Risk-sensitive benchmarked asset management

Abstract

This paper extends the risk-sensitive asset management theory developed by Bielecki and Pliska and by Kuroda and Nagai to the case where the investor's objective is to outperform an investment benchmark. The main result is a mutual fund theorem. Every investor following the same benchmark will take positions, in proportions dependent on his/her risk sensitivity coefficient, in two funds: the log-optimal portfolio and a second fund which adjusts for the correlation between the traded assets, the benchmark and the underlying valuation factors.

Keywords:

• Asset management
• Risk-sensitive stochastic control
• Outperformance
• Dynamic programming
• Benchmark
• Kelly criterion

Acknowledgements

The authors wish to thank Eckhard Platen as well as the two anonymous referees for their insightful comments and suggestions.

Notes

† The case  θ ∈]  − 2, 0 [could be considered. However, we will omit it since it represents a risk-seeking behaviour.

†A matrix M is stable if and only if all of its eigenvalues have negative real parts. See Davis (1977) for details.

‡Let M and K be two n  × n matrices. (M,K) is controllable if the matrix [K MK M ² K  … M ⁿ⁻¹ K] has full rank. See Davis (1977) for details.