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Portfolio Time Consistency and Utility Weighted Discount Rates

Oumar Mbodji, Traian A. Pirvu

Abstract

Merton portfolio management problem is studied in this paper within a stochastic volatility, non constant time discount rate, and power utility framework. This problem is time inconsistent and the way out of this predicament is to consider the subgame perfect strategies. The later are characterized through an extended Hamilton Jacobi Bellman (HJB) equation. A fixed point iteration is employed to solve the extended HJB equation. This is done in a two stage approach: in a first step the utility weighted discount rate is introduced and characterized as the fixed point of a certain operator; in the second step the value function is determined through a linear parabolic partial differential equation. Numerical experiments explore the effect of the time discount rate on the subgame perfect and precommitment strategies.

Portfolio Time Consistency and Utility Weighted Discount Rates

Abstract

Merton portfolio management problem is studied in this paper within a stochastic volatility, non constant time discount rate, and power utility framework. This problem is time inconsistent and the way out of this predicament is to consider the subgame perfect strategies. The later are characterized through an extended Hamilton Jacobi Bellman (HJB) equation. A fixed point iteration is employed to solve the extended HJB equation. This is done in a two stage approach: in a first step the utility weighted discount rate is introduced and characterized as the fixed point of a certain operator; in the second step the value function is determined through a linear parabolic partial differential equation. Numerical experiments explore the effect of the time discount rate on the subgame perfect and precommitment strategies.
Paper Structure (20 sections, 7 theorems, 160 equations, 4 figures)

This paper contains 20 sections, 7 theorems, 160 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Model
  3. The Financial Market
  4. Time Preferences and Risk Preferences
  5. The Expected Utility
  6. Subgame Perfect Strategies
  7. The Value Function
  8. Main Results
  9. The Extended HJB
  10. The Utility Weighted Discount Rate
  11. Fixed Point Characterization of Subgame Perfect Discount Rate
  12. Characterization of The Value Function
  13. Further Interpretation of Subgame Perfect Discount Rate
  14. The Investment Strategy and Discounting
  15. Numerical Analysis
  16. ...and 5 more sections

Key Result

Theorem 5.2

Let $(V, \bar{\pi}, \bar{c})$ solve the extended HJB. Then $(\bar{\pi}, \bar{c})$ is given by eq14_1, eq14_2, $V$ is a value function, and $(\bar{\pi}, \bar{c})$ is a subgame perfect strategy.

Figures (4)

  • Figure 1: The range of $\mathbb{Q}(t,S)$ for hyperbolic discounting is given by the shaded area. The decreasing function is the discount rate $R(t)=\rho(0,t).$
  • Figure 2: Comparing $\pi$ for different volatilities and strategies : $\pi^{PC}=\hat{\pi} = \bar{\pi}=\pi^{TC}$ when the parameters $r, \theta, \sigma$ are all constants. Theoretically, we get $\bar{\pi}=\hat{\pi}=\frac{\theta}{\sigma(1-\gamma)}$ is independent of $z$
  • Figure 3: Study of $\hat{\pi}(t,z), \bar{\pi}(t,z)$ for $\gamma=-5$
  • Figure 4: Study of $\frac{\bar{c}(s)\bar{X}(s)}{ \hat{c}(s)\hat{X}(s)}$ for $\gamma=-5$

Theorems & Definitions (16)

  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Definition 4.1
  • Definition 5.1
  • Theorem 5.2
  • Definition 5.3
  • Theorem 5.4
  • ...and 6 more