The CAPM Debate

This article describes the academic debate about the usefulness of the capital asset pricing model (the CAPM) developed by Sharpe and Lintner. First the article describes the data the model is meant to explain—the historical average returns for various types of assets over long time periods. Then the article develops a version of the CAPM and describes how it measures the risk of investing in particular assets. Finally the article describes the results of competing studies of the model’s validity. Included are studies that support the CAPM (Black; Black, Jensen, and Scholes; Fama and MacBeth), studies that challenge it (Banz; Fama and French), and studies that challenge those challenges (Amihud, Christensen, and Mendelson; Black; Breen and Korajczyk; Jagannathan and Wang; Kothari, Shanken, and Sloan). The article concludes by suggesting that, while the academic debate continues, the CAPM may still be useful for those interested in the long run. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. Most large U.S. companies have built into their capital budgeting process a theoretical model that economists are now debating the value of. This is the capital asset pricing model (the CAPM) developed 30 years ago by Sharpe (1964) and Lintner (1965). This model was the first apparently successful attempt to show how to assess the risk of the cash flow from a potential investment project and to estimate the project’s cost of capital, the expected rate of return that investors will demand if they are to invest in the project. Until recently, empirical tests of the CAPM supported the model. But in 1992, tests by Fama and French did not; they said, in effect, that the CAPM is useless for precisely what it was developed to do. Since then, researchers have been scrambling to figure out just what’s going on. What’s wrong with the CAPM? Are the Fama and French results being interpreted too broadly? Must the CAPM be abandoned and a new model developed? Or can the CAPM be modified in some way to make it still a useful tool? In this article, we don’t take sides in the CAPM debate; we merely try to describe the debate accurately. We start by describing the data the CAPM is meant to explain. Then we develop a version of the model and describe how it measures risk. And finally we describe the results of competing empirical studies of the model’s validity. The Facts Let’s start by examining the facts: the historical data on average returns for various types of assets. We focus on historical average returns because the averages of returns over long time horizons are good estimates of expected returns. And estimating expected returns for different types of assets is a significant part of what the CAPM is supposed to be able to do well. Table 1 provides a summary of the average return history for four types of assets: stocks for large and small firms, long-term U.S. Treasury bonds, and short-term U.S. Treasury bills. For each sample period, we report average annual rates of return. If investors have rational expectations, then the average returns over a fairly long horizon should be a reasonable measure of expected returns. Notice that the historical returns on different types of assets are substantially different. The fact that investors did hold these assets implies that investors would demand vastly different rates of return for investing in different projects. To the extent that the assets are claims to cash flows from a variety of real activities, these facts support the view that the cost of capital is very different for different projects. During the 66-year period from 1926 to 1991, for example, Standard & Poor’s 500-stock price index (the S&P 500) earned an average annual return of 11.9 percent whereas U.S. Treasury bills (T-bills) earned only 3.6 percent. Since the average annual inflation rate was 3.1 percent during this period, the average real return on T-bills was hardly different from zero. S&P stocks, therefore, earned a hefty risk premium of 8.3 percent over the nominally risk-free return on T-bills. The performance of the stocks of small firms was even more impressive; they earned an average annual return of 16.1 percent. To appreciate the economic importance of these differences in annual average, consider how the value of a dollar invested in each of these types of assets in 1926 would have changed over time. As Table 1 shows, by 1991, $1 invested in S&P stocks would be worth about $675, whereas $1 invested in T-bills would be worth only $11. That’s not much considering the fact that a market basket of goods costing $1 in 1926 would cost nearly $8 in 1991. For another perspective, consider what could have been purchased in 1991 if $10 had been invested in each of these assets in 1926. If $10 were invested in small-firm stocks in 1926, by 1991 it would be worth an impressive $18,476. That’s enough to cover one year of tuition in most prestigious universities in the United States. Meanwhile, $10 invested in T-bills would be worth only $110 in 1991, or enough to buy dinner for two in a nice restaurant. Notice in Table 1 that the assets with higher average returns over 1926–91 also had more variable returns. This correspondence suggests that the higher average returns were compensation for some perceived higher risk. For example, small-firm stocks, which yielded the highest return in this period, had the highest standard deviation too. Similarly, in the first two subperiods, 1926–75 and 1976– 80, small-firm stocks had both the highest return and the highest standard deviation. However, something happened in the last subperiod, 1981–91, according to Table 1. Long-term government bonds did extremely well. A dollar invested in Treasury bonds at the end of 1980 would have grown to more than $4 by the end of 1991, which implies a high annual rate of return (14.2 percent). The risk premium (over T-bills) on the S&P 500 for the 1981–91 subperiod was 7.7 percent, not much different from that for the entire sample period. However, during this subperiod, the average annual return on T-bills of 8 percent was substantially more than the average inflation rate of 4.3 percent. This unusual subperiod suggests that the sampling errors for the entire period computed using conventional time series methods (which assume that the entire time series is generated from the same underlying distribution) may overstate the precision with which the sample averages measure the corresponding population expectations. Clearly, though, across all subperiods, the time series of realized returns on these four types of assets are substantially different in both their average and their volatility. This can be seen in another way by examining Chart 1. There we display over the sample period 1926–91 the logarithm of the values of one dollar invested in each asset in January 1926. For example, the values plotted for December 1991 are logarithms of the numbers in Table 1. We plotted the logarithms of the values so they could all be easily displayed together on one chart and compared; the values themselves are vastly different. The chart is intended to further illustrate the great differences in the paths of returns across the four assets. These great differences are unlikely to be entirely accidental. If investors had reasonable expectations in 1926, they would have guessed that something like this would be the outcome 66 years later, but still they were content to invest in portfolios that included all of these different assets. A question that needs to be answered is, In what way are these assets different that makes investors content to hold every one of them even though their average returns are so different? For example, in what way are small-firm stocks different from S&P 500 stocks that makes investors satisfied with an 8.3 percent risk premium (over T-bills) for the latter whereas they require a 12.4 percent risk premium for the former? 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 return 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 αi 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, ∑ n i=0 αiRi). 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 α0, α1, and α2 that maximize