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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The following chart summarizes average monthly returns and variability of returns for all six combinations of strict semiannual rebalancing at the ends of calendar months (Jan-Jul, Feb-Aug, Mar-Sep, Apr-Oct, May-Nov, Jun-Dec) and for buying and holding either SPY or VBMFX over the entire sample period. We assume that the investor can perform rebalancing calculations just before the close for execution at the close. Because each semiannual combination generates the same number of rebalancing trades (35), we ignore SPY rebalancing frictions (thereby treating SPY buy-and-hold somewhat unfairly). We also ignore any costs of reinvesting dividends.

Results show that 60-40 stocks-bonds portfolios are a compromise between return and variability. The differences among the six rebalancing alternatives are small and, given the modest size and seasonal inconsistencies of the sample, very likely due to random variation rather than reliable seasonal effects.

For a different perspective, we compare cumulative values of $100,000 initial investments.

The next chart compares cumulative values of $100,000 initial investments in eight alternatives:

• Buy and hold SPY.
• Buy and hold VBMFX.
• Form six 60-40 SPY-VBMFX portfolios and rebalance semiannually via the alternatives specified above (all plotted in red).

The six trajectories for the 60-40 portfolios are barely distinguishable, with terminal values ranging from $336,000 (Apr-Nov) to $351,000 (Feb-Aug).

Based on terminal values, two of the six 60-40 portfolios beat SPY buy-and-hold, and all six beat VBMFX buy-and-hold. Across the entire sample period, the 60-40 portfolios have higher values than SPY buy-and-hold only 11% of the time, concentrated in the interval after November 2008. The 60-40 portfolios have higher values than VBMFX buy-and-hold 96% of the time. Including costs of rebalancing would debit the cumulative returns for the 60-40 portfolios.

Cumulative return comparisons are generally sensitive to the starting date (in this case, selected based on inception of SPY).

In summary, evidence from simple tests on recent data of semiannual rebalancing of a 60-40 stocks-bonds portfolio suggests that it makes no difference which months an investor uses to rebalance.

Use of different funds to represent stocks and bonds might produce somewhat different results.

A much longer sample period using stock and bond indexes might produce different results, but estimating realistic costs for forming and maintaining portfolios based on indexes is problematic.

Exhaustive mining that picks the ends of any two months of the year (not necessarily six months apart) might discover a rebalancing combination with somewhat better performance statistics. Exhaustive fitting that picks the ends of any two days of the year for rebalancing would probably discover an even better combination. However, data snooping bias escalates with the number of alternatives considered, and correction for this bias would suppress and could eliminate such improvements.

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The following charts summarize the average return by calendar month, the average monthly return by year of the Presidential term and the average monthly return by year of the decade for all three sets of data. Results across the three samples for calendar year and decade decompositions are not remarkably alike, undermining belief in recurring cycles at annual and decade frequencies. Note that:

• The average monthly returns for the entire samples are 0.66%, 0.53% and 0.41% respectively for the S&P 500 Index, the DJIA and the S&P Composite Index, indicating that market returns do vary considerably over long periods.
• Some large differences exist across the samples, despite the 74% overlap between the S&P 500 Index and DJIA sample periods and 43%/59% overlaps between the S&P Composite Index and the S&P 500 Index / DJIA sample periods.
• The samples are small for inferring a Presidential term cycle, consisting of only 16, 20 and 34 Presidential terms. The underlying political process offers some non-empirical basis for belief in this cycle. However, average monthly returns by term year for the non-overlapping part of the S&P Composite Index sample (1871-1928) do not have a peak in year three.
• The samples are extremely small for inferring a decennial cycle, consisting of only about 7, 9 and 14 decades. Given the variability in the data, these samples are too small to support reliable inference.

Postulating nevertheless that annual seasonal, Presidential term and decennial cycles exist, we proceed using the DJIA data as a compromise between sample length and “modernity,” keeping in mind that an investor operating in real time would know only averages to date. In other words, the data used for forecasting should not include any future actual returns.

To test 3-cycle predictive power, we apply the forecasting method described in the introduction above to the DJIA data to predict DJIA returns by month for January 2001 through July 2010 (115 forecasted months based on at least 867 months of historical data).  Forecast granularity must be monthly to exploit an annual seasonal (calendar month) input. For example, to forecast the return for January 2001, we take an average of: (1) the average return for January through 2000; (2) the average monthly return for Presidential term year one up through 1997; and, (3) the average monthly return for the first year of a decade up through 1991. For subsequent months and years, we extend the historical data to incorporate additional actual returns as realized.

The next chart shows the the three DJIA forecast inputs to be averaged for January 2001 through July 2010. The peaks (valleys) in the calendar month inputs are for December-January (September). The peaks in Presidential term year inputs are for term year three. The peak by year of decade is for 2005.

Next, we average the three inputs to calculate a combined 3-cycle monthly forecast.

The next chart compares the monthly 3-cycle DJIA forecasted returns for January 2001 through July 2010 to actual DJIA returns. Unsurprisingly, actual returns are far more volatile than forecasts calculated with averages. The average forecasted and actual monthly returns are 0.58% and 0.08%, respectively (the 2000s have an historically low average monthly return). The usefulness of the relative highs and lows of the forecast is not clear from visual inspection.

The 3-cycle forecasted return is negative for nine of 115 months. During those nine months, the actual return is positive five times and negative four times. The 3-cycle forecasted return is positive for 106 of 115 months. During those 106 months, the actual return is positive 58 times and negative 48 times.

For precision in measuring forecasting power, we recast the data as a scatter plot.

The following scatter plot relates actual to forecasted DJIA monthly returns for January 2001 through July 2010. The Pearson correlation for the relationship is about 0.08 and the R-squared statistic is 0.01, indicating that forecasts explain only 1% of variation in actual results. This weak result suggests that the 3-cycle combination is not materially predictive at a monthly frequency (and undermines belief in the existence of the cycles).

The R-squared statistics for the annual seasonal, the Presidential term and the decennial cycle DJIA forecast inputs separately for this forecast period are 0.01, 0.00 and 0.00, respectively.

In summary, evidence from simple tests of a forecast that straightforwardly combines historical annual seasonal, Presidential term and decennial cycles in stock market behavior does not support a belief that these frequencies usefully predict stock market returns at a monthly horizon.

Other sets of data and other methods of combining the three data frequencies may produce different results.

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iShares Russell 1000 Value Index (IWD) – large capitalization value stocks.
iShares Russell 1000 Growth Index (IWF) – large capitalization growth stocks.
iShares Russell Midcap Value Index (IWS) – mid-capitalization value stocks.
iShares Russell Midcap Growth Index (IWP) – mid-capitalization growth stocks.
iShares Russell 2000 Value Index (IWN) – small capitalization value stocks.
iShares Russell 2000 Growth Index (IWO) – small capitalization growth stocks.

Using monthly dividend-adjusted closing prices for the style ETFs and S&P Depository Receipts (SPY) over the period 8/01-6/10 (107 months, limited by data for IWS/IWP), we find that:

The following chart summarizes average return by calendar month for the six style ETFs and SPY over the sample period. There are some differences in seasonality among these ETFs. For example, growth beats value in October. Large capitalization underperforms in December. However, the sample period is too short (less than nine years) and/or the performance differences among style ETFs by calendar month generally too small for confident inference.

For another perspective, the following table lists the winning and losing style ETFs by calendar month.

In summary, evidence from very limited data suggests that there may be some systematic differences in seasonality among size and value/growth ETFs, but the combination of small sample size and modest magnitude of differences does not support confident belief.

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Materials Select Sector SPDR (XLB)
Energy Select Sector SPDR (XLE)
Financial Select Sector SPDR (XLF)
Industrial Select Sector SPDR (XLI)
Technology Select Sector SPDR (XLK)
Consumer Staples Select Sector SPDR (XLP)
Utilities Select Sector SPDR (XLU)
Health Care Select Sector SPDR (XLV)
Consumer Discretionary Select SPDR (XLY)

Using monthly adjusted closing prices for these exchange traded funds (ETF) since inception, along with contemporaneous data for Standard & Poor’s Depository Receipts (SPY) as a benchmark, for 12/98-6/10 (139 months), we find that:

The following chart shows the average sector performance and the dispersion of sector performances by calendar month over the sample period. Overall, January, February, June, July and September are weak months and March, April, May, November and December are strong months over the sample period. Sector activity is most dispersed in January, February, April, September and October and most compressed in March, May, June, July and August.

How do average returns break down by sector?

The next three charts show the average returns by calendar month, in groups of three, for the nine sector ETFs for January 1999 through June 2010. Each chart also shows the average returns by month for SPY as a broad market benchmark.

The following table lists the sectors with the two highest and two lowest average returns for each calendar month. XLE, XLF and XLK are ranked 1 or 2 in four months. XLK is ranked 8 or 9 in five months, reflective of the bursting of the Internet technology bubble early in the sample period.

In summary, calendar effects may vary across sector ETFs, but with only 11.5 years of data, these results offer only weak hints for calendar-based sector rotation.

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The following chart presents the raw annual (January through December) returns for the S&P 500 index by year for 1950-2009, with shape/color coding to designate the four years of the presidential term cycle. There are 15-16 observations for each cycle year. Visual inspection suggests that years 3 and 4 may be better than years 1 and 2, and that years 1 and 2 are more variable than years 3 and 4. The two best and two of the three worst years all come from year 2. Year 3 has no negative returns, and only three year 4 observations are negative.

To generalize, we compute average returns and standard deviations of returns by year and overall.

The next chart shows the average annual (January through December) return for the S&P 500 index for 1950-2009 for each year of the presidential term cycle and for all 60 full years in the sample. The small squares mark the averages, and the variability ranges span one standard deviation above and below average. The statistics confirm that years 3 is especially attractive. However, after 2008, year 4 no longer clearly beats years 1 and 2, though it is somewhat less volatile. The subsample size of 15 for each cycle year is very small, as is therefore confidence in the results. In other words, a few very contrary future observations could change the statistics substantially.

Are there any interesting quarterly patterns within the annual statistics?

The next chart decomposes S&P 500 index returns by “political quarter” for January 1950 through April 2010. Political quarters derive from the typical election breakpoint of early November, with political quarters therefore offset from calendar quarters by one month. The chart shows that the best political quarter overall is Nov-Jan (consistent with much other calendar effects research), with an average return of 4.2% across all 60 years. The worst political quarter is Aug-Oct, with an average return of -0.1% across all 60 years.

The strongest returns across the presidential term cycle come from Nov-Jan in year 2 through Feb-Apr in year 3; in fact, these are the only two quarters for which the average return is larger than the standard deviation of returns. Note that year 2 is a congressional election year, so November of year 2 brings some level of affirmation or repudiation to the President’s party. Standard deviations for these political cycle quarters range from 3.6% to 9.1%, with Aug-Oct most volatile and Feb-Apr least volatile. Again, subsamples by political quarter are very small, so confidence in these results is very low.

How do stock returns behave immediately around elections?

The next chart is a close-up of average daily S&P 500 index returns from 21 trading days before through 21 trading days after U.S. national elections for the total sample across 1950-2008 and several subsamples. Results for the total sample (light green line) include variability ranges spanning one standard deviation above and below average. The most consistent feature is a tendency to rally from about one week before election through one day after election, perhaps expressing investor relief that the campaign is winding down and/or reduced uncertainty in which party will prevail. As usual for daily return analysis, variability tends to swamp anomaly.

The average daily return for all days in this interval is 0.10%, about three times the average return for all days since the beginning of 1950.

What is the cumulative effect of these daily returns?

The final chart is a close-up of average cumulative S&P 500 index returns from 21 trading days before through 21 trading days after U.S. national elections for the total sample across 1950-2008 and several subsamples. The roughly 2% rally from one week before election through one day after election is consistent. The average return for all two-month periods during 1950-2008 is about 1.3%.

In summary, there appear to be both long-term and short-term connections between the U.S. national election cycle and stock market performance, with presidential term year 3 (1) the best (worst) and a tendency for a brief election-time rally.

However, the subsamples for presidential term year analysis are very small, so confidence in related tendencies is very low.

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The following chart shows average daily S&P 500 Index returns from five trading days before (days -5 to -1) to five trading days after (days 1 to 5) the end of June over the entire sample period, with one standard deviation variability ranges. Results suggest some anomalous but choppy strength around the mid-year point.

To check the reliability of this pattern, we look at two subsamples.

The next chart compares the average daily S&P 500 Index returns from five trading days before (days -5 to -1) to five trading days after (days 1 to 5) the end of June for several subsamples. This chart has no variability ranges and uses a finer vertical scale than the preceding one. Results provide some support for belief in the pattern noted above, but also a hint that the market is adapting to the pattern by shifting it to the left.

To compare any effect of mid-year to the effect of the July 4th holiday, we re-center daily returns on the holiday (which falls one to three trading days after mid-year).

The final chart shows average daily S&P 500 Index returns from five trading days before (days -5 to -1) to five trading days after (days 1 to 5) the 4th of July over the entire 1950-2009 sample period and several subperiods. Inconsistencies in results across subperiods undermine any belief in anomalous market behavior around the holiday.

As above, average daily returns are mostly small compared to the standard deviations of daily returns.

In summary, best guess for the U.S. stock market is a positive bias focused at the mid-year point and no reliable bias around the 4th of July holiday.

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• Returns across all stocks in the sample tend to advance overnight (0.1% per night) and reverse during the trading day (-0.07% per day).
• The overnight price bump derives mostly from high levels of retail buying at the open among stocks that have recently attracted the attention of individual investors. A subsample of stocks with high retail attention and low institutional ownership has an average overnight return (trading day reversal) in the range 0.2% to 0.3% (-0.2% to -0.4%). This attention effect tends to persist across days.
• The overnight price bump is pronounced for stocks that are difficult to value and costly to arbitrage (illiquid, high short interest). A subsample of such stocks has an average overnight return (trading day reversal) in the range 0.4% to 0.6% (-0.4% to -0.7%).
• Average overnight return and trading day reversal are more than twice as large during months with high versus low aggregate investor sentiment.
• The overnight/trading day pattern holds for all styles of stocks, is somewhat stronger on Mondays, and weakens following decimalization in 2001 and expansion of algorithmic trading in 2005.

The following chart, taken from the paper, plots the average ratio of the midquote at various times during the trading day to the closing midquote for all stocks over the entire sample period. Measurements occur at 5-minute intervals over the first and last half hours of the trading day, and at 30-minute intervals over the rest of the trading day. The degree to which these Intraday Price Ratios fall outside the upper and lower 95% confidence intervals (constructed using the standard error of the time series mean across all days) indicates the statistical strength of the mispricing effect.

The chart shows that mispricing is statistically strong during about the first hour of the trading day.

In summary, evidence from a fairly large recent sample of U.S. stocks indicates that traders may be able to suppress trading friction by systematically executing sales at the open and buys later in the trading day.

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The following chart shows the average daily returns for the S&P 500 index, with one standard deviation variability ranges, from 10 trading days before to 10 trading days after ends of all calendar quarters since 1950 (241 quarters). Day -1 is the last trading day of the quarter. The average daily return for all days in the sample is about 0.03%. The chart suggests systematic strength the first few days after ends of quarters bracketed by weakness or doldrums before and after.

As usual, differences between average daily returns and the average daily return for the entire sample are small compared to daily variabilities.

Does a recent subsample confirm an end-of-quarter pattern?

The next chart compares the average daily returns for the S&P 500 Index from 10 trading days before to 10 trading days after ends of all calendar quarters since 1950 and since 1990 (81 quarters). Note that the scale differs from that above. Again, day -1 is the last day of the quarter. Similarities support some belief in the reliability/persistence of an end-of-quarter pattern, but the similarities are imperfect.

Are results different for different calendar quarters?

The next two charts show the average daily returns for the S&P 500 Index from 10 trading days before to 10 trading days after ends of all four calendar quarters since 1950 (60-61 observations for each quarter) and since 1990 (20-21 observations for each quarter). Again, day -1 is the last day of the quarter.

For the entire sample period, some consistency in results across quarters offers modest support for belief in an end-of-quarter effect. Q4 (end of the year) is least like the others, showing earlier positive returns and weaker subsequent returns.

For the recent subsample, results are so noisy that it is difficult to discern any pattern.

Is the overall end-of-quarter pattern simply a turn-of-the-month (TOTM) effect, or vice versa?

The final chart compares the average daily returns for the S&P 500 Index from five trading days before to five trading days after ends of quarters to those for TOTMs, with (725 months) and without (484 months) ends of quarters. Day -1 is the last trading day of the quarter/month. Evidence suggests that ends of quarters may delay the TOTM effect by one or two days and sharpen it.

In summary, evidence suggests some systematic strength the first few days after ends of quarters bracketed by weakness or doldrums before and after, with effects small compared to daily return variability. The fourth quarter pattern is the strongest and most distinctive.

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The following chart summarizes average S&P 500 index returns over the 11 trading days (about half a month) centered on new moons or on full moons over the entire sample period, during the 1990s and during the 2000s. Results are in rough agreement with the conclusion of the study cited above, with intervals centered on new moons outperforming those centered on full moons. The difference for the 1990s is, however, small.

Can we refine the interval of new moon outperformance?

The next chart compares average S&P 500 index returns over the entire sample period for three intervals relative to new or full moons: (1) the five trading days just before full or new moons; (2) the five trading days centered on new or full moons; and, (3) the five trading days just after new or full moons. Results suggest that the outperformance of intervals around the new moon comes from returns after, rather than before, the new moon.

The standard deviation of 11-day returns over the entire sample period is 2.94% (3.43%) centered on new (full) moons, large compared to the difference in average returns.

Might any lunar effects stem from the waxing or waning of the moon rather than new or full moons?

The next chart compares average S&P 500 index returns for the intervals of waxing and waning between new and full moons over the entire sample period, during the 1990s and during the 2000s. Results consistently indicate that the waxing moon (new-to-full) interval on average outperforms the waning moon (full-to-new) interval.

The standard deviation of returns over the entire sample period is 2.86% (3.24%) for waxing (waning) moons, large compared to the difference in average returns.

Can more granular data help explain why intervals around new moons and during waxing moons outperform those around full moons and during waning moons?

The final charts present the average daily detrended S&P 500 index returns from 11 trading days before new and full moons to 11 trading days after new and full moons. The total interval covered in each chart is roughly a month, but mismatches between lunar and monthly cycles introduce differences between them. We detrend by subtracting the average daily return for the entire sample period from the raw average returns for each trading day in the intervals tested. These charts provide some insight into the less granular results above. However, the lack of systematic variation in daily returns casts doubt on the lunar cycle as the explanation of new-full and wax-wane differences in average returns.

Physicist Charles Pennington employs a quite different approach using a Fourier transform, concluding that lunar cycle effects on SPY are, if they exist, very small.

In summary, evidence from simple tests indicates that the U.S. stock market since 1990 performs better on average around new moons than full moons, and during waxing moons than waning moons. However, the levels of relative outperformance are small compared to market variability, so trading these differences is very risky.

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