variance. The parameter, X, is the degree of risk aversion of the investor. This utility function is usually justified as an approximation. Two conditions under which it will accurately represent an investor's behavior are locally where a more general smooth utility function can be approximated by a quadratic function, or globally for an investor with constant relative risk aversion and for which returns are normally distributed. Our view is that the key trade-offs in portfolio construction are likely to be illuminated with this function, that risk aversion is the key parameter to vary, and that the main insights of modern portfolio theory are likely to be robust with respect to alternative utility functions that might be found to be more accurate. This equation is the basis for the mean-variance approach to portfolio optimization. Over time this classic utility function became the basis for the equilibrium theory, which we review in Chapter 4, and the large academic literature now referred to as modern portfolio theory. This mean-variance framework is usually represented graphically as in Figure 3.1, which shows the frontier of efficient portfolios. In this figure the horizontal axis shows portfolio volatility, and the vertical axis shows portfolio expected return. The portfolio frontier is a line or a curve that represents the set of all portfolios with the greatest possible expected return for a give level of volatility. Such portfolios are generally termed "Efficient." Curves of constant utility, termed "indifference" curves, show the trade-offs investors are willing to make in this space between expected return and risk. Increasing utility comes from moving from one such curve to another through generating either higher expected return, lower risk, or both. When portfolios include only risky assets or have other constraints, then the optimal portfolio frontier is likely to be a concave curve as shown in Figure 3.1. If, however, investors are able to borrow and lend freely at a risk-free rate, then the optimal portfolio frontier is a line connecting the risk-free rate with the risky portfolio that has the highest ratio of expected excess return over the risk-free rate per unit of portfolio volatility, a ratio called the Sharpe ratio after Nobel laureate William F. Sharpe. In either case, the portfolio that maximizes utility will be one of the efficient portfolios and thus will lie on the efficient portfolio frontier. For a recent in-depth textbook treatment of modern portfolio theory the interested reader might consult Elton et al. (2002). Clearly one condition for a portfolio to be optimal is that any change in an asset weight must fail to increase utility. This implies that, unless there are binding constraints, small changes in asset weights of an optimal portfolio must increase or decrease expected return per unit of portfolio volatility at the rate given by the slope of the utility indifference curve at the point of tangency to the efficient frontier. Thus when utility is defined as in equation (3.1) our theme from Chapter 2, that for a portfolio to be optimal the ratio of expected excess return to contribution to portfolio risk be the same for all assets, is justified formally as the marginal condition required for this utility function to be maximized. If for any asset this condition is not true, clearly we could, by adjusting that asset weight, increase the utility of the portfolio, contradicting the assumption that the portfolio is optimal. Whatever the measure of portfolio risk, it is important to try to understand what the sources of risk in the portfolio are. Simply knowing the volatility of a portfolio, per se, does not provide any insight into what is creating the risk. Risks