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abnormal returns in the short leg. Specifically, nearly 83.464% (95.48%) of the abnormal re-turn on the long-short PT (LA) portfolio stems from the long portfolio or the undervaluation part, whereas 61.29% of it on the long-short PW portfolio comes from the short portfolio or the overvaluation part. The results of the double sort analysis are consistent with these.
Column Ave indicates that over 90% (55%) of the difference of PT/LA (PW) long-short portfolio returns originate from the subsequent positive (negative) returns of the underval-uation anomalies (overvalunderval-uation anomalies). The evidence suggests that low PT/LA stocks are likely to be underpriced in low sentiment months and then earn larger positive abnormal returns, and high PW stocks are likely to be overpriced in high sentiment months and then achieve less negative abnormal returns in the following month.
4.7 Time-varying prospect theory effect
Stambaugh, Yu, and Yuan (2012) find that the sentiment level significantly affects long-short return spreads for 11 anomalies. The predictive power of investor sentiment comes primarily from the short-leg returns, and an asymmetric effect exists; that is, sentiment makes optimism produce greater mispricing than pessimism. The mechanism is sentiment-driven noise. Traders have strong positive demand for many stocks following high-sentiment periods but do not have correspondingly negative demand following low-sentiment periods because of short-sale constraint. The preliminary evidence in Table 4.9 and the previous analysis, however, suggest that high probability weighting stocks are overpriced in high-sentiment months and high loss aversion stocks are underpriced in low-sentiment months. Following these authors, I examine whether the asymmetry indicates that the investor sentiment could explain the negative PW effect among overvalued stocks or the positive LA effect among undervalued stocks.
To explore in greater depth the relationship between stock market sentiment and ab-normal returns mentioned above, I first conduct a sorting-based portfolio analysis for
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Table 4.10 Prospect theory portfolios during periods of low and high investor sentiment: Excess returns and benchmark-adjusted returns on long-short portfolios
This table presents the average returns following low and high investor sentiment periods, classified based on the median level of Baker and Wurgler’s (2006) sentiment index. Panel A reports average excess returns; Panel B reports average benchmark-adjusted returns, estimated by the following regression:
Ri,t= SLdL,t+ SHdH,t+ βmktM KTt+ βsmbSM Bt+ βhmlHM Lt+ βmomM OMt+ i,t,
where Ri,tis the excess return on either the long leg, the short leg, or the difference in month t; dL,t and dH,tare indicator variables that equal 1 in low- and high-sentiment periods, respectively; SLand SH are the estimates of average returns in low- and high-sentiment periods, respectively. The sample period runs from September 1965 to December 2016. t-statistics appear in parentheses and are adjusted by the heteroskedasticity-consistent standard errors of White (1980).
Long leg Short leg Long-Short
Low High Low Low High Low Low High Low
sentiment sentiment -High sentiment sentiment -High sentiment sentiment -High Panel A: Excess returns
PT 1.632 0.740 0.893 0.715 0.164 0.551 0.917 0.576 0.341
(2.85) (1.58) (1.21) (2.60) (0.53) (1.34) (1.99) (1.52) (0.57)
PW 0.695 0.674 0.021 1.362 -0.463 1.826 -0.667 1.138 -1.805
(1.72) (2.33) (0.04) (2.93) (-0.92) (2.67) (-1.51) (2.74) (-2.98)
LA 1.981 0.382 1.600 0.671 0.284 0.387 1.310 0.097 1.213
(3.28) (0.76) (2.04) (2.52) (0.96) (0.97) (2.59) (0.24) (1.86)
Panel B: Benchmark-adjusted returns
PT 0.727 0.557 0.169 -0.010 -0.224 0.213 0.737 0.781 -0.044
(2.58) (1.82) (0.41) (-0.12) (-2.29) (1.60) (2.49) (2.41) (-0.10)
PW 0.227 0.481 -0.254 0.220 -0.735 0.956 0.007 1.216 -1.209
(1.24) (2.84) (-1.02) (0.91) (-3.69) (3.05) (0.02) (4.41) (-2.81)
LA 0.927 0.177 0.750 0.007 -0.051 0.058 0.920 0.228 0.692
(2.80) (0.51) (1.56) (0.08) (-0.56) (0.46) (2.70) (0.64) (1.41)
sentiment and high-sentiment periods, similar to that in Table 4.2, separately. In Table 4.10, Panel A reports results for excess returns and Panel B reports results for returns adjusted by the four-factor (Fama-French three factor and momentum factor) benchmarks. I expect the PW effect among “overpriced” stocks to be stronger in high-sentiment times, and in contrast, the LA effect among “underpriced” stocks to be stronger in low-sentiment times. Because PT is a comprehensive index computed by probability weighting and loss aversion, finding such sentiment-difference effect existing in PT-based “underpriced” stocks may be difficult.
Panel A of Table 4.10 reveals that PT and LA long-short differences exhibit higher average returns in low-sentiment times and PW long-short difference exhibits higher in high-sentiment times, consistent with previous results. In Panel B, with benchmark-adjustment returns, the results remain similar except long-short portfolio sorted based on PT also obtain
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half the number of Januarys belong to high sentiment (27 out of 51).
I predict the average returns on the PW PW-based short leg should be significantly lower significantly in high- sentiment months than in low- sentiment months, and correspondingly, the average returns on the LA LA-based long leg should be higher significantly in low-sentiment months than in high- low-sentiment months. In Panel A, this hypothesis is strongly supported. The short leg of PW strategy has a lower average excess return in high- sentiment months than in low- sentiment months, which earns 183bps less per month (t statistic: 2.67).
Compared to high- sentiment months, the long leg of LA strategy earns a positive average excess return of 160 bp per month in high- sentiment periods (t statistic: 2.04). In panel Panel B, after adjusting for benchmark exposure, the PW strategy remains the same: result, the spread between high sentiment and low sentiment is 0.956% (t statistic: 3.05). I find weak evidence that indicates the significant difference of in the LA- based long leg between the average return in low -ssentiment and high- sentiment periods, even though it is positive (75 bps) but with a t- statistic of 1.56.
The evidence in Table 4.10 supports the following ratiocination: sentiment-driven over-pricing offers a partial explanation for the PW effect; PW misover-pricing is stronger following high sentiment; and the short-leg has more deficit. Meanwhile, sentiment-driven underpricing seems insufficient, and the market-wide sentiment may not make a significant contribution to the profitability of the long leg of LA strategy because of the impediments to short sale.
Now I transfer my attention to PT and consider the sentiment effects on PT, consisting of all three elements: probability weighting, loss aversion, and concavity/convexity. On one hand, based on the previous examination that included portfolio analysis in Table 4.2 and Table B.1 and Fama-MacBeth regression analysis in Table 4.3, Table 4.4, and Table 4.5, I find PT and LA have predictive power mainly following low sentiment but PW is predictive following high sentiment. I conclude that this phenomenon—that the influence of PT is similar to LA—may be attributed to the large coefficient of λ, which expands the weight of loss aversion while computing PT. On the other hand, the worth of prospect theory value is
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the worth of the parts composing it. Doubtless, PT has the properties of PW and LA, and it could overcome sentiment-driven shortcomings.
The evidence in Table 4.10 supports my assumption. Sentiment has no remarkable effect on PT-sorted long-leg or short-leg returns. In Panel A, the long leg earns 89 bps and the short leg earns 55 bps following low sentiment: they both have higher returns, but the t-statistics are not significant (t statistic: 1.21 & 1.34). After benchmark adjustment, the profitable effect becomes even weaker. The four-factor adjusted return on the long leg and short leg of the PT strategy exhibits nearly 17 and 21 bps difference between low- and high-sentiment periods, respectively, and they don’t have significant t-statistics. Interestingly, I find that the magnitude of low-high sentiment return spreads of the PT strategy always falls in the middle of the other two strategies (PW and LA).
A binary classification of low and high sentiment used in the portfolio analysis above has the advantage of simplicity, but this method is not a powerful test to examine the sentiment sensitivity. Alternatively, following Stambaugh, Yu, and Yuan (2012, 2015), I conduct time-series regressions to verify the prediction predictive power of the Baker and Wurgler’s sentiment index. In Table 4.11, Panel A reports the results of the regressions that using use the percentage excess returns in month t as the dependent variable and the level of the sentiment index in month t − 1 as the independent variable. In Panel B, I also include the three contemporaneous Fama and French three factors (MKT, SMB, HML) and Carhart momentum factor (MOM) as the independent variables, and get obtain the coefficient on St−1to understand the sentiment-driven variation in the benchmark-adjusted returns. I scale the sentiment index to have a mean of zero and a standard deviation of one.
First, I examine the relation between long-short spread and market sentiment. Regarding PT long-short portfolios, I find no evidence to indicate the significant relation between profitability and investor sentiment. The PT strategy has a t-statistic of -0.63 in Panel A and 0.64 in Panel B. For the excess returns, the slope coefficient for the spread of LA is negative with a t-statistic of -1.54, which is significant at a one-tailed 0.1 significance level
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Table 4.11 Prospect theory value and investor sentiment (level): Predictive regres-sions for excess returns and benchmark-adjusted returns on long-short portfolios
Panel A reports estimates of β in the regression:
Ri,t= α + βSt−1+ µt,
and Panel B reports estimates of β in the regression:
Ri,t= α + βSt−1+ βmktM KTt+ βsmbSM Bt+ βhmlHM Lt+ βmomM OMt+ µt,
where Ri,tis the excess return on either the long leg, the short leg, or the difference in month t; St is the level of investor sentiment index of BW (2006). The sample period runs from September 1965 to December 2016. t-statistics are adjusted by the heteroskedasticity-consistent standard errors of White (1980).
Variable ˆb t-Statistic ˆb t-Statistic ˆb t-Statistic Panel A: St−1
and means that a one standard deviation decrease in sentiment is associated with $0.006 of additional long-short monthly profit after adding $1 to each leg of the spread. As for the benchmark-adjusted returns, the coefficient becomes insignificant, which is consistent with the results in Panel B of Table 4.10. It seems as if the sentiment sensitivity of LA is easily affected by risk factors, especially SMB. (In an unreported result, if I add only MKT, HML, and MOM as independent variables in the benchmark-adjusted model, the t-statistic of LA remains -1.54.) The slope coefficients for long-short portfolios of PW are positive in both Panel A and Panel B. The PW strategy has a statistic of 2.86 in Panel A and 2.34 in Panel B, even though after the adjustment of risk exposures, the latter slope is only the half the magnitude of the former. In Panel B, a 1 standard deviation increase in sentiment improves
$0.0055 additional monthly profit in the case of a $1 investment in each leg of the spread.
Secondly, I predict a negative relation between returns on the short leg portfolio of PW and the lagged sentiment index. The evidence in Table 4.11 supports this assumption, no matter to for excess returns or the benchmark-adjusted returns. The PW strategy has a
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t-statistic of -2.72 in Panel A and -2.38 in Panel B. Specifically, after considering the risk factors, increasing sentiment by 1 standard deviation in sentiment accompanied with a 0.37%
lower monthly return on the short-leg portfolio.
Thirdly, to regarding the returns on the long long-leg portfolio of LA, after the adjustment of risk exposures, the slope coefficient changes from -0.88 with a t-statistic of -1.90 to -0.13 with a t-statistic of -0.50. I also use the indicator variable that represents low sentiment or high sentiment as the independent variable to replace the level of sentiment in the regressions.
I report the results in Table I.1 inthe Appendix I Table I.1. In Panel B, the slope coefficient of LA in the column labeled long-leg is negative with a t-statistic of -1.71. The evidence here suggests the that high sentiment would decrease the return on the long-leg portfolio. I attribute this phenomenon to of the low sensitivity of LA to the sentiment resulting from , due to the short-sale constraints and the disposition effect (Shefrin and Statman, 1985). ), the latter Disposition effect meaning thats the investors tend to keep poor-performing stocks.
In my framework, they keep the high loss aversion stocks until the stock market falls into the lowest sentiment horizon. I consider that I could maximize the marginal effect of sentiment while transferring from the high sentiment to the lowest investor sentiment. Based on this consideration, I divide the whole sample period into quartiles according to the sentiment index, and run the time-series regressions using the sample in the first (the lowest) and the third quartile sentiment period. I report the results in Panel C of Table 4.11. The slope coefficient for the short-leg and long-short leg return of LA has a t-statistic of -2.07 and -2.24, respectively, which means thate increasing of sentiment would decrease the returns on the short-leg portfolio and long-short monthly profit. The transformation of the sentiment period enhances the effect of LA.
In sum, the predictive regressions display identical results reported in the approach of portfolio analysis. Specifically, first, regarding the PW portfolio, mispricing mainly originates from overpricing; as a result the mispricing is more prevalent following high sentiment.
Second, regarding the LA portfolio, investor sentiment has little marginal effect on its
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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