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QR43, Introduction to Investments Class Notes, Fall 2004 I. Prices, 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 goods, and the sales proceeds give the company revenue or sales. After paying the workers and set- ting aside money 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, interest 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. Derivative securities (derivatives) make payments that depend • on the prices of other financial assets. The market 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 economics? 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 price financial contracts and instruments. This methodology is applied to value 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.) “Paul Samuelson’s textbook on economics has the following anony- mous quote, “You can make even a parrot into a learned political economist–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.

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