# 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

See all articles by Chi Seng Pun

## Chi Seng Pun

Nanyang Technological University (NTU) - School of Physical and Mathematical Sciences

## Hoi Ying Wong

The Chinese University of Hong Kong (CUHK) - Department of Statistics

Date Written: February 4, 2016

### Abstract

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

Suggested Citation

Pun, Chi Seng and Wong, Hoi Ying, Resolution of Degeneracy in Merton's Portfolio Problem (February 4, 2016). SIAM Journal on Financial Mathematics. 7, 786-811, 2016, Available at SSRN: https://ssrn.com/abstract=2784244 or http://dx.doi.org/10.2139/ssrn.2784244