Robust portfolio selection for individuals: Minimizing the probability of lifetime ruin

Bing Liu; Ming Zhou

doi:10.3934/jimo.2020005

Article Contents

2021, Volume 17, Issue 2: 937-952. Doi: 10.3934/jimo.2020005

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Robust portfolio selection for individuals: Minimizing the probability of lifetime ruin

Bing Liu^1, and
Ming Zhou^2, ,

1.
School of Finance, Nanjing University of Finance and Economics, Nanjing 210023, China
2.
China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China

^* Corresponding author: Ming Zhou
^* Corresponding author: Ming Zhou

Received: August 2018

Revised: June 2019

Early access: January 2020

Published: March 2021

This research is supported by by the National Natural Science Foundation of China (11971506, 11571388), Beijing Social Science Foundation (15JGB046), the MOE Project of Key Research Institute of Humanities and Social Science at Universities (15JJD790036), and the 111 Project (B17050)

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Abstract

Robust portfolio selection has become a popular problem in recent years. In this paper, we study the optimal investment problem for an individual who carries a constant consumption rate but worries about the model ambiguity of the financial market. Instead of using a conventional value function such as the utility of terminal wealth maximization, here, we focus on the purpose of risk control and seek to minimize the probability of lifetime ruin. This study is motivated by the work of [3], except that we use a standardized penalty for ambiguity aversion. The reason for taking a standardized penalty is to convert the penalty to units of the value function, which makes the difference meaningful in the definition of the value function. The advantage of taking a standardized penalty is that the closed-form solutions to both the robust investment policy and the value function can be obtained. More interestingly, we use the "Ambiguity Derived Ratio" to characterize the existence of model ambiguity which significantly affects the optimal investment policy. Finally, several numerical examples are given to illustrate our results.

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Mathematics Subject Classification: Primary: 91B30; Secondary: 91B42.

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Figure 1. Optimal investment policies with respect to the wealth and the model ambiguity

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Figure 5. Ambiguity Derived Ratio with respect to model ambiguity

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Figure 2. Optimal investment policies with respect to the lifetime

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Figure 3. The value function with respect to model ambiguity

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Figure 4. Return rate of the risky asset under robust risk measure ($ \mu = 0.1 $)

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