Inverse S-shaped probability weighting and its impact on investment

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 [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

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 [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

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