Optimization in Finance

In the introductory lectures, we have discussed Markowitz’ theory of mean-variance optimization (MVO) for the selection of portfolios of securities (or asset classes) in a manner that trades off the expected returns and the perceived risk of potential portfolios. Consider assets S1, S2, . . . , Sn (n ≥ 2) with random returns. Let μi and σi denote the expected return and the standard deviation of the return of asset Si. For i 6= j, ρij denotes the correlation coefficient of the returns of assets Si and Sj. Let μ = [μ1, . . . , μn] T , and Q be the n × n symmetric covariance matrix with Qii = σ i and Qij = ρijσiσj for i 6= j. Denoting the proportion of the total funds invested in security i by xi, one can represent the expected return and the variance of the resulting portfolio x = (x1, . . . , xn) as follows: E[x] = x1μ1 + . . .+ xnμn = μ x,