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# Inverse S-shaped probability weighting and its impact on investment

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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.

Mathematics Subject Classification: Primary: 91G10; Secondary: 91B06, 91B16.

 Citation: • • Figure 1.  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

Figure 3.  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

Figure 2.  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

Figure 4.  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

Figure 5.  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

Figure 6.  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

Figure 7.  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

Figure 8.  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 . 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

Figure 9.  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 . The solid line shows the optimal allocation when the mean excess return is $\mu = -0.3$ as estimated by . 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

Figure 11.  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

Figure 10.  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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