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Linear quadratic mean-field-game of backward stochastic differential systems
Inverse S-shaped probability weighting and its impact on investment
1. | Department of SEEM, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China |
2. | College of Management, Mahidol University, Bangkok, Thailand |
3. | Erasmus School of Economics, Erasmus University Rotterdam, Rotterdam, The Netherlands |
4. | Department of IEOR, Columbia University, 500 W. 120th Street, New York, NY 10027, USA |
In this paper we analyze how changes in inverse S-shaped probability weighting influence optimal portfolio choice in a rank-dependent utility model. We derive sufficient conditions for the existence of an optimal solution of the investment problem, and then define the notion of a more inverse S-shaped probability weighting function. We show that an increase in inverse S-shaped weighting typically leads to a lower allocation to the risky asset, regardless of whether the return distribution is skewed left or right, as long as it offers a non-negligible risk premium. Only for lottery stocks with poor expected returns and extremely positive skewness does an increase in inverse S-shaped probability weighting lead to larger portfolio allocations.
References:
[1] |
A. Azzalini,
A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12 (1985), 171-178.
|
[2] |
T. G. Bali, N. Cakici and R. F. Whitelaw,
Maxing out: Stocks as lotteries and the cross-section of expected returns, Journal of Financial Economics, 99 (2011), 427-446.
|
[3] |
N. Barberis and M. Huang,
Stocks as lotteries: The implications of probability weighting for security prices, American Economic Review, 98 (2008), 2066-2100.
|
[4] |
H. Bessembinder,
Do stocks outperform treasury bills?, Journal of Financial Economics, 129 (2018), 440-457.
doi: 10.1016/j.jfineco.2018.06.004. |
[5] |
A. Booij, B. van Praag and G. van de Kuilen,
A parametric analysis of prospect theory's functionals for the general population, Theory and Decision, 68 (2010), 115-148.
doi: 10.1007/s11238-009-9144-4. |
[6] |
B. Boyer, T. Mitton and K. Vorkink,
Expected idiosyncratic skewness, Review of Financial Studies, 23 (2010), 169-202.
|
[7] |
B. H. Boyer and K. Vorkink,
Stock options as lotteries, Journal of Finance, 69 (2014), 1485-1527.
|
[8] |
L. Carassus and M. Rasonyi,
Maximization of nonconcave utility functions in discrete-time financial market models, Mathematics of Operations Research, 41 (2016), 146-173.
doi: 10.1287/moor.2015.0720. |
[9] |
S. H. Chew, E. Karni and Z. Safra,
Risk aversion in the theory of expected utility with rank dependent probabilities, Journal of Economic Theory, 42 (1987), 370-381.
doi: 10.1016/0022-0531(87)90093-7. |
[10] |
J. Conrad, R. F. Dittmar and E. Ghysels,
Ex ante skewness and expected stock returns, Journal of Finance, 68 (2013), 85-124.
|
[11] |
J. Conrad, N. Kapadia and Y. Xing,
Death and jackpot: Why do individual investors hold overpriced stocks?, Journal of Financial Economics, 113 (2014), 455-475.
doi: 10.1016/j.jfineco.2014.04.001. |
[12] |
E. G. De Giorgi and S. Legg,
Dynamic portfolio choice and asset pricing with narrow framing and probability weighting, Journal of Economic Dynamics and Control, 36 (2012), 951-972.
doi: 10.1016/j.jedc.2012.01.010. |
[13] |
H. Fehr-Duda and T. Epper,
Probability and risk: Foundations and economic implications of probability-dependent risk preferences, Annual Review of Economics, 4 (2012), 567-593.
doi: 10.1146/annurev-economics-080511-110950. |
[14] |
W. M. Goldstein and H. J. Einhorn,
Expression theory and the preference reversal phenomena, Psychological Review, 94 (1987), 236-254.
doi: 10.1037/0033-295X.94.2.236. |
[15] |
R. Gonzalez and G. Wu,
On the shape of the probability weighting function, Cognitive Psychology, 38 (1999), 129-166.
doi: 10.1006/cogp.1998.0710. |
[16] |
X. D. He, R. Kouwenberg and X. Y. Zhou,
Rank-dependent utility and risk taking in complete markets, SIAM Journal on Financial Mathematics, 8 (2017), 214-239.
doi: 10.1137/16M1072516. |
[17] |
X. D. He and X. Y. Zhou,
Portfolio choice under cumulative prospect theory: An analytical treatment, Management Science, 57 (2011), 315-331.
|
[18] |
D. Kahneman and A. Tversky,
Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.
doi: 10.21236/ADA045771. |
[19] |
D. Kramkov and W. Schachermayer,
The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Annals of Applied Probability, 9 (1999), 904-950.
doi: 10.1214/aoap/1029962818. |
[20] |
A. Kumar,
Who gambles in the stock market?, Journal of Finance, 64 (2009), 1889-1933.
doi: 10.1111/j.1540-6261.2009.01483.x. |
[21] |
P. K. Lattimore, J. R. Baker and A. D. Witte,
Influence of probability on risky choice: A parametric examination, Journal of Economic Behavior and Organization, 17 (1992), 377-400.
|
[22] |
C. Low, D. Pachamanova and M. Sim,
Skewness-aware asset allocation: A new theoretical framework and empirical evidence, Mathematical Finance, 22 (2012), 379-410.
doi: 10.1111/j.1467-9965.2010.00463.x. |
[23] |
I. P. Natanson, Theory of Functions of a Real Variable, vol. 1, Frederick Ungar, New York, 1955. |
[24] |
V. Polkovnichenko,
Household portfolio diversification: A case for rank-dependent preferences, Review of Financial Studies, 18 (2005), 1467-1502.
|
[25] |
A. Tversky and C. R. Fox,
Weighing risk and uncertainty, Psychological Review, 102 (1995), 269-283.
doi: 10.1037/0033-295X.102.2.269. |
[26] |
A. Tversky and D. Kahneman,
Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297-323.
doi: 10.1007/978-3-319-20451-2_24. |
show all references
References:
[1] |
A. Azzalini,
A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12 (1985), 171-178.
|
[2] |
T. G. Bali, N. Cakici and R. F. Whitelaw,
Maxing out: Stocks as lotteries and the cross-section of expected returns, Journal of Financial Economics, 99 (2011), 427-446.
|
[3] |
N. Barberis and M. Huang,
Stocks as lotteries: The implications of probability weighting for security prices, American Economic Review, 98 (2008), 2066-2100.
|
[4] |
H. Bessembinder,
Do stocks outperform treasury bills?, Journal of Financial Economics, 129 (2018), 440-457.
doi: 10.1016/j.jfineco.2018.06.004. |
[5] |
A. Booij, B. van Praag and G. van de Kuilen,
A parametric analysis of prospect theory's functionals for the general population, Theory and Decision, 68 (2010), 115-148.
doi: 10.1007/s11238-009-9144-4. |
[6] |
B. Boyer, T. Mitton and K. Vorkink,
Expected idiosyncratic skewness, Review of Financial Studies, 23 (2010), 169-202.
|
[7] |
B. H. Boyer and K. Vorkink,
Stock options as lotteries, Journal of Finance, 69 (2014), 1485-1527.
|
[8] |
L. Carassus and M. Rasonyi,
Maximization of nonconcave utility functions in discrete-time financial market models, Mathematics of Operations Research, 41 (2016), 146-173.
doi: 10.1287/moor.2015.0720. |
[9] |
S. H. Chew, E. Karni and Z. Safra,
Risk aversion in the theory of expected utility with rank dependent probabilities, Journal of Economic Theory, 42 (1987), 370-381.
doi: 10.1016/0022-0531(87)90093-7. |
[10] |
J. Conrad, R. F. Dittmar and E. Ghysels,
Ex ante skewness and expected stock returns, Journal of Finance, 68 (2013), 85-124.
|
[11] |
J. Conrad, N. Kapadia and Y. Xing,
Death and jackpot: Why do individual investors hold overpriced stocks?, Journal of Financial Economics, 113 (2014), 455-475.
doi: 10.1016/j.jfineco.2014.04.001. |
[12] |
E. G. De Giorgi and S. Legg,
Dynamic portfolio choice and asset pricing with narrow framing and probability weighting, Journal of Economic Dynamics and Control, 36 (2012), 951-972.
doi: 10.1016/j.jedc.2012.01.010. |
[13] |
H. Fehr-Duda and T. Epper,
Probability and risk: Foundations and economic implications of probability-dependent risk preferences, Annual Review of Economics, 4 (2012), 567-593.
doi: 10.1146/annurev-economics-080511-110950. |
[14] |
W. M. Goldstein and H. J. Einhorn,
Expression theory and the preference reversal phenomena, Psychological Review, 94 (1987), 236-254.
doi: 10.1037/0033-295X.94.2.236. |
[15] |
R. Gonzalez and G. Wu,
On the shape of the probability weighting function, Cognitive Psychology, 38 (1999), 129-166.
doi: 10.1006/cogp.1998.0710. |
[16] |
X. D. He, R. Kouwenberg and X. Y. Zhou,
Rank-dependent utility and risk taking in complete markets, SIAM Journal on Financial Mathematics, 8 (2017), 214-239.
doi: 10.1137/16M1072516. |
[17] |
X. D. He and X. Y. Zhou,
Portfolio choice under cumulative prospect theory: An analytical treatment, Management Science, 57 (2011), 315-331.
|
[18] |
D. Kahneman and A. Tversky,
Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.
doi: 10.21236/ADA045771. |
[19] |
D. Kramkov and W. Schachermayer,
The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Annals of Applied Probability, 9 (1999), 904-950.
doi: 10.1214/aoap/1029962818. |
[20] |
A. Kumar,
Who gambles in the stock market?, Journal of Finance, 64 (2009), 1889-1933.
doi: 10.1111/j.1540-6261.2009.01483.x. |
[21] |
P. K. Lattimore, J. R. Baker and A. D. Witte,
Influence of probability on risky choice: A parametric examination, Journal of Economic Behavior and Organization, 17 (1992), 377-400.
|
[22] |
C. Low, D. Pachamanova and M. Sim,
Skewness-aware asset allocation: A new theoretical framework and empirical evidence, Mathematical Finance, 22 (2012), 379-410.
doi: 10.1111/j.1467-9965.2010.00463.x. |
[23] |
I. P. Natanson, Theory of Functions of a Real Variable, vol. 1, Frederick Ungar, New York, 1955. |
[24] |
V. Polkovnichenko,
Household portfolio diversification: A case for rank-dependent preferences, Review of Financial Studies, 18 (2005), 1467-1502.
|
[25] |
A. Tversky and C. R. Fox,
Weighing risk and uncertainty, Psychological Review, 102 (1995), 269-283.
doi: 10.1037/0033-295X.102.2.269. |
[26] |
A. Tversky and D. Kahneman,
Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297-323.
doi: 10.1007/978-3-319-20451-2_24. |











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