# American Institute of Mathematical Sciences

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, doi: 10.3934/jimo.2018130
##### References:
 [1] Y. Ait-Sahalia, J. Cacho-Diaz and R. J. Laeven, Modeling financial contagion using mutually exciting jump processes, Journal of Financial Economics, 117 (2015), 585-606. [2] G. Bakshi, C. Cao and Z. Chen, Empirical performance of alternative option pricing models, Journal of Finance, 52 (1997), 2003-2049. doi: 10.1111/j.1540-6261.1997.tb02749.x. [3] O. E. Barndorff-Nielsen and N. Shephard, Econometrics of testing for jumps in financial economics using bipower variation, Journal of Financial Econometrics, 4 (2006), 1-30. [4] D. S. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, Review of Financial Studies, 9 (1996), 69-107. doi: 10.1093/rfs/9.1.69. [5] J. Cai, X. Chen and M. Dai, Portfolio selection with capital gains tax, recursive utility, and regime switching, Management Science, 64 (2017), 2308-2324. [6] R. M. Dammon, C. S. Spatt and H. H. Zhang, Optimal consumption and investment with capital gains taxes, Review of Financial Studies, 14 (2001), 583-616. [7] R. M. Dammon, C. S. Spatt and H. H. Zhang, Optimal asset location and allocation with taxable and tax-deferred investing, Journal of Finance, 59 (2004), 999-1037. [8] M. Dai, H. Liu, C. Yang and Y. Zhong, Optimal tax timing with asymmetric long-term/short-term capital gains tax, Review of Financial Studies, 28 (2015), 2687-2721. doi: 10.1093/rfs/hhv024. [9] S. R. Das and R. Uppal, Systemic risk and international portfolio choice, Journal of Finance, 59 (2004), 2809-2834. [10] B. Eraker, Do stock prices and volatility jump? Reconciling evidence from spot and option prices, Journal of Finance, 59 (2004), 1367-1403. doi: 10.1111/j.1540-6261.2004.00666.x. [11] B. Eraker, M. Johannes and N. Polson, The impact of jumps in volatility and returns, Journal of Finance, 58 (2003), 1269-1300. doi: 10.1111/1540-6261.00566. [12] M. F. Gallmeyer, R. Kaniel and S. Tompaidis, Tax management strategies with multiple risky assets, Journal of Financial Economics, 80 (2006), 243-291. [13] Y. Hong and X. Jin, Semi-analytical solutions for dynamic portfolio choice in jump-diffusion models and the optimal bond-stock mix, European Journal of Operational Research, 265 (2018), 389-398. doi: 10.1016/j.ejor.2017.08.010. [14] S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101. [15] S. S. Lee and P. A. Mykland, Jumps in financial markets: A new nonparametric test and jump dynamics, Review of Financial Studies, 21 (2007), 2535-2563. [16] R. Levaggi and F. Menoncin, Optimal dynamic tax evasion: A portfolio approach, Journal of Economic Behavior and Organization, 124 (2016), 115-129. [17] J. Liu, F. A. Longstaff and J. Pan, Dynamic asset allocation with event risk, Journal of Finance, 58 (2003), 231-259. [18] M. Marekwica, Optimal tax-timing and asset allocation when tax rebates on capital losses are limited, Journal of Banking and Finance, 36 (2012), 2048-2063. [19] R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2. [20] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. [21] H. Nagai, H-J-B equations of optimal consumption-investment and verification theorems, Applied Mathematics and Optimization, 71 (2015), 279-311. doi: 10.1007/s00245-014-9258-0. [22] J. Pan, The jump-risk premia implicit in options: Evidence from an integrated time-series study, Journal of Financial Economics, 63 (2002), 3-50. doi: 10.1016/S0304-405X(01)00088-5. [23] L. C. G. Rogers, Optimal Investment, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-3-642-35202-7. [24] I. B. Tahar, H. M. Soner and N. Touzi, The dynamic programming equation for the problem of optimal investment under capital gains taxes, SIAM Journal on Control and Optimization, 46 (2007), 1779-1801. doi: 10.1137/050646044. [25] I. B. Tahar, H. M. Soner and N. Touzi, Merton problem with taxes: Characterization, computation, and approximation, SIAM Journal on Financial Mathematics, 1 (2010), 366-395. doi: 10.1137/080742178. [26] G. Tauchen and H. Zhou, Realized jumps on financial markets and predicting credit spreads, Journal of Econometrics, 160 (2011), 102-118. doi: 10.1016/j.jeconom.2010.03.023. [27] L. Wu, Jumps and dynamic asset allocation, Review of Quantitative Finance and Accounting, 20 (2003), 207-243.

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##### References:
 [1] Y. Ait-Sahalia, J. Cacho-Diaz and R. J. Laeven, Modeling financial contagion using mutually exciting jump processes, Journal of Financial Economics, 117 (2015), 585-606. [2] G. Bakshi, C. Cao and Z. Chen, Empirical performance of alternative option pricing models, Journal of Finance, 52 (1997), 2003-2049. doi: 10.1111/j.1540-6261.1997.tb02749.x. [3] O. E. Barndorff-Nielsen and N. Shephard, Econometrics of testing for jumps in financial economics using bipower variation, Journal of Financial Econometrics, 4 (2006), 1-30. [4] D. S. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, Review of Financial Studies, 9 (1996), 69-107. doi: 10.1093/rfs/9.1.69. [5] J. Cai, X. Chen and M. Dai, Portfolio selection with capital gains tax, recursive utility, and regime switching, Management Science, 64 (2017), 2308-2324. [6] R. M. Dammon, C. S. Spatt and H. H. Zhang, Optimal consumption and investment with capital gains taxes, Review of Financial Studies, 14 (2001), 583-616. [7] R. M. Dammon, C. S. Spatt and H. H. Zhang, Optimal asset location and allocation with taxable and tax-deferred investing, Journal of Finance, 59 (2004), 999-1037. [8] M. Dai, H. Liu, C. Yang and Y. Zhong, Optimal tax timing with asymmetric long-term/short-term capital gains tax, Review of Financial Studies, 28 (2015), 2687-2721. doi: 10.1093/rfs/hhv024. [9] S. R. Das and R. Uppal, Systemic risk and international portfolio choice, Journal of Finance, 59 (2004), 2809-2834. [10] B. Eraker, Do stock prices and volatility jump? Reconciling evidence from spot and option prices, Journal of Finance, 59 (2004), 1367-1403. doi: 10.1111/j.1540-6261.2004.00666.x. [11] B. Eraker, M. Johannes and N. Polson, The impact of jumps in volatility and returns, Journal of Finance, 58 (2003), 1269-1300. doi: 10.1111/1540-6261.00566. [12] M. F. Gallmeyer, R. Kaniel and S. Tompaidis, Tax management strategies with multiple risky assets, Journal of Financial Economics, 80 (2006), 243-291. [13] Y. Hong and X. Jin, Semi-analytical solutions for dynamic portfolio choice in jump-diffusion models and the optimal bond-stock mix, European Journal of Operational Research, 265 (2018), 389-398. doi: 10.1016/j.ejor.2017.08.010. [14] S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101. [15] S. S. Lee and P. A. Mykland, Jumps in financial markets: A new nonparametric test and jump dynamics, Review of Financial Studies, 21 (2007), 2535-2563. [16] R. Levaggi and F. Menoncin, Optimal dynamic tax evasion: A portfolio approach, Journal of Economic Behavior and Organization, 124 (2016), 115-129. [17] J. Liu, F. A. Longstaff and J. Pan, Dynamic asset allocation with event risk, Journal of Finance, 58 (2003), 231-259. [18] M. Marekwica, Optimal tax-timing and asset allocation when tax rebates on capital losses are limited, Journal of Banking and Finance, 36 (2012), 2048-2063. [19] R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2. [20] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. [21] H. Nagai, H-J-B equations of optimal consumption-investment and verification theorems, Applied Mathematics and Optimization, 71 (2015), 279-311. doi: 10.1007/s00245-014-9258-0. [22] J. Pan, The jump-risk premia implicit in options: Evidence from an integrated time-series study, Journal of Financial Economics, 63 (2002), 3-50. doi: 10.1016/S0304-405X(01)00088-5. [23] L. C. G. Rogers, Optimal Investment, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-3-642-35202-7. [24] I. B. Tahar, H. M. Soner and N. Touzi, The dynamic programming equation for the problem of optimal investment under capital gains taxes, SIAM Journal on Control and Optimization, 46 (2007), 1779-1801. doi: 10.1137/050646044. [25] I. B. Tahar, H. M. Soner and N. Touzi, Merton problem with taxes: Characterization, computation, and approximation, SIAM Journal on Financial Mathematics, 1 (2010), 366-395. doi: 10.1137/080742178. [26] G. Tauchen and H. Zhou, Realized jumps on financial markets and predicting credit spreads, Journal of Econometrics, 160 (2011), 102-118. doi: 10.1016/j.jeconom.2010.03.023. [27] L. Wu, Jumps and dynamic asset allocation, Review of Quantitative Finance and Accounting, 20 (2003), 207-243.
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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