The Capital Asset Pricing Model

The Capital Asset Pricing Model А fundamental question in finance is how the risk of an investment should affect its expected return. The Capital Asset Pricing Model (САРМ) provided the first coherent framework for answering this question. The САРМ was developed in the early 1960s Ьу William Sharpe (1964), Jack Treynor (1962),John Lintner (1965а, Ь) andJan Mossin (1966). The САРМ is based оп the idea that not all risks should affect asset prices. In particular, а risk that сап Ье diversified away when held along with other invest­ ments in а portfolio is, in а very real way, not а risk at all. The САРМ gives us insights about what kind of risk is related to return. This paper lays out the key ideas of the Capital Asset Pricing Model, places its development in а historical context, and discusses its applications and enduring importance to the field of finance. Historical Background In retrospect, it is striking how little we understood about risk as late as the 1960s-whether in terms of theory or empirical evidence. Mter all, stock and option markets had been in existence at least since 1602 when shares of the East India Company began trading in Amsterdam (de la Vega, 1688); and organized insurance markets had become well developed Ьу the 1 700s (Bernstein, 1996). Ву 1960, insurance businesses had for centuries been relying оп diversification to spread risk. But despite the long history of actual risk-bearing and risk-sharing in organized financial markets, the Capital Asset Pricing Model was developed at а time when the theoretical foundations of decision making under uncertainty were relatively new and when basic empirical facts about risk and return in the capital markets were not yet known. ■ Andre F. Perold is the George Gund Professor of Finance and Banking, Haruard Business School, Boston, Massachusetts. His e-mail address is (aperold@hbs.edu). Rigorous theories of investor risk preferences and decision-making under uncertainty emerged only in the 1940s and 1950s, especially in the work of von Neumann and Morgenstern (1944) and Savage (1954). Portfolio theory, showing how investors can create portfolios of individual investments to optimally trade off risk versus return, was not developed until the early 1950s Ьу Harry Markowitz (1952, 1959) and Roy (1952). Equally noteworthy, the empirical measurement of risk and return was in its infancy until the l 960s, when sufficient computing power became availaЫe so that researchers were аЫе to collect, store and process market data for the purposes of scientific investigation. The first careful study of returns on stocks listed on the New York Stock Exchange was that of Fisher and Lorie (1964) in which they note: "It is surprising to realize that there have been no measurements of the rates of return on investments in common stocks that could Ье considered accurate and defini­ tive." In that paper, Fisher and Lorie report average stock market returns over different holding periods since 1926, but not the standard deviation of those returns. They also do not report any particular estimate of the equity risk pre­ mium-that is, the average amount Ьу which the stock market outperformed risk-free investments-although they do remark that rates of return on common stocks were "substantially higher than safer alternatives for which data are avail­ aЫe." Measured standard deviations of broad stock market returns did not appear in the academic literature until Fisher and Lorie (1968). Carefully constructed estimates of the equity risk premium did not appear until lbbotson and Sinquefield (1976) puЫished their findings on long-term rates of return. They found that over the period 1926 to 1974, the (arithmetic) average return on the Standard and Poor's 500 index was 10.9 percent per annum, and the excess return over U.S. Treasury bills was 8.8 percent per annum. 1 The first careful study of the historical equity risk premium for UK stocks appeared in Dimson and Brealey (1978) with an estimate of 9.2 percent per annum over the period 1919-1977. In the 1940s and 1950s, prior to the development of the Capital Asset Pricing Model, the reigning paradigm for estimating expected returns presupposed that the return that investors would require (or the "cost of capital") of an asset depended primarily on the manner in which that asset was financed (for example, Bierman and Smidt, 1966). There was а "cost of equity capital" and а "cost of debt capital," and the weighted average of these- based on the relative amounts of debt and equity financing-represented the cost of capital of the asset. The costs of debt and equity capital were inferred from the long-term yields of those instruments. The cost of debt capital was typically assumed to Ье the rate of interest owed on the debt, and the cost of equity capital was backed out from the cash flows that investors could expect to receive on their shares in relation to the current price of the shares. А popular method of estimating the cost of equity this way was the Gordon and Shapiro (1956) model, in which а company's dividends are 1 These are arithmetic average returns. lbbotson and Sinquefield (1976) were also the first to report the term premium оп long-term bonds: 1.1 percent per annum average return in excess of Treasury bills over the period 1926-1974. assumed to grow in perpetuity at а constant rate g. In this model, if а firm's current dividend per share is D, and the stock price of the firm is Р, then the cost of equity capital r is the dividend yield plus the dividend growth rate: r = D/ Р + g. 2 From the perspective of modern finance, this approach to determining the cost of capital was anchored in the wrong place.At least in а frictionless world, the value of а firm or an asset more broadly does not depend on how it is financed, as shown Ьу Modigliani and Miller (1958). This means that the cost of equity capital likely is determined Ьу the cost of capital of the asset, rather than the other way around. Moreover, this process of inferring the cost of equity capital from future dividend growth rates is highly subjective. There is no simple way to determine the market's forecast of the growth rate of future cash flows, and companies with high dividend growth rates will Ье judged Ьу this method to have high costs of equity capital. Indeed, the Capital Asset Pricing Model will show that there need not Ье any connection between the cost of capital and future growth rates of cash flows. In the pre-CAPM paradigm, risk did not enter directly into the computation of the cost of capital. The working assumption was often that а firm that can Ье financed mostly with debt is рrоЬаЫу safe and is thus assumed to have а low cost of capital; while а firm that cannot support much debt is рrоЬаЫу risky and is thus assumed to command а high cost of capital. These rules-of-thumb for incorporating risk into discount rates were ad hoc at best. As Modigliani and Miller (1958) noted: "No satisfactory explanation has yet been provided ...as to what determines the size of the risk [adjustment] and how it varies in response to changes in other variaЫes." In short, before the arrival of the Capital Asset Pricing Model, the question of how expected retums and risk were related had been posed, but was still awaiting an answer. Why Investors Might Differ in Тheir Pricing of Risk Intuitively, it would seem that investors should demand high returns for holding high-risk investments. That is, the price of а high-risk asset should Ье bld sufficiently low so that the future payoffs on the asset are high (relative to the price). А difficulty with this reasoning arises, however, when the risk of an invest­ ment depends on the manner in which it is held. То illustrate, consider an entrepreneur who needs to raise $1 million for а risky new venture. There is а 90 percent chance that the venture will fail and end up worthless; and there is а 10 percent chance that the venture will succeed within а year and Ье worth $40 million. The expected value of the venture in one year is therefore $4 million, or $4 per share assuming that the venture will have а million shares outstanding. Case 1: If а single risk-averse individual were to fund the full $1 million-where 2 The cost of equity capital in this model is the "intemal rate of retum," the discount rate that equates the present value of future cash flows to the current stock price. In the Gordon-Shapiro model, the projected dividend stream is D, D( 1 + g), D( 1 + g)2 ••• The present value of these cash flows when discounted at rate r is D/ ( r - g), which when set equal to the current stock price, Р, estaЫishes r = D/ Р + g. the investment would represent а significant portion of the wealth of that individual-the venture would have to deliver а very high expected retum, say 100 percent. То achieve an expected return of 100 percent оп an investment of $1 million, the entrepreneur would have to sell the investor а 50 percent stake: 500,000 shares at а price per share of $2. Case 11: If the funds could Ье raised from someone who сап diversify across many such investments, the required return might Ье much lower. Consider an investor who has $100 million to invest in 100 ventures with the same payoffs and probabllities as above, except that the outcomes of the ventures are all independent of one another. In this case, the probabllity of the investor sustaining а large percentage loss is small-for example, the probabllity that all 100 ventures fail is а miniscule .003 percent ( = 0.9 100 )-and the diversified investor might consequently Ье satisfied to receive an expected retum of only, say, 10 percent. lf so, the entrepreneur would need to sell а much smaller stake to raise the same amount of money, here 27.5 percent ( = $1.1 million/$4 million); and the investor would рау а higher price per share of $3.64 {= $1 million/275,000 shares). Cases I and II differ only in the degree to which the investor is diversified; the stand-alone risk and the expected future value of any one venture is the same in both cases. Diversified investors face less risk per investment than undiversified investors, and they are therefore willing to receive lower expected retums (and to рау higher prices). For the purpose of determining required returns, the risks of investments therefore must Ье viewed in the context of the other risks to which investors are exposed. The САРМ is а direct outgrowth of this key idea. Diversification, Correlation and Risk The notion that diversification reduces risk is centuries old. In eighteenth-century English language translations of Don Qy,ixote, Sancho Panza advises his master, "It is the part of а wise man to . . . not venture all his eggs in one basket." According to Herblson (2003), the proverb "Do not keep all your eggs in one basket" actually appeared as far back as Torriano's (1666) Соттоп Place of Italian ProverЬs. However, diversification was typically thought of in terms of spreading your wealth across many independent risks that would cancel each other if held in sufficient number (as was assumed in the new ventures example). Harry Markowitz (1952) had the insight that, because of broad economic influences, risks across assets were correlated to а degree. As а result, investors could eliminate some but not all risk Ьу holding а diversified portfolio. Markowitz wrote: "This presumption, that the law of large num­ bers applies to а portfolio of securities, cannot Ье accepted. The retums from securities are too intercorrelated. Diversification cannot eliminate all variance." Markowitz (1952) went оп to show analytically how the benefits of diversifica­ tion depend оп correlation. The correlation between the returns of two assets measures the degree to which they fluctuate together. Correlation coefficients range between -1.0 and 1.0. When the correlation is 1.0, the two assets are perfectly positively correlated. They move in the same direction and in fixed proportions (plus а constant). In this case, the two assets are substitutes for one another. When the correlation is -1.0, the returns are perfectly negatively corre­ lated meaning that when one asset goes up, the other goes down and in а fixed proportion (plus а constant). In this case, the two assets act to insure one another. When the correlation is zero, knowing the return оп one asset does not help you predict the return оп the other. То show how the correlation among individual security returns affects portfo­ lio risk, consider investing in two risky assets, А and В. Assume that the risk of an asset is measured Ьу its standard deviation of return, which for assets А and В is denoted Ьу ил and О"в, respectively. Let р denote the correlation between the returns оп assets А and В; let х Ье the fraction invested in Asset А and у ( = 1 х) Ье the fraction invested in Asset В. When the returns оп assets within а portfolio are perfectly positively correlated (р = 1), the portfolio risk is the weighted average of the risks of the assets in the portfolio. The risk of the portfolio then сап Ье expressed as The more interesting case is when the assets are not perfectly correlated (р < 1). Then there is а nonlinear relationship between portfolio risk and the risks of the underlying assets. In this case, at least some of the risk from one asset will Ье offset Ьу the other asset, so the standard deviation of the portfolio иР is always less than the weighted average of ил and ив· 3 Thus, the risk of а portfolio is less than the average risk of the underlying assets. Moreover, the benefit of diversification will increase the farther away that the correlation р is from 1. These are Harry Markowitz's important insights: 1) that diversification does not rely оп individual risks being uncorrelated,just that they Ье imperfectly correlated; and 2) that the risk reduction from diversification is limited Ьу the extent to which individual asset retums are correlated. If Markowitz were restating Sancho Panza's advice, he might say: lt is safer to spread your eggs among imperfectly correlated baskets than to spread them among perfectly correlated baskets. ТаЫе 1 illustrates the benefits of diversifying across international equity mar­ kets. The tаЫе lists the world's largest stock markets Ьу market capitalization as of December 31, 2003, the comЬination of which we will call the world equity market 3 The portfolio standard deviation, ир, сап Ье expressed in tenns of tЬе standard deviations of assets А and В and tЬeir correlation using tЬе variance fonnula: This expression сап Ье algebraically manipulated to obtain When р = 1, tЬefinal tenn disappears, giving tЬe fonnula in tЬе text. When р < 1, tЬen tЬе size of tЬe second tenn will increase as р declines, and so tЬе standard deviation of tЬе portfolio will fall as р declines. ТаЫе 1 Market Capitalizations and Historical Risk Estimates for 24 Countries, January 1994-December 2003 Лiarket Capitalization ($ Billions, 12/31/03) u.s. Japan UK France Gennany Canada Switzerland Spain Hong Kong Italy Australia China Taiwan Netherlands Sweden South Korea India South Africa Brazil Russia Belgium Malaysia Singapore Mexico $14,266 2,953 2,426 1,403 1,079 910 727 726 715 615 586 513 379 368 320 298 279 261 235 198 174 168 149 123 Capitalization Weight 47.8% 9.9% 8.1% 4.7% 3.6% 3.0% 2.4% 2.4% 2.4% 2.1% 2.0% 1.7% 1.3% 1.2% 1.1% 1.0% 0.9% 0.9% 0.8% 0.7% 0.6% 0.6% 0.5% 0.4% WEMP $29,870 100% S.D. ofWEMP assuming perfect correlation S.D. ofWEMP assuming zero correlation S.D. of Return Beta vs. WEMP Correlation vs. WЕМР 16.1% 22.3% 14.3% 19.3% 21.7% 19.9% 17.1% 21.5% 29.2% 23.9% 18.4% 43.3% 33.0% 19.5% 24.3% 47.7% 26.7% 26.9% 43.6% 76.9% 17.2% 38.6% 28.6% 35.1% 1.00 0.83 0.78 1.00 1.10 1.13 0.73 0.92 1.33 0.90 0.93 1.26 1.15 1.о2 1.25 1.55 0.63 1.09 1.81 2.34 0.65 0.81 1.04 1.40 0.95 0.57 0.83 0.79 0.77 0.87 0.65 0.65 0.70 0.58 0.77 0.45 0.53 0.79 0.78 0.50 0.36 0.62 0.63 0.47 0.58 0.32 0.56 0.61 15.3% 19.9% 8.4% 1.00 1.00 Notes: WEMP stands for World Equity Market Portfolio. S.D. is standard deviation expressed on an annualized basis. Calculations are based on historical monthly returns obtained from Global Financial Data Inc. portfolio, labeled in the tаЫе as WEMP. The capitalization of the world equity market portfolio was about $30 trillion-comprising over 95 percent of all puЫicly traded equities-with the United Statese representing Ьу far the largest fraction. ТаЫе 1 includes the standard deviation of monthly total retums for each country over the ten-year period ending December 31, 2003, expressed on an annualized basis. Assuming that the historical standard deviations and correlations of return are good estimates of future standard deviations and correlations, we can use this data to calculate that the standard deviation of return of the WEMP-given the capi­ talization weights as of December 2003-is 15.3 percent per annum. lf the country returns were all perfectly correlated with each other, then the standard deviation of the WEMP would Ье the capitalization-weighted average of the standard deviations, which is 19.9 percent per annum. The difference of 4.6 percent per annum represents the diversification benefit-the risk reduction stemming from the fact that the world's equity markets are imperfectly correlated. Also shown in ТаЫе 1 is that the standard deviation of the WEMP would Ье only 8.4 percent per annum if the country retums were uncorrelated with one another. The amount Ьу which this figure is lower than the actual standard deviation of 15.3 percent per annum is а measure of the extent to which the world's equity markets share common influences. Portfolio Theory, Riskless Lending and Borrowing and Fund Separation То arrive at the САРМ, we need to examine how imperfect correlation among asset returns affects the investor's tradeoff between risk and return. While risks comblne nonlinearly (because of the diversification effect), expected retums com­ Ьine linearly. That is, the expected retum оп а portfolio of investments is just the weighted average of the expected retums of the underlying assets. Imagine two assets with the same expected return and the same standard deviation of return. Ву holding both assets in а portfolio, one obtains an expected return оп the portfolio that is the same as either one of them, but а portfolio standard deviation that is lower than any one of them individually. Diversification thus leads to а reduction in risk without апу sacrifice in expected return. Generally, there will Ье many comblnations of assets with the same portfolio expected return but different portfolio risk; and there will Ье many comblnations of assets with the same portfolio risk but different portfolio expected return. Using optimization techniques, we сап compute what Markowitz coined the "efficient frontier." For each level of expected return, we сап solve for the portfolio combl­ nation of assets that has the lowest risk. Or for each level of risk, we сап solve for the comblnation of assets that has the highest expected return. The efficient frontier consists of the collection of these optimal portfolios, and each investor сап choose which of these best matches their risk tolerance. The initial development of portfolio theory assumed that all assets were risky. James Tobln (1958) showed that when investors сап borrow as well as lend at the risk-free rate, the efficient frontier simplifies in an important way. (А "risk-free" instrument pays а fixed real return and is default free. U.S. Treasury bonds that adjust automatically with inflation-called Treasury inflation-protected instru­ ments, or TIPS-and short-term U.S. Treasury Ьills are considered close approxi­ mations of risk-free instruments.) То see how riskless borrowing and lending affects investors' decision choices, consider investing in the following three instruments: risky assets М and Н, and the riskless asset, where the expected returns and risks of the assets are shown in ТаЫе 2. Suppose first that you had the choice of investing all of your wealth in just опе of these assets. Which would you choose? The answer depends оп your risk tolerance. Asset Hhas the highest risk and also the highest expected return. You would choose ТаЫе2 How Riskless Boпowing and Lending Mfect Investors' Choices Riskless asset Asset М Asset Н Expected retum Risk (S.D.) 5% (r1) 0% 20% (ам) 40% (ан) 10% (Ем) 12% (Ен) Asset Н if you had а high tolerance for risk. The riskless asset has no risk but also the lowest expected return. You would choose to lend at the risk-free rate if you had а very low tolerance for risk. Asset М has an intermediate risk and expected return, and you would choose this asset if you had а moderate tolerance for risk. Suppose next that you can borrow and lend at the risk-free rate, that you wish to invest some of your wealth in Asset Н and the balance in riskless lending or borrowing. If х is the fraction of wealth invested in Asset Н, then 1 - х is the fraction invested in the risk-free asset. When х < 1, you are lending at the risk-free rate; when х > 1, you are borrowing at the risk-free rate. The expected return of this portfolio is (1 - х) r1 + хЕн, which equals r1 + х(Ен - r1), and the risk of the portfolio is ха-н· The risk of the portfolio is proportional to the risk of Asset Н, since Asset Н is the only source of risk in the portfolio. Risk and expected return thus both comЬine linearly, as shown graphically in Figure 1. Each point on the line connecting the risk-free asset to Asset Н represents а particular allocation ( х) to Asset Н with the balance in either risk-free lending or risk-free borrowing. The slope of this line is called the Sharpe Ratio-the risk premium of Asset Н divided Ьу the risk of Asset Н: The Sharpe Ratio of Asset Н evaluates to 0.175 (= (12 percent - 5 percent)/ 40 percent) and all comЬinations of Asset Н with risk-free borrowing or lending have this same Sharpe Ratio. Also shown in Figure 1 are the risks and expected returns that can Ье achieved Ьу comЬining Asset М with riskless lending and borrowing. The Sharpe Ratio of Asset М is 0.25, which is higher than that of Asset Н, and any level of risk and return that can Ье obtained Ьу investing in Asset Н along with riskless lending or borrow­ ing is dominated Ьу some comЬination of Asset М and riskless lending or borrow­ ing. For example, for the same risk as Asset Н, уои can obtain а higher expected return Ьу investing in Asset М with 2:1 leverage. As shown in Figure 1, the expected return of а 2:1 leveraged position in Asset М is 15 percent (that is, (2 Х 10 per­ cent) - (1 Х 5 percent)), which is higher than the 12 percent expected return of Asset Н. If you could hold only one risky asset along with riskless lending or borrowing, it unamЬiguously would Ье Asset М. Being аЫе to lend and borrow at the risk-free rate thus dramatically changes Figure 1 Combining а Risky Asset with Risk-Free Lending and Borrowing 16 15 14 13 12 i:: .... 11 11) .... 10 -с, 11) ComЬinarions of asset М involving risk-free lending ComЬinarions of asset М involving risk-free --1-- 9 8 7 6 5 4 3 2 2: l Leverage ComЬinations of asset Н involving risk-free lending о 5 10 15 20 25 30 Risk (standard deviation) 35 ComЬinations of asset Н involving risk-free borrowing 40 45 50 our investment choices. The asset of choice-if you could choose only one risky asset-is the one with the highest Sharpe Ratio. Given this choice of risky asset, you need to make а second decision, which is how much of it to hold in your portfolio. The answer to the latter question depends оп your risk tolerance. Figure 2 illustrates the approach in the case where we сап invest in comЬina­ tions of two risky assets, М and Н, plus riskless lending and borrowing. The correlation between the returns of assets М and Н is assumed to Ье zero. In the figure, the curve connecting assets М and Н represents all expected return/ standard deviation pairs that сап Ье attained through comЬinations of assets М and Н. The comЬination of assets М and Н that has the highest Sharpe Ratio is 74 percent in Asset М and 26 percent in Asset Н (the tangency point). The expected return of this comЬination is 10.52 percent, and the standard deviation is 18.09 percent. The Sharpe Ratio evaluates to 0.305, which is consideraЫy higher than the Sharpe Ratios of assets М and Н (0.25 and 0.175, respectively). Investors who share the same estimates of expected return and risk all will locate their portfolios оп the tangency line connecting the risk-free asset to the frontier. In particular, they all will hold assets М and Н in the proportions 74/26. The optimal portfolio of many risky assets сап Ье found similarly. Figure 3 offers а general illustration. Use Markowitz's algorithm to obtain the efficient frontier of portfolios of risky assets. Find the portfolio оп the efficient frontier that has the highest Sharpe Ratio, which will Ье the point where а ray stretching up from the risk-free point isjust tangent to the efficient frontier. Then, in accordance with your risk tolerance, allocate your wealth between this highest Sharpe Ratio portfo­ lio and risk-free lending or borrowing. This characterization of the efficient frontier is referred to as "fund separa­ tion." Investors with the same beliefs about expected returns, risks and correlations all will invest in the portfolio or "fund" of risky assets that has the highest Sharpe Figure 2 Efficient Frontier with Two Risky Assets 15 14 13 Highest Sharpe Ratio portfolio (74% in asset М, 26% in asset Н) 12 .... 11 11) .... 10 11) / / / AssetH 9 8 6 5 4 Risk-free о 5 10 15 25 20 35 30 40 45 Risk (standard deviation) Figure 3 Efficient Frontier with Many Risky Assets Highest Sharpe Ratio portfolio � Efficient frontier of risky assets • • • • • • • • • • • • • • • Individual assets Risk (standard deviation) Ratio, but they will differ in their allocations between this fund and risk-free lending or borrowing based оп their risk tolerance. Notice in particular that the composition of the optimal portfolio of risky assets does not depend оп the investor's tolerance for risk. Market-Detennined Expected Returns and Stand-Alone Risk Portfolio theory prescribes that investors choose their portfolios оп the effi­ cient frontier, given their beliefs about expected returns and risks. The Capital Asset Pricing Model, оп the other hand, is concerned with the pricing of assets in equilibrium. САРМ asks: What are the implications for asset prices if everyone heeds this advice? In equilibrium, all assets must Ье held Ьу someone. For the market to Ье in equilibrium, the expected return of each asset must Ье such that investors collectively decide to hold exactly the supply of shares of the asset. The Capital Asset Pricing Model will tel1 us how investors determine those expected returns-and thereby asset prices-as а function of risk. In thinking about how expected return and risk might Ье related, let us ask whether, as а rule, the expected return оп an investment could possiЫy Ье а function of its stand-alone risk (measured Ьу standard deviation of return). The answer turns out to Ье "по." Consider the shares of two firms with the same stand-alone risk. If the expected return оп an investment was determined solely Ьу its stand-alone risk, the shares of these firms would have the same expected return, say 10 percent. Any portfolio comЬination of the two firms would also have an expected return of 10 percent (since the expected return of а portfolio of assets is the weighted average of the expected returns of the assets that comprise the portfolio). However, if the shares of the firms are not perfectly correlated, then а portfolio invested in the shares of the two firms will Ье less risky than either one stand-alone. Therefore, if expected return is а function solely of stand-alone risk, then the expected return of this portfolio must Ье less than 10 percent, contra­ dicting the fact that the expected return of the portfolio is 10 percent. Expected returns, therefore, cannot Ье determined solely Ьу stand-alone risk. Accordingly, any relationship between expected return and risk must Ье based оп а measure of risk that is not stand-alone risk. As we will soon see, that measure of risk is given Ьу the incremental risk that an asset provides when added to а portfolio, as discussed in the next section. Improving the Sharpe Ratio of а Portfolio Suppose you were trying to decide whether to add а particular stock to your investment portfolio of risky assets. If you could borrow and lend at the risk-free rate, you would add the stock if it improved the portfolio's Sharpe Ratio. It turns out there is а simple rule to guide the decision-a rule that сап Ье derived Ьу understanding the two special cases: 1) when the additional stock is uncorrelated with the existing portfolio, and 2) when the additional stock is perfectly correlated with the existing portfolio. The rule will lead us directly to the equilibrium risk-return relationship specified Ьу the Capital Asset Pricing Model. In what follows, it will Ье helpful to think in terms of "excess return," the return of an instrument in excess of the risk-free rate. The expected excess return is called the risk premium. Adding а Stock that is Uncorrelated with the Existing Portfolio When should а portfolio Ье diversified into an uncorrelated stock? If the excess returns оп the stock and existing portfolio are uncorrelated, adding а small amount of the stock has almost no effect on the risk of the portfolio. 4 At the margin, therefore, the stock is а substitute for investing in the risk-free asset. Including the stock will increase the portfolio's Sharpe Ratio if the stock's expected return Es exceeds the risk-free rate r.t Said another way, the additional stock should Ье included in the portfolio if its risk premium Es - r1 is positive. Adding а Stock that is Perfectly Correlated with the Existing Portfolio If the stock and portfolio excess returns are perfectly correlated, investing in the stock becomes а substitute for investing in the portfolio itself. То see this, recall that а perfect correlation means that the stock and the portfolio excess returns move together in а fixed ratio plus а constant. The fixed ratio is called beta, denoted Ьу /3, and the constant is called alpha, denoted Ьу а. In other words, the excess return of the stock is equal to alpha plus beta times the excess return of the portfolio. It also follows that the expected excess return of the stock is alpha plus beta times the expected excess return on the portfolio-that is, Es - r1 = а + /3 (Ер r1). The constant alpha is therefore given Ьу the difference between the risk premium of the stock and beta times the risk premium of the portfolio. Since the stock and the portfolio move together in а fixed proportion, beta is given Ьу the ratio of stock to portfolio standard deviations of excess return: /3 = U" s/ U'p, Compare now an investment of $1 in the stock with the following "mimicking" strategy: invest $/3 in the portfolio and the balance $(1 - /3) in the risk-free asset, assuming that /3 < 1. For example, if beta is 0.5, then investing $0.50 in the portfolio and $0.50 in the riskless asset is а strategy that will gain or lose 0.5 percent of excess return for every 1 percent gain or loss in the portfolio excess return. The excess return of the mimicking strategy is beta times the excess return of the portfolio. The mim­ icking strategy will behave just like the stock up to the constant difference alpha. The mimicking strategy can Ье thought of as а "stock" with the given beta but an alpha of zero. Similarly, if /3 > 1, the mimicking strategy involves investing $/3 in the portfolio of which $(/3 - 1) is borrowed at the riskless rate. For example, if beta is 3, the mimicking portfolio involves investing $3 in the portfolio of which $2 is borrowed at the risk-free rate. This strategy will gain or lose 3 percent of excess return for every 1 percent gain or loss in the portfolio excess return. Again, the mimicking strategy will behave just like the stock up to the constant difference alpha. When should а stock Ье added to the portfolio if its return is perfectly correlated with that of the portfolio? Since, up to the constant alpha, the stock is just а substitute for the portfolio, adding $1 of the stock to the portfolio amounts to owning $/3 more of the portfolio. But owning more of the portfolio Ьу itself does not change its Sharpe Ratio. Therefore, adding the stock will increase the portfolio's Sharpe Ratio if the 4 Assume that you have $1 ofwealth invested in the portfolio. Then, adding an investment of$x in shares of the stock increases the portfolio variance to lТ} + х2 lТ�, where lТ} is the variance of the portfolio and х2 lТ� is the variance of the additional stock, weighted Ьу the number of dollars invested in the stock. Remember, the variance of а comЬination of uncorrelated risks equals the sum of the variances of the individual risks. The increase in portfolio risk (standard deviation as well as variance) is proportional to х2 , which implies that the change in portfolio risk is negligiЫe for small х. The $х needed to purchase the shares сап соте from holding less of the risk-free asset or Ьу borrowing at the risk-free rate. stock's expected excess retum exceeds that of the mimicking portfolio. This occurs if а > О or equivalently if Е5 - r_r > /3(Ер - r_r), meaning that the stock's risk premium must exceed beta times the portfolio risk premium. Тhе General Case: Adding а Stock that is lmperfecdy Correlated with the Existing Portfolio Suppose next that the retums on the stock and the portfolio are correlated to some degree (О< р < 1). In this case, the stock's retum can Ье separated into а retum component that is perfectly correlated with the portfolio and а retum component that is uncorrelated with the portfolio. Since the standard deviation of the stock is crs, the standard deviation of the perfectly correlated component of the stock's retum is pcr5.5 Thus, the beta of the perfectly correlated component of the stock's excess retum to the portfolio's excess retum is given Ьу the ratio of standard deviations: /3 = pcrs/ crpAs just discussed, the component of the stock's return that is perfectly corre­ lated with the portfolio is а substitute for the portfolio itself and can Ье mimicked through an investment of /3 in the portfolio and (1 - /3) in the riskless asset. The component of the stock's excess return that is uncorrelated with the portfolio can, at the margin, Ье diversified away and will thus have no effect on the risk of the portfolio. This component of return can Ье mimicked through an investment in the risk-free asset. We can therefore conclude that adding the stock to the portfolio will improve the Sharpe Ratio if the stock's risk premium exceeds the sum of the risk premia of the two mimicking portfolios: /3 (Ер - r1) for the perfectly correlated return component and zero for the uncorrelated return component. This insight estaЫishes а rule for improving the portfolio. Adding а marginal share of stock to а portfolio will increase the portfolio's Sharpe Ratio if the stock's alpha is positive, that is, if its risk premium satisfies Conversely, selling short а marginal share of the stock will increase the portfolio's Sharpe Ratio if the alpha is negative, Е5 - r1 < /3 (Ер - r1). The portfolio has the highest attainaЫe Sharpe Ratio if Е5 - r1 = /3(Ер - r1) for every stock-that is, if the risk premium for each stock is equal to beta times the risk premium for the portfolio as а whole. Тhе Capital Asset Pricing Model The rule for improving the Sharpe Ratio of а portfolio allows us to derive the Capital Asset Pricing Model in а straightforward and intuitive way. We begin with four assumptions. First, investors are risk averse and evaluate their investment portfolios 5 The correlation coefficient р is the "R" in "R-squared"-the fraction of the stock's variance that is attributaЫe to movements in the portfolio. If р < О, the standard deviation of the perfectly correlated component is lplas. solely in tenns of expected retum and standard deviation of retum measured over the same single holding period. Second, capital markets are perfect in several senses: all assets are infinitely divisiЫe; there are no transactions costs, short selling restrictions or taxes; infonnation is costless and availaЫe to everyone; and all investors can Ьопоw and lend at the risk-free rate. Third, investors all have access to the same investment opportunities. Fourth, investors all make the same estimates of individual asset ex­ pected retums, standard deviations of retum and the coпelations among asset retums. These assumptions represent а highly simplified and idealized world, but are needed to obtain the САРМ in its basic fonn. The model has been extended in many ways to accommodate some of the complexities manifest in the real world. But under these assumptions, given prevailing prices, investors all will detennine the same highest Sharpe Ratio portfolio of risky assets. Depending on their risk tolerance, each investor will allocate а portion of wealth to this optimal portfolio and the remainder to risk-free lending or boпowing. Investors all will hold risky assets in the same relative proportions. For the market to Ье in equilibrium, the price (that is, the expected retum) of each asset must Ье such that investors collectively decide to hold exactly the supply of the asset. If investors all hold risky assets in the same proportions, those proportions must Ье the proportions in which risky assets are held in the market portfolio-the portfolio comprised of all availaЫe shares of each risky asset. In equilibrium, therefore, the portfolio of risky assets with the highest Sharpe Ratio must Ье the market portfolio. If the market portfolio has the highest attainaЫe Sharpe Ratio, there is no way to obtain а higher Sharpe Ratio Ьу holding more or less of any one asset. Applying the portfolio improvement rule, it follows that the risk premium of each asset must satisfy Е5 - r1 = {3 (Ем - r1), where Е5 and Ем are the expected return on the asset and the market portfolio, respectively, and {3 is the sensitivity of the asset's return to the return on the market portfolio. We have just estaЫished the Capital Asset Pricing Model: In equilibrium, the expected return of an asset is given Ьу This formula is the one that Sharpe, Treynor, Lintner and Mossin successfully set out to find. It is the relationship between expected return and risk that is consistent with investors behaving according to the prescriptions of portfolio theory. If this rule does not hold, then investors will Ье аЫе to outperfonn the market (in the sense of obtaining а higher Sharpe Ratio) Ьу applying the portfolio improvement rule, and if sufficiently many investors do this, stock prices will adjust to the point where the САРМ becomes true. Another way of expressing the САРМ equation is Sharpe Ratio of Asset S = р Х Sharpe Ratio of the Market Portfolio. 6 Using the fact that that f3 = рст 8/ им, the equation Е5 = r1 + {3(Ем - r1) (Es - r1)/и5 = р(Ем - r1)/им, which is the expression in the text. 6 сап Ье rearranged to give In other words, in equilibrium, the Sharpe Ratio of any asset is no higher than the Sharpe Ratio of the market portfolio (since р :5 1). Moreover, assets having the same correlation with the market portfolio will have the same Sharpe Ratio. The Capital Asset Pricing Model tells us that to calculate the expected return of а stock, investors need know two things: the risk premium of the overall equity market Ем - r1 (assuming that equities are the only risky assets) and the stock's beta versus the market. The stock's risk premium is determined Ьу the component of its return that is perfectly correlated with the market-that is, the extent to which the stock is а substitute for investing in the market. The component of the stock's return that is uncorrelated with the market can Ье diversified away and does not command а risk premium. The Capital Asset Pricing Model has а number of important implications. First, perhaps the most striking aspect of the САРМ is what the expected return of an asset does not depend on. In particular, the expected return of а stock does not depend on its stand-alone risk. It is true that а high beta stock will tend to have а high stand-alone risk because а portion of а stock's stand-alone risk is deteпnined Ьу its beta, but а stock need not have а high beta to have а high stand-alone risk. А stock with high stand-alone risk therefore will only have а high expected return to the extent that its stand-alone risk is derived from its sensitivity to the broad stock market. Second, beta offers а method of measuring the risk of an asset that cannot Ье diversified away. We saw earlier that any risk measure for determining expected returns would have to satisfy the requirement that the risk of а portfolio is the weighted average of the risks of the holdings in the portfolio. Beta satisfies this requirement. For example, if two stocks have market betas of 0.8 and 1.4, respec­ tively, then the market beta of а 50/50 portfolio of these stocks is 1.1, the average of the two stock betas. Moreover, the capitalization weighted average of the market betas of all stocks is the beta of the market versus itself. The average stock therefore has а market beta of 1.0. On а graph where the risk of an asset as measured Ьу beta is on the horizontal axis and return is on the vertical axis, all securities lie on а single line-the so-called Securities Market Line shown in Figure 4. If the market is in equilibrium, all assets must lie on this line. If not, investors will Ье аЫе to improve upon the market portfolio and obtain а higher Sharpe Ratio. In contrast, Figure 3 presented earlier measured risk on the horizontal axis as stand-alone risk, the standard deviation of each stock, and so stocks were scattered over the diagram. But remember that not all of the stand-alone risk of an asset is priced into its expected return, just that portion of its risk, ри 8, that is correlated with the market portfolio. Third, in the Capital Asset Pricing Model, а stock's expected retum does not depend on the growth rate of its expected future cash flows. То find the expected retum of а company's shares, it is thus not necessary to carry out an extensive financial analysis of the company and to forecast its future cash flows. According to the САРМ, а11 we need to know about the specific company is the beta of its shares, а parameter that is usually much easier to estimate than the expected future cash flows of the firm. Figure 4 Тhе Securities Market Line (SМL) SML Market In equilibrium, all assets plot оп the SML Ем - 7i = slope of SМL ''' } Beta of market = 1.0 Beta 1s the САРМ Useful? The Capital Asset Pricing Model is an elegant theory with profound implications for asset pricing and investor behavior. But how useful is the model given the idealized world that underlies its derivation? There are several ways to answer this question. First, we can examine whether real world asset prices and investor portfolios conform to the predictions of the model, if not always in а strict quantitative sense, and least in а strong qualitative sense. Second, even if the model does not describe our current world particularly well, it might predict future investor behavior-for example, as а conse­ quence of capital market frictions being lessened through financial innovation, im­ proved regulation and increasing capital market integration. Third, the САРМ can serve as а benchmark for understanding the capital market phenomena that cause asset prices and investor behavior to deviate from the prescriptions of the model. Suboptimal Diversification Consider the САРМ prediction that investors all will hold the same (market) portfolio of risky assets. One does not have to look far to realize that investors do not hold identical portfolios, which is not а surprise since taxes alone will cause idiosyncratic investor behavior. For example, optimal management of capital gains taxes involves early realization of losses and deferral of capital gains, and so tахаЫе investors might react very differently to changes in asset values depending on when they purchased the asset (Constantinides, 1983). Nevertheless, it will still Ье а positive sign for the model if most investors hold broadly diversified portfolios. But even here the evidence is mixed. On one hand, popular index funds make it possiЫe for investors to obtain diversification at low cost. On the other hand, many workers hold concen­ trated ownership of company stock in employee retirement savings plans and many executives hold concentrated ownership of company stock options. One of the most puzzling examples of suboptimal diversification is the so- called home Ьias in international investing. In almost all countries, foreign own­ ership of assets is low, meaning that investors tend to hold predominantly home country assets. For example, in 2003, foreign ownership accounted for only 10 percent of puЫicly traded U.S. equities and 21 percent of puЫicly traded Japanese equities.Japanese investor portfolios therefore deviate significantly from the world equity market portfolio: they own the vast majority of their home country equities, but only а tiny share of U.S. equities. Ву contrast, and as shown in ТаЫе 1, an investor holding the world equity market portfolio would Ье invested 48 percent in U.S. equities and only 10 percent inJapanese equities. Why is suboptimal diversification so pervasive? Common explanations are that obtaining broad diversification can Ье costly, in terms of direct expenses and taxes, and that investors are subject to behavioral Ьiases and lack of sophistication. None of these reasons, if valid, would mean that the САРМ is not useful. The САРМ tells us that investors рау а price for being undiversified in that they are taking risks for which they are not being compensated. Thus, there exists the potential for port­ folio improvement, which in turn creates opportunities for investor education and financial innovation. Indeed, foreign ownership of equities in many countries has more than douЫed over the last 20 years, most likely due to the increased availabllity of low-cost vehicles to invest globally and greater investor appreciation of the need for diversification. Investors today seem to Ье much better diversified than in decades past, а trend that appears likely to continue. Perfonnance Measurement One of the earliest applications of the Capital Asset Pricing Model was to performance measurement of fund managers (Treynor, 1965; Sharpe, 1966; Jensen, 1968). Consider two funds, А and В, that are actively managed in the hope of outperforming the market. Suppose that the funds obtained returns of 12 per­ cent and 18 percent, respectively, during а period when the risk-free rate was 5 percent and the overall market returned 15 percent. Assume further that the standard deviation of funds А and В were 40 percent per annum and 30 percent per annum, respectively. Which fund had the better performance? At first glance, fund А had greater risk and а lower return than fund В, so fund В would appear to have been the better performing fund. However, we know from the САРМ that focusing on stand-alone risk is misleading if investors can hold diversified portfolios. То draw а firmer conclusion, we need to know how these funds are managed: Suppose that fund А consists of а high-risk but "market­ neutral" portfolio that has long positions in some shares and short positions in others, with а portfolio beta of zero. Fund В, on the other hand, invests in selected high beta stocks, with а portfolio beta of 1.5. Instead of investing in funds А and/ or В, investors could have held corre­ sponding mimicking or ''Ьenchmark" portfolios. For fund А, since its beta is zero, the benchmark portfolio is an investment in the risk-free asset; for fund В, the benchmark is а position in the market portfolio leveraged 1.5:1 with borrowing at the risk-free rate. The benchmark portfolios respectively would have returned 5 percent and 20 percent (= 5 percent + 1.5 Х (15 percent - 5 percent)). Fund Table 3 Evaluating Portfolio Managers with the САРМ Riskless asset Market portfolio FundA FundB Return Risk (S.D.) 5% 15% 12% 18% 0% 20% 40% 30% Beta о.о 1.0 о.о 1.5 Alpha 0% 0% 7% -2% А thus outperformed its benchmark Ьу 7 percent, while fund В underperformed its benchmark Ьу 2 percent, as shown in ТаЫе 3. In terms of the САРМ framework, funds А and В had alphas of 7 percent and -2 percent, respectively, where alpha is the difference between а fund's perfor­ mance and that predicted given the beta of the fund. Appropriately risk adjusted, fund A's performance (alpha = 7 percent) exceeded that of fund В (alpha = -2 percent). An investor who held the market portfolio would, at the margin, have obtained а higher return Jor the same risk Ьу allocating money to fund А rather than to fund В. 7 The key idea here is that obtaining high retums Ьу owning high beta stocks does not take skill, since investors can passively create а high beta portfolio simply through а leveraged position in the market portfolio. Obtaining high retums with low beta stocks is much harder, however, since such performance cannot Ье replicated with а passive strategy. Investors therefore need to assess performance based on retums that have been appropriately risk adjusted. The САРМ provides а clear framework for thinking about this issue. Тhе САРМ and Discounted Cash Иоw Analysis According to the САРМ, the appropriate discount rate for valuing the ex­ pected future cash flows of а company or of а new investment project is determined Ьу the risk-free rate, the market risk premium and the beta versus the market of the company or project. Accuracy in estimating these parameters matters greatly for real world decisionmaking since, for long-dated cash flows, an error in the discount rate is magnified manyfold when calculating the net present value. Beta is usually estimated with use of linear regression analysis applied to historical stock market returns data. Beta can in many circumstances Ье accurately measured this way even over а relatively short period of time, provided that there is sufficient high-frequency data. When the company or project being valued is not publicly traded or there is no relevant retum history, it is customary to infer beta from comparaЫe entities whose betas can Ье estimated. But measurement issues can arise even if the availabllity of market retums data is not an issue, for example when the covariance with 7 This assumes that the beta of the overall portfolio is held constant-by holding more of the market portfolio if money is allocated to fund А and less of the market portfolio if money is allocated to fund В. the market is time varying and when local stock market indexes are used as proxies for the broad market portfolio because the latter is not well specified. The hardest of all parameters to estimate is usually the market risk premium. The historical risk premium is estimated from the average of past returns and, unlike variance-related measures like beta, average returns are very sensitive to the beginning and ending level of stock prices. The risk premium must therefore Ье measured over long periods of time, and even this may not Ье sufficient if the risk premium varies over time. None of these measurement questions poses а proЫem for the САРМ per se, however. The market risk premium is common to all models of cash flow valuation, and its estimation needs to Ье performed regardless of the difficulty of the task. Provided that the САРМ is the "right" model, beta too needs to Ье estimated, irrespective of difficulty. Extensions of the САРМ The Capital Asset Pricing Model has been extended in а variety of ways. Some of the best-known extensions include allowing heterogenous beliefs (Lintner, 1969; Mer­ ton, 1987); eliminating the possibility of risk-free lending and borrowing (Вlack, 1972); having some assets Ье nonmarketaЫe (Mayers, 1973); allowing for multiple time periods and investment opportunities that change from one period to the next (Merton, 1973; Breeden, 1979); extensions to international investing (Solnik, 1974; Stulz, 1981; Adler and Dumas, 1983); and employing weaker assumptions Ьу relying оп arЬitrage pricing (Ross, 1976). In most extensions of the САРМ, по single portfolio of risky assets is optimal for everyone. Rather, investors allocate their wealth differentially among several risky portfolios, which across all investors aggregate to the market portfolio. То illustrate, consider the International Capital Asset Pricing Model. This model takes into account that investors have consumption needs particular to the country in which they are resident. Thus, British investors will worry about the purchasing power of pounds while American investors worry about the purchasing power of dollars, which means that British and American investors will differently assess the incremental contribution that any particular asset makes to portfolio risk. As а result, they will hold somewhat different portfolios. 8 In the basic САРМ, investors care about only one risk factor-the overall market. In this international version of the model, they are also concerned about real currency fluctuations. This insight leads to а model of expected returns involving not only the beta of an asset versus the overall market, but also the betas of the asset versus currency movements and any other risk that is viewed differently Ьу different investor segments. Almost all variants of the САРМ have а multi-beta expression for expected 8 British investors who own American assets will hedge а portion of their real pound/dollar exchange rate exposure Ьу borrowing in dollars and lending in pounds, and American investors who own British assets will hedge а portion of their real dollar/pound exchange rate exposure Ьу borrowing in pounds and lending in dollars. British and American investors thus will lend to and borrow from each other, and they will have opposite exposures to the dollar/pound exchange rate. return. They are derived from the same basic notions: 1) investors will hold portfolios that are optimized given their specific needs, constraints and risk pref­ erences; 2) in equilibrium, asset prices reflect these demands; and 3) assets with high expected returns are those that are correlated with any risk that а significant group of investors has been unaЫe to eliminate from their portfolios. Whether the basic САРМ or one of its multifactor extensions is the "correct" model of asset prices is ultimately an empirical question, one that is discussed in detail Ьу Fama and French in their companion paper in this journal. Initial tests of the САРМ Ьу Black, Jensen and Scholes (1972) and Fama and MacBeth (1973) supported the theory in that high beta stocks were found to have had higher returns than low beta stocks. However, the relationship between beta and average returns was not as steep as indicated Ьу the theoretical Securities Market Line. Since this early work, а vast body of research has looked for additional risk factors that affect expected returns. Most notaЫy, Fama and French (1992) find that adding а "value" factor and а "size" factor (in addition to the overall market) greatly improves upon the explanatory power of the САРМ. The pervasiveness of these findings in follow-up research across time and other countries provides strong evidence that more than one systematic risk factor is at work in determining asset prices. However, the value and size factors are not explicitly about risk; at best, they are proxies for risk. For example, size per se cannot Ье а risk factor that affects expected returns, since small firms would then simply comblne to form large firms. Another criticism of the Fama-French findings is that their value effect is based on giving equal weight to small and large companies and is much stronger than observed in capitalization-weighted value indexes. Until the risks that underlie the Fama-French factors are identified, the forecast power of their model will Ье in doubt and the applications will Ье limited. Conclusion The Capital Asset Pricing Model is а fundamental contribution to our under­ standing of the determinants of asset prices. The САРМ tells us that ownership of assets Ьу diversified investors lowers their expected returns and raises their prices. Moreover, investors who hold undiversified portfolios are likely to Ье taking risks for which they are not being rewarded. 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