QR43, Introduction to Investments Class Notes, Fall 2004 I. , Portfolios, and Arbitrage

A. Introduction

1. What are investments? You pay now, get repaid later. But can you be sure how much you will be repaid? Time and uncertainty are essential. Real investments, such as machinery, require an input of re- sources today and deliver an output of resources later. Financial investments,suchasstocksandbonds,areclaimstotheoutput produced by real investments. For example: A company issues shares of stock to investors. The proceeds of the share issue are used to build a factory and to hire workers. The factory produces , and the sales proceeds give the company revenue or sales. After paying the workers and set- ting aside to replace the machinery and buildings when they wear out (depreciation), the company is left with earnings.The company keeps some earnings to make new real investments (re- tained earnings), and pays the rest to shareholders in the form of dividends. Later, the company decides to expand production and needs to raisemoremoneytofinance its expansion. Rather than issue new shares, it decides to issue corporate bonds that promise fixed pay- ments to bondholders. The expanded production increases revenue. However, payments on the bonds, and the setting aside of

1 money to repay the principal (amortization), are subtracted from revenue. Thus earnings may increase or decrease depending on the success of the company’s expansion. Financial investments are also known as capital assets or finan- cial assets. The markets in which capital assets are traded are known as capital markets or financial markets. Capital assets are commonly divided into three broad categories:

Fixed-income securities promise to make fixed payments in the • future. Equities (stocks or shares) give their owners a share of the • profits of a company. If a company has no debt, then the equity is a direct claim on the real assets of the company. securities (derivatives) make payments that depend • on the prices of other financial assets.

The for fixed-income securities is sometimes divided into the money market (payments will be made within one year) and the capital market (at least some payments will made after one year).

2 Functions of financial markets:

Shift the timing of consumption so that it need not coincide • with the timing of income. Allocate risk (e.g. share it among many people, or concentrate • it on people who are particularly willing to bear it). Allocate society’s resources to the most productive real invest- • ments (e.g. by separating the ownership and management of companies). Aggregate the information of many market participants, thereby • revealingittoothers.

3 2. What is ? Classic definition by Lionel Robbins: Economics is the study of “the allocation of scarce resources”.

3. What is finance? Stephen Ross: “Finance is a subfield of economics distinguished by both its focus and its methodology. The primary focus of fi- nance is the workings of the capital markets and the supply and the pricing of capital assets. The methodology of finance is the use of close substitutes to financial contracts and instruments. This methodology is applied to instruments whose charac- teristics extend across time and whose payoffs depend upon the resolution of uncertainty.” (“Finance”, essay in Peter Newman, Murray Milgate, and John Eatwell eds. New Palgrave Dictionary of Economics and Finance, Stockton Press, New York, 1992.) “’s textbook on economics has the following anony- mous quote, “You can make even a parrot into a learned political –all he must learn are the two words ‘supply’ and ‘de- mand’.” By contrast,... the intuition of finance is the absence of arbitrage. To make the parrot into a learned financial economist, he only needs to learn the single word ‘arbitrage’.” (“The Interrela- tions of Finance and Economics: Theoretical Perspectives”, Amer- ican Economic Review, May 1987, 29—34.)

4 B. Returns, Portfolios, and Indexes.

1. Measuring returns over one period. When you make a financial investment, you expect to get a re- turn. How should we measure return? Consider a stock that you buy today. Suppose you buy one share for $Pt. You sell it one period later for $Pt+1. (At this stage we are not making any assumptions about the time interval. A “period” can be any length of time from a second or less to a millennium or more.)

The payoff is the sales proceeds Pt+1.Ithasunitsofdollars. We want a measure of return that does not depend on how much you invest initially. That is, we want to measure the rate of return or return per dollar invested:

Pt+1 (1 + Rt+1)= gross simple return, Pt or

Pt+1 Pt+1 Pt Rt+1 = 1= − net simple return. Pt − Pt

5 This is also the way we measure return or interest on a bank deposit. If the bank offers 3% simple interest per year, $1 invested today becomes $1.03 in a year. The gross simple return is 1.03 and the net simple return is 0.03 = 3%. Since this return is (almost) free of risk, we often write it as Rf : (1 + Rf )=1.03, Rf =0.03. Note that returns, unlike payoffs, are natural numbers. They do not have units of dollars. A limited liability asset is one whose price is never negative. If you hold a limited liability asset, the worst that can happen is that the price goes to zero, in which case the gross simple return is zero and the net simple return is 1= 100%. − − Almost all financial assets have limited liability, but exceptions include

Agreement to provide insurance, e.g. through Lloyds of London • Ownership of a company with legal liabilities, e.g. asbestos or • toxic waste problems.

6 2. Measuring returns over many periods. Whatifyouholdanassetformorethanoneperiod,sayfortwo periods? The gross cumulative return is

Pt+2 Pt+2 Pt+1 =     . Pt Pt+1 Pt     The two-period gross cumulative return is the product of two successive one-period gross returns. Returns multiply over time. This is called compounding.

If we consider a bank deposit paying interest Rf , each dollar 2 T invested is worth (1 + Rf ) after two years, and (1 + Rf ) after T years. This is called compound interest.

7 Sometimes, it is more convenient to work with a log return. The log return is the natural logarithm of the gross simple return:

Pt+1 rt+1 =log  =log(Pt+1) log(Pt). Pt −   The lower-case r denotes the log return as opposed to the simple return. Because logs convert multiplication to addition, and division to subtraction, the one-period log return is just the change in the log price. Also, the log return over 2 periods is just the sum of the 1-period log returns:

Pt+2 log   =log(Pt+2) log(Pt) Pt −   =log(Pt+2) log(Pt+1)+log(Pt+1) log(Pt) − − = rt+1 + rt+2.

The log return is always well defined for an asset with limited liability, because such an asset has a gross simple return that is positive.

8 3. Portfolios. Instead of just buying one share, you might split your money among several shares. For example, you might buy 2 shares of stock 1, each costing P1t,and3sharesofstock2,eachcostingP2t. The total cost of this portfolio is 2P1t +3P2t.

Next period your portfolio is worth 2P1,t+1 +3P2,t+1.Yourgross simple return is

2P1,t+1 +3P2,t+1 (1 + Rp,t+1)= 2P1t +3P2t 2P 3P = 1,t+1 + 2,t+1 2P1t +3P2t 2P1t +3P2t

2P1t P1,t+1 3P2t P2,t+1 =     +     2P1t +3P2t P1t 2P1t +3P2t P2t         P1,t+1 P2,t+1 = w1t   + w2t   P1t P2t    

= w1t(1 + R1,t+1)+w2t(1 + R2,t+1), where w1t is the share of your wealth invested in stock 1 at time t, and w2t =1 w1t is the share of your wealth invested in stock 2 at − time t. The portfolio return is a weighted average of the returns on the individual stocks, where the weights are the shares of wealth invested in each stock.

9 This principle holds more generally with any number of stocks in the portfolio:

(1 + Rp,t+1)=w1t(1 + R1,t+1)+...+ wnt(1 + Rn,t+1)

n = wit(1 + Ri,t+1), iX=1 n where i=1 wit =1, or with net returns, P n Rp,t+1 = witRi,t+1. iX=1 Sometimes, it is convenient to include a bank account with return Rf as the first possible investment. Then the portfolio return would be

n Rp,t+1 = w1tRf + witRi,t+1. iX=2

10 4. Stock indexes. Indexes are just portfolios that are thought to be representative of the general stock market. Each index is described by the number of shares included, the identities of the included shares, and the weights placed on each included share. Alternative weighting schemes:

Equal-weighted. wi =1/n. This requires trading every period • to rebalance the portfolio back to equal weights.

Price-weighted. wi = Pi/(P1 + ...Pn).Example:DowJones • Industrial Average.

Value-weighted. wi = Vi/(V1 +...Vn),whereVi is total market • value of company i, Vi = PiMi where Mi is the number of shares outstanding. Example: Standard and Poor’s 500 (S&P 500). Free-float-weighted. Same as value-weighted except that we • exclude from the calculation of market value any shares that are privately held or held by the government, and thus are unavailable for trading. Example: Morgan Stanley Capital International (MSCI) indexes of foreign stocks.

11 5. Dividends. So far, we have assumed that all returns come through a higher price when you sell than when you buy. That is, all returns take the form of capital gains. In practice, shares also pay dividends, so the payoff on one share sold after one period is Pt+1 + Dt+1,where Dt+1 is the dividend at time t +1that you receive as the owner of one share purchased at time t. This timing convention captures the fact that you must buy a share in advance (before the “ex-dividend date”) in order to receive the dividend. To handle this, we modify the definition of the return on a share to

Pt+1 + Dt+1 1+Rt+1 = . Pt Subtracting one to get the net return,

Pt+1 + Dt+1 Pt Pt+1 Pt Dt+1 Rt+1 = − = − + . Pt Pt Pt Thereturnisnowthesumoftwocomponents:thecapital gain or price return (Pt+1 Pt)/Pt,andthedividend-price ratio or dividend − yield Dt+1/Pt. Even with dividends, returns still compound in the same way as before if dividends are reinvested in the stock. (When you buy a mutual fund, you can arrange to reinvest all dividends automati- cally in the fund.)

12 6. Nominal vs. real returns. So far, we have measured all prices in dollars. Thus we have defined returns in nominal dollar terms. But if the purchasing power of the dollar changes over time, it is more meaningful to correct returns for changes in the value of the dollar. For this purpose we use a price index,saytheConsumerPrice Index or CPI. If the index is 100 in one year and 103 the next, this means that 103 dollars are needed in the second year to buy the same basket of goods that cost only 100 dollars in the first year. We say that the inflation rate is πt =(103 100)/100 = 0.03 = 3%.To − calculate the purchasing power of an asset in any year, we divide its dollar value by the CPI for that year. The gross real return on a stock that pays no dividends is now

P /CP I P /P t+1 t+1 = t+1 t Pt/CP It CPIt+1/CP It (1 + R ) = t+1 (1 + πt+1)

Rt+1 πt+1 =1+ − (1 + πt+1)

1+Rt+1 πt+1. ≈ −

The net real return is approximately Rt+1 πt+1, the nominal return − less the inflation rate. This approximation is quite accurate except in conditions of extremely high inflation.

13 7. Leverage. If you can borrow money from the bank as well as deposit money there, you can invest more aggressively in a stock. Suppose you start with $100. You borrow $50 from the bank at an of Rf , and buy $150 worth of stock with return Rt+1.Afterone period, your portfolio is worth

50(1 + Rf ) + 150(1 + Rt+1). − The negative first term is the amount you owe the bank, and the positive second term is the value of your stock. The return on the portfolio is

50(1 + Rf ) + 150(1 + Rt+1) (1 + Rp,t )=− +1 100

= 0.5(1 + Rf )+1.5(1 + Rt+1). − This is just the usual formula for portfolio return in terms of portfolio weights, except that the portfolio weight on the riskless asset is negative because you have borrowed money instead of de- positing it. The negative portfolio weight on the riskless asset en- ables the portfolio weight on the stock to be greater than one. In this example, we say that leverage is “1.5 to one”. If the bank lends you $100 and you start with $100, then you can buy $200 of stock with $100 of wealth and we say that leverage is “2 to 1”.

14 Leverage is dangerous because it can lead to bankruptcy (a gross return less than zero, so your wealth is exhausted) even if the stock you hold has limited liability. In theory, we could imagine that a bank might lend you money even if you have no money of your own to invest. Suppose the bank lends you $50 and you invest this in the stock. Your portfolio is initially worth zero, and after one period it is worth

50(1 + Rf )+50(1+Rt+1)=50(Rt+1 Rf ). − − The portfolio value is the initial investment in the stock times the excess return on the stock (the difference between the stock return and the riskfree interest rate). This type of portfolio, with no money down (a zero initial value) is known as an arbitrage portfolio. The return on an arbitrage portfolio is undefined because to calculate it we would have to divide by the initial value of zero. However we can talk about the dollar return per dollar invested in the stock; this is just the excess return on the stock over the riskless interest rate, the difference between the stock return and the riskless rate.

15 8. Shorting stocks. It is also possible to borrow stocks rather than money from the bank. Suppose you arrange to borrow a stock and promise to return it next period. Assuming the stock pays no dividend during the period, there should be very little cost to this arrangement because the stock’s owner will still earn the return on the stock when he gets it back next period. If there is a dividend, you must pass it on to the owner of the stock. Suppose you start with $100. You borrow $50 of stock 1, sell it for $50, and buy $150 worth of stock 2. After one period, your portfolio is worth

50(1 + R1,t+1) + 150(1 + R2,t+1). − The negative first term is the cost of buying back the same num- ber of shares of stock 1 to return to the person who lent them to you, and the positive second term is the value of your position in stock 2. The return on the portfolio is

50(1 + R1,t+1) + 150(1 + R2,t+1) (1 + R )=− p,t+1 100

= 0.5(1 + R1,t+1)+1.5(1 + R2,t+1). − This is just the usual formula for portfolio return in terms of portfolio weights, except that the portfolio weight on stock 1 is negative because you borrowed it and sold it instead of buying it. We say that you shorted or took a short position in stock 1. Shorting stock1enablestheportfolioweightonstock2tobegreaterthan one.

16 What is the purpose of shorting? Shorting a stock is profitable if the price of that stock falls; thus you should short stocks that you think are overvalued. What is the effect of shorting? Shorting, like selling, puts down- ward pressure on stock prices. This makes shorting unpopular with corporate executives and politicians. There are several practical difficulties with shorting stock. One is that the owner of the stock has the right to ask for its return at anytime,notjustattheendofafixedperiodoftime.Anotheris that not all investors make their stocks available for lending, and some stocks may be in limited supply for this purpose in which case shorting becomes expensive. In general, the regulations governing shorting are cumbersome, perhaps because shorting is unpopular politically.

17 C. Arbitrage and State Prices

1. Arbitrage. An arbitrage opportunity is a profit opportunity that is riskless in the sense that

The opportunity never loses money today or tomorrow. • The opportunity makes money today or it may make money • tomorrow.

Note that the money you make is random, but always positive. Any individual who prefers more wealth to less will want to take advantage of an arbitrage opportunity. Examples:

A package of 12 pencils sells for $1 today, 1 pencil sells for • 10 cents today, there are no costs of packaging or unpackag- ing pencils. Buy 12 pencils, unpackage them, sell them again today. Profit today is $0.20, and nothing happens tomorrow. A package of 12 pencils sells for $1 today, and money can be • borrowed at 5% interest. Tomorrow 12 pencils will sell for either $1.05 or $1.20. Borrow $1 today, use the money to buy 12 pencils. Sell the pencils tomorrow, pay off the loan with $1.05, keep $0.00 or $0.15.

Note that the second example involves borrowing. To exploit the arbitrage opportunity, you have to construct an arbitrage portfolio with no money down.

18 Financial theory assumes that there are no arbitrage opportuni- ties, because any opportunities that arise are eliminated so quickly that we never observe them. This assumption allows us to relate the prices of some assets to the prices of other assets. To explore this further, we need a simple way to think about time and uncertainty. We use the formal device of “states of the world” at different dates. For simplicity, imagine that there are two dates (today and tomorrow), and that all assets make a single payoff tomorrow.(Thiscanbemademorerealisticifwethinkof the assets as investment strategies; we invest today and sell out tomorrow, even if the assets continue to exist in the more distant future.)

Xij is the payoff tomorrow in state i on a share of asset j. Recall that a payoff is the dollar amount that a share will pay tomorrow.

Asset 1 Asset 2

State 1 X11 X12 State 2 X21 X22

Pj is the price today of a share of asset j:

Asset 1 Asset 2 Price P1 P2

19 Tomorrow

State 1

% 1 Today

% 2 State 2 2. Redundant assets. Redundant assets have payoffs that can also be obtained by com- bining other assets. Example: Asset 1 Asset 2 Icecream Swimsuit shares shares State 1: Sun 1 2 State 2: Rain 0 0 Here asset 2 has payoffs that are always twice the payoffs of asset 1, Xi2 =2Xi1 in every state i. Thus asset 2 is redundant given asset 1 (and asset 1 is redundant given asset 2).

We may always have redundant assets. We must have them when the number of assets exceeds the number of states. Example: Asset 1 Asset 2 Asset 3 State 1 1 0 3 State 2: 0 2 4

In this example, Xi3 =3Xi1 +2Xi2 in every state i.Withtwo states, if we start with two assets that are not redundant, a third asset must always be redundant.

20 The assumption of finance theory that there are no arbitrage opportunities immediately tells us how redundant assets must be priced.

Law of One Price: Redundant assets are priced as linear combi- nations of other assets.

In the first example, P2 =2P1 because Xi2 =2Xi1 in every state i. If not, you could form an arbitrage portfolio that is long the cheap asset and short the expensive asset, and make a sure profit. This would be an arbitrage opportunity. In the second example, we must have P3 =3P1 +2P2.

How does this relate to practice in the financial markets? Real- world “arbitrageurs” look for investment strategies that deliver profits with very high probability. But they can never truly rule out the possibility of loss. Examples:

Royal Dutch and Shell, shares of the same company traded in • the UK and Netherlands stock markets. On-the-run and off-the-run Treasury bonds, issued by the US • Treasury with almost identical cash flows.

21 HowimpressiveistheLawofOnePrice? Larry Summers: Consider “a field of economics which could but does not exist: ketchup economics. There are two groups of re- searchers concerned with ketchup economics. Some general econo- mists study the market for ketchup as part of the broader eco- nomic system. The other group is composed of ketchup locatedintheDepartmentofKetchupwheretheyreceivemuch higher salaries than do general economists.... Ketchup economists have an impressive research program, focusing on the scope for ex- cess opportunities in the ketchup market. They have shown that two quart bottles of ketchup invariably sell for twice as much as one quart bottles of ketchup except for deviations traceable to trans- actions costs.... Financial economists like ketchupal economists... are concerned with the interrelationships between the prices of dif- ferent financial assets. They ignore what seems to many to be the more important question of what determines the overall level of asset prices.” (“On Economics and Finance”, Journal of Finance July 1985, 633—635.)

22 3. Complete markets. Markets are complete if assets exist, or can be constructed, that pay $1 in state i and $0 in all other states, for each state i. These assets are sometimes called Arrow-Debreu securities,and their prices are called Arrow-Debreu prices or state prices. We will write the state price for state i as Si.

Example 1: The original assets are Arrow-Debreu securities. Asset 1 Asset 2 Icecream Umbrella shares shares State 1: Sun 1 0 State 2: Rain 0 1

HerethetwoprimitiveassetsarethemselvestheArrow-Debreu securities for states 1 and 2. We have S1 = P1 and S2 = P2.

Example 2: Arrow-Debreu securities can be constructed from the original assets. Asset 1 Asset 2 Icecream Umbrella shares shares State 1: Sun 1 -1/2 State 2: Rain -1/2 1

23 You can create a portfolio by investing Q1 in asset 1 and Q2 in asset 2. Set Q1 =4/3, Q2 =2/3. Then the payoff on the portfolio in state 1 is

4 2 1 Q1X11 + Q2X12 =( 1) + ( )=1. 3 × 3 × −2 The payoff in state 2 is

4 1 2 Q1X21 + Q2X22 =( )+( 1) = 0. 3 × −2 3 × So this portfolio of the original assets gives the Arrow-Debreu security for state 1. The state price is just the cost of the portfolio:

4 2 S = P + P . 1 3 1 3 2

Similarly, to construct the Arrow-Debreu security for state 2, you choose Q1 =2/3, Q2 =4/3. You can check that this portfolio has a payoff of 0 in state 1, and 1 in state 2. The price of the portfolio is the state price for state 2:

2 4 S = P + P . 2 3 1 3 2

Provided we have no redundant assets, we can always solve this system of equations whenever the number of assets equals the num- berofstates.Hencemarketsarecompleteifthereareasmany non-redundant assets as states.

24 If we have a complete set of state prices, any other more complex asset is redundant. This means that we can use the to find its value. We can think of any asset j as a bundle of payoffstobemadein different states. The payoff X1j to be made in state 1 is equivalent to X1j units of the Arrow-Debreu security for state 1, each of which pays $1 in state 1. Thus the cost of the payoff X1j in state 1 must be S1X1j. Similarly, the cost of the payoff X2j in state 2 must be S2X2j. Adding up across states 1 and 2, we get

Pj = S1X1j + S2X2j, or in general with many states i,

Pj = SiXij. Xi This way of looking at things is the basis of much financial in- novation (“slicing and dicing”).

The assumption of finance theory that there are no arbitrage opportunities tells us something important about state prices.

Restriction on state prices under complete markets: State prices are positive.

If not, if there is an Arrow-Debreu security with a zero or nega- tive price, buy it! You pay nothing today, or even get money today; you cannot lose money in any state tomorrow and you get money in one state. This is an arbitrage opportunity.

25 4. Incomplete markets. What if markets are not complete? Then we cannot solve for a unique set of state prices. Many state prices might satisfy Pj = i SiXij and thus could be consistent with observed asset prices. P Example: Asset 1 State 1 1 State 2 1/2

Suppose the price of this asset is P1 =1. Thisisconsistent with S1 =2/3 and S2 =2/3,since1=(2/3 1) + (2/3 1/2). × × But it is also consistent with S1 =3/4 and S2 =1/2,since1= (3/4 1) + (1/2 1/2). The price of only one asset does not give × × us enough information to figure out the underlying state prices.

Even in this case, however, the assumption of finance theory that there are no arbitrage opportunities tells us that

Restriction on state prices under incomplete markets:Youcan find state prices, all of which are positive, that satisfy Pj = i SiXij and thus are consistent with observed asset prices. P

If not, we can construct an asset that has a negative price, no negative payoffs in any state, and a positive payoff in at least one state. This is an arbitrage opportunity. In the example, you can always find positive state prices unless there is a negative price for asset 1. A negative price for asset 1 is an arbitrage opportunity.

26 5. Dynamic trading. How likely is it that markets are complete? At first sight this might seem extremely implausible. But: Pay no attention to irrelevant dimensions of uncertainty (i.e. • those which have no effect on asset payoffs). Summarize uncertainty using an “event tree” with many peri- • ods.

Example: Consider tossing a coin twice and counting the number of heads. There may be 2, 1, or 0 heads so there are 3 final states. For each coin toss, suppose there is one “head” asset that costs $0.50 and pays $1 if the toss is heads, and $0 otherwise; and one “tail” asset that costs $0.50 and pays $1 if the toss is tails, and $0 otherwise. To get a payoff only if there are 2 heads, invest $0.25 at the first toss in half a unit of the head asset. If the first toss is a head, reinvest your winnings in one unit of the head asset. To get a payoff only if there are 0 heads, use the tail asset in the same way. To get a payoff only if there is 1 head, invest $0.25 at the first toss in each asset. Take the winnings from whichever asset pays off, and invest them in the other asset. The lesson: Dynamic trading in 2 assets can achieve complete markets for more than 2 final states. In fact, you only need as many assets as there are branches leaving each node of the event tree.

27 2 ¼ H ½

H ½ T ½

1 ½ H ½ T ½

T ½ 0 ¼ A final note. In all this discussion we have said nothing about the probabilities of the different states. The theory of arbitrage (as opposed to the practice) is concerned only with what can happen, not what is likely to happen. πi is the probability of state i oc- curring tomorrow, given what we know today. We assume π1 > 0, π2 > 0,andπ1 + π2 =1:

Probability

State 1 π1 State 2 π2 =1 π1 −

28