One of the most important models in finance is the Capital Asset Pricing Model (CAPM).
The CAPM is an equilibrium model that assumes that all investors compose their asset portfolio on the basis of a trade-off between the expected return and the variance of the return on their portfolio. This implies that each investor holds a so-calledmean variance efficient portfolio, a portfolio that gives maximum expected return for a given vari- ance (level of risk). If all investors hold the same beliefs about expected returns and (co)variances of individual assets, and in the absence of transaction costs, taxes and trading restrictions of any kind, it is also the case that the aggregate of all individual portfolios, themarket portfolio, is mean variance efficient. In this case it can be shown that expected returns on individual assets are linearly related to the expected return on the market portfolio. In particular, it holds that21
E{rjt−rf} =𝛽jE{rmt−rf}, (2.77) where rjt is the risky return on assetjin periodt,rmt is the risky return on the market portfolio and rf denotes the riskless return, which we assume to be time invariant for simplicity. The proportionality factor𝛽jis given by
𝛽j= cov{rjt,rmt}
V{rmt} (2.78)
and indicates how strong fluctuations in the returns on assetjare related to movements of the market as a whole. As such, it is a measure of systematic risk (or market risk).
Because it is impossible to eliminate systematic risk through a diversification of one’s portfolio without affecting the expected return, investors are compensated for bearing this source of risk through a risk premiumE{rmt−rf}>0. Accordingly, (2.77) tells us that the expected return on any risky asset, in excess of the riskless rate, is proportional to its ‘beta’.
In this section, we consider the CAPM and see how it can be rewritten as a linear regression model, which allows us to estimate and test it. In Subsection 2.6.3 we use the CAPM to analyse the (fraudulent) returns on Bernard Madoff’s investment fund.
21Because the data correspond to different time periods, we index the observations byt,t=1,2, . . . ,T, rather thani.
k k A more extensive discussion of empirical issues related to the CAPM can be found in
Berndt (1991) or, more technically, in Campbell, Lo and MacKinlay (1997, Chapter 5) and Gouriéroux and Jasiak (2001, Section 4.2). More details on the CAPM can be found in finance textbooks, for example Elton, Gruber, Brown and Goetzmann (2014, Chapter 13).
2.7.1 The CAPM as a Regression Model
The relationship in (2.77) is anex anteequality in terms of unobserved expectations.
Ex post, we only observe realized returns on the different assets over a number of periods. If, however, we make the usual assumption that expectations are rational, so that expectations of economic agents correspond to mathematical expectations, we can derive a relationship from (2.77) that involves actual returns. To see this, let us define the unexpected returns on assetjas
ujt=rjt−E{rjt}, and the unexpected returns on the market portfolio as
umt=rmt−E{rmt}. Then, it is possible to rewrite (2.77) as
rjt−rf =𝛽j(rmt−rf) +𝜀jt, (2.79) where
𝜀jt=ujt−𝛽jumt.
Equation (2.79) is a regression model, without an intercept, where𝜀jt is treated as an error term. This error term is not something that is just added to the model, but it has a meaning, being a function of unexpected returns. It is easy to show, however, that it satisfies some minimal requirements for a regression error term, as given in (A7). For example, it follows directly from the definitions ofumtandujtthat it is mean zero, that is, E{𝜀jt} =E{ujt} −𝛽jE{umt} =0. (2.80) Furthermore, it is uncorrelated with the regressor rmt−rf. This follows from the definition of𝛽j, which can be written as
𝛽j= E{ujtumt} V{umt} (note thatrf is not stochastic), and the result that
E{𝜀jt(rmt−rf)} =E{(ujt−𝛽jumt)umt} =E{ujtumt} −𝛽jE{u2mt} =0.
From the previous section it then follows that OLS provides a consistent estimator for𝛽j. If, in addition, we impose assumption (A8) that 𝜀jt is independent ofrmt−rf and assumptions (A3) and (A4) stating that 𝜀jt does not exhibit autocorrelation or heteroskedasticity, we can use the asymptotic result in (2.74) and the approximate distributional result in (2.76). This implies that routinely computed OLS estimates, standard errors and tests are appropriate by virtue of the asymptotic approximation.
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ILLUSTRATION: THE CAPITAL ASSET PRICING MODEL 41
2.7.2 Estimating and Testing the CAPM
The CAPM describes the expected returns on any asset or portfolio of assets as a func- tion of the (expected) return on the market portfolio. In this subsection, we consider the returns on three different industry portfolios while approximating the return on the market portfolio by the return on a value-weighted stock market index. Returns for the period January 1960 to December 2014 (660 months) for the food, consumer durables and construction industries were obtained from the Center for Research in Security Prices (CRSP). The industry portfolios are value weighted and are rebalanced once every year.
While theoretically the market portfolio should include all tradeable assets, we shall assume that the CRSP value-weighted index is a good approximation. The riskless rate is approximated by the return on 1-month treasury bills. Although this return is time vary- ing, it is known to investors while making their decisions. All returns are expressed in percentage per month.
First, we estimate the CAPM relationship (2.79) for these three industry portfolios.
We thus regress excess returns on the industry portfolios (returns in excess of the risk- less rate) upon excess returns on the market index proxy, not including an intercept. This produces the results presented in Table 2.3. The estimated beta coefficients indicate how sensitive the value of the industry portfolios are to general market movements. This sen- sitivity is relatively low for the food industry, but fairly high for construction: an excess return on the market of, say, 10% corresponds to an expected excess return on the food and construction portfolios of 7.6 and 11.7%, respectively. It is not surprising to see that the durables and construction industries are more sensitive to overall market movements than is the food industry. Assuming that the conditions required for the distributional results of the OLS estimator are satisfied, we can directly test the hypothesis (which has some economic interest) that𝛽j=1 for each of the three industry portfolios. This results int-values of−10.00, 2.46 and 7.04, respectively, so that we reject the null hypothesis for each of the three industries. Because the intercept terms are suppressed, the goodness- of-fit measures in Table 2.3 are uncentredR2s as defined in (2.43). Some regression pack- ages would nevertheless reportR2s based on (2.42) in such cases. Occasionally, this can lead to negative values.
As the CAPM implies that the only relevant variable in the regression is the excess return on the market portfolio, any other variable (known to the investor when making his or her decisions) should have a zero coefficient. This also holds for a constant term. To check whether this is the case, we can re-estimate the above models while including an intercept term. This produces the results in Table 2.4. From these results, we can test the
Table 2.3 CAPM regressions (without intercept) Dependent variable:excess industry portfolio returns
Industry Food Durables Construction
excess market return 0.755 1.066 1.174
(0.025) (0.027) (0.025)
uncentredR2 0.590 0.706 0.774
s 2.812 3.072 2.831
Note: Standard errors in parentheses.
k k Table 2.4 CAPM regressions (with intercept)
Dependent variable:excess industry portfolio returns
Industry Food Durables Construction
constant 0.320 −0.120 −0.027
(0.110) (0.120) (0.111)
excess market return 0.747 1.069 1.174
(0.025) (0.027) (0.025)
R2 0.585 0.705 0.772
s 2.796 3.072 2.833
Note: Standard errors in parentheses.
validity of the CAPM by testing whether the intercept term is zero. For food, the appro- priatet-statistic is 2.92, which implies that we reject the validity of the CAPM at the 5%
level. The point estimate of 0.320 implies that the food industry portfolio is expected to have a return that is 0.32% per month higher than the CAPM predicts. The 95% confi- dence interval for this ‘abnormal return’ is given by (−0.106%, 0.535%). Note that the estimated beta coefficients are very similar to those in Table 2.3 and that theR2s are close to the uncentredR2s.
The R2s in these regressions have an interesting economic interpretation.
Equation (2.79) allows us to write
V{rjt} =𝛽j2V{rmt} +V{𝜀jt},
which shows that the variance of the return on a stock (portfolio) consists of two parts:
a part related to the variance of the market index and an idiosyncratic part. In economic terms, this says that total risk equals market risk plus idiosyncratic risk. Market risk is determined by𝛽jand is rewarded: stocks with a higher𝛽jprovide higher expected returns because of (2.77). Idiosyncratic risk is not rewarded because it can be eliminated by diversification: if we construct a portfolio that is well diversified, it will consist of a large number of assets, with different characteristics, so that most of the idiosyncratic risk will cancel out and mainly market risk matters. TheR2, being the proportion of explained vari- ation in total variation, is an estimate of the relative importance of market risk for each of the industry portfolios. For example, it is estimated that 58.5% of the risk (variance) of the food industry portfolio is due to the market as a whole, while 41.5% is idiosyn- cratic (industry-specific) risk. Because of their largerR2s, the durables and construction industries appear to be better diversified.
Finally, we consider one deviation from the CAPM that is often found in empirical work: the existence of a January effect. There is some evidence that, ceteris paribus, returns in January are higher than in any of the other months. We can test this within the CAPM framework by including a dummy in the model for January and testing whether it is significant. By doing this, we obtain the results in Table 2.5. Computing thet-statistics corresponding to the January dummy shows that for two of the three industry portfolios we do not reject the absence of a January effect at the 5% level. For the food industry, however, the January effect appears to be negative and statistically significant at the 5%
level (with at-value of−2.47). Consequently, the results do not provide support for the existence of a positive January effect.
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ILLUSTRATION: THE CAPITAL ASSET PRICING MODEL 43
Table 2.5 CAPM regressions (with intercept and January dummy) Dependent variable:excess industry portfolio returns
Industry Food Durables Construction
constant 0.400 −0.126 −0.077
(0.114) (0.126) (0.116)
January dummy −0.971 0.081 0.605
(0.393) (0.433) (0.399)
excess market return 0.749 1.069 1.173
(0.024) (0.027) (0.025)
R2 0.589 0.705 0.773
s 2.786 3.074 2.830
Note: Standard errors in parentheses.
2.7.3 The World’s Largest Hedge Fund
The Capital Asset Pricing Model is commonly used in academic studies to evaluate the performance of professional money managers. In these cases, the intercept of the CAPM is interpreted as a risk-adjusted performance measure. A positive intercept, typically referred to as ‘alpha’, reflects superior skill or information of the investment manager.
For example, Malkiel (1995) uses the CAPM to evaluate the performance of all equity mutual funds that existed in the period 1971–1991 and finds that, on average, mutual funds have a negative alpha (i.e. a negative estimated intercept), and that the proportion of funds with a significantly positive alpha is very small. Malkiel concludes that mutual funds tend to underperform the market, which is consistent with the idea that financial markets are very efficient.
Hedge funds typically challenge this view and argue that they can produce excess performance (positive alpha). Unfortunately, the performance data for hedge funds are less readily available than for mutual funds, over shorter histories, and are potentially subject to manipulation or even fraudulent. Bollen and Pool (2012) examine whether the presence of suspicious patterns in hedge fund returns raises the probability of fraud.
One of their potential red flags is a low correlation of hedge fund returns with standard asset classes.
In this subsection we illustrate this by considering the returns produced by Bernard Madoff. A former chairman of the board of directors of the NASDAQ stock market, Madoff used to be a well-respected person on Wall Street. Madoff Investment Securi- ties was effectively running one of the largest hedge funds in the world. Many years in a row, the returns reported by Madoff were incredibly good. However, already in 1999, Harry Markopolos, who presented evidence of the Madoff Ponzi scheme to the Securities and Exchange Commission (SEC), suspected that the Madoff returns were not real and that the world’s largest hedge fund was a fraud. Despite the many red flags, the SEC did not uncover the massive fraud.22One of the red flags described by Markopolos is that Madoff’s returns had a correlation of only 0.06 with the S&P 500, whereas the supposed
22See Markopolos (2010) for an account of how Markopolos uncovered Madoff’s scam, years before it actually fell apart.
k k Table 2.6 CAPM regression (with intercept) Madoff’s returns
Dependent variable:excess returns Fairfield Sentry Ltd.
Variable Estimate Standard error t-ratio
constant 0.5050 0.0467 11.049
excess market return 0.0409 0.0107 3.813 s=0.6658 R2=0.0639 R̄2=0.0595 F=14.54
split-strike conversion strategy should feature a correlation close to 0.50. We consider the returns on Fairfield Sentry Ltd, which was one of the feeder funds of Madoff Invest- ment Securities. Even a simple inspection of the return series produces some suspicious results. Over the period December 1990–October 2008 (T=215), the average monthly return was 0.842% with a surprisingly low standard deviation of only 0.709%. More- over, the number of months with a negative return was as low as 16, corresponding to less than 7.5% of the periods. In comparison, during the same period the stock market index produced a negative return in 39% of the months.
We shall now investigate to what extent the CAPM is able to explain Madoff’s returns, realizing that large positive intercept terms, that is, large alphas, are excluded by the CAPM and quite unlikely in practice. To do so we regress the excess returns on Fairfield Sentry upon a constant and the excess returns on the market portfolio. The results are given in Table 2.6.
Indeed, the Madoff fund has an extremely low exposure to the stock market, with an estimated beta coefficient of only 0.04. This is confirmed by the extremely lowR2 of 6.4%. The fund also produces a high intercept term of 0.505%per month, with a sus- piciously hight-ratio of 11.05, corresponding to a very narrow 95%confidence interval of (0.415%, 0.595%). This suggests that Madoff’s fund was able to reliably outperform the market by 5.0 to 7.1%per year. Despite the fact that the CAPM explains very lit- tle of the variation in Madoff’s returns, the estimated standard deviation of the error term,s, is as low as 0.67%. Apparently, both the systematic risk of the fund is low, as well as its idiosyncratic risk, but nevertheless its returns are very high. From many perspectives, the returns on this fund were too good to be true, and in fact they were not real either.
On December 10, 2008, Madoff’s sons told authorities that their father had confessed to them that Madoff Investment Securities was a fraud and ‘one big lie.’ Bernard Madoff was arrested by the FBI on the following day. In 2009, he was sentenced to 150 years in prison.