# Equilibrium Conclusion

The persistence of the ideas inherent in the CAPM demonstrate its intuitive appeal, notwithstanding that its foundations are becoming increasingly shaky. Alternative models have been proposed, yet no conclusive empirical evidence of their superiority and/or efficacy has been provided. However, when considering applying the CAPM to asset pricing decisions at a macro-level, cognisance should be made of the following.

A common thread between all these models is the linear relationship between expected return and one of the two risk variables. The plurality of papers find a positive trade-off between risk and return. However, few define this as linear.

For econometric reasons the majority of tests of the CAPM rely solely on grouped data, ignoring the behaviour of individual assets. However, it is the behaviour of individual assets that the theory is trying to explain. Individual asset deviations from linearity may cancel out in the formation of portfolios. Also, if the underlying process is similar to the linear relationship expected, grouping equities will make minor deviations more difficult to detect. This can provide evidence to support the acceptance of the proposed model even if it is false. Deviations of an individual asset from exact linearity may cancel out when placed together in a portfolio.

A close examination of the monthly (grouped) behaviour of estimated and actual risk premia indicates a close association between risk and return, in line with the Sharpe-Lintner CAPM. However, such a conclusion may be drawn from a variety of risk/return relationships which are not strictly linear, or linear only in part. This questions the importance of conclusions such as those reached by Ward & Saunders [1976]. They found that the ranking of portfolios conformed to expectations, when sorted in terms of average *β* and variance. Kothari, Shanken & Sloan [1995], when arguing in favour of *β*, state that in months where markets fall steeply, high *β* equities substantially under perform low *β* equities.

The definition of *β* is important. Its origin is the Sharpe-Lintner CAPM (Sharpe [1964] and Lintner [1965a]), where *β* is the market risk coefficient. This is defined within a single-period equilibrium pricing model based on very specific assumptions. As the CAPM was extended to include multiple factors, market risk and thus *β* were mostly retained. These additional factors were often not based on explicit equilibrium models, but were selected on the basis of their explanatory power in empirical tests. The paper by Fama & French [1992] is a good example of this. Therefore, one definition of *β *in general use is that of the (unobservable) market risk coefficient for single and multiple factor equilibrium models.

Estimated *β*s, based on regression techniques of varying sophistication, soon became widely available. Also, early empirical tests supporting the CAPM suggested that *β* was indeed the only factor of relevance, 'the' *β* of an equity being a measure widely reported upon by financial intermediaries.^{47} However, these empirical *β*s were often assumed to be estimates of the `true' CAPM *β*, with the nature of *β* as a risk measure lost on many of its users. This was pointed out by Roll in his series of papers on the lack of empirical content of the CAPM.

An alternative measure of *β *is the estimated risk parameter of a security in a particular portfolio, estimated using a relevant 'market' proxy. For this *β* there is no pretence of equilibrium pricing.^{48} Therefore, a *β* estimate of a certain equity based, for example. on the FT-SE 350 is often also interpreted as the relevant *β* for the portfolio of any investor. This is only reasonable if an investor's portfolio is similar to the FT-SE 350.

Therefore, *β* may provide a useful measure of risk when restricted to the analysis of similar investments e.g. within the confines of an equity portfolio. Under the assumption of the multivariate normal distribution of returns, the 'market model' results are obtained for any combination of investments within that portfolio. Optimisation techniques, as applied to index fund construction, are a good example of an application which utilises this relationship. Remaining within the structure of an asset class, *β* may provide a useful measure of risk, with its interpretation consistent with the model's theory. However, due to the absence of an empirically detectable relationship between average returns and *β*s across asset classes, *β* is of no practical value for a variety of applications, including the construction and performance evaluation of multi-asset portfolios. This is particularly true of an investment universe that includes real estate.

Also, due to the quality and availability of data, the majority of the research has been confined to analysis of stock markets. Therefore, unless the market proxy used is highly correlated with the `true' market portfolio, they violate one of the CAPM's primary assumptions, that of the existence and use of a market portfolio (Roll [1977]). As Kandel & Stambaugh [1987] conclude, if the market portfolio has a correlation of at least 0.9 with the market proxy^{49} employed, the Sharpe-Lintner CAPM may be rejected. Also, by selecting a commonly used market proxy - the FT All-Share, or the S&P 500 - an investor ignores that proportion of the world's wealth which is held in real estate; approximately 50%.^{50} As discussed in Appendix A, the accuracy of these indices is also brought into question.

In addition, there is evidence to support either total or systemic skewness of returns being priced. It suggests that investors express an aversion to variance, and a preference for negative skewness. This result is consistent with the arguments in section 5.3.6 - that investors are more concerned with downside than upside risk - and questions the assumption of quadratic utility. However, in practice the assumption of quadratic utility is seldom used, since it implies satiation and increasing absolute risk-aversion.^{51} This, combined with the globalisation of financial markets and the boom in derivatives, places increasing importance on the assumption of the multivariate normal distribution of returns.

The downside-variance model was an attempt to accommodate the notion that investors require a target rate-of-return, as they are concerned exclusively with downside risk. However, the downside-variance model was shown to be equivalent to the CAPM, even when the target rate-of-return chosen was not the mean i.e. MSV, but an arbitrary point (Nantell & Price [1979]). Combined with the assumption of risk-neutrality above the target rate-of-return, this marked the death of the downside-variance model. However proof of their equivalence, again relies on the assumed multivariate normal distribution of returns.

Therefore, in order to apply the CAPM to investment decisions that include real estate, consideration must be given to the validity of assuming that returns from real estate investment are normally distributed. The following chapter investigates the validity of this assumption.

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^{47}*β* estimates became available from the London Business School in 1979; they had been available in the US since the mid-1960's.

^{48}Its estimation is onerous, and often leads to misleading results. For example, see section 5.3.7.

^{49}In Kandel & Stambaugh [1987] study, they used a value-index of NYSE and AMEX equities. For the last two sub-periods (2nd July 1969 to 1st October 1975, and 8th October 1975 to 23rd December 1981) a correlation of 0.7 was sufficient to reject the CAPM.

^{50}Section 4.5.6 further considers the practical difficulties of defining a market proxy.

^{51}As discussed in section 3.3.2.