Some problems with the Markowitz mean-variance model

Harry Markowitz’s mean-variance model for portfolio choice posits a linear relationship between the return of a portfolio and the returns of its component securities. This linear relationship does not hold in an ex post setting when monthly or quarterly returns are used. 1 The Standard Portfolio Selection Model Harry Markowitz begins Mean-Variance Analysis in Portfolio Choice and Capital Markets (Markowitz [1987]) with a description of the Standard Mean-Variance Portfolio Selection Model: an investor is to choose fractions p1, p2, . . . , pn invested in n securities subject to constraints n ∑ j=1 pj = 1, pj ≥ 0 (j = 1, . . . , n) . (1.1) We suppose that the returns this period on individual securities r1, r2, . . . , rn are jointly distributed random variables, and the return on the portfolio is r = n ∑ j=1 rjpj . (1.2) The expected (mean) return of the portfolio as a whole is r̄ = n ∑ j=1 r̄jpj , (1.3) where r̄j = E(rj) (j = 1, . . . , n) (1.4) and the E is the expected value operator. The variance of return on the portfolio is