 from the past, take r as given from the rest of the world, and take Y1 and

Y2 as given, but to solve for the two values of consumption (endogenous variables) we need more information in the form of a second equation from ...

Step 2: Consumption Plans

With consumption smoothing we simply assume C1 = C2. With utility maximization we solve this problem:

C Y L = U(C ) + βU(C ) + λ C + 2 − (1 + r)A − Y − 2 (2.24) 1 2 1 1 + r 0 1 1 + r

Diﬀerentiate with respect to each of C1 and C2 to give:

0 U (C1) + λ = 0, (2.25a) and 1 βU 0(C ) + λ = 0. (2.25b) 2 1 + r (Of course diﬀerentiating with respect to λ gives us the budget, already known from step 1.) Combining these two conditions to remove λ gives the Euler equation we’ve already seen: 0 0 U (C1) = β(1 + r)U (C2). (2.16)

In this recipe we’ll use the log-utility version (2.19).

Step 3: Solve for Consumption(s)

Working with the Euler equation (2.19) and the budget (2.23) to solve for C1 and C2 gives us ﬁrst: 1 Y C = (1 + r)A + Y + 2 . (2.26) 1 1 + β 0 1 1 + r

26 Be sure you can also ﬁnd C2: β(1 + r) Y C = (1 + r)A + Y + 2 . (2.27) 2 1 + β 0 1 1 + r

These are called consumption functions. Here they describe household saving behaviour and how that depends on the real interest rate and the path of income.

Step 4: Remember National Accounting Deﬁnitions

With no investment or government spending the trade balance is given by Y − C. Capital ﬂows (international borrowing or lending) are given by the current account balance:

1 Y CA = rA + Y − C = rA + Y − (1 + r)A + Y + 2 . (2.28) 1 0 1 1 0 1 1 + β 0 1 1 + r

We now can study this for the eﬀects of changes in r or in Y1 and Y2.

Be sure you can ﬁnd CA2 also. A simple check is available in numerical problems:

A0 + CA1 + CA2 = 0. (2.29)

2.4 Results

To make predictions about the current account and capital ﬂows, we simply follow the recipe. For example, suppose we use consumption-smoothing instead of the log-utility Euler equation. Then the solution is:

1 + r Y C = (1 + r)A + Y + 2 , (2.30) 1 2 + r 0 1 1 + r so that 1 + r Y CA = rA + Y − (1 + r)A + Y + 2 (2.31) 1 0 1 2 + r 0 1 1 + r and one can work out what happens in period 2 also.

Prediction 1: Temporary changes in income have larger predicted eﬀects on capital ﬂows than permanent ones.

To see this result, go to the CA (2.31) and read oﬀ the total eﬀect of an increase in

Y1. That is a temporary change. Next, suppose that Y2 rises at the same time, by the

27 same amount. That is a permanent change. The CA rises in the ﬁrst case. It does not change in the second case.

The logic is from the permanent-income hypothesis. For a permanent change, consumption rises by as much as income so the CA does not change.

A hurricane or typhoon is an example of a temporary, negative output shock. The theory predicts a country should borrow to smooth out the eﬀect on consumption. It does that by running a current-account deﬁcit.

Notice that a key feature of this shock is that it is country-specific. A shock must have a differential impact on countries to affect capital flows. One might wonder whether a pandemic has this feature. Suppose that it results in a sharper fall in income in one country (because of the fall in labour supply), relative to others. Then the theory again predicts that country’s CA balance would fall.

So far we implicitly assume the goods produced and consumed are identical. Suppose that they diﬀer and P is the relative price of output (export) goods in terms of consumption (import) goods. The national budget now is: P Y C (1 + r)A + P Y + 2 2 = C + 2 . (2.32) 0 1 1 1 + r 1 1 + r

Prediction 2: A temporary improvement in the terms of trade has a larger eﬀect than a permanent one does.

You can see this results using the same method we used above. Perhaps this explains some of the large capital outﬂows from Norway, Saudi Arabia, and other energy-exporters.

We also can use our simple model to predict the eﬀects of interest-rate changes on a small, open economy’s borrowing or lending. As an exercise, let us use the log-utility Euler equation, so that 1 Y C = (1 + r)A + Y + 2 (2.26) 1 1 + β 0 1 1 + r and 1 Y CA = rA + Y − (1 + r)A + Y + 2 . (2.28) 1 0 1 1 + β 0 1 1 + r

28 Prediction 3: I think you can see that dCA1/dr depends on the sign of A0. (From intermediate macroeconomics remember that the income and substitution eﬀects of a change in interest rates work in the same direction for a borrower but opposite directions for a lender, so the theory may make a sharp prediction — increased saving — only for borrowers.)

So far we have ignored investment, but of course CA = S − I, so anything that affects I will affect capital flows. You can see the theory in Schmitt-Grohé,Uribe, and Woodford’s chapter 3, but we’ll focus on the logic using things we already know. To start, simply think back to intermediate macroeconomics again and make a list of some economic variables that affect aggregate investment...

Prediction 4: A negative response of I to r makes it more likely that the CA will increase in response to an increase in r.

Try drawing a saving-investment diagram so see this result. What will happen to Greece’s CA deﬁcit if r rises? Will this be true of every country?

Prediction 5: A country converging to a neighbouring country will run a current-account deﬁcit. (e.g. Estonia, Latvia, Poland)

Case Study: Estonia and Latvia. Estonia and Latvia grew rapidly from 1995 to 2010. They have stable currencies pegged to the euro, and they lie near rich countries. Estonia at that time was Europe’s biggest per capita recipient of foreign investment. What would our theory predict about these countries’ current accounts? It seems that they would be predicted to borrow in order to converge, and they did. From 1995 to 2010 each country’s current account deﬁcit averaged 10% of GDP. It makes sense that they should import technology and investment to catch up. (It is interesting that both countries also ran large government budget surpluses. Their current account deﬁcits were due to booming investment, not to low savings rates.)

Prediction 6: A country-speciﬁc productivity shock will lead to a decrease in the current account balance. (e.g. a discovery of iron ore in Western Australia)

2.5 Fiscal Policy

29 Let us try to predict the eﬀect of changes in government spending on international

capital ﬂows. We treat G1 and G2 as exogenous. And for simplicity we work with an endowment economy so Y = C + G + NX. Then notice that we can simply use our previous work but replace Y with Y − G. For example, with C1 = C2 (consumption- smoothing):

1 + r Y − G CA = rA + Y − G − (1 + r)A + Y − G + 2 2 (2.33) 1 0 1 1 2 + r 0 1 1 1 + r

Prediction 7: Temporary increases in government spending reduce the current account balance, while permanent ones have no eﬀect.

Again the logic stems from the permanent-income hypothesis. Historically, wars are example of temporary, sharp increases in G, and they often were accompanied by current- account deﬁcits. Why does the IMF recommend/require reductions in public spending as a condition for loans to highly-indebted countries?

What about changes in taxes? Let us look at the present-value version of the government’s budget: G T G + 2 = (1 + r)A + T + 2 . (2.34) 1 1 + r g0 1 1 + r In the two-period model tax revenue T must pay for government spending and retire debt. Notice we have assumed revenue is collected in lump-sum form. We write it as T not τY for example.

Prediction 8: The timing of taxes does not aﬀect national saving or the current account.

You can see this simply by inspecting the expression for CA1 (2.33). Taxes do not appear there, so any combination of T1 and T2 leads to the same capital flows. This is an example of Ricardian equivalence, the idea that the timing of tax collections does not matter for national saving. If the government reduces T1 (and runs a budget deficit) it must raise T2. The present-value budget of the private sector is unchanged and so their consumption is unchanged. Thye save the entire tax cut. National saving is unchanged because the reduction in government saving is offset by the increase in private saving.

30 Why might Ricardian equivalence be unrealistic? How will departures from it aﬀect the predictions?

Some departures you might think of include finite lives, borrowing constraints, or myopia. Usually these lead to the revised prediction that a tax cut will lead to a fall in national saving. This is sometimes called the ‘twin-deficit’ hypothesis, which is the idea that a budget deficit causes a current-account deficit. Do you see this correlation in the cross-section of The Economist tables? There are countries like Japan, Italy, and Singapore that tend to have budget deficits and current-account surpluses. How would you test this hypothesis while controlling for government spending?

2.6 Large Economies

Studying large economies takes a bit of work but is important. Among other things this work tells us where the world real interest rate r comes from. Usually we simplify by studying a two-country world, and imagine the two countries to be Europe and the US or China and the US.

In your course in intermediate macroeconomics you may recall studying the two- country model with two savings-investment diagrams. Denote the foreign country with a ∗ just as in chapter 1. The key feature is that one country’s lending equals the other country’s borrowing so that:

∗ CAi + CAi = 0 (2.35) or equivalently:

∗ ∗ Si + Si = Ii + Ii , (2.36) for i = 1, 2. In the savings-investment diagrams one adjusts the horizontal line for r up and down until this conditions holds.

The exercises at the end of the chapter let you practice the calculations using some numerical examples and a bit of algebra. They have budgets and consumption plans just like the small open economy case. But there is another variable to solve for (r) so another equation is needed. That equation is (2.35).

31 But, there is a simple trick to implementing that condition. If the two countries have the same utility function note that the Euler equation holds in world consumption. Equivalently, add up the Euler equations for the two countries. Then note that in an endowment economy ∗ ∗ ∗ Ci + Ci = Yi + Yi − Gi − Gi (2.37) for i = 1, 2. Everything on the right-hand side is given so this determines world consumption. We then use the Euler equation to solve for r and proceed.

Now we can predict how shocks (e.g. to US or European ﬁscal policy) change r and capital ﬂows.

Prediction 9: The larger the country, the smaller the effect of a given change in domestic fiscal policy on its current-account balance. The reason: r also changes for large economies. To see the logic remember that for the closed (we hope) world economy only r would change. For large economies the effects are scaled somewhere between those for a closed, world economy (all r) and a small open economy (all CA).

As our last exercise in this chapter, let us think about the effects of a differential pandemic shock that affects large economies. Suppose that the fall in current income Y1 is larger in the US and Europe than it is in China and Japan. What are your predictions for capital flows?

What’s next in the theory? Our theoretical model has been deliberately simple, to allow us to derive the predictions quickly while still capturing the key inﬂuences. One natural extension is to allow for uncertainty, so that plans depend on expected or forecasted future income. A second extension is to allow for many time periods (and an inﬁnite horizon for the country). The direction of our predictions wouldn’t change with this feature, but the numerical values of our predictions would change.

2.7 Global Imbalances

Now that we have some background in understanding international capital ﬂows, let us brieﬂy look at one of the enduring policy issues of the last 20 years or so. This is the

32 issue of global imbalances. The history is that the US in particular has had a large current- account deﬁcit each year, averaging several hundred billion dollars. Its net foreign debt is now roughly 20% of GDP. At the same time China and Germany have had persistent surpluses.

The simplest way to see further detail on the history is to do a web search under images for current account balances. Look for a stacked bar chart that is up to date, then track the histories of the US, China, Germany, and Japan.

Next, pause to think about how you might study the causes. If you begin by studying savings and investment, it turns out that S is unusually low in the US and high in China. (In other words, this is not due to booming I in the US.) So economists then have turned to trying to explain these differences in turn. For China, they sometimes suggest high private saving occurs to allow spending on health in old age, or that high government saving was a reaction to the Asian financial crisis of 1997–1998. For the US, they sometimes point to the twin-deficit hypothesis we saw in section 2.5.

What about the effects? There are two main effects that commentators have been concerned about. First, the high savings rates in some Asian and European countries have sometimes been called a ‘savings glut’. That lowered the world real interest rate, r, by shifting the world savings curve out. In turn, some economists have argued that the low interest rate led investors to take on excessive risk in order to maintain the returns on their portfolios and that this risk-taking caused the financial crisis of 2007–2008. If that chain of reasoning is correct then that would indeed suggest future imbalances should be a great concern.

Second, the US current-account deficit of course involves a trade deficit, which has led export industries and US policymakers to argue that it reflects unfair trade practices. For example, successive US Treasury Secretaries have suggested that the US Congress might declare China a ‘currency manipulator’, a declaration which can trigger tariffs on US imports from China. And of course the federal administration from 2016–2020 did apply such tarriffs. For economists, then, the concern is that persistent current-account deficits may trigger protectionist measures. These are typically poor ways of raising revenue or

33 addressing the distributive consequences of trade and also don’t really help to reduce the trade deﬁcit. One reason is that they reduce the demand for the foreign currency and so lead to an appreciation of the domestic currency (the USD) which reduces NX.

Recently the current-account deficit in the US and the surplus in China has been shrinking. There are several possible explanations including (a) different inflation rates across countries leading to a real depreciation for the US (see chapter 5 below); (b) changes in China’s management of its currency (also discussed in chapter 5); and (c) increased US private saving, which has risen during the last two US recessions. So, perhaps the issue of global imbalances will gradually disappear.

Further Reading

For a more detailed derivation of the two-period model please see chapters 1-3 of International Macroeconomics by Stephanie Schmitt-Groh´e,Mart´in Uribe, and Michael Woodford.

On tax havens see Antonio Coppola, Matteo Maggiori, Brent Neiman, and Jesse Schreger (2020) ‘Redrawing the map of global capital ﬂows: The role of cross-border ﬁnancing and tax havens,’ NBER working paper 26855. And on the same topic look up Gabriel Zucman’s web page for his research on this topic, such as his 2013 article ‘The missing wealth of nations: Are Europe and the U.S. net debtors or net creditors?’ Quarterly Journal of Economics 128, 1321–1364.

Exercises

2.1. An economist writes: “Norway’s large current account surplus suggests that Norwe- gians expect oil prices to fall in the long run.” To assess this claim, consider a two-period model of a small open economy. For simplicity, suppose there is no investment and no government. Norway’s national budget constraint is:

E P Y C (1 + r)A + P Y + 1 2 2 = C + 2 . 0 1 1 1 + r 1 1 + r

34 Net foreign assets initially are A0 = 10. The world interest rate is r = 0.05. The quantity of output is Y1 = Y2 = 100. Households plan to smooth consumption over time so that C1 = C2. The oil price, taken as given by Norway, is P1 = 1 currently. It follows this pattern over time: P2 = ρP1 + 2,

where 2 is a random component with a mean of zero that is uncorrelated with P1. The parameter ρ ∈ [0, 1].

(a) Find a formula for current consumption C1. (Use numbers wherever you know them.)

(b) Find a formula for the first-period value of the current account balance, CA1. (Use numbers wherever you know them.) (c) In this model, what effect does the persistence (in this case expected persistence) in the oil price have on how much Norway adds to its net foreign assets? Briefly comment on whether the economist’s claim makes sense given this economic theory.

2.2. This question uses a small-open-economy model to study two events that might aﬀect the current account of Lithuania. We use a two-period model, with no uncertainty and no investment. Lithuania has net foreign assets A0 = 10. Suppose that Y1 = 100 and Y2 = 105. Households smooth consumption so that C1 = C2. The world real interest rate is r = 0.01.

(a) If G1 = G2 = 30 then solve for the current account in the ﬁrst period: CA1. (b) If the world real interest rate instead took the value 0.03 then what would the value of CA1 be?

(c) Now suppose again that r = 0.01 and G1 = 30. The government is considering increasing G2. Find a formula that predicts CA1 for any value of G2. Then ﬁnd a formula for CA2.

2.3. This question uses a two-period, endowment model of a small, open economy to study the predicted eﬀects of ﬁscal policy changes on international borrowing. Suppose the economy has international debt A0 = −10 and that this is entirely private debt. Also, Y1 = Y2 = 100, G1 = G2 = 20, and r = 0.05. Households smooth consumption so that C1 = C2. There is no investment: I1 = I2 = 0. Taxes are collected at proportional rate τ, so that government revenue is τ1Y1 in period 1 and τ2Y2 in period 2. Household and government budgets balance in present-value terms.

(a) Find an expression for the current account in period 1 as a function of the two tax rates.

(b) Suppose that the government cuts τ1 and raises τ2 by just enough to balance its budget in present-value terms. What is the eﬀect on consumption and the current account?

(c) Suppose instead that the government raises G1 to 22 and cuts G2 by just enough to balance its budget in present-value terms. What is the eﬀect on consumption and the current account?

35 2.4. This question uses a two-period model of a small, open economy with no investment to predict some of the eﬀects of a decrease in government spending. Suppose that the country has net foreign assets labelled A0 = −10. Output is given by Y1 = 100 and Y2 = 110. The world interest rate is r = 0.02. Households smooth consumption so that C1 = C2.

(a) Suppose government spending is G1 = G2 = 20. Solve for the value of the current account in each time period.

(b) Suppose that government spending in period 1 falls to G1 = 18. But G2 = 20 still. Solve for the value of the current account in each time period.

(c) Suppose that government spending in period 1 falls to G1 = 18 but now the eﬀect can persist: G2 = 20 − 2ρ, so that ρ ∈ [0, 1] describes the persistence in this change. Find formulas that give the current account in each time period as function of ρ.

2.5. This question asks you to make predictions relating the current account balance to the time path of output, for a small open economy. There are two time periods, with no uncertainty, government spending, or investment. Suppose that initial net foreign assets are zero. The world interest rate is r = 0.04. Households smooth consumption so that C1 = C2.

(a) If Y1 = Y2 = 100 then solve for the current account in each period. (b) To see what theory predicts if output is temporarily higher (for example because of a resource discovery that will quickly be exhausted), suppose Y1 = 120 and Y2 = 100. Find the current acount balance in each period. (c) To see what theory predicts if there is such an event in period 2 but it is known in advance, suppose that Y1 = 100 and Y2 = 120. Find the current acount balance in each period.

2.6. This question investigates what we predict will happen to the US current account balance if its output growth rate increases. Suppose that the world economy consists of two regions, the US (unstarred) and Europe (starred). There are only two time periods and initial net foreign assets are zero. Let us treat each region as an endowment economy, with no investment or government spending and with nonstorable output. In the US, consumption plans follows this pattern: 1 1 = 0.99(1 + r) , C1 C2

∗ ∗ and similarly in Europe for C1 and C2 . ∗ ∗ (a) Suppose Y1 = 100, Y2 = 101, Y1 = 100, and Y2 = 101. Solve for r and for the current account in each time period.

(b) Suppose instead that US growth is expected to be higher, so now suppose Y1 = 100 ∗ ∗ and Y2 = 104 while still Y1 = 100, and Y2 = 101. Solve for r and for the current account in each time period.

36 2.7. This question uses a two-period model with no uncertainty and no investment to predict some of the international eﬀects of an increase in government spending in Japan. ∗ ∗ Suppose that in Japan Y1 = Y2 = 50 and that in the US Y1 = Y2 = 100. For simplicity, suppose the world consists only of these two countries. In both countries households plan consumption to satisfy the Euler equation with log utility and discount factor 0.98, so that in Japan: 1 1 = 0.98(1 + r) , C1 C2 and similarly in the US. Suppose that Japan has net foreign assets given by A0 = 20. ∗ ∗ (a) Suppose that G1 = G2 = 15 and G1 = G2 = 30. Solve for the real interest rate and for the Japanese current account balance in each time period. ∗ ∗ (b) Suppose instead that G1 = 20 while G2 = 15 and G1 = G2 = 30. Solve for the real interest rate and for the Japanese current account balance in each time period.

(c) Would a permanent increase in government spending (so that G1 = 20 and also G2 = 20) have any eﬀects according to this theoretical model?

2.8. Japan recently has begun running a trade deficit. This question studies a way to predict what might happen to capital flows between Japan and the US if growth in Japan slows even further. For simplicity, imagine a two-period, two-country world with no uncertainty, investment, or government spending. Savers in both countries follow the Euler equation: 1 1 = 0.99(1 + r) . C1 C2 ∗ ∗ In the US, Y1 = 100 and Y2 = 103. In Japan, Y1 = 50 and Y2 = 51. Suppose that initial net foreign assets for Japan are A0 = 10. (a) Solve for the real interest rate and describe the capital flows in each time period.

(b) Suppose that, instead, national output in the future time period in Japan is Y2 = 48. Find the capital ﬂows under this scenario. (c) An economist writes: “Slower growth in Japan would help resolve the problem of ‘global imbalances’.” Does the economic theory you have outlined above support this argument?

37 Chapter 3. Sovereign Debt

Chapter 1.1.1 discussed the mysterious absence of capital ﬂows to EMEs from rich countries. What might be some reasons for the small scale of these ﬂows? This chapter studies one possibility: sovereign default risk. Argentina defaulted in 2005, in the form of paying its creditors about 30 cents per dollar of debt. Other countries defaulted during the 1980s and 1930s. Russia defaulted in 1917 and again in 1998. So it is possible that the probability of default discourages lending.

In section 2 of the course, we imagined a small, open economy that can freely lend and borrow at the world interest rate. It treates the rest of the world like a bank account. But international capital markets may not be quite so open, due to the effects of default. Central to research in this area is the idea of sovereign debt; debt contracts which cannot be legally enforced. Such limits to enforcement may explain cross-country differences in rates of return or even non-convergence in economic growth. We’ll examine the main theoretical ideas and some of the evidence. But we can already see the main lesson that stems from the sovereign nature of these debts: It is not that defaults occur (though they do) but that there may not be much lending in the first place.

In this chapter we’ll look at the two main theories that try to explain why sovereign lending occurs, and evidence on those theories. We’ll then change tack and investigate a diﬀerent set of questions: What are the ill eﬀects of accumulated debt and how can the burden of debt be relieved?

3.1 Sanctions

The first idea we’ll look at is a simple one. The argument is that a country that defaults will face sanctions (such as limitations on its trade). Usually this doesn’t consist of seizing assets that are held in the foreign country, for most borrowers don’t have collateral like that. Instead, it takes the form of disrupting access to trade credit. The threat of these sanctions may make the borrower better off paying than defaulting. And the sanctions do not have to directly benefit the lenders for this threat to be effective.

38 This reasoning implies that a country can be better oﬀ the more exposed it is to potential sanctions, since it will then be able to borrow. It is in the borrowers’ interest that the sanctions be severe. This conclusion may seem odd to our colleagues in other disciplines who study international debt.

3.1.1 Theory

Many of the main ideas can be seen in an example in which a country borrows only for consumption insurance (and not for investment projects). Imagine a two-period endowment economy with a single good. A country seeks to maximize

E1 ln(C2), (3.1)

(one of our earlier examples of consumption goals) and it faces an uncertain endowment:

Y2 = 10 ± 2, (3.2)

where each possible outcome has probability 0.5.

Utility is concave so expected utility would be highest (remember Jensen’s inequality?)

if instead it faced a certain income Y2 = 10. In other words ln(10) > 0.5 ln(8) + 0.5 ln(12).

Now suppose there are lenders (banks) who have two key features. First, they are risk- neutral. Thus they are indiﬀerent between earning say 0 or facing 50:50 odds of earning 2 or -2. (We’ll see how this perspective can result from diversiﬁcation in chapter 4.) Second, they are competitive, so they bid their expected earnings to zero.

Imagine the borrower makes a proposal (prospectus) to the lenders. “If my income is 12 I will pay you 2. If my income is 8 you will pay me 2.” Notice that this transfers the risk from the risk-averse party to the risk-neutral one. The lenders are indiﬀerent, while the borrower beneﬁts by eliminating all risk and being guaranteed a net income of 10.

Question: If you were a lender would you accept this proposal? Suppose that you do and then Y2 = 12. Do you think the borrower will pay you 2?

39 You can see that the lender will default in this state of the world and keep the 12. The borrowers know this, so they won’t agree to this contract in the ﬁrst place. Thus the

borrower won’t be able to insure against Y2 = 8.

Now suppose a sanction can be applied to a defaulting sovereign borrower. It loses a

percentage of income, which we’ll suppose is 0.2 or 20% if it defaults. Now if Y2 = 12 then the borrower makes a simple comparison between two numbers. If it pays then it has net income of 10. If it defaults is has net income of 12(1-0.2) = 9.6. Thus it pays. The lenders know this and so will agree to the contract.

Let us call the sanction rate η, where η = 0.20 in our example. Question: What is the minimum value of η that will lead the contract to be signed?

Now let us work out the general case. Suppose that:

Y2 = 10 ± , (3.3)

where each outcome is equally likely. In autarky (i.e. isolation) expected utility is:

EU = 0.5 ln(10 + ) + 0.5 ln(10 − ). (3.4)

The zero-proﬁt condition for the lenders is:

0.5() + 0.5(−) = 0. (3.5)

Under full insurance C2 = 10.

Question: What is an expression for the minimum sanction rate η that supports the contract?

Once again we simply compare paying and defaulting in the good state. For lending to occur: 10 ≥ (10 + )(1 − η), (3.6)

which means that η ≥ . (3.7) 10 +

40 Now let us go back to our original example with = 2. Our rule (3.7) means that η ≥ 0.167 is needed to support the contract. And our ﬁrst numerical example had η = 0.2 and so satisﬁed that criterion. But suppose that some sanction can occur but at a rate less than 0.167. Can the borrower do any better than autarky?

To answer this question, suppose that η = 0.10. Now suppose the borrower makes a diﬀerent proposal to the lenders: “We’ll pay you d in the good state and you pay us d in the bad state.”

We know that a contract with d = 2 won’t be signed. But let us ﬁgure out whether d > 0. Again we do this simply by comparing the two outcomes for the borrower in the good state. If they pay they net 12 − d. If they default they net 12(1 − 0.10). So the larges d possible satisﬁes: 12 − d ≥ 12(1 − 0.10). (3.8)

which gives d = 1.2 as the largest payment. So the borrower nets 10.8 in the good state and 9.2 in the bad state, and reduces risk, though not to zero.

In general, d = η(10 + ). (3.9)

So imagine drawing a graph with η on the horizontal axis and d on the vertical axis. At η = 0 also d = 0. One η reaches the value (3.7) then d = which is full insurance, so the graph is hroizontal from then on. In between these extremes, the graph is a line (3.9) that slopes up: higher sanction rates support higher payments.

The common method in each of these examples has been to simply compare the outcomes for the borrower in the good state, with and without default. But we also can describe the contract as an optimization problem subject to constraints:

max E ln C2 (3.10)

with C2 = 10 + − d in the good state and C2 = 10 − + d in the bad state, subject to

0.5d + 0.5(−d) = 0, (3.11)

41 which is the zero-proﬁt condition, and also to (3.9), which is sometimes called an incentive- compatibility constraint. That means that in the good state:

C2 = 10 + − η(10 + ), (3.12) and in the bad state

C2 = 10 − + η(10 + ). (3.13)

Example: Ecuador

On 14 December 2008 Ecuadorian President Rafael Correa announced that his country would not repay back its remaining external debt. Citing the economic crisis, Correa called the debt “immoral and illegitimate.” The refusal to pay the debt allowed the government to spend more on social programs. The decision of defaulting on the global bonds (2012, 2015 and 2030) was taken by Correa after the official presentation of the final report from the Public Credit Audit Commission (Comision para la Auditoria Integral del Credito Publico, CAIC) audit regarding Ecuador’s foreign debt. Ecuador’s decision to stop payments on the interest on its national debt was its second in two decades, the other coming in 1999, when it defaulted on $10 billion. In 2008, the figure stood at $3.9 billion. An interesting aspect of the default’s timing is that GDP had grown quite rapidly in recent years, in part because Ecuador is a large oil exporter. So this seems to be an example of defaulting in the ‘good’ state. k

Example: Chad

An oil pipeline runs more than 1000 km from southern Chad through Cameroon to the Atlantic Ocean. It was financed by the World Bank and the EU at a cost of $4 billion on condition that a fixed portion of revenue was set aside for poverty-reducing projects. But the dictator of Chad, Idriss Déby, proposed to spend the income on weapons to maintain his regime (for example to defend the border with Sudan). The World Bank applied sanctions in 2006: it suspended its loans and blocked Chad’s offshore bank accounts. Although the sanctions did not lead to the restoration of the original spending plan for the oil revenues, the government of Chad did eventually pay back the two institutional lenders. k

42 3.1.2 Evidence: GDP as a Constraint on Debt

Historically, sanctions were sometimes large: e.g. Mexico 1861, Egypt 1882, Haiti 1915, or even Newfoundland and Labrador 1934. But some lending also occured from Switzerland, which could not practice ‘gunboat diplomacy’. And Italian banks lent money to England in the 1300s. And many European banks lost money lending to Philip II of Spain in the 1500s. Modern sanctions appear to be negligible: e.g. Argentina 2002–2016. But Rose (2005) argues there is an eﬀect of default or renegotiation of debt payments on trade.

Next, we look at an indirect test of the sanctions theory. One of the main predictions of the theory on sanctions is that a country’s borrowing should be constrained by its output. If lenders can apply sanctions at rate η to all borrowers, then they will make larger payments to borrowers with larger GDP. So output may act as a proxy variable for the severity of punishment through trade restrictions.

Lane (2004) tests whether this link between debt and output is found in the data. Of necessity he assumes that all countries in his sample are credit constrained; they’ve borrowed as much as they can. In addition to testing the connection between debt and output, Lane also examines some variables that might explain some of the remaining variation in debt across countries.

He measures debt and output, both per capita, for 87 developing countries from 1970- 1995. GDP and external debt are measured in constant 1987 US dollars, and collected from World Development Indicators. Notice that this is gross, not net, debt, so it excludes any foreign assets that might be used as collateral. His regression is:

Ai,gross Yi = b0 + b1 P opi P opi as a cross-section regression, with data averaged over 1970–1995.

In a cross section of country averages over the entire period, 70 percent of variation in per capita debt is due to variation in per capita output. Only openness — measured as imports plus exports as a share of output — explains part of the remaining variation

43 in debt across countries. That too ﬁts with theory in that an open economy has much to lose from trade sanctions and so is a relatively good credit risk. This link may explain why open economies seem to grow relatively rapidly; they may have ordinary productivity growth but their access to international capital markets enables them to invest more than relatively closed economies.

To check on the evidence for yourself:

(a) Start at /www.gapminder.org/tools.

(b) On the x-axis: economy/incomes and growth/total GDP.

(d) Use logs on both axes.

Do you ﬁnd evidence of a positive relationship in the most recent cross-section or over time?

As Lane notes, it would be interesting to connect this research on quantities with information on prices. Do the debts of relatively open economies yield relatively low premiums over LIBOR?

Perhaps it doesn’t seem surprising that large, open economies can borrow more and do borrow more. But remember from our ﬁrst mystery in section 1.1.1 that this is not what the growth model predicts should drive capital ﬂows. It predicts they go to the projects with the highest marginal products of capital.

3.2 Reputation

A number of authors have argued that sovereign borrowers will repay their debts, even when no sanctions are available, so that they can maintain a reputation for repayment and hence have access to future borrowing. We next investigate this argument.

3.2.1 Theory

Suppose that there are now many periods, with equally likely output values of 10 + and 10 − . The utility function is now time-separable with discount factor β. Here β

44 measures how much the borrowing country’s policymakers care about the future. With β = 0 they don’t care about the future at all while with β = 1 they value the future just as much as the present.

Again the full insurance contract involves paying in the good state and receiving in the bad state. To see if reputation can support this, consider the country’s decision on defaulting. If it defaults, it will do so in a good state. All we do here is make the same comparison that we did in section 3.1 but with utility: We compare the utility under payment with that under default.

Under payment, the utility in each period is ln(10), so perpetual utility is

1 (1 + β + β2 + ...) ln(10) = ln(10). (3.14) 1 − β

Here we have used tool 4 from section 1.3.

Under default, the borrower consumes 10 + this period. But the penalty for default is that the country is cut oﬀ from future insurance, so it cannot smooth consumption. The expected utility in each future period is:

(0.5 ln(10 + ) + 0.5 ln(10 − ), (3.15) which is less than ln(10).

For payment to occur:

1 β ln(10) ≥ ln(10 + ) + (0.5 ln(10 + ) + 0.5 ln(10 − ). (3.16) 1 − β 1 − β

The comparison depends on the value β (not η, which here is zero). If β = 0 then condition (3.16) does not hold. The borrowers do not care about the future, the threat of being unable to borrow again is an empty one, the country would default, and so lending would not occur. But at some value of β (perhaps near 1) the weight attached to the future is high enough that condition (3.16) holds. The exercises ask you to work out an example.

Example: Argentina

45 Argentina defaulted on its debt in 2002. Then in 2005 it made a take-it-or-leave it offer to its debt-holders: agree to accept 35% of their payments due or else receive nothing. Owners of 75% of the defaulted bonds accepted the offer. Then in 2008 President Cristina Fernándezde Kirchner announced that the government of Argentina would consider an offer from the holdouts. As a sovereign borrower, Argentina can continue to ignore this outstanding debt, so why did it entere these negotiations? The answer is that the federal government needed to borrow once again. Reaching an agreement with its previous creditors allowed it to do so, and at a lower interest rate. This is an example of reputation possibly sustaining some sovereign lending. However, history then repeated itself. The federal government of Argentina borrowed again in the 2010s but then faced a default/restructuring again in 2020. k

Before we leave the reputation model, let me add that a doubt about it has been raised by Bulow and Rogoff (1989). Our example assumes that the borrowing country is completely cut off from capital markets if it defaults. But what if it is cut off from borrowing but can still lend? Then it can default, take its payment , and invest it in a foreign country to self-insure. Lenders know of this possibility, so the reputation equilibrium will not occur. In general, the ability of countries with poor credit records to lend abroad may limit their ability to borrow.

The Bulow-Rogoﬀ argument on this is as follows. Suppose the country does default in the good state. Then it can take its windfall gain of and invest it in some third country. It then can sign a new insurance contract with a new insurer, thus guaranteeing it a steady consumption of 1 per period. As an incentive to acquire this insurance it can use its deposit of as collateral. Meanwhile, it also can collect a steady income of r per period and add that to its consumption.

Thus, the country will be strictly better oﬀ than if it paid to the original insurer. But, knowing this, the original insurer will not oﬀer a contract, and so reputation alone cannot sustain any sovereign lending.

3.2.2 Evidence: Reputation in US State Debt

46 English (1996) set out to test the theory. As you know, Bulow and Rogoﬀ had argued that sanctions are necessary to support sovereign lending, and that reputation alone cannot do so. English tested this idea using U.S. state debt from the 1840s. In that decade a variety of states borrowed heavily, often in Britain, to build canals or to fund state banks. (You might recognize why this was not an ideal time to begin to build a canal.) Many defaulted or repudiated their debts. Typically these were the most indebted states.

English ﬁrst showed that these were indeed sovereign debts. The eleventh amendment to the U.S. constitution prevented foreigners or those outside a state from suing a state government. Most bondholders were not state residents.

No sanctions were applied. These would have been very diﬃcult, because of inter- state trade (sometimes called trans-shipment, which can include simply changing the label on a package). That would have allowed states to avoid trade penalties. Nevertheless, most states eventually repaid or renegotiated their debts, and re-entered capital markets. It seems that being cut oﬀ from further lending was enough to bring about eventual repayment, as predicted by the reputation model.

Why didn’t the Bulow-Rogoff argument apply to U.S. states? One possibility is that there were ‘reputational spillovers’; defaulting hurt aspects of a state’s business other than its access to capital. This seems unlikely. A second possibility is that there were limited investments available to states with which to self-insure. This seems unlikely too. A third possibility is that self-insurance requires large surpluses in some years and that these surpluses were politically impossible. English provides some convincing evidence that taxpayers were suspicious of government surpluses. If that suspicion made surpluses impractical, paying off old debts may have been the best way for U.S. states to have access to finance.

3.3 Debt Overhang and Debt Relief

Now we change tack to investigate a diﬀerent set of questions about sovereign debt. Are there dangers to accumulating external debt? And what can be done to relieve the burden of debt?

47 3.3.1 What is Overhang?

One speciﬁc idea that has been widely debated is that past debt may inhibit investment. The idea is that a large inherited debt discourages investment in physical capital, I. There are two possible mechanisms. First, public investment may be discouraged if the proceeds will largely accrue to the lenders. Second, private investment may be discouraged if the investors think the highly-indebted government will need to tax them at a high rate.

This idea is called debt overhang, and it may be one reason why international policymakers have focused on relieving debt (other than simply bailing out western banks).

Debt overhang might be one reason for relatively low investment in Greece since the European debt crisis. But let us look for some more systematic evidence:

(a) Start at /www.gapminder.org/tools.

(b) On the x-axis: for advanced users/advanced debt & trade/debt servicing costs.

(d) Use log on the x-axis and lin on the y-axis.

See if you think these measures can capture the idea. Does running the historical animation provide support for the debt overhang hypothesis?

3.3.2 Debt Buybacks

During the 1980s a secondary market developed in the debt of many countries in Latin America and eastern Europe. At times a debt worth a dollar at face value might trade for less than a dime, due to the low probability of repayment. In chapter 7 we’ll see formally how the probability of repayment is reﬂected in the market price of debt (or in credit default swaps).

These low debt prices led to some suggestions of debt buybacks. In these schemes, foreign donors would give money to indebted countries with which they could buy back their own debt on the secondary market. Debt-for-equity or debt-for-nature swaps were other examples of these proposals.

48 Bulow and Rogoﬀ argued that these schemes did not lower the indebtedness of borrowing countries. To see the problem, ﬁrst consider the possibility that there can be a negative relationship between the face value of debt and the price per unit of debt. A country with relatively little debt outstanding may be viewed by the market as more likely to repay.

Next, imagine a scenario with four actors: a bank, a sovereign government borrower, an NGO, and a vulture fund. The bank lends to the government, which later defaults. It is then shut out of further lending, perhaps for some years. Its outstanding debt trades at a discount to its face value. Suppose that much of it is bought by vulture funds.

An NGO provides funds for a debt buyback. As the debt is purchased and retired, though, the secondary market price rises as the vulture funds sell into the buyback. While the borrower retires some debt, the remaining debt (of lower face value) will have higher market value than it previously had. As a result, the total market value of the outstanding debt may not be much affected by the buyback. Bulow and Rogoff argued that the main effect of a buyback is a transfer from the funding agency to the creditors; debtors may not gain.

In a classic example, Bolivia bought back a face value of $308 million dollars of its debt in March 1988, at a cost of $34 million, mainly supplied by foreign donors. Here is what happened next:

Table 3.1. Bolivia’s March 1988 Debt Buyback Before After Face value of the debt, D $ 670 million $362 million Price, p (per $) 0.06 0.11 Market value $40.2 million $39.8 million

The market value fell by only $400,000, less than 1.2 percent of the amount Bolivia spent. Where did the $34 million go? And how can this problem be avoided?

You might notice that most of the donated money went to northern-hemisphere banks. This experience leads to the lesson that debt reduction schemes must be coordinated across

49 lenders, and tied to concessions on further debt. Otherwise each creditor has an incentive to wait and let others forgive the debt they hold. One such coordination scheme was proposed by the IMF: international bankruptcy arrangements. That was unsuccessful. But meanwhile markets have responded with widespread collective action clauses (CACs) on bonds.

Do you think a CAC might aﬀect the probability of a debt restructuring? Might it aﬀect the interest rate (usually a premium over LIBOR)?

3.3.3 Debt Relief

Debt relief involves forgiving/cancelling sovereign debts. Relief usually should be coordinated, for reasons we have just seen. For many countries, debt is owed to international agencies such as the World Bank and IMF, not to private banks.

How can debt relief help?

(a) It could free up public-sector resources.

(b) It could improve private incentives. e.g. Debt overhang may discourage investment. The tax on project earnings, needed to raise revenue to pay interest on debt, might prevent projects from being started.

Debt relief is not new. Here are some historical episodes:

(a) Brady Plan, 1989: middle-income countries

(b) HIPC Plan, 1996 and after: low-income countries

(Take a moment and look up what HIPC stands for and see a map of the HIPCs.)

Very sadly, most studies of debt relief suggest it may have little eﬀect. For example, Arslanalp and Henry (2006) predicted little eﬀect of the Gleneagles proposal. They tracked this accounting:

Net Resource Transfer = New Lending + Grants + Portfolio Equity (3.17) + Foreign Direct Investment − Debt Service

50 but for the poorest countries there are few private capital ﬂows and most ﬂows go to the public sector. Thus:

Net Resource Transfer = New Lending + Grants − Debt Service. (3.18)

The Gleneagles Plan involved forgiving $55b of debt. That eliminates interest payments of roughly $2b year (that is the ﬂow burden to the debtor countries). This amount is (a) small as a share of G7 GDP: roughly 0.01 percent; (b) larger as a share of borrower GDP: roughly 2-3 percent; but (c) gross lending is much larger so there still is a positive transfer ongoing.

For 2000–2003 averages, the amounts in (3.18) in billions of USD were:

17.7 = 4.5 + 16.0 − 2.8 or in percentages of HIPC GDP:

12.2 = 3.1 + 11.1 − 2.0.

Reducing the debt service cost would have a small eﬀect on the positive, net resource transfer. In the past, grants sometimes have changed in the opposite direction, so that is a reason to focus on the net transfer. The net transfer may not be much larger after debt relief.

Arslanalp and Henry argued that, either way, there are positive net resource transfers, so no debt overhang. For HIPCs there is little sign of private flows, and the response of other public flows is not driven by debt overhang/taxes. The countries involved have seldom attracted much private capital; it is unlikely debt relief will affect that.

Has debt relief ever succeeded? In the Brady Plan private ﬂows responded and the net transfer became positive. Physical capital investment and growth after the plan were higher than (a) before the plan; (b) in a control group of countries.

Chauvin and Kraay (2005) examined the HIPC initiative after 1996 using the same approach to evaluate the progam. They used the ‘diﬀerence in diﬀerences’ method. They

51 found little eﬀect of debt relief on public spending in poor countries, or on growth, or on investment in physical capital.

The book by Easterly (2001) goes further in presenting a discouraging picture of debt relief. He suggests that high-debt countries: noindent (a) Did not experience more bad luck in the form of war or a fall in their export prices and (b) receive debt relief. So the danger is that repeated debt relief creates an incentive for high debt. He thus argues that debt relief: (a) predicts additional new borrowing; and (b) is also associated with faster extraction of non-renewable resources (which suggests discounting the future i.e. low β). His conclusion is that debt is a symptom not a cause.

These readings are all pessimistic. But perhaps there is a stronger case for debt relief if it arises because of the policies of previous, non-democratic regimes (this is sometimes called ‘odious debt’) or is the result of legal but unfair transfer pricing that harms tax collection in EMEs. I am no expert on this topic so I hope you will do some further reading and form your own views.

As this book is written, in 2020, further negotiations on debt relief are taking place, especially focused on African countries. The G20 governments have offerred a ‘debt service suspension initiative’ that would suspend interest payments owed to those governments (usually to their export finance agencies). But take-up of this offer has been quite low, because African governments also now owe money to private creditors. They fear that the stigma of participating in the suspension will hurt their standing with private lenders and so lead to them paying higher interest rates.

In this chapter we’ve focused on sovereign lending i.e. lending to the governments of EMEs. A broader issue in the macroeconomics of international development involves tracking the capital flow to the private sector too. Historically, the overall flow has been subject to reversals, sometimes called sudden stops. The Asian Financial Crisis of 1997– 1998 is a dramtic example of a sudden stop and its effects. We won’t formally model that but it is important to know something about, in part becuase it still influences policymakers today.

52 Further Reading Like much of this book, this chapter was inspired by the classic book for graduate students: Maurice Obstfeld and Kenneth Rogoﬀ (1996) Foundations of International Macroeconomics, MIT Press. I learned this chapter’s theory from their chapter 6. Begin with some of the articles cited in this chapter, in the order cited: Andrew Rose (2005) One reason countries repay their debts: Renegotiation and international trade, Journal of Development Economics 77, 189–206. Philip Lane (2004) Empirical perspectives on long-term external debt, BE Journal of Macroeconomics 4:1:1. William B. English (1996) Understanding the costs of sovereign default: American state debts in the 1840s, American Economic Review 86, 259–275. Jeremy Bulow and and Kenneth Rogoﬀ (1988) The buyback boondoggle. Brookings Papers on Economic Activity 675–698. Philip Lane (2012) The European sovereign debt crisis. Journal of Economic Perspectives 26:3, 49–68. Serkan Arslanalp and Peter Blair Henry (2006) Debt relief. Journal of Economic Perspec- tives 20:1, 207–220. Nicolas D. Chauvin and Aart Kraay (2005) What has 100 billion dollars worth of debt relief done for low-income countries? economics working paper archive William Easterly (2001) The Elusive Quest for Growth. MIT Press, chapter 7.

One way to keep up with cases in sovereign debt is to read the IMF’s free magazine Finance and Development and The Economist. For example:

Jeremy Bulow, Carmen Reinhart, Kenneth Rogoﬀ, and Christoph Trebesch (2020) The debt pandemic. Finance and Development September, 12–16. The Economist January 21st 2017: Boats and a scandal. The Economist March 10th 2018: Rearing its odious head once more. The Economist April 25th 2020: The Peronist and the pandemic. The Economist June 6th 2020: Thanks but no. The Economist October 17th 2020: Relief eﬀorts.

Exercises

3.1. This question studies the rationales for sovereign lending using a numerical example. Suppose a small open economy has utility function ln C and random output Y given by either 12 or 8, each with probability 0.5. It can borrow from risk-neutral lenders who make zero expected proﬁts.

53 (a) Suppose it borrows only once, before knowing the value of output. In the event of a default, it loses ηY (where η is a positive fraction) due to sanctions. What is the lowest value of η that will allow it to avoid all risk? (b) Suppose that the sanction rate in fact is 10%. Find the size of the payment the borrower/sovereign can be expected to make when output is high, or to receive when output is low. (c) Suppose that no sanctions are possible, but that there is repeated lending, and the borrower tries to maximize:

2 ln C1 + βE1 ln C2 + β E1 ln C3 + ...

After a default, though, no more lending ever occurs, as a punishment. Find the lowest value of the discount factor, β, that allows the threat of this punishment to make full risk-sharing possible.

3.2. Imagine a sovereign borrower that tries to maximize the expected value of next year’s utility: E1 ln C2. In autarky it faces risky income, given by:

12 with probability 0.5 Y = 2 8 with probability 0.5

Competitive, risk-neutral lenders are considering a loan to this sovereign borrower. (a) What is the borrower’s expected utility under autarky (i.e. with no borrowing)? What is its expected utility under full insurance? (b) What is the lowest sanction rate, η, that can sustain full insurance if used as a threat?

3.3. This question studies the sanctions-based model of sovereign debt. Suppose that a sovereign borrower in autarky can experience consumption of either 80 or 120, each with probability 0.5. Competitive, risk-neutral lenders can provide consumption insurance, and they can apply a sanction rate η. (a) What is the smallest value of η that will allow full insurance? (b) Compose and carefully label a graph with η on the horizontal axis and the payment made to the lender (in the good state of the world) on the vertical axis (i.e. your graph should thus show the value of the payment for any non-negative value of η). (c) Suppose that U(C) = ln(C) and expected utility thus is EU = 0.5 ln(C) + 0.5 ln(C), where C and C are the values of consumption in the good and bad states respectively. Compose a graph with η on the horizontal axis and the borrower’s expected utility on the vertical axis.

54 3.4. Suppose that a sovereign borrower faces risky consumption that can take on two values, 1 and 1.4, with equal probabilities 0.5. Competitive, risk-neutral lenders can oﬀer to take on this consumption risk and can apply a sanction rate η on realized consumption if a default occurs. (a) What is the minimum value of η that will make complete risk-sharing possible? (b) Suppose the actual sanction rate is η = 0.05. What pattern of payments will the contract call for? (c) Suppose the borrower’s utility function is:

∞ X t E0 β 10 ln(Ct) t=0

If there were no sanctions possible but the sovereign lending relied on the reputation eﬀect (with the borrower permanently isolated after a default) then what is the smallest value of β that would allow sovereign lending?

55 Chapter 4. Measuring Capital Market Integration

Recall that in chapter 2 we assumed that the only assets countries can trade are risk- free bonds. In chapter 3 we thought about some natural limits to sovereign lending. Now we look at the other extreme and imagine that there are a wide range of assets that can be traded between countries. To make this interesting and realistic, we imagine though that the returns on these assets may be risky. The risk is not due to the possibility of sovereign default but instead due to ﬂuctuations in stock or bond markets or exchange rates.

Even though foreign investments are risky, we’ll see that they can be used to reduce the risk associated with domestic investments or with domestic consumption. The general approach that economists have taken to this question involves asking what the world would look like if capital markets were fully integrated, and deriving measures of the degree of integration. Popular indicators include:

(a) savings-investment correlations (over time or across countries)

(b) the diversiﬁcation in portfolios

Whether one looks at how capital markets work or instead at their eﬀects on consumption, it turns out that there is considerable evidence that international diversiﬁcation is far from complete.

4.1 Savings-Investment Correlations

Our ﬁrst indicator of the extent of risk-sharing or capital market integration is the correlation between saving and investment shares of GDP, either in a cross-section or a time series. The basic idea is that, if capital markets are really open, then factors that aﬀect domestic saving or investment should show up in the current account. At the other extreme, for a closed economy the saving and investment witll be perfectly correlated.

Feldstein and Horioka (1980) looked at cross-plots (ran regressions) like this, for a cross-section of countries: Ii Si = β0 + β1 , (4.1) Yi Yi

56 ˆ in data which were averages over 1960–1974. They found values of β1 insignificantly different from one, and significantly greater than zero. They concluded that capital was not very mobile. Subsequent work showed that this correlation remained high for more recent time periods, for other countries, and in time series for individual countries. So the message seemed to be that capital markets were not very open.

Early commentators on Feldstein and Horioka’s work focused on the existence of capital controls during the period of their study. Critics also noted that the inclusion of large countries—like the U.S. and Germany—biases the correlation up because world savings equals world investment. However, the correlation has outlived capital controls, and is found even for small countries.

Another criticism has focused on shocks that affect both savings and investment, even in a small open economy. A positive productivity shock with some persistence is a good example. The textbook of Schmitt-Grohé,Uribe, and Woodford works through an example. If a shock is short-lived enough to affect saving but long-lived enough to affect investment then it might induce a positive correlation, even with integrated bond markets. A concrete example might be a pandemic shock. You might imagine that would affect both savings and investment, which thus would be correlated, even if the capital market is very open.

For these reasons most researchers no longer focus on this indicator (even though it is simple to calculate) but instead look at the measures in the next two sections.

4.2 Home Bias

A second indicator of the integration of capital markets is the degree to which portfolios are diversiﬁed across countries. These portfolios could be held directly by households or indirectly through pension plans. One of the deepest puzzles in international ﬁnance is that there seems to be ‘home bias.’

4.2.1 Basic Portfolio Theory

To see what this means, start with the statistical principle that underlies diversiﬁca- tion. Imagine two returns, r1 and r2 in two diﬀerent countries. A portfolio involves some

57 stock of assets, with a share ω invested at home (country 1) and a share 1 − ω invested in the foreign country (country 2). The return on the portfolio is:

rp = ωr1 + (1 − ω)r2. (4.2)

Now suppose that ω = 0.5. The variance of the return on the portfolio will be given by: 2 2 2 σp = 0.25σ1 + 0.25σ2 + 0.5σ12, where σ12 is the covariance. This formula simply comes from thing 5 in chapter 1.3.

2 Here is a numerical example. Remember that ρ = σ12/(σ1σ2). Suppose that σ1 = 2 σ2 = 4 and that ω = 1/2. Then

2 1 2 1 2 1 1 σ = σ + σ + 2 ρσ1σ2 p 4 1 4 2 2 2 (4.3) = 1 + 1 + 2ρ = 2 + 2ρ.

Suppose that ρ = 0 so there is no correlation between returns in the two countries. Then 2 σp = 2: The variance is cut in half, relative to that of a portfolio completely invested at home. This is the magic of diversiﬁcation. Question: What do you think would happen to the variance if there were three countries? What if there were n countries?

You can see that the variance could be even smaller if ρ < 0. When two returns are negatively correlated one is said to provide a hedge against the risk in the other. On the 2 other hand, though, if ρ = 0.9 then σp = 3.8. If we underestimate ρ between two assets we will overestimate the gains from diversiﬁcation or underestimate the remaining risk.

Let us look at what might determine the weight ω. Suppose that r1 has mean µ1 and

r2 has mean µ2. Thus the expected or mean return on the portfolio is:

µp = ωµ1 + (1 − ω)µ2. (4.4)

Imagine that the portfolio managers are told to maximize expected returns but without taking on too much risk. Let us imagine that they maximize:

2 µp − λσp, (4.5)

58 where the parameter λ measures how averse to risk the managers are. We’ll next work out how this plan leads to a choice of ωopt, the optimal weight on home investments.

Formally, we imagine the portfolio problem of choosing the share in home assets, ω, to maximize:

2 2 2 2 2 µp − λσp = ωµ1 + (1 − ω)µ2 − λ ω σ1 + (1 − ω) σ2 + 2ω(1 − ω)σ12 . (4.6)

Diﬀerentiate with respect to ω, set the derivative to zero, and re-arrange the result to give:

2 µ1 − µ2 + 2λ(σ2 − σ12) ωopt = 2 2 . (4.7) 2λ(σ1 + σ2 − 2σ12) In the special case where the covariance is zero, the answer simpliﬁes to:

2 µ1 − µ2 + 2λσ2 ωopt = 2 2 . (4.8) 2λ(σ1 + σ2)

Our formula (4.7) has more greek letters than the yogurt section of the grocery store. That means we have some interesting scenarios to think about. First, look at expected

returns. Notice that ∂ωopt/∂µ1 > 0: The higher the average return at home the more you

invest there. Also, ∂ωopt/∂µ2 < 0: The higher the average return overseas the less you keep at home.

Second, you can show that ∂ωopt/∂σ1 < 0: The riskier are returns at home the smaller

the share of your investments you keep there. And ∂ωopt/∂σ2 > 0: Risk in foreign returns encourages you to stay home.

Third, the formula shows that ωopt also depends on risk aversion λ and on the covariance σ12. We’ll look at those eﬀects using numerical examples in the exercises.

Question: Suppose that µ1 = µ2 and σ1 = σ2. What is ωopt?

Before we leave the theory, there is one additional feature to discuss. Suppose that the investment goal is slightly diﬀerent from the one outlined above (4.5). Suppose that the investor has uncertain labour income, that has a covariance σ1y with domestic returns and a covariance σ2y with foreign returns. Suppose the manager of the portfolio tries to maximize: 2 µp − λσp − δσpy. (4.9)

59 In other words, there now is a third goal, which is to minimize the covariance between labour income and investment returns. This leads to a hedging motive in the choice of

ω. Our optimal weight ωopt now will reﬂect the two underlying covariances in ways which

are easy to explain economically. For example, if σ1y is positive, that would suggest a low value of ω would be a wise choice. That is like advising a relative who works for a specific firm not to also invest their savings in the stock of the same firm.

4.2.2 Evidence

What is the evidence on these ωs? They tend to be large, which is evidence of home bias. Now the term ‘home bias’ is used in two diﬀerent ways. I think there are two deﬁnitions implicitly in use:

A: A country’s portfolios are home-biased if ω is greater than that country’s share in the world capital market.

In our example above, if the two countries are the same size we then would say there is home bias if ω > 0.5. In actual data, we would say Canadian portfolios are home-biased if they contain more than 2.7% Canadian investments and US portfolios are home-biased if they contain more than 32.6% US investments, according to this deﬁnition. An appealing feature of this deﬁnition is that it does not require any additional measurements.

Even so, measuring home bias for entire countries is not easy. But here is a great example from Coeurdacier and Rey (2012) for 2008:

Table 4.1. Home Bias in Equity Holdings

Source Country Domestic market % share of Share of portfolio in world market capitalization domestic equity (%)

Australia 1.8 76.1 Brazil 1.6 98.6 China 7.8 99.2 Canada 2.7 80.2 Euro Area 13.5 56.7 Japan 8.9 73.5 South Africa 1.4 87.8 South Korea 1.4 88.5

60 Sweden 0.7 43.6 Switzerland 2.3 50.9 United Kingdom 5.1 54.5 United States 32.6 77.2

It is easier to measure these portfolios for public pension funds, for they usually must report on their interntional diversiﬁcation. For some more recent measurements, try looking at the website for the Norwegian oil fund www.nbim.no which has ω = 0 by law. Or look at the site for the CPPIB and rollover the map entries here: www.cppinvestments.com/the-fund/where-we-invest

Example: South Korea’s NPS

The National Pension Service of South Korea (like the CPPIB in Canada) is the country’s largest investor, with assets of several hundred trillion won (or billion dollars), an amount equal to roughly a third of the value of companies listed on domestic stock exchanges. In January 2011 The Economist reported that its new manager wanted to raise the proportion of assets invested abroad from 9.8% in 2010 to 12.6% in 2011, with this share to rise to 30% by 2020. NPS now is beginning to diversify internationally; for example it owns 12% of Gatwick airport outside London. The article emphasized the low recent returns in South Korea and the need to seek higher returns given the country’s age distribution. But remember that diversiﬁcation alone is a good reason to avoid such extreme home bias.|

Now, you might look at table 4.1 and say: “Even if Switzerland represents only 2.3% of the world capital market, perhaps the returns there have a high mean and low variance, so that ω = 0.509 is a good choice for them”. So the other way of deﬁning home bias that might make sense is this:

B: A country’s portfolios are home-biased if ω > ωopt, the value predicted from portfolio theory.

2 The idea here is that if σ1 is very low, for example, then it would not be surprising to ﬁnd a large value of ω. We then would say there is a home bias puzzle only if ω is even greater than we can explain with a portfolio theory.

61 But of course the challenge for this approach is that all countries seem to be home biased. Overall, most studies ﬁnd that there is home bias, by either deﬁnition, though it seems to be slowly declining .

4.2.3 Explanations?

I next invite you to pause before reading on, to think about what might explain home bias or what is missing from our theory in chapter 4.2.1. I don’t know the answer but I suppose that probably a mixture of factors is at work and that this mixture can be diﬀerent for diﬀerent countries.

OK, thank you for pausing. Here is a list of some possible factors to think about. In some cases I also list the counter-argument, in brackets.

1. Information

(a) Perhaps savers have better information about domestic investments. (But can’t money managers learn just as much about foreign investments as domestic ones?)

(b) Perhaps savers are interested in the local, social benefit of their decisions. (But it might seem a bit strange that they forego such large benefits of diversification to meet this goal.)

2. Taxes/Controls

(a) Many countries have either higher taxes or outright controls (such as quantity limits) on how much can be held oﬀshore. (These factors certainly play a role for many countries. I’m not sure how much they aﬀect the wealthy, open economies among whom capital markets are most open. Recall the case of Korea’s NPS which is home-biased even without these restrictions.)

(b) Foreign investments may be less liquid than domestic ones. (But bond portfolios, which generally are quite liquid, also seem to be home-biased.)

(c) There may be sovereign risk on holding foreign debt. (This is certainly true but probably does not apply to the bonds issued by the governments of Switzerland, the UK, or Japan, for example.)

62 (d) In holdings of foreign, private issues there may be political risk, such as the risk of additional taxes applied in the foreign jurisdiction or the risk of expropriation. (But, again, this probably does not explain the home bias within the OECD, though it might explain limited capital ﬂows to some speciﬁc emerging economies.)

3. Statistical Moments

(a) Our portfolio model predicts a high share of home investments if the foreign ones have a high return variance, so high risk. (But in that case the foreign investors should be biased oﬀshore. Yet in practice we observe they are home-biased too.)

(b) So perhaps exchange-rate risk explains why foreign returns seem risky when viewed by us and why our returns seem risky when viewed by them, which would explain home bias. (But pairs of countries with firmly-fixed exchange rates or a common currency also seem to have home bias. And investors can hedge exchange rate risk yet still gain diversification, yet many do not do this.)

(c) Perhaps savers hold home-biased portfolios because they wish to hedge against risk in their labour income. (But this requires that there be a negative covariance between domestic capital income and labour income, which could be investigated for diﬀerent countries. If there is a positive covariance then that would simply deepen the puzzle, because in that case savers should tilt their portfolios even more oﬀshore.)

4. Measurement

(a) Possibly holding shares in home-based firms can provide international diversification, if their earnings are diversified internationally. (But I’m not sure for how many firms that would be true in practice.)

(b) Possibly investors hold exchange traded funds (ETFs) that are locally listed but indexed to some foreign stock markets. (I am not sure in practice how these holdings are counted in measurements of home bias.)

4.3 Consumption Correlations in Seven Steps

A third indicator of capital-market integration is the correlation across countries in consumption (or consumption growth). The focus here is on studying whether international

63 ﬁnancial markets deliver an outcome that diversiﬁes away risk as much as possible. We don’t study the means of doing this—via holding bonds, equity, or making remittance payments—directly. Instead we look at the outcomes. Incidentally, we can use the same approach for regions within a country (this is called intranational economics).

We’ll approach this topic in seven steps.

Step 1: What is risk-sharing?

Pooling income can reduce risk and so raise expected utility, potentially for everyone. What are some social and economic institutions that provide risk-sharing? Asset markets are one such institution, but there are many others.

Step 2: ‘Complete markets’ in economic theory.

Once there are many asset markets solving for a competitive equilibrium (CE) in an economic model can become very diﬃcult. There now are many prices and asset-holdings to calculate, with the exogenous variables random. For example, imagine adding randomness and many assets or time periods to our two-period, two-country problems from chapter 2.

Step 3: The fundamental theorems of welfare economics come to the rescue.

See if you can recall these theorems from microeconomics. The second welfare theorem provides a calculation device that allows us to study what a CE implies for consumption. One reason that this calculation is simple is that it involves no prices, just quantities.

Step 4: Maximizing a sum of utilities gives a Pareto optimum.

Imagine that there are two economies, each of which receives a stream of nonstorable ∗ endowments of a single good. Call these endowments {Yt,Yt }. Each country has log utility. So this is like our two-country model in chapter 2.6 but the incomes are random.

Next imagine assigning or allocating consumption to each country. Suppose we maximize: ∗ ln Ct + ln Ct , (4.10) subject to: ∗ ∗ Ct + Ct = Yt + Yt . (4.11)

64 Then see if you can show that:

Y + Y ∗ C = C∗ = t t . (4.12) t t 2

Because we’ve maximized the sum, we can’t make one element larger without making the other one smaller, which is the deﬁnition of a Pareto optimum.

Step 5: Maximizing a weighted average of utilities gives all of the Pareto optima. One of them is the CE.

Suppose γ is a fraction and we now maximize:

∗ γ ln Ct + (1 − γ) ln Ct , (4.13)

subject to: ∗ ∗ Ct + Ct = Yt + Yt . (4.14)

Then see if you can show that:

∗ ∗ ∗ Ct = γ(Yt + Yt ) and Ct = (1 − γ)(Yt + Yt ). (4.15)

Here γ and 1 − γ are called the welfare weights. This mathematical example traces out the Pareto frontier or the Edgeworth-Bewley box diagram, in case either term is familiar. With this second problem we’re simply allowing for the fact that the countries may diﬀer in size. By choosing γ we can get their shares of world consumption to mimic those in the CE.

Step 6: The key indicator: The correlation between consumptions in diﬀerent locations.

Notice that the examples in step 4 or 5 predict that relative consumption is constant, or that the two consumptions are always in the same proportion to one another. Thus: (a) ∗ ∗ ∗ ρ(Ct,Ct ) = 1 and (b) ρ(Ct,Ct ) ≥ ρ(Yt,Yt ). The theory predicts that the consumption correlation is 1 and that it exceeds the income correlation.

You could try checking on these properties by simulating some random numbers in excel. But let us look at a detailed, concrete example. Again we’ll use thing 5 in chapter 1.3. This will get a bit complicated for one page, so please skip to ahead step 7 if you wish.

65 Suppose that the countries are the same size, so γ = 0.5. Each country has an equal claim on the output of the other country, and so each period the world ‘pie’ is simply divided in 2. Now, let us imagine that each country’s output consists of a common component and an idiosyncratic component. Thus

Yt = zt + t (4.16) ∗ ∗ Yt = zt + t .

∗ Let us suppose that cov(zt, t) = cov(zt, t ) = 0. To make this even simpler, suppose zt, ∗ ∗ t, and t each has a mean of zero. And suppose that t and t have the same variance: 2 σ . This is starting to look complicated, so let us make each variance the same number: 2.

First let us ﬁgure out the variance and covariance of the two incomes. Now

2 2 2 2 σY = σY ∗ = σz + σ = 2 + 2 = 4. (4.17)

That’s because the variance of the sum is the sum of the variances, because there is no ∗ covariance between zt and either t or t . If you recall that everything has a mean of zero, the covariance is: ∗ ∗ σ ∗ = E(Y · Y ) = E (z + )(z + ) Yt,Yt t t t t t t 2 ∗ ∗ = E(zt + ztt + ztt + tt ) (4.18) = 2 + 0 + 0 + 0 = 2.

Thus the cross-country correlation of income is ρY,Y ∗ = 2/4 = 0.5. The incomes are positively but not perfectly correlated.

Now let us look at the consumptions. We already know that

∗ ∗ Ct = Ct = 0.5(Yt + Yt ). (4.12)

Thus the consumption correlation is 1. Since we’ve come this far let us also ﬁgure out the variance of consumption:

2 2 2 2 2 σ = 0.5 σ + 0.5 σ ∗ + 2(0.5)(0.5)σ ∗ C Y Y Yt,Yt = 0.25(4) + 0.25(4) + 0.5(2) (4.19)

= 1 + 1 + 1 = 3.

66 Thus consumption is also less volatile than income.

7. What is the evidence?

Please see Olivei (2000) for a short, readable summary of this logic and of some evidence. Overall, the consumption correlations historically seem to be less than one, and sometimes to be less than the income correlations. Possibly capital markets are much less integrated than we might have supposed. It is easy to assemble evidence for diﬀerent regions, countries or time periods. (And we could look up how to do statistical inference on sample correlation coeﬃcients too.) The exercises ask you to check on some evidence and see if the correlation has risen over time.

Further Reading Here are the articles cited in this chapter, in the order cited: Nicolas Coeurdacier and HélèneRey (2012) Home bias in open economy financial macroeconomics. Journal of Economic Literature 51, 63–115. Giovanni P. Olivei (2000) Consumption risk-sharing across G7 countries. New England Economic Review, March/April 3–14

Exercises

4.1. This question studies the eﬀect of exchange-rate risk on an investment portfolio, using a simple theoretical model. Suppose an investment manager puts a share ω of a portfolio in a domestic investment, with return r. This return has mean 2 and standard deviation 1. The manager then puts a share 1 − ω in a foreign investment with return:

r∗ + ∆s,

where r∗ is the foreign return and ∆s is the rate of depreciation of the domestic currency. The foreign return r∗ also has mean 2 and standard deviation 1. The rate of depreciation has mean 0 (so depreciation cannot be predicted) and standard deviation σ. Suppose that ∗ there are no correlations between r, r , and ∆s. Finally, let us call the portfolio return rp; it has mean µp and standard deviation σp. The manager is instructed to maximize:

2 µp − σp.

67 (a) Find a formula for the optimal portfolio weight. √ (b) Report the optimal share held in the domestic asset ﬁrst if σ = 0 and then if σ = 2. (c) Suppose that there were actually a positive correlation between the foreign return r∗ and the rate of depreciation ∆s. Qualitatively, what eﬀect would that have on the optimal portfolio weight ω?

4.2. Suppose that an investigator observes that the average portfolio has a weight ω = 0.75 in home assets and 0.25 in foreign assets. From other studies, she knows that the degree of risk aversion is λ = 1, so that the average household maximizes:

2 µp − σp,

2 where µp is the mean return on the portfolio and σp is the variance of the return on the portfolio. Domestic returns have mean µ and variance σ2, while foreign returns have mean µ∗ and variance σ2∗. (a) If there is no covariance between domestic and foreign returns, ﬁnd an expression for the optimal portfolio weight. (b) Suppose that, in assessing investment portfolios, we do not know the details of foreign holdings, so we cannot easily estimate the mean and variance of foreign returns. But suppose that we know the mean of the domestic return is 2 and the variance is 4. Find the combinations of foreign return means and variances that are consistent with the observed portfolio being the optimal one.

4.3. Suppose that European investors can invest in Europe and earn returns with mean 0.9 and standard deviation 1. They also can invest in the rest of the world and earn returns with mean 1 and standard deviation 1. The correlation between the two returns is 0. Call ω the weight on home (European) investments in their portfolio, so 1 − ω is the weight on the returns in the rest of the world. Their portfolio return has mean µp and standard deviation σp. The investors seek to maximize:

2 µp − λσp, so that λ measures their aversion to risk. (a) If λ = 1 then what is the optimal weight ω on home investments? (b) Suppose that we observe that ω = 0.75. An economist tries to explain that large value by arguing that investors are over-optimistic about the average return in the home market (i.e. they believe it is higher than 0.9). What must the expected return in Europe be to account for the observed value of ω? (c) Is it possible that, instead, the large observed value of ω is due to a value of λ that is greater than 1?

4.4. Suppose that ﬁnancial investments in developed economies have returns with mean 2 µ1 = 1 and variance σ1 = 1. Meanwhile, investments in emerging economies have returns

68 2 with mean µ2 = 2 and variance σ2 = 2. The correlation between the two returns is zero. Investors in developed economies hold a portfolio share ω in investments in those same economies and a share 1 − ω in emerging-economy investments. (a) If the portfolio managers maximize:

2 µp − λσp, then ﬁnd the optimal ω as a function of λ, which measures how risk-averse they are. (b) If we observe ω = 0.6 then what must λ be? (c) Brieﬂy explain any problems with explaining home bias with high risk aversion.

4.5. Suppose that home investments have a return r1 with mean 1 and variance 1.44. Foreign investments have a return with mean 1.5 and variance 2.25, which is higher than the variance of r1 perhaps due to exchange-rate risk. The correlation between the two returns is 0.5. Suppose investors hold a share ω of their savings in home assets and a share 2 1 − ω in foreign assets. Their portfolio return has mean µp and variance σp. (a) A researcher believes that investors seek to maximize:

2 µp − λσp, and that λ = 0.5. Solve for the predicted, optimal portfolio weights. (b) Suppose that in fact λ = 4. Will it appear to the investigator that the investors are home biased? (c) Does the design of this problem — underestimating risk aversion combined with riskier foreign returns — provide a convincing explanation for home bias in portfolios?

4.6. Suppose that households in Europe and the United States each have the same utility function: C1−α U(C ) = t t 1 − α where α is the coeﬃcient of relative risk aversion. Also suppose that output in Europe is given by: ∗ ∗ Yt = (1 − ω)zt + ηt , and output in the US by Yt = ωzt + ηt,

where zt is a common, world component of output. There is no covariance between any ∗ of the world component and the two country-speciﬁc components of output, ηt and ηt . The parameters ω and 1 − ω simply measure the impact of the world shock on each area. Imagine that the shocks have no persistence, and output cannot be stored from period to ∗ period. The shocks ηt and ηt each have a mean of zero, while zt has a mean labelled µ.

69 (a) Use a Pareto problem (not with equal weights) to ﬁnd the combinations of consumptions that mimic complete sharing of risk. (b) Do the weights on the two countries’ utilities matter to the predicted, international consumption correlation? (c) The two country-speciﬁc shocks have a variance of 1 while the common shock has a variance of 2.Find expressions for the variance of consumption and the variance of income in the home (unstarred) economy.

4.7. Suppose that households in Europe and the United States each have the same utility function: C1−α U(C ) = t t 1 − α where α is the coefficient of relative risk aversion. Also suppose that output in Europe is given by: ∗ ∗ Yt = (1 − ω)zt + ηt , and output in the US by Yt = ωzt + ηt, where zt is a common, world component of output. There is no covariance between any ∗ of the world component and the two country-specific components of output, ηt and ηt . The parameters ω and 1 − ω simply measure the impact of the world shock on each area. Imagine that the shocks have no persistence, and output cannot be stored from period to ∗ period. The shocks ηt and ηt each have a mean of zero, while zt has a mean labelled µ. (a) Use a Pareto problem (not with equal weights) to find the combinations of consumptions that mimic complete sharing of risk. (b) Do the weights on the two countries’ utilities matter to the predicted, international consumption correlation?

70 Part B: Exchange Rates and Asset Prices

71 Chapter 5. Real Exchange Rates

Having studied how capital markets are connected across countries we now turn to study how goods markets are connected internationally. The focus will be on prices: how we can compare them across countries or regions and whether they are converging over time. Taylor and Taylor (2004) provide a great review to be read with this chapter.

Chapter 5.1 begins with some statistical terms and tools we need. Then chapter 5.2 studies purchasing power parity, a 100-year-old idea that is still very interesting to study. Chapter 5.3 then decomposes the real exchange rate in various ways that reveal more about why PPP might not hold.

5.1 Predictability and Persistence

In macroeconomics and ﬁnance we often want to study whether something is predictable or forecastable or how persistent it is. For example, we’ll study these properties for both real exchange rates and nominal exchange rates. A time-series variable is predictable if it is correlated with one or more variables known in an earlier time period. It is persistent if large or small values tend to persist.

5.1.1 First-Order Autoregression

The easiest way to see what these properties mean in practice is to imagine collecting a time series yt, then creating a separate series of the lagged values yt−1. Now imagine creating a scatterplot with yt−1 on the horizontal axis and yt on the vertical axis. Suppose you drew a straight line through the scatter of points and called the intercept µ and the slope ρ. Then the formula for the line would be yt = µ + ρyt−1.

Of course this line won’t ﬁt the points exactly, so let us label the departure up or down of the observation yt with the symbol t. (If you have studied econometrics you will recognize that it is partly about how to estimate µ and ρ to make these departures as small as possible. We’ll just assume that we know those two parameter values exactly and so we won’t writeµ ˆ andρ ˆ in this section.) Then the observations would satisfy:

yt = µ + ρyt−1 + t. (5.1)

72 Using statistical methods to estimate µ and ρ is sometimes called regression, so this equa-

tion (5.1 1) is called an autoregression, because it involves regressing yt on its own, lagged value.

You also can think of this relationship as telling you how to forecast yt+1 given yt. We’ll write that forecast as:

E(yt+1|yt) = Etyt+1 = µ + ρyt. (5.2)

So yt+1 is predictable or forecastable (though the conﬁdence interval could still be wide) as long as the two parameters µ and ρ are not zero.

You might wonder what happened to the term t+1. If statisticians do a good job at

ﬁtting the line to the original scatterplot then the mean of t will equal zero. We’ll also assume that:

Ett+1 = 0 (5.3) so t+1 is itself unpredictable. It will materialize in the actual value of yt+1, but its value cannot be predicted in advance. Overall, then, the original autoregression formula just expresses how a value of y can be broken into two parts: its conditional expectation or forecast and an error term that cannot be predicted. Thus:

Etyt+1 = Et(µ + ρyt + t+1) (5.4) = µ + ρyt.

So much for predictability. Now persistence refers to the precise value of ρ. In economic time series we usually ﬁnd that ρ lies between 0 and 1. The closer it is to 1, the more persistent the series is said to be. To see why this makes sense, lag the original equation

(5.1) and substitute for yt−1, then repeat to ﬁnd:

yt = µ + ρyt−1 + t

2 = µ + ρµ + ρ yt−2 + ρt−1 + t (5.5)

2 2 t = µ(1 + ρ + ρ ...) + t + ρt−1 + ρ t−2 + ... + ρ y0.

This is sometimes called solving a diﬀerence equation backward.

73 The result (5.5) shows that you can think of the value of yt as having evolved from a series of past shocks. The higher the value of ρ, the greater the impact of past shocks on the current value, and so the more persistent the series is said to be. When ρ is near zero

the time series of yt will simply vary randomly around µ. But when ρ becomes a larger, positive fraction, the series will exhibit cycles: persistent runs of large or small values.

To convince yourself this all makes sense, open ExcelTM and be sure you have installed the Analysis Toolpak. Go to the ‘Data’ tab at the top of the screen then to ‘Data Analysis’ on the left top of the next screen. That takes you to a dialog box where you select ‘Random Number Generation’. Create a normal random variable with mean 0 and any standard deviation you like and place perhaps 100 observations in column A of your spreadsheet.

These are your values for t, t = 1, ..., 100.

Next, choose some values for µ an ρ. A value of ρ betwen 0 and 1 is a realistic choice.

Then choose a starting value y1. I suggest choosing y1 = µ/(1 − ρ) because that seems to be where the inﬁnite series at the start of equation (5.5.) is aiming. Place this value in cell B1. Finally, in B2 create a formula by typing: ‘= µ + ρ ∗ B1 + A2’, where you use your numerical values for µ and ρ. That is simply equation (5.1). Then copy this formula and paste it in the rest of the B column down to row 100. Well done! You now have simulated the ﬁrst-order autoregression.

The last step I suggest is to graph your simulated data in a time series plot. And if you experiment with several diﬀerent values of ρ (perhaps by using columns C and D) you will see that the higher the value of ρ the more momentum the series appears to have or the longer the duration of the cycles it displays.

5.1.2 Rules of Forecasting and Rational Expectations

Often we assume that someone besides ourselves is doing the forecasting. For example, we might be studying nominal exchange rates that depend on the forecasts that participants in the foreign exchange market make of some variable yt. Most of the time we assume that they forecast just the way a statistician would, by using the correlation patterns or regressions available in the historical data. Attributing this type of forecasting to them is sometimes called assuming that their expectations are ‘rational’.

74 Rational expectations have four properties that we’ll need to keep track of. We can study those using the ﬁrst-order autoregression. First, forecast errors have a mean of zero:

E t = 0. (5.6)

This property just means that the forecaster has correctly used the past data and ﬁt the regression line. To see why it makes sense, imagine a forecaster saying to you: “I am a good forecaster. My average forecast over the last 300 weeks is just 10% above the average actual value.” You might reply: “Why don’t you revise your forecasts down by that amount?”

The second property we’ve also seen already. Forecast errors should be unpredictable:

E(t+1|t) = 0. (5.7)

Again this is a natural property of good forecasts. If you see runs of positive or negative forecast errors, you would revise your forecasts accordingly. The result is a forecast error that is not autocorrelated, sometimes called white noise.

Third, we also can see how to construct multi-step forecasts. Notice that:

yt+2 = µ + ρyt+1 + t+2 (5.8) = µ + ρ(µ + ρyt + t+1) + t+2, so that 2 Etyt+2 = µ + ρµ + ρ yt (5.9) = µ + ρEtyt+1. This is called the chain rule of forecasting. It just means that you build up multi-step forecasts by induction or compounding the single-step one.

Fourth, the forecast today of what tomorrow’s forecast will be for some event further in the future still is simply today’s forecast:

EtEt+1yt+2 = Etyt+2. (5.10)

This is called the law of iterated expectations (LIE). Again it’s simply a property that rational forecasts must have. To see why this makes sense, imagine a weather forecaster

75 who says: “It’s winter now [t] and I’m predicting a hot summer [t + 2]. But if you ask me again in the spring [t+1] I expect that by then I’ll be predicting a cold summer.” The LIE implies that there won’t be such predictable revisions in forecasts. One’s winter forecast about the summer should be the same as one’s winter forecast of what the spring forecast for the summer will be. This property is a bit arcane but will appear once or twice later in this book.

5.1.3 Random Walks

Sometimes in international macroeconomics and ﬁnance both theory and empirical evidence point to a particular example of the autoregression called a random walk. It is deﬁned by the condition that ρ = 1. In that case,

yt = µ + yt−1 + t. (5.11)

The parameter µ is called the drift in the random walk.

Imagine a random walk with no drift. You can see that:

Etyt+1 = yt. (5.12)

The forecast of next period’s value is just today’s value. But then using the chain rule of forecasting (5.9) gives the forecast at some further horizon h:

Etyt+h = yt, (5.13) for any h > 0. So you cannot improve upon the current value as a forecast for any future value.

A random walk gives us the most possible persistence. Glancing back at equation (5.5) will show you that the impact of past shocks never dies out. If you study econometrics you can see how one could test whether a random walk holds with a hypothesis test of ρ = 1. Informally, our original scatterplot would have a slope of 45 degrees if the series follows a random walk.

76 With a random walk the future level is predicted with the current level, but notice that the change is unpredictable. Again if the drift is zero, imagine the ﬁrst-order autoregression

with yt−1 subtracted from each side:

yt − yt−1 = (ρ − 1)yt−1 + t. (5.14)

If ρ = 1 then the change in the series is purely random and won’t be correlated with

anything. If we constructed a scatter plot with ∆yt on the vertical axis and yt−1 on the horizontal axis, we would ﬁnd no relation or slope evident. In chapter 1.1.3 we saw that

this tends to be true for the log nominal exchange rate st.

In contrast, if ρ < 1 then ρ−1 < 0 and this second scatterplot would slope down. Low levels of the series would tend to be followed by large positive changes and high levels would tend to be followed by low or negative changes. A series with this property is not a random walk but instead is sometimes said to be mean-reverting, because of this property of the second scatterplot. A key question is whether the log real exchange rate is mean-reverting or not.

5.2 PPP

5.2.1 The Real Exchange Rate

Begin with the deﬁnition of the real exchange rate that we saw in chapter 1.1.2:

SP ∗ Q = . (5.15) P

This gives the relative price of goods. The numerator is just the price of goods in the foreign country, converted into the home currency. So the ratio tells us whether prices are higher in the foreign country (Q > 1) or in the home country (Q < 1).

Constructing Q for a pair of countries begins with collecting S, the nominal exchange rate. Let us pause to note three rules for working with exchange rate measurements. First, if the data series is in USD then remember you need to invert it to put it in local currency. In other words, between countries 1 and 2 S1/2 = 1/S2/1. Second, remember that if you have exchange rates against the USD then you can construct them for any

77 pair of countries. For example SCAD/GBP = SCAD/USD × SUSD/GBP . Third, recall that

s is the log exchange rate and notice that ∆s1/2 = −∆s2/1. That simply means that if the CAD is depreciating against the USD then the USD is appreciating against the CAD. These same three rules apply to real exchange rates.

When Q rises that is a real depreciation for the home country (local prices are falling). When Q falls, that is a real appreciation (local prices are rising).

5.2.2 Does Q Predict S?

One of the hypotheses we study is H0 : Q = 1. When the prices are for an individual commodity we call this hypothesis the law of one price (LOP). For baskets of goods (like the ones in the CPI) we call the hypothesis purchasing power parity (PPP).

Remember that if Q < 1 then goods are relatively cheap in the foreign country. So if PPP holds on average we would expect their relative price to rise. One way that can happen is through an appreciation of the foreign currency (depreciation of the home currency): a rise in S. In these circumstances, the foreign currency is sometimes said to be undervalued and so PPP predicts that it will appreciate. (Of course, the prices could adjust too, but exchange rates tend to change more often.) Conversely, if Q > 1 then PPP predicts a depreciation of the foreign currency. As we discussed in chapter 1.1.3, predicting changes in S would be very useful (and presumably lucrative).

The Economist periodically reports Q using the price of a Big Mac (which I am told is a fast-food product) in diﬀerent cities and countries. Please pause and look up their page to see whether the value of Q for that speciﬁc good actually predicts subsequent changes in S, the nominal exchange rate.

As we’ll see later on, the US Treasury has long argued that China’s currency, the renminbi, is undervalued, and that the Chinese government is preventing its currency from appreciating.

5.2.3 International Income Comparisons

Now let us detour for a moment to think about the question of how real income can be compared across countries. Suppose that real GDP per capita is $30,000 in the US and

78 JPY3,000,000 in Japan. The simplest way to compare these is to convert the Japanese value into USD by multiplying it by S. But ﬂoating nominal exchange rates go through large swings, as you can see at the Paciﬁc Exchange Rate service, and we really don’t think relative GDP per capita varies so much from quarter to quarter.

So economists usually use a wiser way to make the comparison: they divide each income per capita by the local price level of the same basket of goods. This way they

measure real income, and compare $30, 000/PUS with JPY3, 000, 000/PJ .

You can see that these two methods will give the same answer if:

PUS S$/JP Y = , (5.16) PJ in other words if Q = 1 and PPP holds.

Let me emphasize that PPP generally does not hold, so the two methods give diﬀerent answers, and the second method is best. Now, to make matters confusing, the second method is sometimes referred to as a PPP-based income comparison. A better name would be a purchasing-power-based comparison.

To see the evidence, go to gapminder tools on the web. Click on the small arrow below the ? symbol on the vertical axis, and then change the y-axis variable to economy/incomes & growth/ GDP/capita (US$, inﬂation-adjusted). Make sure that the variables on both axes are in logs. Good work! You now have the inexpensive, exchange- rate-based measure on the vertical axis and the expensive, PPP-labelled measure on the horizontal axis. Now rollover various countries and you will discover a very important fact: For low-income countries the PPP-labelled measure tends to suggest higher GDP per capita than the USD measure does.

Another way of expressing this fact is that prices tend to be lower in low-income countries. If you estimate the price of bread in Egypt by multiplying the price in the US by the exchange rate you will overestimate it. Thus you will underestimate the level of real GDP per capita. This is sometimes called the Balassa-Samuleson eﬀect, and section 5.3.2 will suggest one way we might try to explain it. Of course notice that the income diﬀerences across countries are still vast, even with this adjustment.

79 Having made the point that prices tend to be lower in low-income countries, let us pause to add an important additional fact. If you ﬁt a regression line to your gapminder plot then sveral African countries (shown in cyan) would lie above that line. There prices are higher than one would expect, given there development indicators, and so there PPP- adjusted income per capita is lower than one would expect. The further reading at the end of this chapter lists an article from The Economist with examples and discussion.

Are you aware of any other regions or areas (perhaps remote ones) that have relatively low incomes yet surprisingly high prices? The further reading section also directs us to a map of the US constructed by the Tax Foundation (2016) that shows GDP per capita in each state adjusted for diﬀerences in purchasing power.

While you are at gapminder you might also run some animations over time. First, check the boxes on the right-hand side for Japan, Singapore, South Korea, and the United States. Then run the animation arrow at the lower left corner of the screen. You will see how the ﬁrst three countries caught up to the US over time, depending on which income measure you use.

Second, you also can use the menus to graph total income for a country (rather than income per capita). I’m not sure that total income matters for many purposes. But running that animation can show you when or whether the size of the economy of India overtook that of the UK or Japan, or the size of the economy of China overtook that of Japan or the US, again depending on which income measure you use.

The challenge in the second method is that we need to measure the prices of the same goods in the two different countries. But for most countries the routine measurements of the general level of prices use an index, like the CPI, that has no natural units. And the CPI is not directly comparable across countries, for example because it may have a different base year in different countries.

5.2.4 Growth Rates

How can we study PPP using price indexes that are generally available then? One

80 way is to look at it in growth rates:

∆q = ∆s + π∗ − π, (5.17) remembering that lower case letters denote logs. Relative PPP would hold if ∆q = 0: whatever the level of q, it would not then change. You can see that, if that hypothesis held, then ∆s = π − π∗. (5.18)

The prediction then would be that two countries with a fixed exchange rate (so ∆s = 0) would have the same inflation rate. And for countries with a floating exchange rate the rate of depreciation would equal the inflation difference. (This is just the Big Mac exercise, but in growth rates.)

For some pairs of countries this works well, especially when the inflation differences are large. For example, Zimbabwe has an astronomical inflation rate and an astronomical rate of nominal depreciation.

5.2.5 Statistical Tests of Long-Run PPP

So far we’ve seen how useful PPP would be: It would help us predict exchange-rate movements and compare incomes across countries. How can we test whether it holds or not? The simplest way to do this is to construct the log real exchange rates, q, for some pairs of countries, and simply graph them. Usually we see that the resulting time-series plot shows large, persistent swings.

A simple way to describe movements in the real exchange rate, or to test for PPP, is to ﬁt a regression like this:

qt = µ + ρqt−1 + t. (5.19)

This is the first-order autoregression from chapter 5.1. The coefficientρ ˆ measures how persistent the swings are. For example, some studies findρ ˆ = 0.85 on average.

This equation is designed to test for a weak version of PPP. As long as ρ < 1 the real exchange rate will tend to slowly return to some long-run average level. That level won’t be Q = 1 or q = 0 necessarily (because of the index number problem), and the speed of

81 return could be quite slow, but there still would be some tendency in that direction. If we ﬁnd that ρ < 1 we say that there is long-run PPP or that the real exchange rate is mean-reverting. If instead ρ = 1 then PPP doesn’t hold even in the long run. The real exchange rate is said to follow a random walk, with no long-run, average level.

Visually, you can test the hypothesis that ρ = 1 using a scatter plot of qt against qt−1, by seeing whether the slope is 1 or less than 1. Or you can run the regression and do a one-sided t-test.

Another way to do this is to subtract qt−1 from both sides of the equation (5.19) to give:

∆qt = µ + (ρ − 1)qt−1 + t. (5.20)

In this second scatter plot, the slope will be zero if long-run PPP does not hold. The slope will be negative if long-run PPP does hold, because then ρ − 1 < 0. The variable on the left-hand side now is the rate of real depreciation. If this scatter plot or regression line slopes down then the real exchange rate is mean-reverting: a low value of qt−1 is likely to be followed by an increase, and a high value by a decrease.

One of the issues involved in the statistical tests is test power. Remember that a test with low power is one that may have diﬃculty rejecting a false null hypothesis. And it turns out that tests for the null that ρ = 1 do suﬀer from low power, for a fairly obvious reason. If in fact ρ = 0.9 then the swings in the real exchange rate will be at very low frequency, so it will take a long span of data (maybe many decades) to conclusively distinguish between ρ = 0.9 and ρ = 1.

What is the evidence? There have been hundreds of econometric studies, for all sorts of countries and time periods. Some find that ρ = 1, but a small majority find that ρ < 1, though it is a large fraction like 0.9 or 0.95. Rather than throwing up our arms and celebrating the success of the hypothesis of long-run PPP, though, we should simply take from these findings that real exchange rates are very persistent and think about how we might explain that fact.

What could explain the limited evidence for PPP? Economists have pointed to things

82 like:

(a) shipping costs;

(b) imperfect competition and pricing to market;

(d) the role of non-traded goods.

The next section decomposes Q to see if that sheds light on some of these candidate explanations.

5.3 Decomposing the Real Exchange Rate

Next we see some leading explanations for the large, persistent swings in real exchange rates that we observe in most countries. To start, let us think of goods and services as being grouped into traded and non-traded groups, labelled T and N. So the domestic CPI is: 1−n n P = PT PN , (5.21) where the superscript n is the share of nontraded goods and services in the domestic consumption basket. Notice that the index number has ‘constant returns to scale’ since the exponents sum to 1. Be sure you see why this makes sense.

With a bit of patience we then can write the real exchange rate as:

SP ∗ Q = P SP ∗1−nP ∗n = T N 1−n n (5.22) PT PN ∗ ∗ n SPT PT PN = × ∗ , PT PT PN where I have kept this as simple as possible by assuming that the share of non-traded goods, n, is the same in both countries. Now, research on international prices can be separated into two streams, depending on which of these two terms it focuses on, so let us look at these terms in turn.

5.3.1 Departures from the Law of One Price

83 You can see that the ﬁrst possible explanation for big swings in Q is that the law of ∗ one price does not hold. If the LOP did hold, then PT = SPT and the ﬁrst term in the decomposition (5.22) would be 1. But what is the evidence on this connection between traded goods prices across countries?

First, when researchers classify goods in the CPI into the T and N categories they can construct the ﬁrst term in our decomposition. Doing that shows that the variance of the LOP departure is at least 50% of the variance of Q. So at least half the mystery of departures from PPP is due to departures from the LOP.

Second, the same conclusion holds when we focus on the persistence of LOP deviations. We can construct the term: SP ∗ T (5.23) PT for any single traded good or category of them, then see how persistent the resulting ratio is. It turns out that it often is very persistent, though not as persistent as Q.

When you tell friends that prices are persistently different in different countries they might reply that this finding is not surprising because of the costs of transportation. For example, groceries are expensive in Yellowknife. So to see whether that also explains international differences, we can imagine using intra-national price differences as a sort of control group.

Engel and Rogers did just this in a classic study of the LOP. They studied 14 goods in 23 US and Canadian cities between 1978 and 1994. First, they converted all prices to USD. Next, they calculated: SP ∗ std dev ∆ln T PT for each good and each city, over time. Notice that this set of statistics thus includes the volatilities of relative prices between cities within the same country. The highest standard deviations were found for fuel, public transit, and women’s apparel.

Finally, they ran a linear regression of these standard deviations on (a) the distance between the two cities and (b) a dummy variable equal to 1 if the two cities were in diﬀerent countries. Crossing the border was associated with higher relative price volatility,

84 and distance was too, so they asked what sort of extra distance would have to apply between two cities in the same country in order to mimic the greater volatility they would have if they were in diﬀerent countries. The answer was 1780 miles, a very large ‘border eﬀect’.

One way to think about these border eﬀects is to take the price ratio (5.23) and rewrite it in logarithms. If the LOP held then in logs:

∗ pT = s + pT , (5.24) so that the local log price would respond 1:1 to changes in the exchange rate or in the foreign price. Researchers often investigate the extent to which these responses do not happen, by running regressions like this:

∗ pT = β0 + β1s + β2pT + β3x, (5.25) where x is anything else that might change over time and aﬀect the local price.

Some studies investigate the identical product sold by the same firm in different loca- ∗ tions or currencies. In this case pT is the log export price and pT is the log price in the producer’s country. The extra variable x is the transport cost. Classic studies along these lines have looked at automobiles, books, and magazines. They usually find that the export price is not simply the producer-country price adjusted for the exchange rate. Instead, there seems to be evidence of pricing to market (PTM). How can producer’s carry that off? Why would they want to do so?

A different group of studies looks at commodities like fruits and vegetables that typi- ∗ cally are sold by different firms in different countries. Now pT might be the US wholesale price and pT the Canadian wholesale or perhaps retail price. These studies are said to measure exchange-rate pass-through (ERPT) and the coefficient β1 measures the rate of

pass-through. Most studies of pass-through ﬁnd that β1 is roughly 0.5, so about half of a change in the exchange rate shows up in local currency prices.

Who cares what the pass-through rate is? It turns out that it matters a lot to monetary policy, especially in smaller, very open economies. There is some evidence that the value of

85 β1 has fallen over time. That means that a given depreciation, say, will have less effect on domestic inflation than it used to. And some economists have suggested that the decline is due to the low overall inflation rate in many countries over the past 20 years. To see this low ERPT in inaction, simply note that the Canadian dollar has at times depreciated persistently against the USD over the past 20 years, for example in the 2008 or 2015. Yet those episodes have not led to bursts of inflation in Canada.

In keeping with this explanation, pass-through also tends to be higher in countries with high inﬂation rates. That means that a change in S in those countries will lead to a relatively large change in PT and hence in domestic inﬂation. In turn, monetary policy may need to closely react to exchange-rate changes therefore.

5.3.2 The Balassa-Samuelson Eﬀect

The second term in our decomposition (5.22) also might explain why price levels diﬀer across countries and PPP does not hold. That term is:

∗ n PT PN ∗ . (5.26) PT PN

To study this term, imagine an economy with sectors N and T and suppose that workers can move between the two sectors, so that the nominal wage is the same in both sectors. Denote output by Y and suppose for simplicity that we ignore capital, so the production function is: Y = AL, (5.27) where A is TFP in this sector. I am omitting the sector subscripts for ease of reading. Let L be the number of workers hired. Proﬁt is given by:

PY − WL = P (AL) − W L. (5.28)

It won’t surprise you that maximizing proﬁt leads to the real wage (as paid by the ﬁrm) being equal to the marginal product of labour:

W W = AN and = AT , (5.29) PN PT

86 which gives us our key result: P A T = N , (5.30) PN AT or in logs:

pT − pN = aN − aT . (5.31)

The simple way to remember this result (5.31) is to ignore the T and N for a moment and simply imagine two goods: opera tickets and hand calculators. Over time, productivity growth in hand calculators has been much greater than productivity growth in opera tickets (though surtitles are great) and so the relative price of hand calculators has fallen over time, while opera tickets of course strike all of us as expensive.

Armed with this result, let us ignore departures from the LOP, just for simplicity. Then the real exchange rate (5.22) becomes:

∗ n AN AT Q = ∗ . (5.32) AN AT This is the Balassa-Samuelson model, which relates international price diﬀerences to productivity diﬀerences.

With four terms on the right-hand side it looks like anything could happen to the real exchange rate. It is helpful to look at two especially interesting examples.

∗ Example 1: AN = AN = 1

In this case, productivity in the nontraded sector is both constant and equal across countries. You can think about how realistic this simplification is. The idea is that one person with scissors administers a haircut in each country. This example usually is used to apply to very different economies, where the difference in productivity in the traded sector is much larger than in the non-traded sector.

In this example, then: A∗ n Q = T . (5.33) AT ∗ Imagine that India is the home country and the US is the foreign country. Then AT < AT and Q > 1: prices are higher in the US than in India. This is the key Balassa-Samuelson

87 result: price levels are lower in poorer countries. We saw the evidence for this pattern in section 5.2.3 above.

We can also predict what may happen to the real exchange rate over time:

∗ ∆q = n(∆aT − ∆aT ). (5.34)

If traded-goods productivity rises in India over time (relative to the value in the US), then India will experience a real appreciation (a fall in Q).|

∗ Example 2: AT = AT

In our second, illustrative example, let us assume that traded-goods productivity is the same in two countries. Then n AN Q = ∗ . (5.35) AN Imagine the home country is Japan and the foreign (starred) country is the US. These are both very rich countries, and they have similar productivity in sectors like manufacturing that are largely traded. But suppose that productivity in the non-traded sector is lower ∗ in Japan than in the US, so AN < AN . In Japan, serving tea or pumping gasoline is more labour-intensive than in the US. You can see that Q < 1: prices will be higher in Japan than in the US. Again this seems to match reality, which is why we study this economic reasoning.|

The main implication of the Balassa-Samuelson model is that productivity diﬀerences show up in relative prices. It is interesting to notice that exactly how these diﬀerences materialize can depend on the policy regime that is in place for the nominal exchange rate. To see this, we shall again ignore LOP departures. And recall that:

∆q = ∆s + π∗ − π. (5.17)

First imagine that two countries have a common currency or a ﬁrmly ﬁxed nominal exchange rate, so that ∆s = 0. Then imagine an emerging economy that is pegged to the USD. Suppose the two economies have the same productivity in sector N (or, a weaker assumption, that there is no growth in their relative productivity in sector N). Then

∗ ∗ ∆q = π − π = n(∆aT − ∆aT ). (5.36)

88 So the theory makes predictions for the inﬂation-diﬀerential across countries.

The same idea would apply for countries that share the Euro. For example, suppose Slovakia is the starred country and Germany is the unstarred one. And suppose that ∗ ∆aT > ∆aT . Slovakia has lower labour productivity in traded goods than Germany does, but its productivity is growing faster (and perhaps gradually catching up). Then the theory predicts π∗ > π and a real appreciation for Slovakia.

To see this process in action search on the web for ‘ECB inflation dashboard’. Then enlarge the map in the top left panel. Rolling your mouse over individual countries shows their inflation rates, and you can even run an animation to see how these differ over time.

Second, imagine a different scenario in which two countries both target inflation, successfully, at the same rate. Then π∗ = π which means that ∆q = ∆s. Thus productivity differences that are reflected in a real appreciation or depreciation will show up in a nominal appreciation or depreciation. The interesting prediction here is that the two countries can have the same monetary policy goals, yet their nominal exchange rate can trend up or down over time.

5.4 China’s Exchange-Rate Policy

One of the most followed real exchange rates in the world is that of China. The reason is one familiar from earlier macroeconomics courses: The evidence is that the value of Q affects the trade balance (net exports), and specifically that an increase in Q (a real depreciation) raises NX. Recall our discussion of global imbalances in chapter 2.7. The US has run a persistent current-account deficit with the world and specifically with China. US policymakers have often argued that one cause of this imbalance was China’s real exchange rate, in turn affected by its management of its nominal exchange rate.

The trilemma is a key concept for understanding the policy framework in China. The country has an independent monetary policy, so the PBOC can and does change its policy tools to inﬂuence the domestic economy. The international capital market is not open, as there are extensive capital controls. Thus you know that China can have a stable exchange

89 rate if it wishes. And remember that the reason policymakers stabilize exchange rates is that they believe this stability promotes trade.

What would the path of nominal exchange rate look like if it was being managed to promote exports from China? First, the level might be one symptom. A country trying to promote exports might depreciate its currency (or peg it at a weak value) for example by buying foreign exchange reserves (USD). And China did indeed accumulate large foreign exchange reserves for many years. Second, the variance might also be a symptom. If the policymakers believe that exchangre-rate volatility discourages trade they may reduce that by design.

Let us look at the historical path of the exchange rate to see whether there is evidence of these patterns. China’s currency is called the renminbi and its main unit is the yuan. The ISO 4217 code is CNY. Please pause to graph its exchange rate against the USD in FRED or PXRS. You will see that it was pegged at 8.28 per USD from 1995–2005. Successive US Treasury Secretaries stated that the RMB was under-valued (judging this not by PPP but by trade balance) and called for a revaluation. They suggested that the US Congress could name China a ‘currency manipulator’ which would trigger protectionist measures.

But the renminbi then gradually appreciated (was revalued) to about 6.05 in January 2014. Since then it has depreciated to around 6.60 RMB/USD and it also has become more volatile. In chapter 2.7 we saw that the trade surplus with the US has declined gradually over the same period. So it is possible that reducing protectionist sentiment in other countries is one reason for the gradual appreciation of the CNY. But the appreciation also might reﬂect tighter domestic monetary policy that has been aimed aimed at domestic objectives.

So does the higher value of the CNY in USD and its greater volatility mean that the exchange rate is no longer being managed? No. The level is not as controversial as it once was, because of the appreciation against the USD and gradual decline in China’s trade surplus. But the exchange rate is still being managed. To see that, go to FRED and graph the broad eﬀective exchange rate for China. The eﬀective exchange rate, also called the

90 trade-weighted exchange rate, is a weighted average of bilateral exchange rate (CNY/USD, CNY/JPY etc) with the weights given by trade shares. (There is also a real version then includes relative prices, so it is an average of Qs.)

Your graph shows that the eﬀective exchange rate has been very stable since 2016. That reﬂects deliberate management. The PBOC seeks to stabilize the renminbi against a basket of 24 currencies, and it succeeds in doing that given the capital controls. Back to the trilemma then: China has capital controls, domestic monetary policy autonomy, and a stable exchange rate.

This history suggest two further issues that you might think about. First, the choices of policymakers no doubt reflect political factors, but they also seem to reflect views about some economic connections. What is the effect of a change in monetary policy on S, on Q, and then on NX? Notice that there are several elements in this causal chain, each of which we can explore some evidence on. What is the effect of the volatility of S on NX? We’ll see some evidence on that connection in chapter 10.

Second, economists often focus more on the question of the appropriate regime or system than on the appropriate level of the exchange rate. China may be in transition to (a) a more ﬂexible exchange rate and (b) more open capital markets. Why do these go together? Some researchers look for precedents in Germany and Japan in the 1950s or Chile and Israel in the 1990s.

Further Reading

Here are the articles cited in this chapter, in the order cited: Alan M. Taylor and Mark P. Taylor (2004) The purchasing power parity debate. Journal of Economic Perspectives 18:4, 135–158. The Economist March 15th 2018: Overpriced: Africa’s economic paradox Tax Foundation (2016) The real value of $100 in each state. https://taxfoundation.org/real-value-100-each-state-2016 Charles Engel and John Rogers (1996) How wide is the border? American Economic Review 86, 1112–1125.

91 Jennine Bailliu and Hafedh Bouakez (2004) Exchange-rate pass-through in industrialized countries. Bank of Canada Review Spring, 19–29. The Economist June 13th 2020: The yuan: The 24-body problem. The Economist October 31st 2020: The yuan: Caveat victor.

Exercises

5.1. This question uses the Balassa-Samuelson model to predict the path of the real exchange rate between India and the US. Suppose that each country’s price index has a share of non-traded goods equal to n = 0.5. Let the US be the home country and India the foreign country. Assume that the law of one price holds for traded goods prices. (a) Find an expression for the real exchange rate as a function of labour productivity in each sector and each country. (b) Suppose that productivity in non-traded goods grows at the same rate in both countries. But traded-goods productivity growth in India is 2% per year faster than in the US. What will happen to the real exchange rate over time? (c) Over the past decade, the Indian rupee has depreciated by an average of 1.25% per year against the USD. According to the model, what was the average inﬂation diﬀerential per year over this time period?

5.2. Suppose that the law of one price holds, and that the real exchange rate between Kenya (the unstarred, home country) and South Africa (the starred, foreign country) is given by: ∗ PT PN 0.5 Q = ∗ , PT PN where the subscript N denotes nontraded goods and the subscript T denotes traded goods. Suppose that the two countries have the same labour productivity in the nontraded sector. But in the traded sector labour productivity in Kenya is two-thirds of the value in South Africa. (a) Use the Balassa-Samuelson model to predict the level of the real exchange rate. In which country are prices lower? (b) Suppose that the growth rate of T -sector labour productivity is 3% per year in Kenya and 1% per year in South Africa. Find the rate of real appreciation or depreciation for Kenya relative to South Africa. (c) The inﬂation rate in Kenya is 7.5% while the inﬂation rate in South Africa is 6%. Predict the rate of nominal appreciation or depreciation of the Kenyan shilling (KES) relative to the South African rand (ZAR).

92 5.3. This question studies the real exchange rate between eastern Europe (the home region) and the western European countries that also are in the euro zone (the foreign, starred region). Suppose the real exchange rate is given by:

∗ ∗0.5 ∗0.5 SP SPT PN Q = = 0.5 0.5 , P PT PN

where T and N denote the traded and non-traded sectors respectively. Suppose that the Balassa-Samuelson model applies (where the LOP holds for traded goods) and that labour productivity in the N-sector is the same in both regions. In the T -sector, labour productivity in the east is half the value in the west, but it is catching up. In growth rates: ˙ ˙ ∗ AT = 4% per year and AT = 1% per year. (a) Where are prices (as measured by the price index) lower and by how much? Find the rate of real appreciation or depreciation in eastern Europe. (b) Consider Slovenia, which is in the euro zone and also in eastern Europe. If western Europe in the euro zone has an inflation rate of 2% per year then what will the inflation rate in Slovenia be? (c) Consider Poland, which has a floating exchange rate against the euro and is in eastern Europe. Its inflation rate is the same as the inflation rate in the western European countries of the European zone (2% per year). What will be happenning to its nominal exchange rate over time?

5.4. The Chinese renminbi has appreciated by 30% against the US dollar during the past 10 years. Over this same time period, cumulative inﬂation has been 10% in the US and 20% in China.

(a) What has been the cumulative (or overall) real appreciation of the renminbi? (b) Suppose that, in each country, the price level has a 50% weight on traded goods prices and a 50% weight on nontraded goods prices. The real exchange rate thus is given by:

∗0.5 ∗0.5 SPT PN Q = 0.5 0.5 , PT PN

where the US is the home country. Suppose that the law of one price holds. Suppose productivity in sector N in the US has grown by 20% and in China by 10%. Suppose productivity in sector T has grown by 20% in the US. What must the cumulative growth rate of productivity in sector T in China have been? (c) If these productivity-growth trends continue but the renminbi no longer appreciates then what would you predict to happen to the inﬂation diﬀerential between China and the United States?

93 Chapter 6. Floating Nominal Exchange Rates

This chapter studies ﬂoating exchange rates focusing on the question of whether one can explain their movements. The main theory for doing that views relative monetary policy as the major cause. We’ll develop the monetary model of the exchange rate in this chapter. Along the way, we’ll learn how to solve a present-value model, in which the exchange rate depends on monetary policy expected in the future. And we’ll see the evidence on this theory and on the random walk pattern discussed in chapter 1.1.3. As we’ll see, the monetary model is successful at explaining trends in exchange rates but not as successful at explaining the ﬂuctuations in exchange rates that still occur when two countries have similar monetary policy frameworks.

6.1 ISO 4217, Data, Volume, and Floating Incidence

For those (like me) with poor pronunciation, I recommend using ISO 4217 codes such as CAD, USD, GBP, JPY, EUR, CNY, CHF. A great place to track recent exchange rates is at the Pacific Exchange Rate Service.

An amazing thing to know about the foreign exchange market is how enormous the transaction volume is. The market is mainly conducted between banks. Every few years the BIS surveys them to ask about their transactions. Here is just one example: Daily trading volume for Canadian institutions is about $100B, or $24T per year, which is roughly 14 times the ﬂow of GDP. And world daily volume is about $3T daily. These enormous values suggest that more than trade in goods and services is involved. Yet swings in the prices have large consequences for income and employment in certain sectors and countries.

If you take a moment to look up which countries ﬂoat you’ll see that it is a diverse group. Also note that the membership changes over time. Some countries like Japan or Canada have ﬂoated since the 1970s, while others have joined then later left this club.

6.2 CIP and UIP

The current exchange rate is called the spot rate, from the spot market. We’ll denote it S and measure it in local currency. Thus an increase is a depreciation. And remember that s ≡ ln S.

94 For many currencies there also is a forward market and price F , with f ≡ ln F . That price is known today but the actual transaction occurs say 3, 6, or 12 months in the future. Who would use a forward contract?

Now consider two diﬀerent ways in which investors could guarantee receiving some local currency one period from now. First, they could simply put their money in a deposit account. Every unit that goes in would yield 1 + i units later, where i is the label for the one-period, nominal interest rate.

Second, they could take a unit of local currency and follow three steps: (a) convert it to 1/S units of foreign currency (e.g. if in Canada S = 1.3157 is the price of a USD then the investor buys 0.75 USD with every CAD); (b) invest their foreign-currency holdings in a deposit account denominated in that currency, which thus gives 1 + i∗ at the end of the time period for every unit invested; and then (c) convert the proceeds back to local currency at the pre-arranged forward rate F .

The gross return on this second strategy is (1 + i∗)F/S. But because both methods guarantee the investors local currency at the same time, and neither involves any risk, they have the same return. This is sometimes called a no-arbitrage condition, and it is: F 1 + i = (1 + i∗) . (6.1) S This relationship is called covered interest parity or CIP. It is called covered because the second strategy is a covered position in the foreign exchange market. That means that the investors are not exposed to any risk of ﬂuctations in the exchange rate because they have locked in the forward rate. Remember that for realistic, low interest rates ln(1 + i) ∼ i. Thus in logarithms CIP is: i = i∗ + f − s. (6.2)

Think of CIP as an identity that links these four variables, so that if you know three of them then you also know the fourth. (There are some recent studies that ﬁnd departures from CIP, but we’ll leave those to be studied by specialists.)

Investors might instead take an uncovered position, where they wait and see at what spot rate they can convert their foreign currency holdings back into local currency. Let

95 us use some time subscripts now, so that future exchange rate is St+1. The gross return ∗ on this position thus is (1 + it )St+1/St. They now are exposed to some risk. If the domestic currency depreciates (so the foreign currency appreciates) while they are holding the foreign currency that will raise their return, while if the domestic currency depreciates that will lower it.

It’s time to see an important hypothesis about these returns. That is uncovered interest parity or UIP: ∗ EtSt+1 1 + it = (1 + it ) . (6.3) St

Notice that UIP includes the expectations operator at time t, denoted Et. So this is an hypothesis about the expected depreciation, not about the actual one. We cannot directly measure EtSt+1, but we’ll see ways of learning about it indirectly, here and in chapter 7.

The log of an expectation is not the expectation of a log. But we’ll ignore that small diﬀerence for now and treat UIP in logarithms as:

∗ it = it + Etst+1 − st, (6.4)

Notice that Et(∆st+1) is the expected rate of depreciation of the domestic currency. A good way to remember UIP is to notice that it implies that the international interest diﬀerential equals the expected rate of depreciation. A high-interest-rate currency is expected to depreciate, according to UIP, and so that equalizes the expected returns.

Combining CIP and UIP gives in levels:

EtSt+1 = Ft, (6.5) or in logs:

Etst+1 = ft, (6.6) so that the forward rate is the expected value, or best predictor, of the future spot rate. This hypothesis is sometimes called unbiasedness, because it implies the forward rate predicts the spot rate correctly on average. We’ll see how to test this hypothesis in section 6.7.

96 6.3 Monetary Model

Now back to trying to explain or perhaps even predict the exchange rate. It is the relative price of two monies, so begin with the money supply M equal to the simplest money demand function, PY , in each country:

M = PY and M ∗ = P ∗Y ∗. (6.7)

Next, assume that PPP holds: SP ∗ 1 = . (6.8) P Of course chapter 5 showed that PPP is at best a long-run theory so building that in here might make this a long-run model too.

Combining these two building blocks gives

s = m − m∗ − (y − y∗), (6.9)

or in growth rates (diﬀerences of logs):

∆s = ∆m − ∆m∗ − (∆y − ∆y∗). (6.10)

The key insight here is that the exchange rate depends on relative monetary policy. (We use the money supply or its log as the indicator of that policy, rather than the interest rate set by the central bank). A country with a persistently high (relative) money growth rate will have a steadily depreciating currency. (This is why we quote S in local currency units: It then rises with other prices during a sustained inflation.) This simple theory is consistent with trends in some contemporary exchange rates, like those of the Turkish lira or Argentine peso. It also fits with historical evidence on hyperinflations, where rapid money growth led to extremely high inflation and also very rapid depreciation.

Now we’ll upgrade our monetary model to the more advanced version. Suppose that money demand also depends on the interest rate:

∗ M = P Y e−αi and M ∗ = P ∗Y ∗e−αi , (6.11)

97 so that at higher interest rates people hold less money. These functions look a bit strange, but they give us log versions that are easy to work with:

mt − pt = −αit + yt (6.12) ∗ ∗ ∗ ∗ mt − pt = −αit + yt .

Notice that variables except the interest rates are in logs and we assume the interest semi-elasticity α is the same across countries. Next we have log PPP:

∗ st − pt + pt = 0, (6.13) and log UIP: ∗ it − it = Etst+1 − st. (6.14)

Please take a moment and ﬁnd that combining these three elements gives:

α 1 s = E s + [(m − m∗) − (y − y∗)]. (6.15) t 1 + α t t+1 1 + α t t t t which shows that the exchange rate can be thought of as an asset price. In this model the price is related to its expected future value (or to the capital gain) and to a ‘fundamental’.

For simplicity, call the fundamental xt:

∗ ∗ xt ≡ (mt − mt ) − (yt − yt ). (6.16) so α 1 s = E s + x . (6.17) t 1 + α t t+1 1 + α t Call this the fundamental equation of the monetary model of the exchange rate. It is a difference equation in that it involves values of s at two different, discrete times. And it captures a realistic feature of any asset: The price today depends on the expected price tomorrow. That feature is here predicted for foreign currency but it also holds for things like gold, stocks, or fine art. But what determines the expected future price? We’re about to find out.

6.3.1 Repeated Substitution/Solving Forwards

98 Solving this equation means isolating st on the left-hand side with only exogenous things on the right-hand side. Start with a numercial example. Suppose that α = 1. Then the fundamental equation (6.17) becomes:

1 1 s = E s + x . (6.18) t 2 t t+1 2 t

Now we solve the diﬀerence equation forwards. Lead all dates/subscripts by 1 time period:

1 1 s = E s + x . (6.19) t+1 2 t+1 t+2 2 t+1

Then take this expression for st+1 (6.19) and substitute it in (6.18):

1 1 s = x + E s t 2 t 2 t t+1 1 1 h1 1 i = x + E x + E s (6.20) 2 t 2 t 2 t+1 2 t+1 t+2 1 1 1 = x + E x + E s . 2 t 4 t t+1 4 t t+2

Notice that the Et+1 operator disappears because of the law of iterated expectations.

Next, maintain your composure and repeat this same process to replace st+2. That gives: 1 1 1 1 s = x + E x + E x + E s . (6.21) t 2 t 4 t t+1 8 t t+2 8 t t+3 Now a pattern emerges. You can see a sequence of x’s running into the future, with declining weights. And there is a remainder term which also has a declining weight as we continue to make these substitutions. Eventually:

∞ 1 X 1j 1j st = Et xt+j + lim Etst+j. (6.22) 2 2 j→∞ 2 j=0

The second term will usually be zero. As long as st+j isn’t very explosive, the fraction in front of it becomes zero so the whole term becomes zero. That leaves the ﬁrst term. It gives our main result. The exchange rate follows a present-value (PV) model in which it is given by the PV of current and expected future fundamentals. Remember that the fundamental (6.16) involves relative monetary policy. Thus our theory predicts that both current and

99 expected future monetary policy matter for the current exchange rate. Information about future m can thus aﬀect today’s s.

We’re not quite ﬁnished solving the model, because the expectations Etxt+j on the right-hand side cannot be observed directly. So we need to model these expectations or forecasts. Let us do that using rational expectations and the ﬁrst-order autoregression from chapter 5.1. I’ll suppose that µ = 0 and ρ = 0.9 so that

xt = 0.9xt−1 + t. (6.23)

j From the chain rule of forecasts that means that Etxt+j = 0.9 xt: We simply compound this pattern forward. Finally, let us use this in our exchange-rate equation (6.22):

1 1 1 s = x + E x + E x + ... t 2 t 4 t t+1 8 t t+2 1 1 1 = x + 0.9x + 0.92x + ... 2 t 4 t 8 t 1 = x (1 + 0.45 + 0.452 + ...) (6.24) 2 t 1 1 = x 2 t 1 − 0.45

= 0.9090 xt.

Now the exchange rate depends on current and expected future fundamentals, but the latter are forecasted with the former so that is the only thing that enters the equation.

6.3.2 Guess-and-Verify

It is good to know that the model implies a present-value relationship, where future monetary policy matters. But, this is our most involved algebra of this book, and even so we solved this problem only for a specific values of α (1), ρ (0.9), and µ (0). Fortunately there is an alternative to this method of repeated substitution. It is called guess-and- verify or the method of undetermined coefficients. To see how this works, we’ll use the same numerical example, so we can also confirm the two methods give the same answer.

The method begins with a guess:

st = kxt, (6.25)

100 where k is a number, an unknown coeﬃcient, that we will try to solve for. If this form

of the answer is correct, then it will also be true that st+1 = kxt+1. Remember also that

Etxt+1 = 0.9xt. Subsitute the guess and this lead-forward implication for the s-terms in the original diﬀerence equation:

1 1 1 1 kx = x + E kx = x + 0.9kx . (6.26) t 2 t 2 t t+1 2 t 2 t

For the two sides of the equation to be equal, the coeﬃcients on xt must be equal, so

1 1 k = + 0.9k, (6.27) 2 2 which gives k = 0.9090 just as we found above.

Easy eh? But where does the guess come from? Their are two rules of thumb to follow.

First, if there is an intercept, µ, in the law of motion for xt then put one in the guess. Second, the number of lags in the guess should be the number in the x-autoregression minus one. With those in mind, let us look at a more general case involving α and µ and ρ. The diﬀerence equation we saw several pages ago:

α 1 s = E s + x . (6.17) t 1 + α t t+1 1 + α t

Guess the solution is:

st = k0 + k1xt, (6.28) where k0 and k1 are unknown coeﬃcients. Using our guess:

1 α k + k x = x + [k + k (µ + ρx )]. (6.29) 0 1 t 1 + α t 1 + α 0 1 t

This takes a moment to rearrange but equating the intercepts and slopes shows that:

1 k = (6.30a) 1 1 + α − αρ

and

k0 = αµk1 (6.30b)

101 so the theory predicts what happens when we regress st on xt: αµ 1 s = + x . (6.31) t 1 + α − αρ 1 + α − αρ t The exercises contain more practice with the guess-and-verify method.

Notice that if xt has a unit root (follows a drifting random walk) then with ρ = 1 the solution specializes to:

st = αµ + xt. (6.32)

So the exchange rates tracks xt but shifted up by the addition of αµ. Random walk in:

Random walk out. This is an interesting case because of the evidence that st does indeed follow a random walk, even though we are not always sure of the identity of xt.

We’ve spent some time on the analysis of the monetary model, so let us recap the economics. First, the model predicts that relative monetary policy matters for movements in the exchange rate. Second, it is forward-looking. If a future change in monetary policy is anticipated (’discounted’) the currency value may not react to the change when it occurs. Or there can be a big reaction to a surprise that aﬀects future policy with no change in policy today.

6.4 Evidence on Floating Exchange Rates

We’ll next look at several types of evidence. The approaches include formal tests of the monetary model but also more general studies of ﬂoating exchange rates. I’ll break this into small subsections so you can easily keep track of the diﬀerent types of research studies.

6.4.1 Cross-Equation Restrictions

To see how to estimate and test the model, imagine running a regression like this:

st = β0 + β1xt + ηt, (6.33) where ηt is an error term. Notice that the monetary model (6.31) predicts that the two coeﬃcients are combinations of µ, ρ, and α. So imagine also running the ﬁrst-order autor- ˆ gression and estimating µ and ρ there. If you record those estimates, that means that β0

102 ˆ and β1 can be unscrambled to give you an estimateα ˆ. The fact that there are two ways to ﬁnd α provides a test. They should give the same result (subject to sampling variability) and if they do that supports the theory.

Another way to think about this is that there are two regressions, (6.33) and the autoregression. That gives two intercepts and two slopes. But there are only three underlying parameters. That restriction can then be tested to see if it holds, by comparing the fit of the regressions with the restriction on with the fit with them off. These restrictions are sometimes called rational expectations cross-equation restrictions. They are not usually supported in econometric studies, which leads us to think about less restrictive looks at the evidence.

6.4.2 A Tournament

To see a more general way to look at the evidence, imagine extending the s-regression

by also including st−1 as a regressor:

st = β0 + β1xt + β2st−1. (6.34)

ˆ Here the idea is that if we’ve really measured the fundamentals correctly then β1 should be ˆ significantly different from 0 and β2 should not be. One can also think of this regression as a contest between xt and st−1 to see which of these can best statistically explain st. Notice the timing difference. Using st−1 is like forecasting while using xt has the advantage of

using information from within the same time period as st.

Analysts have run many version of this regression, using the fundamentals from the monetary model but also other candidates too. The results are discouraging. Usually ˆ ˆ analysts ﬁnd that actually β1 is near 0 and β2 is near one. This captures the random walk

property of st, but that random walk does not seem to be driven by one in xt. Equivalently,

a forecasting tournament ﬁnds st predicts st+1 better than xt+1 does. It is hard to improve on the random walk model even knowing future fundamentals. Often the best forecast is just whatever the current value is.

Sometimes researchers ask a robot to search a database and ﬁnd a candidate x-variable ˆ that leads to a statistically signiﬁcant value β1. As we discussed in chapter 1.1.3, though,

103 there is a possibility of type I error (‘data mining’) in doing this. If one studies 100 possible x’s then 5 of them will appear to be statistically signiﬁcant at the 5% level. I think it is wise to be sceptical of studies that ﬁnd such a candidate, then.

6.4.3 News

A diﬀerent way to test for the importance of monetary fundamentals is based on comparing surprises or unexpected changes in s with those in x, sometimes called ‘news’. To see how the ‘news’ approach works, imagine that ρ = 1 so that

st = αµ + xt. (6.32)

Notice that

Et−1st = αµ + Et−1xt. (6.35)

Therefore, subtracting gives:

st − Et−1st = xt − Et−1xt, (6.36) where the right-hand side is the ‘news’. Researchers use data from surveys of participants in the foreign exchange market to measure the two expectations. Then they see whether the surprises on either side of this equation (6.36) go together. Often they seem to do so, meaning that an announcement or surprise concerning monetary policy leads to a jump in the exchange rate. That pattern suggests that monetary policy is at least one of the things that determines s.

6.4.4 Commodity Currencies

If you work through the exercises or simply follow some currency values informally then you might note that there is a correlation between the exchange rate and commodity prices, for countries that specialize in commodity exports. For example, the Canadian dollar or Norwegian krone is sometimes called a ‘petrocurrency’ for this reason. (This would make the AUD a ferrocurrency and the NZD a lactocurrency.) It turns out that this sort of correlation is perfectly consistent with the monetary model of the exchange rate, though not with our simple, ﬁrst-order autoregression.

104 ∗ To see this consistency, just suppose that xt = mt − mt but that this fundamental doesn’t evolve autonomously following the autoregression but instead reacts endogenously to current macroeconomic conditions. Sometimes these descriptions are called the reaction functions of central banks. Next, suppose that an increase in energy prices in Canada or Norway tends to lead to an expansion in output and hence to inflation via a Phillips curve. Finally, suppose that the inflation-targeting central bank reacts by tightening monetary policy (reducing the growth rate of m) so as to keep inflation on target. You can see from this simple sequence of reasoning that the initial increase in energy prices leads to an appreciation of the currency, because it forecasts a tightening of monetary policy. But this connection follows from the reaction function of monetary policy. More generally, then, anything that helps practitioners forecast future monetary policy will be correlated with today’s exchange rate.

Overall, economists have had success explaining floating exchange rates when there are very large, ongoing differences in monetary policy across countries. They have had less sucess in explaining swings in floating exchange rates between countries like the US, Japan, the UK, Australia, Canada, Switzerland, and so on. Yet there are large swings in the exchange rates between these countries that have serious consequences for employment and income in importing and exporting sectors. One response to this evidence is to study the possibility that the swings are caused by speculation unrelated to macroeconomic fundamentals. We consider this idea next.

6.5 Bubbles

Given some of the negative statistical results, a number of researchers have concluded that speculative bubbles may explain many movements in nominal exchange rates. Bubbles (or peso problems) are essentially omitted variables. One reason they are invoked is that ﬂoating rates seem to be much more volatile than fundamentals, even when signiﬁcant fundamentals can be found. This section looks at the logic of bubbles, the mathematics, the history, and a pitfall in testing for them.

First, the logic. The term ‘bubble’ often is used informally to refer to a high price level. But formally a rational speculative bubble occurs when the price rises (or at least is

105 expected to rise) and this appreciation constitutes a return. What is the minimum rate at which the price must rise? If the bubble might burst then how would that aﬀect the growth rate? Bubbles are similar to Ponzi schemes too. In both cases there is no underlying value and there must be ongoing growth.

Second, the math. Remember the remainder term in our forward solution for the

exchange rate (6.22)? That term went to zero provided Etst+j did not grow faster than [α/(1 + α)]j shrank. But, as we’ve just seen from the logic, a bubble involves growth in the price. It turns out that the diﬀerence equation has multiple solutions if that term is not zero. In particular, adding a term like

1 + α 1 + αt b = b = (6.37) t α t−1 α gives a solution that still satisﬁes the equation. For example, in our original numerical example with α = 1 and ρ = 0.9 :

t st = 0.9090 xt + 2 (6.38) will also be a solution. You can show that by substituting (6.38) into (6.18). But don’t worry if you don’t wish to dwell on the mathematical details.

Third, the history. An oft-cited example of a bubble is the Dutch tulipmania of 1677. Commentators also cite other examples such as the South Sea bubble and the Mississippi bubble, so it is fun to look those up and learn a bit about them. Are there recent examples that come to mind where prices were bubblicious?

Fourth, a pitfall. It can be tricky to detect speculative bubbles because there may be an ‘apparent’ bubble: bubbly price behavior even when no bubble is present, simply because we have misunderstood the market’s forecast of future fundamentals.

For example, imagine a biotech stock price that rises sharply as information spreads about the ﬁrm’s promising research. Then suppose the research fails, and the price falls back. A statistician looking at the price and earnings data years later may falsely conclude a bubble was present. The price rose rapidly with no change in earnings, which looks like a (bursting) bubble.

106 A similar concept sometimes is invoked in current markets: a ‘peso problem’. (The name comes from the peso of Mexico.) This is an extreme event in the fundamentals (relative monetary policy, say) that occurs very rarely. It aﬀects expected future fundamentals,

and so the exchange rate, but may not be observed in the historical sample of xt so that it might appear that s is disconnected from x. For this reason, I think most economists are sceptical about bubbles appearing in currency markets.

6.6 Currency Transaction Tax

A second reaction to the large swings in ﬂoating exchange rates (combined with the enormous volume of transactions) is to set a small tax on foreign exchange transactions. This is a currency transaction tax, sometimes called a Tobin tax after the economist who ﬁrst proposed it in the 1970s. The idea is that every time banks exchange dollars for euros, say, the client pays a very small percentage tax on the transaction, with that amount remitted to the government.

A CTT has two distinct purposes. First, it is intended to discourage speculative transactions back-and-forth between currencies and thus to make exchange rates less volatile or more stable. Second, because the small tax rate would be applied to an extremely large tax base, the tax would raise revenue which then could be applied to international development goals. Thus you may come across the CTT if you or colleagues are involved in debates about funding international development.

Economists have traditionally been sceptical of a CTT for two reasons. First, if a tax must be paid on transfers between bank deposits in diﬀerent currencies then possibly banks and their customers can avoid the tax yet still change currencies by exchanging some other asset like a bond. Thus it could be challenging to monitor and collect the tax and unclear what the base should be. Second, sceptics have argued that the tax would need to be collected in every jurisdiction with a foreign exchange market (New York, Frankfurt, Tokyo, Singapore, etc) otherwise the market would migrate to the location with no tax or to a new location entirely.

Several countries have transactions taxes on particular assets (such as real estate

107 or stocks) but there is no transaction tax that is internationally coordinated. In 2011 the European Commission proposed a ﬁnancial transactions tax (not just on currency transactions) that would apply within the EU and that would direct revenue to the EU budget, not to development goals. I invite you to form your own views on the eﬃcacy of a CTT and on whether it is likely to be implemented.

6.7 Testing UIP or Unbiasedness

One of the building blocks we used in the monetary model of the exchange rate was the UIP condition. In this section we’ll study the evidence on this condition, which leads to the forward premium anomaly, one of the central topics in international ﬁnance.

To start, remember the CIP condition:

∗ Ft (1 + it) = (1 + it ) , (6.1) St

and its approximate log version:

∗ it = it + ft − st. (6.2)

Also remember the UIP hypothesis:

∗ EtSt+1 (1 + it) = (1 + it ) , (6.3) St

and its approximate log version:

∗ it = it + Etst+1 − st. (6.4)

Rearranging the approximate UIP condition gives the expected proﬁts on an uncovered position:

∗ it + Etst+1 − st − it = 0. (6.39)

This equation predicts that the expected proﬁts on the ‘carry trade’ should be zero. In other words, any interest-rate advantage of investing in a high-yielding currency will be

108 expected to be oﬀset by a depreciation in that currency. Combining this with CIP, though, also means that:

Etst+1 = ft. (6.6)

Since we can think of CIP as an identity, this rewriting tests the same hypothesis. Thus UIP also means that the forward rate is the expectation or forecast of the future spot rate.

We cannot observe Etst+1 directly, but just imagine splitting the actual outcome for the future spot rate into the predicted part and the ‘surprise’ or forecast error:

st+1 = Etst+1 + ηt+1, (6.40) so that

Etst+1 = st+1 − ηt+1. (6.41)

Now let us simply use this split to replace Etst+1 in the expected proﬁts condition:

∗ it + st+1 − st − it = ηt+1. (6.42)

And we also can use it in the other version of UIP:

st+1 = ft + ηt+1. (6.43)

Now, ηt+1 is a forecast error. Suppose that this error has exactly the statistical properties that an econometric forecast error would have (as if market participants made forecasts like an econometrician would). In that case, ηt+1 would (a) have a mean value of zero and (b) be uncorrelated with variables known at date t (otherwise forecasters could improve their forecasts).

If forecast errors do indeed have these properties, then UIP can be tested by seeing whether carry-trade proﬁts have an average value of zero and are unpredictable or, equivalently, by seeing whether ft is an unbiased predictor of st+1. We can test this by running a regression like this:

st+1 = β0 + β1ft, (6.44)

109 and seeing whether the intercept is zero and the slope is 1 as equation (6.43) predicts.

Researchers sometimes subtract st from both sides to give:

∗ st+1 − st = ft − st + ηt+1 = it − it + ηt+1. (6.45)

In this version the rate of depreciation is on the left-hand side. When UIP holds, the rate of depreciation is equal on average to the forward premium (or discount) or, equivalently (remember equation (6.2) for CIP), the interest rate diﬀerential. A currency is at a forward

premium if ft < st. Here that means it is expected to appreciate. It is at a forward discount

if ft > st (the foreign currency is more expensive forward than spot so the local currency is less valuable) which, according to the theory, predicts a depreciation.

To test UIP one runs this regression:

∗ st+1 − st = ft − st + ηt+1 = γ0 + γ1(it − it ) + ηt+1. (6.46)

What is the evidence? A classic case study involves the JPY and AUD. If you collect comparable interest rates from Japan and Australia from FRED you will see that often the latter have been well above the former over the past several decades. According to UIP, the AUD should depreciate on average to offset this difference. But if you graph that exchange rate (from Pacific Exchange Rate Service or by typing AUD in a google window) I don’t think you see that pattern overall. In fact, over certain spans the AUD appreciated, offerring investors large profits from this carry trade.

More systematically and broadly, hundreds of studies have run regressions like (6.45) or (6.46), to test for unbiasedness (UIP). Often, they ﬁnd thatγ ˆ1 is signiﬁcantly less than

1. Sometimes they even ﬁnd thatγ ˆ1 is negative, which means that the high-interest-rate currency actually appreciates, on average, rather than depreciating as the theory predicts. There seem to be two possible explanations for these puzzling ﬁndings.

First, our assumption that ηt+1 is like an econometric error could be wrong. Maybe the forward exchange rate does equal the expected future spot rate, but investors make runs of correlated forecast errors.

110 Here is a simple example. Suppose that econometricians ﬁnd that the exchange rate follows a random walk. Thus they ﬁnd that:

st+1 = st + t+1, (6.47) when they run a regression to see how much persistence there is in the spot exchange rate. They then would guess that Etst+1 = st. That means that they think the expected depreciation is 0 and so the interest-rate diﬀerential should be 0 too.

Meanwhile suppose there is a ‘peso problem’ in the foreign exchange market. Investors believe there is some probability of a large depreciation (even though one has not been recently observed by the econometricians) so that Etst+1 − st = υ > 0. (This stems from a similar view of the fundamentals.) The investors will make repeated forecast errors, as each period they expect some depreciation that does not occur.

Meanwhile the interest differential reflects their forecasts, not those of the econome- ∗ tricians, so it − it = Etst+1 − st = υ. Thus there are profits from the carry trade and the persistently higher interest rate in the home currency isn’t offset by depreciations on average.

In this example UIP holds but that hypothesis is tested jointly with the rational

expectations hypothesis about ηt+1. The rational expectations hypothesis does not hold, in that the forecasts don’t coincide with the actual statistical properties of the exchange rate. So the test rejects.

One way around this diﬃculty is to use surveys of professional exchange-rate forecasters. That way we measure the expected future spot rate directly and can compare it to the forward rate. Studies that make this comparison generally ﬁnd that unbiasedness holds, or nearly holds. It’s still puzzling though that forecast errors can be persistent and predictable.

The other possible explanation for the forward premium anomaly is that Etst+1 diﬀers

from ft because the uncovered and covered positions have diﬀerent risk and so have diﬀerent prices and returns. To work out what to look for here, we need to turn to the (interesting)

111 basics of asset-pricing theory, one of the main branches of ﬁnance. We’ll do that in the next chapter and, while we won’t resolve the forward premium anomaly, we’ll make interesting discoveries along the way.

Further Reading Here are two great textbook sources and then a couple of other articles cited in this chapter: Robert Feenstra and Alan Taylor (2017) International Macroeconomics, 4th edition. Macmillan. Chapters 2.1–2.4, 3.2–3.6, and 4.1–4.3 Laurence S. Copeland (2014) Exchange Rates and International Finance, 6th edition. Pear- son. Chapters 3, 5, and 12. Michael B. Devereux and Gregor W. Smith (2020) Testing the present-value model of the exchange rate with commodity currencies. Journal of Money, Credit and Banking forthcoming. North-South Institute (2008) The currency transaction tax: A bold idea for ﬁnancing development. Ottawa: North-South Institute. Kenneth Froot and Richard Thaler (1990) Anomalies: Foreign exchange. Journal of Eco- nomic Perspectives 4:3, 179–192.

Exercises

6.1. Suppose that PPP holds between Turkey (the home country) and the euro area (the foreign, starred country), so that in logs:

∗ st + pt − pt = 0.

Also suppose that UIP holds. Money demand in Turkey is given by:

mt − pt = −0.5it, and in the euro area by ∗ ∗ ∗ mt − pt = −0.5it .

(a) Derive the fundamental equation of the monetary model of the exchange rate. ∗ (b) Suppose that the relative money supply in logs is given by xt ≡ mt − mt , and that

xt = 0.5 + xt−1 + t,

112 where t is white noise with mean zero. Solve for the exchange rate st in terms of the current value of xt.

(c) Find expressions for the international interest diﬀerential and also for ft, the log of the one-period-ahead forward rate.

6.2. This question uses the monetary model of the exchange rate to study the statistical properties of currency ﬂuctuations. Suppose that a ﬂoating exchange rate follows this equation: 1 2 s = x + E s , t 3 t 3 t t+1

where the fundamental variable, xt, in turn evolves like this:

xt = µ + ρxt−1 + t,

and t has a mean of zero and is uncorrelated with xt−1.

(a) Solve for st as a function of xt.

(b) Find the variance of st as a function of the variance of xt.

6.3. This question studies whether the monetary model of the exchange rate can be applied when one country’s money supply is growing persistently relative to another’s. Suppose that the fundamental for Turkey evolves like this:

xt = λ + xt−1 + t,

where t is a mean-zero white noise. Meanwhile, the exchange rate (the lira price of the euro) is related to the fundamental and the expected future exchange rate like this:

1 α s = x + E s . t 1 + α t 1 + α t t+1

(a) Solve for the exchange rate as a function of exogenous variables that are currently observed. (b) Will depreciations in the lira be predictable? (c) If UIP holds then how will the international interest-rate diﬀerential between Turkey and the euro area behave?

6.4. Suppose that PPP holds between Japan (the home country) and the US (the foreign, starred country), so that in logs:

∗ st + pt − pt = 0.

113 Also suppose that UIP holds. Money demand in Japan is given by:

mt − pt + it = 0

and in the US by ∗ ∗ ∗ mt − pt + it = 0.

(a) Derive the fundamental equation of the monetary model of the (log) exchange rate. ∗ (b) Suppose that the relative money supply in logs is given by xt ≡ mt − mt , and that

xt = 0.5 + 0.8xt−1 + t,

where t is white noise with mean zero. Solve for the exchange rate st in terms of the current value of xt. (c) How does the variance of the exchange rate compare to the variance of the fundamentals?

6.5. Suppose that the yen/dollar exchange rate is described by the monetary model of the exchange rate, with: st = 0.5xt + 0.5Etst+1, where st is the logarithm of the exchange rate measured in dollars per yen. For simplicity assume that m∗ = y∗ = y = 0, while:

mt = 0.02 + 0.96mt−1 + t, with t ∼ N(0, 1).

(a) Use the guess-and-verify method to solve for st as a function of mt. (b) According to this theory, is the exchange rate more or less variable than the ‘fundamental’, xt?

(c) Imagine there is a one-time, positive shock to monetary policy, so t = 1 while later values of t+i are zero. Trace out the eﬀect of this shock on the exchange rate over time.

6.6. This question studies the exchange rate between the euro and the Turkish lira. Suppose that money demand in Turkey is given by:

mt = pt − it, where m is the log money supply, p is the log price level, and i is the interest rate. Money demand in the euro area is given by:

∗ ∗ ∗ mt = pt − it .

Also, both UIP and PPP hold.

114 (a) Derive the equation linking the nominal exchange rate, st, to fundamentals and the expected, future value of the exchange rate. (b) Suppose that the money supply in Turkey relative to that in the euro area follows a drifting random walk: ∗ ∗ mt − mt = µ + mt−1 − mt−1 + t, ∗ where t is white noise with mean zero. Solve for st in terms of mt − mt . (c) Will the exchange rate follow a random walk? What will be the value of the international interest diﬀerential?

115 Chapter 7. Asset Pricing

An unaswered question from chapter 6 is whether there is a risk premium in uncovered fx positions. In this chapter we figure out what risk premia are and one way to model them. Our most interesting applications will be to forecasting sovereign defaults (or re- structurings) or future growth and inflation. So the topics won’t be all international but I hope that you find them interesting and useful.

We’ll use two of our mathematical things from chapter 1.3: being aware of Jensen’s inequality (thing 5) and the covariance decomposition (thing 6). You will need your calculator to practice some examples as we proceed. Our main economic tool is the familiar Euler equation, modified to allow for realistic uncertainty about the future. Our first task is to make that modification. You also can skip ahead to section 7.2 if you wish and simply use the result.

7.1 Deriving the Euler Equation

We derived the Euler equation in chapter 2.2 and practiced that again in chapter 2.3. In chapter 2.6 we used it to calculate the world real interest rate. But there was no uncertainty about the future in those models. Let us see what the Euler equation looks like—and how to calculate interest rates—when there is uncertainty.

To make this simple, there are two time periods and no initial assets. Imagine that income Y2 is a random variable that can take on two values Y 2 and Y 2 each with probability 0.5. So there are only two possible outcomes, sometimes called states of the world or simply states.

The household maximizes expected utility:

U(C1) + βE1U(C2) (7.1) by choosing {C1, C2,C2}. In other words, they choose current consumption but also make plans for future consumption in each state. The budget constraints are:

C2 = Y 2 + (1 + r)(Y1 − C1) (7.2) C2 = Y 2 + (1 + r)(Y1 − C1)

116 Now simply notice that the goal is to maximize:

h1 1 i U(C ) + β U(C ) + U(C ) (7.3) 1 2 2 2 2 because the term in square brackets is E1(C2). Then form a Lagrangean: h1 1 i L = U(C ) + β U(C ) + U(C ) 1 2 2 2 2

+ λ[C2 − Y 2 − (1 + r)(Y1 − C1)] (7.4)

+ λ[C2 − Y 2 − (1 + r)(Y1 − C1)]. Aside from the budget constraints, the ﬁrst-order conditions are:

0 U (C1) + λ(1 + r) + λ(1 + r) = 0 1 β U 0(C ) + λ = 0 2 2 (7.5) 1 β U 0(C ) + λ = 0 2 2 If you substitute to remove the two multipliers you see that

0 0 U (C1) = E1β(1 + r)U (C2). (7.6)

Whew! That is our Euler equation in the case with uncertainty. We’ve derived it with just two time periods but it applies to any two adjacent period, usually labelled t and t + 1.

7.2 Asset Pricing using the Euler Equation

Suppose that the underlying attitudes to risk are based on the expected utility model we have already introduced. Consider a two-period model of asset-pricing. With uncertainty we can now think of rates of interest on several different assets, rather than there being only one interest rate. One of our tasks as macroeconomists is to account for interest rates that differ according to risk, maturity, and country, as well as variation over time in the general level of rates. We shall attempt to do that by finding attitudes to risk from an underlying problem of maximizing expected utility.

We shall use an important, general relationship between prices and rates of return:

payoﬀ + resale price 1 + r = , (7.7) price

117 where r is a one-period return. In most of our examples in this section, time ends at period 2 and so there is no resale value to the asset.

A key insight is that an asset is defined by its payoffs. If we are told when and what amounts an asset (contract) pays, then we can calculate its price. How can we do this? The Euler equation tells us something about (1 + r). If we know the payoffs, then we can calculate the price. As a specific example, suppose that U(C) = ln(C), then we have

1 1 = E1β(1 + r) . (7.8) C1 C2

This leads to a recipe for ﬁguring out predictions for interest rates. There are three steps in the recipe:

(a) describe completely the payoﬀs on an asset, for each time and state;

(b) calculate the asset’s price using the payoﬀs and information on consumption, using the Euler equation;

7.3 A Discount Bond

For simplicity let us assume that there are only two possible states of the world in period 2. To see how the recipe works, consider an asset which costs h1 in period 1 and pays 1 unit in period 2 no matter which state occurs. This is sometimes called a riskless asset, for obvious reasons. It also is called a zero-coupon bond or a discount bond (because h1 usually is less than 1). Its rate of return is known in period 1 (because it pays 1 in period 2 in each state) and, rearranging (7.7), that is:

1 r = − 1. (7.9) h1

For this bond the payoﬀ is 1 and the resale price is zero.

Suppose that consumption C can take on two values: 1 and 2. Whatever the value in period 1 there is a 60% chance of that continuing in period 2 and a 40% chance of

118 switching. Suppose that we are in state 1 in period 1 (so C1 = 1). For the riskless asset the Euler equation is:

0 0 h1U (C1) = E1U (C2)β · 1. (7.10)

Also suppose that U(C) = ln(C) and β = 0.95. Then

h h0.6 0.4i 1 = .95 · + = 0.76, 1 1 2 (7.11) 1 r = − 1 = 0.315 h1 using the probabilities from our two-state example. Thus we’ve found both the price and the rate of return. Notice that we were careful to calculate E(1/C2) which is not 1/E(C2).

This model bond return is not random, yet we observe changes over time in the interest rate on bonds. So, as you might guess, we attribute that to variation in the initial state. Bond prices are just forecasts, so for the bond price to vary, the forecast must vary depending on the initial state. That means that constant consumption growth, for example, won’t give an interesting pattern in bond returns.

If we begin in the high state with C1 = 2, then the bond price is:

h h0.6 0.4i 1 = 0.95 + = 1.33, (7.12) 2 2 1 so that the interest rate is −0.248. We get these unrealistic values because we have used integers for consumption, for simplicity. In reality consumption growth rates are not this enormous and so interest rates aren’t either.

The two starting values are equally likely, so the average bond price is 0.5(0.76) + 0.5(1.33) = 1.045. And the average interest rate is 0.5(0.315) + 0.5(−0.248) = 0.0335. Then please note carefully that the average bond return is not the inverse of the average bond price, minus one. That is an example of Jensen’s inequality.

We end this section with two economic observations. First, faster consumption growth raises r. You can see that directly from the Euler equation or from the numerical examples. Conversely, then, low expected consumption growth gives a low r.

119 Second, variability in consumption growth lowers r. Imagine moving the two possible values for C2 further apart. That will raise E1(1/C2) as you can see from a numerical example, from drawing a chord across a rectangular hyperbola, or from Jensen’s inequality.

That means it will raise h1 and lower r. In the background, the economic mechanism is that this variability leads to more saving, which lowers the interest rate.

7.4 Equities and Risk Premia

We can use exactly the same method for assets which have more complicated patterns of payoﬀs that may depend on the state. For example consider an asset with price h1

which pays 1 in state 1 and zero in state 2, and an asset with price h1 with the opposite payoﬀs. These are sometimes called contingent claims, because their returns depend on the state which occurs in period 2. They also are referred to as Arrow-Debreu basis securities because their returns span the space of possible outcomes in period 2. It turns out that knowing their prices is suﬃcient to tell us the prices of more complicated assets, because we can write the latter as combinations of the basis securities. For example, holding the riskless asset is equivalent to holding both the AD assets, so its price is simply the sum of the prices of the AD assets.

The key is to note that our Euler equation links 1 + r (for any asset) to consumption. But this does not mean that their rates of interest are equal! In fact in the two-period model with only two states we can easily solve for their rates of return and explain the pattern of interest rates across assets, given the current state.

So what are the prices of the AD securities? By the same recipe, starting from C1 = 1:

1 1 h1 = E1β [payoﬀ], C1 C2 (7.13)

h1 = .95 · [1 · 0.6/1 + 0 · 0.4/2] = 0.57.

Likewise,

h1 = .95 · [0 · 0.6/1 + 1 · 0.4/2] = 0.19. (7.14)

Notice that h1 + h1 = h1. This is an example of arbitrage: if two portfolios give the same payoﬀs in all states then they have the same price.

120 We found the bond return, starting in the low state, to be 0.315. What about the returns on the two equities? First consider the stock which pays oﬀ if C2 = 1. We know that its price is h1 = 0.57. If in fact C2 = 1 then the payoﬀ is 1 and the return is 0.754.

But if C2 = 2 then the payoﬀ is zero, and the return is −1.00. Hence the return is random, and we usually study the expected return, which here is 0.6(0.754) + 0.4(−1.0) = 0.0524. What is the expected return on the other asset?

In our calculation of the AD security prices you might suspect that h1 > h1 simply because the former asset pays off in the low state, we have assumed we are in state 1 in period 1, and there is some persistence to states. You can think of one asset as a stock in a countercyclical industry (say a firm of bailiffs and auctioneers) and the other as a stock in a procyclical industry, which pays off in booms but not in recessions.

Exercise: Find the asset prices h1 > h1 if the economy is in the high state in period 1. Thus ﬁnd the unconditional means of the prices and rates of return. Does the premium on defensive stocks vary over the business cycle (i.e. depending on the initial state)?

In fact, the unconditional mean price of the defensive stock is higher than that of the procyclical stock, even though the payoffs are completely symmetric. Why? The rationale is that holding the former asset is more desirable because it pays off (can be redeemed) in the state in which income is low and hence the marginal utility of an additional unit of consumption is high. Thus it provides some insurance and commands a higher price than the equity which pays off in the good state. This gives a well-defined meaning to risk; it is not simply variance.

The differences between asset prices that result from differences in risk are called risk premiums (or premia). A very useful tool for describing them is the covariance decomposition, which we saw as thing 6 in chapter 1.3. Using the definition of a covariance, we showed that for two random variables, say x and y:

E(xy) = E(x)E(y) + cov(x, y). (7.15)

Now imagine that we have a two-period asset, with price h and payoﬀ or dividend in period 2 d2, which in general will be random. We know already that with log utility the

121 price satisﬁes: C1 h = βE1 d2. C2 The right-hand side is the expectation of a product, so using the covariance decomposition gives: C1 C1 h = βE1 E1(d2) + βcov( , d2). (7.16) C2 C2

It is obvious that assets with higher average payoﬀs should have higher prices, so let

us assume that this asset has an average payoﬀ of one: E1(d2) = 1. Then the ﬁrst term is simply the bond price, and C1 h = h1 + βcov( , d2). (7.17) C2

Here the second term is the risk premium. If the asset’s payoﬀ is high when C2 is high then the covariance is negative and the asset has a lower price than the riskless bond.

Conversely, if the asset tends to pay oﬀ when C2 is low then it will have a positive risk premium.

This result is called the consumption capital asset pricing model (CCAPM). It suggests that idiosyncratic risk is not priced; in other words, it is covariance between payoﬀs and marginal utility that matters, not the variance of payoﬀs. For example, if d2 were random but unrelated to C2, the theory predicts that the asset should have the same price as the riskless bond.

The risk premiums also can be expressed in returns, as opposed to prices, and we’ll see some examples below and in the exercises. One of the most well-known is the equity premium, the diﬀerence between average returns on stocks and on bonds. Historically, there has been an equity premium of several percentage points. According to the theory, higher returns (lower prices) on equity are explained by the positive covariance between dividends and consumption. However, when economists directly calculate this covariance using historical time series, they predict a premium much smaller than the observed one. This is the so-called equity premium puzzle.

One way to raise the risk premium predicted by the theory is to increase risk aversion (the examples here use log utility). We’ll see examples of this eﬀect in the exercises.

122 Intuitively, if people are more risk averse then equilibrium features higher expected returns on equity to compensate them for bearing risk. But it seems that explaining the scale of historical equity premia requires unrealistically high risk aversion.

7.5 Bond Default Risk

As an application of the same CCAPM reasoning, let us next look at the interest rate on a bond with some default risk. This bond pays 1 with probability 1 − λ and 0 with probability λ, so that λ is the probability of a complete default. The idea here is that that the issuer (for example a sovereign borrower) may default with some probability λ, as viewed by the market.

Call the price of this bond h1J , where J stands for junk, as a memory aid. Call the payoﬀ d2. Thus 1 + r = d2/h1J and so

d2 h1J = βC1E1 . (7.18) C2

For simplicity, assume cov(d2, 1/C2) = 0, so that

1 h1J = βC1E1d2E1 = h1E1d2 = h1[1(1 − λ) + 0λ] = h1(1 − λ). (7.19) C2

As long as λ > 0 we ﬁnd h1J < h1. The ﬁrst lesson here is that default risk lowers the bond price. As analysts we can read this in reverse and try to estimate λ from observations on {h1, h1J }.

Notice that the return now is random also: ( 1 − 1 w.p. 1 − λ r = h1J (7.20) J 0 − 1 w.p. λ. h1J

Most of the time this investment will earn a high return (because h1J is lower than h1) but then λ × 100% of the time it will earn a return of -100%. Ouch.

If you use the distribution (7.2) to calculate the expectation of this random variable rJ

you will ﬁnd that E1rJ = r, which is the second lesson we can draw. Perhaps surprisingly, the expected return on the bond with default risk is equal to the riskless return on the

123 discount bond. This simply reﬂects the fact that the risk premium is zero, because there is no covariance.

This example has a complete default, where the payoﬀ is 0 with probability λ. We can also calculate more realistic examples, where a default means a payoﬀ of 0.7, say, a

30% haircut instead of a 100% one. In that case h1J won’t be as low and so rJ won’t be as high in the good state.

If you think about this section or practice with some of the exercises, you might notice that h1J could be low either because a default is likely or because the market expects a very low payoﬀ if a default occurs. In general, we won’t be able to distinguish these two possibilities from the bond price alone. A small chance of a big default and a big chance of a small default can have similar eﬀects on the bond price.

You might remember the vulture funds that we met in chapter 3. They are the entities that buy (and possibly resell) sovereign debt when the possibility of default rises and the price falls. They are speculating that the market has overestimated λ. Using your calculator, you can see that a small decrease in λ would result in a large capital gain. Conversely, if the secondary market price falls, that indicates that the market’s view of λ has risen: a default is more likely. An interesting study by Michael Greenstone (2007) tracked the price of Iraq’s government debt during the US surge of 2007. He found that the price fell, suggesting the market thought a default was more likely after the surge began than before.

Let me note one last point on sovereign debt returns. I had always thought that treating default and consumption as uncorrelated was empirically realistic, so that there was no risk premium and E1rJ = r as we saw above. But recent research now suggests otherwise. Joseﬁn Meyer, Carmen Reinhart, and Christoph Trebesch (2019) studied returns on portfolios of sovereign bonds from 1815 to 2016 and found that average returns were greater than those on bonds issued by the US or UK governments. The premium was 2.61% over this full span. The premium was 5.83% over 1995–2016, a span that includes defaults by Argentina in 2001 and Ecuador in 2008. In the CCAPM these ﬁndings imply that there is indeed a covariance term, so that investing in sovereign bonds is more like

124 investing in the stock market than we may have supposed.

7.6 Nominal Bonds

Most assets are priced in currency (say dollars), not units of consumption, and their payoﬀs or redemption values are in dollars too. Let us next brieﬂy see how the theory deals

with this realistic feature. Consider a one-period nominal bond with price H1 in dollars today, and payoff one dollar next year. Let P denote the general price level. The bond’s price satisfies: H1 C1 $1 = E1β . (7.21) P1 C2 P2 Here we are simply using our original bond-pricing equation but writing the real price as the nominal price deflated by the price level and the real payoff as the nominal payoff deflated by the future price level. Thus

P1 H1 = h1E1 + cov, (7.22) P2 or 1 1 1 = E + cov. (7.23) 1 + i 1 + r 1 1 + π If the covariance term is zero then this is the exact version of the Fisher relation between a nominal interest rate, a real interest rate, and expected inﬂation. Taking logs gives approximately: i ≈ r + E1π.

Notice that if the expectation of P2 rises then H1 falls (and i rises). One of the themes of this chapter is that asset prices involve forecasts, so observing them can help us forecast as analysts or researchers. In this case we notice that bond prices are sensitive to news about inﬂation (or deﬂation) and so may provide an early warning system.

7.7 The Term Structure of Interest Rates

Our theory also can be used to see what information about the economy might be contained in the prices of assets which differ in maturity, not in payoff risk or default probability. Imagine now that there are three periods, and two assets with the following payoffs:

125 Period 1 2 3

S Cashﬂow −H1 1 L −H1 1

S The ﬁrst asset is our original riskless bond. It costs h1 and pays 1 a period later. The L second asset costs h1 and pays 1 two periods later. We might call these short and long bonds.

s s The short-term interest rate is i1 = (1/H1 ) − 1, just as we saw earlier. What about the long-term interest rate? To see how to report that, suppose your bank account has a balance of 100. With no deposits or withdrawals two years later the balance is 121. What is the interest rate? If this makes sense to you then I hope you agree that, if you spend L H1 this year for a redemption value of 1 two years from now, then implicitly the return per year satisﬁes: L L 2 H1 (1 + i1 ) = 1. (7.24) This is called the yield-to-maturity or just the yield on the long bond. It is reported per S year so that it is on the same scale as i1 . But notice that you cannot access this return for just the ﬁrst year or just the second year but rather only for both years.

The pattern of interest rates on assets that diﬀer only by maturity, graphed against maturity, is called the term structure (sometimes a curve ﬁtted to them is called the yield curve). We think of monetary policy as acting on the short end of the term structure and investment decisions as depending on the longer end, so understanding how they are connected is crucial to monetary policy. We’ll also see that the term structure can be a valuable tool for forecasting.

Continuing with log utility, the bond prices satisfy: HS 1 1 1 1 = E β , P C 1 C P 1 1 2 2 (7.25) L H1 1 2 1 1 = E1β . P1 C1 C3 P3 For the long bond price, notice that we take the payoﬀ in period 3, deﬂate it by P3, weight it by marginal utility, and then discount back to the present by multiplying by β2. We’re simply using the Euler equation across this two-period span.

126 As for the yields, the gross nominal interest rate on the short bond is simply

S 1 1 1 + i1 = S = . (7.26) H1 E1βC1P1/C2P2 The gross nominal return on the long bond satisﬁes

L 2 1 1 (1 + i1 ) = L = 2 . (7.27) H1 E1β C1P1/C3P3 where we recall (7.24) and so include the exponent.

S L To see how the term structure {i1 i1 } reflects forecasts, let us pause and look at the differences between the two formulas (7.26) and (7.27). First, suppose that, for some reason, investors expect P3 to be very high. In other words, they expect a burst of inflation from period 2 to period 3. If you raise P3 in the yield formula (7.27) you will also raise L L S i1 . Thus, an upward-sloping yield curve, with i1 > i1 , can reflect a forecast of increasing inflation. Conversely, a downward-sloping yield curve (sometimes called an inverted yield curve) would reflect a forecast of a decline in the inflation rate or perhaps even a deflation.

Second, looking at the formulas shows you that the same logic holds for C3. So an upward-sloping yield curve also could reﬂect forecasts of more rapid real growth. Con- versely, suppose that C3 is expected to be very low. Then marginal utility at time 3 will be high, the price of a bond which pays oﬀ at time 3 will be high and so its interest rate will be low and the term structure will slope down. Thus the term structure also may forecast recessions.

Third, the term structure also may forecast future short-term interest rates. To see this, take the long bond price and apply the covariance decomposition. The price of the long bond is: L 2 C1P1 H1 = E1β C3P3 C1P1 C2P2 = E1β β (7.28) C2P2 C3P3 S S βC1P1 βC2P2 = H1 E1H2 + cov( , ). C2P2 C3P3 The risk premium (the covariance term) usually is small in practice because there is very little persistence or autocorrelation in consumption growth. So, ignoring that term, this

127 expression (7.28) relates the long bond price to current and expected future short bond S s prices. Suppose that H1 = 0.95 and you have the expectation E1H2 = 0.94. Then the product is 0.893, which should be approximately the long bond price. The intuition is that you can assure yourself of a dollar in period 3 by buying a long bond, or by rolling over short bonds. The two methods should have similar prices. They do not have identical prices because of the risk premium. For example, buying the long bond exposes you to risk that the price of the short bond in period 2 will be higher than you expected.

We also can express the predictions in terms of interest rates, instead of bond prices. If we ignore the risk premium then we get

1 2 1 1 L = S E1 S . (7.29) 1 + i1 1 + i1 1 + i2

L S S S This shows that if i1 > i1 then E1i2 > i1 also. An upward-sloping yield curve predicts an increase in the short yield. Overall, then, an upward-sloping curve predicts faster growth, or higher inﬂation, and either of these would be reﬂected in a higher short-term interest rate. Predicting any of these things would be very useful, and collecting the yields is easy to do, so this is a promising forecasting tool.

7.8 Forecasting with the Term Structure

To see whether the slope of the yield curve really helps us forecast, we usually run a linear regression, like this:

L S πt − πt−1 = α0 + α1(it−2 − it−2) + t, (7.30) or like this:

L S ∆ ln Ct − ∆ ln Ct−1 = β0 + β1(it−2 − it−2) + t. (7.31)

Notice the timing. If t is counting years, then we’re checking to see if the slope of the yield curve two years ago predicted a change in inﬂation or consumption growth between the subsequent two years. These regressions are easy to run and have a good track record. Of course other maturities are also used in practice.

128 If you found that one of these regressions fit well, then the idea would be to use the estimated coefficients to make forecasts from today. For inflation for example:

L S πˆt+2 − πˆt+1 =α ˆ0 +α ˆ1(it − it ). (7.32)

To check on the interest-rate forecasts begin with their formula (7.29) where we’ve already ignored any possible risk premium. Next, take logarithms and completely ignore Jensen’s inequality: iS + E iS iL ≈ 1 1 2 . (7.33) 1 2 So as an approximation, long bond yields are the arithmetic average of current and expected short bond yields. This is sometimes called the expectations hypothesis of the term structure. If the current 1-year rate is 1% and the 2-year rate is 1.8% then the forecast for next year’s 1-year rate is is 2.6%. This is a very helpful memory aid.

Rearranging (7.33) suggests another forecasting regression:

S S L S it − it−1 = δ0 + δ1(it−1 − it−1) + t, (7.34)

where we would predict δ1 = 2. An inverted yield curve predicts a decline in short rates. This prediction does not hold up as well but again is very easy to update.

Sometimes researchers who are trying to forecast the short-term interest rate combine the information in the yield curve with the time-series pattern in the short rate itself. Here is an example, which uses the idea of cross-equation restrictions that we saw in chapter 6.4.1. First suppose the short-rate follows a ﬁrst-order autoregression like this:

S S it = 0.4 + 0.8it−1 + et. (7.35)

In this case, the short-rate is mean reverting, and it long-run average value of 2. To see S S this, take averages (unconditional expectations) of both sides: E(it ) = 0.4 + 0.8E(it ) + 0 S so E(it ) = 0.4(1 − 0.8) = 2. (For our ﬁrst-order autoregression the long-run mean is µ/(1 − ρ).)

129 Now let us solve for the two-period yield using the expectations hypothesis of the term structure. That gives:

iS + E iS iS + (0.4 + 0.8iS) iL = t t t+1 = t t = 0.2 + 0.9iS. (7.36) t 2 2 t

Thus L S S S it − it = 0.2 − 0.1it = −0.1(it − 2) (7.37)

S so the yield curve slopes up (down) when it < (>)2. Researchers would estimate equations (7.35) and (7.36) as a system, because they both involve the same parameters.

To remind yourself of what the yield curve looks like, I suggest an online search for the phrase ‘3D yield curve’ which may show this curve for the US and track it over time to produce a three-dimensional surface. That graph also shows you when inversions have occurred and whether they tended to precede recessions.

7.9 Risk Premia in the Foreign Exchange Market

We’re ﬁnally in a position to look at risk premiums in an uncovered position in the ∗ foreign exchange market. The nominal return on an uncovered position is (1 + i )S2/S1. Remember our recipe involves inserting the return on an investment into the Euler equation. That gives us: ∗ C1 (1 + i ) S2 1 = E1β . (7.38) C2 1 + π S1 Next, we simply apply the one statistical tool in this chapter, the covariance decomposition: C1 1 ∗ S2 C1 1 ∗ S2 1 = E1β E1(1 + i ) + cov β , (1 + i ) . (7.39) C2 1 + π S1 C2 1 + π S1

By now you probably recognize that the ﬁrst term is simply the price of a domestic, nominal bond, so let us make that substitution: 1 ∗ S2 C1 1 ∗ S2 1 = (1 + i )E1 + cov β , (1 + i ) . (7.40) 1 + i S1 C2 1 + π S1 Now you might recognize the ﬁrst term as just the UIP expression. So UIP holds only if the covariance (or risk premium) is zero. In other words, a non-zero covariance term might explain why UIP does not hold.

130 Imagine that

∗ S2 (1 + i )E1 > 1 + i, (7.41) S1 so that the expected, risky return from going oﬀshore is greater than the safe, onshore return. That means that ﬁrst term is greater than 1, so the covariance term must be negative for the two terms to add up to 1.

A negative value for the covariance in turn reﬂects a positive covariance between the exchange-rate depreciation and inﬂation or between depreciation and real consumption growth. One interpretation is that, while one is holding the foreign currency, a domestic appreciation (a fall in S2) is especially painful if there also turns out to be a recession occurring when one converts one’s foreign-currency earning back into domestic currency. Thus the return on the foreign-currency position must be higher to compensate for this risk. I realize that this is a somewhat convoluted story, so I simply report it while urging you not to memorize it.

While this overall model of risk premia has some success empirically, it has not had much success in explaining the forward premium anomaly. When we measure the risk premium as a covariance using aggregate, macroeconomic data we generally ﬁnd that it is not large enough or variable enough to act as an omitted variable that explains why unbiasedness doesn’t hold. Perhaps our underlying model of consumption risk is not the right one for measuring the premia. Meanwhile the rejection of UIP is still a mystery.

Postscript: While we’ve just looked at forward currency contracts, we also could learn about how markets work by studying forward commodity contracts, or futures contracts. Futures contracts in currencies or commodities are similar to forward contracts, except that they trade in a market (as opposed to being written with a bank) so that the price can vary over time as the maturity date approaches. The world’s ﬁrst futures market was the Dojima rice market in Osaka, which began trading in the 1730s during the Tokugawa Shogunate. Ito (in Keizai Kenkyu 1993, in Japanese) and Hamori, Hamori, and Anderson (in the Journal of Futures Markets, 2001) have studied whether the futures price of rice was a good predictor of the eventual spot price by using modern statistical methods. They

131 ﬁnd a strong correlation, but that unbiasedness did not hold. Apparently there were risk premiums in the rice market, a conclusion also reached by researchers who study today’s commodity and currency markets.

Further Reading Here are two articles cited in this chapter: Michael Greenstone (2007) Is the ‘Surge’ working? Some new facts. MIT Department of Economics Working Paper No. 07-24 Joseﬁn Meyer, Carmen Reinhart, and Christoph Trebesch (2019) Sovereign bonds since Waterloo. NBER working paper 25543 or see www.nber.org/digest-2019-04

Exercises

7.1. Suppose that asset returns satisfy this Euler equation:

1 1 = Et0.94(1 + rt+1) Ct Ct+1

where rt+1 denotes the return from period t to period t + 1 and C is real consumption in the US. For simplicity suppose there is no inﬂation. Suppose that in any period C can take on two values, 1.00 and 1.02, each with probability 0.5. There is no autocorrelation in consumption.

(a) Call ht the price of a one-period bond that pays 1 in all states of the world in period t + 1. Find the interest rate on this bond ﬁrst when Ct = 1 and second when Ct = 1.02.

(b) Call hjt the price of a one-period bond (issued by a sovereign government in an emerging economy) that pays 1 in period t+1 with probability 1−λ and pays 0.8 with probability λ. The event of this partial default is uncorrelated with the value of consumption in period t + 1. Solve for the expected return on this bond ﬁrst when Ct = 1 and second when Ct = 1.02. (c) How could an analyst estimate the value of the default probability λ?

7.2. Suppose that asset prices for US investors satisfy the following Euler equation:

1 1 = Et0.98(1 + rt+1) , Ct Ct+1

where rt+1 is a real return from period t to period t + 1. Imagine that there is no inﬂation.

132 (a) Suppose that, viewed from period t, the ratio Ct+1/Ct in the US can take on the values 1 (in a recession) or 1.03 (in a boom) each with probability 0.5. Find the price and interest rate for a one-period, discount bond. (b) Analysts study a one-period bond issued by a sovereign borrower, Spain, and believe that it will pay 1 with probability 1 − λ and 0.4 with probability λ. They also believe that a Spanish government default (or restructuring) is uncorrelated with US consumption growth. If they observe that the price of the sovereign bond is 0.75 then what will they conclude is the value of λ, the probability of a debt restructuring? (c) Suppose that in fact there is a negative covariance between US consumption growth and default by this sovereign borrower. What qualitative eﬀect will that feature have on the analysts’ conclusion about the probability of default?

7.3. Suppose that in the US treasury bonds pay one next year in all states of the world u and have price h1 that satisﬁes this Euler equation:

u C1 h1 = E1β . C2

Also suppose that gross, consumption growth C2/C1 can take on two values—1.03 and 1.01—with equal probability. Assume that β = 0.99. (a) Solve for the price and interest rate on a US treasury bond. a (b) Suppose that Argentina begins to issue new discount bonds with price h1 and payoﬀ a d2. The payoﬀ equals 1 with probability 1 − λ and 0.7 with probability λ. A default (i.e. a payment of 0.7) occurs independently of consumption growth in the US. Show whether the price of these bonds depends on the default probability λ. Show whether the expected return on these bonds depends on the default probability λ. (c) Now suppose that a default is in fact more likely when there also is slow growth in the US, because there is a world business cycle. Would this pattern to defaults raise or lower the price of Argentina’s bonds?

7.4. Suppose that asset returns satisfy this Euler equation: 1 1 = E10.94(1 + r) C1 C2 where r denotes the real return from period 1 to period 2 and C is real consumption in the world. Suppose that C1 = 1 and that C2 can take on two values, 1.00 and 1.02, each with probability 0.5. (a) Solve for the world real interest rate, on a one-period, real, discount bond that is free of default risk.

(b) Now suppose there is inﬂation, with P1 = 1 and P2 = 1.03. Solve for the world real interest rate and nominal interest rate on a one-period, nominal discount bond that is free of default risk.

133 (c) Now imagine an emerging market debt issuer, whose nominal discount bonds pay 1 with probability 1 − λ and 0.9 with probability λ. Will this debt have a higher expected, real return than the asset you studied above?

7.5. This question studies government bond prices and returns in Europe. Suppose that ﬁnancial asset returns satisfy this equation:

1 1 = Et0.98(1 + rt+1) , Ct Ct+1 where rt+1 is the return from time t to time t + 1. Suppose that Ct = 1 and Ct+1 = 1.03 or 1, each with probability 0.5. There is no inﬂation. (a) Find the price of a one-period, riskless, discount bond (i.e. an asset that will pay 1 next period in all states of the world). Also ﬁnd the return on this bond. (b) Now imagine a bond issued by the government of Portugal that pays 1 with probability 0.9 and pays 0.3 with probability 0.1. Thus, there is a 10% chance of a debt restructuring. Assume the event of this partial default is uncorrelated with the outcome for European consumption. Find the price of this bond. Find the return when there is no default.

(c) Suppose that the partial default in Portugal was more likely to occur when there was slow consumption growth in Europe. What eﬀect would that correlation have on the price of the second bond and on its return?

134 Part C: Monetary Policy

135 Chapter 8. Currency Wars

This part of the book turns to discussing monetary policy. Now, we know from the open-economy trilemma that this is the same thing as exchange-rate policy, in the absence of capital controls. Or is it? In this brief chapter we’ll look at some case studies of policy changes that were driven by movements in exchange rates. Then we’ll see whether foreign exchange intervention can somehow sidestep the trilemma.

8.1 Case Studies

The case studies all have an underlying theme. From chapter 5 we know that a decrease in s tends to lead to a decrease in q (a real appreciation), for PPP does not hold. From previous courses in macroeconomics recall that a fall in q tends to lead to a fall in exports, by making them more expensive. And from chapter 6 we know that there is evidence that an increase in m tends to raise s. Thus, when s falls, for whatever reason, ﬁrms and workers in export sectors may ask for expansionary, domestic monetary policy to counteract that appreciation. Depending on the political system and the mandate of the central bank, that may aﬀect policy.

We also saw in chapter 6 that a domestic appreciation could be caused by loose monetary policy in a foreign country: a decrease in m∗. A situation in which m∗ rises and then m rises to prevent a ﬂoating exchange rate from depreciating is sometimes called a currency war, though there is no precise deﬁnition for this dramatic term.

The first case study refers to Brazil. During the 2007–2009 financial crisis the US Federal Reserve implemented expansionary monetary policy. Thus m∗ increased. The real appreciated against the USD, with an effect on exporters in Brazil. The Brazilian authorities responded in two ways. First, they publicly objected to the loose monetary policy in the US, because of these effects on the rest of the world. Second, they applied some taxes on purchases of domestic assets (capital inflows) to try to limit the demand for reals. The taxes were applied in late 2009 then increased in 2011.

The objections of policymakers in Brazil and other EMEs did not lead to a change in US monetary policy. But the taxes did have an eﬀect, I think. If you graph the USD/BRL

136 exchange rate you will see that the timing of the taxes is associated with a turn in the path of the exchange rate, as the real began to depreciate. In fact, the policy may have been too effective, and caused a large depreciation with possible effects on domestic inflation via ERPT. As time passed other shocks of course occurred though. One lesson may be that it is very difficult to fine-tune the management of an exchange rate using taxes or controls.

The second case study refers to Switzerland. In this case the increase in m∗ was the work of the ECB, not the Fed, in response to the European debt crisis in 2011. The Swiss franc (CHF) appreciated against the euro, in other words s fell. In Switzerland this appreciation was certainly viewed as harmful to exporters, and newspapers reported stories of customers switching away from Swiss goods to those of other countries. The Swiss trade minister was reported as saying that buying a clock from China would be like buying chocolate from Germany, which shows that policymakers were under some stress.

The Swiss National Bank (SNB) responded by announcing a floor to S at 1.2 CHF/EUR. If you graph this exchange rate you will see that the floor applied from 2011 to 2015. You can think of the policy as simply promising increases in m to keep S above this level. In chapter 9 we’ll see how a monetary authority can work backwards from a target for the exchange rate to a target for monetary policy. And we’ll also see there that a commitment to future monetary policy can have an effect on today’s exchange rate.

In 2015 the ﬂoor was discontinued for two reasons. First, the SNB was concerned that relatively expansionary monetary policy in Switzerland would lead to inﬂation. Second, the euro by then had depreciated against some other currencies so an appreciation of the CHF against the euro would not necessarily imply an appreciation against many other countries’ currencies. In 2015 S quickly dropped to 1.1, around which it has varied since.

The third case study refers to Japan. To study this case, load the JPY/USD in the Paciﬁc Exchange Rate Service, for 2000–2020. You will notice an appreciation of the yen from 2008 to 2011 (again perhaps caused by m∗) followed by a depreciation until 2015. You might also ask yourself whether you see any dramatic outliers in the time series. Next, zoom in to the span from 2010 to 2012 and again see if there are outliers. Repeat the exercise by graphing the exchange rate for 2011 only. Finally, zoom in on one month:

137 March 2011. What happened on 11 March 2011?

You should see that the JPY appreciated sharply for several days after March 11th. In contrast to the ﬁrst two cases, this high-frequency appreciation was not caused by an increase in m∗ (or news about that). We’re never sure what causes exchange-rate movements but I suspect that appreciation was due to a sudden demand for yen by domestic investors, redeeming some of their foreign assets to meet emergency needs.

Policymakers in Japan were concerned that the appreciation would harm exports. In response to the sudden appreciation, there was a coordinated intervention in the foreign exchange market by a number of central banks (including the Bank of Canada). Roughly speaking this means that they sold yen and purchased foreign currencies, to try to reduce the value of the yen. Section 8.2 discusses this tool in more detail.

Back to the history. You’ll notice that the exchange-rate did a U-turn on March 17th, as the appreciation was reversed. So perhaps the intervention was eﬀective. On the other hand, if you zoom back out and look at the graph for all of 2011 or out to 2012 then you will notice that the yen appreciated further later on. Its peak value was at around 76 JPY/USD in October 2011, much appreciated from the value near 79 on March 17th.

These three case studies suggest a list of four ways that a central bank can resist a currency appreciation:

(a) apply taxes or controls to limit purchases of domestic assets (Brazil)

(b) loosen current and expected future monetary policy (Switzerland)

(d) try to inﬂuence foreign monetary policy.

Now, these histories might make it seem that a foreign monetary expansion is a bad thing for the domestic economy, because it causes a domestic appreciation and then a decline in exports. But remember that importers might benefit. And, the exchange rate is not the only variable affected by a change in foreign monetary policy. It is important to remember (from intermediate macroeconomics) that a foreign monetary expansion also has a spillover effect on the domestic economy (certainly including the export sector) as

138 the foreign economy draws in more imports. For example, the monetary expansions of the Fed after 2007 and the ECB after 20010 no doubt prevented deeper recessions in those areas and hence also had an expansionary eﬀect in trading partners. Bernanke’s (2016) short essay reminds you of this channel.

Another perspective on the history is that a currency war (or competitive depreciations) is not necessarily a bad outcome when inﬂation is low and countries are in recessions. Perhaps at such a time it is simply another term for a monetary expansion that is eventually coordinated across countries. Frankel (2019) observes that alternative political responses to the concerns of the export sector, such as tarriﬀs, may be more damaging.

And ﬁnally, remember that PPP may hold in the long run. That is another way of reminding yourself that monetary policy probably cannot permanently alter the trade balance.

8.2 Intervention

We saw that central banks used foreign exchange market intervention in March 2011. To see how that works, visualize a central bank’s balance sheet:

Assets Liabilities

bonds (B) monetary base (M) reserves (R)

Reserves are assets denominated in foreign currency, typically USD, EUR, or JPY but also GBP, CAD, CHF, or AUD.

Imagine a central bank trying to prevent an appreciation. From supply-and-demand reasoning, it is natural to think that buying reserves will help with that goal. An increased demand for foreign currency will increase its value and lower the value of the domestic currency.

Unsterilized intervention involves such purchases (or sales) of R. This is simply an open-market operation from introductory economics. The central bank buys an asset and pays for it by adding to the balaces of ﬁnancial institutions at the central bank. R increases

139 and M increases. As with any monetary expansion, our theory predicts that S will rise, so the currency depreciates or is prevented from appreciating.

Sterilized intervention involves purchases (or sales) of R oﬀset by sales or purchases of B, so that M does not change. In this case the overall size of the balance sheet does not change, but the composition of assets does. Again reserves are bought to try to prevent an appreciation and sold to try to prevent a depreciation. If you are ever in a country where this is tried, you may ﬁnd that the news media report on a loss of reserves in ‘defending’ an exchange rate. Remember that the loss is equal to the purchase of domestic-currency- denominated assets.

The key question here is: Does sterilized intervention have any effect? It seems to offer a way around the trilemma, by influencing the exchange rate without changing monetary policy. There are two theories of how sterilized intervention might work. The first is called (a) the portfolio balance effect. I believe this is simply another name for the supply-and- demand reasoning above. But in advanced economies the reserves of the central bank are quite small compared to the overall size of the foreign exchange market (seen in chapter 6.1), so I am sceptical.

The second is called (b) the signalling eﬀect. Here the idea is that the intervention has no direct eﬀect on the exchange rate, but it reveals that the central bank would like a depreciation and so is more likely to loosen monetary policy in the future. As we saw in chapter 6, any variable that predicts future monetary policy can be correlated with the current exchange rate.

Testing for an effect of intervention requires data on central bank transactions that are released (perhaps with a delay) and are recoreded at high-frequency (such as daily). Researchers then use econometric methods to see if intervention smooths out movements in the exchange rate. That may have happened in Japan in 2011. They also see if intervention predicts later changes in monetary policy. Overall, there is limited evidence that sterilized intervention is effective. And central banks in advanced economies tend not to practice it. But Fratzscher et al (2019) study a wide range of countries from 1995 to 2011 and conclude there are some significant effects.

140 We conclude with two questions. First, if intervention does not have much effect then is there anything a monetary authority can do to limit fluctuations in a floating exchange rate? Of course the answer is that they can apply taxes or controls on some kinds of foreign exchange transactions. These tools have been used by Brazil (as noted above), Chile, Malaysia, and a range of other countries.

Second, this chapter has mostly focussed on concerns about exchange-rate appreciation. Is there any reason to be concerned about depreciation? There are several possible answers. A nominal depreciation may be a real one and so may harm importers, and make imports of key materials or equipment more expensive. If there is significant ERPT it may lead to inflation, something central banks usually try to control. And a depreciation may also reduce wealth and cause bankruptcy via a valuation effect, if firms have liabilities in foreign currency.

Among advanced economies, fluctuations in exchange rates have not been as controversial recently as they were at times in the past. That might be because USD/EUR/JPY exchange rates have been relatively stable in recent years, a situation sometimes called ex- tended Bretton Woods II. That stability might occur because monetary policies are quite similar in most AEs. But around the world, monetary authorities often still are concerned with these fluctuations. In this part of the book we see a range of methods to limit those: foreign exchange intervention here in chapter 8, fixed exchange rates (target zones) in chapter 9, and currency unions in chapter 10. It also is important though to remind yourself of the important things that are lost when the exchange rate is not floating, namely either an independent monetary policy or international capital mobility.

Further Reading

Here are the articles cited in this chapter:

Ben S. Bernanke (2016) What did you do in the currency war? Brookings Institution blog www.brookings.edu/blog/ben-bernanke/2016/01/05/what-did-you-do-in-the- currency-war-daddy/

141 Jeffrey Frankel (2019) The currency manipulation game is afoot — but that’s better than a trade war. The Globe and Mail 13 August 2019. Marcel Fratzscher, Oliver Gloede, Lukas Menkhoff, Lucio Sarno, and Tobias Stöhr(2019) When is foreign exchange intervention effective? Evidence from 33 countries. American Economic Journal: Macroeconomics 11(1), 132–156.

142 Chapter 9. Fixed Exchange Rates and Target Zones

To continue our study of exchange-rate regimes and monetary policy, we’ll next learn more about fixed exchange rates. These regimes vary a lot in their durability, with some lasting for many years and others being temporary. Some countries officially have fixed or pegged exchange rates while, as we’ll see in chapter 10, others report that they float but actually seem to stabilize their exchange rates as if they were fixed. In order to see why a country might adopt this system and also why it might abandon it, we need to study what the system implies for monetary policy and interest rates.

First in section 9.1 we’ll see how ﬁxed exchange rates work, and what guides or constrains monetary policy under this system. Then in section 9.2 we’ll look at what can go wrong (or how pegs meet their end). Chapter 10 studies the broad issue of why this regime is adopted in the ﬁrst place.

9.1 Operating a Fixed Exchange-Rate System

To see how to operate the system, we begin with a perfectly ﬁxed system then later study the situation in which the exchange rate is allowed to vary within some range.

9.1.1 Monetary Reaction

Recall from the open-economy trilemma that adopting a fixed exchange rate requires devoting monetary policy to that goal (unless there are capital controls.) So the main lesson of managing a fixed exchange rate concerns how monetary policy has to react to shocks. Let us use the monetary model of the exchange rate to illustrate the reactions. Remember that: P S = P ∗ M exp(−αi∗)Y ∗ = (9.1) M ∗ exp(−αi)Y M Y ∗ = exp[α(i − i∗)] M ∗ Y Let us write this in logs, keep track of time subscripts, and also impose the fixed exchange rate: S = S and so s = s:

∗ ∗ ∗ s = (mt − mt ) + (yt − yt) + α(it − it ). (9.2)

143 Finally, we can invert or rearrange this to give a rule for domestic monetary policy:

∗ ∗ ∗ mt = s + mt + (yt − yt ) − α(it − it ). (9.3)

Lesson 1: The level of the ﬁxed exchange rate or peg determines the stance of domestic monetary policy. Choosing a value for s implies a value for m. That consideration is important for countries that are joining or adopting a ﬁxed exchange rate. In equation

(9.3) imagine that s is below the current ﬂoating exchange rate st. Then monetary policy will need to be tightened (m will need to fall) to engineer an appreciation and reach that goal.

Lesson 2: Domestic monetary policy has to shadow foreign monetary policy. You can see that m moves 1:1 with m∗. If the foreign monetary authority tightens monetary policy then the domestic authority must do the same, to avoid a depreciation.

Lesson 3: Monetary policy also has to respond to shocks to domestic money demand. Notice that m moves 1:1 with y. If there is a domestic recession then the currency will tend to depreciate, unless there is an oﬀsetting contraction in monetary policy.

Lesson 4: Monetary policy may need to respond to a loss of credibility in the peg. To

see this lesson, ﬁrst suppose that the peg is completely credible, so that Etst+1 = s. Then from UIP we know that i = i∗. As a result, the last term in the reaction function is zero. What happens if the peg is imperfectly credible and speculators think there is a possibility ∗ that Etst+1 > s? Then from UIP it − it > 0 and so from (9.3) be higher, so there must be an oﬀsetting tightening of monetary policy to maintain the peg.

Another way to see this result is to combine UIP with the rule (9.3):

∗ ∗ mt = s + mt + (yt − yt ) − α(Etst+1 − s). (9.4)

You can read oﬀ directly that an expected depreciation — Etst+1 > s — necessitates an even tighter current stance for monetary policy.

Lesson 5: Reserves generally respond to shocks. Suppose that the central bank holds two assets, foreign exchange reserves R and domestic bonds B (sometimes called domestic

144 credit). Its balance sheet then is: M = B + R. (9.5)

A sure-fire way to maintain a peg is to maintain sufficient reserves to allow buying and selling to make the market exchange rate equal to the official rate. In levels then:

∗ M Y ∗ S = ∗ exp[α(i − i )] M Y (9.6) (B + R) Y ∗ = exp[α(i − i∗)] M ∗ Y which means that we can solve for the necessary reserves:

SM ∗Y R = − B (9.7) Y ∗ exp[α(i − i∗)]

We then can see how reserves need to respond to the variables on the right-hand side, just as in the previous lessons. (I have not written this in logs because the log of B + R is not equal to the sum of their logs.)

Lesson 6: The central bank must sterilize changes in domestic credit. Suppose that the central bank increases B, to lend to the government or to assist the banking system. Then we can read oﬀ that is must decrease R 1:1 with the increase in B in order to maintain the ﬁxed exchange rate. This transaction is called sterilization.

Lesson 7: The level of reserves needs to be suﬃcient for the central bank to respond to shocks without running out of reserves. You can see that if R = 0 that it cannot respond to shocks.

One way the central bank can make sure it has a large stock of reserves is by borrowing to ﬁnance them. It can issue domestic debt and use the proceeds to purchase reserves. One of the striking developments in international ﬁnance in the past several decades has been the large-scale accumulation of reserves by central banks in emerging economies.

9.1.2 Target Zones

Most fixed exchange rates are not really fixed but involve fluctuations within a band called a target zone. Historical examples include the Bretton Woods bands against the dollar and the ERM bands in western Europe which ranged from ±2.25 percent to ±15

145 percent during the 1980s and 1990s. To see current examples, go to the Pacific Exchange Rate Service then select the plot interface and choose the exact time period to begin in 2000. Choose the euro as the base currency and then the Danish krone (pl: kroner) as the target currency. Finally, look at a second plot with the USD as the base currency and the Hong Kong dollar as the target currency.

You will notice that these plots do not resemble the ﬂoating exchange rates we’ve seen previously, yet they are not perfectly horizontal lines either. In theses countries the authorities keep s between bounds of s and s using monetary policy. We’ll next see how this system works and discover why a monetary authority might adopt it.

Most formal models of target zones use the simple monetary model of the exchange rate. They generally use some continuous-time mathematics with which you may be unfamiliar, so to avoid this entry barrier, we work with a simple, discrete-time example.

Imagine that the authorities can control the fundamental so that it is a constant plus some small variation:

xt = µ + ηt. (9.8)

Suppose for example that ηt has a uniform distribution centred at zero and between −δ and +δ. Its probability density function is box-shaped. That way it is bounded from above and below, unlike a normal density for example. Suppose that ηt is white noise (remember that term from chapter 5?) so that Etxt+j = µ ∀j ≥ 1. And each period xt is uniformly distributed between µ − δ and µ + δ.

Remember the difference equation from the monetary model: α 1 s = E s + x . (6.17) t 1 + α t t+1 1 + α t Just as in chapter 6 we can solve this two ways. First, we can solve the difference equation forwards and replace Etxt+j by µ. Then we find: 1 α α 2 s = [x + E x + E x + ... t 1 + α t 1 + α t t+1 1 + α t t+2 1 1 α α = x + (µ + µ + ...) (9.9) 1 + α t 1 + α 1 + α 1 + α x αµ = t + . 1 + α 1 + α

146 It certainly takes some patience to ﬁgure out the power series in (9.9). So let us also solve the diﬀerence equation the second way, by the guess-and-verify method. Suppose that we guess:

st = k0 + k1xt. (9.10)

Then st+1 = k0 + k1xt+1 which means that

Etst+1 = k0 + k1Etxt+1 = k0 + k1µ. (9.11)

Now substitute our guess (9.10) for st and its implication (9.11) for Etst+1 in (6.17):

α 1 k + k x = (k + k µ) + x (9.12) 0 1 t 1 + α 0 1 1 + α t

Making sure that the intercepts and slopes are the same on both sides of this equation gives us: α 1 s = µ + x . (9.13) t 1 + α 1 + α t Well done! We’ve completed our algebra and the two methods give the same answer.

For future reference notice that we also discovered that Etst+1 = µ. The midpoint of the range for x is also the midpoint of the range for s. So we can use that as a check or fast solution method in numerical examples, just by plugging that into (6.17).

Putting bounds on the fundamental will lead to bounds on the exchange rate. Suppose

that xt is at the top of its range so xt = µ + δ. Then st = s = µ + δ/(1 + α). And if

xt = µ − δ then st = s = µ − δ/(1 + α). So, speaking informally, the monetary authority keeps s in a box by keeping x in a box.

As long as α > 0 then 1/(1 + α) is a fraction. But that means that the range of s is narrower than the range of x and the variance of s is less than the variance of x. Credibility in controlling the fundamentals pays oﬀ in a narrower range for s than for x. The logic is that the authorities are credibly controlling all future fundamentals, and not just the current ones.

This results tells us something about why a country might adopt a target zone in the ﬁrst place. It allows some variation in x and in fact a surprising amount of such variation,

147 while still keeping x within a fairly narrow range. So this is just an intermediate point in the trilemma. The monetary authority partly stabilizes the exchange rate and it has ∗ limited monetary policy autonomy in that mt need not shadow mt perfectly.

So have we now found a way to magically reduce the volatility of the exchange rate? No. The volatility goes into interest rates. This sounds like a bad thing, but another way to describe this outcome is again to say that the central bank has some policy independence under a target zone, something which is not possible with a perfectly fixed exchange rate. Suppose that the band is ±2 percent wide. Then suppose we are at the lower edge of the band. Then the maximum expected depreciation is 4 percent. Now recall the UIP condition. Since the depreciation could occur rapidly, large, temporary interest-rate differentials are possible in short-term interest rates quoted at annual rates. For example, 3-month rates could display a 12 percent differential.

9.1.3 Testing Target-Zone Credibility

There are several ways to test whether a target zone is credible. Since we don’t have much conﬁdence in any speciﬁc model of the fundamental, most tests of target zone models focus on the unconditional properties of the exchange rate or on the relation between the exchange rate and interest rates.

One way to test the model is to regard the forward exchange rate as the expected future spot rate, and see if:

j s ≥ ft ≥ s (9.14)

j at any maturity j. You can see that this jointly tests the unbiasedness hypothesis (ft =

Etst+j) and the credibility of the exchange-rate band.

A second way is based on uncovered interest rate parity. At the top of the band, s can only fall (appreciate), so the local interest rate should be below the foreign one. At the bottom of the band, s can only rise (depreciate) so the local interest rate should be above the foreign one. Thus we can graph the interest diﬀerential against the exchange rate. We should see a negative slope.

148 We do see this pattern for the Hong Kong dollar or Danish krone. But historically, this pattern is not so clear for every target zone. Sometimes the scatter plot of i − i∗ against s is a cloud, not a downward sloping line, and forward rates often lie outside the band. This is evidence that the foreign exchange market does not believe the target zone will endure.

So a realistic modification is to add the possibility of realignments, which are shifts up in the target zone (somestimes called devaluations). That is realistic for most target zones, like the Bretton Woods ones, where the GBP was devalued several times and the DM was revalued. Realignment risk may explain why forward rates lie outside the band and why interest differentials don’t match the position of the spot rate within the band. But realignment models are difficult to test because realignments are relatively infrequent. If we have time we’ll set up an example and see how it works.

Of course, the forward rate might lie outside the target zone for the spot rate not because investors expect a discrete, one-time realignment but because they expect the fixed exchange rate to be abandoned and the currency to float and depreciate. One reason there is not much ongoing research on target zones is that most of them do not last long. The Danish peg to the euro or the Hong Kong peg to the dollar are exceptions. In the next section, we’ll study speculative attacks or crises in fixed rate systems.

9.2. Speculative Attacks

Some fixed exchange rates do not last very long. And we’ve seen that the statistics from target zones may reflect expected ends to these arrangements. Often they end with an exchange-rate crisis or speculative attack, defined as a sudden depreciation of the exchange rate. We next see two reasons why these crises might occur.

9.2.1 First Generation Model

The first type of speculative attack arises endogenously in a situation with fiscal dominance, where the central bank is monetizing debt while also operating a fixed exchange rate. We’ll solve the forward-looking version of the model of this type of attack. This

149 involves some mathematics so please don’t worry if you decide to skip it and please don’t memorize it either.

Imagine that the fundamental is simply mt for simplicity. We’ll also ignore the narrow target zone. Start with the central bank’s balance sheet. Its liabilities are M while its assets are government bonds, B and foreign exchange reserves, R. Remember that:

M = B + R. (9.5)

Now imagine that the ﬁscal authority pressures the central bank to buy its bonds. For that reason, B grows steadily:

Bt − Bt−1 = bt − bt−1 = µ + ηt. (9.15) Bt−1

This involves an average growth rate µ; it also is called a drifting random walk. Thus there is an upward trend in the fundamentals in contrast to the target zone model of section 9.1.2.

We know that the ﬁxed exchange rate requires:

s = m. (9.16)

To freeze m at this level, the central bank runs down its reserves to maintain the peg. As b rises, r falls so m is constant, due to sterilization.

Eventually the central bank runs out of reserves. Suppose that the currency then

ﬂoats. With zero reserves, mt = bt, so the exchange rate is given by:

1 α s = [b + E b + ...] t 1 + α t 1 + α t t+1 1 α = [b + (b + µ) + ...] (9.17) 1 + α t 1 + α t

= αµ + bt.

Where did this equation come from? It is simply the ﬂoating exchange rate solution (6.32) we found in chapter 6 when the fundamental follows a drifting random walk.

150 Suppose that the ﬂoat begins at time T . From our ﬂoating solution (9.17) that means that

sT = αµ + bT . (9.18)

Now we use the principle that there cannot be predictable jumps in the exchange rate, for those would imply inﬁnite, instantaneous, capital gains. So at the time of the switch, the

ﬂoating rate must be equal to the ﬁxed rate: st = s.

2 To simplify the mathematics, I shall assume that ση = 0 so that there is no uncertainty and we can ignore the error term ηt. That means that

bT = bt + µ(T − t). (9.19)

Thus,

s = bt + µ(T − t) + αµ. (9.20)

We can rearrange this to ﬁnd how long the ﬁxed exchange rate will last:

s − b − αµ T − t = t . (9.21) µ

What is the eﬀect of µ on this duration?

Also recall that s = m = ln(B + R). That means that:

ln(B + R ) − b − αµ T − t = t t t , (9.22) µ so you can see that the lifespan also depends on the initial reserves of foreign exchange,

Rt.

Our next result is the surprising and interesting one. We shall show that reserves vanish gradually but only up to a point. Before they gradually reach zero there arrives a day on which they are bought out and jump to zero in a speculative attack.

We can rearrange the diﬀerence equation of the monetary model in this form:

st = mt + α(Etst+1 − st). (9.23)

151 Then, at the end of the ﬁxed exchange rate regime at time T :

s = sT = mT + α(ET sT +1 − sT ). (9.24)

Consider the terms in this equation. When a float begins, the rate of change in s (the last term) jumps up from 0 to µ. But we have argued that s cannot jump. Thus mT must jump down, reflecting a sudden loss of remaining reserves. We shall draw a diagram to show the behaviour of each of these variables over time. It will appear that the speculative attack causes the end of the fixed exchange rate, whereas we know in this formal example that the underlying cause is the inconsistency between two goals of monetary policy.

Some studies use the approximation m = b + r, where lower-case letters denote logs. In that case the life expectancy of the ﬁxed exchange rate is:

b + r − b − αµ r − αµ T − t = t t t = t . (9.25) µ µ

Here is a numerical example using that approximation. Suppose that bond holdings grow like this:

bt+1 = 1 + bt. (9.26)

Currently, the exchange rate is ﬁxed at rate s = m. And suppose that α = 2. If the central bank runs out of reserves, then the exchange rate will ﬂoat, and follow:

st = 2 + bt. (9.27)

Imagine that currently the exchange rate is fixed at s = 60, with bt = 40 and rt = 20. Let us figure out how many periods the fixed exchange rate will last. Using equation (9.25):

r − αµ 20 − 2 T − t = t = = 18. (9.28) µ 1

Let us also calculate the value of bT at which a speculative attack occurs and what reserves the central bank loses on that day. Bond holdings begin at 40 and grow by 1 each day so after 18 periods bT = 58. Reserves will drop suddenly by 2 at that point.

152 Peru’s experience from 1985–1986 is an example of a speculative attack caused by fiscal dominance. There were large, ongoing budget deficits, financed in part by bond sales to the central bank. Reserves fell while the exchange rate was fixed, then stabilized once the exchange rate for the sol was allowed to float. Sadly, the growth in domestic credit was so rapid that hyperinflation developed in Peru later in the 1980s.

9.2.2 Second Generation Model

The ﬁrst-generation model described so far has a couple of very useful features. First, it shows that a sudden loss of reserves (a speculative attack) isn’t pathological, but stems from the logic of our standard, monetary model. Second, it reminds us that inconsistent goals of the central bank may doom a ﬁxed exchange rate.

But that model of speculative attacks has been criticized because (a) central banks can tighten monetary policy (reduce m) to defend a fixed exchange rate and (b) they can borrow reserves and, in general (c) it seems inappropriate to model the authorities in the mechanistic, dual way that is adopted in the first-generation model. And the empirical record shows some pegs ending without loose fiscal policy.

So more recent models of speculative attacks start by giving an objective function to the central bank. First, the bank wants exchange-rate stability, either to promote trade or as a way to control inﬂation. And there is another important reason why a central bank might not want a devaluation. If domestic ﬁrms are are liability dollarized, then a devaluation may be contractionary.

Second, though, the bank wishes to avoid a recession created by a monetary contraction and high interest rates. The bank places weights on both of these goals. As we shall see, they may conﬂict.

Remember Lesson 4 from section 9.1.1? We saw there that an expectation of a depreciation will raise the current interest rate (from UIP) and also raise the exchange rate if not counteracted by monetary policy.

Under this scenario the central bank could then maintain the peg only by contracting the money supply. But remember, from your previous macroeconomics courses and the

153 related empirical evidence, that a monetary contraction (especially an unexpected one) can cause a fall in output and a rise in unemployment. If this occurs, the central bank may abandon the ﬁxed exchange rate rather than accept these costs of contractionary policy.

But logically, then, an expected depreciation or devaluation may be self-fulfilling. If investors expect an end to the peg and begin buying the foreign currency in anticipation of that, the central bank may well abandon the peg rather than defend it. But if investors expect the peg to be defended then the interest-rate differential will not widen in the first place.

The classic example of a second-generation speculative attack occurred in Europe in the early 1990s. At that time, western European countries participated in a target zone called the exchange rate mechanism (ERM) of the European monetary union (EMU), a precursor to the euro. German interest rates rose, in part because of the ﬁscal shock associated with reuniﬁcation. Central banks in other countries had to then tighten their monetary policies to maintain their exchange rates. But speculators reckoned that these central banks could not stomach the costs of tightening monetary policy.

Speculators thus came to expect a depreciation. They sold the pound, lira, and Swedish krona and bought deutschmarks. And interest rates rose dramatically in the UK, Italy, and Sweden because of the anticipated devaluation and due to the tightening of monetary policy as the authorities defended their currencies. Eventually, all three countries left the ERM. Wider bands were established for some countries, while the pound ﬂoated, as it has done ever since.

One of the distinctive features of this exchange-rate crisis was the role played by large investors. A small number of very large funds speculated against the pound, for example. That made it easier for the market to coordinate its expectations of a devaluation or ﬂoat.

Another distinctive feature was the very large interest rate diﬀerentials that arose. Remember that if you expect a 2% devaluation within a month that the annualized rate of return on an uncovered position would be 26.8%. So a very large value for the short-term, domestic interest rate would result. That is exactly what happened in the UK and Sweden.

154 The second-generation perspective offers two more lessons. First, there may be no early warning system (unlike the first-generation model where budget deficits persist and reserves erode). The fixed exchange rate can go on indefinitely if there is no speculation, but it may fall apart quickly once some speculation begins.

Second, it is important to remember that the ﬁxed exchange rate ends because of a choice and not simply because of the large size of speculators. The barrier to its continuing is not an absence of tools available to the central bank but rather the bank’s unwillingness to stomach the costs of tight monetary policy (‘buying back the base’).

9.3 Exchange-Rate Crises in EMEs

One of the most striking facts about exchange-rate crises (sharp devaluations) is that they are associated with large output losses and economic disruptions, especially in emerging markets.

An exchange-rate crisis often is accompanied by a default or a banking crisis. For example, if there is a sovereign default then ﬁnancial capital may ﬂee a country’s bond market and so its currency may naturally depreciate. Or, a banking crisis may lead to the expectation that the central bank will act as the lender of last resort, bail out the banks, and hence abandon the peg. Thus the central bank’s known concern with the solvency of private banks may act like the concern for output or employment in the second-generation model of speculative attacks.

Unfortunately, an exchange-rate crisis may not just be a joint symptom of some deeper problem but also act as a cause of further troubles. For example, a central bank that tries to defend a ﬁxed exchange rate using high interest rates may cause a recession, just as any very tight monetary policy would do. Or, if the central bank abandons the peg and ﬂoats then banks with foreign-currency liabilities may begin to fail. This mechanism seems to have been an important part of the Asian crisis of the late 1990s.

Studying exchange-rate crises suggests several ways of preventing them. One is to borrow reserves or to self-insure by holding very large reserves. A second one is to ﬂoat,

155 in the hope of avoiding sudden, large devaluations. A third one is to have some capital controls. Each of these approaches of course comes with a cost of its own.

A final aspect of exchange-rate crises that you should be aware of is contagion. That term is used to describe the fact that a crisis often seems to spread from one country to others. This pattern has led researchers to investigate whether contagion occurs because investors suddenly become aware of dangers common to several markets or, instead, whether there is some feature of the first crisis that affects the fundamentals in those other countries. This is an important area of research because, unfortunately, exchange-rate crises will probably continue to occur in the future.

Further Reading

For reading on exchange-rate crises I warmly recommend Paul Blustein’s books The Chastening (about the Asian ﬁnancial crsis) and And the Money Kept Rolling In (and Out) (about the crisis in Argentina).

Exercises

9.1. Suppose that in the target zone between the Hong Kong and the US dollars the log price of the USD in HKD is given by:

1 2 s = x + E s , t 3 t 3 t t+1

∗ where xt ≡ mt − mt . Also suppose that xt can be described as a discrete random variable. It can take on the values 1 or 3, each with probability 0.5, and is independently distributed over time.

(a) Find the two possible values of st.

(b) Find the variance of xt and the variance of st. (c) If UIP holds, find the two possible values of the international interest rate differential, ∗ it − it . Also find the value of the log forward exchange rate ft.

9.2. This question studies the scope for monetary policy autonomy in Hong Kong that may be provided by allowing some movements in the exchange rate. Suppose that the log

156 nominal exchange rate, st, follows this model:

∗ st = 0.5(mt − mt ) + 0.5Etst+1, where m is the log Hong Kong money supply and m∗ is the log US money supply. Also suppose that the Hong Kong Monetary Authority sets the money supply so that:

∗ mt = mt + ηt, where ηt is random and can take on two values, 0 and k, each with probability 0.5. (a) Suppose that we observe the exchange rate takes on the value 0.5 or the value 1.5. What is the value of k? (b) If UIP holds then what are the possible values for the international interest-rate differential? ∗ (c) Find the variance of the fundamental (i.e. mt − mt ) and the variance of the exchange rate.

9.3. This question studies a system with bounds on the exchange-rate change each year. Suppose that PPP and UIP hold. Money demand is given by:

M = exp(−i), P and similarly in the foreign country:

M ∗ = exp(−i∗). P ∗

Relative monetary policy evolves this way:

Mt Mt−1 ∗ = ∗ exp(t), Mt Mt−1

and t is uniformly and continuously distributed between -1 and 1, with no persistence from year to year.

(a) Solve for the log nominal exchange rate, st, in terms of current fundamentals and the expected future log nominal exchange rate, Etst+1.

(b) Solve for the log nominal exchange rate, st, in terms of current fundamentals. (c) How will the domestic inﬂation rate behave relative to the foreign inﬂation rate?

9.4. Suppose that Denmark operates its target zone with the fundamental xt = 10 + t, where t is a discrete random variable that can take on the following values (with

157 probabilities in brackets): −1 (3/8), 0 (2/8), and 1 (3/8). This distribution is the same every time period. Also suppose the price of the euro in Danish kroner follows:

st = 0.5xt + 0.5Etst+1.

(a) Find the conditional mean and variance of xt+1. (b) Find the distribution for the exchange rate. (c) If UIP holds then what is the one-period-ahead forward exchange rate? What is the distribution of the krone-euro interest rate diﬀerential?

9.5. Imagine that a central bank operates a target zone for its currency. Its exchange rate satisﬁes: 1 α s = x + E s . t 1 + α t 1 + α t t+1 It manages the fundamentals so that they are uniformly distributed between µ − and µ + : xt ∼ U(µ − , µ + ).

(a) Solve for the exchange rate. How is the range of values for s related to the range of values for x? (b) If uncovered interest rate parity holds then what is the value of the log of the forward exchange rate, ft? (c) Could there ever be proﬁts to the ‘carry trade’ when the relevant exchange rate is managed this way?

9.6. This question studies the first generation model of speculative attacks. Recall that M = B + R and suppose that the central bank’s holdings of government bonds evolve over time like this: Bt = exp(µ) · Bt−1. Reserves have a floor at zero. After a float begins at time T , the log exchange rate will evolve like this: sT = bT + αµ

The exchange rate is currently ﬁxed at st = s = ln M. (a) Find the duration of the ﬁxed exchange rate and show it is a decreasing function of µ, the drift in bond purchases by the central bank. (b) What foreign exchange reserves will the central bank lose in the speculative attack that must occur?

9.7. This question studies the life expectancy of a ﬁxed exchange rate using the ﬁrst generation model of a speculative attack. Suppose that the exchange-rate fundamentals are given simply by m, the logarithm of the money supply, with m = ln M = ln(B + R).

158 The log of the central bank’s bond holdings, b ≡ ln B grows like this:

bt = µ + bt−1.

The exchange rate currently is ﬁxed at a value s = m. Once reserves are exhausted at time T , the currency will ﬂoat, with a value given by:

sT = µ + bT .

(a) Find an expression for the life-span of the ﬁxed exchange rate, T − t. (b) How does this life-span depend on the growth rate of central bank’s bond-holdings, µ, and on the initial level of foreign exchange reserves, Rt?

159 Chapter 10. Currency Unions and Exchange-Rate Regime Choice

Why do some countries have floating exchange rates while others have fixed ones or even share a common currency? And why do countries switch from one system to another? As economists who are interested in international finance, we would like to explain both the positive aspects of this question — who does what and why — and the normative aspects — what system countries should adopt. This note chapter some recent research that sheds light on the choice of exchange-rate regime.

10.1 Arguments for Exchange-Rate Stability

We’ll start with the normative arguments for exchange-rate stability.

10.1.1 Stability 1: Trade Promotion

One of the classic arguments for exchange-rate stability is that it promotes trade. What is the evidence on this eﬀect? Most of the empirical studies use a regression model called the gravity equation. In the gravity equation, trade volume or value (the left- hand variable) is positively related to the national national income of each country and negatively related to the distance between them (so these are the explanatory variables). With these regressors included, researchers then include other variables, like indicators of the exchange-rate regime, so as to measure their eﬀect.

Rose (2000) used a large panel of data to show that countries with a common currency trade more than three times as much as those without one, controlling for other influences on trade such as distance or language. Reducing exchange rate volatility also has a positive effect on trade, but this effect is much smaller. A currency union seems to have a much greater effect than simply reducing exchange-rate volatility towards zero.

Rose’s results had a large impact because they were much greater than any previous estimates of the effect of fixed exchange rates or common currencies on trade. The time- series work on exchange-rate volatility and trade had produced very little evidence of any relation. But Rose used cross-sectional variation in trade, with a gravity model to estimate effects on trade volume. Rose also controlled for land borders, language, trade agreements,

160 and colonial links. (The gravity model also has been used to study trade between Canadian provinces.) The cross-section includes 186 countries, dependencies, overseas departments, colonies and so on, and is studied in several diﬀerent years.

Rose’s work has been criticized from a variety of angles. One possible criticism is that there is simultaneity bias (or reverse causation): countries that trade a lot decide to form currency unions and countries that break up currency unions reduce their trade for other reasons.

Klein and Shambaugh (2006) studied a wide range of exchange-rate regimes and found smaller eﬀects than Rose did. A direct peg appeared to increase trade by 21% and a currency union by 38%.

A key question is whether these results forecast what is happening in the gradually expanding Euro area. Many of the countries in currency unions in Rose’s historical sample are very small countries. But there now is enough data to assess the effect of EMU on trade. Statistical work by Baldwin (2007) has found that the scale of the effect found by Rose does not apply to the Euro area. What makes his estimates possible is the existence of a control group—the UK, Sweden, and Denmark—that did not adopt the euro. He finds an effect on trade within the euro area of only about 9% as a result of the common currency.

A different way to look at the effect of the exchange-rate regime on goods markets involves studying prices instead of quantities. As we saw in chapter 5, there is some evidence that LOP deviations are smaller when the exchange rate is fixed than when it is floating.

10.1.2 Stability 2: Monetary Discipline

The other traditional argument for exchange-rate stability is that it makes sense as a way of controlling inflation, especially if domestic monetary institutions are weak. As you know, fixing the nominal exchange rate with open capital markets means importing the foreign inflation rate (except for Balassa-Samuelson effects). An emerging-market country with high inflation might be able to lower inflation with relatively low costs in lost output if

161 it pegs to the currency of a low-inflation country or even adopts that currency. You might recall from intermediate macroeconomics that a monetary authority can lower inflation with smaller loss of output if it has credibility, so that expected inflation falls along with actual inflation.

Of course, a peg can be badly run, as we’ve seen. So a fixed exchange rate is not necessarily more stable than a floating one: it may simply replace frequent small changes in the currency’s value with periodic large ones. In the past two decades, the credibility- based case for fixed rates has been weakened by events like speculative attacks on a number of fixed exchange rates. Perhaps there isn’t a credibility gain and lower inflation or lower costs of disinflation if the monetary authority can choose to abandon the peg. Conversely, a country could instead float and target inflation like Brazil and Mexico do.

Some economists have suggested a currency board as a way to have a stable exchange rate with more credibility than a traditional ﬁxed one. This system is half-way between a ﬁxed exchange rate and a monetary union. It restricts the assets of the central bank to exclude domestic credit. Foreign exchange reserves equal the monetary base. Thus the base could be bought back readily to limit speculation. In practice some currency boards hold less than 100% reserves, while other (like HKSAR’s) hold considerably more. The gold standard in a way was a currency board system, using gold rather than a foreign currency. The most prominent example in recent years was the board in Argentina from 1991 to 2002.

As the Argentine example shows, currency boards do not solve the problem of an inconsistency between goals of the central bank. They can be abandoned. They also are expensive to establish, since they require large purchases of foreign currency (or a large contraction of the domestic money stock).

Overall, then, is there any evidence that fixing the exchange rate, setting up a currency board, or joining a currency union (or dollarizing) leads to lower inflation? Yes. For emerging economies (but not for industrialized ones) there is some statistical evidence that a stable exchange rate is associated with lower inflation.

10.1.3 Stability 3: Avoiding Valuation Eﬀects

162 Recent history suggests a third reason why exchange-rate stability might be a good idea. Many countries have a currency mismatch in their foreign assets and liabilities. In particular, they have net liabilities denominated in US dollars, a situation known as liability dollarization.

In this situation, a depreciation of the domestic currency obviously leads to a large increase in the value of debt, measured in the domestic currency. Remember the valuation effects we discussed in chapter 2? This fall in wealth can exert a contractionary effect on the economy. First, households may reduce consumption spending due to their lower net wealth. Second, firms may go bankrupt or have limited collateral and so contract investment spending. These effects seem to have been especially large during the Asian financial crisis of 1997–1998. For example, exporters were not able to take advantage of their depreciated currencies because they could not obtain financing. Please take a moment and look up real GDP for Indonesia and Korea in FRED. You will see severe depressions in the late 1990s, partly due to this effect.

But, rather than fixing the exchange rate would it not be simpler for firms and governments simply to borrow in their own currency? Doing so would reduce this valuation effect. But traditionally many countries could not borrow in their own currencies because lenders believed they might inflate away the value of the debt. This pattern seems to be declining, with more countries able to borrow in domestic currency rather than dollars, but it has not gone away altogether. Meanwhile, then, stabilizing the exchange rate may be a way to minimize these valuation effects and hence their magnification of the effects of shocks.

10.2 Arguments for Floating

Next, let us look at the arguments in favour of floating or, equivalently, at some of the costs associated with a fixed exchange rate or a currency union. The traditional arguments for floating exchange rates have been (a) that they allow autonomy in monetary policy and (b) that they serve as a shock-absorber or expenditure-switching mechanism.

10.2.1 Floating 1: Policy Autonomy and Country-Speciﬁc Shocks

163 A central reason for having a ﬂoating exchange rate is that it allows autonomy in domestic monetary policy. The choice of regime is part of monetary policy, and so one reason it matters is that prices are sticky. Just imagine that there is a country-speciﬁc shock and that monetary policy can hasten adjustment to the shock and minimize a recession. For example, if there is a fall in export demand the central bank may reduce its policy interest rate rather than wait for price adjustment. In that case the domestic currency would depreciate. The central bank may not necessarily be aiming for a depreciation, but it has to allow that as a consequence of its expansionary monetary policy.

For this ﬂexibility to be a good idea there must be country-speciﬁc shocks or the foreign central bank must be unskilled at identifying and reacting to shocks and the domestic central bank must be skilled in these areas.

Of course if the country in question is the centre or leader of a fixed exchange rate system then it can respond to the shock on its and the exchange rate will not move. The centre country does have policy autonomy. One way to make a fixed exchange rate system more palatable for countries in the periphery then is to have some sort of policy cooperation. For example, both the centre country and the home country could reduce interest rates in response to a recession in the home country. Alternately, the system could allow the home country to devalue its currency and set a new fixed exchange rate. But cooperation in fixed exchange rate systems has been difficult to sustain.

Economists have used some econometrics to try to judge how important country- specific shocks are. That involves measuring shocks somehow and then looking at their correlation across countries. They’ve also looked at this question indirectly, by studying output volatility and specifically checking whether it is lower for countries with floating exchange rates. The idea is that low output volatility would show that central banks with floating exchange rates have put their policy autonomy to good use. The evidence on this is somewhat mixed, though.

10.2.2 Floating 2: Expenditure Switching

Even if there is not an activist central bank, changes in currency values can be stabilizing on their own. For example, suppose again that there is a fall in the demand for exports.

164 That means that demand for the home currency also will fall and so it will depreciate. But that means that the foreign-currency price of our exports will fall, if they are priced in the home currency, which will cushion the blow to our export industries. In this example the floating exchange rate acts as a shock absorber through its expenditure-switching effect. (See if you can predict how a floating exchange rate would react to other shocks such as a burst of foreign inflation.)

Here is a second example, that again shows how a ﬂexible exchange rate can substitute for ﬂexible prices. Suppose that prices in the US and Canada are set, and then a shock occurs in the form of a drought in Canada. The only way the real exchange rate can move— signalling a shortage of Canadian goods—is via the nominal exchange rate. Although the Canadian dollar price of our exports does not rise, the nominal exchange rate falls (the Canadian dollar appreciates) so that the U.S. price rises and U.S. consumption of Canadian goods falls.

But notice that this argument assumes 100% pass-through. As we saw in chapter 5, there is evidence of local-currency pricing and very limited pass-through for developed countries. Thus, the relative price may not change even if the nominal exchange-rate changes. This evidence may not be an argument for fixing, but it does seem to weaken one of the traditional arguments for floating. In contrast, ERPT is much greater in less- developed countries. They might then be able to use the nominal exchange-rate as a shock absorber, and so it might make more sense for them to float.

So arguing for the expenditure-switching advantage of a floating exchange rate requires that there be significant exchange-rate pass-through. But remember from section 10.1.3 that it also requires that the stabilizing effect of expenditure switching isn’t dwarfed by a destabilizing valuation effect. Thus a key question is which effect of a depreciation is larger: the positive effect on exports and output via the real exchange rate or the negative effect on output via the valuation effect. The answer may vary over countries and over time too. (Of course, if a country had assets in dollars and liabilities in local currency then the valuation effect of a depreciation would be stabilizing, and would enhance the case for a flexible exchange rate.)

165 10.2.3 Floating 3: Seigniorage

Remember an advantage of a fixed exchange rate or currency union is that it imports foreign (or union-level) monetary policy and so can help reduce inflation. But of course the converse is that a country with a fixed exchange rate that pegs to the currency of a low-inflation country cannot use the inflation tax as a variable or large source of revenue. And a country that adopts a foreign currency (rather than joining a multilateral currency union) collects no seigniorage revenue from issuing currency. For most countries with well-run public policies and well-developed taxation systems these are not large sources of revenue, though, so losing the inflation tax or seigniorage revenue does not play a central role in debates about exchange-rate policy.

10.3 Varieties of Exchange-Rate Stability

So far this chapter has listed and discussed the main arguments for exchange-rate stability and then the main arguments for ﬂoating. We next need to pause and add some interesting and important details about possible diﬀerences among ways of stabilizing the exchange rate.

10.3.1 Are Currency Unions Diﬀerent from Fixed Exchange-Rate Systems?

In discussing the beneﬁts of exchange-rate stability we so far haven’t distinguished carefully between unilaterally pegging to a foreign currency and joining a multilateral currency union. Here are some diﬀerences. A currency union may:

(a) promote trade more than a ﬁxed exchange rate does (as discussed in chapter 10.1.1);

(b) be harder to leave than a ﬁxed exchange rate (which may give added credibility beneﬁts as in 10.1.2 but may also explain why Denmark has not adopted the euro even though it pegs to it);

(d) share seigniorage revenue;

(e) involve labour mobility or ﬁscal transfers.

The main message to remember here is that these are signiﬁcant diﬀerences between the two frameworks.

166 What regions should form a currency union then? A currency union may have the benefit of increasing trade but, as you know, one of the main costs is the loss of monetary policy autonomy. For example, during the European debt crisis of 2009–2013 Iceland was able to speed its recovery by engineering a real depreciation via a nominal one, as you’ll see if you plot the Icelandic króna/euroexchange rate. In contrast, Spain (which uses the euro) could not have a nominal depreciation and so experienced a real depreciation via price deflation, a more prolonged process sometimes called ‘internal devaluation.’

But there are two ways that a currency union potentially can reduce the costs associated with the loss of monetary policy autonomy. The ﬁrst one is labour mobility. If Spain receives a negative shock but Spanish workers can migrate to the rest of the Euro area, that will reduce the costs of not being able to respond to the shock with monetary policy with an independent, Spanish monetary policy.

A second mechanism that might allow adjustments involves ﬁscal transfers. If a region or country within the union is hit by a negative shock but the ﬁscal system transfers resources there (for example through unemployment insurance) then that too may reduce the costs and substitute for an independent monetary policy and separate currency.

The possibility (e) that a currency union may involve labour mobility or fiscal transfers, is central to the idea of an optimum currency area (OCA). This phrase describes a collection of regions or countries for which the benefits of a common currency exceed the costs. Overall, a collection of regions is more likely to form an OCA if it trades intensely, has symmetric shocks, allows labour mobility, and has a system of fiscal transfers between regions.

Economists have compared these criteria within some older currency unions, like the United States, and some new ones, like the euro area. Krugman (2012) provides a great discussion of this topic. For example, he notes that there was labour mobility out of Massachusetts after it experienced a local recession in its tech sector. And there were large ﬁscal transfers to Florida (via unemployment insurance) after its construction sector contracted sharply during the ﬁnancial crisis. The evidence suggests that the euro members are much further than US states from satisfying the OCA criteria. For example, in the

167 case of Spain there is no Euro-area-wide ﬁscal transfer system.

Advocates of currency unions sometimes mention the possibility that the criteria can be endogenous in the sense that forming a currency union brings about changes that make the criteria satisfied. Their argument is that a common currency increases trade and so raises the benefits of avoiding currency transaction costs. Section 10.1.1 outlined some of the empirical evidence which suggests the effect of the euro on trade has been relatively small. Moreover, it’s also possible that a currency union leads to greater regional specialization in production, which would make shocks less symmetric, not more.

10.3.2 Are Fixed Exchange Rates Still Viable?

Some commentators have argued that fixed exchange rates aren’t really possible any- more, because they are vulnerable to speculative attacks. This view is sometimes called the ‘missing middle’. Obstfeld and Rogoff (1995) addressed this question in an influential paper called ‘The mirage of fixed exchange rates.’

According to Obstfeld and Rogoff fixing is feasible even with enormous hedge funds. All you need is enough reserves to buy back the base, and a central bank may borrow to do that. They tabulate reserves and base for a range of countries and show how this could occur. ‘Buying back the base’ is just another way of saying that a monetary authority can contract the money supply to maintain the fixed exchange rate.

So the real problem with fixed exchange rates is competing objectives, as we saw in chapter 9. A central bank that tries to defend a fixed exchange rate (at least when there are no capital controls) may have to endure the effects of a spike in interest rates. The central banks of Sweden and the UK balked at these costs in the early 1990s, and their central banks made huge capital losses. Target zones and crawling pegs have also at times been abandoned because of these competing objectives.

The conclusion: A ﬁxed exchange rate is still possible even with open capital markets and large speculative funds.

168 10.4 Actual Regime Choices

So far in this chapter we’ve looked at the normative question of what countries should do (and a bit about what they can do). Now it’s time to discuss what the positive question of what they actually do.

Start with Obstfeld and Rogoff’s (1995) article once again. They argued that countries could fix the exchange rate but that few did so. As of the date they wrote, they reported that few countries had fixed for more than five years. The rest were small principalities or tourism economies.

It seems that a fixed exchange rate had huge costs in Thailand and Indonesia during the Asian financial crisis, as did the currency board in Argentina, so that many countries float now. For example, Korea, Chile, Brazil, Russia, Argentina, and Poland float. Ecuador is dollarized. Europe has a monetary union. So the ‘missing middle’ is not a technical requirement but it does seem to be a valid observation about the choices countries make. Many countries seem to either float or adopt a currency union or dollarize.

But, it turns out that there are differences between what countries report that they do and what they actaully do. Specifically, Calvo and Reinhart (2002) showed that many countries say that they float, but do not. They call this situation the ‘fear of floating’.

They studied 39 countries from 1970 to 1999, and constructed frequency distributions of monthly exchange-rate changes. They found that many countries that were officially floating had little variation in their nominal exchange rate. For example, the distributions were much more concentrated than the corresponding ones for U.S., Japan, or Australia, say. They found that, even for Chile and Singapore, there was an 88% probability that monthly changes were within ±2.5%. And this stability did not depend on whether the country has officially declared a float or managed float with the IMF. They also found that this stability did not arise because these countries faced fewer shocks.

Why are so many countries afraid to ﬂoat? We’ve seen several reasons in this course and summarized the main ones in section 10.1. First, they may think that exchange-rate stability promotes trade. Second, they may think that it will help control inﬂation. EMEs

169 tend to have higher pass-through than developed countries, which means that depreciations lead to inflation. Third, if liabilities are dollarized then floating may be very costly at times. Fourth, some of these countries have capital controls, so they still may have some monetary policy autonomy even with an unofficially fixed exchange rate.

Now, it turns out that trying to figure out what countries actually do gets even more complicated. That is because of the phenomenon of dual exchange rates, in which a country has capital controls and then both an official rate and an unofficial, market one. Please pause and ask yourself why a country would have such a system. The answer again seems to be that it reflects conflicting goals. They want a highly valued currency to make vital imports affordable. They want ongoing monetary expansion because they need seigniorage as a source of government revenue.

In these systems the official rate can be stable but the unofficial or parallel rate can be rising (an ongoing depreciation). Reinhart and Rogoff (2004) studied 153 countries after 1946, using monthly data and specifically tracked these parallel, market exchange rates. They found that

(a) Dual rates are very common, even today.

(b) De facto floats were common even under the Bretton Woods arrangements. Only for the G-3 was there a dramatic change in exchange-rate behaviour in the early 1970s. Conversely, countries that were officially floating during the 1980s often followed crawling pegs.

(d) Freely falling exchange rates—a new category with annual inﬂation above 40%—were more common than freely ﬂoating ones.

It can be challenging to keep track of these classifications so let me summarize three key conclusions. First, be a bit sceptical about official classifications as reported to the IMF. They may not reflect what countries actually do or the importance of a parallel exchange rate.

Second, both ‘fear of ﬂoating’ studies of exchange-rate changes and studies of dual

170 rates suggest that pegged or ﬁxed exchange rates remain widely adopted.

Third, the fact that official and actual arrangements may be different means one should be very careful in assembling empirical evidence on the effects of exchange-rate regimes. For example, we began this chapter by noting that studies of the gravity model of trade may include a dummy variable that keeps track of currency unions or fixed exchange rates. But some officially floating countries might actually have monetary policies and exchange-rate variation that are very similar to those of countries that are officially fixed.

This third conclusion brings us to our last topic. Even though classifying exchange- rate regimes is challenging, there are many interesting studies that try to assess whether the regime matters for some macroeconomic indicator like the volume of trade, the inﬂation rate, or the volatility of business cycles. These studies usually use cross-section or panel regressions to see whether these variables are correlated with some an indicator of the exchange-rate regime.

It is difficult to find evidence of differences in macroeconomic performance across (official) regimes. And one reason for that may be that the official regimes are misleading, as we’ve just seen. But it turns out that it is difficult to find large effects even when the regimes are very clear.

Rose (2011) gives us three fascinating examples. First, Singapore and Hong Kong are very similar economies yet the the Singapore dollar floats while the Hong Kong dollar lies in a very stable target zone against the USD. Second, Sweden, Denmark, and Finland are similar economies with similar growth and inflation, yet Sweden floats, Denmark operates a target zone, and Finland uses the euro. Third, Panama is dollarized while in Costa Rica the colónexchange rate follows a crawling peg against the USD. In each of these cases the macroeconomic outcomes seem similar across the regimes. In each case the monetary authority and private citizens probably believe that their system is a good one. But it is possible that exchange-rate regimes don’t really matter.

Further Reading

171 Here are the articles in this chapter, in the order cited: Andrew K. Rose (2000) One money, one market: The effect of common currencies on trade. Economic Policy 15, 9–45. Michael Klein and Jay Shambaugh (2006) Fixed exchange rates and trade, Journal of International Economics 70, 359–383. Richard Baldwin (2007) Trade effects of the Euro: a comparison of estimators, Journal of Economic Integration 22, 780–818. Paul Krugman (2012) Revenge of the Optimum Currency Area. NBER Macroeconomics Annual 27 439–448. Maurice Obstfeld and and Kenneth Rogoff (1995) The mirage of fixed exchange rates. Journal of Economic Perspectives 9:4, 73–96. Guillermo Calvo and Carmen Reinhart (2002) Fear of floating. Quarterly Journal of Eco- nomics 117, 379–408. Carmen Reinhart and Kenneth Rogoff (2004) The modern history of exchange rate arrangements: A reinterpretation. Quarterly Journal of Economics 119, 1–48. Andrew K. Rose (2011) Exchange rate regimes in the modern era: Fixed, floating, and flaky. Journal of Economic Literature 49, 652–672.

172 Answers to Exercises

2.1. (a) The budget constraint gives:

ρ100 10.5 + 100 + = C (1.95238) 1.05 1 so C1 = 56.5975 + 48.78ρ.

(b) The ﬁrst-period current account is:

CA1 = rA0 + p1Y1 − C1 = 0.5 + 100 − C1 = 43.90 − 48.78ρ.

(c) Clearly the higher the value of ρ the lower the value of the current account. For example, if ρ = 1 then CA1 = −4.88. If, instead, ρ = 0.5 then CA1 = 19.51. Thus the claim makes sense according to the theory.

2.2. (a) We use:

Y 1 G A (1 + r) + Y + 2 = C[1 + ] + G + 2 . 0 1 1 + r 1 + r 1 1 + r

Using the baseline values gives 214.06 = C(1.99) + 59.7 so that C = 77.57. Thus

CA1 = rA0 + Y1 − C − G1,

gives CA1 = 0.1 + 100 − 77.57 − 30 = −7.47. (b) The same method gives 212.24 = 1.97C + 59.13 so C = 77.72. Thus

CA1 = 0.3 + 100 − 77.72 − 30 = −7.42.

C = 92.49 − 0.497G2.

Then we simply use this in the deﬁnition of the current account:

CA1 = 0.1 + 100 − 30 − 92.49 + 0.497G2 = −22.39 + 0.497G2.

Finally, we know A0 = 10 so

CA2 = −A0 − CA1 = −10 − CA1.

173 2.3. (a) The household budget is:

100(1 − τ ) 1 1.05(−10) + 100(1 − τ ) + 2 = C 1 + , 1 1.05 1 1.05

and the current account is given by:

CA1 = rA0 + Y1 − G1 − C1 = −0.5 + 100 − 20 − C1.

(Many people gave incorrect mixtures of the national, private, and government constraints. Be sure you know what each of the three looks like.)

So substituting the ﬁrst equation in the second one to replace C1 gives the current account as a function of the tax rates. (b) Notice that the tax rates enter the current account as:

τ 100 τ 100 + 2 , 1 1.05 but, with no initial government debt, that equals the present value of government spending: 20 20 + . 1.05 So a re-arrangement of the two tax rates has no eﬀect on the overall present value, or on consumption, or on the current account. Ricardian equivalence holds here because there is perfect foresight and the tax changes do not directly aﬀect output.

(b) With that background, we might as well assume each tax rate is 20%, so C1 = 74.636 and CA1 = 5.864. In contrast if G1 rises to 22 and G2 falls there is no change in the present value of tax revenue so again no change in consumption, but now there is a direct effect on the current account, which becomes CA1 = 3.864. Obviously then CA2 rises from 4.136 to 6.136. The change in the timing of government spending affects the rate at which external debt is paid off.

2.4. (a) We know that:

110 G (1.02)(−10) + 100 + = 197.643 = C [1.98] + G + 2 1.02 1 1 1.02

So now if G1 = G2 = 20 then C1 = C2 = 79.82. Thus

CA1 = 0.02(−10) + 100 − 20 − 79.82 = −0.02,

and so CA2 = 10.02.

(b) Now with x = 2 we have C1 = C2 = 80.83. Thus

CA1 = 0.02(−10) + 100 − 18 − 80.83 = 0.97,

174 and so CA2 = 9.03. (Notice that a temporary decline in government spending leads to an increase in the current account balance.) (c) Now the constraint is: 20 − 2ρ 197.643 = C (1.98) + 18 + . 1 1.02 Rearranging gives: 1 20 − 2ρ C = [197.643 − 18 − ]. 1 1.98 1.02 And then CA1 = 81.8 − C1, and CA2 = 10 − CA1. (If you are patient you can consolidate and simplify these formulas.)

2.5. (a) Clearly C1 = C2 = 100. The current account is zero in each period.

(b) Now C1 = C2 = 110.198. Thus CA1 = 120−110.198 = 9.802 and CA2 = 0.04(9.802)+ 100 − 110.198 = −9.802.

(c) Now C1 = C2 = 109.81. Thus CA1 = 100−109.81 = −9.81 and so CA2 = 0.04(−9.81)+ 120 − 109.81 = 9.81. (You may round your numbers slightly diﬀerently in each case.)

2.6. The Euler equation holds for each country and so holds for the world. Thus 202 0.99(1 + r) = , 200 so r = 0.0202 (with some rounding allowed). It is clear there is no reason for lending here, ∗ ∗ so C1 = C1 = 100, C2 = C2 = 101, and the current account is zero in each time period and country. (b) The same method now gives r = 0.035. Next, for the US we combine the Euler equation and the budget constraint: 104 C + 0.99C = 100 + , 1 1 1.035 ∗ so C1 = 100.745. Thus in Europe C1 = 99.255. The US borrows: CA1 = −0.745.

Thus in the second period CA2 = 0.745 so that A0 + CA1 + CA2 = 0: the US pays oﬀ its loan by running a surplus.

2.7. (a) Clearly there is no variation in consumption over time, because this is an endowment economy. So with β = 0.98 then r = 2.04%. To ﬁnd consumption in Japan: 50 15 20(1.0204) + 50 + = C + C β + 15 + , 1.0204 1 1 1.0204

175 which gives C1 = 45.307 and C2 = 45.307 using the Euler equation and rounding (not consumption smoothing). Thus

CA1 = 0.0204(20) + 50 − 15 − 45.307 = −9.899,

and CA2 = −10.101 so that terminal debt is zero. Notice that the two current account values have to oﬀset A0, so that property is available as a check. (b) Combine the two Euler equations:

∗ ∗ C2 + C2 = (C1 + C1 )0.98(1 + r),

so 150 − 45 = (150 − 50)0.98(1 + r), so that r = 7.14%. Now the budget in Japan is:

50 20(1.0714) + 50 + = C (1.98) + 20 + 15/1.0714, 1.0714 1 so that C1 = 42.47. Now the Euler equation gives C2 = 44.59. Thus

CA1 = 0.0714(20) + 50 − 20 − 42.47 = −11.042.

Thus CA2 = −8.958. The increase in government spending widens the current account deﬁcit. (c) A permanent increase would not change the world interest rate from part (a). It would lower C1 and C2 to 40.307. There would be no eﬀect on the current account relative to part (a).

2.8. (a) Summing the Euler equations gives:

154 = 0.99(1 + r)(150) so r = 3.70%. That means that in Japan:

51 C (1.99) = 1.0370(10) + 50 + , 1 1.037

∗ ∗ so C1 = 55.05 so C1 = 94.95. In period 2, C2 = 56.52 so C2 = 97.48. Thus CA1 = −4.68 and CA2 = −5.32. Japan, naturally, can run down its foreign assets.

(b) The same methods give r = 1.68%. Then C1 = 53.96 and C2 = 54.32. Thus CA1 = −3.79 and so CA2 = −6.21. Here Japan slows down the rate at which it runs down its foreign assets; its trade and current account deﬁcits are initially smaller. (c) No. In this example we see that slower growth reduces the US trade balance, so it does not act in the direction of reducing so-called global imbalances.

176 3.1. (a) In the good state the borrower will be tempted to default. If it defaults its consumption will be 12(1 − η) while if it pays its consumption will be 10. Setting these equal gives η = 0.166 which is the lowest sanction rate that will allow full risk-sharing. (b) Again we examine the borrower’s decision in the good state of the world, which determines what contract the lender will agree to. In a default, the borrower now gets 12(1 − 0.10) = 10.8. The borrower cannot be better oﬀ defaulting than paying, otherwise the borrower would not pay. So the maximum payment the borrower can be called on to make in the good state (or receive in the bad state) is 1.2. (c) As background ln 10 = 2.30258, ln 8 = 2.07944, and ln 12 = 2.48491. After a default, therefore, the borrower has expected utility that is the average that is the average of the latter two numbers, namely 2.28217. So the borrower compares expected utility from defaulting to that from retaining 12 this year but bearing risk in the future. Setting these streams equal gives the value of β that we are looking for:

2.30258(1 + β + β2 + ...) = 2.48491 + 2.28217(β + β2 + ...)

The result is β = 0.8993. Any discount factor greater than or equal to this means the borrower is concerned enough about the future that the threat of loss of reputation and risk in the future will enforce payment today.

3.2. (a) Under autarky expected utility is 2.282 and under full insurance it is 2.303. (b) The key inequality is: 12(1 − η) < 10 so that η > 0.167.

3.3. (a) Clearly the sanction rate must be high enough that defaulting in the good state is no better than paying, so: 120 − 20 = 120(1 − η) so η = 0.167. (b) Let us call y the payment (since it goes on the vertical axis). The same principle gives

120 − y = 120(1 − η) so y = 120η. This starts at 0, rises along this line to a height of 0.167 and isflat thereafter. (c) At η = 0, EU = 0.5(4.38) + 0.5(4.79) = 4.585. At η = 0.167, EU = ln(100) = 4.605. Now let us look at an intermediate point. At η = 0.0833 we find y = 10 so the values of consumption are 90 and 110 and EU = 4.595. These two differences are the same, so EU rises along a straight line.

3.4. (a) We set 1.4(1 − η) = 1.2

177 which gives η = 0.143 or 14.3%. (b) Now 1.4(1 − 0.05) = 1.4 − P gives P = 0.07 (rather than 0.2). The borrower pays 0.07 in the good state and receives 0.07 in the bad state. (c) Note that 10 ln(1.4) = 3.36, 10 ln(1) = 0, and 10 ln(1.2) = 1.823 so the average, in autarky, of the ﬁrst two values is 1.68. We then get:

β 1 3.36 + 1.68 = 1.823 1 − β 1 − β

This gives β = 0.916. Only a discount factor above this will prevent default and so the absence of lending.

4.1. (a) The objective is to choose ω to maximize:

2 2 2 2 µp − σp = ω2 + (1 − ω)2 − ω (1) − (1 − ω) (1 + σ ).

The ﬁrst order condition is:

0 = −2ω + (2 − 2ω)(1 + σ2), so 1 + σ2 ω = . 2 + σ2

(b) If σ = 0 then ω = 0.5 and the portfolio is√ divided equally between the two countries so as to minimize the variance. If instead σ = 2 then ω = 0.75, so the share held in the domestic asset is much higher. (c) Such a correlation would raise the variance of the overall return on the foreign investment, r∗ + ∆s, so it would raise the optimal share held in the home investment, ω.

4.2. (a) The optimal portfolio is:

µ − µ∗ + 2σ2∗ ω = . 2(σ2 + σ2∗)

(b) Thus: 2 − µ∗ + 2σ2∗ 0.75 = . 2(4 + σ2∗) which simpliﬁes to 4 + µ = 0.5σ2∗.

178 These are the combinations that might explain the observed portfolio. (Incidentally, you can see that they are positively related for obvious reasons, and that the same mean and variance as in the domestic returns will not ﬁt with the portfolio model.)

4.3. (a) We ﬁnd that: 2 µ1 − µ2 + λ2σ2 ω = 2 2 , 2λ(σ1 + σ2) so that 0.9 − 1 + 2 ω = = 0.475. 4

(b) We ﬁnd that: 2 − 1 + 2 0.75 = , 4 so the perceived average return at home must be 2. (b) I don’t think raising λ will account for home bias here. It will simply raise the weight on the variance term. That is minimized at ω = 0.5 so emphasizing that term will not move us towards ω = 0.75. Alternately, if you try to reverse-engineer from ω = 0.75 as in part (b) you ﬁnd a negative value fo the implied λ.

4.4. (a) We ﬁnd that: µp = ω(1) + (1 − ω)2 = 2 − ω and that 2 2 2 σp = ω 1 + (1 − ω) 2. Thus 2 2 µp − λσp = 2 − ω − λ(2 − 4ω + 3ω ) and the derivative wrt ω is: −1 − λ(−4 + 6ω) = 0 so the formula is 4λ − 1 2 1 ω = = − . 6λ 3 6λ

(b) I get λ = 2.488. (c) Two of the problems are that (a) people in the other country tend to be home biased too, yet they presumably are risk averse also, and (b) some of the risk from overseas investments (from the exchange rate) can be hedged). In our example, (c) even an inﬁnite value of λ only gives ω = 0.667: our function might be wrong but this suggests the fact that high risk is associated with high expected returns in the risky market may limit this explanation too.

179 4.5. (a) The general problem is to maximize:

1.5 − 0.5ω − λ[ω21.44 + (1 − 2ω + ω2)2.25 + (2ω − 2ω2)0.5 · 1.2 · 1.5].

which gives: 2.7λ − 0.5 ω∗ = . 3.78λ When λ = 0.5 we ﬁnd ω∗ = 0.45. (b) When λ = 4 then ω∗ = 0.68. so it may appear to the investigator that ω∗ is surprisingly high: a case of home bias. (c) No. One can estimate the degree of risk aversion. If the added risk is due to exchange rates that can be avoided by hedging the portfolio at low cost. And there is home bias even for pairs of countries with a ﬁxed exchange rate or common currency. If the added risk is intrinsic, then the setup does not explain why foreign residents also are home-biased.

4.6. (a) The Pareto problem is to maximize:

C1−α C∗1−α ω + (1 − ω) 1 − α 1 − α

subject to C + C∗ = Y + Y ∗. The ﬁrst-order conditions give

1 ω α C = C∗ = kC∗, 1 − ω so Y + Y ∗ C = 1 + k and k C∗ = (Y + Y ∗). 1 + k

(b) No, regardless of the value of ω (and hence k) the correlation is 1. (c) The variance of consumption is k2(2). The variance of income is ω2(2) + 1.

4.7. (a) The Pareto problem is to maximize:

C1−α C∗1−α ω + (1 − ω) 1 − α 1 − α

subject to C + C∗ = Y + Y ∗. The ﬁrst-order conditions give

1 ω α C = C∗ = kC∗, 1 − ω

180 so Y + Y ∗ C = 1 + k and k C∗ = (Y + Y ∗). 1 + k

(b) No, regardless of the value of ω (and hence k) the correlation is 1.

5.1. (a) ∗ 0.5 AN AT Q = ∗ AN AT

(b) ˙ ˙ India ˙ USA Q = 0.5(AT − AT ) = 0.5(2) = 1 so, this will be a real depreciation for the US or appreciation for India, at a rate of 1% per year. (c) If the rupee depreciated by 1.25% per year yet we predict a real appreciation of 1% per year then inﬂation in India is predicted to have been 2.25% higher per year than in the US.

5.2. (a) ∗ ∗ AT 0.5 AT 0.5 Q = = ∗ = 1.2247. AT 0.667AT Thus prices are lower in Kenya. (b) In growth rates: ˙ ˙ ∗ ˙ Q = 0.5(AT − AT ) = 0.5(−2) = −1. Thus Q will be falling: a real appreciation for Kenya at 1% per year. (c) Now Q˙ = S˙ + π∗ − π, so:

S˙ = π − π∗ = Q˙ = 7.5 − 6 − 1 = 0.5, so a 0.5% per year depreciation of the KES against the ZAR.

5.3. (a) The assumptions in the preamble tell us that:

∗ AT 0.5 0.5 Qt = = 2 = 1.414 AT

so prices are lower in the east. Also

˙ ˙ ∗ ˙ Q = 0.5(AT − AT ) = 0.5(−3) = −1.5

181 and the east experiences a real appreciation of 1.5% per year. (b) The inﬂation rate in Slovenia will be 3.5% per year. (b) The zloty will appreciate by 1.5% per year.

5.4. (a) Obviously: Q˙ = S˙ + π∗ − π, so with the US as the home country:

Q˙ = 30 + 20 − 10 = 40,

which is a 40% real depreciation of the USD or appreciation of the CNY. (b) Under these assumptions, and using the Balassa-Samuelson model:

˙ ˙ ∗ ˙ ˙ ˙ ∗ Q = 0.5(AT − AT ) + 0.5(AN − AN ) so that ˙ ∗ ˙ ∗ 40 = 0.5(AT − 20) + 0.5(20 − 10) = 0.5AT ,

so A˙ T = 90%. (c) For Q to keep rising with no change in S means the inflation differential must widen: much higher inflation in China would be predicted.

6.1. (a) With α = 0.5 the fundamental equation is: 2 1 s = x + E s t 3 t 3 t t+1

(b) Using either solution method with this drifting random walk gives:

st = αµ + xt = 0.25 + xt.

∗ (c) Notice that Etst+1 = 0.5 + st = 0.75 + xt = ft and it − it = µ.

6.2. (a) The solution is: 2µ 1 s = + x t 3 − 2ρ 3 − 2ρ t

(b) Obviously: 1 2 σ2 = σ2. s 3 − 2ρ x

(c) Using the chain rule of forecasting: 2µ 1 2µ 1 E s = + E x = + [µ + µρ + ρ2x ]. t t+2 3 − 2ρ 3 − 2ρ t t+2 3 − 2ρ 3 − 2ρ t

182 6.3. (a) The guess-and-verify method gives:

st = αλ + xt

(b) The exchange rate also will follow a drifting random walk (not a pure one; changes have a predictable part: the drift). So the average depreciation rate is predictable, but not the variation around that. The forecast of the depreciation is Etst+1 − st = λ. (c) It is then easy to see that the interest diﬀerential will be λ.

6.4. (a) 1 1 s = (m − m∗) + E s t 2 t t 2 t t+1

(b) st = 0.417 + 0.833xt

(c) The variance of s is 0.693 times the variance of x, so perhaps this ﬁts with the volatility of exchange rates in the data.

6.5. (a) st = 0.019 + 0.96mt

(b) The variance of s is 0.962 times the variance of x, so it is less variable. (c) At time t, m rises by 1 so s rises by 0.96, a depreciation. The persistence parameter is 0.96, so I think each subsequent value is 0.96 times the previous value. This is called the impulse response function, by the way.

6.6. (a) 1 1 s = (m − m∗) + E s t 2 t t 2 t t+1

(b) ∗ st = µ + mt − mt

(c) Yes, the exchange rate is just a random walk plus a constant so it will follows a (drifting) random walk too. The interest diﬀerential will be µ.

7.1. (a) The bond price is: 0.5 0.5 h = 0.94C [ + ]. t t 1 1.02

When Ct = 1 then ht = 0.93 so r = 7.5%. When Ct = 1.02 then ht = 0.9493 so r = 5.3% (b) We know that hjt = ht[1 − 0.2λ]

183 There are now two possible values for the interest rate for each starting value of Ct, but the expected returns are the same as in part (a).

1 1 0.8 (1 − 0.2λ) 1 E[ − 1] = (1 − λ)[ − 1] + λ[ − 1] = = − 1. hjt hjt hjt hjt ht

h λ = 5[1 − jt ]. ht

7.2. (a) 1 1 h = 0.98[0.5 + 0.5 ] = 0.9657. 1 1 1.03 so the interest rate is 3.55% (b) We ﬁnd that h1S = 0.75 = h1[(1 − λ)1 + λ0.4], so λ = 0.3722 or 37.22%. (c) That correlation would give a risk premium and would lower the price of the Spanish government bonds. (Be sure to show the covariance term explicitly.) Thus if we observe a price of 0.75 even with this eﬀect now present, λ, the default probability, must be lower than the value in part (b).

7.3. (a) The price is 0.9706 so the return is 3.03%. a (b) The price is h1 = 0.9706[1 − 0.3λ] which obviously falls with a rise in λ. The expected gross return is: 1 · (1 − λ) 0.7 · λ 1 E(1 + r) = a + a = u , h1 h1 h1 so that does not depend on λ.

7.4. (a) The real bond price is 0.93078 so the real interest rate is 7.44%. (b) The nominal bond price is:

P1 1 0.93078 H1 = E10.94 = = 0.9037 P3 C2 1.03 so the nominal interest rate is 10.66%. The real interest rate is unchanged, and 1 + i 1 + r = . 1 + π

184 (c) If there is no correlation between default and C2 then the expected, real return will be the same on this bond as on the bond in part (b). (We also see inﬂation is not random, so that does not yield a covariance term either). Only if there is a covariance (a risk premium) will the expected returns diﬀer.

7.5. (a) The price is ht = 0.9657 so the return is r = 3.55%.

(b) The price is hpt = 0.9657[0.9(1)+0.1(0.3)] = 0.898 so the interest rate is rp = 11.358%. (c) A positive covariance between the payoﬀ and consumption would lower the price of the bond, as seen in the covariance decomposition. That would then raise the return, when a default does not occur. The average or expected return now would also be higher than in cases (a) and (b).

9.1. (a) We know that the expected value of s is the expected value of x, which is 2. Thus:

1 4 s = x + . t 3 t 3

When x = 1 we find s = 5/3 = 1.667. When x = 3 we find s = 7/3 = 2.333 (b) The mean of x is 2 and the variance is 1. To find the variance of s we can use the two values, or work from the formula in part (a).

1 2 σ2 = σ2 = 0.111. s 3 x

(c) The log forward rate is just ft = Etst+1 = 2. So the two possible values for the interest diﬀerential come from: ∗ it − it = Etst+1 − st = 2 − st, which can be 0.333 or -0.333.

9.2. (a) The monetary model gives:

s = 0.5(m − m∗) + 0.5(k/2).

So if m − m∗ = k then s = 0.75k while if m − m∗ = 0 then s = 0.25k. Thus since s = 1.5 or 0.5 we must have k = 2. (b) The expected future log exchange rate is 1 so the interest rate diﬀerential is 0.5 or -0.5 from UIP. (c) The variance of m − m∗ is 1. The variance of s is 0.25.

9.3. (a) This is a simple version of the monetary model, so:

∗ st = 0.5(mt − mt ) + 0.5Etst+1.

185 (b) It is easy to see that: ∗ st = mt − mt , is the solution. (c) Thus ∗ ∗ pt − pt = mt − mt , and so ∗ πt − πt = t. The inﬂation diﬀerential will be uniform between -1 and +1, and with no persistence.

9.4. (a) The conditional mean is 10. The conditional variance is simply the variance of the shock which is 0.75. (b) The exchange rate is st = 0.5xt + 5 so its three possible values are 9.5 (3/8), 10 (2/8), and 10.5 (3/8). (c) The expected future exchange rate is 10, which is the forward rate. The interest-rate diﬀerential has possible values 0.5 (3/8), 0 (2/8), and -0.5 (3/8).

9.5.(a) 1 α s = x + µ. t 1 + α t 1 + α The range for x is 2 so the range for s is: 2 . 1 + α (b) UIP implies that the forward rate is the expected future spot rate, which is simply µ. (c) If the target zone is credible and UIP holds then there can be no expected profits from the carry trade. Actual profits could be realized randomly (though bounded, as one can show). But if the target zone is not completely credible then larger profits are possible in particular time periods, even if they still average zero if UIP holds. Only if UIP fails to hold and the target zone is not credible can there be persistent profits from the carry trade.

9.6.(a) s − b − αµ T − t = t , µ which is a decreasing function of µ. (b) I think the loss of reserves is αµ.

9.7. (a) s − b − µ T − t = t µ (b) Calculus shows this is increasing in R and decreasing in µ.

186