As discussed, the CAPM is based on the theory of mean-variance portfolio choice developed by Hicks [1946], Markowitz [1952; 1959] and Tobin [1958], and relies on the following assumptions: - investors' preferences can be described by a (derived) utility function over the mean and variance of the return on an individual's portfolio,
*U*(*μ*,*σ*^{2}) - preferences are such that higher means and lower variances are favoured,*U*1 > 0,*U*2 < 0, and that*U*is concave; - a risk-free asset is available, which can be freely bought and sold;
- there are no restrictions on short sales;
- securities are traded within a competitive market, with no taxes or transaction costs; and
- each asset is infinitely divisible.
The investor's decision problem is then: where r is the risk-free asset's return, _{f} is the mean vector and μ the variance-covariance matrix of the asset returns. The choices of portfolio weights are unconstrained in the above problem; the budget constraint is imposed by making the percentage held in the risk-free asset the residual. Differentiating equation (5.1) with respect to x, and setting the derivative equal to zero, gives: Σwhere is a measure of an investor's risk tolerance. The second-order condition is met since is always negative; a result of the concavity of Equation (5.2) illustrates Tobin's two-fund separation theorem [1958]; that the optimal mix of risky assets in the investor's portfolio depends only on the parameters defining the joint distribution of asset returns, as perceived by the investor. Investor preferences. as measured by Portfolios satisfying equation (5.2) for some positive value of |