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      As noted, mean-variance optimizers can be error maximizers. Therefore, erroneous forecasts of the mean, variance, and covariance can result in extreme portfolio weights, with a resulting portfolio concentration in a few assets with estimated high means and estimated low volatilities. Most institutions view such concentrated positions as unacceptable speculation on the validity of the forecasted mean and volatility.

      Although higher frequency of observed data can improve the accuracy of the estimated variance and covariance, for most alternative assets, high-frequency data is not available. More important, the assets whose prices cannot be observed with high frequency tend to be illiquid, and the reported quarterly returns are based on appraisals such as those used in real estate and private equity. These prices tend to be smoothed and therefore can substantially understate the variance and covariance of returns. Because volatility and covariance are key inputs in the optimization process, asset classes with low estimated correlation and volatility receive relatively large weights in the optimal portfolio. If smoothing has caused the reported volatility and correlation of an asset to substantially underestimate the true volatility, then a traditional mean-variance optimizer would overweight the asset. In this case, and to prevent extremely large allocations to assets with smoothed returns, the time series of returns may be unsmoothed, as discussed in Chapter 15, before being added to the optimization routine. But unsmoothing is imperfect, and other issues with accurately forecasting volatility and correlation remain.

      1.8.9 Data Issues for Large-Scale Optimization

      The problems with covariance estimation include the potentially large scale of the inputs required. This is typically not an issue when working at the asset class level, at which the investor may consider 10 asset classes for inclusion in the portfolio. However, optimizing an equity portfolio selected from a universe of 500 stocks has very large data requirements. A 500-asset optimization problem requires estimates of covariance between each pair of the 500 assets. Not only does this problem require n(n − 1)/2, or 124,750 covariance estimates, but it is also difficult to be confident in these estimates, especially when there are too many to analyze individually. Also, notice that to estimate 124,750 covariance terms, we need more than 124,750 observations, or more than 10,000 years of monthly data or more than 340 years of daily data.

      The problem of needing to calculate thousands of covariance estimates can be reduced with factor models. Rather than estimating the relationships between each pair of stocks in a 500-stock universe, it can be easier to estimate the relationship between each stock and a limited number of factors. While some investors simply choose to estimate the single-factor market model beta of each stock, others use multifactor models. To see how a factor model can reduce the data requirement, suppose the return on each asset can be expressed as a function of one common factor and some random noise:

      (1.23)

      where F is the common factor and a_i, b_i are the estimated parameters. It is assumed that for two different assets, the error terms ϵi and ϵj are uncorrelated with each other. Under this assumption, the covariance between two assets is given by:

      (1.24)

      This means that to estimate the covariance matrix of 500 assets, we need 500 estimates of b_i and one estimate of Var[F].

      1.8.10 Mean-Variance Ignores Higher Moments

      A problem that is especially acute with alternative investments is that the mean-variance optimization approach considers only the mean and variance of returns. This means that the optimization model does not explicitly account for skewness and kurtosis. Investors' expected utility can be expressed in terms of mean and variance alone if returns are normally distributed. However, when making allocations to alternative investments and other investments with nonzero skewness and nonzero excess kurtosis, portfolio optimizers tend to suggest portfolios with desirable combinations of mean and variance but with highly undesirable skewness and kurtosis. In other words, although mean-variance optimizers can identify the efficient frontier and help create portfolios with the highest Sharpe ratios, they may be adding large and unfavorable levels of skewness and kurtosis to the portfolio.

      For example, two assets with returns that have the same variance may have very different skews. In a competitive market, the expected return of the asset with the large negative skew might be substantially higher than that of the asset with the positive skew to compensate investors willing to bear the higher downside risk. A mean-variance optimizer typically places a much higher portfolio weight on the negatively skewed asset because it offers a higher mean return with the same level of variance as the other asset. The mean-variance optimizer ignores the unattractiveness of an asset's large negative skew and, in so doing, maximizes the error.

      There are three common ways to address this complication. First, as we saw earlier, it is possible to expand our optimization method to account for skewness and kurtosis of asset returns. Second, we can continue with our mean-variance optimization but add the desired levels of the skewness and kurtosis as explicit constraints on the allowed solutions to the mean-variance optimizer, such as when the excess kurtosis of the portfolio returns is not allowed to exceed 3, or when the skewness must be greater than –0.5. A problem with incorporating higher moments in portfolio optimization is that these moments are extremely difficult to predict, as they are highly influenced by a few large negative or positive observations. In addition, in the second approach, a portfolio with a desired level of skewness or kurtosis may not be feasible at all. Finally, the analyst may choose to explicitly constrain the weight of those investments that have undesirable skew or kurtosis. For example, the allocation to a hedge fund strategy that is known to have large tail risk (e.g., negative skew) might be restricted to some maximum weight.

      1.8.11 Other Issues in Mean-Variance Optimization

      The results from a mean-variance optimization can be extremely sensitive to the assumptions, as small changes in the mean return or covariance matrix (i.e., the set of all variances and covariances) can lead to enormously different prescribed portfolio weights. The high sensitivity of portfolio optimizers to the input data has led to approaches that attempt to harness the power of optimization to identify diversification potential without generating extreme portfolio weights. In addition, in most cases, portfolio managers want to adjust the historical estimates to reflect their views about the estimated parameters going forward. For instance, a portfolio manager may want to incorporate her view that the health care sector is likely to do better than indicated by its historical track record. Perhaps the most popular modification to account for views and obtain reasonable estimates of weights is described by Black and Litterman.

      The first problem addressed by the Black-Litterman approach is the tendency of the user's estimates of mean and variance to generate extreme portfolio weights in a mean-variance optimizer. Note that if a security truly and clearly offered a large expected return, low risk, and high diversification potential, then demand for the security would drive its price upward and its expected return downward until the demand for the security equaled the quantity available. In competitive markets, securities prices tend toward offering a perceived combination of risk and return in line with other assets.

      The key to understanding the Black-Litterman approach is to understand that if a security offers an equilibrium expected return, then the demand for the asset will equal the supply. Further, the optimal allocation of the asset into every well-diversified portfolio will be equal to the weight of the asset in the market portfolio (i.e., the market weight). Thus, an equilibrium expected return for a security is the expected return that causes the optimal weight of that security in investor portfolios to equal its market weight.

      This observation means that if the portfolio manager has no views about the future performance of a particular asset class, then its market weight should be used. For instance, a market-cap-weighted portfolio of global equities would be optimal. However, since market cap weights are not well defined for some asset classes, the Black-Litterman approach will need to be adjusted for application to alternative assets.

      The primary

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