Construction of Risk-Averse Enhanced Index Funds

We propose a partial replication strategy to construct risk-averse enhanced index funds. Our model takes into account the parameter estimation risk by defining the asset returns and the return covariance terms as random variables. The variance of the index fund return is forced to be below a low-risk threshold with a large probability, thereby limiting the market risk exposure of the investors and the moral hazard associated with the wage structure of fund managers. The resulting stochastic integer problem is reformulated through the derivation of a deterministic equivalent for the risk constraint and the use of a block decomposition technique. We develop an exact outer approximation method based on the relaxation of some binary restrictions and the reformulation of the cardinality constraint. The method provides a hierarchical organization of the computations with expanding sets of integer-restricted variables and outperforms the Bonmin and the Cplex 12.1 solvers. The method can solve very large (up to 1000 securities) instances, converges fast, scales well, and is general enough to be applicable to problems with buy-in threshold constraints. Cross-validation tests show that the constructed funds track closely and are consistently less risky than the benchmark on the out-of-sample period.