Portfolio Optimization and the Capital Asset Pricing Model: A Matrix Approach

Actuaries are acquainted with the basic ideas of Modem Portfolio theory and the Capital Asset Pricing Model (CAPM). Briefly, portfolios are formed by weighting risky assets with varying means, variances. and covariances. Each portfolio can be plotted in the X-Y plane by its total return, with the standard deviation as the x-coordinate and the mean as the y-coordinate. It is plausibly asserted that the resulting subspace of returns has an envelope, which is called the efficient frontier. The efficient frontier contains the returns which offer the greatest mean for a given standard deviation, or the least standard deviation for a given mean, and therefore would correspond to portfolios chosen by perfectly informed and rational investors. However, when a riskless asset is introduced, one point on the efficient frontier becomes preferable to the others, the point at which a line becomes tangent to the efficient frontier. Since this point is optimal, it will be chosen by all informed and rational investors. which is lo say that it will correspond to the portfolio of an efficient market. This article shows how the aforementioned argument can be made rigorous through fairly simple matrix algebra, which will foster a deeper understanding of and appreciation for the theory. Moreover, the article offers an easy method for determining the optimal, or market, portfolio. Finally, there will be a few remarks as to why CAPM theory may falter under empirical testing.