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Date Submitted: 04/05/2013 05:46 PM
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.
3
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)...