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Inverse Sshaped 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 Sshaped probability weighting influence optimal portfolio choice in a rankdependent 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 Sshaped probability weighting function. We show that an increase in inverse Sshaped 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 nonnegligible risk premium. Only for lottery stocks with poor expected returns and extremely positive skewness does an increase in inverse Sshaped 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), 171178. 
[2] 
T. G. Bali, N. Cakici and R. F. Whitelaw, Maxing out: Stocks as lotteries and the crosssection of expected returns, Journal of Financial Economics, 99 (2011), 427446. 
[3] 
N. Barberis and M. Huang, Stocks as lotteries: The implications of probability weighting for security prices, American Economic Review, 98 (2008), 20662100. 
[4] 
H. Bessembinder, Do stocks outperform treasury bills?, Journal of Financial Economics, 129 (2018), 440457. 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), 115148. doi: 10.1007/s1123800991444. 
[6] 
B. Boyer, T. Mitton and K. Vorkink, Expected idiosyncratic skewness, Review of Financial Studies, 23 (2010), 169202. 
[7] 
B. H. Boyer and K. Vorkink, Stock options as lotteries, Journal of Finance, 69 (2014), 14851527. 
[8] 
L. Carassus and M. Rasonyi, Maximization of nonconcave utility functions in discretetime financial market models, Mathematics of Operations Research, 41 (2016), 146173. 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), 370381. doi: 10.1016/00220531(87)900937. 
[10] 
J. Conrad, R. F. Dittmar and E. Ghysels, Ex ante skewness and expected stock returns, Journal of Finance, 68 (2013), 85124. 
[11] 
J. Conrad, N. Kapadia and Y. Xing, Death and jackpot: Why do individual investors hold overpriced stocks?, Journal of Financial Economics, 113 (2014), 455475. 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), 951972. doi: 10.1016/j.jedc.2012.01.010. 
[13] 
H. FehrDuda and T. Epper, Probability and risk: Foundations and economic implications of probabilitydependent risk preferences, Annual Review of Economics, 4 (2012), 567593. doi: 10.1146/annureveconomics080511110950. 
[14] 
W. M. Goldstein and H. J. Einhorn, Expression theory and the preference reversal phenomena, Psychological Review, 94 (1987), 236254. doi: 10.1037/0033295X.94.2.236. 
[15] 
R. Gonzalez and G. Wu, On the shape of the probability weighting function, Cognitive Psychology, 38 (1999), 129166. doi: 10.1006/cogp.1998.0710. 
[16] 
X. D. He, R. Kouwenberg and X. Y. Zhou, Rankdependent utility and risk taking in complete markets, SIAM Journal on Financial Mathematics, 8 (2017), 214239. 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), 315331. 
[18] 
D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263291. 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), 904950. doi: 10.1214/aoap/1029962818. 
[20] 
A. Kumar, Who gambles in the stock market?, Journal of Finance, 64 (2009), 18891933. doi: 10.1111/j.15406261.2009.01483.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), 377400. 
[22] 
C. Low, D. Pachamanova and M. Sim, Skewnessaware asset allocation: A new theoretical framework and empirical evidence, Mathematical Finance, 22 (2012), 379410. doi: 10.1111/j.14679965.2010.00463.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 rankdependent preferences, Review of Financial Studies, 18 (2005), 14671502. 
[25] 
A. Tversky and C. R. Fox, Weighing risk and uncertainty, Psychological Review, 102 (1995), 269283. doi: 10.1037/0033295X.102.2.269. 
[26] 
A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297323. doi: 10.1007/9783319204512_24. 
show all references
References:
[1] 
A. Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12 (1985), 171178. 
[2] 
T. G. Bali, N. Cakici and R. F. Whitelaw, Maxing out: Stocks as lotteries and the crosssection of expected returns, Journal of Financial Economics, 99 (2011), 427446. 
[3] 
N. Barberis and M. Huang, Stocks as lotteries: The implications of probability weighting for security prices, American Economic Review, 98 (2008), 20662100. 
[4] 
H. Bessembinder, Do stocks outperform treasury bills?, Journal of Financial Economics, 129 (2018), 440457. 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), 115148. doi: 10.1007/s1123800991444. 
[6] 
B. Boyer, T. Mitton and K. Vorkink, Expected idiosyncratic skewness, Review of Financial Studies, 23 (2010), 169202. 
[7] 
B. H. Boyer and K. Vorkink, Stock options as lotteries, Journal of Finance, 69 (2014), 14851527. 
[8] 
L. Carassus and M. Rasonyi, Maximization of nonconcave utility functions in discretetime financial market models, Mathematics of Operations Research, 41 (2016), 146173. 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), 370381. doi: 10.1016/00220531(87)900937. 
[10] 
J. Conrad, R. F. Dittmar and E. Ghysels, Ex ante skewness and expected stock returns, Journal of Finance, 68 (2013), 85124. 
[11] 
J. Conrad, N. Kapadia and Y. Xing, Death and jackpot: Why do individual investors hold overpriced stocks?, Journal of Financial Economics, 113 (2014), 455475. 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), 951972. doi: 10.1016/j.jedc.2012.01.010. 
[13] 
H. FehrDuda and T. Epper, Probability and risk: Foundations and economic implications of probabilitydependent risk preferences, Annual Review of Economics, 4 (2012), 567593. doi: 10.1146/annureveconomics080511110950. 
[14] 
W. M. Goldstein and H. J. Einhorn, Expression theory and the preference reversal phenomena, Psychological Review, 94 (1987), 236254. doi: 10.1037/0033295X.94.2.236. 
[15] 
R. Gonzalez and G. Wu, On the shape of the probability weighting function, Cognitive Psychology, 38 (1999), 129166. doi: 10.1006/cogp.1998.0710. 
[16] 
X. D. He, R. Kouwenberg and X. Y. Zhou, Rankdependent utility and risk taking in complete markets, SIAM Journal on Financial Mathematics, 8 (2017), 214239. 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), 315331. 
[18] 
D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263291. 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), 904950. doi: 10.1214/aoap/1029962818. 
[20] 
A. Kumar, Who gambles in the stock market?, Journal of Finance, 64 (2009), 18891933. doi: 10.1111/j.15406261.2009.01483.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), 377400. 
[22] 
C. Low, D. Pachamanova and M. Sim, Skewnessaware asset allocation: A new theoretical framework and empirical evidence, Mathematical Finance, 22 (2012), 379410. doi: 10.1111/j.14679965.2010.00463.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 rankdependent preferences, Review of Financial Studies, 18 (2005), 14671502. 
[25] 
A. Tversky and C. R. Fox, Weighing risk and uncertainty, Psychological Review, 102 (1995), 269283. doi: 10.1037/0033295X.102.2.269. 
[26] 
A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297323. doi: 10.1007/9783319204512_24. 
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