# Initial Tests of the Standard CAPM

# Initial Tests of the Standard CAPM

## Douglas [1969] and Lintner [1965]

The early tests of the standard CAPM focused on the ability of the market to price the risk of securities. The main hypothesis tested was whether the model could describe observed security returns. Douglas [1969] calculated the annual rate-of-return on each of 616 equities for the period 1947-1963. For both the arithmetic and geometric means Douglas ran regressions of the mean rate-of-return on the standard deviation, variance and variance-squared of returns (linear and quadratic terms), finding a significant relationship between the means and all the variables in the regression. Douglas explained the non-linearities by the difference between borrowing and lending rates on individuals with different risk attitudes.

Douglas then replicated earlier work by Lintner [1965a], computing *β* coefficients for equity returns (1926-1960) and regressed average returns for an asset *i*, using a variant of (5.16):

where is the residual from the regression used to estimate the coefficient of return - - in equation (5.14). Under the assumptions of the CAPM, this represents diversifiable risk and thus has an expected value of zero. However, the regression coefficient for

was much smaller, and the intercept term larger, than expected. The regression coefficient of the residual risk was also significantly different from zero. Douglas attributed these results to a bias caused by the large *ex post *returns for the period studied.

Douglas also regressed average returns, for seven sub-periods of five and four year lengths within 1926-1960, on the variance of the return for a security, and the covariance of that security with the average return for the sample. He found that the regression coefficient of the variance was significantly positive. Also, that that for the covariance was not significantly different from zero. He recognised there were several problems in studying these effects, one being that the relevant covariance is that with a (unobservable) efficient portfolio. He suggested regressing returns on a number of indices. However, a preliminary analysis did not yield the expected results. Douglas distinguished the ability of the market to price the risk of an equity from its ability to price covariances of returns. Evidence of the former was found in the study, while there was no evidence that fund managers used covariance information to form efficient portfolios. Douglas expressed the hope that with improved computing abilities, covariance information would be exploited. In general, these studies were taken as early evidence that the standard CAPM was inadequate.

## Portfolio Management

Nevertheless, the principles of the CAPM were employed by Jensen [1968] to evaluate the performance of fund managers. Jensen used a variation of equation (5.14), regressing the portfolio excess returns on market excess returns, and also included an intercept to allow for 'non-average' fund managers. Jensen assumed *β* stationarity for each fund i.e. the managers attempting to maintain a target risk level. The data consisted of returns on 115 unit trusts^{9} for the period 1955-1964. The following regression model was used:

*R _{pt}* is the observed return for fund

*p*in period

*t*. Jensen found that the average

*β*for the funds was 0.84, indicating that the managers were generally cautious. The

*α*'s ranged from -0.0805 to 0.0582 with a mean value (net of expenses) of -0.011. No individual fund performed significantly better than would be expected from random asset selection. However, he qualified his results. Firstly, the analysis did not account for the liquidity required by funds, which therefore could not be fully invested. Secondly, that there was no measurement of the social value of the service provided by the fund managers, that is minimising the risk borne by shareholders. Nevertheless this was not a popularstudy; Farrar

^{10}commented:

'The theory of random walks so permeates America's intellectual heartland that it would be astonishing to learn from one of its products that any financial manager provides a socially useful function.'

Performance evaluation is further discussed in section 7.4 and section 4.5.

## Proxy Measurements

At this point, Roll [1969] had been examining the difficulties involved in using proxies for the risk-free rates and the market index. His objectives were to show the effect of measurement error on the exogenous variables in equations (5.14) and (5.15), and to derive and test a procedure for judging the adequacy of the current model. If the CAPM were valid, then equation (5.15) should hold for the measured risk-free and market returns as well. Measurement errors were expected in two areas:

- the market return because it only includes equity assets; and
- the risk-free rate, because the maturity used does not always match an investor's horizon.

Inflation and tax rates also affect the risk-free asset's returns. Roll distinguished between the unobserved risk-free rate *R _{f}* and the observed rate

*r*:

_{f}A similar equation was constructed for the market return, and the following reduced form was tested:

Again, the *β*s were assumed to be stationary. The market index used was the Dow- Jones composite average. The rates were taken from Treasury bills with one week to maturity - for 792 weeks^{11} commencing 4^{th} October, 1949 - divided into four sub- samples each of 198 weeks. Roll found that the risk coefficients were non-stationary and attributed this to one or more of the following:

- the invalidity of Sharpe-Lintner model;
- the true coefficients changing over time; or
- the non-stationarity of the true process generating
*R*and_{f}*R*._{m}

The squared correlation coefficients estimated from these relationships, *θ ^{2}_{m} *and

*θ*were found to lie outside the [0,1] range; this can be explained if

^{2}_{f}*γ*is non-zero because of an excess return on Treasury bills. Roll then allowed an intercept term and found the estimates in the [0,1] range. He concluded that a relaxation of the model's assumptions was necessary to conform with empirical studies.

_{0}## Examination of Biases

Another test of the equilibrium condition was examined by Miller & Scholes [1972]. They first used data from the Centre for Research in Security Prices ("CRSP") to replicate Lintner's [1965a] results on equity returns for the period 1926-1960. They found that the average rate of return for each company depended significantly on terms not included within the simple regression model used. They therefore attempted to explain the apparent rejection of the CAPM in two ways. Firstly, through bias from misspecification of the *β*-estimating equation, and secondly bias from the variables used to measure risk and return. The first test concerned the risk-free rate. In particular, given the CAPM relationship over any period of time, Miller & Scholes demonstrated that *β* in equation (5.14) will be an asymptotically biased estimate of *β*, unless either the market portfolio return is independent of the level of interest rates, or there is no variation in the interest rate during the sample period. Consequently, the tests used by Lintner [1965a] and Douglas [1969] would result in biased estimates. However, although the correlation between market returns and Treasury bills was negative, it was not significantly so, and they did not find that accounting for *r _{f}* would change Lintner's conclusions. The second possible reason proposed was non-linearity in the risk-return relationship, but the inclusion of a quadratic term in equation (5.16) did not reduce the unexplained excess returns. The third proposal was heteroscedasticity of the error terms. This was also rejected.

The examination of measurement error bias in the variables proved more fruitful. The measurement errors in the *β* estimates used in equation (5.16) were shown to bias the *β *coefficient's estimate towards zero. They were also shown to distort the coefficient of the residual variance term in equation (5.18). Both of these effects would explain Lintner's and Douglas's results. The third reason examined was the misspecification of an index for the market. They showed that theoretically any correlation between the index used and the true index, of less than one, will inflate *γ*_{1} in equation (5.18), and would upwardly bias the coefficient on the variance of returns. However, because the use of expanded indices with bond components did not alter the basic results, they concluded that the choice of index was not the cause. The final effect examined was skewness. The association of large variances with large returns would tend to inflate the coefficient of the variance and bias the coefficient of *β* downwards. This was confirmed through simulation. However, they did not test whether returns were proportional to non-diversifiable risk. Miller & Scholes provided a thorough analysis of how the rejection of the CAPM, implied by the regressions to date, could be explained by the violation of the assumptions of the CAPM. However, they did not uncover the source of the observed results, and the CAPM remained under scrutiny.

The main conclusion drawn from Miller & Scholes results was that the use of sample *β*s, rather than population *β*s, could result in a significant downward bias in the risk premium associated with *β*, and an increased intercept that is significantly different from zero. Hence, measurement error seemed to be a potentially pernicious problem in estimating the risk premium associated with *β*. As a result, three main approaches were developed to alleviate the problems caused by measurement error in *β*s. However, it is interesting to note that due to the poor quality and availability of data, the majority of papers testing the validity of the CAPM confine their analysis to stock markets. This violates one of the CAPM's primary assumptions, that of the existence and use of a market portfolio:

*Forming portfolios upon which to undertake two-pass regression analyses.*- The advantage of this approach is that the measurement error in the portfolio
*β*s is likely to be much smaller than the measurement error in individual security*β*s. This is because much of the measurement error in the individual securities' estimated*β*s will cancel in forming portfolios. However, the disadvantage of this approach is the potential loss of information in discarding individual security information. The results of two important papers which follow this approach (Black, Jensen & Scholes [1972] and Fama & MacBeth [1973]) will be discussed in the next section.

- The advantage of this approach is that the measurement error in the portfolio
*Using measurement error correction procedures, such as those discussed in Litzenberger & Ramaswamy [1979], to alleviate bias arising in two-pass regression procedures when estimated βs of individual securities are used in the analyses.*- The advantage of this approach is that it makes use of the information contained in individual security returns. The disadvantage of the measurement error correction procedures is that the theoretical justification for the procedures relies on asymptotics (as the sample size grows to infinity) and hence the approach is not guaranteed to be superior in finite samples. Litzenberger & Ramaswamy's results will be discussed in section 5.3.6.

*Implementing multivariate techniques which simultaneously estimate βs and factor risk premia*.- These procedures will be briefly discussed in section 5.3.9. In particular, the results of Gibbons [1982] and Stambaugh [1982] will be examined. An advantage of multivariate procedures is that econometrically they are the cleanest. A disadvantage is that they require, for practical reasons, the formation of portfolios which again results in the loss of information on individual security returns and possible over-fitting of known anomalies.

____________________________

^{9}Unit trusts are the UK equivalent of open-ended mutual funds in the US.

^{10}in The Economist March 7^{th}, 1992.

^{11}Roll had 793 weeks of data; one was 'lost due to relative changes.'