Daniel l. Rubinfeld

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166 Part 2 Producers, Consumers, and Competitive Markets asset Something that provides a flow of money or services to its owner. risky asset Asset tha t provides an uncertain flow of money or services to its owner. Most people are risk averse. Given a choice, they prefer fixed monthly incomes to those 'Nhich, though equally large on average, fluctuate random.ly from month to month. Yet manv of these same people will im-est all or part of their savings in stocks, bonds, -and other assets that carry some risk. Why do risk· averse people invest in the stock market and thereby risk losing part or all of their uweshl1entslO How do people decide how much risk to bear 'when making investments and planning for the future To ans,,,'er these questions, we must examule the demand for risky assets. Assets An asset is somethillg that provides a flow of mOlley or services to its OlUner A home, an aparhl1ent building, a savings account, or shares of General Motors stock are all assets. A home, for example, provides a flow of housing services to its owner, and if the owner did not ,\'ish to live there, could be rented out, thereby providing a monetary flow. Likewise, apartments in an apartment building can be rented out, providing a flow of rental income to the owner of the building. A savings account pays mterest (usually every day or every month), which is usuallv reinvested Ul the account. -The monetary flow that one receives from asset ownership can take the form of an explicit payment, such as the rental income from an apartment buildulg: Every month, the landlord receives rent checks from the tenants. Another form of explicit payment is the dividend on shares of common stock: Every three months the mvner of a share of General Motors stock receives a quarterly dividend payment. But sometimes the monetary flow from ownership of an asset is implicit: It takes the form of an increase or decrease in the price or value of the asset. An increase in the value of an asset is a capital gain, a decrease a capital loss. For example, as the population of a city grows, the value of an aparhnent building may increase. The owner of the buildulg will then earn a capital gain beyond the rental income. The capital gain is unrealized until the buildmg is sold because no money is actually received until then. There is, howe\'er, an implicit monetary flow because the building could be sold at any time. The monetary flow from owning General Motors stock is also partly implicit. The price of the stock chanaes from dav to dav, and each time it does, owners gauL or lose. b J_ Risky and Riskless Assets A risky asset provides a monetary flow that is at least in part random. In other words, the monetary flow is not known 'with certainty in advance. A share of General Motors stock is an obvious example of a risky asset: You cannot know 'whether the price of the stock will rise or fall over time, nor can you even be sure that the company will continue to pay the same (or any) dividend per share. Although people often associate risk with the stock market, most other assets are also risky. 10 );lost Americans have at least some monev ll1vested in stocks or other risky assets, though often indirectly. For example, many people \\'ho l~old full-time jobs ha::e share: ll1 pension funds ~nde~ wntten m part by their own salary contnbutlOns and lI1 part by employer contnbutlOns. Usuall, such funds are ll1\"ested partly in the stock market An apartment buildulg is one example. You cannot know how much land values will rise or faIt whether the buildmg will be fully rented all the time, or even whether the tenants will pay their rents promptly. Corporate bonds are another example-the issumg corporation could go bankrupt and fail to pay bond owners their interest and principal. Even long-term U.s. government bonds that mature in 10 or 20 years are risky. Although it is highly unlikely that the federal O"overnment will go bankrupt, the rate of mflation could tmexpectedly increase ~nd make future interest payments and the eventual repayment of principal worth less Ul real terms, thereby reducmg the value of the bonds. In contrast, a riskless (or risk-free) asset pays a monetary flow that is known with certaulty. Short-term U.s. government bonds-called Treasurv bills-are riskless, or almost riskless. Because these bonds mature m a few mo~ths, there is very little risk from an unexpected mcrease Ul the rate of mflation. You can also be reasonably confident that the U.s. government will not default on the bond (i.e., refuse to pay back the holder when the bond comes due). Other examples of riskless or almost riskless assets include passbook savings accounts and shortterm certificates of deposit. Asset Returns People buy and hold assets because of the monetary flows they provide. To compare assets with each other, it helps to thmk of this monetary flow relative to an asset's price or value. The return on an asset is the totall1lol1etary flow it yieldsincluding capital gains or losses-as a fraction of its price. For example, a bond worth 51000 today that pays out $100 this year (and every year) has a return of 10 percent. l1 If an apartment buildmg was worth 510 million last year, mcreased in value to $11 million this year, and also provided rental income (after expenses) of $0.5 million, it would have yielded a return of 15 percent over the past year. If a share of General Motors stock was worth $80 at the beguming of the year, fell to $72 by the end of the year, and paid a dividend of $4, it will have yielded a return of - 5 percent (the dividend yield of 5 percent less the capital loss of 10 percent). When people mvest their savulgs m stocks, bonds, land, or other assets, they usually hope to earn a return that exceeds the rate of mflation, so that by delaying consumption, they could buy more Ul the future than they could by spendma all their income now. Thus we often express the return on an asset in real-i.e~ inflation-adJllsted-terms. The real return on an asset is its simple (or nomulal) return less the rate of inflation. For example, with an annual inflation rate of 5 percent, our bond, aparhnent building, and share of GM stock have yielded real returns of 5 percent, 10 percent, and -10 percent, respectively. . versus Because most assets are risky, an mvestor cannot know m advance what returns they will yield over the commg year. For example, our apartment building might have depreciated in value Th~ pr~ce of a bnd often changes durmg the course of a year. If the bond appreciates (or depreciat7 s ) m \ alue durmg the year, Its return will be greater (or less) than 10 percent. In addition, the def ~mhon ot return given above should not be confused with the "internal rate of return," which is .ometimes used to compare monetary flows occurrmg over some time. We discuss other return measures in Chapter 15, when we deal w-ith present discounted \"alues. Lfba'''1i"f'F 5 Choice Under Uncertainty 67 riskless (or risk-free) asset Asset that provides a flow of monel' or services that is known with certainty. return Total monetary flow of an asset as a fractiorl of its price. real return Simple (or nominal) return on an asset, less the rate of mflation.
168 Part 2 Producers, Consumers, and Competitive Markets 5 Choice Under Uncertainty 69 Common stocks (S&P 500) Long-term corporate bonds U.S. Treasury bills REAL RATE OF RETURN(%) 9.5 2.7 0.6 RiSK (STANDARD DEVIATION, %) 20.2 8.3 3.2 Treasury bills, only in stocks, or in some combination of the h\'o. As we will see, this problem is analogous to the co.nsumer's problem of allocating a budget betw'een purchases of food and clothmg. Let's denote the risk-free return on the Treasury bill by Rr. Because the rehlrn 's risk free, the expected and actual returns are the same, In addition, let the ~xpected return from investing in the stock market be RIJI and the actual rehm1 be l' The achlal rehlrn is risky At the time of the investment decision, we know th~ set of possible outcomes and the likelihood of each, but we do not know what particular outcome will occur, The risky asset will have a higher expected return than the risk-free asset (RIJI > Rr). Otherwise, risk-averse investors \\'ould buy only Treasury bills and no stocks would be sold. expected return Return that an asset should earn on a\'erage actual return Return that an asset earns, instead of appreciating, and the price of GM stock might ha\'e risen instead of falling. Howe\'er, we can still compare assets by looking at their expected returns. The expected return on an asset is the expected ('ollie of its tetl/m, i.e., the return that it should earn on average, In some years, an asset's actual return may be much higher than its expected return and in some years much lower. Over a long period, howe\'er, the average return should be close to the expected return. Different assets ha\'e different expected rehlrns. Table 5.8, for example, shows that while the expected real return of a U.s. Treasury bill has been less than 1 percent, the expected real return on a group of representative stocks on the New York Stock Exchange has been more than 9 percentY Why \·vould anyone buy a Treasury bill when the expected return on stocks is so much higher Because the demand for an asset depends not just on its expected return, but also on its risk: Although stocks have a higher expected return than Treasury bills, they also carry much more risk. One measure of risk, the standard deviation of the real annual return, is equal to 202 percent for com,mon stocks, 8.3 percent for corporate bonds, and only 3.2 percent for u.s. Treasury bills. The numbers in Table 5.8 suggest that the higher the expected return on an im'estment, the greater the risk ilwolved. Assuming that one's irwestments are well di\'ersified, this is indeed the case. 13 As a result, the risk-averse im'estor must balance expected return against risk, We examine this trade-off in more detail in the next section, and Return Suppose a woman wants to im'est her savings in two assets-Treasury bills, vvhich are almost risk free, and a representative group of stocks.l~ She must decide how much to invest in each asset. She might, for instance, invest only in To determine how much money the investor should put in each asset, let's set b equal to the fraction of her savings placed in the stock market and (1 - b) the fraction used to purchase Treasury bills. The expected return on her to_tal portfolio, R,,, is a weighted average of the expected return on the two assets: b (5.1) Suppose, for example, that Treasury bills pay 4 percent (R r = .04), the stock market's expected return is 12 percent (RIJI = ,12), and b = l/i Then Rp = 8 percent. How risky is this portfolio One measure of its riskiness is the standard deviation of its return. We will denote the stillldord deviotioll of the risky stock market investment by CTIJI' With some algebra, we can show that the stoHdord deviotioll of the portfolio, CT p (with one risky and one risk-free asset) is the fraction of the portfolio invested in the risky asset times the standard deviation of that asset: 16 The investor's Choice Problem (5.2) We have still not determined how the investor should choose this fraction b. To do so, we must first show that she faces a risk-rehlrn trade-off analogous to a consumer's budget line, To identify this trade-off, note that equation (5.1) for the expected rehlrn on the portfolio can be rewritten as In §3.2, we explain how a budget line is determined from an individual's income and the prices of the available goods. 12 For some stocks, the expected return is higher, and for some it is lower. Stocks of smaller companies (e,g, some of those traded on the NASDAQ) ha\'e higher expected rates of return-and higher return standard de\'iations. 13 It is lIolldil'crsifiablc risk that matters. An indi\'idual stock may be \'en' risk\' but still ha\'e a low expected return' because most of the risk could be dh'ersified ~,,'ay by'holdi;1g a large number of such stocks. Z\olldil'crsifiablc risk, which arises from the fact that indi\'idual stock prices are correlated with the o\'erall stock market, is the risk that remains e\'en if one holds a diversified portfolio of stocks. We discuss this point in detail in the context of the Capital Asset Pricillg ivIodd in Chapter Ii 14The easiest way to im'est in a representati\'e group of stocks is to buy shares in a IlIlItllll! flilld, Because a mutual fund invests in many stocks, one effecti\'ely buys a portfolio. 15 The expected value of the sum of two variables is the sum of the expected values Therefore R" = E[bl'",l + E[(l - b)Rrl = bE[r",] + (1 ~' , b)Rr . l1R" (1 b)R. , To see why, we observe from footnote 4 that we can write the variance of the portfolio return as u~ = E[br", + (1 - b)Rr R,, Substituting equation (5,1) for the expected return on the portfolio, R", we ha\'e up E[bl'", + (1 - b)Rj - bR", (1 b)Rrf E[b(r" - Rm)f = b 2 u;, Because the standard deviation of a random variable is the square root of its \'ariance, u" bu""
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PART 1 Introduction: Marl
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PART 1 surveys the scope of microec
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few years ago, General Mills decide
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Price Dollars per 7 I Unit of Outpu
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Daniel l. Rubinfeld

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