Time-consistent lifetime portfolio selection under smooth ambiguity

Luyang Yu; Liyuan Lin; Guohui Guan; Jingzhen Liu

doi:10.3934/mcrf.2022023

Article Contents

2023, Volume 13, Issue 3: 967-987. Doi: 10.3934/mcrf.2022023

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Time-consistent lifetime portfolio selection under smooth ambiguity

1.
China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China
2.
Department of Statistics and Actuarial Science, University of Waterloo, Canada
3.
School of Statistics, Renmin University of China, Beijing 100874, China

^*Corresponding author: Jingzhen Liu
^*Corresponding author: Jingzhen Liu

Received: November 2021

Revised: April 2022

Early access: May 2022

Published: September 2023

The corresponding author is supported by National Natural Science Foundation of China (11771466, 11901574)

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Abstract

This paper studies the optimal consumption, life insurance and investment problem for an income earner with uncertain lifetime under smooth ambiguity model. We assume that risky assets have unknown market prices that result in ambiguity. The individual forms his belief, that is, the distribution of market prices, according to available information. His ambiguity attitude, which is similar to the risk attitude described by utility function $ U $, is represented by an ambiguity preference function $ \phi $. Under the smooth ambiguity model, the problem becomes time-inconsistent. We derive the extended Hamilton-Jacobi-Bellman (HJB) equation for the equilibrium value function and equilibrium strategy. Then, we obtain the explicit solution for the equilibrium strategy when both $ U $ and $ \phi $ are power functions. We find that a more risk- or ambiguity-averse individual will consume less, buy more life insurance and invest less. Moreover, we find that the Tobin-Markowitz separation theorem is no longer applicable when ambiguity attitude is taken into consideration. The investment strategy will change with the characteristics of the decision maker, such as risk attitude, ambiguity attitude and age.

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

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Figure 1. Effect of $ \alpha $ on consumption

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Figure 2. Effect of $ \alpha $ on premium

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Figure 3. Effect of $ \alpha $ on investment

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Figure 4. Effect of $ \alpha $ on investment share

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Figure 5. Effect of $ \gamma $ on consumption

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Figure 6. Effect of $ \gamma $ on premium

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Figure 7. Effect of $ \gamma $ on investment

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Figure 8. Effect of $ \gamma $ on investment share

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Figure 9. Effects of $ \sigma $ and $ \gamma $ on risk investment

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Figure 10. Effects of $ \sigma $ and $ \alpha $ on risk investment

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Figure 11. Effects of $ \mu $ and $ \gamma $ on risk investment

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Figure 12. Effects of $ \mu $ and $ \alpha $ on risk investment

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Table 1. Values of the parameters

Text interpretation	Symbol	Value
risk-free interest rate	$ r $	0.03
discount rate	$ \beta $	0.04
mortality hazard function	$ \mu(t) $	$ \frac{e^{\frac{t-86.3}{9.5}}}{9.5} $
premium rate	$ \eta(t) $	$ 1.1\mu(t) $
income	$ i(t) $	$ 50000e^{0.04t} $
retirement time	$ T $	40
volatility of risky assets	$ \sigma_1 $, $ \sigma_2 $	2.5%
distribution of $ \lambda_1 $	$ N(u_1, \Sigma_1) $	$ N(0.1, 0.3) $
distribution of $ \lambda_2 $	$ N(u_2, \Sigma_2) $	$ N(0.1, 0.5) $
initial wealth	$ X(0) $	20000
risk aversion	$ \gamma $	2
ambiguity aversion	$ \alpha $	2

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