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with Ivol. Table 4.7 reports the full specifications, showing that the related prospect theory element exerts the significant main effect in Table 4.6 in corresponding sentiment periods.
Specifically, I examine the full model that includes PT or LA in low-sentiment periods and includes PW in high- sentiment periods.
In Table 4.7, the significant effect of PT*Size (0.051), PT*Ilq 0.018), and PT*Ivol (-0.027) or LA*Size (0.065), LA*Ilq (-0.029), and LA*Ivol (-0.029) indicates that the predictive power of PT and LA is greater among stocks that are speculative and with high transaction cost to arbitrage (small size, high illiquidity, high volatility) or with high arbitrage risk (high volatility). 7
4.5 Double sorts
For each period, to control the non-linear effects of other known predictors of stock returns associated with firm characteristics and risk, I conduct a series of double-sort tests. Specif-ically, each month, I first sort stocks into quintiles based on the control variables. Then within each quintile, I sort stocks based on PT, PW, or LA into quintiles again. Thus, I have 25 portfolios for each time period. The returns of the portfolios for period t are denoted rijt, where i indicates the group of the first-sort control variable and j indicates that of the second prospect theory element related variable (PT, PW, or LA). For each PT, PW, or LA quintile, j=1,. . . ,5, I compute the average return for each quintile as follows:
rjt = r1jt+ ... + r5jt
5 (4.1)
The return differences of low minus high portfolios are computed as follows:
r1t− r5t = (r11t− r15t) + ... + (r51t− r55t)
5 (4.2)
7 I report the results of limits to arbitrage in other periods in Table G.1 in Appendix G.
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I take control of the variables, including Size, BM, Beta, Ilq, Svar, Ivol, Min, Skew, and Mom. I report the four-factor alphas on the value-weighted basis of the portfolios in Table 4.8. The results indicate that in low- (high-) sentiment periods, all the differences of low minus high PT or LA (PW) portfolios are positive and significant. Among the differences of low minus high PT or LA (PW) portfolios in high- (low-) sentiment, three or five (six) of the nine are not significant, respectively. In other words, these results are consistent with what I find in Table 4.3, Table 4.4, and Table 4.5 by using Fama-MacBeth regression analysis.
The predictive power remains significant even when I take control of the nonlinear effects of other predictive variables of returns. 8
Table 4.8 Double sorts: Value-weighted
First, I sort stocks into quintiles on a control variable basis in each month (Size, BM, Beta, Ilq, Svar, Ivol, Min, Skew, Mom). Second, I sort stocks into quintiles based on corresponding prospect theory value (PT in Panel A, PW in Panel B, LA in Panel C) in each quintile. Finally, I take an average of the returns of the five respective prospect theory value portfolios (PT, PW, and LA) across the five control variable quintiles and report the four-factor alphas on a value-weighted basis of the five portfolios associated with PT, PW, and LA as well as that of the low minus high long-short portfolio following different sentiment periods. PT is the prospect theory value of a stock’s historical return distribution including the following elements: diminishing sensitivity, loss aversion, and probability weighting. PW and LA are the prospect theory value of a stock’s historical return distribution, consisting of probability weighting and loss aversion, respectively. Size is the natural log of market capitalization at the end of the previous month. BM is the natural log of book-to-market ratio. Beta is the stock’s beta using five years of monthly return data. Ilq is Amihud’s (2002) measure of liquidity. Svar is the semivariance of a stock in month t − 1. Ivol is the volatility of the stock’s daily idiosyncratic returns over month t − 1 as in Ang et al. (2006).
Min is the negative of the minimum daily return in month t − 1 as in Bali, Cakici, and Whitelaw (2011). Skew is the skewness of monthly returns over the previous five years. Mom is the 11-month cumulative return during months t − 12 through t − 2. I sort the sample period into low and high levels of investor sentiment, classified based on the median level of Baker and Wurgler’s (2006) sentiment index. The sample period runs from September 1965 to December 2016. t-statistics appear in parentheses.
Control Size BM Beta Ilq Svar Ivol Min Skew Mom
Panel A: Returns of portfolios of stocks sorted on control variables and PT Low sentiment
Low 1 0.556 0.507 0.302 0.510 0.485 0.364 0.450 0.390 0.380
(4.07) (3.18) (1.77) (3.71) (3.10) (2.38) (2.96) (2.11) (2.40)
2 0.293 0.177 0.128 0.176 0.096 0.106 0.026 0.135 -0.012
(3.49) (1.73) (1.41) (2.12) (0.96) (0.99) (0.26) (1.23) (-0.13)
3 0.145 0.110 0.068 0.006 0.045 -0.055 0.087 0.114 0.063
(2.01) (1.45) (0.92) (0.07) (0.46) (-0.57) (0.92) (1.40) (0.76) Continued on next page
8 I report the four-factor alphas on the equal-weighted basis of the portfolios in Table H.1 in Appendix H.
In Table H.1, I find the existence of PT/PW/LA effect in both low and high sentiment periods; however, after considering the control variables, the results of the equal-weighted approach are the same as that of the value-weighted approach to calculate the returns of portfolios.
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Table 4.8 – Continued from previous page
Control Size BM Beta Ilq Svar Ivol Min Skew Mom
4 0.099 0.086 -0.094 -0.041 0.026 -0.009 0.062 -0.011 0.026 (1.48) (1.19) (-1.42) (-0.52) (0.28) (-0.11) (0.78) (-0.16) (0.35) High 5 0.020 0.045 -0.018 -0.113 -0.120 -0.126 -0.053 -0.008 -0.041 (0.31) (0.59) (-0.25) (-1.47) (-1.46) (-1.45) (-0.67) (-0.12) (-0.57)
1-5 0.536 0.461 0.320 0.623 0.605 0.489 0.503 0.398 0.420
(3.75) (2.53) (1.71) (3.90) (3.51) (2.84) (2.94) (2.02) (2.35) High sentiment
Low 1 0.277 0.428 0.289 0.159 0.216 0.257 0.276 0.207 0.299
(1.89) (2.48) (1.80) (1.12) (1.48) (1.71) (1.89) (1.15) (1.96)
2 0.201 0.074 0.243 0.109 0.143 0.030 0.090 0.182 0.140
(2.45) (0.65) (2.35) (1.25) (1.25) (0.26) (0.77) (1.63) (1.26)
3 0.210 0.212 0.051 0.130 0.164 0.121 0.244 0.058 0.177
(3.13) (2.48) (0.65) (1.85) (1.57) (1.26) (2.60) (0.63) (2.20)
4 0.133 0.125 0.034 0.081 0.060 -0.082 0.019 0.002 0.161
(2.02) (1.70) (0.44) (1.08) (0.66) (-0.92) (0.22) (0.03) (2.31) High 5 0.109 -0.031 -0.046 -0.036 -0.173 -0.212 -0.142 -0.202 0.046 (1.51) (-0.38) (-0.58) (-0.50) (-1.74) (-2.14) (-1.60) (-2.77) (0.57)
1-5 0.168 0.459 0.335 0.195 0.389 0.469 0.418 0.409 0.253
(1.14) (2.22) (1.84) (1.22) (2.20) (2.55) (2.42) (1.92) (1.37) Panel B: Returns of portfolios of stocks sorted on control variables and PW
Low sentiment
Low 1 0.501 0.276 0.107 0.337 0.199 0.127 0.171 0.100 0.161
(4.62) (2.95) (1.13) (3.31) (1.83) (1.08) (1.60) (0.99) (1.64)
2 0.197 0.147 0.044 0.054 0.070 -0.027 0.066 0.147 0.056
(2.49) (1.81) (0.59) (0.66) (0.73) (-0.29) (0.74) (1.73) (0.70)
3 0.152 0.078 0.016 0.031 0.001 0.055 0.044 0.040 0.059
(2.23) (0.96) (0.22) (0.39) (0.01) (0.61) (0.50) (0.49) (0.75) 4 0.146 0.042 -0.004 -0.039 -0.110 -0.063 -0.105 0.093 -0.070 (1.90) (0.41) (-0.05) (-0.47) (-1.06) (-0.60) (-1.06) (1.12) (-0.76) High 5 0.005 0.143 0.031 -0.054 -0.026 -0.072 0.036 -0.031 -0.233 (0.05) (0.94) (0.23) (-0.48) (-0.19) (-0.49) (0.25) (-0.24) (-1.72)
1-5 0.495 0.133 0.076 0.391 0.225 0.199 0.135 0.130 0.394
(3.29) (0.69) (0.43) (2.54) (1.20) (1.03) (0.68) (0.69) (2.17) High sentiment
Low 1 0.370 0.235 0.169 0.251 0.301 0.125 0.244 0.240 0.276
(3.58) (2.39) (1.93) (2.61) (2.46) (0.97) (2.11) (2.40) (2.86)
2 0.338 0.145 0.132 0.203 0.124 0.077 0.116 0.075 0.137
(4.35) (1.92) (1.67) (2.62) (1.28) (0.79) (1.24) (0.86) (1.87)
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Table 4.8 – Continued from previous page
Control Size BM Beta Ilq Svar Ivol Min Skew Mom
3 0.200 0.097 -0.056 0.080 0.030 -0.056 -0.003 -0.179 0.103 (2.58) (1.17) (-0.76) (1.01) (0.32) (-0.63) (-0.03) (-1.97) (1.25) 4 0.101 -0.143 -0.024 -0.041 -0.138 -0.194 -0.089 -0.163 0.034 (1.30) (-1.36) (-0.25) (-0.55) (-1.25) (-1.79) (-0.80) (-1.77) (0.33) High 5 -0.162 -0.204 -0.286 -0.317 -0.471 -0.297 -0.437 -0.168 -0.560 (-1.26) (-1.55) (-2.30) (-2.80) (-3.78) (-2.26) (-3.41) (-1.38) (-4.76)
1-5 0.531 0.439 0.456 0.569 0.772 0.423 0.681 0.408 0.836
(3.27) (2.66) (2.97) (3.59) (4.56) (2.30) (4.00) (2.27) (5.08) Panel C: Returns of portfolios of stocks sorted on control variables and LA
Low sentiment
Low 1 0.533 0.588 0.378 0.435 0.341 0.381 0.358 0.356 0.545
(3.57) (3.04) (2.00) (2.89) (2.01) (2.31) (2.06) (1.89) (2.89)
2 0.328 0.297 0.144 0.261 0.181 0.166 0.134 0.088 0.144
(3.68) (2.52) (1.39) (2.92) (1.47) (1.30) (1.11) (0.77) (1.34)
3 0.139 0.125 0.117 -0.032 0.144 0.090 0.172 0.155 0.029
(1.86) (1.45) (1.44) (-0.41) (1.41) (0.93) (1.79) (1.79) (0.33)
4 0.060 0.107 0.013 -0.068 0.093 -0.029 0.059 -0.013 0.025
(0.86) (1.38) (0.19) (-0.91) (1.01) (-0.32) (0.69) (-0.17) (0.31) High 5 0.058 0.060 -0.029 -0.038 -0.071 -0.075 -0.002 0.017 -0.023 (0.87) (0.81) (-0.42) (-0.49) (-0.84) (-0.87) (-0.03) (0.24) (-0.33)
1-5 0.475 0.528 0.407 0.474 0.412 0.456 0.361 0.339 0.567
(2.84) (2.45) (1.95) (2.63) (2.18) (2.49) (1.86) (1.67) (2.72) High sentiment
Low 1 0.236 0.186 0.211 0.147 0.237 0.241 0.277 0.211 0.217
(1.50) (0.97) (1.11) (0.95) (1.54) (1.51) (1.67) (1.14) (1.27)
2 0.178 0.241 0.135 0.090 0.120 -0.012 0.097 0.158 0.096
(2.02) (1.97) (1.17) (0.97) (1.03) (-0.10) (0.83) (1.38) (0.78)
3 0.156 0.185 0.173 0.144 0.137 0.034 0.158 0.052 0.250
(2.22) (2.00) (2.03) (1.97) (1.42) (0.33) (1.64) (0.60) (2.97)
4 0.189 0.131 0.034 0.055 0.008 -0.038 0.012 0.071 0.095
(3.00) (1.62) (0.44) (0.76) (0.08) (-0.45) (0.14) (0.99) (1.22) High 5 0.177 -0.011 -0.044 0.052 -0.102 -0.129 -0.070 -0.157 0.133 (2.39) (-0.15) (-0.53) (0.64) (-1.09) (-1.31) (-0.81) (-2.18) (1.79)
1-5 0.059 0.197 0.255 0.095 0.339 0.370 0.347 0.367 0.084
(0.35) (0.90) (1.18) (0.53) (1.90) (1.94) (1.84) (1.72) (0.42)