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
1.8.5 Hurdle Rate for Mean-Variance Optimization
An interesting implication of mean-variance portfolio optimization when there is a riskless asset is that the benefits of diversification can be shown to cause low-return assets to be desirable for inclusion in a portfolio. It is easy to show that the addition of a new asset to an already optimal portfolio will improve its risk-return properties (i.e., increases the expected utility) if the expected rate of return on this new asset exceeds a hurdle rate. The hurdle rate is an expected rate of return that a new asset must offer to be included in an already optimal portfolio. An asset being considered for addition to a portfolio should be included in the portfolio when the following expression is satisfied:
(1.21)
Here, E[RNew] is the expected rate of return on the new asset, E[Rp] is the expected rate of return on the optimal portfolio, Rf is the riskless rate, and βNew is the beta of the new asset with respect to the optimal portfolio.7
Equation 1.21 states that the addition of the new asset to an optimal portfolio will improve the risk-return properties of the portfolio if the expected excess rate of return on the new asset exceeds the expected excess rate of return on the optimal portfolio times the beta of the new asset. If the new asset satisfies Equation 1.21, then its addition to the optimal portfolio will move the efficient frontier in the northwest direction. For example, if the beta of the new asset is zero, then the new asset will improve the optimal portfolio as long as its expected rate of return exceeds the riskless rate. If the new asset has a negative beta, then it could improve the optimal portfolio even if its expected rate of return is negative. In other words, assets that can serve as hedging instruments could have negative expected returns and still improve the performance of a portfolio.
APPLICATION 1.8.5
Suppose that an investor is using mean-variance optimization, the expected annual rate of return of an optimal portfolio is 16 %, and the riskless rate is 1 % per year. What is the hurdle rate for a new asset that has a beta of 0.5 with respect to the optimal portfolio?
Given the formula of Equation 1.21, the hurdle rate would be:
What if the new asset has a beta of –0.3, which means that it can hedge some of the portfolio's risk?
In this case, even if the new asset is expected to lose some money (i.e., less than 3.5 %), its addition to the optimal portfolio could still improve its risk-return properties.
1.8.6 Issues in Using Optimization
We have already seen that even in the case of three risky asset classes and the riskless rate, reasonable estimates of the weights could not be obtained unless short sale restrictions were imposed, and even in that case, no allocation to the MSCI World Index was recommended. This lack of allocation to an asset class was driven mostly by the portfolio's high volatility and relatively low return. In other words, using the past as an unbiased forecast of the future and using global equities as an asset class would have meant no allocation would be made to equities, and the portfolio would have focused on the remaining two assets. In practice, implementing an optimization method (mean-variance optimization, in particular) for portfolio allocation decisions raises major challenges.
1.8.7 Optimizers as Error Maximizers
Portfolio optimizers are powerful tools for finding the best allocation of assets to achieve superior diversification, given accurate estimates of the parameters of the return distributions. When the mean-variance method is used, there is a need for accurate estimates of expected returns and the variance-covariance matrix of asset returns. However, portfolio optimizers that use historical estimates of the return distributions have been derogatorily called “error maximizers” due to their tendency to generate solutions with extreme portfolio weights. For example, very large portfolio weights are often allocated to the assets with the highest mean returns and lowest volatility, and very small portfolio weights are allocated to the assets with the lowest mean returns and highest volatility. It is then argued that assets with the highest estimated means are likely to have the largest positive estimation errors, whereas assets with the lowest estimated means are likely to have the largest negative estimation errors. Hence, mean-variance optimization is likely to maximize errors. Therefore, if an analyst overstates mean returns and understates volatility for an asset, then the weights that the model recommends are likely to be much larger than an institutional investor would consider reasonable. Further, other assets are virtually omitted from the portfolio if the analyst supplies low estimates of mean returns and high estimates of volatility.
A typical attempt to use a mean-variance optimization model for portfolio allocation is this: (1) The portfolio manager supplies estimates of the mean return, volatility, and covariance for all assets; (2) the optimizer generates a highly unrealistic solution that places very large portfolio weights on what are considered the most attractive assets, with high mean return and low volatility, and zero or minuscule portfolio weights on what are considered the least attractive assets, with low mean return and high volatility; and (3) the portfolio manager then modifies the model by adding constraints or altering the estimated inputs – including mean, variance, and covariance – until the resulting portfolio solutions appear reasonable.
The problem with this process is that the portfolio weights become driven by the subjective judgments of the analyst rather than by the analyst's best forecasts of risk and return. The remaining sections discuss a variety of challenges that emanate from the tendency of mean-variance portfolio optimizers to select extreme portfolio weights.
It is important to point out that while higher-frequency data tends to improve the accuracy of the estimated variance-covariance matrix, it will do nothing to improve the accuracy of the estimated means; only a longer history has the potential to do so. To see this, assume that we have five years of annual data on the price of an asset. The annual rate of return on the asset is calculated to be Rt + 1 = ln (Pt + 1/Pt). Now consider an estimate of the average return using the observed four annual returns:
(1.22)
Notice that all the intermediate prices cancel out, and only the first and the last prices matter. This result will not change even if one could use daily or even high-frequency data. The accuracy of the mean depends on the length of data and not on the frequency of the observations.
This observation regarding mean accuracy leads to the following dilemma. To obtain accurate estimates of the mean, it is necessary to have a very long history of prices. However, firms, industries, and economies go through drastic changes over long periods, and it would be highly unlikely that all observed prices would have come from the same distribution. In other words, of all the estimated parameters, the estimated mean is most likely to be the least accurate, yet it is the one with the most influence on the outputs of the mean-variance optimization.
The final difficulty in deriving estimates of return and risk for each asset class is that return and risk are nonstationary, meaning that the levels of risk and return vary substantially over time. Therefore, the true risk and return over one period may be substantially different from the risk and return of a different period. Thus, in addition to traditional estimation errors for a stationary process, estimates for security returns may include errors from shooting at a moving target.
1.8.8