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Concluding Remarks

With academics debating the value of the CAPM, what are companies that now use it in their capital budgeting process to do? Maybe nothing different. Obviously, capital budgeting decisions were made before there was a CAPM, and they can be made again without it. But the data seem to suggest that those who choose to use the CAPM now despite the academic debate will actually not be getting worthless advice. Recall our Chart 5, where we plotted the return/beta relationship for four types of assets over a period as long as 66 years. The result was more-or-less a positively sloped, straight line, just as the CAPM predicts. As we saw, that straight-line relationship breaks down over shorter time periods, and academics continue to debate why that is so. But for now, for those interested in the longer view, the CAPM still seems to have something to offer.

a The authors thank Jaeuk Khil for research assistance and Gordon Alexander, V. V. Chari, David Marshall, David Runkle, and especially John Boyd for helpful comments.

For a discussion of how corporate managers use models like the CAPM, see the box displayed later in the article. According to the description given by the source for these data, Ibbotson Associates 1992; the common stock returns are based on Standard & Poor's composite index. This index-includes 500 stocks now, but it included only 90 stocks before March 1957: For the period 1926-81, the small-firm stock index consisted of stocks in the smallest quintile of firms in terms of their market value of equity (their share price times shares outstanding) listed in the New York Stock Exchange; the portfolio composition is rebalanced once-every five years. Starting in 1982, the small-firm stock index corresponds to the Dimensional Fund Advisors' Small Company Fund. For the period 1926-76, the total returns on long-term U.S. government bonds are from the Center for Research in Security Prices (CRSP) at the University of Chicago Graduate School of Business. Each year one bond portfolio is constructed with a maturity of about 20 years. For the period 1977-91, data from the Wall Street Journal are used to compute the total returns on bond funds. For U.S. Treasury bill returns, data from the CRSP U.S. government bond file are used through 1976. Data reported in the Wall Street Journal are used for the period thereafter Table 1 also reports the implied annual returns that would produce those dollar values. To calculate these returns, we flint calculate the value of the asset if $1 were invested in the first year of the period. We then raise that value to the power of 1 over the number of years for the period and subtract 1. Note that we could have started our derivations of equation (10) by using the fact that investors trade off expected returns (that is, means) and risk (that is, variances) when making portfolio decisions. Suppose that investors are indifferent between assets that yield the same return/risk ratios: that is, suppose that (dR/dxi)/(dS/dxii)|z = 0, x[sub p] = 1is constant for all i, where R = xiRi + xpRp - x1,R0. Then (ERi-R0) betaiimust be constant; hence, equation (10) holds. For a discussion on computing the sampling errors associated with fire estimates of the coefficients in the cross-sectional regression, see Shahken 1992 and Jagannathan and Wang, forthcoming. For a description of better alternatives for econometric evaluation of the CAPM that rely on either the method of maximum likelihood or the generalized method of moments, see Gibbons 1982, Stambaugh 1982, Shahken 1985, Mackinlay and Richardson 1991, and Jagannathan and Wang 1993. Support the relation being estimated is yt = gammaxt + ur. If we observe Xt = xt + vt rather than just xt where vt is measurement error uncorrelated with xt then the least squares estimate for gamma will be biased toward zero. The larger is the variance of vt, the greater is the bias. These figures are from Fama and French's (1992) regressions of individual NYSE stocks on beta for 1941-90. Chari, Jagannathan, and Ofer (1988) also point out this bias in the Compustat Compustat claims that it rarely adds more than two years of back data when it adds a firm to its list. In view of this, in their follow-up article, Fama and French (1993) omit the flint two years of data, but they still find that average returns are strongly related to the book-to-market equity ratio in the cross section. Hence, the reason for this effect is still unknown.

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Table 1 Financial Asset Returns and Inflation During 1926-91 Legend for Chart:A - Type of Calculation and Time PeriodB - Stocks, S& 500 C - Stocks, Small FirmsD - U.S. Treasury, BondsE - U.S. Treasury, Bills F - Consumer Price Index A B C D E F Annual Rate of Return[a]Average 1926-91 11.94% 16.05% 4.94% 3.64% 3.11% 1926-75 10.89 14.71 3.14 2.30 2.29 1976-80 14.17 35.55 2.27 7.51 8.85 1981-91 15.71 13.27 14.14 7.98 4.25Variability 1926-91 20.22% 31.02% 7.62% .94% 2.01% 1926-75 21.46 33.59 5.38 .61 2.15 1976-80 14.48 25.87 11.16 .82 1.14 1981-91 16.31 18.25 12.41 .75 .97 Result of $1 invested Start of PeriodValue at End of Period 1926-91 $675.59 $1,847.63 $20.95 $11.01 $7.67 1926-75 73.86 109.34 4.29 3.14 3.08 1976-80 1.92 4.89 1.09 1.45 1.55 1981-91 4.81 3.53 4.33 2.40 1.59 Implied Annual Growth Rate[c] 1926-91 10.38% 12.07% 4.72% 3.70% 3.14% 1926-75 8.99 9.84 2.95 2.32 2.22 1976-80 13.95 37.35 1.68 7.77 9.21 1981-91 15.34 12.14 14.24 8.28 4.33

a The annual rate of return is the assets monthly return multiplied by 12.

b The variability of the return is its standard deviation multiplied by the square root of 12.

c 'The implied annual growth rate is calculated by this formula: (Value of $1 at end of period)1/ n-1, where n is the number of years in the period.

Source: Ibbotson Associates 1992

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According to the CAPM, expected returns vary across assets only because the assets' betas are different. Hence, one way to investigate whether the CAPM adequately caprares all important aspects of reality is to test whether other asset-specific characteristics can explain the cross-sectional differences in average returns that are unrelated to cross-sectional differences in beta. To do this, additional terms are added to equation (15):

(16) rp = gamma0 + gamma1 bp + gamma2 psip + epsilonp.

The vector psip in (16) corresponds to additional factors assumed to be relevant for asset pricing. In empirical evaluations of the CAPM, researchers want to know if gamma2 = 0 holds -- at is, if beta is the only characteristic that matters.

Classic Support

One of the earliest empirical studies of the CAPM is that of Black, Jensen, and Scholes (1972). They find that the data are consistent with the predictions of the CAPM, given the fact that the CAPM is an approximation to reality just like any other model.

Black, Jensen, and Scholes (1972) use all of the stocks on the NYSE during 1931--65 to form 10 portfolios with different historical beta estimates. They regress average monthly excess returns on beta. Chart 4 shows their fitted relation between beta and the average excess monthly return (where the risk-free asset is the 30-day T-bill) for these 10 portfolios and a proxy for the total market portfolio. The average monthly excess rerum on the market proxy used in the study is 1.42 percent. The estimated slope for the resulting regression line is 1.08 percent instead of 1.42 percent as predicted by the CAPM. The estimated intercept is 0.519 percent instead of zero as predicted by the CAPM. The t-statistics that Black, Jensen, and Scholes report indicate that the slope and the intercept of their regression line are significantly different from their theoretical values.

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This does not necessarily mean that the data do not support the CAPM, however. As Black (1972,1993) points out, these results can be explained in two plausible ways. One is measurement and model specification error that arises due to the use of a proxy instead of the actual market portfolio. This error biases the regression line's estimated slope toward zero and its estimated intercept away from zero.[6] The other plausible explanation is simpler: if no risk-free asset exists, then the CAPM does not predict an intercept of zero. In fact, Black, Jensen, and Scholes (1972) conclude that the data are consistent with Black's (1972) version of the model [equation (11)].

To illustrate the empirical method used in the Black, Jensen, and Scholes study, let's evaluate the CAPM using the sample data on stocks, bonds, and bills that we described earlier. In Chart 5, we plot the average returns of those assets for the period from 1926 to 1991 against their estimated betas. These estimates of beta--as well as those for the subperiods -- are reported in Table 3. We also fit a straight line to the data by running a linear regression. Notice in Chart 5 that the relation between average return and beta is very close to linear and that portfolios with high (low) betas have high (low) average returns. This positive relationship is consistent with the CAPM prediction and the findings reported by Black, Jensen, and Scholes.

Another classic empirical study of the CAPM is by Fama and MacBeth (1973). They examine whether there is a positive linear relation between average return and beta and whether the squared value of beta and the volatility of the return on an asset can explain the residual variation in average returns across assets that is not explained by beta alone. Using return data for the period from 1926 to 1968, for stocks traded on the NYSE, Fama and MacBeth find that the data generally support the CAPM.

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Challenges

The CAPM thus passed its first major empirical tests. In 1981, however, a study suggested that it might be missing something. A decade later, another study suggested that it might be missing everything, and the debate about the CAPM's value was on.

• What About Firm Size?

Banz (1981) tests the CAPM by checking whether the size of the firms involved can explain the residual variation in average returns across assets that is not explained by the CAPM's beta. Banz challenges the CAPM by showing that size does explain the cross-sectional variation in average returns on a particular collection of assets better than beta. He finds that during the 1936-75 period, the average return to stocks of small finns (those with low values of market equity) was substantially higher than the average return to stocks of large firms after adjusting for risk using the CAPM. This observation has become known as the size effect.

Banz (1981) uses a procedure similar to the portfolio-grouping procedure of Black, Jensen, and Scholes (1972). The assets are first assigned to one of five subgroups, based on their historical betas. Stocks in each of the subgroups are then assigned to five further subgroups, based on the market value of the firms' equities. This produces total of 25 portfolios. Portfolios are updated at the end of each year. Banz uses finns on the NYSE and estimates the cross-sectional relation between return, beta, and relative size -- that is, our equation (16) with psip equal to the relative size of the pth portfolio. With this procedure, then, in (16), gamma0 is the rate of return for a portfolio with beta equal to zero, and gamma1 and gamma2 are risk premiums for beta and size risks, respectively.

Banz (1981) reports estimates for gamma0 - R0 and gamma1 -(Rm-R0), where R0 and Rm - R0 are the intercept and the slope predicted by the CAPM. The idea is to report deviations from theory. Theory predicts that gamma0 = R0, gamma1 = Rm R0 and gamma2 = 0. If deviations from theory are statistically significant (if the t-statistics are large in absolute value), then Banz would conclude that the CAPM is misspecified. For the entire period, 1936-75, Banz obtains the following estimates (and t-statistics): gamma0 - R0 = 0.0045 (2.76), gamma1 -(Rm-R0) = -0.00092 (-1.0), and gamma2 = -0.00052 (-2.92), where Rm is a measure of the market return. Because the t-statistic for gamma2 is large in absolute value, Banz concludes that the size effect is large and statistically significant. The fact that the estimate for gamma2 is negative implies that stocks of firms with large market values have had smaller returns on average than stocks of small firms. From these results, relative size seems to be able to explain a larger fraction of the cross-sectional variation in average rerum than beta can.

To assess the importance of these results, Banz (1981) does one additional test. He constructs two portfolios, each with 20 assets. One portfolio contains only stocks of small finns, whereas the other contains only stocks of large firms. The portfolios are chosen in such a way that they both have the same beta. Banz finds that, during the time period 1936-75, the small-firm portfolio earned on average 1.48 percent per month more than the large-firm portfolio, and the differences in returns are statistically significant. Thus, the CAPM seems to be missing a significant factor: firm size.

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• Is Beta Dead?

The general reaction to Banz's (1981) finding that the CAPM may be missing some aspect of reality was, Of course: since the CAPM is only an abstraction from reality, expecting it to be exactly right is unreasonable. While the data may show some systematic deviations from the CAPM, these are not economically important enough to reject it. This view has been challenged by Fama and French (1992). They show that Banz's finding may be economically so important that it questions the validity of the CAPM in any economically meaningful sense.

Fama and French (1992) estimate the relation in equation (16) for the period from July 1963 to December 1990 with psip equal to size. They group stocks for firms listed on the NYSE, AMEX, and NASDAQ (the National Association of Securities Dealers Automated Quotations) into 10 size classes and then into 10 beta classes, for a total of 100 portfolios. They obtain estimates of gamma1 = -0.37 with a t-statistic of -1.21 and gamma2 = -0.17 with a t-statistic of -3.41. Furthermore, even when they include only beta in the regression equation [equation (15)], they do not find a significantly positive slope; their estimate for gamma1 is -0.15 with a standard error of 0.46. However, the size effect is significant with or without betas. Thus, their estimates indicate that, for a large collection of stocks, beta has no ability to explain the cross-sectional variation in average returns, whereas size has substantial explanatory power.

Fama and French (1992) also consider the ability of other attributes to account for this cross-sectional variation. When they include the ratio of the book value of a firm's common equity to its market value as an explanatory variable in addition to size, they find that this ratio can account for a substantial portion of the cross-sectional variation in average returns. In fact, book-to-market equity appears to be more powerful than size.

What is so surprising about these results is that Fama and French (1992) use the same procedure as Fama and MacBeth (1973) but reach a very different conclusion: Fama and MacBeth find a positive relation between return and risk, and Fama and French find no relation at all. Fama and French attribute the different conclusions to the different sample periods used in the two studies. Recall that Fama and MacBeth (1973) use stock returns for 192668, whereas Fama and French (1992) use stock returns for 1963-90. When Fama and French rerun their regressions for 1941-65, they find a positive relationship between average return and beta.

The sensitivity of the conclusions to the sample period used can be illustrated using our four-asset data set. Suppose we repeat the exercise of Chart 5 for several subperiods. In Chart 6, we plot the average returns of our four types of assets for the first subperiod, 1926-75, against their estimated betas. A straight line is fit to the data by running a linear regression. Notice that Chart 6 is very similar to Chart 5, which includes the entire sample period. In both charts, we see a positive, linear relationship between average return and beta.

For the subperiods 1976-80 and 1981-91, however, we do not see that relationship. Consider first the plot in Chart 7 for the period 1976-80. In these years, small-firm stocks gave an usually higher return of 35.6 percent while the S& 500 gave only a more-usual 14.2 percent. Meanwhile, Treasury bills did much better than usual, and Treasury bonds did worse. Consider next the plot in Chart 8 for the period 1981-91. Notice that the small-firm effect disappeared in this period. The S& 500 stocks yielded an average return of 15.7 percent, and the return on small stocks was only 13.3 percent. Yet the two types of assets have approximately the same beta value. This fact is counter to the prediction of the CAPM. Thus, although we find empirical support for the CAPM over a long horizon (1926-91 or 1926-75), there are periods in which we do not find it.

The evidence against the CAPM can be summarized as follows. First, for some sample periods, the relation between average rerum and beta is completely flat. Second, other explanatory variables such as firm size (market equity) and the ratio of book-to-market equity seem to do better than beta in explaining cross-sectional variation in average asset returns.

Responses

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Because this method uses estimates of the expected return and beta, the relation being examined using data is not (10) or (11) but rather

(15) rp = gamma0 + gamma1bp + epsilonp

where rp is an estimate of the expected excess return on portfolio p (the difference between the rerum on the portfolio and the rerum on a risk-free asset); bp is an estimate of beta for portfolio p; gamma1 is the market price of risk, the risk premium for bearing one unit of beta risk; gamma0 is the zero-beta rate, the expected rerum on an asset which has a beta of zero; and epsilonp is a random disturbance term in the regression equation. Black, Jensen, and Scholes (1972) use time series data on returns to construct a sample average forte (re = sigmaT, sub t = 1r=p, t/T, where rp, t is the excess rerum at time t). However, there are problems with the standard errors on gamma0 and gamma1 obtained by a least squares regression of average excess returns on estimated betas. Therefore, Black, Jensen, and Scholes suggest computing the standard errors of the parameters in the cross-sectional regression in the following way: First run a cross-sectional regression for each period for which data on returns are available. This procedure generates a time series of parameter estimates. Then use the standard deviation of the estimated time series of parameters as the standard error of the parameter in the cross-sectional regression.[5]

For the original Sharpe (1964) and Lintner (1965) version of the CAPM, gamma0 should be equal to zero and gamma1 should be equal to the risk premium for the market portfolio. For the Black (1972) version of the CAPM, given in equation (11), gamma0 is not necessarily equal to zero. If we take a parameter estimate and divide by its standard error, we can construct a t-statistic for that coefficient. If the absolute value of the t-statistic is large (greater than 2), then the coefficient is said to be statistically different from zero. Usually, empirical tests of the CAPM are based on the t-statistics of the coefficients in the regression equation (15).

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The Tests

Now we want to see how the CAPM measures up to the data. As we shall see, there's some debate about that.

Methods

If expected returns and betas were known, then all we would have to do to examine the empirical support for the CAPM is to plot the return and beta data against each other. Unfortunately, neither of these is known. We have to form estimates of them to use in empirical tests. We do this by assuming that sample analogs correspond to population values plus some random noise. The noise is typically very large for individual assets, but less for portfolios. To understand why noise creates problems, notice that two portfolios with measured betas that are very different could well have the same population betas if the measurement error is very large. The objectives are to have sufficient dispersion in asset betas and to measure this dispersion sufficiently precisely.

Black, Jensen, and Scholes (1972) came up with a clever strategy that creates portfolios with very different betas for use in empirical tests. They estimate betas based on history (by regressing historical returns on a proxy for the market portfolio), sort assets based on historical betas, group assets into portfolios with increasing historical betas, hold the portfolios for a selected number of years, and change the portfolio composition periodically. As long as historical betas contain information about population betas, this procedure will create portfolios with sufficient dispersion in betas across assets.

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The Model

The CAPM was developed, at least in part, to explain the differences in risk premium across assets. According to the CAPM, these differences are due to differences in the riskiness of the returns on the assets. The model asserts that the correct measure of riskiness is its measure--known as beta--and that the risk premium per unit of riskiness is the same across all assets. Given the risk-free rate and the beta of an asset, the CAPM predicts the expected risk premium for that asset. In this section, we will derive a version of the CAPM. In the next section, we will examine whether the CAPM is actually consistent with the average rerum differences.

To derive the CAPM, we start with the simple problem of choosing a portfolio of assets for an arbitrarily chosen investor. To set up the problem, we need a few definitions. Let R0 be the return (that is, one plus the rate of return) on the risk-free asset (asset 0). By investing $1, the investor will get $R0 for sure. In addition, assume that the number of risky assets is n. The risky assets have returns that are not known with certainty at the time the investments are made. Let alphai be the fraction of the investor's initial wealth that is allocated to asset i. Then Ri is the return on asset i. Let Rm be the return on the entire portfolio (that is, sigman, sub i=0) alphai Ri). Here Ri is a random variable with expected value ERi and variance var(Ri), where variance is a measure of the volatility of the return. The covariance between the return of asset i and the return of asset j is represented by cov(Ri, Rj). Covariance provides a measure of how the returns on the two assets, i and j, move together.

Suppose that the investor's expected utility can be represented as a function of the expected return on the investor's portfolio and its variance. In order to simplify notation without losing generality, assume that the investor can choose to allocate wealth to three assets: i = 0, 1, or 2. Then the problem is to choose fractions alpha0, alpha1, and alpha2 that maximize

(1) V(ERm,var(Rm)))

subject to

(2) alpha0 + alpha1 + alpha2 = 1

(3) ERm = alpha0R0 + alpha1ER1 + alpha2ER2

(4) var (Rm) = alpha2, sub 1 var (R1) + alpha2, sub 2 var (R2) + 2 alpha1 alpha2 cov (R1, R2.

The objective function V is increasing in the expected return, deltaV/deltaERmm > 0; decreasing in thee variance of the return deltaV/delta var(R[sub m) < 0; and concave. These properties imply that there is a trade-off between expected returns and the variance of returns. The constraint n equation (2) ensures that the fractions sum to 1. Equations (3) and (4) follow from the definition of the rate of return on the wealth portfolio of the investor, Rm.

Substituting 1 - alpha1 - alpha2 for alpha0 in equation (1) and taking the derivative of V with respect to alpha1 and alpha2 yields the following conditions that must hold at an optimum:

(5) (ER1 - R0) V1 + 2[alpha1 var(R1) + alpha2 cov (R1, R2) V2 = 0

(6) (ER1 - R0) V1 + 2[alpha2 var(R1) + alpha2 cov (R1, R2) V2 = 0

where Vj is thee partial derivative of V with respect to its jth argument, for j = 1, 2. Now consider multiplying equation (5) by alpha1 and equation (6) by alpha2 and summing the results:

(7) [alpha1 (ER1 - R0 + alpha2 (ER2 -R0) V1 + 2 {alpha1 var (R1) + alpha2 cov (R1, R2) + alpha2 [alpha2 var (R2 + alpha1 cov (R1, R[sub 2)]} V2 = 0

Using the definitions of ERmmm and var(Rm), we can write this more succinctly:

(8) (ERm - R0 V1 + 2var (Rm) V2 = 0.

The expressions in (5), (6), and (8) can all be written as explicit functions of the ratio V2/V1, and then the first two expressions [from (5) and (6)] can be equated to the third [from (8)]. This yields the following two relationships:

(9) ERi - R0 = [cov(Rj, Rm)var(Rm)](ERm- R0)

for i = 1, 2. In fact, even for the more general case, where n is not necessarily equal to 2, equation (9) holds. Let cov(Ri, Rm)/var(Rm) be the beta of asset i, or betai. Then we have

(10) ER1 = R0 + (ERm - R0) Betai

for all i = 1, . . . , n.

A portfolio is said to be on the mean-variance frontier of the return/variance relationship if no other choice of weights alpha0, alphaj (for j = 1, 2, . . . , n) yields a lower variance for the same expected return. The portfolio is said to be on the efficient part of the frontier if, in addition, no other portfolio has a higher expected rerum. The optimally chosen portfolio for the problem in equations (1)-(4) has this property. In fact, equation (10) will continue to hold if the return Rm is replaced by the return on any mean-variance efficient portfolio other than the risk-free asset.

Note that the return Rm in (10) is the return for one investor's wealth portfolio. But equation (10) holds for every mean-variance efficient portfolio, and V need not be the same for all investors. A property of mean-variance efficient portfolios is that portfolios of them are also mean-variance efficient. If we define the market portfolio to be a weighted sum of individual portfolios with the weights determined by the fractions of total wealth held by individuals, then the market portfolio is mean-variance efficient too. Therefore, an equation of the form given by (10) also holds for the market portfolio.

In fact, equation (10) with Rm equal to the return on the market portfolio is the key relation for the CAPM. This relation implies that all assets i have the same ratios of reward, measured as the expected return in excess of the risk-free rate (ERi - R0), to risk (betai). This is consistent with the notion that investors trade off return and risk.

In specifying the problem of a typical investor [in (1)-(4)], we assumed that a risk-free asset is available. If we drop this assumption and set alpha0 = 0 from the start, then we obtain a slightly different relationship between return and risk than is given in (10). In particular, Black (1993) shows that without a risk-free asset, expected returns on the risky assets satisfy this relationship:

(11) ER1 = ERz + (ERm - Rz)Betai

where Rz is the return on a zero-beta portfolio [that is, cov(RzRm = 0], Rm is the return on the market portfolio, and betai = cov (rj, Rm/var (Rm).

We now provide an interpretation of beta in (10) or (11) as a measure of the asset's contribution to portfolio risk. Consider a portfolio p of assets that earns return Rp and has standard deviation Sp = (var Rp)1/2. Let the standard deviation of an arbitrary asset i be Si and the covariance between asset i's return and that of the portfolio be Ci,p. Now consider a new portfolio with xi invested in asset i, -xi invested in the risk-free asset, and xp invested in the original portfolio. That is, consider modifying the portfolio of an investor who currently holds xp in portfolio p by borrowing $xi and investing it in asset i. The standard deviation of the new portfolio is then

(12) S = (x2, sub i S2, sub i + x2, sub p S2, sub p + 2xi xp Ci, p)1/2.

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Note that the derivative of S with respect to x[sub i] is

(13)dS/dxi = (xiS2, sub ii + xpCi,p t,)/S.

This derivative measures how much the standard deviation (or risk) of the whole portfolio changes with a small change in the amount invested in asset i. If we evaluate this derivative at xi = 0 and Xp = I, then we find that

(14) dS/dxi|x i = 0, x[sub p - 1]] = Ci,p/Sp = (Ci,p/S2, sub p)Sp = betai Sp.

Notice that dS/dxi = betaiSp. That is, at the margin, an additional dollar invested in asset i (by borrowing the dollar) increases the standard deviation of the portfolio by betaiSp and not by Si.[4] Since Sp does not depend on the particular asset i, betai measures the relevant risk up to a scale multiple. In other words, when assets are held in a portfolio, the right measure of the increase in the portfolio risk due to an additional dollar of investment in the asset is the beta of the asset, not the volatility of its return.

To see this more clearly, consider the following example. Suppose an investor is holding $1,000 in a portfolio that includes stocks of all firms listed on the New York Stock Exchange (NYSE) and the American Stock Exchange (AMEX), where the investment proportions are the same as the relative market capitalization of the stocks of the firms. Suppose that all dividends are reinvested in that portfolio. Now suppose that the investor borrows $1 and invests in stocks of one of the randomly selected 11 firms listed in Table 2. There we report the sample means and the sample standard deviations of the monthly percentage rates of return for these 11 stocks along with their sample betas, computed with respect to the index of all stocks on the NYSE and AMEX (the total portfolio): We also report there the change in the total portfolio's standard deviation with a $1 increase in the holdings of any of the stocks. If, as we have found above, dS/dxi = betaiSp,, then we should observe that across stocks those changes (deltaS/deltaxi) are a scale multiple of the betas for the 11 stocks. Chart 2 plots the incremental standard deviation, deltaS/deltaxi, against the beta for each asset i. Notice that the points lie on a positively sloped straight line; that is, the beta of an asset does measure the incremental risk. Chart 3 plots delta S/deltaxi against Si. Notice that this relationship has no particular pattern; that is, the volatility of the return on the asset is not the right measure of its riskiness.

When the CAPM assumptions are satisfied, everyone in the economy will hold all risky assets in the same proportion. Hence, the betas computed with reference to every individual's portfolio will be the same, and we might as well compute betas using the market portfolio of all assets in the economy. The CAPM predicts that the ratio of the risk premium to the beta of every asset is the same. That is, every investment opportunity provides the same amount of compensation for any given level of risk, when beta is used as the measure of risk.