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Asset-liability management with Epstein-Zin utility under stochastic interest rate and unknown market price of risk

Wilfried Kuissi-Kamdem

TL;DR

The paper addresses asset-liability management with Epstein–Zin recursive utility under partial information, where the market price of risk is unobservable and a terminal liability is present. It develops a reduction to the observable filtration via stochastic filtering, yielding a coupled forward-backward SDE (FBSDE) that admits an explicit solution for the optimal consumption and investment strategies. Under mild integrability and Malliavin differentiability conditions, the authors obtain closed-form expressions for the optimal controls and a tractable representation of the value function, along with a Clark-Ocone-type representation of the investment strategy. Finally, they quantify the welfare loss from ignoring learning about the market price of risk, highlighting the economic importance of partial information in ALM with stochastic rates and liabilities.

Abstract

This paper solves a consumption-investment choice problem with Epstein-Zin recursive utility under partial information--unobservable market price of risk. The main novelty is the introduction of a terminal liability constraint, a feature directly motivated by practical portfolio management and insurance applications but absent from the recursive utility literature. Such constraint gives rise to a coupled forward-backward stochastic differential equation (FBSDE) whose well-posedness has not been addressed in earlier work. We provide an explicit solution to this FBSDE system--contrasting with the typical existence and uniqueness results with no closed-form expressions in the literature. Under mild additional assumptions, we also establish the Malliavin differentiability of the solution allowing the optimal investment strategy to be expressed as a conditional expectation of random variables that can be efficiently simulated. These results allows us to obtain the explicit expressions of the optimal controls and the value function. Finally, we quantify the utility loss from ignoring learning about the market price of risk, highlighting the economic significance of partial information.

Asset-liability management with Epstein-Zin utility under stochastic interest rate and unknown market price of risk

TL;DR

The paper addresses asset-liability management with Epstein–Zin recursive utility under partial information, where the market price of risk is unobservable and a terminal liability is present. It develops a reduction to the observable filtration via stochastic filtering, yielding a coupled forward-backward SDE (FBSDE) that admits an explicit solution for the optimal consumption and investment strategies. Under mild integrability and Malliavin differentiability conditions, the authors obtain closed-form expressions for the optimal controls and a tractable representation of the value function, along with a Clark-Ocone-type representation of the investment strategy. Finally, they quantify the welfare loss from ignoring learning about the market price of risk, highlighting the economic importance of partial information in ALM with stochastic rates and liabilities.

Abstract

This paper solves a consumption-investment choice problem with Epstein-Zin recursive utility under partial information--unobservable market price of risk. The main novelty is the introduction of a terminal liability constraint, a feature directly motivated by practical portfolio management and insurance applications but absent from the recursive utility literature. Such constraint gives rise to a coupled forward-backward stochastic differential equation (FBSDE) whose well-posedness has not been addressed in earlier work. We provide an explicit solution to this FBSDE system--contrasting with the typical existence and uniqueness results with no closed-form expressions in the literature. Under mild additional assumptions, we also establish the Malliavin differentiability of the solution allowing the optimal investment strategy to be expressed as a conditional expectation of random variables that can be efficiently simulated. These results allows us to obtain the explicit expressions of the optimal controls and the value function. Finally, we quantify the utility loss from ignoring learning about the market price of risk, highlighting the economic significance of partial information.

Paper Structure

This paper contains 11 sections, 8 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and problem formulation
  3. The financial market
  4. The partial information framework
  5. The Epstein-Zin utility maximisation problem with partial information
  6. Main results
  7. Reduction to the observable filtration
  8. Solution to the optimisation problem
  9. Utility loss
  10. Integrability of $\alpha\varphi$
  11. Proof of Proposition \ref{['Sufficient conditions for the assumptions to hold']}

Key Result

Proposition 3.1

Let the conditional mean-variance pair $(m_{t},v_{t})_{t\in[0,T]}$ be defined as in Conditional characteristics of R. Then, $(m_{t},v_{t}),~t\in[0,T]$, solves the system where $m_{0}=\mathbb{E}[R_{0}],v_{0}=\mathbb{E}[(R_{0}-m_{0})^{2}]$ and the $\mathbb{R}^{2}$-valued process $I=(I_{t}^{1},I_{t}^{2})_{t\in[0,T]}$, called the innovation process, given by is a two dimensional Brownian motion unde

Figures (1)

  • Figure 1: Welfare loss $L$. Both figures use $\delta=0.08$. The left panel uses $\gamma=5$, and the right panel takes $\psi=1.5$. The solid lines represent the cases where $K=500$ and the dashed lines the cases where $K=0$.

Theorems & Definitions (20)

  • Definition 2.1
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • proof
  • Proposition 3.8
  • ...and 10 more