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 ai, bi 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 bi 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 innovation of the Black-Litterman approach is that it allows the investor to blend asset-specific views of each asset's expected return with views that would be consistent with market weights in a market equilibrium model.
Whereas some asset allocators employ advanced techniques such as the Black-Litterman approach to reduce the sensitivity of the weights to the expected risks and returns, a much larger number of asset allocators choose to add additional constraints to the optimization model to circumvent the difficulties and sensitivities of mean-variance optimization. Common additional constraints include the following:
The last constraint, limits on portfolio weights, is the most popular. These constraints can prescribe upper or lower weights, outside of which the asset allocator will not invest. The portfolio optimizer is forced to generate weights within those ranges. In practice, however,
