# The Case for β

# The Case for β

Reinganum [1981] cited the empirical evidence against the CAPM as a reason to re-examine whether an equity's *β* is a important determinant of equilibrium price. Specifically, he studied whether equities with different ß estimates experienced systemically different returns. He analysed daily returns on equities over the period 1962-1979, and monthly returns for the period 1934-1979 Estimates of *beta* and portfolio groups were formed as in Fama & MacBeth [1973]. If an equity was delisted during the holding period, any funds returned were held in cash until the end of the adjustment period. Portfolios were revised annually for the daily data and every five years for the monthly data. In addition the market models' estimates of *β* (Scholes-Williams's and Dimson's) were calculated with twenty lags and five leads.

For the daily return data, the return on the portfolio decreased with increasing portfolio *β*s. Employing Hotelling's *T ^{2}*-statistic which accounts for correlation between the portfolio returns, Reinganum found that the differences were also significant. However, the returns were also skewed and leptokurtic. Using both the Scholes-Williams and Dimson estimates similar trends were observed, but the returns were also indistinguishable from each other.

When monthly data was used, the market model estimator resulted in monthly returns that increased with the estimated *β* portfolio. However, using Hotelling's *T ^{2}*-statistic the null hypothesis of identical mean returns could not be rejected. Unfortunately,

*β*s from the Scholes-Williams and Dimson models were not calculated. Using a value-weighted NYSE index as opposed to the former equal-weighted index, he found a statistically significant relationship between returns and

*β*s. The non-stationarity of the

*β*s was addressed by examining the mean differences between the monthly returns of the high and low

*β*portfolios for nine sub-periods, for both the equal and value-weighted indices. For the most part the mean difference did not exceed two standard errors.

The final concern of Reinganum was the implication of non-normality of returns on the interpretation of the test statistics. The portfolios were skewed and leptokurtic for both the monthly and daily returns. The *β* estimates were compared using Friedman's rank test. Under the null hypothesis every ranking is equally likely, and each set of monthly returns is independent but may not be identically distributed. The tests detected systemic tendencies. For the equal and market weighted indices, the null hypothesis could not be rejected. Therefore, the monthly rankings were indistinguishable from random rankings. Reinganum concluded that the *β* estimates are not systemically related to returns, and that the returns of high *β* equities are not significantly different from those of low *β* equities. He also suggested that the CAPM may lack significant empirical content.

Clare & Thomas [1994] looked at the average monthly mean return, standard deviation and market *β* of 56 portfolios on the ISE over the period January 1978 to December 1990. They also found that the mean return of the portfolios did not vary systemically with portfolio *β*s.

## Roll [1977]

## Before Roll's critique, tests of the CAPM were carried out on the assumption that the following questions had to be answered:

- did high
*β*portfolios earn more than low*β*portfolios?; and - did zero-risk portfolios earn the same as the risk-free rate?

Roll [1977]^{30} argued that these were inappropriate questions, and demonstrated mathematically that they were equivalent to asking the question:

- is the proxy market portfolio mean-variance efficient?

A 'revolution' in empirical work on the CAPM was triggered by Roll's critique of previous studies. Roll called into question both what was being tested and the methodology used. His argument was based on three main points.

- There is only a single, testable hypothesis associated with either the Sharpe-Lintner CAPM or Black's generalisation. This hypothesis is that 'the market portfolio is mean-variance efficient'. Other implications of the model, such as the linearity of expected returns in
*β*coefficients, follow from market portfolio efficiency, and are not independently testable. - Using a proxy for the market portfolio induces the possibility of both type I errors (incorrectly rejecting the null hypothesis) and type II errors (failing to reject the null hypothesis). This results from the possibility of the proxy being mean-variance efficient when the market is not, or the proxy might be inefficient when the market portfolio is efficient. This problem is not accounted for by choosing proxies that have a high correlation with the market, as the efficiency characteristics of the two portfolios can nevertheless be very different. Therefore:
- 'the theory is not testable unless the exact composition of the true market portfolio is known'; and
- 'all individual assets are included in the sample.'

- Moreover, even if everybody could agree on the composition of the market portfolio, testing for the (
*ex ante*) proxy's efficiency by using common regression techniques to verify the existence of a linear relationship between returns and*β*coefficients is unsatisfactory, since:- 'The two-parameter theory does not make a prediction about parameter values, but only about the (linear) form of the cross-sectional relationship. Thus, econometric procedures designed to obtain accurate parameter estimates are not very useful.'; and
- 'The widely-used portfolio grouping procedure can support the theory even when it is false. This is because individual asset deviations from exact linearity can cancel out when placed together in a portfolio.'

The immediate response to Roll's critique was largely a rejection of what was interpreted as a nihilistic message.^{31} Mayers & Rice [1979], for example, maintained that these criticisms:

'impose extremely severe criteria on empirical work that few, if any, econometric studies can meet'

and that

`proxies must be used constantly to test all types of economic theories.'

Therefore, their conclusion was that:

'... in an ideal world, these problems would not exist - and we would certainly support the creation of such a world, were it costless - but it provides little justification for rejecting studies done in the world in which we now live.'^{32}

However, Roll's critique changed the empirical work on the CAPM in two ways. It promoted the development of a new statistical methodology for testing the mean- variance efficiency of a given index,^{33} and was the basis of attempts to test for the efficiency of the (unobservable) market portfolio, conditional on an assumption about the correlation between the proxy being used and the true market portfolio (Kandel & Stambaugh [1987], Shanken [1987]). These approaches are considered in section 5.3.9.

## Fama & French [1992]

Fama & French [1992] argue that the cross-section of average returns on US equities^{34} shows little relationship to either the market *β*'s of the Sharpe-Lintner asset-pricing model, or the consumption *β*'s of the intertemporal asset-pricing model of Breeden [1979], *inter alios*.^{35} On the other hand, variables that have no special standing in asset-pricing theory show reliable power to explain the cross-section of average returns. The list of empirically determined average-return variables^{36}** **includes size,^{37} gearing, earnings/price ratio, and book-to-market equity.^{38}

Their main results can be summarised as follows:

- The risk premium on
*β*is insignificantly different from zero when it is included by itself. - The risk premium on
*β*is also insignificantly different from zero when it is included as an explanatory variable. - Size appears to be negatively related to returns when it is included by itself or when it is included with
*β*. - Another important determinant of average returns appears to be the natural logarithm of the ratio of the book value of equity to the market value of equity. This ratio is positively related to returns regardless of whether it is included byitself or included with, for instance, net asset value.
- Average returns also appear to be positively related to companies earnings/priceratios, when that ratio is positive. The earnings/price effect, however, is not as strong as the size or book-to-market effects, becoming only marginally significant when the other two variables are included.

Black [1992; 1993] argued that any observed empirical effect which cannot be justified on *a priori* theoretical grounds is most likely the result of 'data mining'^{39} further discussed in section 4.5.5 - noting that:

'The size factor seems priced only because of the pervasive influence of data mining.'

Black further argued that:

'The book-to-market equity factor is priced because markets are sometimes inefficient, not because it is related to an economic factor that investors care about.'

Kothari, Shanken & Sloan [1995, p. 186], using an alternative data set, found that:

'... book-to-market equity is at best weakly related to average stock return.'

Their results suggest important differences between their data set S&P industry level data covering 1947-1987 - and that from Compustat examined by Fama & French. The authors suggest that the strong relationship found by Fama & French between the ratio of the book value of equity to the market value of equity and average returns maybe due to a survivorship bias. This is further discussed in section 4.5. Their results suggest that companies which are included in CRSP but not Compustat over the period 1963-89 period have a substantially lower average return, lending some credence to this hypothesis.

Kothari, Shanken & Sloan concluded their paper by summarising the case for and against *β*. In favour they made the following points:

- if the entire history of security returns in the US is used, the estimated average compensation for
*β*risk is 0.47% per month and is close to being significant. Moreover, the estimated compensation is not significantly different from the actual average risk premium^{40}of 0.76% per month; - if the study were to stop in 1982, the support for
*β*would be overwhelming. The last ten years, which have not been favourable for*β*'s explanatory role for returns, are an aberration; and - a close examination of the monthly behaviour of the estimated and actual premiums indicates a close association between the two variables, in line with the Sharpe-Lintner CAPM. For example, in months where the market falls steeply, high
*β*equities substantially underperform low*β*equities.

Conversely, against the case for *β*, they state that:

- even with sixty years of data on returns, spanning many generations of money managers, the t-statistic for the estimated average compensation for
*β*risk is a modest 1.84; - the most recent period, more representative of current experience, provides much weaker support for
*β*. The strongest support for*β*, however, emerges from periods many years before Markowitz's ideas on how to form equities into efficient portfolios; and - including other variables may substantially diminish the role of
*β*in explaining security returns.

____________________________

^{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 indi- cation 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.

^{30}Fama [1976] seemed to be aware of the insufficiency of previous empirical work (including his own) when he summarised the current state of tests of the CAPM by observing that ... 'In truth, all we can really say at this time is that the literature has not yet produced a meaningful test of the Sharpe-Lintner hypothesis.'

^{31}For example, see Ross [1978a].

^{32}See Roll [1979] for a reply.

^{33}See paragraph one, under Roll [1977] 5.3.8.

^{34}Fama & French's [1992] data sample included all non-financial equities which satisfied the following criteria:

- Equities returns that were listed on the NYSE, AMEX, or NASDAQ with return data available from the CRSP database - NYSE and AMEX data was used for the period 1962-1989, NASDAQ data from 1973.
- Financial accounting data as available from the merged Compustat annual industrial files - Compustat data was available on the 1962-89 period.
- They used the following estimation procedure:
- Two-pass Fama & MacBeth cross-sectional regression technique;
- Cross-sectional regressions run on individual equity returns;
- Individual
*β*estimates based on the 'portfolio*β*' of the particular equity- Equities first sorted on the basis of size, into ten size-decile portfolios;
- Each size-portfolio was then sorted on the basis of pre-ranking
*β*s into ten*β*-decile portfolios. Portfolio returns were the equally-weighted monthly returns of the equity in the portfolio. Portfolio returns were regressed on the value-weighted CRSP benchmark returns (and lagged returns) to obtain portfolio ßs. Equities were assigned the*β*of the portfolio.

- They used the following estimation procedure:

^{35}See for example, Reinganum [1981] and Breeden, Gibbons & Litzenberger [1989].

^{36}For example see Banz [1981], Basu [1983], Lakonishok & Shapiro [1984; 1986], Rosenberg, Reid & Lanstein [1985] and Bhandari [1988].

^{37}Market equity - equity price times number of shares in issue.

^{38}The ratio of the book value of a company's ordinary equity, to its market value.

^{39}Lo & MacKinlay [1990] present a rigorous argument that the size effect may be attributable to what they call, 'data snooping'.

^{40}*r _{m}* -

*r*.

_{f}