FX Premia from the Markets

Mi Wu∗

This Draft: November 20, 2017

Abstract

This paper proposes a two-country term structure model of joint behavior of bond markets and foreign exchange (FX) markets. With information extracted from local bond markets of G10 countries, the model is able to repro- duce the uncovered interest parity (UIP) puzzle as observed in the FX market. Bond factors explain up to 50% of the variations in movements at a one-year horizon and over 90% for some countries at a five-year horizon. For currency excess returns, the model-implied time-varying risk pre- mia deliver higher explanatory power than the interest rate differentials. These findings quantify how closely the FX market and the bond markets are inte- grated. The empirical findings also reveal heterogeneity between - and funding-currency countries in terms of the risk exposure to the transitory shocks of the bond markets.

∗PhD Candidate, 4-339 Simon Business School, University of Rochester, Rochester, NY, USA 14620. Email: [email protected]; Tel: +1 415-279-6090. I am grateful to my advisor and committee members, Prof. Robert Ready, Prof. Ron Kaniel, and Prof. Robert Novy-Marx, for their invaluable advice. I also thank Prof. Olga Itenberg, Prof. John Long, Prof. Yan Bai, Prof. Andreas Stathopoulos, Prof. Jerry Warner, Dr. Leon Cui and all seminar participants at Simon Business School and University of Sydney Business School for helpful discussions and comments. All remaining errors are mine.

1 1 Introduction

In theory, currency trades over any length horizon can be conducted through the bond markets. When there is no arbitrage, capital gains from the interest rate differential between two countries’ sovereign bonds should be eliminated by the relative depreci- ation between the two (uncovered (UIP)). As covered interest parity (CIP) generally holds1, UIP also provides the economic foundation for the forward unbiasedness hypothesis, postulating that the is an unbiased forecast for the future spot rate. However, a large literature finds the relation violated in the data. High interest rates are usually associated with increasing currency values, which lead to the (in)famous UIP failure (or the forward premium puzzle), and the success of various carry trade strategies in the market.

Fama (1984) is among the first to rationalize such observation by incorporating a foreign exchange (FX) risk premium to the UIP model. Recognizing forward rates are usually biased predictors of future spot exchange rates, Fama (1984) and others consider a risk-adjusted UIP, in which the expected exchange rate changes comprise the forward premium and a time-varying FX risk premium. Capturing the FX risk premium accurately is therefore at the heart of understanding the UIP failure puzzle.

To measure the FX risk premium empirically, I propose a quantitative approach to construct a proxy for the FX risk premium between any two countries using the risk perception information extracted from the bond markets alone. The approach arises from the strong connection between the bond and the foreign exchange markets.

More specifically, when there is no arbitrage, a pricing kernel capturing the effect of factors determines the prices of all securities in the economy.

1Akram, Rime, and Sarno (2008) conclude that CIP holds at daily and lower frequencies. This relation was violated during the financial crisis in the fall of 2008 (Baba and Packer, 2009), an extreme time period that is not included in this study.

2 Therefore, any that drives the term structure of bond prices should also be able to explain the price fluctuations in the foreign exchange markets, i.e., the spot exchange rate movements. If the bond market risk perception is accurately measured, and the bond and the currency markets are highly integrated, a FX risk premium constructed with bond market risk factors should help explain and forecast exchange rate fluctuations and currency excess returns in the foreign exchange markets.

The first step is to capture investors’ valuation of the risk from the bond markets of various countries. Cochrane and Piazzesi (2005) find that a single risk factor

(henceforth the CP factor) composed of a linear combination of yields and forward rates captures all of the variations in one-year expected excess returns for bonds of one- to five-year maturities in the US market. To study the , I verify the existence of this single risk factor for the other bond markets.

The nine countries whose currencies are the most traded in the FX market (other than the US Dollars) are Australia, Canada, Germany, Japan, New Zealand, Norway,

Sweden, Switzerland, and United Kingdom. Together with the US, these countries comprise the list of G10-currency (henceforth G10) countries. G10 countries are all developed market countries, and their nominal interest rates and exchange rates are not exposed to high inflation fluctuations in general. The results from my analysis confirm that the CP factor exists for these bond markets and captures over 95% of the economically interesting variation in one-year expected bond excess returns across one- to ten-year maturities for all G10 countries. Since it is well established in the literature that the first three yield curve principal components provide a good representation of the cross-sectional variation of any country’s yield curves, I augment the orthogonalized CP factor to the three yield risk factors as an unspanned risk factor in the term structure model. Interest rates are fitted by yield factors only

3 under the risk-neutral distribution of the state vector, yet risk associated with the

CP factor is still embedded in the FX risk premia under the physical distribution.

Under the no-arbitrage condition, and assuming complete markets, exchange rates between any two countries are modeled as the ratio of their stochastic discount factors (SDFs), where each country’s SDF captures the effect of all four risk factors extracted from its local bond market.

The fitted exchange rate movements from the two-country term structure model are able to reproduce the UIP failure, or the forward premium puzzle, observed in historical data. In the short term, the fitted exchange rate movements follow the trend of the real exchange rates, but with a lower volatility. As the horizon becomes longer, the fitted exchange rate movements gradually match up with the actual data. According to the model, the exchange rate fluctuations are driven by three components: (1) the difference in the level of interest rates between two countries, (2) a nonlinear time-varying risk premium that captures the effect of risk factors extracted from bond markets, and (3) the difference in shocks to the prices of the . Equipped with the two-country term structure model estimation, the time-varying risk premium can be quantified for each country pair. Comparing the explanation power for the exchange rate fluctuations, it is clear that the time-varying risk premium is the main driving force behind exchange rate movements. At the one- year horizon, risk premia extracted from bond markets help explain up to 50% of the variations in exchange rate movements, and up to over 90% at a five-year horizon.

Next, I explore the model implication for currency excess returns in the FX markets. Famous anomalies such as carry trade and dollar carry trade refer to the fact that lending in high-interest currencies and borrowing in low-interest curren- cies generate profitable returns. Therefore, positive interest rate differentials are

4 not eliminated by the depreciation of the currency. Instead, the high interest rate currency appreciates. Furthermore, when currency excess returns are regressed on interest rate differentials, although the sign of the coefficient agrees with the carry trade implications, the R2 is close to 0. This suggests that a large portion of the variations in the currency excess returns is not explained linearly by the interest rates. In the two-country term structure model I employ, the currency excess return is composed of the compensations for two layers of risks: (1) the interest rate uncer- tainty and (2) the . The first term is captured by the yield term premium and the second term by the nonlinear time-varying risk premium extracted from the bond market. The empirical analysis shows that the nonlinear risk premium is again the key driver behind the variations of the currency excess returns. For the one-month horizon, the time-varying risk premium extracted from the bond markets has an R2 up to 7.3% higher than the interest rate differentials in explaining the cur- rency excess returns and over 50% higher for some countries at a one-year horizon.

Finally, the empirical results also point to heterogeneity in risk exposure between the investment currencies and the funding currencies of carry trade strategies. For investment currencies that are featured in the long leg of carry trades, i.e., currencies of Australia, Canada, New Zealand and Norway, the risk premia constructed by the bond market information can successfully explain the fluctuations in the FX market; while for funding currencies such as Japanese Yen and Swedish Krona, ex-post ex- change rate movements and currency excess returns seem to be more disconnected from the risk evaluations in the bond markets. The finding of heterogeneity in risk exposure is consistent with the empirical investigation by Lustig et al. (2011) and

Colacito, Croce, Gavazzoni and Ready (2015), among others.

Related Literature This paper contributes to the growing macro-finance literature

5 on the joint dynamics of bond and foreign exchange markets. Short-term interest rates, yield curve factors and other macroeconomic fundamentals have been used to explain exchange rate fluctuations; however, there is no consensus on the conclusion so far. Diez de los Rios (2009) studies four developed country pairs using risk-free interest rates only as the state variable for his continuous-time term structure model.

The paper finds that the model estimates provide better forecasts than the random walk model in explaining the movements of the British pound and the Canadian dollar against the US dollar, but worse forecasts than the random walk model for

German and Swiss currencies. In contrast, Inci and Lu (2004) find that interest rate factors alone are insufficient to determine exchange rate fluctuations for the UK and

Germany (against the US) in their term structure model with unobserved factors.

Dong (2006) estimates a no-arbitrage term structure model using the US–Germany data and finds that macroeconomic fundamentals, namely output gap and inflation, drive about 70% of the of forecasting the conditional mean of exchange rate changes. More recently, Bauer and de Los Rios (2012) develop a multi-country model with yield factors, macroeconomic fundamentals, and exchange rate changes as state variables. They find unspanned macroeconomic variables are important drivers of foreign exchange risk premia and expected changes in exchange rates (when certain restrictions are imposed in the estimation). Yung (2015) develops a two-country framework with yield curve factors to explain exchange rate movements for seven developed countries. Her results suggest that, from 1999 to 2014, interest rate factors help explain more variations than the interest rate differentials in the exchange rate movements, especially for long horizons. My paper is closely related to these works.

Instead of studying the macro influences on exchange rate movements, I focus more on bond and foreign exchange market risk integration. The set of risk factors in my

6 model not only captures the variations of yield curves, but also includes information embedded in bond excess returns. After all, bond excess returns carry the most relevant information regarding investors’ risk perception.

In order to accurately capture the risk premia from each country’s local bond market, I critically extend Cochrane and Piazzesi’s (2005) results to the other nine

G10 countries. Several other papers have also studied the CP factor internation- ally. Hellerstein (2011) constructs a global return forecasting factor that is the GDP weighted average of each country’s local CP factor. She finds that the forecasting faactor has information not spanned by the traditional level, slope and curvature factors of the yield curves. My paper confirms this finding, but for each country individually. Dahlquist and Hasseltoft (2009) is another example. They find that the local and global CP factors are jointly significant in explaining a country’s bond excess returns, where the global factor is the US CP factor. This suggests that vari- ations in bond excess returns are driven by country-specific factors and a common global factor. Shocks to the US bond risk premia seem to be particularly important determinants for international markets. Such finding justifies constructing coun- try pairs with the US as the common domestic country in my two-country term structure model, so that both CP factors from the US and from another country contribute to the time-varying risk premia in their foreign exchange market. Other papers that make use of international CP factors include Kessler and Scherer (2009), which focuses on portfolio construction methodologies in bond markets; and Koijen,

Lustig and Nieuwerburgh (2017), who find that the CP factor is able to price the cross-section of US stock returns.

The results of my paper show that the risk premium from local bond markets not only alleviates the UIP failure documented by Hansen and Hodrick (1980) and

7 Fama (1984), but also serves as a major force behind currency excess returns, when compared to interest rate differentials. Naturally, this relates the paper to findings in the carry trade literature. Backus, Foresi and Telmer (2001) are among the first to link bond markets to currency markets using an affine model to see if it accounts for the forward premium anomaly. Chen and Tsang (2012, 2013) study the predictability of yield curve factors for three-month ahead currency excess returns and exchange rate innovations for six country pairs. Ang and Chen (2010) explore several different specifications for the affine model and test the model implications for various yield curve factors’ predictability for currency excess returns. The main difference between these papers and mine lies in the structure of the model-implied risk premia. In my model, the risk factors are nonlinear to the time-varying FX risk premia, while previous research usually assumes a linear relation. Lustig, Roussanov and Verdelhan

(2011, 2014) examine the predictability of short-term interest rates through a cross- sectional portfolio analysis, and I compare the predictability of the FX risk premia with their results in the out-of-sample section. Although most of the carry trade literature focus on short-term strategies, my paper studies the effect of the bond risk premia in the FX market over a long horizon (one month to five years). Lustig,

Stathopoulos and Verdelhan’s (2016) paper is another example that looks at long- term bond excess returns among G10 countries. However, their research focuses on long-term bond risk premia parity. They find no significant differences in long-term government bond risk premia in dollars across G10 countries, in contrast to the large differences in risk premia at short maturities. Further, they argue that under the no arbitrage assumption, such findings suggest nominal exchange rates are stationary in levels, which is contrary to the academic consensus.

The paper is organized as follows. Section 2 proposes a two-country term struc-

8 ture model. Section 3 discusses the method of model estimation. Section 4 presents the data and verifies the existence of the CP factor for G10 countries. Section 5 reports the key empirical results from the model. Section 6 concludes.

2 A Two-Country Term Structure Model

In order to study the bond market influences on the changes in exchange rate and the currency excess returns at various horizons, I fit a two-country affine term structure model for each country pair to decompose these two variables numerically into an interest rate part and a risk premium part. The model is an extension of the typical single-country term structure model (proposed by Dai and Singleton (2002), Ang and

Piazzesi (2003) and many others). Compared to its predecessors, a two-country term structure model can not only fit the yield curves and quantify bond term premia, but also capture the effect of bond market risk factors on the changes of exchange rates and the FX excess returns in between two countries.

2.1 State Variables

Macro-finance affine term structure model literature can be roughly divided into two categories, where the main difference resides in the composition of the state variables.

One strand of the research believes that the macro factors that determine bond prices are fully spanned by the current yield curve.2 On the other hand, more recently,

Joslin, Priebsch and Singleton (2014) (JPS), Wright (2011) and others consider the possibility that some factors in a term structure model could be unspanned. An

2Examples of such literature include, but are not limited to, Ang and Piazzesi (2003), Ang, Dong, and Piazzesi (2007), Rudebusch and Wu (2008), Smith and Taylor (2009), Bikbov and Chernov (2010), Joslin, Singleton and Zhu (2011). Many equilibrium models with long-run risks also have expected consumption growth and inflation spanned by yields, such as Bansal, Kiku and Yaron (2012), and Bansal and Shaliastovich (2013).

9 important feature of the yield curves in most developed countries is that they are well captured by a low-dimensional yield factor model. In order to avoid the downside of overparameterizing the risk-neutral distribution of the state vector3, yet still utilize the predictive content of CP factor for excess returns, I follow JPS’s approach to construct a model where the pricing of bonds is described by a 3-dimensional yields- only state vector, and CP factor is augmented as a non-linearly spanned fourth factor in the physical distribution, but not in the risk-neutral distribution. The estimation results and empirical findings reported in the paper will be mainly based on this model. Meanwhile, I also discuss a yields-only fully spanned model in the model setup section to outline the difference between the two.4

Each country pair in the model consists of a “domestic” country and a “foreign” country. I take the perspective of a US investor. Hence, the “domestic” economy is considered to be the US for all country pairs (denoted by US), while the “foreign” economy is one of the other nine G10 countries (denoted by i). For most of the developed countries, the cross maturity of bond yields follows a low dimensional factor model. Among G10 countries, the data shows that the first three components can account for more than 99.5% of the cross-sectional variations of the yields. This translates into high accountability for all the cross-sectional variations in the yields by the first three or four principal components of the variance-covariance matrix of the yield curves (Joslin, Priebsch and Singleton, 2014). For the completely spanned model (denoted by M3F below), without loss of generality, I rotate the risk factors to be the first N principal components of the yield curves5, and employ them as the state variables, where N is set to be 3. These factors are observable, country-specific,

3Duffee (2010) and Joslin, Singleton and Zhu (2011) document that model-implied Sharpe ratios for certain bond portfolios are implausibly large when the number of yield principal component factors is as low as 4 for macro-finance affine term structure models. 4Estimation results are available upon request. 5This rotation follows the approach in Joslin, Singleton and Zhu (2011).

10 (k),i and are extracted from each country i’s yield curves yt such that

i i (1,...,m),i yPt = W · yt , (2.1) where k = 1, . . . , m is the periods-to-maturity of the bonds, W i is an N × m matrix of loadings for each factor on country i’s yields of different maturities. Following the literature, the first three principal components of the yield curves are named level, slope, and curvature, based on the shape of the plots of these three factors. Given all the factors are linearly extracted from the yields, this model therefore only consists of risks spanned by the bond yields.

Cochrane and Piazessi (2005) find that a single tent-shaped factor, which is a linear combination of forward rates of the cross-maturity of bonds, predicts excess returns on one-to five-year maturity bonds with R2 up to 0.44 for the US, higher than the predictability of level, slope and curvature combined. This factor forecasts stock returns and has an important component that is unrelated to the movements of the first three principal components of the yield curves. I extend their approach to investigate all the other nine G10 countries, and verify that a tent-shaped factor of forward rates exists for each of these countries during the span of my data (Jan

1990 to May 2009), and bond excess return predictions are superior using this risk factor than the conventional yield factors (See in Section 4.1, Figure 1 and Table 2 for details). To reflect such findings in the term structure model, in my main model setting (MCP ), I augment the 3 yield curve factors in the fully spanned model with

i the bond excess return forecasting factor (CP factor, CPt ), where 1) the factors for interest rates fitting are still restricted to level, slope, and curvature extracted linearly from the yield curves, such that the yield curves are not over-fitted with unnecessary risk factors; and 2) the important component unrelated to the 3 conventional yield factors in the CP factor is accounted for as a fourth state variable in the risk premia,

11 mirroring its predictability for the excess returns in bonds and stocks. To extract

i the additional information from this fourth state variable, I project CPt onto the

i yield factors yPt for each country i at each time t,

i i i i i i CP rest = CPt − P roj CPt |levelt, slopet, curvaturet , (2.2)

i and use the orthogonalized CP factor CP rest together with the three yield curve factors to capture the variations in the risk premium.

i To summarize, let the vector of state variables for country i in time t be Zt , where its full dimension is Nf . For the fully spanned model M3F , the dimension of the vector of the state variables is the same as the number of the risk factors for

i i yields fitting, i.e. Nf = N = 3, and Zt = yPt . In contrast, for model MCP , the

i i0 i 0 i state variables Zt = [yPt CP rest] , while the yields fitting factors remain to be yPt

i only. Hence, the dimension of Zt for each country at each time point is four, i.e.

Nf = 4, while only the first N = 3 principal components are used to fit the yields. In a two-country term structure model, the comparison of the results from these two model settings helps to explore the effect of CP factor not only in the local bond markets, but across borders in the foreign exchange markets as well.

2.2 The Pricing Kernel and the Stochastic Discount Factors

The assumption of no arbitrage opportunity guarantees the existence of a pricing kernel for an economy. It represents the probability-weighted cost of receiving a state-contingent payoff sometime in the future. Specifically, for assets denominated in

i currency i, the nominal pricing kernel in that currency, denoted by ξt(Z), corresponds to the marginal value of a unit of currency i delivered at time t in the state of the

i i i world Z. The growth rate of the nominal pricing kernel Mt+1 = ξt+1/ξt is the

i Stochastic Discount Factor (SDF), which prices any payoff Pt+1 at time t + 1 such

12 i that the price of the asset at time t, Pt , satisfies

i P  i i  Pt = Et Mt+1Pt+1 , (2.3)

P where Et stands for the expected values under the physical or actual probability measure P. The SDF adjusts for the risks in discounting the value of any tradable asset from t + 1 back to t. In comparison, under the risk neutral probability Q, the discounted price of the asset does not need to be adjusted for the risks by the nominal SDF due to risk neutrality:

i i −rt Q  i  Pt = e Et Pt+1 . (2.4)

Moreover, if the economy is assumed to have a complete market for state-contingent

i claims, the SDF Mt+1 is the unique solution to the pricing equation (2.3). The SDF

i Mt+1 and the associated pricing relation in (2.3) are the basis of modern theories for bond pricing. Since the payoffs of bonds are deterministic, the absence of arbitrage implies restrictions on the time series and the cross-sectional properties of bond

(n),i yields. Following Cochrane and Piazzesi (2005, 2008), for country i, let Pt be the price of an n-period to maturity zero-coupon bond at time t, and hence the yield of

(n),i (n),i such a bond at t is yt = − ln Pt /n. Following equation (2.3), the bond price (n),i Pt is the risk-adjusted discounted value of its next period payoff, h i (n),i P i (n−1),i Pt = Et Mt+1Pt+1 , (2.5)

(n−1),i where Pt+1 is the market price of an (n − 1)-period to maturity bond at t + 1. For

i the short term risk free interest rate rt, i.e., the yield of a 1-period bond of country i, at time t, since the payoff in the next period is deterministic and normalized to 1, it is simply the log expected value of the 1-period SDF under P:

i P  i  rt = − ln Et Mt+1 . (2.6)

13 i Further assume the nominal SDF Mt+1 is conditionally log-normal for each country i, and its functional form follows:   1 0 0 M i = exp −ri − Λi Λi − Λi P,i , (2.7) t+1 t 2 t t t yP,t+1

i i where Λt is the market price of risk in country i, rt is the short term risk free rate

P,i defined in (2.6), yP,t+1 ∼ N (0,IN ) is the future shock to the yield factors under

i P. Assume further the law of motion of each country’s vector of state variables Zt follows an unconstrained vector autoregression (VAR) under P:

i i i i P,i P,i Zt+1 = µ + Φ Zt + ΣZ Z,t+1, (2.8) or more specifically,         yP i µi Φi yP i  t+1   yP   yP   t  P,i P,i = + + ΣZ Z,t+1, (2.9)  i   i   i   i CP rest+1 µCP ΦCP CP rest

    Φi Φi Φi yP yP,yP yP,CP i i where   =   , with ΦyP being an N × Nf matrix, and ΦCP  i   i i  ΦCP ΦCP,yP ΦCP,CP P,i  an (Nf − N) × Nf matrix; Z,t+1 ∼ N 0,INf is the innovation to all state variables

i P,i Zt , while the risks yP,t+1 in (2.7) are the first N innovations from the above uncon-

P,i P,i P,i0 strained VAR (equation (2.8)); ΣZ is a lower triangular matrix, such that ΣZ ΣZ

i is the Nf ×Nf variance-covariance matrix of shocks to Zt . Meanwhile, under the risk

i neutral distribution Q, yield factors yPt , for each country i, are assumed to follow an autonomous Gaussian VAR,

i Q,i Q,i i Q,i Q,i yPt+1 = µ + Φ yPt + ΣyP yP,t+1, (2.10)

Q,i Q,i Q,i Q,i0 where yP ∼ N (0,IN ), ΣyP is a lower triangular matrix such that ΣyP ΣyP is the

i N × N variance-covariance matrix of shocks to the yield factors yPt only. According to the diffusion invariance principle (Girsanov, 1958), changing the distributions

14 from P to Q is essentially a shift in the mean and not the variance of the factors, i.e.,

P Q ΣyP = ΣyP = ΣyP . Therefore, when the state variables coincide with the N yield factors as in the conventional yield factor term structure models (fully spanned model

P,i i M3F ), ΣZ is the same as ΣyP , both with N × N for the dimension. In contrast, for the augmented model MCP , yield factors are only the first N elements of the

Q,i P,i i P,i state variables, and ΣyP (= ΣyP = ΣyP ) is the upper N × N block of ΣZ . Hence, the forecasts of future yield factors are conditioned on the full set of Nf risk factors

i in Zt —both yield factors themselves and the CP factor. The SDF also adjusts for risks embedded in all risk factors, even though only future shocks to yP i show up in equation (2.7) for SDF.

With above assumptions and the absence of arbitrage assumption, it can be shown

(n),i (n),i that for any country i, the (log) price, pt (= ln Pt ), of an n-period bond (n > 0),

i is an affine function of the yield factors yPt :

(n),i i i0 i pt = An + Bn yPt . (2.11)

i i where the loadings An and Bn are restricted to satisfy internal consistency conditions

i i of bonds, for all n > 0. An is a scalar, Bn is an N × 1 vector, and they are functions of the parameters governing the Q pricing distribution of yields. Since the yield of (n),i (n),i such a bond is yt = −pt /n, it follows that

(n),i i i0 i yt = −an − bnyPt , (2.12)

i i i An i Bn where an = n , bn = n . Given (2.12), the short term interest rate for each country i i is also linear in yPt . Assume

i i i0 i rt = ρ0 + ρ1 yPt , (2.13)

i i i i i i the relation in (2.6) suggests that for n = 1, A1 = a1 = −ρ0, and B1 = b1 = −ρ1.

i i For bond maturities n > 1, the internal consistency conditions restrict An and Bn to

15 follow the recursions below6:

0 1 0 0 Ai = −ρi + Ai + Bi µQ,i + Bi ΣQ,iΣQ,i Bi , (2.14) n+1 0 n n 2 n yP yP n

0 i i Q,i i Bn+1 = −ρ1 + Φ Bn. (2.15)

The difference between the actual and the risk-neutral pricing distribution is the adjustment for the market prices of the risks, which captures investors’ valuations of

1 unit of each risk at various times. Following Duffee (2002), the market prices of risks for each country i, at time t, are constructed by the difference in the drifts of

i yPt under P (in (2.9)) and under Q (in (2.10)):

i i i i Q,i i Λ0(Zt ) = µyP (Zt ) − µ (yPt ), (2.16)

  i i i i Λ (Z ) = Φ (Z ) − Q,i i , (2.17) 1 t yP t Φ (yPt ) 01×(Nf −N)

i i i i where Λ0(Zt ) is an N × 1 vector, and Λ1(Zt ) is N × Nf . The time-variant market

i prices of risk in (2.7) are affine in state variables Zt ,

i i,−1 i i i Λt = ΣyP Λ0 + Λ1Zt , (2.18)

i with Λt being an N × 1 vector. Under the fully spanned model M3F , each element

i i of Λt captures the price of the risks from yPt , namely, level, slope, and curvature, at each time t respectively. Under the augmented model MCP , even though the only priced risks in Treasury markets are those from the yield factors, investors’ risk

i tolerance Λt is also influenced by the information contained in the CP factor at each

i time point. Therefore agents’ SDF cannot be represented in terms of yPt alone, and the CP factor could shift the prices of yield risks from those in a completely linearly spanned model.

6See Appendix A.

16 2.3 Change in Exchange Rate and Currency Excess Returns

In a two-country setting, if an asset’s return is denominated in a foreign currency i

(other than USD), (2.3) is equivalent to the following relation regarding the asset’s

i gross return Rt+1, P  i i  1 = Et Mt+1Rt+1 . (2.19)

Alternatively, under “no arbitrage” assumption, the returns can also be converted

US into US dollars and be valued with the US (domestic) SDF Mt+1. Let the nominal

i spot exchange rate between the US and the foreign country i be St at time t, which is the US dollar price of one unit of the foreign currency i ($/i). The converted dollar return of the asset in foreign currency i is therefore: i US St+1 i Rt+1 = i Rt+1. (2.20) St Equation (2.19) and (2.20) together suggests  Si  P  i i  P US t+1 i Et Mt+1Rt+1 = Et Mt+1 i Rt+1 , (2.21) St

i i US St+1 the equality of which is satisfied if Mt+1 = Mt+1 i , under the assumption of com- St plete markets for currencies and state-contingent claims. Therefore, as shown by

Backus, et al. (2001), among others, the exchange rate change is the ratio of the

SDFs in the domestic and the foreign country, i i St+1 Mt+1 i = US , (2.22) St Mt+1 i i i i i or, in natural logarithm, let ∆st→t+1 = st+1 −st, with st = ln St, and mt+1 = ln Mt+1 for each countries i and the US, i i US ∆st →t+1 = mt+1 − mt+1. (2.23)

Given the functional form of the SDF in (2.7), the model implied one-period log exchange rate change is:

1  0 0   0 0  ∆si = rUS − ri + ΛUS ΛUS − Λi Λi + ΛUS P,US − Λi P,i . (2.24) t →t+1 t t 2 t t t t t yP,t+1 t yP,t+1 17 To extend to k periods into the future, for k > 1, since the k-period SDF from today Qk is the product of all of the future 1-period SDFs: Mt→t+k = j=1 Mt+j−1→t+j, the k-period log exchange rate change is the difference between the (log) k-period SDFs of foreign country i and the US:

k  X 1  0 0  ∆si = rUS − ri  + ΛUS ΛUS − Λi Λi t →t+k t+j−1 t+j−1 2 t+j−1 t+j−1 t+j−1 t+j−1 j=1 (2.25)   US0 P,US i0 P,i  + Λt+j−1yP,t+j − Λt+j−1yP,t+j .

Equation (2.24) and (2.25) demonstrate that in the two-country term structure model, the change in exchange rates comprises three parts: 1) the interest rate differentials between the two countries; 2) the FX risk premia; and 3) the difference in the yield factor shocks to the investors’ valuation of the risks between the two countries. The literature has verified that Uncovered Interest (Rate) Parity (UIP) does not hold in the data, suggesting that the movements in the exchange rates cannot be explained by the change in the interest rate differentials alone. Fama

(1984) proposes that a time-varying risk premium contributes to the deviation from

UIP, which is consistent with the second component of the model implied change in exchange rates.

As shown in section 2.2, the market price of risks for each country, Λt, prices the risks from the yield factors yPt in both model M3F and MCP ; while for the latter,

Λt also reflects information on investors’ risk tolerance regarding the CP factor. In (2.24) and (2.25), yield factors from the domestic and the foreign bond markets enter the FX risk premium term in ∆st→t+k(k ≥ 1) nonlinearly, which departs from the standard methodology of linearly projecting the movements in exchange rates onto the yield curve information or other macroeconomic fundamentals. Moreover, only innovations to the risk factors extracted from the bond markets are involved in the

18 model-implied change in exchange rates – no additional white noise is presented to account for its movements. With the aid from a term structure model estimation, it is possible to disentangle the impact on the overall movements of the exchange rates from various components. By integrating the local bond markets and the foreign exchange market, I can quantify the amount of fluctuations in the foreign exchange market that is accounted for by the bond market information alone. The more integrated the two markets are, and the more accurately bond risk premia capture investors’ real perspective on risks, the better can we explain Forex market movements.

Furthermore, the change in exchange rates is a crucial component for predicting currency excess returns in the FX market. The 1-period currency excess return is the return, in excess of the risk free rate, from investing in a foreign currency denominated bond for 1 period in time, and converting the proceeds back to dollars at the end of the period. Hence, the currency excess return for a domestic investor with foreign currency i can be expressed as:

P  i  P  i US i  Et cert→t+1 = Et rt − rt + ∆st→t+1 , (2.26) together with (2.24), the expected 1-period currency excess return is reduced to the

FX risk premium differential only. Since the price of risk Λt for each country is

US i linear in all risk factors, the model suggests that the risk factors Zt and Zt affect the currency excess returns non-linearly,

1  0 0  EP ceri  = ΛUS ΛUS − Λi Λi . (2.27) t t→t+1 2 t t t t

Since the short term interest rates are assumed to be linear in the yield curve risk factors, the currency excess returns therefore are also implied to be quadratic to short term interest rates. In the standard carry trade strategy, currency dealers sort currencies according to short term interest rates to borrow in low interest rate

19 currencies, and lend in high interest currencies. Empirical studies usually project currency excess returns linearly on short term interest rate differentials to account for its contribution to the carry trade anomaly. Here, the term structure model suggests that short term interest rates are closely related to 1-period currency excess returns, but non-linearly. Currency excess returns could be better explained and predicted, if the FX risk premium for each country pair is properly quantified at time t.

Similar to the extension for the change in exchange rates, the annualized currency excess return for a k-period horizon is: 1 h   i EP ceri  = EP k y(k),i − y(k),US + ∆si , (2.28) t t→t+k k t t t t→t+k

(k) where yt is the annualized yield on a k-period to maturity bond in country i or the US. Combining (2.28) and (2.25), the k-period expected currency excess return

(annualized) becomes   P  i  (k),i (k),US Et cert→t+k = yt − yt k  1 X 1  0 0  + EP rUS − ri  + ΛUS ΛUS − Λi Λi k t t+j−1 t+j−1 2 t+j−1 t+j−1 t+j−1 t+j−1 j=1   US0 P,US i0 P,i  + Λt+j−1yP,t+j − Λt+j−1yP,t+j . (2.29)

(k),i The yield yt of a long (k > 1) maturity bond for country i is often interpreted as the average of expected short term rates over the horizon, plus a yield term premium, which captures the uncertainty of the future interest rates at the long

(k),i 1 k P i (k),i end: yt = k Σj=1Et (rt+j−1) + tpt . Rearranging equation (2.29), the k-period expected currency excess return is decomposed into two parts of risk compensations: the difference in premia (or the yield term premia) between the

20 domestic and the foreign country, and the FX risk premia differentials over the next k periods:

" k # " k # (k),i 1 X (k),US 1 X EP ceri  = y − EP ri  − y − EP rUS  t t→t+k t k t t+j−1 t k t t+j−1 j=1 j=1 | {z } Yield Term Premia k 1 X  0 0  + EP ΛUS ΛUS − Λi Λi 2k t t+j−1 t+j−1 t+j−1 t+j−1 j=1 | {z } F X Risk P remia k 1 X  0 0  + EP ΛUS P,US − Λi P,i k t t+j−1 yP,t+j t+j−1 yP,t+j j=1 k  (k),i (k),US 1 X  0 0  = tp − tp + EP ΛUS ΛUS − Λi Λi t t 2k t t+j−1 t+j−1 t+j−1 t+j−1 j=1 k 1 X  0 0  + EP ΛUS P,US − Λi P,i . k t t+j−1 yP,t+j t+j−1 yP,t+j j=1 (2.30)

This decomposition is intuitive. For a horizon of k periods, a trading position of buying and shorting bonds in two markets is exposed to two layers of risks: 1) the uncertainty of yield curve movements in these two markets over the next k periods, and 2) the uncertainty of future exchange rate fluctuations. With the two-country term structure model, I can quantify the risk compensation for each layer of the two risks, and see which one of the two drives the currency excess returns.

3 Maximum Likelihood Estimation

I follow Joslin, Priebsch, and Singleton (2014) in the maximum likelihood estimation for the two-country term structure model. The parameters to be estimated are

i i P,i Q,i Q,i i i Θ = {µ , Φ , ΣZ , µ , Φ , ρ0, ρ1}, where i is referred to the US and one of the other G10 countries in a country pair. The results of Joslin, Singleton, and Zhu

21 i (2011) allow the Q distribution of yield factors yPt to be uniquely parameterized

i Q,i Q,i Q,i by {ΣyP , λ , r∞ }, where λ denotes the N-dimension vector of ordered nonzero

Q,i Q,i eigenvalues of Φ matrix, and r∞ is the long-run mean of rt under the risk neutral probability measure7. Under P measure, as suggested by Joslin, Le and Singleton

i (2013), since the state vector Zt for each country i, including the US, is observed, the maximum likelihood estimates of µi and Φi can be obtained from fitting a VAR to

i P,i Zt , as in (2.8) and (2.9), and ΣZ is parameterized by the lower triangular Cholesky factorization of the variance-covariance matrix from the VAR for each country in a country pair.

Joslin, Priebsch, and Singleton (2014) points out that an important property of the log-likelihood function in question is the complete separation of the parameters

{µi, Φi}, governing the conditional mean of the risk factors, from those governing risk-neutral pricing of the bond yields and yield risk factors. In particular, the

OLS estimators of {µi, Φi} are invariant to the imposition of restrictions on the

Q probability measure of the risk factors and the pricing of yields. In detail, the conditional log likelihood function is the sum of P and Q conditional log likelihood, such that the model-implied yields and changes in exchange rates plus i.i.d. Gaussian measurement errors are equal to their counterparts in the observed data:

˜ Q Q,i Q,US Q,i Q,US i US f(Ot|Ot−1; Θ) = f (Ot|Zt; λ , λ , r∞ , r∞ , ΣyP , ΣyP ) (3.1) P i US i US P,i P,US + f (Zt|Zt−1; µ , µ , Φ , Φ , ΣZ , ΣZ ),

where Ot is a vector of observed yields from the domestic country (the US), the foreign

7 Q,i Q,i Joslin, Singleton and Zhu (2011) show that {λ , r∞ } are rotation invariant, therefore they are independent of the choice of pricing factors. As in Joslin, Priebsch and Singleton (2014), the model i here is first reparameterized with a latent state vector Xt , which follows a VAR with zero intercept Q,i Q,i 0 i and a diagonal slope coefficient matrix, I + λ , and the short term interest rates rt = r∞ + i Xt , where i is a vector of ones. The likelihood is maximized with respect to these parameters, and Q,i Q,i i i then the parameters for the observed pricing factors under Q, {µ , Φ , ρ0, ρ1}, can be recovered i Q,i Q,i through a mapping from the estimated {ΣyP , λ , r∞ }. The mapping is outlined in their paper.

22 country i, and the change in exchange rate in between the two countries: Ot =  0 obs,US obs,i obs , and Z is a vector of domestic and foreign risk factors: yt yt ∆st t  0 Z = US i . Assuming the state vector for each country is conditionally t Zt Zt Gaussian, the log likelihood function under P is:

N US + N i f P(Z |Z ; µi, µUS, Φi, ΦUS, ΣP,i, ΣP,US) = f f log (2π) t t−1 Z Z 2 T (3.2) 1 0 1 X h 0 i + log|Σ Σ |+ (Z − E (Z )) · (Σ Σ )−1 · (Z − E (Z )) , 2 Z Z 2 t t−1 t Z Z t t−1 t t=1 where T is the total number of time series observations, Et−1(Zt) = µ + ΦZt, for  0  P,US P,US 0 ΣZ ΣZ 0 country i and the US respectively, and ΣZ Σ =  . The Z 0  P,i P,i  0 ΣZ ΣZ cross-country covariance is shut down to relieve the computational complexity8. On the other hand, the log likelihood under Q distribution is:

T 1 X e2 f Q(O |Z ; λQ,i, λQ,US, rQ,i, rQ,US, Σi , ΣUS) = t t t ∞ ∞ yP yP 2 Σ2 t=1 e (3.3) 1 1 + (J T − N T ) log(2π) + (J T − N T ) log(Σ2), 2 2 e where e2 is the squared errors between the observed and model implied yields and t   obs,US US y − yˆt  t   obs,i i  changes in exchange rate, with et =  y − yˆ . The errors are assumed to be  t t    obs ˆ ∆st − ∆st conditionally independent of their lagged values and satisfy the consistency condition

q PT 2 obs  9 t=1 et T US i P rob W · yt = yPt|yPt = 1 . Finally, Σe = T (JT −N T ) , with J = J + J , and N T = N US + N i. J is the number of observed variables, and N is the number of yield pricing factors for each country.

8A model without such restriction is also estimated, and the results are not substantially differ- ent. 9See Joslin, Singleton and Zhu (2011) for details.

23 4 Data

4.1 Pricing Factors from Bond Markets

The yield curve data I use in this paper for G10 countries are constructed by Jonathan

Wright (2011). The zero-coupon yields are monthly from January 1990 to May 2009 for Australia, Canada, Germany, Japan, New Zealand, Switzerland, UK and US, from

December 1992 to May 2009 for Sweden, and from January 1998 to May 2009 for

Norway. Wright’s original dataset covers yields of maturities for 3 month, 6 month,

9 month... up to 10 years (40 in total), but in this paper, I follow the majority of the literature and pick 14 maturities, namely, 3 month, 6 month, 9 month, 1 year, 2 year...10 year, to construct the yield factors and CP factors for the ten G10 countries.

All yield curve data are annualized in percentage, end-of-month, and denominated in each country’s local currency.

The yield factors for each country are constructed by extracting the first three principal components from that country’s yield curves, as detailed in equation (2.1).

In order to construct the time series for CP factor, following the methodology in

Cochrane and Piazzesi (2005, 2008), I apply the yield curve data for zero-coupon bonds at annually spaced maturities for all countries. For the US only, Cochrane and Piazzesi (2005) use the Fama and Bliss (1987) unsmoothed data (henceforth

FB) on 1- to 5-year-to-maturity zero coupon bond prices. Later in their 2008 paper, they incorporate longer maturities in the analysis by using the Gurkaynak, Sack and

Wright (2007) data (henceforth GSW). GSW data consists of a fitted function which smoothes across maturities, in contrast to the FB unsmoothed approach. But the differences between the two for the forward rate data are quite small on most dates.

The disadvantage of the GSW type of smoothed yield curve data is that it will cause

24 a multi-collinearity problem when projecting the bond excess returns onto forward rates of different maturities. One way to mitigate the multi-collinearity problem is by reducing the number of regressors. However, Cochrane and Piazzesi (2008) shows that the R2 reduces from 0.38 to 0.26 for the US data when the regressors reduce from

(1) (2) (3) (4) (5) forward rates of 1- to 5-year-to-maturity bonds, i.e., ft = [ft , ft , ft , ft , ft ], (1) (3) (5) to 1-, 3- and 5-year-to-maturity rates, i.e., ft = [ft , ft , ft ], for the Fama-Bliss 1 P10 (n) regression rx¯ t+1 = 9 n=2 rxt+1 = α + βft + t+1, where rx¯ t+1 is the average of one-year bond excess returns on the bonds of 2- to 10- year to maturity; the (log) forward rate for an n−year to maturity bond at time t is the difference in price for

(n) an (n − 1)−year to maturity bond and an n−year to maturity bond at t: ft = (n−1) (n) pt − pt . Wright (2011) dataset, used to construct the pricing factors here, is a direct extension, to the other nine G10 countries, of the GSW smoothed yield curve construction methodology. Therefore, it will be exposed to the same multi- collinearity problem when running the Fama-Bliss regressions.

The purpose of constructing CP factor for each country in this paper is to best characterize the bond excess returns, or, a typical investor’s valuation of the risk in the country’s bond market. To achieve that, I need to first verify that Cochrane and Piazzesi’s (2005) result applies to the other 9 developed countries, i.e., there is a single risk factor (CP factor) that explains almost all of the variations in expected bond excess returns. One straight-forward way to show the existence is to plot the regression coefficients of one-year bond excess returns for two- to ten-year to maturity bonds on forward rates, (n) rxt+1 = α + βft + t+1, (4.1)

(n) (n−1) (n) (1) h (1) (3) (5)i where rxt+1 = pt+1 − pt − yt , n = 2, 3..., 10; and let ft = ft , ft , ft , due to the potential multi-collinearity problem from the smoothed yield curve data. If the coefficients on the forward rates for bonds of different maturities display a common

25 shape, it suggests that the expected bond excess returns with two- to ten-year to maturity all move together, and hence a single risk factor exists to account for all the

h (n) (n) (n)i variations. Figure 1 shows the slope coefficients β = β1 , β3 , β5 as a function of the maturities n for all G10 countries. The pattern during the time period I study is clear: for almost all countries, with New Zealand being a border line case, there is a function of forward rates that forecasts holding period returns at all maturities, with longer maturities having greater loadings on the same function. The last graph in Figure 1, labeled “G9”, is for an artificial country or a bond portfolio that takes an equal-weighted average of 9 non-US G10 countries’ yields across all maturities. From the graph it’s obvious that the same function that forecasts bond excess returns of different maturities exists for this artificial G9 country as well. The results coincides with what Cochrane and Piazzesi’s (2005) findings for the US bond market, which justifies constructing the bond market risk factor, CP factor, for each of the G10 countries, and for the artificial country “G9”, to be included in the term structure model as a state variable.

Now that the existence of CP factor for each G10 country is verified, in order to best characterize the risk premia in the bond market, I switch to use all the forward

(1) (2) (3) (4) (5) rates from one- to five-year of maturity, ft = [ft , ft , ft , ft , ft ], in regression (4.1) to construct the CP factor. Table 1 shows the percentage of the variance of

Table 1: Eigenvalues of Expected Bond Excess Returns: Fraction of Variance (%)

Australia Canada Germany Japan New Zealand Norway Sweden Switzerland UK US 1st(CP Factor) 99.61 97.61 99.12 98.40 99.37 99.54 98.69 97.87 95.61 95.57 2nd 0.38 2.32 0.84 1.54 0.63 0.45 1.30 2.03 4.32 4.40 3rd 0.01 0.06 0.03 0.06 0.00 0.01 0.01 0.09 0.07 0.02

Notes: This table reports the fraction of the variance of the expected bond excess returns that is accounted for by the largest eigenvectors or each G10 country. Three largest factors are reported. Factors smaller than three mostly accounts for 0% of the total variation.

26 expected bond excess returns that is accounted for by the 3 largest factors. The

first factor, i.e., CP factor, accounts for over 97% of the variation in the expected returns for most G10 countries, and about 96% for UK and the US. In Figure 1, New

Zealand is a border line case for displaying a tent-shape for the slope coefficients of the forward interest rates. Statistics in Table 1 relieves any concerns arising from it, given that CP factor accounts for 99.37% of the variations in bond risk premia for New Zealand. Hence, almost nothing is changed when imposing a single- factor model on each country that has CP factor to track all the movements in its expected bond excess returns, or risk premia in the bond market. In addition,

Cochrane and Piazzesi (2005) find that for the US, CP factor is not fully spanned by level, slope and curvature factors of the yield or forward curves to explain the bond excess returns. Examining the predictive power of CP factor distinguishes from decomposing the yield curves, and then testing the predictability of each yield curve factors. Table 2 compares the ability and significance of the four risk factors in explaining the average holding period returns across 1- to 5-year-to-maturity bonds for the other 9 G10 countries, and the artificial country/portfolio G9. The level, slope and curvature factors in Table 2 are extracted from the forward interest rate curves of each country, following Cochrane and Piazzesi (2005). Since a forward curve is merely a linear transformation of its corresponding yield curve, the same regressions run with yield curve factors generate extremely similar results as those shown here.

From the table, it is clear that R2 from CP factor alone is usually higher than the three forward/yield curve factors combined. When putting all four factors together in one regression, yield factors’ explanation power is usually subsumed by CP factor.

27 4.2 Foreign Exchange Markets

The data used in foreign exchange analysis consists of a panel of 9 foreign currencies that are foreign relative to the US. End-of-month spot exchange rates are downloaded from Bank of England, and 1-month forward exchange rates are from Datastream.

The exchange rates are defined as the USD price of 1 unit of foreign currency. When the exchange rates become higher, US dollar depreciates. The sample period for the spot and the forward exchange rates is restricted by the availability of Wright’s

(2011) yield curve data, from January 1990 to May 2009. For the model including

CP factor as a state variable, since CP factor is constructed from 1-year bond excess returns, the sample period is shortened by 1 year, and ends in May 2008. Avoiding the European Debt crisis in 2009, and some part of the financial crisis in the US in

2008, this data sample excludes, at least partially, the possibility that any predictive power is completely driven by the crises that swept through most developed countries during that time period. In addition, recent paper by Du, Tepper and Verdelhan

(2017) finds that not only did the covered interest rate parity fall apart during the

financial crisis, the arbitrage spread still has not closed down as of today. Since the model derivation for exchange rates and short-term currency returns depends on the assumption of no-arbitrage, excluding the data after the crisis to avoid any potential regime change is a cautious . Finally, the Euro series starts in January 1999, so I splice the data for German Deutsche Mark with the data for Euro since then.

Germany is the country with the largest GDP among countries in the Eurozone, hence it is used as a representative of all Eurozone countries. All together, the data set contains currencies of: Australia (AUS), Canada (CA), Germany (DE), New Zealand

(NZ), Norway (NO), Sweden (SE), Switzerland (CH), United Kingdom (UK), and the US. Inspired by Dollar Carry discussed in Lustig, Roussanov and Verdelhan

28 (2014), I also take 1/9 of each yield curve of the 9 non-US countries at each time t to put together a bond portfolio, or an artificial country “G9”, which represents the whole foreign developed world from the perspective of a US investor. It is created to examine if the US plays a special role in the international foreign exchange market.

Since all exchange rates are in the unit of USD, the time series of the exchange rate between the artificial country “G9” and the US is simply the average exchange rates among 9 non-US G10 countries, for each time t.

29 Figure 1: Existence of CP Risk Factor in the Bond Market

(a) CA (b) CH (c) DE

(d) NO (e) SE (f) UK

(g) AUS (h) NZ (i) JP

(j) US (k) G9

(n) Notes: Each figure plots the slope coefficients β (y-axis) of the regression rxt+1 = α + βft + t+1 as a function of the maturities (x-axis, in years), for bonds that are n = 2, 3, ..., 10 years to maturity, h (1) (3) (5)i with ft = ft , ft , ft . G9 refers to the artificial country/bond portfolio that has the average of nine G10 countries’ yields, excluding the US.

30 Table 2: Average Bond Excess Returns Explained by Risk Factors

rx¯ rx¯ rx¯ Country Country Country (1) (2) (3) (1) (2) (3) (1) (2) (3) CP Factor 0.30 0.31 0.24 0.31 0.32 0.30 t-value [2.18] [4.49] [1.02] [5.95] [4.15] [4.93] level 0.02 0.33 0.05 0.21 -0.01 0.31 t-value [0.12] [3.37] [0.21] [1.39] [-0.07] [1.88] slopeCA -0.17 0.27CH 0.40 1.70DE -0.09 0.63 t-value [-0.26] [0.50] [0.29] [5.32] [-0.29] [1.81] curvature -0.21 -0.14 -0.98 -3.09 0.26 -3.77 t-value [-0.16] [-0.11] [-0.45] [-3.51] [0.19] [-2.11] R2 0.26 0.26 0.19 0.37 0.37 0.35 0.41 0.41 0.30 CP Factor 0.23 0.30 0.39 0.30 0.31 0.31 t-value [2.86] [7.70] [6.51] [11.43] [1.31] [4.11] level 0.09 0.30 -0.11 0.34 0.01 0.40 t-value [0.42] [1.33] [-0.42] [1.57] [0.04] [4.08] slopeNO 0.85 2.83SE 0.05 -0.32UK 0.04 -0.40 t-value [1.06] [7.03] [0.09] [-0.47] [0.19] [-1.05] curvature -0.91 -2.24 -2.29 5.65 0.21 -0.61 t-value [-0.95] [-1.57] [1.58] [3.66] [0.20] [-0.77] R2 0.50 0.49 0.41 0.41 0.41 0.30 0.35 0.35 0.30 CP Factor 0.23 0.31 0.18 0.30 0.25 0.31 t-value [1.08] [8.50] [1.36] [8.89] [2.84] [13.95] level 0.16 0.66 0.37 0.85 0.07 0.33 t-value [0.36] [7.86] [1.11] [8.00] [0.57] [6.30] slopeAUS 0.19 0.37NZ 0.03 -0.26JP -0.10 -0.80 t-value [0.21] [0.35] [0.06] [-0.45] [-0.24] [-3.04] curvature -0.67 -1.55 -0.23 -0.58 -0.90 -3.55 t-value [-0.37] [-1.18] [-0.15] [-0.36] [-1.35] [-2.84] R2 0.45 0.45 0.44 0.52 0.51 0.51 0.61 0.60 0.56 CP Factor 0.35 0.30 t-value [1.84] [8.30] level -0.06 0.40 t-value [-0.21] [5.37] slopeG9 0.13 -0.49 t-value [0.15] [-0.68] curvature -1.35 4.12 t-value [-0.48] [3.85] R2 0.41 0.41 0.33

Notes: This table compares the coefficients, t-value and R2 for the three regressions whose dependent variables are the average bond excess returns across 5 maturities (1 year to 5 years) for each country i, and independent variables are (1) CP factor, level, slope, and curvature factors extracted from forward interest rate curves; (2) CP factor alone; and (3) forward factors alone. Same comparison is also made between CP factor and yield factors. Since a forward curve is a linear transformation from the corresponding yield curve, the results are very similar.

31 5 Empirical Results

In theory, any risk factor affecting the prices of sovereign bonds potentially is an im- portant part of the pricing kernel, and has the ability to predict changes in exchange rates and the currency excess returns in between two countries. As demonstrated in the Data section, CP factor alone can predict the bond risk premia with a higher R2 than the three yield factors combined for all G10 countries, and it is not spanned by level, slope and curvature. Hence, the main estimation results discussed in the paper focus on the augmented two-country term structure model with CP factor,

10 i.e., MCP .

5.1 Parameter Estimates and Fit of the Model

The full set of parameter estimates from fitting this two-country term structure model for each of the foreign-domestic country pair, i.e., the nine non-US G10 countries and the artificial country/portfolio “G9” vis-à-vis the US, is reported in the appendix

(Table B.1, B.2 and B.3). All parameters are annualized. For all countries, the first- order autocorrelation coefficients Φi on the diagonal are close to one and statistically significant for all risk factors. The level factor is the most persistent over time, with its autocorrelation above 0.96 for all countries. The CP factor is of a much less persistent process compared to the yield factors, with its autocorrelation coefficient around 0.7-0.8 for most countries, and at its lowest at 0.5 for New Zealand. The off-diagonal elements are all close to zero and insignificant. The lower triangular

Cholesky factorization of the estimated variance-covariance matrix for each country

P,i are close to that from an unrestricted VAR, which is used as the initial guess for ΣZ in estimation. Finally, the last columns in Table B.1, B.2 and B.3 show the eigenvalues

10 Estimation results for the fully-spanned model, M3F , are available upon request.

32 of the transition matrix under risk-neutral probability Q. The eigenvalues are ranked from high to low, and are all close to 1. It suggests that shocks to the 3 yield factors are permanent over time.

For a two-country model, yields and changes in exchange rate are fitted simulta- neously for each country pair. Table 3 summarizes the fit of the yield curves for each

Table 3: Model MCP : Fit of the Yield Curves

Country RMSE

Canada 0.2611 Switzerland 0.1334 Germany 0.0855 Norway 0.095 Sweden 0.0816 UK 0.1542 Australia 0.2902 New Zealand 0.1389 Japan 0.0517 G9 0.0787

Notes: This table shows the root mean square fitting error for each of the foreign country i, which is the square root of the average squared difference between actual yields and the fitted yields from the estimated two-country term structure model in annualized percentage points. foreign country i. The table lists out the root mean square error (RMSE) of yields for each non-US G10 country, and the average country portfolio G9. The average

RMSE is around 10 basis points. The best yield fit is for Japan-US country pair, whose RMSE is 0.05. The worst fit is for Canada and Australia, whose RMSE are above 0.2. Compared to other affine term structure models in the literature that focus on fitting the yields only, the fitting for the yield curves from the two-country model here is competitive for most of the G10 countries. The fitting for Canada’s and Australia’s yields seems to be slightly compromised for the goodness of fit of the changes in exchange rates. Table 4 reports Wald statistics of the test for the rele- vance of CP factor under P measure. The null hypothesis is that lags of CP factor

33 Table 4: Relevance of Augmented CP Factor under P Probability Measure

Country Chi-sq

Canada 963.11 Switzerland 819.18 Germany 483.93 Norway 299.34 Sweden 377.65 UK 1560.30 Australia 302.31 New Zealand 275.63 Japan 451.39 G9 798.64

Notes: This table shows the Wald statistics of the test whose null hypothesis is that all elements of the transition i matrix ΦZ for country i that represents the effect of CP factor on yields and changes in exchange rates are jointly zero. does not contribute in any way to the evolution of yields. It is strongly rejected for all countries, confirming that CP factor does help to predict future interest rates, and should be included in the state vector as in model MCP . Meanwhile, to see the goodness of fit for changes in exchange rates, the model implied 1-month (log) changes in exchange rates are plotted against the observed data in Figure 2 and 3 . For 1-month series, the trend of model-implied data follows the observed data in general, however the volatility of the observed data is much higher. For fitted changes in exchange rates over longer horizons, as the horizon becomes longer, the volatility also starts to match up more with the observed data, and the fitting error gets smaller. This is clear from Table 5, where the observed k-month changes in exchange rates of each country pair (country i vis-à-vis the US) are projected onto the data implied by model MCP : obs,i i i fit,i i i ∆st →t+k = α + β ∆st→t+k(Θ ) + ηt+k, (5.1) for the horizon k (in month) of 1 month to 5 years. Adjusted R2 for the 1-month horizon is around 5% on average, with the highest at 6.1% for UK, and the lowest

34 for Sweden and Japan at around 2%. The coefficient β is close to 1, and significant at at least 95% confidence level for all countries. As the horizon gets longer, there is a general increase in R2. At the 5-year horizon, the highest R2 goes up to 93% for Australia, followed by Canada and New Zealand at 83%. Sweden and Japan are the two exceptions with lower values for R2, estimated coefficient β, and significance level for the goodness of fit. This behavior is not model specific to MCP for these two countries. For example, when the widely used model M3F in the literature is estimated with the same data – omitting CP factor and employing 3 yield principal components as the state variables – adjusted R2 for measuring the goodness of fit of changes in exchange rates is either worse than, or comparable to the results from MCP

2 in Table 5, for all countries at all horizons. With model M3F , Sweden’s adjusted R goes from 2.9% for 1-month horizon to 8% for 5-year horizon, and Japan’s R2 ranges from 1.2% from 1-month horizon to 15.6% for 5-year horizon. This is very close to the results from model MCP . Meanwhile, for the other 7 countries, MCP always offers a higher R2 on the short end. As a lot of earlier researches have pointed out, the exchange rate fluctuations at shorter horizons are much more difficult to predict than at longer horizons. Therefore, an improvement in R2 on the short end from model

MCP suggests that information carried by CP factor from the bond market helps predict the movements in the exchange rates better. Finally, to account for small- sample bias, over-lapping bias and potential correlation and heteroskedasticity in the error term, robust (White), Hansen-Hodrick and Newey-West (reported) standard errors are estimated. Newey-West standard errors are more conservative overall. I apply Newey-West correction to the standard errors with the optimal number of lags following Andrew (1991), which is the horizon in month k + 1. The statistical significance for the coefficient β at 1-month horizon is at least at 95% confidence

35 level for all countries, regardless of the method used to estimate standard errors.

5.2 Uncovered Interest Parity and Forward Premium Puzzle

Uncovered interest parity (UIP) is a parity condition stating that the difference in interest rates between two countries is equal to the expected change in exchange rates between the countries’ currencies, if there is no arbitrage. When domestic interest rate increases, and rational expectations hold, the domestic currency has to depreciate to eliminate any arbitrage opportunities. Therefore, for a domestic and foreign (i) country pair, the regression

(k) (k),i i ∆st →t+k = αUIP + βUIP (yt − yt ) + ηt+k (5.2)

jointly tests the null hypothesis of UIP and rational expectations: H0 : αUIP =

0, βUIP = 1. However, as discussed at great length in the literature, the uncovered interest rate parity usually fails. Not only the coefficient is far from 1, it is even negative. The explanation power of the interest rate differential is very low as well.

The survey by Froot and Thaler (1990), for instance, finds an average estimate for

βUIP of -0.88. Chinn and Meredith (2004) document that this result holds for more recent periods extending up to 2000.

The forward premium puzzle is closely related to the failure of uncovered interest

k,i parity. In theory, the forward rate Ft at time t to buy 1 unit of currency i at

i time t + k should be the same as the expected spot exchange rate Et(St+k), given rational expectations. Hence, by the same logic behind UIP, covered interest parity

(CIP) states that forward discount (in log terms) is equal to the interest rate (yield)

k,i i (k) (k),i differential between the two countries, ft − st = yt − yt . CIP is empirically confirmed for most countries in the literature for the sample period in my study11.

11Akram, Rime and Sarno (2008) study high frequency deviations from covered interest parity

36 Therefore, the forward discount and hence the interest rate differential, should be an unbiased predictor of the ex post change in the spot rate. The puzzle is the

finding that the forward premium usually points in the wrong direction for the ex post movement in the spot rate.

In this section, I examine if the two-country term structure model with CP factor can generate exchange rates that reproduce the same puzzles found in the historical data. In Table 6, for columns numbered “(1)”, a standard UIP test as in (5.2) is conducted for 9 G10 countries vis-à-vis the US for horizons from 1 month to 2 years12, where the dependent variables are the observed (log) changes in exchange

k,i i rate, and the independent variables are observed forward discounts (ft − st) for (k),US (k),i k = 1 month, and interest rate differentials (yt − yt ) for k = 3 months and upwards. For the 1-month horizon, the coefficient βUIP is negative for all countries except the UK. It is close to -2 for Japan and significant, adjusted for serial correlation and heteroskedasticity. The average βUIP across the 9 G10 countries for a 1-month horizon is -0.75, close to Froot and Thaler ’s (1990) findings in the global market.

As the horizon approaches 2 years, the UIP puzzle is alleviated for Canada and

UK, but not the others. The explanation power of the interest rate differential remains extremely low at all horizons, with Japan’s adjusted R2 being the highest cross-sectionally at 0.265 for a 2-year horizon. Columns numbered “(2)” in Table

6 replicate the UIP puzzle with the model-implied interest rate differentials from estimating MCP : 1  ∆sobs,i = αi + βi Σk rUS − ri  + ηi , (5.3) t →t+k MCP MCP k j=1 t+j−1 t+j−1 t+k

(CIP) , and find that CIP holds at daily and lower frequencies, except for the extreme episodes of the financial crisis in the fall of 2008. Du, Tepper and Verdelhan (2017) find deviations from CIP have not disappeared for G10 countries since the financial crisis in 2008. 12Results for longer horizons, up to 5 years, are also available. As the horizon approaches 5 years, the UIP puzzle is slightly alleviated in general.

37 where the dependent variable is again the observed change in exchange rate, and the independent variable is the sum of model-implied short term interest rate differentials between the domestic country (US) and the foreign country i, for k periods from time t. Note the independent variable for regression (5.3) is not exactly the same as that for standard UIP in (5.2) for longer horizons (k > 1 month). The standard UIP uses yield differentials from bonds of k - period to maturity to predict k periods ahead exchange rate movements, while the interest rate differential from the term- structure model comes from the mapping of SDFs. In addition, the sum of short term interest rate differentials over time is different from long-term yield differentials, given the existence of the yield term premium. Therefore, I restrict the comparisons for horizons up to 2 years here such that the yield premium remains negligible. The idea is to evaluate if the interest rate differentials implied by estimating the term structure model MCP can reproduce the infamous UIP/forward premium puzzle. Comparing the reported statistics in columns (1) and (2) in Table 6, the signs on β match between the real and the model-implied UIP regressions for all countries and all horizons up to 2 years, which confirms that the model truly can reproduce the UIP/forward premium puzzle. However, the model-implied regression generates a more negative and more significant coefficient for 7 out of 9 countries, especially for horizons on the short end. UK and Canada are two exceptions. For the sample period, UK’s interest rate differential coefficients for all horizons stay positive, though not significant.

Canada has positive coefficients from the 3-month horizon and upwards. In both cases, model-implied interest differentials produce similar coefficients.

38 5.3 FX Risk Premia

There are several reasons why the UIP puzzle might still exist when capital is per- fectly mobile, given that the covered interest parity holds. Intensive research has been devoted to study if rational expectation hypothesis is invalid, and issues of econometric implementation. But perhaps the most natural explanation for why the forward premium predicts the wrong direction of spot exchange movements is that a risk premium drives a wedge between expected changes and actual changes. How to model the risk premium is the challenge13. Assuming no-arbitrage, a pricing kernel captures the effect of systematic factors which determine the prices of all securities in the economy. Yield factors and CP factor are shown to be able to capture bond market risk premia well. Since the FX risk premium is determined by the differences in conditional volatilities of risk factors affecting the pricing kernels in each country, the risk factors that help predict local bond risk premia should theoretically predict

FX risk premia as well. I empirically test this hypothesis for the two-country term structure model MCP here.

So far the evidence suggests that estimating model MCP helps to generate yields and exchange rates that fit closely to the data, which also replicate the UIP/forward premium puzzle observed in the foreign exchange market. According to the model setup (see Equation (2.24) and (2.25)), the change in exchange rates comprises three parts: 1) the interest rate differentials between the two countries; 2) the FX risk premia; and 3) the difference in the shocks to the investors’ valuation of the risks between the two countries. Fama (1984) proposes that a time-varying risk premium contributes to the deviation from UIP. Equipped with the two-country term structure model estimation, I can numerically decompose the exchange rate movements to

13Engel (1996) provides a survey on the topic.

39 extract the risk premia, which is the second component in (2.24) and (2.25). The risk premia term is constructed non-linearly with prices of the risk evaluated for all four risk factors that help to capture bond market risk premia. Table 7 compares the standard UIP test and the model-implied regression, which includes the estimated risk premium term extracted from the bond markets, in addition to the interest rate differential term, to explain the observed exchange rate movements:

1  ∆sobs,i = αi + βi Σk rUS − ri  t →t+k MCP IRD j=1 t+j−1 t+j−1 k (5.4)   1  0 0  + βi Σk ΛUS ΛUS − Λi Λi + ηi , RP 2k j=1 t+j−1 t+j−1 t+j−1 t+j−1 t+k with the horizon k ranging from 1 month up to 1 year14. According to the model

i implication, both interest rate differential coefficient βIRD and the risk premia coeffi-

i cient βRP should be 1. This test helps to evaluate to what extent, the risk premium implied by model MCP , can help solve the UIP/forward premium puzzle. Three observations are worth highlighting when the model-implied risk premium term is incorporated. First, the coefficient βIRD on the interest rate differential term is either corrected to the right sign (from negative to positive), or become less negative and less significant across the countries for the 1-month horizon. As the horizon approaches 1 year, the coefficient is close to 1 for Canada, Switzerland,

Norway, UK and Australia; Germany and New Zealand have βIRD that is close to zero, negative yet insignificant; Sweden and Japan have negative coefficients close to

-2, but Japan is the only country whose negative interest rate differential coefficient remains significant. This coincides with the finding in Table 5, where the model

fitting of the exchange rates, measured by R2, is the lowest for Sweden and Japan

14Results for higher horizons (up to 5 years) are also available but not displayed in the paper to save space. The results for longer horizons do not deviate from the conclusions discussed in the paper.

40 for this sample period. Second, the model-implied risk premium clearly captures most of the exchange rate fluctuations. Coefficient on the risk premium term βRP is positive and significant for all countries for the 1-month horizon, and it quickly approaches 1 as suggested by the model when the horizon becomes longer, again, with the exception of Sweden and Japan. Finally, the adjusted R2 increases significantly when the model-implied risk premium is included. The explanation power for the exchange rate movements improves by up to 50.4% (from 0.3% to 50.7% for Norway) at 1-year horizon. When the horizon goes up to 5 years (results not shown to save space), the adjusted R2 increases to over 90% for Australia and New Zealand, over

80% for Canada and Norway, 64% for Japan, around 50% for Switzerland, Germany, and UK, and 26% for Sweden, while the average R2 from standard UIP remains under

10%. Therefore, the risk premium implied by the SDF dynamics is a key driver of exchange rate fluctuations, which is consistent with the explanation of the UIP puzzle by Fama (1984) and Hansen and Hodrick (1983). It is important to emphasize that the innovations in the model setup are only for the risk factors extracted from the bond markets. There is no random shock to drive the changes in exchange rate in the

FX market. The findings in Table 7 therefore suggest that two countries’ yield curves and local bond market risk premia contain a large portion of information regarding the exchange rate movements.

Given the success in improving the explanation power for exchange rate move- ments, the natural next step is to study the currency excess returns in the foreign exchange market with the help of the two-country term structure model estimation.

The 1-period currency excess return is the return, in excess of the risk free rate, from investing in a foreign currency denominated bond for 1 period, and converting the proceeds back to dollars at the end of the period. Hence, the 1-period currency ex-

41 cess return is the sum of forward discount, or interest rate differential (when Covered

Interest Parity holds), and the change in exchange rate over the period. If Uncov- ered Interest Parity holds, the gain from a higher interest rate should be exactly canceled out by the depreciation of the currency, and hence the expected currency excess return goes to zero. However, as documented by a large body of research, currency anomalies prevail in the FX market, such as carry trade and dollar carry trade. The carry trade anomaly refers to the fact that lending in currencies that have high interest rates while borrowing in currencies that have low interest rates is a profitable trading strategy. The same is true for “dollar carry trade” anomaly, a strategy where investors go long in all foreign currencies when the world average interest rate is higher than the US interest rate, and short all foreign currencies when the average interest rate is lower than the US rate. To study what contributes to the excess returns realized in these profitable trading strategies, I first project ob- served currency excess returns for each country pair on the interest rate differential as suggested by the standard carry trade strategy:

obs,i i i  (k),i (k),US i cert →t+k = αcarry + βcarry yt − yt + µt+k, (5.5)

and compare the results to those from the model MCP implied regression as in (2.30):

  cerobs,i = αi + βi tp(k),i − tp(k),US t →t+k MCP tp t t   (5.6) 1  0 0  + βi Σk ΛUS ΛUS − Λi Λi + µi , RP 2k j=1 t+j−1 t+j−1 t+j−1 t+j−1 t+k

(k),i (k),i 1 k i where the yield term premium tpt = yt − k Σj=1rt+j−1, for k = 1, 3, ..., 60 months. For horizons k > 1, the model-implied regression is based on the idea that a currency trade over k periods in time entails two layers of risks: the interest rate uncertainty and the foreign exchange risks. Currency excess return therefore should be the required compensations for bearing these two risks combined. For k = 1

42 month, the yield and the short-term interest rate coincide, therefore the yield term premium term drops out. It follows that for the short-term carry trade, currency excess returns should only be affected by risk factors non-linearly through the FX risk premium term.

The importance of the risk premium extracted from the bond markets in ex- plaining movements of exchange rates suggests its potential to account for currency excess return fluctuations. In Table 8, I compare the effect on the currency ex- cess returns from interest rate differentials and the estimated risk premia extracted from the bond markets. Standard carry trade strategy implies that currency returns between country i and the domestic country covary positivley with the difference between country i’s interest rates and the domestic interest rates. Columns labeled

“Carry” in Table 8 report the coefficient estimates, adjusted R2, and standard errors robust to serial correlation, heteroskedasticity and over-lapping bias, for such a rela- tion in the sample period I study. The results confirm the findings in the literature that higher interest rates are usually associated with higher currency excess returns,

i especially on the short end of the horizon. βcarry is positive for all countries at a 1-month horizon. Canada, Sweden, Australia and Japan have a coefficient close to 2 in value, and statistically significant. This result agrees with the fact that Canadian and Australian dollars are known to be the investment currencies in the carry trade, while Japanese Yen and Swedish Krona are the funding currencies. However, R2 is extremely low for all countries at the short end of the horizon, ranging from 0% for

UK and New Zealand to 2.9% for Japan and Sweden. This suggests that most of the variations in 1-month currency excess returns are not accounted for by the change in forward discount, or interest rate differentials. Columns labeled “RP” in Table

8 report the estimates from regressing the observed currency excess returns on the

43 model-implied risk premia. They exhibit higher adjusted R2 estimates for all coun- tries at a 1-month horizon, ranging from 3% for Japan to 7.3% for Germany. The adjusted R2 grows substantially as the horizon becomes longer for the model-implied risk premia regression among most countries, with Japan and Sweden as two excep- tions again. At a 5-year horizon, risk premia extracted from bond markets account for around 90% of the currency return variations for Australia and New Zealand, but

i only around 10%-20% for Japan and Sweden. The risk premia coefficient βRP is close to 1 and significant at 99% confidence level at a 1-month horizon for all countries.

i In fact, βRP remains significant and close to 1 throughout all length of the horizon cross-sectionally, just as how the model suggests.

Table 9 and Table 10 incorporate the yield term premium for interest rate risks, in addition to FX risk premia, as outlined by the model. The yield term premium is the difference between long-run yields and the average expected future short-term interest rate. Hence, for shorter horizons, this term would be small and negligible, but it becomes more important as the horizon gets longer. The regression results agree with such intuition. The coefficient associated with the yield term premium,

i βtp, is not significant for almost all countries when the length of the horizon is under 2 years. At longer end of the horizon, it starts to gain importance for about two-thirds of the G10 countries. Notice that there is no results shown here for the 1-month horizon, since the yield term premium is zero for the shortest horizon, and the model-implied regression for a 1-month horizon is estimated and shown in column

(2) of Table 8, where the risk premia coefficient is highly significant and delivers a higher R2 than the interest rate differentials in explaining the currency excess

i return variations. For horizons above 1 month, the risk premia coefficient βRP is mostly close to 1 and significant for all countries, as the model suggests. However,

44 again, Sweden displays a different pattern from every other country in this study,

2 whose βRP becomes insignificant beyond a 1-year horizon. R of the model-implied regression for Sweden also remains low (below 20%) throughout the horizons, while it goes up to around 90% for Canada, Norway, Australia and New Zealand at a

5-year horizon. This is consistent with the earlier findings that the model-implied risk premia extracted from the bond markets can not well capture the variations in

Swedish Krona. For Japanese Yen, the risk premia coefficient is significant and varies between 0.5 and 0.8, but the adjusted R2 is the second lowest among 9 countries, especially on the short end of the horizon. It is also the only currency whose excess returns can be better explained by interest rate differentials alone.

Overall, the two-country term structure model MCP is able to explain the varia- tions in the foreign exchange market with information extracted from bond markets for most G10 currencies. It works especially well for countries featured in the long leg of carry trade strategies, i.e. Australia, New Zealand, Canada and Norway. These countries’ currencies are known as the investment currencies in the FX market. On the other hand, the variations of the funding currencies in carry trade strategies, i.e. Japanese Yen and Swedish Krona, cannot be well explained by the risk premia constructed from the bond market information. Such finding points to heterogene- ity in risk exposure between the investment and the funding currencies, which is consistent with the empirical investigation by Lustig et al. (2011) and Colacito,

Croce, Gavazzoni and Ready (2015). The empirical results from this paper confirm a close integration of the bond market risk premia and the FX market risk premia among investment currency countries, and also demonstrate a disconnection between the two markets among funding currency countries. What fundamentals drive such heterogeneity across countries are open for further research.

45 Figure 2: Model MCP : Fit of 1-month Change in Exchange Rates

(a) CA (b) CH (c) DE

(d) NO (e) SE (f) UK

(g) AUS (h) NZ (i) JP

(j) G9 Notes: Each figure plots the model implied 1-month (log) change in exchange rate data compared to the observed counterpart for each non-US G10 country. G9 refers to the artificial country/bond portfolio that has the average of nine non-US G10 countries’ yields. The data plotted is monthly but annualized.

46 Figure 3: Model MCP : Fit of 1-month Change in Exchange Rates

(a) CA (b) CH (c) DE

(d) NO (e) SE (f) UK

(g) AUS (h) NZ (i) JP Notes: Each figure plots the model implied 1-month (log) change in exchange rate data compared to the observed counterpart for each non-US G10 country in scatter plots. On the y-axis are the model fitted (log) changes in exchange rate for each country, and on the x-axis are the observed (log) changes in exhcnage rates. The data plotted is monthly but annualized.

47 Table 5: Model MCP : Goodness of Fit for Changes in Exchange Rates

(log) Change in Exchange Rate Country 1 month 3 month 6 month 9 month 1 year 2 year 3 year 4 year 5 year CA β 0.755*** 0.805*** 0.834*** 0.859*** 0.876*** 1.044*** 1.142*** 1.111*** 1.092*** s.e. (0.221) (0.210) (0.191) (0.173) (0.169) (0.188) (0.198) (0.160) (0.122) R2 0.047 0.133 0.226 0.309 0.364 0.554 0.738 0.804 0.829 CH β 0.834*** 0.985*** 0.769*** 0.691** 0.848*** 1.003*** 1.069*** 1.285*** 1.453*** s.e. (0.247) (0.235) (0.256) (0.302) (0.310) (0.284) (0.344) (0.390) (0.300) R2 0.048 0.120 0.094 0.097 0.158 0.237 0.296 0.423 0.458 DE β 0.872*** 0.934*** 0.700*** 0.591** 0.652** 0.567 0.817** 1.173*** 1.384*** s.e. (0.225) (0.224) (0.242) (0.263) (0.290) (0.463) (0.410) (0.265) (0.190) R2 0.058 0.129 0.101 0.085 0.105 0.060 0.132 0.291 0.351 NO β 0.711** 0.783*** 0.793*** 0.857*** 0.910*** 0.936*** 0.826*** 0.514** 0.388 s.e. (0.285) (0.211) (0.158) (0.142) (0.127) (0.113) (0.170) (0.234) (0.320) R2 0.043 0.156 0.298 0.399 0.500 0.639 0.552 0.218 0.078 SE β 0.511** 0.471* 0.470* 0.475 0.454 0.429 0.336 0.279 0.420** s.e. (0.239) (0.248) (0.277) (0.300) (0.308) (0.278) (0.242) (0.184) (0.190) R2 0.021 0.047 0.080 0.098 0.098 0.096 0.061 0.039 0.070 UK β 0.869*** 1.023*** 0.878*** 0.738*** 0.675*** 0.609*** 0.549*** 0.498*** 0.458*** s.e. (0.243) (0.280) (0.266) (0.241) (0.239) (0.195) (0.177) (0.141) (0.115) R2 0.061 0.200 0.236 0.237 0.248 0.295 0.343 0.388 0.429 AUS β 0.816*** 0.974*** 1.072*** 1.186*** 1.264*** 1.335*** 1.264*** 1.210*** 1.246*** s.e. (0.196) (0.201) (0.188) (0.176) (0.172) (0.159) (0.120) (0.105) (0.089) R2 0.051 0.189 0.358 0.490 0.587 0.760 0.850 0.898 0.934 NZ β 0.713*** 0.812*** 0.787*** 0.810*** 0.867*** 1.042*** 1.113*** 1.119*** 1.099*** s.e. (0.199) (0.199) (0.181) (0.187) (0.198) (0.186) (0.149) (0.123) (0.096) R2 0.047 0.155 0.221 0.271 0.326 0.552 0.718 0.805 0.832 JP β 0.476** 0.493** 0.368* 0.281 0.266 0.312 0.447 0.402** 0.324*** s.e. (0.210) (0.215) (0.211) (0.231) (0.266) (0.343) (0.302) (0.200) (0.122) R2 0.017 0.050 0.041 0.034 0.034 0.054 0.139 0.172 0.154 G9 β 0.690*** 0.769*** 0.656*** 0.612*** 0.649*** 0.813*** 0.797*** 0.795*** 0.827*** s.e. (0.203) (0.189) (0.173) (0.193) (0.203) (0.180) (0.167) (0.140) (0.120) R2 0.048 0.133 0.163 0.186 0.229 0.415 0.520 0.601 0.658

obs,i i i fit,i i i Notes: In-sample regression coefficient estimates for ∆st→t+k = α + β ∆st→t+k(Θ ) + ηt+k, k =1 month, ..., 60 obs,i months. ∆st→t+k is the observed (log) change in exchange rate in USD between time t and t + k for country pair i fit,i i vis-à-vis the US. ∆st→t+k(Θ ) is the model implied change in exchange rate, i.e., the difference between country i’s and the domestic (US) SDF. Standard errors in parentheses (s.e.) are Newey-West standard errors, with lags equal to k + 1. The significance level of the coefficients is marked with * if p<0.10, ** if p<0.05, and *** if p<0.01. R2 are adjusted R2. G9 is the equal-weighted portfolio of currencies formed by nine non-US G10 country’s currencies, therefore its exchange rate at time t in USD is the average of the 9 non-US countries’ exchange rates at time t.

48 Table 6: Model MCP : Replication of UIP

(log) Change in Exchange Rate (observed) 1 month 6 months 1 year 2 years Country (1) (2) (1) (2) (1) (2) (1) (2) CA β -0.933 -0.167 0.059 0.349 0.228 0.481 0.333 0.788 s.e. (0.827) (0.581) (0.661) (0.627) (0.843) (0.770) (0.990) (0.941) R2 -0.000 -0.004 -0.004 -0.000 -0.003 0.009 0.002 0.039 CH β -0.485 -1.479 -0.795 -1.151 -1.006 -0.893 -1.818* -0.998 s.e. (1.252) (1.449) (0.956) (0.923) (0.958) (0.787) (0.941) (0.887) R2 -0.003 0.004 0.008 0.025 0.032 0.034 0.154 0.081 DE β -0.645 -1.100 -0.696 -1.057 -1.105 -1.097 -1.450 -1.205 s.e. (1.177) (1.314) (1.007) (0.969) (1.097) (0.946) (1.097) (0.956) R2 -0.002 0.001 0.005 0.021 0.038 0.049 0.100 0.104 NO β -0.451 -1.408 -0.237 -1.330 -0.633 -1.364 -1.531 -1.037 s.e. (1.460) (1.467) (1.329) (1.173) (1.382) (1.082) (0.986) (0.997) R2 -0.007 -0.000 -0.007 0.038 0.003 0.067 0.112 0.060 SE β -1.206 -2.182* -0.724 -2.102* -1.387 -2.489** -1.330 -2.556** s.e. (0.845) (1.249) (1.234) (1.131) (1.149) (1.087) (0.984) (1.188) R2 0.005 0.017 0.004 0.106 0.051 0.231 0.078 0.301 UK β 0.325 0.439 1.312 0.607 1.476 0.964 0.889 0.850 s.e. (1.747) (1.835) (1.585) (1.350) (1.057) (0.793) (0.925) (0.907) R2 -0.004 -0.004 0.021 0.003 0.054 0.036 0.034 0.043 AUS β -1.347 -2.738** -0.614 -2.598** -0.908 -2.752* -0.966 -3.084* s.e. (1.274) (1.152) (1.664) (1.219) (1.438) (1.468) (1.212) (1.685) R2 0.000 0.020 -0.000 0.106 0.012 0.164 0.025 0.262 NZ β -0.152 -2.216* -0.728 -2.412* -1.066 -3.068** -1.531 -4.369*** s.e. (1.762) (1.314) (2.119) (1.226) (2.167) (1.357) (1.816) (1.170) R2 -0.004 0.008 -0.001 0.057 0.008 0.119 0.030 0.285 JP β -1.858* -2.279** -2.488*** -2.428*** -2.641*** -2.367*** -2.365*** -2.358*** s.e. (0.985) (1.096) (0.796) (0.814) (0.716) (0.715) (0.728) (0.701) R2 0.010 0.013 0.103 0.101 0.246 0.225 0.265 0.343

Notes: The columns numbered with “(1)” report the estimated coefficient, standard error, and adjusted R2 for the obs,i i i (k),US (k),i i UIP regression with observed exchange rate and yield data: ∆st→t+k = αUIP + βUIP (yt − yt ) + ηt+k, k =1 month, ..., 60 months. Columns numbered “(2)” report the same statistics for the UIP replication regression obs,i i i h 1 k  US i i i with model MCP implied data: ∆s = α + β Σ r − r + η ,k =1 month, ..., 60 t→t+k MCP MCP k j=1 t+j−1 t+j−1 t+k obs,i months. ∆st→t+k is the observed (log) change in exchange rate in USD between time t and t + k for country pair i (k),i vis-à-vis the US. yt is the annualized yield for country i with k-month to maturity. Only 1 month, 6 months, 12 k  US i  months and 24 months results are reported here to save space. Σj=1 rt+j−1 − rt+j−1 is the sum of model-implied short term interest rate differentials between the US and country i, for k periods ahead from time t. Standard errors in parentheses (s.e.) are Newey-West standard errors, with lags equal to k + 1. The significance level of the coefficients is marked with * if p<0.10, ** if p<0.05, and *** if p<0.01. All variables are annualized.

49 Table 7: Model MCP : Decomposition of Change in Exchange Rate

(log) Change in Exchange Rate (observed) 1 month 6 months 1 year (1) (2) (1) (2) (1) (2) Country UIP IRD RP UIP IRD RP UIP IRD RP CA β -0.933 0.516 0.757*** 0.059 1.215** 0.835*** 0.228 1.375** 0.878*** s.e. (0.827) (0.627) (0.221) (0.661) (0.552) (0.189) (0.843) (0.588) (0.160) R2 -0.000 0.043 -0.004 0.227 -0.003 0.376 CH β -0.485 0.113 0.797*** -0.795 0.392 0.719** -1.006 1.025 0.883*** s.e. (1.252) (1.513) (0.266) (0.956) (1.153) (0.283) (0.958) (0.954) (0.340) R2 -0.003 0.045 0.008 0.091 0.032 0.155 DE β -0.645 0.247 0.853*** -0.696 0.039 0.656*** -1.105 -0.136 0.567* s.e. (1.177) (1.293) (0.233) (1.007) (1.123) (0.251) (1.097) (1.135) (0.292) R2 -0.002 0.055 0.005 0105 0.038 0.121 NO β -0.451 0.412 0.693** -0.237 0.889 0.802*** -0.633 1.580** 0.988*** s.e. (1.460) (1.395) (0.290) (1.329) (0.747) (0.152) (1.382) (0.627) (0.169) R2 -0.007 0.036 -0.007 0.292 0.003 0.507 SE β -1.206 -1.350 0.431* -0.724 -1.412 0.369 -1.387 -1.915 0.284 s.e. (0.845) (1.341) (0.241) (1.234) (1.249) (0.269) (1.149) (1.355) (0.301) R2 0.005 0.030 0.004 0.152 0.051 0.265 UK β 0.325 1.540 0.877*** 1.312 1.671 0.888*** 1.476 1.691** 0.680*** s.e. (1.747) (1.769) (0.253) (1.585) (1.312) (0.274) (1.057) (0.851) (0.225) R2 -0.004 0.058 0.021 0.244 0.054 0.290 AUS β -1.347 -0.431 0.719*** -0.614 0.690 1.033*** -0.908 0.994 1.235*** s.e. (1.274) (1.065) (0.182) (1.664) (0.850) (0.177) (1.438) (0.6676) (0.160) R2 0.000 0.051 -0.000 0.356 0.012 0.586 NZ β -0.152 -0.461 0.665** -0.728 -0.199 0.732*** -1.066 -0.464 0.780*** s.e. (1.762) (1.441) (0.214) (2.119) (1.394) (0.197) (2.167) (1.520) (0.221) R2 -0.004 0.046 -0.001 0.225 0.008 0.340 JP β -1.858* -1.505 0.419** -2.488*** -1.920** 0.251 -2.641*** -2.211*** 0.072 s.e. (0.985) (1.178) (0.200) (0.796) (0.885) (0.162) (0.716) (0.746) (0.183) R2 0.010 0.025 0.103 0.116 0.246 0.224

Notes: The columns marked (1) “UIP” report the estimated coefficient, standard error, and adjusted R2 for the UIP regression with observed change in exchange rate for the dependent, and yield differentials as the indepen- obs,i i i (k),US (k),i i dent variable: ∆st→t+k = αUIP + βUIP (yt − yt ) + ηt+k, k = 1 month, ..., 60 months. Columns obs,i numbered “(2)” report estimates for the regression based on model MCP implied decomposition: ∆st→t+k = h  i h  0 0 i αi + βi 1 Σk rUS − ri + βi 1 Σk ΛUS ΛUS − Λi Λi + ηi , k = 1 MCP IRD k j=1 t+j−1 t+j−1 RP 2k j=1 t+j−1 t+j−1 t+j−1 t+j−1 t+k obs,i month, ..., 60 months. ∆st→t+k is the observed (log) change in exchange rate in USD between time t and (k),i t + k for country pair i vis-à-vis the US. yt is the annualized yield for country i with k-month to maturity. k  US i  Σj=1 rt+j−1 − rt+j−1 , labeled by IRD, is the sum of model-implied short term interest rate differentials between 1 k  US0 US i0 i  ths US and country i, for k periods ahead from time t. 2k Σj=1 Λt+j−1Λt+j−1 − Λt+j−1Λt+j−1 , labeled by RP, is the average of k periods model-implied risk premia differentials between domestic and foreign countries. Standard errors in parentheses (s.e.) are Newey-West standard errors, with lags equal to k + 1. The significance level of the coefficients is marked with * if p<0.10, ** if p<0.05, and *** if p<0.01. Only 1 month, 6 months, and 12 months results are reported here to save space. All variables are annualized.

50 Table 8: Model MCP : Bond Risk Premia in Explaining Currency Excess Returns

Currency Excess Returns 1 month 3 month 6 month 9 month 1 year (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) Country Carry RP Carry RP Carry RP Carry RP Carry RP CA β 1.933** 0.799*** 1.060* 0.813*** 0.941 0.820*** 0.728 0.820*** 0.772 0.818*** s.e. (0.827) (0.208) (0.575) (0.194) (0.661) (0.180) (0.777) (0.170) (0.843) (0.170) R2 0.013 0.056 0.009 0.145 0.014 0.238 0.011 0.311 0.017 0.348 CH β 1.485 0.879*** 2.048 0.996*** 1.795* 0.828*** 1.898** 0.760*** 2.006** 0.831*** s.e. (1.252) (0.228) (1.266) (0.214) (0.956) (0.225) (0.923) (0.249) (0.958) (0.250) R2 0.006 0.062 0.035 0.150 0.057 0.150 0.099 0.174 0.128 0.235 DE β 1.645 0.923*** 1.490 0.965*** 1.696* 0.787*** 1.899* 0.716*** 2.105* 0.762*** s.e. (1.177) (0.200) (1.161) (0.190) (1.007) (0.209) (1.045) (0.232) (1.097) (0.245) R2 0.009 0.073 0.017 0.157 0.049 0.152 0.092 0.152 0.135 0.182 NO β 1.451 0.754*** 0.285 0.805*** 1.237 0.820*** 1.115 0.869*** 1.633 0.897*** s.e. (1.460) (0.265) (1.766) (0.195) (1.329) (0.150) (1.467) (0.129) (1.382) (0.111) R2 0.001 0.058 -0.007 0.192 0.017 0.362 0.019 0.470 0.062 0.560 SE β 2.206*** 0.617*** 1.529 0.584** 1.724 0.594** 1.982 0.605** 2.387** 0.585** s.e. (0.845) (0.225) (1.162) (0.225) (1.234) (0.246) (1.228) (0.261) (1.149) (0.265) R2 0.029 0.036 0.017 0.084 0.043 0.139 0.079 0.168 0.144 0.172 UK β 0.675 0.850*** -0.940 0.979*** -0.312 0.828*** -0.406 0.673*** -0.476 0.586*** s.e. (1.747) (0.211) (1.753) (0.244) (1.585) (0.223) (1.241) (0.195) (1.057) (0.197) R2 -0.002 0.063 0.004 0.200 -0.003 0.233 -0.001 0.226 0.002 0.221 AUS β 2.347* 0.866*** 1.589 0.988*** 1.614 1.062*** 1.538 1.153*** 1.908 1.215*** s.e. (1.274) (0.187) (1.684) (0.188) (1.664) (0.174) (1.622) (0.161) (1.438) (0.153) R2 0.009 0.070 0.010 0.231 0.023 0.411 0.032 0.540 0.066 0.631 NZ β 1.152 0.769*** 0.840 0.839*** 1.728 0.825*** 1.794 0.847*** 2.066 0.894*** s.e. (1.762) (0.185) (1.956) (0.187) (2.119) (0.172) (2.262) (0.177) (2.167) (0.186) R2 -0.002 0.061 -0.001 0.182 0.017 0.261 0.025 0.317 0.041 0.373 JP β 2.858*** 0.576*** 3.390*** 0.601*** 3.488*** 0.520** 3.708*** 0.462** 3.641*** 0.457** s.e. (0.985) (0.204) (0.985) (0.210) (0.796) (0.206) (0.738) (0.214) (0.716) (0.231) R2 0.029 0.030 0.090 0.081 0.187 0.091 0.332 0.101 0.384 0.113

Notes: The columns marked (1) and “Carry” report the estimated coefficient, standard error, and adjusted R2 for obs,i i i (k),i (k),US i the standard carry trade regression: cert→t+k = αcarry + βcarry(yt − yt ) + ηt+k, k = 1 month, ..., 60 months. Columns numbered “(2)” with label “RP” report estimates for the regression based on model MCP implied h  0 0 i risk premia: cerobs,i = αi + βi 1 Σk ΛUS ΛUS − Λi Λi + ηi , k = 1 month, ..., 60 t→t+k MCP RP 2k j=1 t+j−1 t+j−1 t+j−1 t+j−1 t+k obs,i months. Dependent variable cert→t+k is the observed currency excess return between time t and t + k for country obs,i (k),i (k),US i (k),i pair i vis-à-vis the US: cert→t+k = yt − yt + ∆st→t+k; yt is the annualized yield for country i with 1 k  US0 US i0 i  k-month to maturity; 2k Σj=1 Λt+j−1Λt+j−1 − Λt+j−1Λt+j−1 is the monthly average of k periods model-implied risk premia differentials between the domestic and the foreign country. Standard errors in parentheses (s.e.) are Newey-West standard errors, with lags equal to k + 1. The significance level of the coefficients is marked with * if p<0.10, ** if p<0.05, and *** if p<0.01. Only 1 month, 3 months, 6 months, 9 months and 12 months results are reported here to save space, but results are available up to 5-year horizon. All variables are annualized.

51 Table 9: Model MCP : Explaining Currency Excess Returns (3 Months to 1 Year)

Currency Excess Returns

3 months 6 months 9 months 1 year Country TP RP TP RP TP RP TP RP CA β 5.026* 0.786*** 2.585 0.804*** 2.554 0.833*** 3.299** 0.876*** s.e. (2.646) (0.192) (1.590) (0.168) (1.548) (0.152) (1.362) (0.143) R2 0.159 0.252 0.339 0.423 CH β -5.653 0.946*** -2.545 0.794*** 0.643 0.777*** 1.882 0.916*** s.e. (7.968) (0.206) (5.147) (0.233) (3.303) (0.277) (2.387) (0.295) R2 0.152 0.152 0.171 0.250 DE β -4.370 0.956*** 2.802 0.802*** 3.947 0.797*** 3.679 0.914*** s.e. (8.191) (0.189) (5.929) (0.214) (3.512) (0.246) (2.510) (0.272) R2 0.155 0.152 0.171 0.215 NO β -3.029 0.795*** 0.190 0.819*** 0.573 0.867*** 1.075 0.902*** s.e. (3.480) (0.192) (1.683) (0.152) (1.336) (0.130) (1.249) (0.111) R2 0.191 0.357 0.467 0.568 SE β -10.353 0.440* 0.929 0.617** 2.831 0.726** 3.532 0.804** s.e. (8.163) (0.245) (4.750) (0.247) (3.999) (0.288) (3.033) (0.310) R2 0.094 0.135 0.176 0.195 UK β -10.417 0.848*** -4.082 0.771*** -1.103 0.645*** -0.085 0.582*** s.e. (7.109) (0.216) (3.198) (0.196) (1.827) (0.180) (1.449) (0.191) R2 0.230 0.253 0.227 0.217 AUS β -1.939 0.970*** 2.337 1.089*** 1.181 1.171*** 0.768 1.231*** s.e. (5.548) (0.200) (3.218) (0.181) (2.490) (0.171) (1.793) (0.164) R2 0.228 0.411 0.539 0.630 NZ β 2.370 0.837*** 3.217 0.822*** 2.166 0.856*** 1.139 0.906*** s.e. (3.404) (0.186) (2.680) (0.169) (2.525) (0.183) (2.294) (0.198) R2 0.181 0.273 0.326 0.375 JP β 8.027 0.580*** 3.347 0.521** 3.525 0.503** 2.822 0.527** s.e. (9.238) (0.218) (6.329) (0.208) (4.158) (0.214) (2.963) (0.231) R2 0.084 0.091 0.110 0.126

Notes: This table reports the estimates for model MCP implied currency excess return decomposition regression:   h  0 0 i cerobs,i = αi +βi tp(k),i − tp(k),US +βi 1 Σk ΛUS ΛUS − Λi Λi +µi , where the t→t+k MCP tp t t RP 2k j=1 t+j−1 t+j−1 t+j−1 t+j−1 t+k yield term premium for country i and horizon k is the difference between long-term yield and short-term interest (k),i (k),i 1 k i rate:tpt = yt − k Σj=1rt+j−1, and k = 3, ..., 12 months. The columns marked “TP” report the estimates for i the yield term premium differential term, βtp. Columns labeled “RP” report estimates for the model MCP implied i obs,i risk premia coefficient βRP . Dependent variable cert→t+k is the observed currency excess return between time t and obs,i (k),i (k),US i t + k for country pair i vis-à-vis the US: cert→t+k = yt − yt + ∆st→t+k. Standard errors in parentheses (s.e.) are Newey-West standard errors, with lags equal to k + 1. The significance level of the coefficients is marked with * if p<0.10, ** if p<0.05, and *** if p<0.01. All variables are annualized.

52 Table 10: Model MCP : Explaining Currency Excess Returns (2 Years to 5 Years)

Currency Excess Returns

2 years 3 years 4 years 5 years

Country TP RP TP RP TP RP TP RP

CA β 3.095*** 1.071*** 2.776*** 1.186*** 2.845*** 1.234*** 2.717*** 1.303*** s.e. (0.506) (0.125) (0.310) (0.101) (0.189) (0.077) (0.235) (0.073) R2 0.640 0.813 0.910 0.923 CH β 2.802*** 1.221*** 3.035*** 1.485*** 3.098*** 1.827*** 1.989** 1.667*** s.e. (0.715) (0.295) (0.444) (0.281) (0.700) (0.367) (0.768) (0.415) R2 0.439 0.581 0.665 0.541 DE β 2.565** 1.123*** 2.565*** 1.452*** 2.799*** 1.879*** 3.362** 2.367*** s.e. (1.128) (0.334) (0.530) (0.207) (0.800) (0.348) (1.571) (0.720) R2 0.207 0.332 0.483 0.484 NO β 2.309*** 0.990*** 2.275*** 1.020*** 1.993*** 0.924*** 1.961*** 0.919*** s.e. (0.540) (0.083) (0.351) (0.111) (0.325) (0.183) (0.096) (0.092) R2 0.768 0.808 0.724 0.854 SE β 1.360 0.703 0.153 0.378 -1.202 -0.232 -2.413*** -0.879* s.e. (1.952) (0.491) (1.860) (0.586) (1.697) (0.646) (0.740) (0.489) R2 0.143 0.062 0.022 0.093 UK β 1.405 0.613*** 1.524*** 0.587*** 1.492** 0.573** 0.911 0.435 s.e. (0.862) (0.206) (0.460) (0.196) (0.610) (0.226) (1.088) (0.337) R2 0.253 0.280 0.258 0.189 AUS β -0.588 1.191*** -0.258 1.108*** 0.488* 1.124*** 0.892*** 1.198*** s.e. (1.096) (0.144) (0.422) (0.098) (0.287) (0.103) (0.204) (0.077) R2 0.799 0.892 0.916 0.937 NZ β -1.896 0.981*** -1.761*** 1.001*** -0.387 1.042*** 0.263 1.063*** s.e. (1.170) (0.150) (0.454) (0.092) (0.373) (0.092) (0.745) (0.067) R2 0.648 0.827 0.849 0.855 JP β 1.836 0.652** 1.516 0.786** 1.517* 0.765** 1.840* 0.776** s.e. (1.565) (0.283) (1.277) (0.364) (0.812) (0.343) (0.968) (0.361) R2 0.170 0.244 0.246 0.197

Notes: This table reports the estimates for model MCP implied currency excess return decomposition regression:   h  0 0 i cerobs,i = αi +βi tp(k),i − tp(k),US +βi 1 Σk ΛUS ΛUS − Λi Λi +µi , where the t→t+k MCP tp t t RP 2k j=1 t+j−1 t+j−1 t+j−1 t+j−1 t+k yield term premium for country i and horizon k is the difference between long-term yield and short-term interest (k),i (k),i 1 k i rate:tpt = yt − k Σj=1rt+j−1, and k = 24, 36, ..., 60 months. The columns marked “TP” report the estimates i for the yield term premium differential term, βtp. Columns labeled “RP” report estimates for the model MCP implied i obs,i risk premia coefficient βRP . Dependent variable cert→t+k is the observed currency excess return between time t and obs,i (k),i (k),US i t + k for country pair i vis-à-vis the US: cert→t+k = yt − yt + ∆st→t+k. Standard errors in parentheses (s.e.) are Newey-West standard errors, with lags equal to k + 1. The significance level of the coefficients is marked with * if p<0.10, ** if p<0.05, and *** if p<0.01. All variables are annualized.

53 6 Conclusion

This paper looks into two related markets, namely, the international bond markets and the foreign exchange market, and inspect the connection between the two. I propose an arbitrage-free two-country term structure model, which integrates the two markets by setting the exchange rate between the two countries to be the ratio of their SDFs. In order to best capture the risk evaluations in SDFs from the bond market, I first verify the existence, and then extract Cochrane and Piazzesi’s single bond excess return forecasting factor for each of the G10 countries. The results show that the factor explains more than 95% of the variations in bond excess returns for all countries, and it is not subsumed by yield curve factors. Together with level, slope and curvature, CP factor is used as the fourth risk factor in the model, and it is not linearly spanned by the yield curve factors. The time-varying risk premium is a nonlinear function of these bond market risk factors. For the sample period I study, the empirical results in the paper show that the model-implied risk premia can help explain much more variations in the exchange rate movements and currency excess returns than the interest rate differentials alone for most G10 countries. The evidence is especially strong for Canada, Norway, Australia and New Zealand, but weak for Sweden and Japan. The bond market and the foreign exchange market are therefore quite highly integrated for the investment currency countries in carry trade strategies, but not so much for the funding currency ones. The next step is to investigate the out of sample predictability of this international affine term structure model for exchange rate fluctuations and currency excess returns.

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60 A Appendix: Bond Pricing in Affine Term Struc-

ture Models

The price of an n-month to maturity zero-coupon bond is given by

(n)  0  Pt = exp An + BnyPt , (A.1) and the SDF governs the pricing of the bond from one period to the next such that

(n+1) h (n)i Pt = Et Mt+1Pt+1 , (A.2)

0 0 1 P 0 −rt− 2 ΛtΛt−ΛtyP,t+1 where Mt+1 = e , and the short term interest rate rt = ρ0 + ρ1yPt.

Given −1 Q Λt = ΣyP (Λ0 + Λ1Zt) , (A.3)

and plugging all functional forms above into (A.2), (An,Bn) solve the first-order difference equations

0 1 0 0 A − A = B µQ + B ΣQ ΣQ B − ρ , (A.4) n+1 n n 2 n yP yP n 0

0 Q Bn+1 = −ρ1 + Φ Bn, (A.5)

subject to the initial conditions A0 = 0, B0 = 0.

61 Table B.1: Parameters of Model MCP for non-US G10 Countries (Part 1)

i P,i i i Q,i Country µ Φ ΣZ ρ0 ρ1 eig(Φ ) CA level 0.5199 0.9688 -0.0346 -0.0610 -0.0020 1.1493 -0.0004 0.3003 0.9980 s.e. 0.3829 0.0103 0.0472 0.1732 0.0200 0.0187 slope 0.0037 -0.0070 0.9480 -0.2438 -0.0027 0.0589 0.4665 0.4180 0.9170 s.e. 0.1567 0.0042 0.0193 0.0709 0.0082 0.0054 0.0031 curvature 0.0376 0.0031 0.0080 0.8427 0.0018 -0.1133 0.0709 0.2017 0.5774 0.8679 s.e. 0.0806 0.0022 0.0099 0.0365 0.0042 0.0031 0.0012 0.0008 CP factor -2.1755 0.0278 -0.0508 -0.7056 0.8692 -0.5275 0.6367 -0.1048 1.6130 s.e. 0.6049 0.0163 0.0746 0.2736 0.0317 0.0217 0.0089 0.0045 0.0466 CH level 0.3284 0.9831 -0.1047 0.1008 -0.0098 0.6697 -0.0011 0.3609 0.9864 s.e. 0.4458 0.0067 0.0460 0.1311 0.0151 0.0063 slope 0.3789 0.0032 0.9464 0.0925 0.0089 0.0453 0.3107 -0.3413 0.9488 s.e. 0.2090 0.0031 0.0216 0.0615 0.0071 0.0021 0.0014 curvature 0.0103 0.0007 0.0153 0.8097 0.0009 -0.0430 -0.0771 0.1862 0.4869 0.9388 s.e. 0.1371 0.0021 0.0141 0.0403 0.0046 0.0014 0.0007 0.0006 CP factor -7.1735 0.0127 0.3172 -0.3202 0.7752 -0.0684 -1.2611 -0.4453 1.2706 s.e. 1.2287 0.0185 0.1267 0.3614 0.0416 0.0124 0.0070 0.0038 0.0482 DE level 0.2777 0.9863 -0.0030 -0.2263 -0.0071 0.6901 -0.0015 0.3117 0.9860 s.e. 0.2381 0.0067 0.0320 0.1150 0.0097 0.0067 slope -0.0351 -0.0015 0.9786 0.2139 -0.0024 0.1187 0.2866 -0.3479 0.9760 s.e. 0.1070 0.0030 0.0144 0.0517 0.0044 0.0023 0.0014 curvature -0.0159 0.0027 -0.0006 0.8451 -0.0034 -0.0920 -0.0545 0.1864 0.5796 0.8825 s.e. 0.0742 0.0021 0.0100 0.0358 0.0030 0.0016 0.0007 0.0007 CP factor -3.6794 0.0193 0.2201 -1.6141 0.7737 -0.3651 -0.9979 0.9648 2.5617 s.e. 1.0133 0.0285 0.1360 0.4895 0.0413 0.0204 0.0097 0.0069 0.1219 NO level -1.3016 0.9715 -0.0860 -0.3488 -0.0265 1.3174 0.0004 0.3892 0.9980 s.e. 1.9847 0.0242 0.1299 0.3047 0.0203 0.0245 slope 0.4039 0.0111 0.8619 -0.0660 0.0170 -0.1281 0.4421 0.4626 0.9880 s.e. 0.6933 0.0085 0.0454 0.1064 0.0071 0.0063 0.0030 curvature 0.6085 0.0045 0.0346 0.9051 0.0030 -0.0451 -0.0587 0.1459 0.4427 0.9780 s.e. 0.2465 0.0030 0.0161 0.0378 0.0025 0.0022 0.0008 0.0004 CP factor -45.6548 0.0436 -2.1692 2.1731 0.6716 -0.3592 2.7432 -1.2111 2.3189 s.e. 5.7361 0.0699 0.3753 0.8807 0.0587 0.0504 0.0216 0.0070 0.2050 SE level -0.5671 0.9836 -0.0675 -0.3168 -0.0417 0.8200 0.0002 0.2658 0.9910 s.e. 0.2705 0.0081 0.0538 0.1188 0.0094 0.0095 slope 0.1788 0.0009 0.9509 -0.2006 0.0115 -0.1393 0.2984 0.4622 0.9810 s.e. 0.1086 0.0033 0.0216 0.0477 0.0038 0.0029 0.0015 curvature 0.1244 0.0017 0.0234 0.9295 0.0048 -0.0948 -0.0279 0.1438 0.4849 0.9710 s.e. 0.0576 0.0017 0.0115 0.0253 0.0020 0.0016 0.0006 0.0004 CP factor -6.2314 0.0279 -0.9542 -1.5167 0.7817 -1.4690 1.2459 0.6129 3.1881 s.e. 1.2451 0.0374 0.2479 0.5467 0.0431 0.0332 0.0137 0.0069 0.2015 UK level -0.0299 0.9687 -0.0674 -0.0995 -0.0325 0.9351 0.0001 0.3137 0.9980 s.e. 0.4564 0.0080 0.0348 0.1203 0.0214 0.0124 slope -0.1662 -0.0016 0.9684 -0.1431 -0.0055 -0.0674 0.4056 0.4457 0.9880 s.e. 0.2007 0.0035 0.0153 0.0529 0.0094 0.0039 0.0024 curvature -0.0026 0.0044 0.0166 0.9014 0.0054 -0.1509 -0.0355 0.1532 0.5221 0.9780 s.e. 0.1064 0.0019 0.0081 0.0280 0.0050 0.0025 0.0009 0.0007 CP factor -4.0051 0.0289 -0.0142 -1.1325 0.8444 -1.1071 0.0122 0.3590 0.8950 s.e. 0.7166 0.0126 0.0547 0.1889 0.0336 0.0172 0.0061 0.0039 0.0305 AUS level -1.2996 0.9613 -0.0294 -0.0377 -0.0586 1.0237 -0.0002 0.2706 0.9878 s.e. 0.8286 0.0091 0.0451 0.1707 0.0208 0.0148 slope -0.1911 -0.0060 0.9466 -0.2715 -0.0078 -0.1380 0.3922 0.4460 0.9762 s.e. 0.3365 0.0037 0.0183 0.0693 0.0084 0.0045 0.0024 curvature 0.4982 0.0022 0.0083 0.7582 0.0138 -0.1835 -0.0528 0.1752 0.5321 0.8874 s.e. 0.2098 0.0023 0.0114 0.0432 0.0053 0.0033 0.0011 0.0009 CP factor -12.8867 0.0390 -0.1101 -1.8161 0.6871 -1.2506 0.3686 -0.4410 1.8308 s.e. 1.8540 0.0204 0.1009 0.3818 0.0465 0.0267 0.0100 0.0061 0.0742

Notes: Parameter estimates and asymptotic standard errors for country i as the foreign country, US as the domestic

i i0 i 0 P,i country, with state vector Zt = [yPt CP rest] for model MCP . All parameters are annualized. µ and ΣZ are scaled by ×100.

62 Table B.2: Parameters of Model MCP for non-US G10 Countries (Part 2)

i P,i i i Q,i Country µ Φ ΣZ ρ0 ρ1 eig(Φ ) NZ level -3.3305 0.9621 -0.0220 -0.0714 -0.0769 1.0650 0.0001 0.3029 0.9980 s.e. 1.6370 0.0116 0.0498 0.1793 0.0287 0.0160 slope -1.1756 -0.0015 0.9340 -0.1650 -0.0198 0.0253 0.5248 0.4357 0.9238 s.e. 0.8076 0.0057 0.0246 0.0884 0.0142 0.0056 0.0039 curvature 0.3167 0.0023 -0.0047 0.7323 0.0056 -0.0248 0.0642 0.2647 0.6106 0.7789 s.e. 0.4204 0.0030 0.0128 0.0460 0.0074 0.0029 0.0015 0.0011 CP factor -30.2288 0.0400 -0.2763 0.3819 0.4909 -1.2234 1.2523 -0.6838 1.0485 s.e. 3.3081 0.0234 0.1007 0.3623 0.0580 0.0264 0.0129 0.0059 0.0655 JP level 0.0590 0.9818 -0.0555 0.1181 -0.0333 0.6212 -0.0009 0.3000 0.9980 s.e. 0.1376 0.0056 0.0391 0.1694 0.0138 0.0055 slope 0.0510 0.0020 0.9631 0.1788 0.0025 0.1173 0.2508 -0.3259 0.9758 s.e. 0.0613 0.0025 0.0174 0.0755 0.0062 0.0019 0.0011 curvature 0.0187 0.0021 0.0019 0.7954 -0.0003 -0.1025 -0.0138 0.0994 0.5268 0.7966 s.e. 0.0318 0.0013 0.0090 0.0391 0.0032 0.0011 0.0004 0.0003 CP factor -1.7489 0.0270 0.2404 -3.1457 0.7088 -0.9101 -0.7584 0.2814 1.5054 s.e. 0.4289 0.0176 0.1217 0.5281 0.0431 0.0133 0.0061 0.0031 0.0530

Notes: Parameter estimates and asymptotic standard errors for country i as the foreign country, US as the domestic

i i0 i 0 P,i country, with state vector Zt = [yPt CP rest] for model MCP . All parameters are annualized. µ and ΣZ are scaled by ×100.

Table B.3: Parameters of Model MCP for Portfolio G9

i P,i i i Q,i Country µ Φ ΣZ ρ0 ρ1 eig(Φ ) G9 level 0.1371 0.9769 -0.0361 -0.3048 -0.0144 0.5999 -0.0004 0.2987 0.9801 s.e. 0.2439 0.0059 0.0382 0.1380 0.0099 0.0051 slope -0.0095 -0.0027 0.9613 -0.2389 -0.0002 -0.1168 0.2188 0.4038 0.9701 s.e. 0.1008 0.0025 0.0158 0.0570 0.0041 0.0016 0.0009 curvature -0.0281 0.0023 0.0108 0.9145 -0.0012 -0.0854 0.0052 0.0792 0.5521 0.9601 s.e. 0.0474 0.0012 0.0074 0.0268 0.0019 0.0009 0.0003 0.0002 CP factor -4.1119 0.0440 -0.0595 -1.3451 0.8250 -0.8610 0.4695 0.2971 1.7650 s.e. 0.8298 0.0202 0.1300 0.4693 0.0338 0.0133 0.0055 0.0026 0.0589

Notes: Parameter estimates and asymptotic standard errors for artificial country/portfolio “G9” as the foreign country, which takes 1/9 of the yields of each 9 non-US G10 countries across all maturities, and the US as the domestic country,

i i0 i 0 P,i with state vector Zt = [yPt CP rest] for model MCP . All parameters are annualized. µ and ΣZ are scaled by ×100.

63