Histogram Models for Robust Portfolio Optimization

This paper presents numerical experiments solving complex robust portfolio optimization problems. The models we study are motivated by realistic considerations, and are in principle combinatorially difficult; however we show that using modern optimization methodology one can solve large, real-life cases quite efficiently. We consider classical mean-variance problems [M52], [M59] and closely related variants (see Section 1.2 for background). These problems require, as inputs, estimates of expectations and variances of asset returns. These estimates are computed from inherently noisy time series, possibly leading to nontrivial errors. Furthermore, and possibly more critically, a portfolio construction technique that hews too closely to previously observed data patterns may result in portfolios that are dangerously exposed to unexpected data behavior, such as a suddenly realized correlation pattern among certain assets. Such considerations naturally lead to a desire for “robustness”, albeit in an informal sense. Stochastic Programming (see [BL97], and [ZM98] for applications to portfolio optimization) is an approach for handling uncertainty that has received a significant amount of attention; typically a stochastic programming approach proceeds by constructing an explicit probability distribution for data and optimizing some stochastic variant of the a priori objective function, e.g. expected value, or value-at-risk. From an informal robustness standpoint the reliance on an explicit probability distribution may be problematic. In contrast, Robust Optimization is a formal approach for optimization under uncertainty that does not rely on a stochastic model of the distribution of data. Often one can view a robust optimization model as a two-stage game; in the first stage the decision maker chooses values for decision variables, while in the second stage an adversary (or “nature”) picks values for data parameters from a given uncertainty set – the process is adversarial in that the worst-case data distribution will be chosen. Robust optimization is sometimes criticized for being overly conservative. At the same time, a robust optimization approach only guards against data realizations that are allowed by the given uncertainty model, while potentially becoming very vulnerable to realizations outside of the realm of the model. One can guard against this particular problem by (in some sense) enlarging the uncertainty set, but this of course may make us even more conservative, possibly in an unexpected way. Furthermore, in an abstract sense, a robust optimization model tends to give the same weight to all possible data realizations, which may be unrealistic in practice. This difficulty is reduced in so-called ambiguous chance-constrained models, but not completely removed.