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October  2019, 15(4): 1809-1830. doi: 10.3934/jimo.2018124

## Uncertain portfolio selection with mental accounts and background risk

 1 Business School, Central University of Finance and Economics, Beijing 100081, China 2 Guanghua School of Management, Peking University, Harvest Fund Management Co., Ltd, Beijing 100871, China

* Corresponding author: Hao Di

Received  July 2017 Revised  May 2018 Published  October 2019 Early access  August 2018

Fund Project: The second author is supported by CPSF (2017M611513).

In real life, investors face background risk which may affect their portfolio selection decision. In addition, since the security market is too complex, there are situations where the future security returns cannot be reflected by historical data and have to be given by experts' estimations according to their knowledge and judgement. This paper discusses a portfolio selection problem with background risk in such an uncertain environment. In the paper, in order to reflect different attitudes towards risk that vary by goal in one portfolio investment, we apply mental accounts to the investment. Using uncertainty theory, we propose an uncertain portfolio selection model with mental accounts and background risk and provide the determinate form of the model. Moreover, we discuss the shape and location of efficient frontier of the subportfolios with background risk and without background risk. Further, we present the conditions under which the optimal aggregate portfolio is on the efficient frontier when return rates of security and background asset are all normal uncertain variables. Finally, a real portfolio selection example is given as an illustration.

Citation: Li Xue, Hao Di. Uncertain portfolio selection with mental accounts and background risk. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1809-1830. doi: 10.3934/jimo.2018124
##### References:
 [1] G. J. Alexander and A. M. Baptista, Portfolio selection with mental accounts and delegation, Journal of Banking and Finance, 36 (2011), 2637-2656. [2] S. Aramonte, M. G. Rodriguez and J. Wu, Dynamic factor Value-at-Risk for large heteroskedastic portfolios, Journal of Banking and Finance, 37 (2013), 4299-4309. [3] A. M. Baptista, Portfolio selection with mental accounts and background risk, Journal of Banking and Finance, 36 (2012), 968-980.  doi: 10.1016/j.jbankfin.2011.10.015. [4] R. Castellano and R. Cerqueti, Mean-variance portfolio selection in presence of infrequently traded stocks, European Journal of Operational Research, 234 (2014), 442-449.  doi: 10.1016/j.ejor.2013.04.024. [5] S. Das, H. Markowitz, J. Scheid and M. Statman, Portfolio optimization with mental accounts, Journal of Financial and Quantitative Analysis, 45 (2010), 311-334.  doi: 10.1017/S0022109010000141. [6] S. Das, H. Markowitz, J. Scheid and M. Statman, Portfolios for investors who want to reach their goals while staying on the mean-variance efficient frontier, Journal of Wealth Management, (2011), 1-7. [7] C. Gollier, The Economics of Risk and Time, MIT Press, Cambridge, 2001. [8] X. X. Huang, Portfolio Analysis: From Probabilistic to Credibilistic and Uncertain Approaches, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11214-0. [9] X. X. Huang, Mean-risk model for uncertain portfolio selection, Fuzzy Optimization and Decision Making, 10 (2011), 71-89.  doi: 10.1007/s10700-010-9094-x. [10] X. X. Huang, A risk index model for portfolio selection with returns subject to experts' evaluations, Fuzzy Optimization and Decision Making, 11 (2012), 451-463.  doi: 10.1007/s10700-012-9125-x. [11] X. X. Huang, Mean-variance models for portfolio selection subject to experts' estimations, Expert Systems with Applications, 39 (2012), 5887-5893.  doi: 10.1016/j.eswa.2011.11.119. [12] X. X. Huang and H. Di, Uncertain portfolio selection with background risk, Applied Mathematics and Computation, 276 (2016), 284-296.  doi: 10.1016/j.amc.2015.12.018. [13] H. H. Huang and C. P. Wang, Portfolio selection and portfolio frontier with background risk, North American Journal of Economics and Finance, 26 (2013), 177-196.  doi: 10.1016/j.najef.2013.09.001. [14] X. X. Huang and H. Y. Ying, Risk index based models for portfolio adjusting problem with returns subject to experts' evaluations, Economic Modelling, 11 (2012), 451-463.  doi: 10.1007/s10700-012-9125-x. [15] C. H. Jiang, Y. K. Ma and Y.B An, An analysis of portfolio selection with background risk, Journal of Banking and Finance, 34 (2010), 3055-3060. [16] D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-292. [17] B. D. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-39987-2. [18] B. D. Liu, Why is there a need for uncertainty theory?, Journal of Uncertain Systems, 6 (2012), 3-10. [19] B. D. Liu, Uncertainty Theory, 4nd edition, Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-540-39987-2. [20] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [21] H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Wiley, New York, 1959. [22] F. Menoncin, Optimal portfolio and background risk: an exact and an approximated solution, Insurance Mathematics and Economics, 31 (2002), 249-265.  doi: 10.1016/S0167-6687(02)00154-3. [23] H. S. Rosen and S. Wu, Portfolio choice and health status, Journal of Financial Economics, 72 (2004), 457-484. [24] R. H. Thaler, Mental accounting and consumer choice, Marketing Science, 4 (1985), 199-214. [25] L. M. Viceira, Optimal portfolio choice for long-horizon investors with nontradable labor income, Journal of Finance, 56 (2001), 433-470.

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##### References:
 [1] G. J. Alexander and A. M. Baptista, Portfolio selection with mental accounts and delegation, Journal of Banking and Finance, 36 (2011), 2637-2656. [2] S. Aramonte, M. G. Rodriguez and J. Wu, Dynamic factor Value-at-Risk for large heteroskedastic portfolios, Journal of Banking and Finance, 37 (2013), 4299-4309. [3] A. M. Baptista, Portfolio selection with mental accounts and background risk, Journal of Banking and Finance, 36 (2012), 968-980.  doi: 10.1016/j.jbankfin.2011.10.015. [4] R. Castellano and R. Cerqueti, Mean-variance portfolio selection in presence of infrequently traded stocks, European Journal of Operational Research, 234 (2014), 442-449.  doi: 10.1016/j.ejor.2013.04.024. [5] S. Das, H. Markowitz, J. Scheid and M. Statman, Portfolio optimization with mental accounts, Journal of Financial and Quantitative Analysis, 45 (2010), 311-334.  doi: 10.1017/S0022109010000141. [6] S. Das, H. Markowitz, J. Scheid and M. Statman, Portfolios for investors who want to reach their goals while staying on the mean-variance efficient frontier, Journal of Wealth Management, (2011), 1-7. [7] C. Gollier, The Economics of Risk and Time, MIT Press, Cambridge, 2001. [8] X. X. Huang, Portfolio Analysis: From Probabilistic to Credibilistic and Uncertain Approaches, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11214-0. [9] X. X. Huang, Mean-risk model for uncertain portfolio selection, Fuzzy Optimization and Decision Making, 10 (2011), 71-89.  doi: 10.1007/s10700-010-9094-x. [10] X. X. Huang, A risk index model for portfolio selection with returns subject to experts' evaluations, Fuzzy Optimization and Decision Making, 11 (2012), 451-463.  doi: 10.1007/s10700-012-9125-x. [11] X. X. Huang, Mean-variance models for portfolio selection subject to experts' estimations, Expert Systems with Applications, 39 (2012), 5887-5893.  doi: 10.1016/j.eswa.2011.11.119. [12] X. X. Huang and H. Di, Uncertain portfolio selection with background risk, Applied Mathematics and Computation, 276 (2016), 284-296.  doi: 10.1016/j.amc.2015.12.018. [13] H. H. Huang and C. P. Wang, Portfolio selection and portfolio frontier with background risk, North American Journal of Economics and Finance, 26 (2013), 177-196.  doi: 10.1016/j.najef.2013.09.001. [14] X. X. Huang and H. Y. Ying, Risk index based models for portfolio adjusting problem with returns subject to experts' evaluations, Economic Modelling, 11 (2012), 451-463.  doi: 10.1007/s10700-012-9125-x. [15] C. H. Jiang, Y. K. Ma and Y.B An, An analysis of portfolio selection with background risk, Journal of Banking and Finance, 34 (2010), 3055-3060. [16] D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-292. [17] B. D. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-39987-2. [18] B. D. Liu, Why is there a need for uncertainty theory?, Journal of Uncertain Systems, 6 (2012), 3-10. [19] B. D. Liu, Uncertainty Theory, 4nd edition, Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-540-39987-2. [20] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [21] H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Wiley, New York, 1959. [22] F. Menoncin, Optimal portfolio and background risk: an exact and an approximated solution, Insurance Mathematics and Economics, 31 (2002), 249-265.  doi: 10.1016/S0167-6687(02)00154-3. [23] H. S. Rosen and S. Wu, Portfolio choice and health status, Journal of Financial Economics, 72 (2004), 457-484. [24] R. H. Thaler, Mental accounting and consumer choice, Marketing Science, 4 (1985), 199-214. [25] L. M. Viceira, Optimal portfolio choice for long-horizon investors with nontradable labor income, Journal of Finance, 56 (2001), 433-470.
Efficient frontier of optimal subportfolio
Efficient frontier of optimal subportfolios with and without background risk when $0<\alpha<0.5$
Efficient frontier of aggregate portfolio and portfolio without mental accounts
Efficient frontier of optimal subportfolios and aggregate portfolio
Twenty candidate securities from the Shanghai Stock Exchange
 Security $i$ Name Code Security $i$ Name Code 1 Shanghaijichang 600009 11 Shoukaigufen 600376 2 Nanjingyinhang 601009 12 Zhejianglongsheng 600352 3 Bohuizhiye 600966 13 Chunqiuhangkong 601021 4 Huaxiayinhang 600015 14 Beifangxitu 600111 5 Zhaoshangyinhang 600035 15 Sananguangdian 600703 6 Zhongguoshihua 600028 16 Zhonghangziben 600705 7 Zhongguoliantong 600050 17 Anxinxintuo 600816 8 Tongfanggufen 600100 18 Pengboshi 600804 9 Nanshanlvye 600219 19 Zhongchuanfangwu 600685 10 Guangdayinhang 601818 20 Fenghuotongxin 600498
 Security $i$ Name Code Security $i$ Name Code 1 Shanghaijichang 600009 11 Shoukaigufen 600376 2 Nanjingyinhang 601009 12 Zhejianglongsheng 600352 3 Bohuizhiye 600966 13 Chunqiuhangkong 601021 4 Huaxiayinhang 600015 14 Beifangxitu 600111 5 Zhaoshangyinhang 600035 15 Sananguangdian 600703 6 Zhongguoshihua 600028 16 Zhonghangziben 600705 7 Zhongguoliantong 600050 17 Anxinxintuo 600816 8 Tongfanggufen 600100 18 Pengboshi 600804 9 Nanshanlvye 600219 19 Zhongchuanfangwu 600685 10 Guangdayinhang 601818 20 Fenghuotongxin 600498
Uncertain return rates of 20 securities
 Security $i$ $\xi_i$ Security $i$ $\xi_i$ 1 $N(0.0720,0.1028)$ 11 $N(0.2360,0.4620)$ 2 $N(0.0608,0.0700)$ 12 $N(0.2120,0.4580)$ 3 $N(0.1498,0.7890)$ 13 $N(0.2008,0.5930)$ 4 $N(0.1560,0.1800)$ 14 $N(0.2606,0.3970)$ 5 $N(0.0780,0.1100)$ 15 $N(0.2937,0.5132)$ 6 $N(0.1256,0.1890)$ 16 $N(0.2682,0.3140)$ 7 $N(0.1396,0.3080)$ 17 $N(0.2926,0.4870)$ 8 $N(0.1602,0.2340)$ 18 $N(0.1946,0.2780)$ 9 $N(0.0970,0.1590)$ 19 $N(0.2210,0.3150)$ 10 $N(0.0868,0.0952)$ 20 $N(0.2460,0.4050)$
 Security $i$ $\xi_i$ Security $i$ $\xi_i$ 1 $N(0.0720,0.1028)$ 11 $N(0.2360,0.4620)$ 2 $N(0.0608,0.0700)$ 12 $N(0.2120,0.4580)$ 3 $N(0.1498,0.7890)$ 13 $N(0.2008,0.5930)$ 4 $N(0.1560,0.1800)$ 14 $N(0.2606,0.3970)$ 5 $N(0.0780,0.1100)$ 15 $N(0.2937,0.5132)$ 6 $N(0.1256,0.1890)$ 16 $N(0.2682,0.3140)$ 7 $N(0.1396,0.3080)$ 17 $N(0.2926,0.4870)$ 8 $N(0.1602,0.2340)$ 18 $N(0.1946,0.2780)$ 9 $N(0.0970,0.1590)$ 19 $N(0.2210,0.3150)$ 10 $N(0.0868,0.0952)$ 20 $N(0.2460,0.4050)$
Optimal subportfolios and aggregate portfolio with background risk
 Retirement Leisure Aggregate $2$ 0.9888 0.3891 0.6889 $4$ 0.0112 0.6109 0.3111 Expected return 6.19% 11.90% 9.045%
 Retirement Leisure Aggregate $2$ 0.9888 0.3891 0.6889 $4$ 0.0112 0.6109 0.3111 Expected return 6.19% 11.90% 9.045%
Optimal subportfolios and aggregate portfolio without background risk
 Retirement Leisure Aggregate $2$ 0.7941 0.1944 0.4943 $4$ 0.2059 0.8056 0.5057 Expected return 8.04% 13.75% 10.895%
 Retirement Leisure Aggregate $2$ 0.7941 0.1944 0.4943 $4$ 0.2059 0.8056 0.5057 Expected return 8.04% 13.75% 10.895%
Optimal subportfolios and aggregate portfolio with background risk
 Retirement Leisure Aggregate $2$ 0.9888 0.0000 0.4926 $4$ 0.0112 0.5484 0.4144 $16$ 0.0000 0.4516 0.0930 Expected return 6.19% 20.67% 13.43%
 Retirement Leisure Aggregate $2$ 0.9888 0.0000 0.4926 $4$ 0.0112 0.5484 0.4144 $16$ 0.0000 0.4516 0.0930 Expected return 6.19% 20.67% 13.43%
Loss of efficiency when VaRU is misspecified
 $MIS$ -5% -10% -15% -20% -25% -30% $\frac{E'-E_2}{E_1}$ -4.61% -11.29% -18.89% -27.59% -37.67% -49.49%
 $MIS$ -5% -10% -15% -20% -25% -30% $\frac{E'-E_2}{E_1}$ -4.61% -11.29% -18.89% -27.59% -37.67% -49.49%
Random return rates of 20 securities
 Security $i$ $\eta_i$ Security $i$ $\eta_i$ 1 $\textrm{N}(0.0796,0.1460)$ 11 $\textrm{N}(0.2000,0.3240)$ 2 $\textrm{N}(0.0676,0.1440)$ 12 $\textrm{N}(0.2270,0.2360)$ 3 $\textrm{N}(0.2730,0.7580)$ 13 $\textrm{N}(0.2860,0.5470)$ 4 $\textrm{N}(0.1560,0.1720)$ 14 $\textrm{N}(0.3020,0.3150)$ 5 $\textrm{N}(0.1610,0.1400)$ 15 $\textrm{N}(0.4140,0.3570)$ 6 $\textrm{N}(0.0819,0.1400)$ 16 $\textrm{N}(0.3250,0.4260)$ 7 $\textrm{N}(0.1250,0.1670)$ 17 $\textrm{N}(0.3050,0.3000)$ 8 $\textrm{N}(0.1570,0.2810)$ 18 $\textrm{N}(0.3530,0.3000)$ 9 $\textrm{N}(0.0830,0.2450)$ 19 $\textrm{N}(0.2380,0.3090)$ 10 $\textrm{N}(0.0498,0.0965)$ 20 $\textrm{N}(0.1990,0.2670)$
 Security $i$ $\eta_i$ Security $i$ $\eta_i$ 1 $\textrm{N}(0.0796,0.1460)$ 11 $\textrm{N}(0.2000,0.3240)$ 2 $\textrm{N}(0.0676,0.1440)$ 12 $\textrm{N}(0.2270,0.2360)$ 3 $\textrm{N}(0.2730,0.7580)$ 13 $\textrm{N}(0.2860,0.5470)$ 4 $\textrm{N}(0.1560,0.1720)$ 14 $\textrm{N}(0.3020,0.3150)$ 5 $\textrm{N}(0.1610,0.1400)$ 15 $\textrm{N}(0.4140,0.3570)$ 6 $\textrm{N}(0.0819,0.1400)$ 16 $\textrm{N}(0.3250,0.4260)$ 7 $\textrm{N}(0.1250,0.1670)$ 17 $\textrm{N}(0.3050,0.3000)$ 8 $\textrm{N}(0.1570,0.2810)$ 18 $\textrm{N}(0.3530,0.3000)$ 9 $\textrm{N}(0.0830,0.2450)$ 19 $\textrm{N}(0.2380,0.3090)$ 10 $\textrm{N}(0.0498,0.0965)$ 20 $\textrm{N}(0.1990,0.2670)$
Optimal subportfolios and aggregate portfolio with background risk when security return rates are random variables
 Retirement Leisure Aggregate $1$ 0.0939 0.1185 0.1062 $2$ 0.1979 0.1257 0.1618 $3$ 0.0000 0.0022 0.0011 $4$ 0.0000 0.0647 0.0324 $5$ 0.0000 0.0766 0.0383 $6$ 0.0748 0.1222 0.0985 $7$ 0.0000 0.0816 0.0408 $8$ 0.0000 0.0393 0.0197 $9$ 0.0359 0.0694 0.0527 $10$ 0.5976 0.2041 0.4009 $11$ 0.0000 0.0234 0.0117 $12$ 0.0000 0.0229 0.0114 $13$ 0.0000 0.0057 0.0029 $19$ 0.0000 0.0147 0.0074 $20$ 0.0000 0.0287 0.0144 Expected return 6.01% 10.42% 8.215%
 Retirement Leisure Aggregate $1$ 0.0939 0.1185 0.1062 $2$ 0.1979 0.1257 0.1618 $3$ 0.0000 0.0022 0.0011 $4$ 0.0000 0.0647 0.0324 $5$ 0.0000 0.0766 0.0383 $6$ 0.0748 0.1222 0.0985 $7$ 0.0000 0.0816 0.0408 $8$ 0.0000 0.0393 0.0197 $9$ 0.0359 0.0694 0.0527 $10$ 0.5976 0.2041 0.4009 $11$ 0.0000 0.0234 0.0117 $12$ 0.0000 0.0229 0.0114 $13$ 0.0000 0.0057 0.0029 $19$ 0.0000 0.0147 0.0074 $20$ 0.0000 0.0287 0.0144 Expected return 6.01% 10.42% 8.215%
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