# American Institute of Mathematical Sciences

September 2018, 8(3&4): 679-706. doi: 10.3934/mcrf.2018029

## 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

* Corresponding author

Received  November 2017 Revised  May 2018 Published  September 2018

Fund Project: Xue Dong He acknowledges financial support from the General Research Fund of the Research Grants Council of Hong Kong SAR (Project No. 14225916). Xun Yu Zhou acknowledges financial supports through start-up grants at both University of Oxford and Columbia University, and research funds from Oxford-Nie Financial Big Data Lab, the Oxford-Man Institute of Quantitative Finance, and East China Normal University

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.

Citation: Xue Dong He, Roy Kouwenberg, Xun Yu Zhou. Inverse S-shaped probability weighting and its impact on investment. Mathematical Control & Related Fields, 2018, 8 (3&4) : 679-706. doi: 10.3934/mcrf.2018029
##### 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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##### 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.
Comparative inverse S-shape. A family of probability weighting functions, $w(z) = az^2 + \big(1-a/2\big) z, z\in [0, 0.5]$, $w(z) = 1-w(1-z), z\in(0.5, 1]$, are plotted for three values of $a$, 0, $-1$, and $-2$, in dash-dotted, solid, and dashed lines, respectively. As $a$ becomes more negative, the probability weighting function becomes more inverse S-shaped
Optimal dollar amount $\theta^*$ invested in the risky asset with respect to different degrees of inverse S-shape of the probability weighting function. The utility function is the exponential one in (10) with $\beta = 1$. The probability weighting function is given by (9), so $1-\gamma$ represents the degree of inverse S-shape of the probability weighting function. We set $\delta = 1$. The excess return of the risky asset $R$ follows a skew-normal distribution with mean $\mu = 6\%$ and standard deviation $\sigma = 20\%$. The skewness takes three values $-0.5$, $0$, and $0.5$, corresponding to the solid line, dashed line, and dash-dotted line, respectively
Probability density function of the excess return $R$ of the risky asset when $R$ follows a skew-normal distribution. The mean and standard deviation of $R$ are set to be $\mu = 6\%$ and $\sigma = 20\%$, respectively, and the skewness of $R$ takes three values: $-0.5$, $0$, and $0.5$, corresponding to the probability density functions in the left, middle, and right panes, respectively
Optimal dollar amount $\theta^*$ invested in the risky asset with respect to different degrees of inverse S-shape of the probability weighting function. The utility function is the exponential one in (10) with $\beta = 1$. The probability weighting function is given by (9), so $1-\gamma$ represents the degree of inverse S-shape of the probability weighting function. We set $\delta = 1$. The excess return of the risky asset $R$ follows a skew-normal distribution with mean $\mu = 1\%$ and standard deviation $\sigma = 20\%$. The skewness takes three values $-0.5$, $0$, and $0.5$, corresponding to the solid line, dashed line, and dash-dotted line, respectively
Probability density function of the excess return $R$ of the risky asset when $R$ follows a skew-normal distribution. The mean and standard deviation of $R$ are set to be $\mu = 1\%$ and $\sigma = 20\%$, respectively, and the skewness of $R$ takes three values: $-0.5$, $0$, and $0.5$, corresponding to the probability density functions in the left, middle, and right panes, respectively
Optimal dollar amount $\theta^*$ invested in the U.S. stock market as a function of the degree of inverse S-shape of the probability weighting function. The utility function is the exponential one in (10) with $\beta = 1$. The probability weighting function is given by (9), so $1-\gamma$ represents the degree of inverse S-shape of the weighting function. We set $\delta = 1$. The excess return of the risky asset $R$ follows a skew-normal distribution with mean $\mu = 6.5\%$, standard deviation $\sigma = 17.6\%$ and skewness of $-0.6$, based on historical data of excess returns for the U.S. stock market (1962--2016). The solid line shows the optimal allocation when the skewness is $-0.6$ as in the historical data, while the dotted line shows the optimal allocation when skewness is $0$ for comparison sake
Optimal dollar amount $\theta^*$ invested in Apple as a function of the degree of inverse S-shape of the probability weighting function. The utility function is the exponential one in (10) with $\beta = 1$. The probability weighting function is given by (9), so $1-\gamma$ represents the degree of inverse S-shape of the weighting function. We set $\delta = 1$. The excess return of the risky asset $R$ follows a skew-normal distribution with mean $\mu = 29.5\%$, standard deviation $\sigma = 70.5\%$ and skewness of $0.9$, based on historical data of excess returns for the stock of the company Apple (1980--2016). The solid line shows the optimal allocation when the skewness is $0.9$ as in the historical data, while the dotted line shows the optimal allocation when skewness is $0$ for comparison sake
Optimal dollar amount $\theta^*$ invested in one randomly selected U.S. stock as a function of the degree of inverse S-shape of the probability weighting function. The utility function is the exponential one in (10) with $\beta = 1$. The probability weighting function is given by (9), so $1-\gamma$ represents the degree of inverse S-shape of the weighting function. We set $\delta = 1$. The excess return of the risky asset $R$ follows a skew-normal distribution with mean $\mu = 11.3\%$, standard deviation $\sigma = 82.0\%$ and skewness of $0.99$ (the highest feasible value), based on the annual excess return distribution when one U.S. listed stock is picked randomly and held for one year, from [4]. The solid line shows the optimal allocation when the skewness is $0.99$ as in the historical data, while the dotted line shows the optimal allocation when skewness is $0$ for comparison sake
Optimal dollar amount $\theta^*$ invested in a portfolio of U.S. lottery stocks as a function of the degree of inverse S-shape of the probability weighting function. The utility function is the exponential one in (10) with $\beta = 1$. The probability weighting function is given by (9), so $1-\gamma$ represents the degree of inverse S-shape of the weighting function. We set $\delta = 1$. The excess return of the risky asset $R$ follows a skew-normal distribution with mean $\mu = -0.3\%$, standard deviation $\sigma = 27.5\%$ and skewness of $0.33$, based on a portfolio of U.S. lottery stocks described in [20]. The solid line shows the optimal allocation when the mean excess return is $\mu = -0.3$ as estimated by [20]. The other lines show the portfolio allocation for other levels of expected return: $\mu = -1\%$ (dashed line), $\mu = 1\%$ (dashed-dotted line) and $\mu = 3\%$ (dotted line), while keeping $\sigma$ and skewness constant
Probability density function of the excess return $R$ of the risky asset when the gross return $R+1$ follows a log skew-normal distribution. The mean and standard deviation of $R$ are set to be $\mu = 6\%$ and $\sigma = 20\%$, respectively, and the skewness takes three values: $-0.5$, $0$, and $0.5$, corresponding to the probability density functions in the left, middle, and right panes, respectively
Optimal percentage allocation $\theta^*$ to the risky asset with respect to different degrees of inverse S-shape of the probability weighting function. The utility function is the power one in (12) with $\beta' = 1$. The probability weighting function is given by (9), so $1-\gamma$ represents the degree of inverse S-shape of the probability weighting function. We set $\delta = 1$. The gross return of the risky asset $R+1$ follows a log skew-normal distribution, and the mean $\mu$ and standard deviation $\sigma$ of $R$ are set to be $6\%$ and $20\%$, respectively. The skewness of $R$ takes three values $-0.5$, $0$, and $0.5$, corresponding to the solid line, dashed line, and dash-dotted line, respectively
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