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 3 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. 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) vector whose i” element is Ri. Similarly, M will denote the (n x 1) vector whose I . * element is pi = E(Ri ). Let X denote the symmetric (n x n) matrix whose (i, j)” element is qj = CoV(Ri , Rj ). In matrix terms, M = E(R), and C = Vat+(R) = E((R-M)(R-M)‘), where ’ is the symbol for transposition. ’ We will write R [M , C] as shorthand for ‘R is a random vector with mean M and variance C. ’ We will not make any assumptions as to the probability distribution of R; in particular, R need not be multivariate normal. Our portfolio contains the assets in some proportion. Let Sz denote the (n x 1) vector whose elements represent the weights of the assets according to their market values; in other words, Q represents the allocation of the portfolio’s market value among its components. For Sz to be a true allocation vector its elements must sum to unity, or J,‘Q = 1, where Jn is an (n x 1) vector of ones. The elements of R are real numbers, whether positive or negative. Negative weights are feasible in that assets may be borrowed or sold short. The portfolio (R, Q) has overall return characteristics: its mean is E(alR) = SYM, and its variance is Var(iz’R) = E({Q’R R’M)(R’R R’M)‘) = E(nl(R M){R MJ’Q) = fi’E(m M}{R M}‘)S2 = R’CR. Therefore, R’R [R’M , n’Z%Tj. Since the (1 x 1) matrix R’ZQ is a variance, it must be greater than or equal to 0, irrespective of Q. A symmetric matrix Z such that R’CQ is nonnegative for all R is said to be nonnegative definite. Every real-valued random vector with finite covariances must have a nonnegative definite variance. If C has the additional property that ! XX? = 0 only if ! 2 = 0, then C is said to be positive dejnite. If the variance of a random vector is positive