Mean-Variance Analysis and the Diversification of Risk

Harry W. Markowitz in the 1950’s developed mean-variance analysis, the theory ofcombining risky assets so as to minimize the variance of return (i.e., risk) at any desiredmean return. The locus of optimal mean-variance combinations is called the efficientfrontier, on which all rational investors desire to be positioned.Actuaries see diagrams of efficient frontiers in their finance readings. Perhaps they areaware 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 morethan two assets. But the minimum-variance combination of assets as a function ofexpected return has a simple matrix formulation; and the derivation of this formula is3well within the grasp of actuaries. From this follows the formula for the efficient frontier.