William T. Ziemba
Alumni Professor of Financial Modeling andStochastic Optimization (Emeritus),University of British Columbia
Distinguished Visiting Research AssociateSystemic Risk Center, London School of Economics
In the static portfolio theory case, suppose there are n assets, i = 1, …, n, with random returns ξ1, …, ξn. The return on asset i, namely ξi, is the capital appreciation plus dividends in the next investment period such as monthly, quarterly, or yearly or some other time period. The n assets have the distribution F(ξ1, …, ξn) with known mean vector ξ = (ξ1, …, ξn) and known n × n variance-covariance matrix Σ with typical covariance σij for i ≠ j and variance for i = j. A basic assumption (relaxed in section 6) is that the return distributions are independent of the asset weight choices, so F ≠ φ(x).
A mean-variance frontier is
where e is a vector of ones, x = (x1, …, xn) are the asset weights, K represents other constraints on the x, and w0 is the investor's initial wealth.
When variance is parameterized with δ > 0, it yields a concave curve, as in Figure 7.1(a). This is a Markowitz (1952, 1987, 2006) mean-variance efficient frontier and optimally trades off mean, which is desirable, with variance, ...