Resolution of Degeneracy in Merton's Portfolio Problem

The Merton problem determines the optimal intertemporal portfolio choice by maximizing the expected utility and is the basis of modern portfolio theory in continuous-time finance. However, its empirical performance is disappointing. The estimation errors of the expected rates of returns make the optimal policy degenerate, resulting in an extremely low (or unbounded) expected utility value for a high-dimensional portfolio. We further prove that the estimation error of the variance-covariance matrix leads to the degenerated policy of solely investing in the risk-free asset. This study proposes a constrained $\ell_1$-minimization approach to resolve the degeneracy in the high-dimensional setting and stabilize the performance in the low-dimensional setting. The proposed scheme can be implemented with simple linear programming and involves negligible additional computational time, compared to standard estimation. We prove the consistency of our framework that our estimate of the optimal control tends to be the...

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