國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
leg portfolio. The LA portfolio is not sensitive to change of sentiment because of short-sale impediments and disposition effect. Third, with regard to the combination of probability weighting and loss aversion, the PT effect is not sensitive to change in level of investor sentiment. 9
4.8 Characteristics of prospect theory portfolios
Based on previous analyses, I have found that the types of prospect theory value of a stock could predict its subsequent return in the cross section following different sentiment periods.
I conclude that the predictive power of prospect theory comes mainly from its probability weighting during high sentiment periods and from loss aversion during low sentiment periods.
The mechanism is as follows: some investors will assign higher values to high probability weighting stocks following high-sentiment periods and lower values of high loss aversion stocks following low-sentiment periods. Regarding investment behavior, in high-sentiment months, investors are prone to tilt portfolios toward stocks whose distribution of past returns is attractive under probability weighting; as a result these stocks are overvalued and ultimately earn low subsequent returns. In low-sentiment months, investors sell portfolios including stocks whose past return distribution is uninviting under loss aversion, causing these stocks to be undervalued and to earn high subsequent returns.
In this section, I view in detail the characteristics of low-PT/PW/LA and high-PT/PW/LA stocks. Overall, prospect theory values are positively associated with past returns. The prob-ability weighting element is positively correlated with skewness, and the loss aversion element
9 The results show that the PW portfolio will be more sensitive to sentiment and respond peacefully, but the LA will respond quickly to very low-sentiment levels. Based on this discussion, I expect that the negative effect of PW on return will be adjusted quickly and the negative effect caused by LA will be adjusted slowly. Thus, compared to the PW long-short portfolio, the return continuation of the LA long-short portfolio would last longer. I present the average subsequent three months’ excess returns on both a value-weighted and equal-weighted basis of portfolios of stocks sorted on PT, PW, and LA following different sentiment periods. The results in Table J.1 in Appendix J suggest that in low-sentiment periods, the return continuation of the LA/PT long-short portfolio is significant but that of the PW long-short portfolio is not in high-sentiment periods, which is consistent with my expectations.
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
is negatively associated with standard deviation and susceptible to negative returns. Thus, I expect stocks with high PT, PW, or LA values to be stocks with high past returns. High-PW stocks have high past skewness, and high-LA stocks have low past volatility and large minimum return.
Specifically, I sort stocks into deciles based on each prospect theory value (PT, PW, LA) each month. In each decile, I take the average of each characteristic and then repeat this calculation in each month in the period from September 1965 to December 2016. Finally, I compute the time-series average of each decile-month data and obtain the decile-level characteristics.
I report results in Table 4.12. I first examine the speculation noted above. Each measure of past returns (LtRev, Mom, Rev) increases monotonically from PT/PW/LA decile 1 to PT/PW/LA decile 10. As expected, past skewness also increases from PW decile 1 to decile 10, and this increase centered mainly on the last three deciles. The skewness in PW decile 10 triples that in PW decile 7. PT also contains the probability weighting element; as a result PT decile 10 also has the highest past skewness. I find no evidence that indicates the existence of an increase pattern for skewness in LA portfolios; conversely, in general past skewness decreases from LA decile 1 to LA decile 10. As for past volatility, Svar, and Ivol, they decrease monotonically from LA decile 1 to decile 9 and increase slightly in decile 10.
The volatility from PT decile 1 to decile 10 also present a similar pattern; however, Ivol has a tendency to descend first and ascend in succession. Barberis et al. (2016) connect idiosyncratic volatility with skewness. I believe that idiosyncratic volatility comoves with skewness to some extent; for example, the correlation between Ivol and Skew is 0.34 in Panel B of Table 4.1. If I decompose PT into PW and LA, evidence suggests that idiosyncratic volatility is more positively correlated with the maximum and the minimum of past returns than with past skewness.
Next, I investigate further about the variation of maximum and minimum of past returns from decile 1 to decile 10 based on different prospect theory elements. PT and LA are similar
‧
Table 4.12 Characteristics of different prospect theory value portfolios
I sort stocks into ten portfolios based on related prospect theory values (PT, PW, LA) and compute the average of the characteristics listed in the first row of this table across all stocks in each portfolio in each month. I then take time-series average of these mean characteristic values for each portfolio. Panel A reports the characteristics of PT portfolios, Panel B reports the characteristics of PW portfolios, and Panel C reports the characteristics of LA portfolios. All variables are defined in Table 4.1. The sample period runs from September 1965 to December 2016.
Portfolios PT PW LA Size Ilq Max Min Svar Ivol Mom LtRev Skew Beta Rev BM
Panel A: Characteristics of PT portfolios
1 -12.519 0.266 -10.051 3.460 11.814 11.070 9.047 15.092 4.437 -11.624 -0.440 0.504 1.352 -1.389 0.071 2 -9.357 1.063 -7.220 4.224 5.709 8.248 6.802 8.335 3.238 1.484 -0.091 0.498 1.289 0.302 -0.166 3 -7.963 1.308 -6.004 4.592 3.650 7.140 5.898 6.212 2.767 6.570 0.136 0.497 1.225 0.835 -0.242 4 -6.934 1.449 -5.104 4.936 2.440 6.355 5.274 4.941 2.438 10.081 0.339 0.482 1.162 1.001 -0.317 5 -6.092 1.557 -4.377 5.217 1.747 5.820 4.817 4.132 2.209 12.461 0.536 0.473 1.113 1.276 -0.384 6 -5.343 1.665 -3.757 5.474 1.294 5.375 4.476 3.557 2.025 15.028 0.725 0.469 1.071 1.436 -0.439 7 -4.639 1.782 -3.171 5.726 0.980 5.058 4.193 3.108 1.886 17.982 0.932 0.459 1.036 1.625 -0.520 8 -3.924 1.990 -2.623 5.956 0.748 4.806 3.972 2.778 1.775 20.882 1.182 0.479 1.013 1.851 -0.611 9 -3.082 2.433 -2.047 6.145 0.661 4.741 3.852 2.649 1.729 26.009 1.540 0.553 1.015 2.222 -0.741 10 -1.012 5.381 -1.271 6.011 0.920 5.930 4.584 3.906 2.122 40.473 2.791 1.075 1.221 3.153 -1.124 Panel B: Characteristics of PW portfolios
1 -10.124 -1.124 -7.134 4.943 4.911 6.907 6.104 8.071 2.794 -10.811 -0.247 0.021 0.992 -1.546 0.142 2 -7.330 -0.162 -4.999 5.602 2.198 5.118 4.503 4.098 2.012 1.299 0.082 0.134 0.941 0.085 -0.129 3 -6.362 0.272 -4.276 5.770 1.726 4.845 4.223 3.475 1.877 6.187 0.292 0.182 0.943 0.578 -0.259 4 -5.928 0.637 -4.002 5.738 1.718 4.988 4.265 3.483 1.910 9.278 0.450 0.255 0.983 0.964 -0.339 5 -5.708 1.017 -3.903 5.628 1.804 5.299 4.443 3.693 2.008 12.125 0.607 0.331 1.038 1.252 -0.410 6 -5.551 1.448 -3.859 5.475 2.060 5.709 4.723 4.124 2.155 15.647 0.781 0.421 1.095 1.543 -0.493 7 -5.526 1.985 -3.972 5.173 2.540 6.313 5.137 4.827 2.382 18.732 0.975 0.551 1.189 1.850 -0.568 8 -5.438 2.724 -4.107 4.899 3.172 7.068 5.626 5.702 2.653 23.076 1.253 0.721 1.289 2.186 -0.678 9 -5.303 3.937 -4.458 4.512 3.910 8.173 6.336 7.099 3.055 28.185 1.541 1.012 1.405 2.576 -0.777 10 -3.604 8.168 -4.927 3.993 5.952 10.140 7.569 10.163 3.787 35.630 1.915 1.862 1.624 2.824 -0.963 Panel C: Characteristics of LA portfolios
1 -11.780 1.899 -10.617 3.107 14.666 12.229 9.740 17.070 4.888 -9.561 -0.500 0.984 1.428 -1.181 0.003 2 -9.010 1.971 -7.521 3.868 6.292 8.947 7.217 9.171 3.509 2.785 -0.142 0.796 1.339 0.423 -0.154 3 -7.808 1.782 -6.148 4.407 3.347 7.433 6.065 6.395 2.871 7.402 0.095 0.658 1.258 0.853 -0.231 4 -6.895 1.678 -5.171 4.776 1.997 6.483 5.355 4.928 2.477 10.358 0.313 0.571 1.188 1.109 -0.287 5 -6.115 1.598 -4.376 5.146 1.268 5.741 4.787 3.912 2.175 12.784 0.513 0.496 1.118 1.268 -0.357 6 -5.412 1.550 -3.701 5.483 0.853 5.211 4.371 3.247 1.957 14.916 0.704 0.441 1.063 1.389 -0.417 7 -4.748 1.571 -3.095 5.797 0.612 4.808 4.047 2.785 1.792 17.223 0.911 0.395 1.019 1.564 -0.496 8 -4.067 1.655 -2.508 6.095 0.421 4.485 3.762 2.389 1.651 19.944 1.147 0.361 0.980 1.731 -0.584 9 -3.307 1.923 -1.863 6.375 0.299 4.349 3.619 2.205 1.583 24.671 1.543 0.346 0.979 2.115 -0.745 10 -1.726 3.260 -0.623 6.689 0.199 4.854 3.949 2.601 1.720 38.812 3.067 0.437 1.123 3.038 -1.206
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
with each other. To At maximum, stocks in PT/LA -decile 1 have the largest maximum, and generally speaking, the maximum appears to be a decreasing trend from decile 1 to decile 10. It This seems to contradict with the argument that investors prefer the stocks with high past returns. Nevertheless, loss aversion plays a vital role in constructing these two values. Shifting the focus to the minimum, I find that the large maximum accompanied with theby a small minimum. Put differentlyIn other words, the negative utility of minimum offsets the positive utility of maximum because of, due to the loss aversion effect. PW portfolios exhibit the other scenescenario. To At maximum, stocks in PW -decile 10 have the largest maximum, even though they also have the smallest minimum. Interestingly, I also find that the dispersion of minimum in PW deciles is smaller than that in PT/LA deciles (3.35% vs 5.20%/6.12%, if I just compare only decile 1 and decile 10, the comparison will be 1.47% vs 2.96%/5.80%). I attribute this difference to the γ and δ that construct probability weighting. The former represents the lottery-type demand, and the latter represents the insurance-type demand. The lottery-type demand accounts for the PW decile 10, which with largest maximum return is attractive to investors; and the insurance-type demand requires the minimum not to be too small.
In order to further verify this idea, I plot the time-series average of each portfolio’s past 60 months’ returns from the lowest to the highest, using the same sorting method described above; but this time the decile 1 and the decile 10 portfolios are of interest. For each month, I rank the past 60 months’ excess returns of each stock from the most negative to the most positive in each decile. Then I take the average of return in corresponding order and compute the monthly decile’s returns of past 60 months from minimum to maximum.
Finally, I average the time-series data of decile 1 and decile 10 and obtain each decile’s return distribution.
Figure 4.2 plots the return distribution of the long portfolio (decile 1) and short portfolio (decile 10) based on the same prospect theory element (PT, PW LA) using three subplots. (I also plot the return distribution for the same decile based on prospect theory elements using
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 4.2 Return distribution over the past 60 months of long- and short- portfolio
This figure plots the time series average of each portfolio’s past 60 months’ return distribution. I sort stocks into ten portfolios from bottom to top by PT, PW, and LA in each month, respectively, and extract the stocks in portfolio 1 (low value) and portfolio 10 (high value). I rank these stocks’ past 60 months’ excess returns (return in excess of the market) from low (most negative) to high (most positive) and compute the mean of all stocks’ excess returns of the same previous month in the same portfolio as each portfolio’s past 60 months’ return distribution. Figure 4.2a,
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
two subplots in Figure K.1 in the Appendix.) Figure 4.2a shows the return distribution of extreme portfolios based on PT. PT decile 10 has higher returns than PT decile 1 at both ends of the curves. With regard to PW-based portfolios, in Figure 4.2b, PW decile 10 has lower returns on the left side but much higher returns on the right side of the curves than PW decile 1. Regarding LA-based portfolios, Figure 4.2c suggests that even though the LA decile 1 has a high curve on the right side of the junction, it also has a long flat low curve on the left side. LA decile 10 seems more valuable than LA decile 1 to a prospect theory individual in a low-sentiment period because of its limited losses.
In summary, in low-sentiment periods, stocks in LA decile 1 are those that trade off past return, minimum (negative past return), and volatility in a way that is maximally repellent to prospect theory investors. This result is also consistent with Barberis and Huang (2001), who suggest that loss aversion and narrow framing form individual stock accounting. Investors are loss averse over individual stock fluctuations, and prior performance affects the degree to which they suffer from a loss. During high-sentiment periods, stocks in PW decile 10 are those that trade off past return, maximum (positive past return), and skewness in a way that is maximally appealing to prospect theory investors.
‧
國立 政 治 大 學