Mean-Variance Analysis and the Diversification of Risk

Mean-Variance Analysis and the

Diversification of Risk

by Leigh H. Halliwell

Mean-Variance Analvsis and the Diversification of Risk

Leigh J. Halliwell

ABSTRACT

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 Injlation

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.

Mr. Halliwell is an Associate of the Casualty Actuarial Society and a member of the

American Academy of Actuaries. Since April of 1993 he has been the Chief Actuary of

the Louisiana Workers’ Compensation Corporation in Baton Rouge, LA. Prior to that he

worked at the National Council on Compensation Insurance in Boca Raton, FL.

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1) PORTFOLIOS AS MATRICES

We have a portfolio of n assets, the return of the iti asset, Ri, being a random variable

with mean pi. We will let R denote the (n x 1)...