# Market Sampling

# Market Sampling

Not only were there persistent uncertainties as to the source of divergences from the CAPM, but there were still questions as to whether the data used had any systemic problems that contributed to, or detracted from, the inconsistencies found. As almost all the research was conducted on a common data set (CRSP database) and over similar time periods, most results would share these problems. Two of the problems studied were the influence of market trends during the period studied, and the effect of non-synchronous trading on estimations of *β*.

## Bull and Bear Markets

The earliest empirical research on the CAPM was undertaken during a bullish market period, the 1940's through to the 1960's. Haugen & Heins [1975] studied the implications of the data sampling period. In particular, using Fama's specification of the regression model, they showed that the regression coefficient of risk on return, ^{γ}_{β}, could be expressed as:

where the first bracketed term is the true risk-return trade-off, and the second is the expectation error. They proposed that in a bull market, results exceed expectations, and ^{γ}_{β} is overestimated.^{28} Haugen & Heins claim the solution is to assume stationarity and to sample large numbers of observations over long time periods. Their sample portfolios consisted of 114 portfolios with 25 equities each using equal (dollar) weights. Monthly returns were calculated for the period 1926-1971 and the sub-period 1946-1971. Haugen & Heins confirmed that the residual variance was small relative to the portfolios' variances as a result of diversification. Their results implied that the rates-of-return in 1946-1971 failed to meet expectations. Examining the time-period problem, Haugen & Heins showed that as market performance during any period exceeded that of the previous ten-year period, the coefficient of the regression of return on *σ* was negative. Haugen & Heins concluded that over the long run, portfolios with a relatively small variance in monthly returns yield relatively larger returns than those predicted to have a greater risk-return trade-off.

The behaviour of equities in bull and bear markets was also investigated by Kim & Zumwalt [1979]. They found that investors require an additional premium to assume the downside risk, and are willing to accept a discount for the upside risk. Chen [1982] validated the downside risk premium but found the bull market risk to be statistically insignificant. This supports those arguments discussed in section 5.3.6 - that investors are more concerned with downside than upside risk - and also questions the assumption of quadratic utility.

## Estimating β

The availability of daily returns data in 1977 introduced an additional econometric problem in tests of the CAPM (Scholes & Williams [1977]). Price changes occurred at distinct, random intervals removing the possibility of accurately calculating returns and thus biasing *β* estimates. Given the nature of the reported returns, they showed that extremes in trading activity, either very frequent or very infrequent,^{29} result in upwardly biased intercepts, in equation (5.14), and downwardly biased *β*s. The opposite effect was found in periods of average trading activity. They developed consistent estimators -

- for the coefficients in equation (5.14) as follows:

where

, and are the OLS estimates of *β*. The independent variables used were:

- the market return for the day before it was realised;
- the market return on the day it was realised; and
- the day immediately following.

and is the estimation period value of the first-order autocorrelation coefficient of the market return.

Scholes & Williams also show these to be equivalent to instrumental-variable estimators, using a moving sum of rates-of-return as instruments. Daily returns were used from equities on the NYSE and American Stock Exchange ("AMEX") during the period 1963-1975. The consistent estimators were compared with the traditional estimates and the direction of the biases were as expected.

The same type of problem was studied by Dimson [1979], who believed that the Scholes-Williams method was impaired, as prices that are not preceded or followed by a trade in an adjacent time period cannot be used. Fowler & Rorke [1983], however, show this not to be the case, and provide a variant of Dimson's procedure which yields results identical to those of Scholes & Williams.

According to Levy and Sarnat [1984, p. 442]:

'Some betas are stable and do not change much from period to period, but most betas are quite unstable, so that historical betas are not very good predictors of future market data'.

A tendency of *β*s to move towards 1 has been found by Blume [1971; 1975], who claims that equities with a *β* greater than 1 tend to decrease towards 1 over time, while equities with *β*s less than 1 tend to increase towards 1. He adjusts an equity's historical *β* in an attempt to obtain a better estimation of its future *β*. However, Blume demonstrated that the *β*s of undiversified investments were still unstable. This is important, when considering the difficulty of compiling a diversified real estate portfolio, and will be further discussed in section 2.4.3.

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^{28}Haugen & Heins's [1975] proposal also applied to a bear market, where results fall below expectations, and *γ _{β}* underestimated.

^{29}Note that throughout this thesis reference is made to literature covering trading and/or market activity, mood or volume. Terms used to describe these include abnormal, frequent, high, infrequent, large, low, normal, reasonable, small, and can be preceded by superlatives such as very. Little indication is made in the papers of the relativity of these terms to what is e.g. normal, or frequent. No attempt has been made to interpret the degree of any such relation. Note that in a protracted period of little or no activity, a day of non-stop trading might be described as abnormal, whereas during other periods that day may be regarded as normal.