Mean-Variance Portfolio Choice

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, σ2) - preferences are such that higher means and lower variances are favoured, U1 > 0, U2 < 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 x is a vector of portfolio weights, rf is the risk-free asset's return, μ is the mean vector and Σ 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:

where

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 U and the variance-covariance matrix of asset returns, Σ, being always positive.

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 θ, only affect the balance of the optimal portfolio between how much to place in the risk-free asset and risky assets. 

Portfolios satisfying equation (5.2) for some positive value of θ are said to be mean- variance efficient. They are characterised by having the greatest expected return for a given level of variance and, simultaneously, the smallest variance for a given expected return.