Non-exponential discounting portfolio management with habit formation

Jingzhen Liu; Liyuan Lin; Ka Fai Cedric Yiu; Jiaqin Wei

doi:10.3934/mcrf.2020019

Article Contents

2020, Volume 10, Issue 4: 761-783. Doi: 10.3934/mcrf.2020019

This issue Previous Article Maximal discrete sparsity in parabolic optimal control with measures Next Article Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach

Non-exponential discounting portfolio management with habit formation

1.
School of Insurance, Central University of Finance and Economics, Beijing, 100081, China
2.
Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, 999077, China
3.
School of Statistics, East China Normal University, Shanghai, 200241, China

^* Corresponding author: Liyuan Lin
^* Corresponding author: Liyuan Lin

Received: April 2019

Revised: January 2020

Early access: March 2020

Published: October 2020

This work was supported by the National Natural Science Foundation of China (Grant No. 11771466, 11301559 and 11601157)

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Abstract

This paper studies the portfolio management problem for an individual with a non-exponential discount function and habit formation in finite time. The investor receives a deterministic income, invests in risky assets, buys insurance and consumes continuously. The objective is to maximize the utility of excessive consumption, heritage and terminal wealth. The non-exponential discounting makes the optimal strategy adopted by a naive person time-inconsistent. The equilibrium for a sophisticated person is Nash subgame perfect equilibrium, and the sophisticated person is time-consistent. We calculate the analytical solution for both the naive strategy and equilibrium strategy in the CRRA case and compare the results of the two strategies. By numerical simulation, we find that the sophisticated individual will spend less on consumption and insurance and save more than the naive person. The difference in the strategies of the naive and sophisticated person decreases over time. Furthermore, if an individual of either type is more patient in the future or has a greater tendency toward habit formation, he/she will consume less and buy less insurance, and the degree of inconsistency will also be increased. The sophisticated person's consumption and habit level are initially lower than those of a naive person but are higher in later periods.

Keywords:

Non-exponential discounting,
Habit formation,
Optimal portfolio,
Optimal insurance,
Extended Hamilton-Jacobi-Bellman equation.

Mathematics Subject Classification: Primary: 91B08, 91B51; Secondary: 91G80.

Citation:

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Figure 1. The impact of $ k $ on $ \bar{A} $ and $ \hat{A} $

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Figure 2. The impact of $ k $ in the degree of inconsistency

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Figure 3. The impact of habit on $ \bar{A} $ and $ \hat{A} $

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Figure 4. The impact of habit on the degree of inconsistency

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Figure 5. Strategy for individual with or without habit

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Figure 6. Example of $ \bar{H}(t)-\hat{H}(t) $ and $ \bar{c}(t)-\hat{c}(t) $

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