# American Institute of Mathematical Sciences

October  2019, 15(4): 1937-1953. doi: 10.3934/jimo.2018130

## Optimal investment and consumption in the market with jump risk and capital gains tax

 1 College of Finance and Statistics, Hunan University, Changsha 410079, China 2 School of Management, Zhejiang University, Hangzhou 310085, China

* Corresponding author: Weidong Xu. Tel.: +86-0571-88206867

Received  March 2018 Revised  May 2018 Published  August 2018

This paper investigates the problem of dynamic investment and consumption in a market, where a risky asset evolves with jumps and capital gains are taxed. In addition, the investor's behavior of tax evasion is taken into account, and tax evasion is subject to penalty when it is uncovered by audits. Using dynamic programming approach, we derive an analytical solution for an investor with the CRRA utility. We find the following: (1) jumps in the risky asset do not affect the optimal tax evasion strategy; (2) jump risk lessens the optimal fraction of wealth in the risky asset; (3) tax evasion can be reduced by increasing the fine and/or the frequency of tax audits; (4) the effects of the jumps, audits and penalty on the optimal consumption are determined by the degree of risk aversion of the investor.

Citation: Yong Ma, Shiping Shan, Weidong Xu. Optimal investment and consumption in the market with jump risk and capital gains tax. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1937-1953. doi: 10.3934/jimo.2018130
##### References:

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##### References:
Dynamic optimal investment and consumption
The effect of jump intensity of asset price on optimal investment and consumption
The effect of jump mean of asset price on optimal investment and consumption
The effect of tax policy on optimal investment and consumption ($\gamma>1$)
The values of the parameters in the base case
 Parameter Value Parameter Value Current time $t=0$ Terminal time $T=20$ Expected return $\mu=0.08$ Volatility $\sigma=0.2$ Riskless interest rate $r=0.04$ Risk aversion coefficient $\gamma=2.5$ Discount factor $\rho=0.04$ Relative weight $\chi=1$ Audit intensity $\lambda_2=0.1$ Punishment intensity $\alpha=0.08$ Tax rate on riskless asset $\tau_G=0.27$ Tax rate on risky asset $\tau=0.235$ Jump intensity $\lambda_1=0.15$ Log jump mean $\mu_J=0$ Jump volatility $\sigma_J=0.5$
 Parameter Value Parameter Value Current time $t=0$ Terminal time $T=20$ Expected return $\mu=0.08$ Volatility $\sigma=0.2$ Riskless interest rate $r=0.04$ Risk aversion coefficient $\gamma=2.5$ Discount factor $\rho=0.04$ Relative weight $\chi=1$ Audit intensity $\lambda_2=0.1$ Punishment intensity $\alpha=0.08$ Tax rate on riskless asset $\tau_G=0.27$ Tax rate on risky asset $\tau=0.235$ Jump intensity $\lambda_1=0.15$ Log jump mean $\mu_J=0$ Jump volatility $\sigma_J=0.5$
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