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Merton's Problem with Recursive Perturbed Utility

Min Dai, Yuchao Dong, Yanwei Jia, Xun Yu Zhou

Abstract

The classical Merton investment problem predicts deterministic, state-dependent portfolio rules; however, laboratory and field evidence suggests that individuals often prefer randomized decisions leading to stochastic and noisy choices. Fudenberg et al. (2015) develop the additive perturbed utility theory to explain the preference for randomization in the static setting, which, however, becomes ill-posed or intractable in the dynamic setting. We introduce the recursive perturbed utility (RPU), a special stochastic differential utility that incorporates an entropy-based preference for randomization into a recursive aggregator. RPU endogenizes the intertemporal trade-off between utilities from randomization and bequest via a discounting term dependent on past accumulated randomization, thereby avoiding excessive randomization and yielding a well-posed problem. In a general Markovian incomplete market with CRRA preferences, we prove that the RPU-optimal portfolio policy (in terms of the risk exposure ratio) is Gaussian and can be expressed in closed form, independent of wealth. Its variance is inversely proportional to risk aversion and stock volatility, while its mean is based on the solution to a partial differential equation. Moreover, the mean is the sum of a myopic term and an intertemporal hedging term (against market incompleteness) that intertwines with policy randomization. Finally, we carry out an asymptotic expansion in terms of the perturbed utility weight to show that the optimal mean policy deviates from the classical Merton policy at first order, while the associated relative wealth loss is of a higher order, quantifying the financial cost of the preference for randomization.

Merton's Problem with Recursive Perturbed Utility

Abstract

The classical Merton investment problem predicts deterministic, state-dependent portfolio rules; however, laboratory and field evidence suggests that individuals often prefer randomized decisions leading to stochastic and noisy choices. Fudenberg et al. (2015) develop the additive perturbed utility theory to explain the preference for randomization in the static setting, which, however, becomes ill-posed or intractable in the dynamic setting. We introduce the recursive perturbed utility (RPU), a special stochastic differential utility that incorporates an entropy-based preference for randomization into a recursive aggregator. RPU endogenizes the intertemporal trade-off between utilities from randomization and bequest via a discounting term dependent on past accumulated randomization, thereby avoiding excessive randomization and yielding a well-posed problem. In a general Markovian incomplete market with CRRA preferences, we prove that the RPU-optimal portfolio policy (in terms of the risk exposure ratio) is Gaussian and can be expressed in closed form, independent of wealth. Its variance is inversely proportional to risk aversion and stock volatility, while its mean is based on the solution to a partial differential equation. Moreover, the mean is the sum of a myopic term and an intertemporal hedging term (against market incompleteness) that intertwines with policy randomization. Finally, we carry out an asymptotic expansion in terms of the perturbed utility weight to show that the optimal mean policy deviates from the classical Merton policy at first order, while the associated relative wealth loss is of a higher order, quantifying the financial cost of the preference for randomization.
Paper Structure (15 sections, 7 theorems, 59 equations)

This paper contains 15 sections, 7 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Market Environment and Investment Objective
  4. Preference for Randomization
  5. Recursive Perturbation Utility
  6. Theoretical Analysis
  7. Gaussian Randomization
  8. When is the Optimal Randomized Policy Unbiased?
  9. An Asymptotic Analysis on $\lambda$
  10. Conclusions
  11. Different Temperature Schemes
  12. Constant Temperature
  13. Wealth-Dependent Temperature
  14. CARA Utility
  15. A BSDE Perspective

Key Result

Proposition 1

If $\gamma\in (0,1)$, then the problem with the objective functional objective_functional_no_recursive, subject to the wealth dynamics controlled_system, is ill-posed with an infinite optimal value.

Theorems & Definitions (8)

  • Proposition 1
  • Definition 1
  • Proposition 2
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Theorem 4