# Preferences and the Distribution of Returns

As outlined, the original derivation of the CAPM was based on the assumption that investors' preferences can be defined over the mean and variance of a portfolio's distribution of returns. Both Markowitz  and Tobin  showed that this would be consistent with expected utility maximisation under von Neumann-Morgenstern  by assuming quadratic utilities or the multivariate normal distribution of returns. Tobin [1958, p. 76] further remarked that ...

' ...a two-parameter family of probability distributions and declining utility of return ... '

would be sufficient. However, it became clear that this was not the case, and that further restrictions would have to be imposed.

Feldstein , for example, showed that log normal distributions, while defined by two parameters, would not be sufficient for the CAPM to hold. The important missing requirement was closure under linear combinations; any linear combination of random returns must also share a distribution from the same family.

The characterisation of distributions that imply mean-variance - or more generally median-dispersion - utilities was completed by Chamberlain . Chamberlain showed that given more than two assets and the availability of a risk-free asset, a necessary and sufficient condition for mean-variance analysis with arbitrary preferences was that the joint distribution of asset returns is elliptical i.e. that random returns are linear transformations of a spherically distributed random vector. An interesting property of this family of distributions is that it allows the distribution of asset returns to be bounded above and below. One of the criticisms of the CAPM is that it requires assets to have unlimited liability. Although true for the normal case, the above shows that this criticism is not generally valid.

More generally, Ross [1978b] showed that the CAPM would follow even when mean-variance analysis is not applicable, as long as two-fund separation holds. Given the presence of a risk-free asset, for two-fund separation and the CAPM to hold requires that the following condition exist. That a (random) pair , with and such that the random vector of returns, i;, can be expressed as: The 'separating distributions' in equation (5.6) include the elliptical distributions as a special case.

Either quadratic utilities or a 'separating distribution' of asset returns is a sufficient condition for the CAPM to hold. However, all assumptions must hold in the absence of further suppositions on the distribution of returns or investors' tastes. A limited amount of work has been undertaken on the characterisation of weaker, but paired, restrictions on both return distributions and preferences. Merton , however, has shown that if asset returns belong to the family of compact distributions, the CAPM holds as the decision-making horizon shrinks to zero, as outlined in Samuelson . Samuelson & Merton  proposed conditions under which mean-variance analysisis an approximation of expected utility maximisation.

In practice, the assumption of quadratic utility is seldom used, since it implies satiation and increasing absolute risk-aversion. The assumptions placed on the distribution of returns is therefore an important one, however, it becomes difficult to make in the presence of derivative securities with truncated payoff distributions. Hence, the result of the CAPM for all securities is vulnerable to the assumptions placed on the distribution of individual asset returns.