Mean-Variance Analysis and the Diversification of Risk

Harry W. Markowitz in the 1950’s developed mean-variance analysis, the theory of combining risky assets so as to minimize the variance of return (i.e., risk) at any desired mean return. The locus of optimal mean-variance combinations is called the efficient frontier, on which all rational investors desire to be positioned. Actuaries see diagrams of efficient frontiers in their finance readings. Perhaps they are aware that efficient frontiers are parabolic. However, no mathematics is ever presented, so actuaries would be at a loss to derive an efficient frontier for problems involving more than two assets. But the minimum-variance combination of assets as a function of expected return has a simple matrix formulation; and the derivation of this formula is well within the grasp of actuaries. From this follows the formula for the efficient frontier. This paper will present the mathematical theory of the efficient frontier. Then the theory will be illustrated by deriving the efficient frontier of a portfolio of stocks, treasury bonds, and treasury bills, as discussed in Ibbotson’s Stocks, Bonds, Bills, and Inflation 1994 Yearbook. Also shown will be how to determine the mix of annual statement items which minimizes risk-based capital. An appendix will delve into the theory more deeply.