Jamal Munshi, Sonoma State Univesity, 1994
Statistical Identity as Economic Theory
Much of what is known as Modern Portfolio Theory follows from the observation by Markowitz (1955) that when two risky assets are combined their standard deviations are not additive if the returns from the two assets are not perfectly positively correlated. When a portfolio of risky assets is formed, the standard deviation risk of the portfolio is less than the additive amount and this reduction in risk is known as diversification.
The computation of risk reduction as proposed by Markowitz is tedious. For an n-asset portfolio one would construct an nxn variance-covariance matrix containing n*n/2 - n terms. To get around this problem, Sharpe (1963) proposed a computationally efficient method of solving Markowitz's equations which requires that a portfolio of all assets that may be chosen for a portfolio be known. If the sum portfolio is known then only the n covariances between each risky asset and the sum portfolio are needed to make the portfolio computations. He developed a simple relationship for the contribution of each risky asset to the standard deviation risk of the sum portfolio.
If the sum portfolio has a standard deviation of Sm and a risky asset has a standard deviation Si and covariance with the sum portfolio of Sim, then it can be shown that for Gaussian distributions the contribution of the i-th asset to the Sm is
which may be written as either
Setting Sim/Smm = Bi the regression coefficient for asset i returns against sum portfolio returns, Ri=f(Rm) and Sim/(SiSm) to Ri the correlation coefficient between the ith asset and the sum portfolio we can write that the unique contribution of asset i to the standard deviation of the sum portfolio as
Sharpe chose the first form and his Bi or beta became the cornerstone of the capital asset pricing model or CAPM. In an economy in which returns are Gaussian and independent; and in which standard deviation risk is priced; and in which investors hold portfolios similar to the sum portfolio, it can be shown that investors will only price the additive component of standard deviation risk (called the systematic risk) of a risky asset. The remainder of the risk, called 'diversifiable risk', simply cancels out on addition to the sum portfolio.
These relationships are graphically depicted in Figures 2, 3, and 4 for a simulated economy that meets the above conditions. Figures 3 and 4 show that as long as standard deviation risk is priced in the economy, higher systematic risk will be associated with higher returns while there will be no relationship between diversifiable risk and returns. Figure 2 shows that since Sm is a constant for all assets, it can be removed from the coordinate without any loss of the risk-return relationship information in Figure 3.
I emphasize at this point that these relationships are statistical identities and not economic theory. The only economic theory we have used so far are: (1) standard deviation risk is priced in the economy, (2) the sum portfolio is observable, and (3) investors hold well diversified portfolios similar to the sum portfolio. It is therefore only these aspects of the theory that may be put to empirical tests.
Yet, most research on the CAPM have tended to test the statistical identities instead and because of poor data collection and interpretation and model mis-specification have tended to produce contradictory and confusing results (Fama and macbeth 1973, Fama and French 1992, Haugen and Baker 1993, Shukla and Trzcinka 1991).
Let us now examine the Fama and French (1992) paper in some detail. The authors collect beta and returns data over a moving window of time from 1962 to 1989 and find no relationship between beta and returns. We would expect to see such a relationship only if the market volatility Sm were constant. As you can see from Figure 9, this is not the case. Therefore no conclusion may be drawn from their test. What we would like to investigate with the same data is whether Ri are positively related to BiSm.
But even these are mere algebraic relationships and not economic theory. The real question is whether standard deviation risk is priced. And the corollary questions are; how should the standard deviation be measured? and is the S&P500 index an adequate proxy for the sum portfolio? In Figure 10 we show that a moving 12-month standard deviation risk appears to be priced by the S&P500 index. But this illusion quickly disappears under either of two conditions. First, we find that the monthly S&P500 index returns have a strong first order correlation of 25% (Figure 11). When this autocorrelation is removed from the data the risk pricing relationship disappears (Figure 12).
The risk relationship also disappears when the S&P500 series is reversed. Figure 5 shows the reveresed series, Figure 6 shows the moving standard deviation of the reversed series, and Figure 7 shows the apparent absence of the risk relationship evident in Figure 10 . These results might be explained by the algebraic relationship between capital gains and standard deviation risk. As long as the market is going up at an increasing rate, the algebra will dictate a positive relationship between standard deviation risk and capital gains return. The shape of Figure 8 indicates that this is the case with the historical data we have at our disposal and financial theory based on this data may be an artifact of its peculiar shape.
The Multicolinearity Problem
When Fama and French (1992) removed the size effect from the data they may have also removed that which they intended to measure - the beta effect. A failure to find a beta effect in the residuals of the size effect does not imply an absence of a beta effect since size and beta are related. There are better ways to find the unique contributions of the two correlated variables. For example, one might first regress beta against size and use these residuals as the unique contribution of beta and then regress size against beta and use those residuals as the unique contribution of size. Alternately, one might look for orthogonal principal components of size and beta.
An added complication in asset pricing research is that some of the empirical models include the PE ratio (PE = stock price over accounting earnings) as an explanatory variable in addition to the risk measure beta. But PE too contains a risk measure. Current financial theory interprets PE as a combination of two effects. Ceteris paribus higher perceived risk would lower the PE ratio and higher perceived growth would raise the PE ratio. Empirical studies are complicated by a high co-linearity between PE and beta. There are other problems with asset pricing studies that have to do with the time series nature of the data and the methods by which the concept of risk is rendered; but until the multicolinearity problem is resolved the regression models of the empiricists will remain unstable and generate spurious and conflicting results.
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