Resolution of Degeneracy in Merton's Portfolio Problem
SIAM Journal on Financial Mathematics. 7, 786-811, 2016
22 Pages Posted: 26 May 2016 Last revised: 17 Mar 2017
Date Written: February 4, 2016
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 ℓ1 - minimization approach to resolve the degeneracy. 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 true one. We also derive the rate of convergence. Simulation studies are provided to verify the finite-sample properties. An empirical study using S&P 500 component stock data demonstrates the superiority of the proposed approach.
Keywords: High-Dimensional Portfolio, Merton's Problem, Expected Utility Maximization, Constrained $\ell_1-$Minimization, Dantzig Selector, Sparsity
JEL Classification: C44, C61, G11
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