Methodology and Empirical Results
4.2 Portfolio analysis
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elements. Panel B indicates that PT and LA are highly positively correlated in the cross section, whereas the correlations between PW and LA are relatively low. This indicates that PT is similar to LA, and PW and LA represent two different aspects of mental representation:
probability weighting and loss aversion.
The correlations between the measure of prospect theory value and other stock-level characteristics show that PT, PW, and LA display different patterns. Specifically, PT and LA have a strong positive relation with measures of past returns (Mom, LtRev ) and a strong negative relation with a measure of past volatility (Ivol ), whereas PW is less positively related to LtRev and positively related to past volatility. High-PW stocks tend to have higher past skewness (Skew ), higher maximum positive stock return (Max ), and lower minimum negative stock return (Min); but high-LA stocks represent an inverse pattern. PT or LA stocks also tend to have higher market capitalizations. The correlation analyses also show a negative relation between PT or LA and book-to-market ratio (BM ), beta (Beta), and illiquidity (Ilq); however, PW exhibits a converse relation with these variables. Preliminary evidence suggests that loss aversion captures mainly the effect of PT; PT, PW, and LA stocks have different return distributions; and different market cap categories probably explain the various relations between PT, PW, or LA and other stock-level characteristics. For example, small-cap stocks have less liquidity and are more volatile.
4.2 Portfolio analysis
At the beginning of each month, I sort stocks into deciles based on PT, PW, LA, and CC. I then calculate the average value-weighted and equal-weighted return of each decile portfolio over the subsequent month. The excess return is the return in excess of the risk-free rate;
the four-factor alpha is the return adjusted by the three factors from Fama and French 1993 and by a momentum factor; the five-factor alpha is the return adjusted by the three Fama-French factors, the momentum factor, and the Pastor and Stambaugh (2003) liquidity factor
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(Carhart, 1997; Fama and French, 1993; Pástor and Stambaugh, 2003).
I perform univariate sort portfolio analyses based on PT, PW, LA, and CC to start my investigation of the cross-sectional relation between prospect theory values and expected stock returns. Each month, beginning in September 1965 and ending in December 2016, I sort all stocks in the sample into deciles based on an ascending sort of PT, PW, LA, and CC. I then calculate the average value-weighted return of each decile portfolio over the subsequent month and obtain time-series monthly returns of each decile. First, I compute the average return of each decile in the entire sample period, which I then I sort into low and high levels of investor sentiment 1, classified based on the median level of Baker and Wurgler’s (2006) sentiment index—and compute the average return in each decile over these two investor sentiment periods. I report the average value-weighted excess return for each of the ten decile portfolios as well as Carhart’s (1997) factor alpha and alpha of the four-factor model augmented by the Pastor and Stambaugh (2003) liquidity four-factor in the sample period and in the various sentiment periods in Table 4.2. I long the first decile portfolio and short the 10th decile portfolio, reporting the difference in the farthest right column. 2
Table 4.2 Portfolio analysis: Value-weighted
This table presents average monthly excess returns, four-factor alphas and five-factor alphas on the value-weighted basis of portfolios of stocks sorted on PT, PW, LA and CC in the sample period and the following sentiment periods in Panel A, Panel B, Panel C, and Panel D, respectively. Each month, all stocks in the CRSP sample are sorted on the portfolios based on the corresponding prospect theory value. Then I sort the whole period into low and high levels of investor sentiment as classified based on the median level of Baker and Wurgler’s (2006) sentiment index. I report the average excess return for each of the 10 decile portfolios and long the first decile portfolio and short the decile 10 portfolio as well as Carhart’s (1997) four-factor alpha and alpha of the four-factor model augmented by the Pastor and Stambaugh (2003) liquidity factor following different sentiment periods. The sample period runs from September 1965 to December 2016 except in the case of five-factor alpha, where it starts in January 1968 because of the availability constraint of the liquidity factor. t-statistics appear in parentheses.
1 2 3 4 5 6 7 8 9 10 1-10
Panel A: Returns of portfolios of stocks sorted on PT Whole periods
Excess return 1.185 0.621 0.783 0.799 0.583 0.630 0.621 0.522 0.525 0.439 0.746 (3.20) (2.08) (3.16) (3.43) (2.80) (3.09) (3.38) (2.87) (2.98) (2.12) (1.76) Four-factor alpha 0.642 0.088 0.295 0.312 0.128 0.133 0.139 0.010 0.068 -0.117 0.759 (3.33) (0.68) (2.97) (3.54) (1.54) (1.78) (2.15) (0.17) (1.10) (-1.73) (3.64) Five-factor alpha 0.633 0.050 0.267 0.290 0.112 0.129 0.154 0.024 0.081 -0.124 0.757
Continued on next page
1 I define low- (high-) sentiment period in time t as the sentiment index less (greater) than the median in time t − 1.
2 I present the equal-weighted return and alpha in Table A.1 in Appendix A.
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Table 4.2 – Continued from previous page
1 2 3 4 5 6 7 8 9 10 1-10
(3.16) (0.37) (2.60) (3.20) (1.32) (1.70) (2.34) (0.36) (1.29) (-1.82) (3.52) Low sentiment
Excess return 1.632 0.786 0.938 0.845 0.692 0.710 0.566 0.514 0.698 0.715 0.917 (2.84) (1.72) (2.44) (2.40) (2.18) (2.31) (2.06) (1.90) (2.78) (2.60) (1.44) Four-factor alpha 0.680 -0.030 0.266 0.189 0.103 0.102 -0.020 -0.082 0.096 -0.060 0.740 (2.41) (-0.16) (1.92) (1.68) (0.89) (1.01) (-0.22) (-0.99) (1.14) (-0.69) (2.49) Five-factor alpha 0.694 -0.034 0.239 0.169 0.105 0.082 -0.008 -0.051 0.124 -0.048 0.742 (2.31) (-0.17) (1.63) (1.44) (0.88) (0.81) (-0.08) (-0.60) (1.44) (-0.55) (2.37)
High sentiment
Excess return 0.740 0.456 0.630 0.752 0.474 0.551 0.675 0.529 0.353 0.164 0.576 (1.58) (1.19) (2.00) (2.46) (1.75) (2.04) (2.76) (2.17) (1.43) (0.53) (1.03) Four-factor alpha 0.641 0.203 0.277 0.413 0.109 0.118 0.271 0.068 0.033 -0.130 0.771 (2.41) (1.10) (1.97) (3.04) (0.92) (1.11) (3.07) (0.74) (0.37) (-1.31) (2.61) Five-factor alpha 0.600 0.130 0.260 0.389 0.092 0.139 0.285 0.063 0.042 -0.155 0.755 (2.25) (0.71) (1.84) (2.85) (0.77) (1.31) (3.21) (0.69) (0.46) (-1.56) (2.54) Panel B: Returns of portfolios of stocks sorted on PW
Whole periods
Excess return 0.685 0.618 0.667 0.564 0.614 0.480 0.468 0.573 0.603 0.447 0.238 (2.76) (3.27) (3.68) (3.14) (3.28) (2.49) (2.16) (2.29) (2.11) (1.30) (0.56) Four-factor alpha 0.353 0.206 0.190 0.088 0.074 -0.074 -0.105 0.014 0.061 -0.255 0.609 (2.90) (2.74) (2.60) (1.24) (1.03) (-1.03) (-1.26) (0.14) (0.49) (-1.55) (2.73) Five-factor alpha 0.315 0.219 0.210 0.122 0.057 -0.057 -0.158 -0.008 0.044 -0.222 0.538 (2.50) (2.88) (2.83) (1.69) (0.78) (-0.77) (-1.85) (-0.07) (0.34) (-1.30) (2.32)
Low sentiment
Excess return 0.695 0.514 0.640 0.710 0.777 0.654 0.695 0.885 1.227 1.362 -0.667 (1.72) (1.74) (2.33) (2.65) (2.86) (2.39) (2.29) (2.74) (3.27) (2.92) (-1.08) Four factor alpha 0.286 0.048 0.061 0.160 0.086 -0.060 -0.133 0.018 0.285 0.124 0.163
(1.65) (0.46) (0.61) (1.53) (0.86) (-0.61) (-1.21) (0.14) (1.67) (0.50) (0.50) Five factor alpha 0.252 0.057 0.086 0.223 0.080 0.000 -0.210 0.001 0.299 0.199 0.053 (1.37) (0.53) (0.84) (2.07) (0.78) (0.00) (-1.86) (0.00) (1.68) (0.75) (0.15)
High sentiment
Excess return 0.674 0.720 0.694 0.419 0.453 0.307 0.243 0.262 -0.017 -0.463 1.138 (2.33) (3.06) (2.92) (1.75) (1.75) (1.13) (0.78) (0.69) (-0.04) (-0.92) (1.96) Four factor alpha 0.379 0.318 0.279 -0.033 0.035 -0.084 -0.048 0.046 -0.091 -0.576 0.955 (2.33) (3.20) (2.72) (-0.35) (0.35) (-0.79) (-0.39) (0.32) (-0.50) (-2.79) (3.45) Five factor alpha 0.344 0.335 0.301 -0.009 0.025 -0.108 -0.079 0.021 -0.146 -0.568 0.912 (2.11) (3.36) (2.92) (-0.10) (0.25) (-1.01) (-0.64) (0.14) (-0.81) (-2.73) (3.28) Panel C: Returns of portfolios of stocks sorted on LA
Whole periods
Excess return 1.179 0.726 0.729 0.721 0.709 0.637 0.621 0.597 0.539 0.477 0.702 (3.00) (2.26) (2.56) (2.92) (3.21) (3.16) (3.22) (3.31) (3.00) (2.39) (1.59) Four-factor alpha 0.554 0.172 0.203 0.191 0.262 0.173 0.126 0.060 0.022 -0.022 0.576 (2.50) (1.20) (1.75) (1.93) (3.02) (2.34) (1.90) (0.93) (0.34) (-0.35) (2.48) Five-factor alpha 0.541 0.154 0.183 0.193 0.242 0.177 0.144 0.060 0.021 -0.019 0.560 (2.34) (1.04) (1.53) (1.91) (2.75) (2.37) (2.11) (0.90) (0.31) (-0.30) (2.33)
Low sentiment
Excess return 1.981 1.089 0.979 0.817 0.855 0.664 0.643 0.655 0.683 0.671 1.310 (3.28) (2.19) (2.26) (2.16) (2.58) (2.16) (2.18) (2.44) (2.63) (2.51) (1.98) Four factor alpha 0.866 0.254 0.221 0.082 0.239 0.072 0.052 0.036 0.058 -0.045 0.911 (2.58) (1.22) (1.42) (0.61) (2.07) (0.73) (0.57) (0.38) (0.66) (-0.54) (2.62) Five factor alpha 0.900 0.240 0.234 0.119 0.237 0.083 0.078 0.047 0.058 -0.021 0.920 (2.49) (1.09) (1.42) (0.86) (2.03) (0.85) (0.85) (0.48) (0.64) (-0.25) (2.49)
High sentiment
Excess return 0.382 0.365 0.481 0.627 0.565 0.609 0.600 0.539 0.396 0.284 0.097 Continued on next page
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Table 4.2 – Continued from previous page
1 2 3 4 5 6 7 8 9 10 1-10
(0.76) (0.89) (1.30) (1.96) (1.93) (2.33) (2.40) (2.23) (1.59) (0.96) (0.17) Four factor alpha 0.323 0.079 0.197 0.285 0.244 0.232 0.157 0.067 -0.028 0.028 0.295 (1.11) (0.40) (1.13) (1.94) (1.89) (2.18) (1.70) (0.75) (-0.29) (0.31) (0.96) Five factor alpha 0.276 0.055 0.143 0.252 0.223 0.232 0.175 0.060 -0.023 0.009 0.266 (0.94) (0.28) (0.83) (1.71) (1.71) (2.16) (1.90) (0.66) (-0.23) (0.10) (0.86) Panel D: Returns of portfolios of stocks sorted on CC
Whole periods
Excess return 0.815 0.712 0.760 0.672 0.631 0.618 0.540 0.507 0.468 0.616 0.199 (2.42) (2.83) (3.69) (3.51) (3.39) (3.33) (2.86) (2.60) (2.18) (2.22) (0.46) Four-factor alpha 0.303 0.269 0.270 0.184 0.128 0.064 -0.032 -0.033 -0.079 0.087 0.216 (1.61) (2.26) (2.94) (2.15) (1.61) (0.81) (-0.43) (-0.43) (-0.98) (0.79) (0.99) Five-factor alpha 0.250 0.264 0.272 0.190 0.161 0.048 -0.044 -0.047 -0.128 0.075 0.175 (1.27) (2.17) (2.92) (2.19) (1.98) (0.59) (-0.56) (-0.60) (-1.58) (0.67) (0.77)
Low sentiment
Excess return 1.127 0.766 0.683 0.701 0.637 0.773 0.743 0.732 0.710 1.034 0.093 (2.14) (1.83) (2.08) (2.33) (2.21) (2.76) (2.71) (2.66) (2.42) (2.92) (0.15) Four-factor alpha 0.408 0.198 0.119 0.119 0.038 0.086 0.023 0.033 -0.080 0.118 0.290 (1.56) (1.11) (0.91) (0.95) (0.32) (0.74) (0.23) (0.32) (-0.75) (0.82) (0.97) Five-factor alpha 0.398 0.214 0.116 0.129 0.096 0.072 0.008 0.024 -0.143 0.150 0.248 (1.41) (1.16) (0.86) (1.00) (0.76) (0.62) (0.07) (0.23) (-1.34) (1.01) (0.78)
High sentiment
Excess return 0.505 0.658 0.836 0.643 0.625 0.463 0.338 0.284 0.227 0.201 0.304 (1.20) (2.34) (3.34) (2.71) (2.66) (1.91) (1.30) (1.03) (0.73) (0.47) (0.51) Four-factor alpha 0.223 0.303 0.373 0.182 0.170 -0.002 -0.090 -0.121 -0.067 0.116 0.108 (0.82) (2.03) (3.11) (1.66) (1.73) (-0.02) (-0.82) (-1.07) (-0.57) (0.71) (0.34) Five-factor alpha 0.128 0.293 0.388 0.202 0.190 -0.001 -0.081 -0.126 -0.095 0.056 0.072 (0.47) (1.95) (3.22) (1.84) (1.92) (-0.01) (-0.73) (-1.10) (-0.80) (0.35) (0.23)
For portfolios formed by sorting on PT (PT portfolio), the average monthly excess returns of the value-weighted 1-10 portfolios of 0.917% in low sentiment periods or 0.576% in high sentiment periods per month are economically large but not statistically significant. However, the abnormal returns of the 1-10 portfolios relative to four- and five-factor models are positive and statistically significant. Changing from the four-factor model to the five-factor model does not much alter the alpha. I focus my discussion on the four-factor adjusted returns (FF4 alpha), the return adjusted by the three Fama-French factors and by a momentum factor.
A strategy of buying low-PT stocks and shorting high-PT ones earns a very significant FF4 alpha of 0.740% (t-statistic=2.49) in low sentiment periods and 0.771% (t-statistic=2.61) in high sentiment periods, which is nearly 8.88% or 9.25% per annum.
For portfolios formed by sorting on PW (PW portfolio), the excess return, the FF4
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alpha, and the FF5 alpha (alphas relative to the four-factor model augmented with the liquidity factor) of the difference portfolio are not all statistically significant in low-sentiment periods. In contrast, in high-sentiment periods, they are all statistically significant and of economically important magnitude. The stocks in the lowest PW-decile, have a significantly positive FF4 alpha of 0.379% per month, and those in the highest PW-decile, have a negative alpha of −0.576%. The PW 1-10 portfolio produces an FF4 alpha of 0.955% per month (t-statistic=3.44), which is nearly 11.46% annually.
The results of the portfolio analysis using LA as the sort variable run counter to the results generated by the PW-sorted portfolios. The relation between LA and expected stock returns appears stronger in low investor sentiment periods. For example, the FF4 alpha of the LA-sorted decile portfolios decreases, even though so not monotonically, from 0.866%
per month for the first decile portfolio to −0.045% per month for the 10th decile portfolio.
The difference in returns of 0.911% per month with a t-statistic of 2.62 is highly statistically significant, which is over 10.93% per annum. Similar to the situation of PW portfolios in low sentiment periods, the excess return, the FF4 alpha, and the FF5 alpha of the LA 10-1 portfolio are all indistinguishable from zero in high sentiment periods.
In the sample period, the FF4 alpha and the FF5 alpha on the PT/PW/LA long-short portfolio remain significant, and in terms of magnitude, they are approximately the average of its corresponding value in two different sentiment periods. This result suggests the effect of prospect theory in the sample period is caused by its effect in different periods. For value-weighted portfolios formed by sorting on CC, no matter in the sample period, in low-sentiment periods, or in high-low-sentiment periods, the relation between CC and expected stock returns is insignificant, which is consistent with previous literature stating that diminishing sensitivity is not an important element in investors’ decisions on the stock market.
Figure 4.1 reports a graphic view of the results in Table 4.2. It plots the value-weighted FF4 alphas on the ten PT-decile portfolios, ten PW-decile portfolios, and ten LA-decile portfolios in low- and high-sentiment periods in Figure 4.1a, Figure 4.1b, and Figure 4.1c,
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Figure 4.1 Performance of deciles in different sentiment periods: Value-weighted
I sort stocks into ten portfolios from bottom to top by PT, PW, and LA in each month and calculate each decile’s return over the following month on a value-weighted basis. Then I sort the sample period into low and high levels of investor sentiment and compute Carhart’s (1997) four-factor alpha for these ten deciles by using the average of time-series returns following different periods. I plot the results in Figure 4.1. The Figure 4.1a is for PT; the Figure 4.1b is for PW; the Figure 4.1c is for LA. The vertical axis is the percentage monthly alpha; the horizontal axis
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respectively. Figure 4.1a shows that FF4 alphas decrease in both low- and high-sentiment periods but not monotonically. Meanwhile, the FF4 alphas of the PT 1-10 portfolio are driven mainly by the significant positive abnormal returns of the low-PT stocks. Figure 4.1b and Figure 4.1c show that the decreasing pattern of PW portfolios or LA portfolios exists only in high- sentiment periods or low-sentiment periods, respectively. These two figures make another clear point that the FF4 alphas of the PW 1-10 portfolio are driven mainly by the significant negative abnormal returns of the high-PT stocks, and the LA 1-10 portfolio entirely driven by the substantial positive returns of the low-LA stocks. The decreasing patterns of PT portfolios and LA portfolios are similar in low-sentiment periods, which may be attributed to the fact that PT and LA are highly correlated. Later, I will look in greater detail at the origin of the FF4 alphas of the long-short portfolio and examine whether overvaluation or undervaluation contributes mainly to the FF4 alphas.
To sum up, Table 4.2 shows that probability weighting works mainly in high-sentiment periods, reflecting negative abnormal returns of high-PW stocks; loss aversion works pri-marily in low-sentiment periods, indicating positive abnormal returns of low-LA stocks; and no matter in low- or high-sentiment periods, the PT 1-10 portfolio produces a positive FF4 alpha. However, applying a dynamic strategy that holding the PW 1-10 portfolio in high-sentiment periods and the LA 1-10 portfolio in low-high-sentiment periods generates the FF4 alpha of 11.20% per annum. The magnitude is over 2% higher than 9.07%, the FF4 alpha of holding the PT 1-10 portfolio statically. 3
3 I examine Fama-French three-factor alphas, Fama-French five-factor alphas, and six-factor alphas (+mo-mentum factor) on the value-weighted basis of portfolios of stocks sorted on PT, PW, and LA following different sentiment periods in Table B.1 in Appendix B. If I consider only the Fama-French three-factor model or five-factor model, the results suggest that no matter PT, PW, or LA little effect on the return of long-short portfolio occurs. After adding the momentum factor, the six-factor alphas remain similar to the previous results, proving that the momentum factor is first, too pervasive and important to ignore and second, plays a vital role in the prospect theory effect.
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