The average value of HSNSI for the sample is 22.7%, with standard deviation 25.9% (high volatility). The high for the sample is 67.8% (on 11/20/06), and the low is -36.1% (on 9/20/08). Assuming normal variable distributions, the sample is large enough to test predictive power with reasonable reliability for short-term stock returns (a few trading days or weeks), but is not large enough to test HSNSI reliably as a long-term stock market indicator. The sample may be biased due to selection and clustering, because Mark Hulbert may be more likely to cite HSNSI, and cite it frequently, when either HSNSI or stock market behavior is extreme.

The following chart superimposes the HSNSI sample on a plot of the S&P 500 Index. The blue dashed line is the average HSNSI for the sample. As noted, HSNSI has high variability. Visual inspection suggests that HSNSI, like many other sentiment indicators, tends to be high (low) during market advances (declines).

To add precision to that observation, we examine how HSNSI relates to future S&P 500 Index returns.

The following scatter plot relates the return on the S&P 500 Index over the next five trading days to HSNSI. The Pearson correlation for the two series is -0.12, indicating a slight tendency for the stock market to be relatively weak (strong) when HSNSI is high (low). The R-squared statistic is 0.02, indicating that variation in HSNSI explains 2% of stock market returns over the next five days.

Using future return intervals of 21 and 63 trading days yields R-squared statistics of 0.00 and 0.00, respectively, for the relationship between HSNSI and future stock market returns.

Note that the sample may have some selection/clustering biases at this point. Clustering leads to some overlap of five-day test intervals, effectively reducing sample size and perhaps overweighting extreme values of HSNSI. Clustering could also confound a trading rule that specified entry/exit based on values of HSNSI. What happens if we alleviate clustering?

To eliminate measurement interval overlap and alleviate clustering bias, we winnow sample points so that no two surviving points lie within the same return interval.

The next chart depicts the relationship between the return on the S&P 500 Index over the next five trading days to HSNSI for a winnowed sample (175 surviving sample points), ordered from lowest to highest HSNSI. Ordering helps determine whether some ranges of HSNSI might be more useful than others. Note that the horizontal axis is not time-sequential. The Pearson correlation between the two series is -0.18, again indicating some tendency for the stock market to be relatively weak (strong) when HSNSI is high (low). The R-squared statistic indicates that HSNSI explains about 4% of stock market returns over the next week.

About half the explanatory power of HSNSI in this case comes from the large peak at the left side (from Fall 2008).

Using future return intervals of 21 and 63 trading days on appropriately winnowed samples yields R-squared statistics of 0.06 (with negative correlation) and 0.00, respectively, for the relationship between HSNSI and future stock market returns. Winnowed sample sizes are only 68 and 29, respectively, so these results have limited reliability.

For a different perspective that does not assume linearity in the relationship, we evaluate future stock market returns by winnowed tercile of HSNSI values.

The final chart summarizes average S&P 500 Index returns over the next five, 21 and 63 trading days by tercile (ordered third) of the winnowed HSNSI distributions for the entire sample period. Average returns for all intervals of five, 21 and 63 trading days during the sample period are 0.1%, 0.3% and 1.1%, respectively. Results generally support a belief that high (low to moderate) values of HSNSI indicate relatively low (high) future stock market returns.

Note that the terciles for the 21-day and 63-day future returns are very small, 23 and 10, respectively. One or two new observations could change results for these intervals substantially.

Note also that this breakdown, which uses all data for the sample period, does not represent an outcome of realistic trading during the period. A realistic scenario would require a continuous sample of HSNSI over an extended period, a real trading vehicle with frictions and a trading rule based solely on values of HSNSI known by a trader to date.

See “Purifying Stock Market Sentiment Indicators” for a summary of a study finding that poll-based sentiment indicators, including HSNSI, are proxies for past stock market behavior and that they add little or no value to indicators based purely on price.

HSNSI is, in fact,  more strongly related to past than future stock returns. Pearson correlations between HSNSI and S&P 500 Index returns over the past five, 21 and 63 trading days (unwinnowed) are 0.35, 0.60 and 0.68, respectively. When the stock market advances (declines), HSNSI tends to rise (fall). For returns over the future five, 21 and 63 trading days (unwinnowed), correlations are -0.12, -0.03 and -0.03, respectively. In other words, it is much easier to predict the future value of HSNSI based on past stock returns than it is to predict future stock returns based on the value of HSNSI.

In summary, evidence from simple tests indicate that, while results are mixed, the Hulbert Stock Newsletter Sentiment Index may have some power to predict future stock market returns over short horizons.

See below for Mark Hulbert’s comments on a prior iteration of this analysis.

See Guru Grades for a snapshot of the accuracies of various experts in forecasting the direction of the U.S. stock market, including links to detailed individual evaluations. We do not include Mark Hulbert in the snapshot because the above review is fundamentally different from the reviews of guru commentaries.

On 10/31/08, Mark Hulbert wrote:

Thanks for your rigorous analysis of the Hulbert Stock Newsletter Sentiment Index. I am sorry you had such limited data with which to conduct your analysis, which — as you conceded — greatly reduces the potential power of any test results. We have found–using our data set back to 1985, and correcting for various statistical biases, including the ones you mention–markedly greater statistical significance than you found from your more limited sample.

Response:

The sample used derives from the nearly [now well over] 200 mentions of the Hulbert Stock Newsletter Sentiment Index in MarketWatch columns over the past 6+ [now 8+] years. It may be that there is some skew in the published data, or that the predictive power of the index varies over time (and has diminished). If you have made public your study of the index, we will be glad to cite it. Or, if you make public the complete dataset, we will be glad to run analyses on the complete set, as well as subsets to test robustness over time.

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• The estimated equity risk premium depends on when researchers collect measurements. In general, estimates were low until 1920, high in the 1920s and high again post-World War II (with a dip in the 1970s).
• The estimated equity risk premium depends on where researchers collect measurements, with estimates relatively low (high) in developed (emerging) markets. Average estimates vary from a low for Canada (3.95%) to a high for the Asian Tigers (13.1%).
• The estimated equity risk premium depends on how researchers collect and process measurements. The average of arithmetic (geometric) mean estimates is 6.37% (4.46%). The average of ex post (ex ante) estimates is 6.03% (4.48%). The average of estimates based on Treasury bills (bonds) as the risk-free rate is 6.07% (5.26%).
• Estimates of the equity risk premium tend to be higher for sample periods and countries with higher economic volatility. A one standard deviation increase in the volatility of Gross Domestic Product indicates a 1.7% increase in estimated equity risk premium.
• Nominal interest rates relate negatively to the estimated equity risk premium. A 1% increase in interest rates indicates a 0.5% decrease in the estimate.

The following chart, taken from the paper, shows the distribution of equity risk premium estimates for all 535 observations, along with some distribution statistics (the distribution is not normal). The average of all estimates is 5.73%. Out of the 535 estimates, 24 (48) of are negative (greater than 10%).

In summary, estimates of the reward for risking equity investment vary with sample period, market and calculation method. Estimates tend to be higher when GDP is volatile and nominal interest rates are low.

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In performing analyses, we make the following assumptions:

• Start with an initial investment of $10,000.
• Use of dividend-adjusted values of SPY assumes reinvestment of dividends.
• For the Mojena Market Timing strategy, trade between SPY and cash at the close on the first trading day after the dates (Sundays) specified. Trading at the open would be a feasible alternative.
• For the 10-month SMA benchmark strategy, invest in SPY (cash) when the S&P 500 Index is above (below) its 10-month SMA at the end of the month. Assume this strategy can anticipate signals sufficiently to trade at the monthly close with the signals. This benchmark strategy generates 17 switches between stocks and cash over the sample period (not counting the initial position), compared to 30 switches for the Mojena Market Timing strategy.
• The return on cash is the T-bill yield.
• Assume one-way trading friction (transaction fee plus bid-ask spread) is 0.1% of the balance for the Mojena Market Timing strategy trades and the 10-month SMA strategy trades. Ignore trading frictions for reinvestment of dividends.
• Ignore tax implications of trading.

The following chart compares the cumulative values of $10,000 initial investments for the Mojena Market Timing strategy and for the buy-and-hold and 10-month SMA benchmark strategies over the period 1/29/93 through 8/27/10. The cumulative values of the Mojena Market Timing strategy, the buy-and-hold benchmark strategy and the 10-month SMA benchmark strategy are $36,751, $32,910 and $58,646, respectively. The Mojena Market Timing strategy sometimes lags, sometimes tracks and sometime beats the buy-and-hold benchmark strategy. It generally lags the 10-month SMA benchmark strategy.

Since cumulative value analyses are sensitive to start and stop dates, we look at annual returns.

The next chart compares the calendar year returns for the Mojena Market Timing strategy and the two benchmark strategies, with 1993 a partial year commencing 1/29/93 and 2010 a partial year ending 8/27/10. The Mojena Market Timing strategy beats the buy-and-hold (10-month SMA) strategy during eight (six) of 18 years.

For additional perspective, we look at some return series statistics.

The following table provides the geometric mean annual returns, arithmetic mean annual returns and standard deviations of annual returns for the Mojena Market Timing strategy and the two benchmark strategies over the entire sample period (treating 1993 and 2010 as complete years). The geometric mean is arguably the better estimator of long-term strategy performance, while the arithmetic mean is arguably the better estimator of next year’s return. Results indicate that the Mojena Market Timing strategy beats the buy-and-hold benchmark strategy but loses to the 10-month SMA benchmark strategy for long-term performance. The 10-month SMA strategy wins for all three statistics.

These results for the Mojena Market Timing strategy differ from those presented by Professor Mojena for 1990-2009 because sample periods are different and perhaps because of trade modeling assumptions and SPY tracking error relative to the S&P 500 Index. Professor Mojena presents other return statistics for the 1990-2009 period.

For additional insight into how the Mojena Market Timing strategy differs from buy-and-hold, we relate annual returns using a scatter plot.

The following scatter plot relates annual returns for the Mojena Market Timing strategy and the buy-and-hold benchmark strategy over the entire sample period (again treating 1993 and 2010 as complete years). The R-squared statistic for the relationship is 0.87, indicating that SPY annual returns explain 87% of Mojena Market Timing strategy annual returns. Results indicate that the Mojena Market Timing strategy generates some alpha relative to SPY for the sample period and that it tends to underperform (outperform) an advancing (declining) stock market. Its performance relative to the market over a given sample period therefore depends on the mix of bull and bear states during that period.

In summary, evidence from simple tests over a modest sample period supports a belief that the Mojena Market Timing strategy tends to outperform a buy-and-hold benchmark strategy (depending on the mix of bull and bear states) but may not outperform a simple long-term trend following strategy.

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Jim Rohrbach’s recent trading record lists 46 switches between stocks and cash during 3/24/03 through 12/31/09, starting with a BUY on 3/24/03 and ending with a BUY on 11/17/09. In analyzing this record, we make the following assumptions:

• Start with an initial investment of $10,000 (per Jim Rohrbach’s analysis).
• For Jim Rohrbach’s strategy, trade between SPY and cash at the close on the trade dates specified.
• Consider two benchmarks:
  • Buy and hold SPY.
  • Time SPY monthly using the 10-month simple moving average (SMA) of the S&P 500 Index by investing in SPY (cash) when the index is above (below) its 10-month SMA at the end of the month. Assume this strategy can anticipate signals sufficiently to trade at the monthly close with the signals. This benchmark strategy generates only five switches between stocks and cash over the sample period (at the ends of April 2003 into stocks, July 2004 into cash, October 2004 into stocks, November 2007 into cash and June 2009 into stocks).
• Use of dividend-adjusted values of SPY assumes reinvestment of dividends. (Jim Rohrbach’s analysis appears to ignore SPY dividends.)
• The return on cash is the T-bill yield. (Jim Rohrbach’s analysis appears to assume substantially higher returns on cash for some periods.)
• Assume one-way trading friction (transaction fee plus bid-ask spread) is 0.1% of the balance for Jim Rohrbach’s strategy trades and the 10-month SMA strategy trades. Ignore trading frictions for reinvestment of dividends.
• Assume an annual debit of $395 the last trading day of each year for Jim Rohrbach’s strategy to cover his fee. (Note that this fee is material for the assumed investment size.)
• Ignore tax implications of trading.

The following chart compares the cumulative values of $10,000 initial investments for Jim Rohrbach’s strategy and for the buy-and-hold and 10-month SMA benchmark strategies over the period 3/24/03 through 12/31/09. As modeled, Jim Rohrbach’s strategy lags both benchmarks all or most of the time. However, the sample period is short (especially for the 10-month SMA benchmark strategy).

Excluding Jim Rohrbach’s annual fee increases the terminal value of his strategy to nearly that of the buy-and-hold benchmark. Said differently, Jim Rohrbach’s strategy performance is more attractive compared to buy-and-hold (about the same terminal values with lower volatility) for an initial investment amount much larger than $10,000.

Since cumulative value analyses are sensitive to start and stop dates, we also look at annual returns.

The next chart compares the calendar year returns for Jim Rohrbach’s strategy and the two benchmark strategies, with 2003 a partial year commencing 3/24/03.  Jim Rohrbach’s strategy beats the buy-and-hold (10-month SMA) strategy only in 2008 (2003). Again, the annual fee of $395 materially degrades the performance of Jim Rohrbach’s strategy.

In summary, evidence from straightforward tests on a fairly small sample does not support a belief that Jim Rohrbach’s timing approach (including service fees) beats simple benchmarks.

See Jim Rohrbach’s reactions to the original review (which was substantially different).

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The following chart depicts the behaviors of the S&P 500 Index and WLI over the entire sample period. The shapes are generally similar, but the degree to which WLI leads or lags the stock market is not obvious. Because stock market behavior is an input to the WLI, the two series should display some similarity.

Does the alternative visualization of a scatter plot reveal more about the relationship between WLI and stocks?

The following scatter plot relates the S&P 500 Index to WLI over the entire sample period, with an exponential best-fit curve. The data indicate that the two series generally move up and down together. However, clustering of data suggests that the relationship sometimes changes (switches regimes).

Where might the regimes be within the sample period?

The next scatter plot breaks the stocks-WLI relationship into four regimes: (1) the declining stock market of 3/01-9/02; (2) the advancing stock market of 10/02-10/07; (3) the declining stock market of 11/07-2/09; and, (4) the advancing stock market of 3/09-8/10. There is a positive relationship between the stock market and WLI during both bull and bear regimes, but bull regimes appears to be more conservative (lower stock price for a given level of leading indicators). Possible explanations are:

• When the stock market is advancing (declining), inputs to WLI tend to be optimistic (pessimistic).
• Investors accept a lower (higher) level of risk to the economy when the stock market is advancing (declining) in anticipation of reversal.

The number of data points on this plot and the prior one likely overstates reliability of any inferences, because some inputs to WLI probably do not vary weekly.

For a more sensitive view of the relationship between WLI and stocks, we compare weekly changes.

The next scatter plot relates weekly change in the S&P 500 Index during the week before WLI release to change in WLI across the entire sample period. The Pearson correlation between these two series is 0.30, and the R-squared statistic is 0.09, confirming that WLI and stocks tend to move in the same direction and suggesting that weekly changes in WLI explain 9% of the variation in contemporaneous stock market behavior.

But does WLI lead the stock market, or vice versa?

The next chart shows the Pearson correlations for various WLI-stocks lead-lag relationships, ranging from stocks lead WLI by 13 weeks (-13) to WLI leads stocks by 13 weeks (13). As noted above, the coincident correlation (0 weeks lead-lag) between weekly changes in WLI and weekly changes in the S&P 500 Index is 0.30. There is some indication that stocks may lead WLI by a few weeks, but these indications are all weaker than the coincident correlation. There is no evidence that WLI leads stocks, especially given that ECRI releases WLI preliminary/revised readings with one-week/two-week lags.

To test whether WLI exhibits any cumulative and exploitable predictive power for stocks, we relate weekly change in WLI (as revised) to the change in the S&P 500 Index from initial release to four weeks later, from initial release to 13 weeks later and from initial release to 26 weeks later. The Pearson correlations for these three relationships are 0.05, 0.09 and 0.08, respectively. These relationships are all small but positive, possibly indicative of some momentum. In contrast, the correlation between the weekly change in WLI and the change in the S&P 500 Index during the four weeks prior to WLI release is 0.37, again suggesting that WLI lags rather than leads the stock market.

Might there be some non-linearity in the WLI-stocks relationship that yields useful prediction of stock returns?

The final chart summarizes average weekly S&P 500 Index returns by quintile of change in WLI after both preliminary and final release weeks (assuming revisions in WLI are not material). Lack of any patterns across the range of smallest (most negative) to largest (most positive) changes in WLI again suggest no useful predictive power for stock returns.

In summary, evidence from simple tests suggest that ECRI WLI movements coincide with or slightly trail stock market behavior, offering little or no short-term trading intelligence.

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Each of the Hussman funds “varies its exposure to market fluctuations – from neutral to aggressive – based on the unique return/risk characteristics of each Market Climate”. Performance of a fund with respect to some index, such as the S&P 500 Index, therefore derives from some combination of:

1. Stock selection (including dividends)
2. General policy to hedge exposure of the selected portfolio to market fluctuations
3. Hedging adjustments based on expected returns for varying market climates (market timing)
4. Trading frictions and fund fees

If the hedging adjustments are effective, the fund portfolio should be on average more (less) sensitive to fluctuations of the broad stock market when the market advances (declines).

As a test of the effectiveness of HSGFX hedging adjustments, the following scatter plot relates the adjusted weekly returns of HSGFX to those of the S&P 500 Index separately when the weekly index return is negative (red) and positive (green) over the entire sample period. The chart shows both the equation for the best-fit line and the the R-squared statistic for each of these two scatters. Results show that the exposures of the HSGFX portfolio to market fluctuations in aggregate are very similar when the broad market is falling and rising. In fact, the portfolio is a bit more sensitive to market fluctuations when the market is falling (slope or beta of 16.7% and R-squared of 8.6%) than when it is rising (slope or beta of 15.9% and R-squared of 6.4%). In other words, efforts to time the market by adjusting the level of hedging may be slightly harming rather than enhancing long-run performance.

Note that the y-intercept or alpha is positive (slightly negative) when the broad market is falling (rising), indicating that benefit of stock selection derives mostly from intervals of broad market weakness (portfolio stock holdings are operationally defensive). Splitting the sample period into two approximately equal subperiods indicates that nearly all the alpha of HSGFX comes from the older half and that most of the beta comes from the newer half.

Results are a little different using the Russell 2000 Index as a benchmark, with HSGFX a little more sensitive to Russell 2000 Index fluctuations when the index is rising than when it is falling.

For comparison, we replicate this analysis for a conventional, unhedged mutual fund.

The next scatter plot relates the adjusted weekly returns of Fidelity Magellan (FMAGX) to those of the S&P 500 Index separately when the weekly index return is negative (red) and positive (green) over the entire sample period. Results show that the exposure of the FMAGX portfolio to market fluctuations in aggregate is about the same whether the broad market is rising or falling, and that it is far more sensitive to these fluctuations than is the HSGFX portfolio. The R-squared statistics indicate that broad market fluctuations explain about 90% of the variation in FMAGX returns. Note that the y-intercept or alpha is little different from zero when the market is falling or rising.

For another perspective on the role of hedging, we look at average weekly returns.

The final chart compares average weekly adjusted returns for HSGFX, FMAGX, S&P Depository Receipts (SPY) and the iShares Russell 2000 Index (IWM) when the weekly index return is negative (red), positive (green) and overall (black) for the entire sample period. The chart shows that, while FMAGX closely mimics SPY behavior, HSGFX on average avoids market weakness but also misses most of market strength. When the mix of up and down market intervals tilts substantially toward down (up) intervals, the average HSGFX weekly return tends to beat (lag) that of the broad market.

HSGFX has done relatively well during the 508 weekly returns in the sample, encompassing 268 (52.8%) up weeks and 240 down weeks (47.2%). Note that the hedging approach suppresses portfolio volatility considerably. The standard deviations of weekly returns are 1.2%, 3.0%, 2.7% and 3.4%, respectively, for HSGFX, FMAGX, SPY and IWM over the entire sample period. Metrics that adjust returns for volatility (e.g., Sharpe ratio) therefore substantially favor HSGFX over the index and unhedged mutual funds.

During the older (newer) half of the sample period, the average weekly return for HSGFX is 0.27% (0.03%).

HSGFX is arguably more like a market-neutral equity hedge fund than an equity mutual fund and its appropriate benchmark therefore the average performance of such hedge funds. However, hedge funds are inaccessible to many investors. See “Testing a Market Neutral Equity Mutual Fund” for comparison of HSGFX to an available market neutral mutual fund over a recent period.

In summary, while hedging has generally been advantageous for equity investing over the past decade, evidence from simple tests does not convincingly support a belief that John Hussman successfully times the stock market via hedging adjustments based on his assessments of market valuation and market action.

These findings are consistent with those described in “Hedge Fund Success: Timing or Stock Picking?” and “Mutual Fund Stock Selection vs. Market Timing” that funds generate alpha through stock picking and not market timing.

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The following chart compares the coincident S&P 500 index and net blogger sentiment over the past 217 weeks (there were no surveys for 10 weeks during the sample period). On these visually comparable scales, blogger sentiment is generally more volatile than the stock market. The fairly large week-to-week swings in net blogger sentiment suggest either that the bloggers are very sensitive to changes in market conditions, or that participation in polling varies considerably across weeks. The average net blogger sentiment over the entire sample period is +4%.

For a more precise test of the relationship, we look at poll-to-poll changes in net blogger sentiment versus associated stock market returns.

The following scatter plot relates poll-to-poll changes in net blogger sentiment to weekly changes in the S&P 500 index for concurrent intervals over the entire sample period. If bloggers as a group react to what just happened in the stock market, a best-fit line would run from the lower left to the upper right. Based on 206 poll-to-poll changes, there is some support for this hypothesis. The Pearson correlation for these two series is 0.23. The R-squared statistic for the relationship is 0.05, indicating that the change in the stock market over the past week explains 5% of the change in blogger sentiment during that week.

How well does net blogger sentiment predict future stock returns?

The next scatter plot relates the 4-week future change in the S&P 500 index to net blogger sentiment over the entire sample period.

• If net blogger sentiment forecasts stock market behavior, a best-fit line would run from the lower left to the upper right.
• If net blogger sentiment is a contrary indicator for stock market behavior, a best-fit line would run from the upper left to the lower right.
• If net blogger sentiment does not predict stock market behavior at all, the plot would show no pattern and a best-fit line would be flat.

Based on 203 observations, the data indicate that bloggers in aggregate cannot predict the direction of the stock market. The Pearson correlation for the distribution is -0.17, and the R-squared statistic is 0.03. Blogger sentiment explains (in a contrarian direction) 3% of the variation in stock returns over the next month.

For a different perspective and a test of linearity, we look at returns by quintiles sorted on net blogger sentiment.

The final chart shows the average 4-week future return for the S&P 500 Index by quintile of net blogger sentiment over the entire sample period. Each quintile represents only 40-41 observations. Results generally confirm those from the preceding regression, but the middle quintile makes the relationship unsystematic.

In summary, analysis of Ticker Sense Blogger Sentiment Poll results indicates that aggregate blogger sentiment is perhaps, like many sentiment indicators, somewhat contrarian with respect to future stock market behavior.

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• Based on annual returns, sampled mutual funds beat the S&P 500 Index 38.2% of the time, with the frequency of outperformance varying considerably over time (see the first chart below).
• Over the entire sample there are 454 (3) outperformance streaks of at least five (ten) calendar years. The longest streak is 15 years.
• Actual mutual fund performance exhibits significantly more streaks than generated by “no-skill” simulations (see the second chart below). Moreover, in aggregate, funds with outperformance streaks of at least seven years beat the S&P 500 Index 55.0% of the time, compared to 38.2% for all funds.
• Adding a skill component to the simulation based on a normal distribution of skill with a mean of 0% and a standard deviation of approximately 16% mimics empirical data.
• The empirical basis for the simulation, and the length of the longest outperformance streaks, make it unlikely that fund styles explain the indication of skill.

The following chart, taken from the paper, shows the frequency of outperformance by calendar year for all funds sampled. The average frequency of outperformance across all years is 38.2%. There is considerable variation in outperformance rates over the 1962-2008 sample period.

The next chart, also from the paper, compares the empirical and simulated frequencies of outperformance streaks (log scale) over the entire sample period, with one standard deviation variability ranges for simulation outcomes. It shows that outperformance streaks are systematically more prevalent in actual data than in 10,000 “no-skill” simulation trials. In other words, some factor other than pure luck is material in generating the distribution of outperformance streaks among mutual funds.

In summary, evidence from analysis of market outperformance streaks among actively managed U.S. mutual funds indicates that fund manager skill is material to fund performance.

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The following chart shows the S&P 500 Index average daily returns over the three trading days before (LD-3 to LD-1) and the three trading days after (LD+1 to LD+3) Labor Day for the entire 1950-2009 sample period, with one standard deviation variability ranges. The average daily return for all trading days in the sample is 0.032%. Results suggest some strength on the Friday before Labor Day and no notably abnormal returns the other five trading days. Volatility is relatively high the day after Labor Day.

As usual for daily data, noise generally dominates signal (standard deviations are large compared any indicated abnormalities in daily returns).

To check the stability of the pre-holiday peak, we next look at a “modern” subsample.

The next chart compares S&P 500 Index average daily returns over the three trading days before and after Labor Day in the entire sample to those for a 1990-2009 subsample (20 observations). This chart shows no variability ranges and uses a finer vertical scale than the preceding chart. There are noticeable differences, with the peak shifted from the day before to the day after Labor Day. All days except the day after Labor Day are more negative after 1990.

Volatility is relatively high the day after Labor Day for both the entire sample and the subsample.

Some of the differences between the overall sample and the subsample are noise. Could the shift in peaks relate to the turn- of-the-month effect (whether Labor day comes early or late within its one-week window)?

The final chart compares S&P 500 Index average daily returns over the three trading days before and after Labor Day for the 17 observations when Labor Day occurs earliest (9/1/ or 9/2) and the 17 observations when Labor Day occurs latest (9/6 or 9/7) over the entire 1950-2008 sample period. The patterns are similar. It appears that the turn-of-the-month effect does not substantially influence results.

In summary, best guess is that any anomalous U.S. stock market behavior around Labor Day is strength one trading day before or one trading day after the holiday, with high volatility on the latter day, but noise generally dominates.

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• Gross risk-adjusted (for market, size and book-to-market) returns of the momentum-reversal strategies are significantly higher than those of a traditional momentum strategy (see the chart below). Over the entire sample period, annualized average three-factor gross alphas for the momentum-reversal strategies range as high as 21.1%, compared to 4.8% for the traditional momentum strategy.
• Among the comparison momentum-reversal strategy variations, short comparison intervals are more effective in screening for potential reversals than long comparison intervals.
• The convex-concave strategy screens stocks for potential reversals more precisely than most of the comparison strategy variations.
• Results are generally robust for subperiods, ranges of market capitalization and deletion of low-priced and small stocks.

The following chart, constructed from data in the paper, compares the average monthly three-factor gross alphas of the following strategies over the entire 1965-2009 sample period:

• A traditional momentum strategy which is long (short) the tenth of stocks with the highest (lowest) returns over the past 12 months, with a skip month, reformed monthly.
• A convex-concave momentum-reversal strategy that is long (short) those stocks within the traditional winner (loser) portfolios that exhibit concave (convex) profiles as defined above, reformed monthly.
• A set of comparison momentum-reversal strategies that is long (short) those stocks within the traditional winner (loser) portfolios for which the geometric mean return over the past n<11 months is less (greater) than the geometric mean return over the past 11 months, reformed monthly. For example “Comparison 2″ compares the geometric mean return over the past two months to that over the past 11 months.

Results indicate that both the concave-convex and the comparison momentum-reversal strategies substantially outperform the traditional momentum strategy. Results across the comparison momentum-reversal variations suggest that acceleration of returns in recent months is most indicative of pending reversal.

In summary, evidence indicates that investors may be able to enhance the performance of momentum strategies by using price acceleration metrics to screen out the past winning and losing stocks most susceptible to reversal.

Note that this study does not consider trading frictions that would be incurred in implementing the strategies considered.

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