# CHAPTER 7

# Portfolio Optimization: Theory and Practice

**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

## STATIC PORTFOLIO THEORY

In the static portfolio theory case, suppose there are *n* assets, *i* = 1, …, *n*, with random returns *ξ*_{1}, …, *ξ*_{n}. The return on asset *i*, namely *ξ*_{i}, 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*(*ξ*_{1}, …, *ξ*_{n}) with known mean vector *ξ* = (*ξ*_{1}, …, *ξ*_{n}) and known *n* × *n* variance-covariance matrix Σ with typical covariance *σ*_{ij} 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* = (*x*_{1}, …, *x*_{n}) are the asset weights, *K* represents other constraints on the *x*, and *w*_{0} 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, ...