Uncertain portfolio selection with mental accounts and background risk

Li Xue; Hao Di

doi:10.3934/jimo.2018124

Article Contents

2019, Volume 15, Issue 4: 1809-1830. Doi: 10.3934/jimo.2018124

This issue Previous Article Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints Next Article Capital-constrained supply chain with multiple decision attributes: Decision optimization and coordination analysis

Uncertain portfolio selection with mental accounts and background risk

Li Xue^1, and
Hao Di^2, ,

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
* Corresponding author: Hao Di

Received: June 30, 2017

Revised: April 30, 2018

Early access: August 2018

Published: July 2019

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

Abstract Full Text(HTML) Figure(4) / Table(8) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 90A09; Secondary: 90B50.

Citation:

Full Text(HTML)

Figure 1. Efficient frontier of optimal subportfolio

Download: Full-size image PowerPoint slide

Figure 2. Efficient frontier of optimal subportfolios with and without background risk when $0<\alpha<0.5$

Download: Full-size image PowerPoint slide

Figure 3. Efficient frontier of aggregate portfolio and portfolio without mental accounts

Download: Full-size image PowerPoint slide

Figure 4. Efficient frontier of optimal subportfolios and aggregate portfolio

Download: Full-size image PowerPoint slide

Table 1. 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

| Show Table

DownLoad: CSV

Table 2. 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	$N(0.2460,0.4050)$

| Show Table

DownLoad: CSV

Table 3. 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%

| Show Table

DownLoad: CSV

Table 4. 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%

| Show Table

DownLoad: CSV

Table 5. 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%

| Show Table

DownLoad: CSV

Table 6. 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%

| Show Table

DownLoad: CSV

Table 7. 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	$\textrm{N}(0.1990,0.2670)$

| Show Table

DownLoad: CSV

Table 8. 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%

| Show Table

DownLoad: CSV

Related Papers

Cited by

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, 2^nd 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, 4^nd 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.