Optimizing Allocations

The classical approach to allocation optimization discussed in the second part of the book assumes that the distribution of the market is known. The samplebased allocation, discussed in the previous chapter, is a two-step process: first the market distribution is estimated and then the estimate is inputted in the classical allocation optimization problem. Since this process leverages the estimation error, portfolio managers, traders and professional investors in a broader sense mistrust these two-step "optimal" approaches and prefer to resort to ad-hoc recipes, or trust their prior knowledge/experience. In this chapter we discuss allocation strategies that account for estimation risk within the allocation decision process. These strategies must be optimal according to the evaluation criteria introduced in the previous chapter: in other words, the overall opportunity cost of these strategies must be as low as possible. The main reasons why estimation risk plays such an important role in financial applications is the extreme sensitivity of the optimal allocation function to the unknown parameters that determine the distribution of the market. In Section 9.1 we use the Bayesian approach to estimation to limit this sensitivity. We present Bayesian allocations in terms of the predictive distribution of the market, as well as the classical-equivalent Bayesian allocation, which relies on Bayes-Stein shrinkage estimators of the market parameters. The Bayesian approach provides a mechanism that mixes the positive features of the prior allocation and the sample-based allocation: the estimate of the market is shrunk towards the investor’s prior in a self-adjusting way and the overall opportunity cost is reduced. In Section 9.2 we present the Black-Litterman approach to control the extreme sensitivity of the optimal allocation function to the unknown market parameters. Like the Bayesian approach, the Black-Litterman methodology makes use of Bayes’ rule. In this case the market is directly shrunk towards the investor’s prior views, rather than indirectly through the market parameters. We present the theory in a general context, performing the computations explicitly in the case of normally distributed markets. Then we apply those symmys.com