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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}/P_t). 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.