Please prove 3.15 Lagrange multipliers under short selling Given a target value * for the mean return of the portfolio, the weight vector w of an efficient portfolio can be characterized by Weff = arg min w w subject to w7 = , w T w1 = 1, (3.9) W when short selling is allowed. The method of Lagrange multipliers leads to the optimization problem minw,, {w Ew-2A(w*)2(w11)}. Differentiating the objective function with respect to w, , and y and setting the derivatives equal to 0 leads to the system of linear equations w = +y1, w = *, w1 = 1, W (3.10) which has the explicit solution Weff = - {-11 - - + .(- - -11)}/D when is nonsingular, where -1 (3.11) A = 1 = 1 B= C=11, D=BCA. The variance of the return on this efficient portfolio is 3.2 Markowitz's portfolio theory 71 off = (B-2A+C)/D. (3.12) The * that minimizes off is given by minvar= 4P (3.13) which corresponds to the global MVP with variance minvar = 1/C and weight vector Wminvar = 1/C. (3.14) For two MVPs with mean returns p and g, their weight vectors are given by (3.11) with * = p, q, respectively. From this it follows that the covariance of the returns rp Cov(rp, rq)= and ra given by - 2 (-2) (~2) + 4) D (3.15) Combining (3.15) and (3.13) shows that Cov(rminvar, rq) = 1/C for any MVP q whose return is denoted by rq.