**13**

**MEAN-VARIANCE PORTFOLIO CHOICE**

*I*n this chapter, we provide a detailed and rigorous introduction to the most well-known theory of portfolio selection: mean-variance portfolio theory. We start from the simple case of two risky assets to understand the basic problem, then generalize to an arbitrary number of risky assets. After deriving the mean-variance frontier both graphically and analytically, we apply the theory to a realistic example. We also discuss some practical issues associated with the implementation of mean-variance analysis. Finally, we consider some advanced issues.

## 13.1 TWO RISKY ASSETS

Consider the investment of $1 in two risky assets, asset 1 and asset 2, whose random returns next period are *r*_{1} and *r*_{2}, respectively. In this section, we assume that the expected returns, variances, and covariances are known, and focus on the portfolio selection problem. In practice, these variables are unknown, and have to be estimated using historical data. An illustration will be given in Section 13.4.3.

### 13.1.1 Portfolio Return

If we let *w*_{1} and *w*_{2} be the percentage of the portfolio invested in asset 1 and asset 2 respectively, then the total percentage allocated to the two assets must be equal to one, that is,

This equation is known as the *budget constraint*. For example, if the investor allocates 40% of the portfolio to asset 1 (i.e., *w*_{1} = 0.4), that means the investor allocates ...