–0.65
–1.67
–0.16
______________________________
Figure 6.4. Growth of $1 for the Equally Weighted Quartile Portfolios Based on Harris Poll RQs, April 2000–August 2017 (log scale)
The annual return differences in Table 6.7 are qualitatively and quantitatively similar to the results reported in Statman et al. (2008) , who studied Fortune ’s most admired companies between September 1982 and September 2006, although Statman et al. split the stocks into two groups (“spurned” and “admired”) rather than quartiles and varied the reconstitution time period.
The last two rows in Table 6.7 show Jensen’s annualized alpha and its t -statistic. We used the equally weighted returns for all stocks across the quartiles as the benchmark and ran simple regressions to get Jensen’s annualized alpha. The annualized alphas for the lowest and highest poll values are far apart.
Figure 6.4 shows the superior performance of Quartile 4, stocks with the lowest RQ.
Table 6.8 presents various performance summary statistics for the cap-weighted four quartiles. The last two rows show the alphas against the Carhart four factors. Overall, the results are consistent with Table 6.7 . A difference is that Carhart’s alpha for Q1 is the highest (α = 3.80%). A close look shows that Q1 is loaded with a negative HML (high book/market minus low book/market, the value factor) coefficient (β = –0.32, t -statistic = –6.40) and negative momentum coefficient (β = –0.09, t -statistic = –2.98), and Q4 is loaded with a positive HML coefficient (β = 0.39, t -statistic = 7.28) and positive momentum coefficient (β = 0.01, t -statistic = 0.37). The other two factor loadings (R m – R f and SMB, or small minus big, the size factor) are similar for Q1 and Q4. Q4 is apparently capturing some value effect, which is consistent with the notion that a previously popular company moves toward being a value company when it is getting less popular and a previously unpopular company moves toward being a growth company when it is getting more popular.
Overall, based on company reputation, the equally weighted composites are consistent with both popularity and the risk–return paradigm. However, the cap-weighted composites are not monotonic, and in our judgment, they are moderately consistent with the popularity theory but lack a clear relationship between return and risk. Given the relatively small number of companies in the composites, the market-cap composites may be heavily influenced by a small number of stocks. For both the equally weighted and market-cap weighted composites, the quartiles with the worst company reputations (unpopular stocks) produced the highest geometric returns.
______________________________
Table 6.8. Summary Statistics of Cap-Weighted Quartile Portfolios Based on Harris Poll RQ Rankings, April 2000–August 2017
Measure
Q4
(lowest poll value)
Q3
Q2
Q1
(highest poll value)
Geometric mean (%)
8.49
–0.07
3.41
6.07
Arithmetic mean (%)
9.74
1.25
4.86
7.37
Standard deviation (%)
15.18
16.03
16.59
15.58
Sharpe ratio
0.532
–0.018
0.196
0.367
Skewness
–0.303
–0.607
–0.551
–0.572
Carhart’s alpha (%)
2.91
–3.74
–0.72
3.80
t -stat. of alpha
1.46
–1.71
–0.41
2.04
Tail Risk (Coskewness)
Kraus and Litzenberger (1976) showed that investors dislike stocks with returns with high tail risk as measured by negative coskewness with the market. Thus, stocks with more negative coskewness tend to have higher expected returns. Negative coskewness means that the security in question contributes negatively to the skewness of the market portfolio, so the security is expected to experience large losses when the market falls. Harvey and Siddique (2000) suggested that the stocks with negative coskewness should command a higher expected return than those with positive coskewness because negatively skewed returns are less desirable. 43 Indeed, by studying past returns, the authors showed that coskewness has been economically important and has earned a risk premium of, on average, 3.6% per year for US stocks.
In this section, we report empirical analyses similar to those of Harvey and Siddique (2000) but focused on the subsequent 20-year period—basically an out-of-sample test of their analysis. 44
Starting in January 1991 and using the first 60 months of returns, we sorted the universe of all US stocks from lowest or most negative coskewness (least popular) to highest or most positive coskewness (most popular), and we assigned companies to one of four quartiles. We averaged monthly returns starting from January 1996 with equal weights for each quartile. We updated the quartiles monthly.
Table 6.9 presents various performance summary statistics for the equally weighted quartiles. Quartile 4 consists primarily of the lowest, or most negative, coskewness companies, and Quartile 1 consists of the highest, or most positive, coskewness companies. We believe that investors prefer companies that have high positive coskewness. Thus, along this dimension of popularity, Q1 represents the most popular stocks and Q4 represents the least popular.
In Table 6.9 , as we move from left to right from Q4 of companies with the lowest or most negative coskewness (least popular companies) to Q1 of companies with the highest or most positive coskewness (most popular companies), we see that companies with lower coskewness monotonically produced superior arithmetic and geometric mean returns. This additional return came with nearly equal standard deviations (inferring no relationship between risk and return), so the Sharpe ratios are monotonically decreasing from Q4 to Q1. Table 6.9 thus confirms the empirical findings of Harvey and Siddique (2000) in an out-of-sample sense.
The annualized Jensen’s alphas for Q1 and Q4, both statistically significant at the 5% level, support the popularity thesis.
Figure 6.5 shows the growth of a $1 investment in each of the four quartiles. The lowest coskewness portfolio outperformed the highest coskewness portfolio.
Table 6.10 presents various summary performance statistics for the cap-weighted four quartiles. The last two rows show the alphas against the Carhart four factors. In general, the results are consistent with Table 6.9 , but the t -statistics are less significant.
______________________________
Table 6.9. Summary Statistics of Equally Weighted Quartile Portfolios Based on Coskewness, January 1996–August 2017
Measure
Q4
(most negative coskewness)
Q3
Q2
Q1
(least negative or most positive coskewness)
Geometric mean (%)
12.16
9.67
9.57
8.21
Arithmetic mean (%)
13.39
10.85
10.79
9.39
Standard deviation (%)
14.74
14.59
14.87
14.75
Sharpe ratio
0.741
0.579
0.564
0.476
Skewness
–0.611
–0.672
–0.638
–0.572
Jensen’s alpha (%)
2.22
–0.22
–0.47
–1.49
t -stat. of alpha
2.84
–0.60
–1.23
–2.23
Figure 6.5. Growth of $1 for Equally Weighted Quartile Portfolios Based on Coskewness, January 1996–A
ugust 2017 (log scale)
______________________________
Table 6.10. Summary Statistics of Cap-Weighted Quartile Portfolios Based on Coskewness, January 1996–August 2017
Measure
Q4
(most negative coskewness)
Q3
Q2
Q1
(least negative or most positive coskewness)
Geometric mean (%)
10.83
8.10
8.57
6.15
Arithmetic mean (%)
12.09
9.23
9.78
7.38
Standard deviation (%)
15.04
14.40
14.86
15.18
Sharpe ratio
0.644
0.477
0.498
0.333
Skewness
–0.638
–0.569
–0.715
–0.527
Carhart’s alpha (%)
2.43
–0.16
0.18
–1.70
t -stat. of alpha
1.86
–0.16
0.18
–1.72
Overall, based on coskewness, both the equally weighted composites and cap-weighted composites are consistent with popularity and are inconsistent with the risk–return paradigm.
Lottery Stocks
A number of empirical studies have shown that portfolios held by individual investors are often underdiversified, containing fewer than five stocks, on average. Example studies are Odean (1999) , Mitton and Vorkink (2007) , Kumar (2007) , and Goetzmann and Kumar (2008) . Statman (2004) called the situation the “diversification puzzle” because this underdiversification choice is inconsistent with portfolio theory—that is, by the simple act of diversifying, investors should be able to obtain an equivalent expected return at a much lower expected standard deviation or, conversely, a much higher expected return at an equivalent expected standard deviation.
The key to understanding this puzzle is investors’ preferences for positive skewness, as described by Xiong and Idzorek (forthcoming 2019) . A considerable body of empirical evidence shows that individual investors prefer lottery-like stocks (see, for example, Barberis and Huang 2008 ). Lottery-like stocks have a relatively small probability of a large payoff. From a popularity perspective, this strong preference for lottery-like stocks suggests that lottery-like stocks are popular . Indeed, Mitton and Vorkink (2007) found that lottery-like stocks are relatively popular with underdiversified investors.
No single definition of what represents a lottery-like stock or company is accepted. Bali, Cakici, and Whitelaw (2011) used the maximum daily return over the past month as a proxy for lottery stocks. They performed portfolio-level analyses and company-level cross-sectional regressions that showed a negative and significant relationship between the maximum daily return over the past one month (MAX) and expected stock returns. Average raw and risk-adjusted return differences between stocks in the lowest and highest MAX deciles exceeded 1% per month. These results were robust to controls for size, value, momentum, short-term reversals, liquidity, and skewness. This evidence suggests that investors are willing to pay more for stocks that exhibit extreme positive returns; thus, these stocks exhibit lower returns in the future.
For our analysis, we once again formed quartiles on the basis of a proxy for a dimension of popularity—in this case, the strong preference for lottery-like payoffs. We used the same stock universe as we did in the section on tail risk (coskewness). We also used the measure of lottery-like behavior of stocks presented in Bali et al. (2011) —that is, the average return associated with the five best days during the prior month. Starting in January 1991, we formed quartile portfolios every month from February 1991 to August 2017 by sorting stocks on the basis of the average return associated with each stock’s five best days during the prior month (MAX-5). Quartile 4 is the portfolio of stocks with the lowest MAX-5 over the past one month (the least popular stocks). Conversely, Quartile 1 is the portfolio of stocks with the highest MAX-5 over the past one month (the most popular stocks).
Consistent with our prior analyses, Table 6.11 presents summary performance statistics for the four quartiles sorted on MAX-5.
In Table 6.11 , moving from left to right from Q4, stocks with the lowest MAX-5 (least popular stocks), to Q1, stocks with the highest MAX-5 (most popular stocks), we see that stocks with lower MAX-5 ratings monotonically produced superior Sharpe ratios. The monotonic increase in standard deviation is also clearly seen across the quartiles, and the highest MAX-5 quartile has the highest volatility. This finding is consistent with the well-known low-volatility anomaly, in that the least popular stocks have the lowest volatility. It indicates that high-volatility stocks have lottery-like payoffs, so they underperform low-volatility stocks on a risk-adjusted basis . The widely separated annualized Jensen’s alphas for Q4 and Q1 are statistically significant at the 5% level. Our results confirm the results of Bali et al. (2011) on a risk-adjusted basis, although our result for the arithmetic mean return for the lowest MAX-5 quartile is lower than that for the highest MAX-5 quartile.
Figure 6.6 shows the growth of a $1 investment in each of the four quartiles. So, contrary to the popularity hypothesis, the highest MAX-5 daily return portfolio outperformed the lowest MAX-5 daily return portfolio, even though the predictions of the popularity hypothesis do hold up on a risk-adjusted basis.
______________________________
Table 6.11. Equally Weighted Quartile Summary Statistics Based on the Average of MAX-5 Daily Returns, February 1991–August 2017
Measure
Q4
(lowest MAX-5 daily returns)
Q3
Q2
Q1
(highest MAX-5 daily returns)
Geometric mean (%)
11.63
12.53
12.16
13.35
Arithmetic mean (%)
12.18
13.67
13.95
17.38
Standard deviation (%)
9.98
14.17
17.74
26.67
Sharpe ratio
0.937
0.763
0.625
0.541
Skewness
–0.970
–0.832
–0.617
0.061
Jensen’s alpha (%)
4.42
1.73
–1.53
–4.42
t -stat. of alpha
4.38
1.87
–3.67
–2.29
Figure 6.6. Growth of $1 for Equally Weighted Quartile Portfolios Based on Average of MAX-5 Daily Returns, February 1991–August 2017 (log scale)
______________________________
Table 6.12. Summary Statistics for Cap-Weighted Quartile Portfolios Based on the Average of MAX-5 Daily Returns, February 1991–August 2017
Measure
Q4
(lowest MAX-5 daily returns)
Q3
Q2
Q1
(highest MAX-5 daily returns)
Geometric mean (%)
9.46
9.26
10.32
8.40
Arithmetic mean (%)
10.16
10.3
12.21
12.69
Standard deviation (%)
11.26
14.24
18.42
27.73
Sharpe ratio
0.656
0.534
0.510
0.356
Skewness
–0.651
–0.465
–0.491
–0.310
Carhart’s alpha (%)
0.53
–0.62
–0.60
&nb
sp; –2.00
t -stat. of alpha
0.53
–0.70
–0.57
–0.85
Table 6.12 presents various summary statistics for the market cap– weighted four quartiles. The last two rows show the alphas for the Carhart four factors. Overall, the results are consistent with those presented in Table 6.11 , but the t -statistics are lower, indicating less statistical significance.
Overall, based on lottery-like stocks (sorts on average of MAX-5 daily returns), the equally weighted composites and cap-weighted composites are consistent with both popularity and the risk–return paradigm; the less popular quartiles monotonically produced better Sharpe ratios, yet higher returns always came with more risk.
Conclusion
In Table 6.13 , we have consolidated the previous analyses and our assessment of how the results are or are not consistent with the popularity framework and/or risk–return paradigm. We found 10 out of 10 of the analyses to be at least somewhat consistent with the popularity framework, while 5 out of 10 are consistent with the more-risk/more-return paradigm.
______________________________
Table 6.13. Summary Results and Consistency with Asset Pricing Framework
Table 6.14. Correlation Analysis
A reasonable question to ask is how these various potential dimensions of popularity relate to the Fama–French SMB and HML factors, which we obtained from the French Data Library. 45 To approach an answer, for each of the analyses in this chapter, we created a “factor” series by subtracting the monthly return of the popular quartile portfolio from that of the unpopular quartile portfolio, where applicable. (For the moat factor, we subtracted the returns of the wide-moat portfolio from those of the no-moat portfolio.) Then, based on the maximum common time period, July 2002 through August 2017, we calculated the correlation matrix of the five characteristics that we studied plus SMB and HML. Panel A of Table 6.14 displays the equally weighted results, and Panel B displays the market cap–weighted results.
Popularity Page 13
