Robust portfolio choice problems with relative regret and learning
Andrew Lim
Department of Industrial Engineering & Operations Research
University of California, Berkeley
Wednesday, November 14, 2007
4:30 - 5:30 PM
Terman Engineering Center, Room 453
Abstract:
Expected returns for risky assets are notoriously difficult to
estimate. This causes significant problems in classical portfolio
selection models (such as the Markowitz model) which assume that
expected returns (and variances) for the risky assets are available to
the investor. Indeed, it has been documented that optimization on the
basis of incorrect estimates can actually magnify the estimation
errors and give rise to extremely poor out-of-sample performance. In
this talk, I will discuss some recent work on portfolio choice
problems with model uncertainty. We formulate this problem in the
framework of relative regret, which is a departure from the usual
approach to robust portfolio selection that is based on worst case
expected utility/mean-variance objectives. The relative regret
objective evaluates a portfolio by comparing the resulting wealth to
that of a (benchmark) investor who behaves optimally given knowledge
of the model. In this regard, portfolios that perform well under the
relative regret objective are those that perform well relative to all
such benchmarks over the entire family of alternative models. (In
contrast, the solution of the usual worst case objective is the
portfolio that is optimal for the "most pessimistic" model,
irregardless of its performance under more "optimistic" alternatives).
We consider two situations: portfolio selection problems where
learning is possible and problems where it is not. The optimal
relative regret portfolio has an intuitive characterization in both
cases. In particular, the solution in the multi-period problem with
learning has a Bayesian structure (this is not imposed, but comes out
endogenously), and also involves the family of wealths associated with
each of the benchmark investors. The optimal policy uses a weighted
version of the posterior which is obtained by tilting the classical
posterior distribution using a likelihood ratio that is specified in
terms of the family of benchmark wealths. The tilting favors models
that have done well for the realized time series of asset returns.
Interpretations of the optimal solution from the perspective of loss
functions will also be discussed.
Operations Research Colloquia: http://or.stanford.edu/oras_seminars.html