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It turns out that a similar graph will be obtained even if the number of asset classes is greater than one. In that case, the graph will be referred to as the efficient frontier. The efficient frontier is the set of all feasible combinations of expected return and standard deviation that can serve as an optimal solution for one or more risk-averse investors. Put differently, no portfolio can be constructed with the same expected return as the portfolio on the frontier but with a lower standard deviation, or, conversely, no portfolio can be constructed with the same standard deviation as the portfolio on the frontier but with a higher expected return.

Example: In this example, the set of risky asset classes is expanded to three. The necessary information is provided in Exhibit 1.7. The figures are estimated using monthly data in terms of USD. The annual riskless rate is assumed to be 1 %. Note that these estimates are typically adjusted to reflect current market conditions. This example is meant to illustrate an application of the model.

Using the optimal solution that was displayed in Equation 1.18, the optimal weights of a portfolio consisting of the three risky assets and one riskless asset can be calculated for different degrees of risk aversion. The results are displayed in Exhibit 1.8.

A number of interesting observations can be drawn from these results. First, notice that the optimal weights are not very realistic. For example, for every degree of risk aversion, the optimal portfolio requires us to take a short position in the MSCI World Index. Second, the optimal investment in the HFRI index exceeds 100 % for some degrees of risk aversion considered here. Third, unless the degree of risk aversion is increased beyond 40, the optimal portfolio requires some degree of leverage (i.e., negative weight for the Treasury bills). Finally, the bottom two rows display annual mean and annual standard deviation of the optimal portfolios. These represent points on the efficient frontier.

EXHIBIT 1.7 Statistical Properties of Three Risky Asset Classes

Source: Bloomberg and authors' calculations.

EXHIBIT 1.8 Optimal Weights and Statistics for Different Degrees of Risk Aversion

Source: Authors' calculations. (Note that because of rounding errors, the weights do not add up to one.)

As we just saw, mean-variance optimization typically leads to unrealistic weights. A simple way to overcome this problem is to impose limits on the weights. For instance, in the example just provided, we can impose the constraint that the weights must be nonnegative. Unfortunately, when constraints are imposed on the weights, a closed-form solution of the type presented in Equation 1.18 can no longer be obtained, and we must use a numerical optimization package to solve the problem.6

If we repeat the example but impose the constraint that weights cannot be negative, the resulting optimal portfolios will reflect those displayed in Exhibit 1.9.

It can be seen that the weight of the MSCI World Index is constantly zero for all degrees of risk aversion. This means that portfolios that are on the efficient frontier in this case do not have any allocation to the MSCI World Index. Another important point to consider is that the optimal portfolios do not have the same attractive risk-return properties. By imposing a constraint, the resulting portfolios are not as optimal as they were when there were no constraints.

The mean-variance optimization approach discussed in this section can be presented in different forms. The Appendix at the end of this book provides two alternative methods that have appeared in the literature. The advantages of the approach presented in this section are twofold. First, as Equations 1.18 and 1.19 showed, simple closed-form solutions can be obtained when there are no constraints. Second, the approach can be easily expanded to take into account preferences for higher moments of the probability distributions of asset returns. This will turn out to be important for our purpose, as alternative investments tend to have return distributions that deviate significantly from the normal distribution; therefore, their higher moments will be of interest to investors.

EXHIBIT 1.9 Optimal Weights without Short Sale Constraints