N°Years: the number of years of history of a financial instrument.

N°Trades: the number of trades.

N°Trades 1y: the average number of trades that has been done in a calendar year.

Avg Trade Duration (days): it is the average duration, in calendar days, of a single trade.

Total[R]: it is the total return in percentage.

Total profit: it is the total profit, in dollars, starting from a capital of 100.

Avg [R] annualized: it's the average return of the trades, in percentage, annualized.

Avg [R] 1y: it is the average yearly return of the trades.

Stdev [R] 1y: it's the standard deviation of the yearly returns.

RR 1y: the Reward / Risk or RR, is the ratio between the yearly average return and the standard deviation of the yearly returns

Winning % 1y: it is the percentage of the positive yearly returns.

Avg [R]: it is the average return of the trades.

"Stdev [R]: it is the standard deviation of the trades.

RR: Reward / Risk or RR, is the ratio between the average return and the standard deviation.

Winning %: it is the percentage of winning trades.

PF: the Profit Factor, or PF, is an index of the quality of trading, which evaluates, with a number, the relationship between the risks assumed and the results. It is computed by dividing the sum of the profits by the sum of the losses.

Stability: it's the stability of the equity line of the backtest. It can goes from 0% (min stability) to 100% (max stability). An high stability means that the equity have had a steady linear rise over time.

Sharpe: the 'Sharpe ratio' is a metric to evaluate the risk-adjusted return of the anomaly. It indicates how well the anomaly has performed in comparison to a 'Risk-Free' rate of return. It is computed as the ratio between the anomaly yearly excess return over the risk free (US 3m T-Bill rate) and the standard deviation of the yearly anomaly's returns.

Sortino: the Sortino ratio' is a metric to evaluate the risk-adjusted return of the anomaly. It differs from the Sharpe ratio just in the denominator: it only considers the standard deviation of the downside risk, rather than that of the entire (upside + downside) risk.

Avg Dwn: the average drawdown of the equity line of the backtest. The lower is the average drawdown, the closer the equity line has been to its all-time highs.

Max Dwn: it is the maximum drawdown of the equity line.

Max Dwn / Avg [R] 1y: the ratio between the maximum drawdown of the anomaly and its average yearly return. It expresses how many years it could take to recover from a drawdown equal to the maximum historical drawdown.

Z-Score streaks: the Z-Score streaks measures how it is likely that our streaks of trades (consecutive wins and consecutive loss) are random or not. It fluctuates between -3 to +3, but sometimes, can go above and below these levels. A Z-score value of 0 means that we are dealing with completely random results. A positive Z-score means that a profitable position is likely to be followed by a losing one, while a losing one should probably be followed by a winning one, so the probability of long winning and losing streaks is low. Instead, a negative Z-score means that a profitable position is likely to be followed by more profitable positions, and a losing position uses to be followed by more losing positions, it means that winning or losing streaks are probable.
E.g., if the last trade were a winning one, we can expect that the following one will be: (1) if Z-Score is near to +3 ==> losing (2) if Z-Score is near to 0 ==> 50% losing, 50% winning (3) if Z-Score is near to -3 ==> winning.

C-VaR: "the CVaR (or 'Conditional Value at Risk' or 'expected shortfall') is the average of the worst centile of the daily returns. In other words, it is a measure of risk since it is an average of the worst daily historical returns. CVaR is derived by taking the average of the “extreme” losses in the tail of the distribution of possible returns, beyond the value at risk (VaR) cutoff point. C-VaR is used in portfolio optimization for effective risk management.

Excess Metrics (anomalies)

Exc. Avg [R] ann.: it is the difference between the 'average gross return annualized' of the anomaly and the 'other trades' one.

Exc. Avg [R]: it is the difference between the 'average gross return' of the anomaly and the 'other trades' one.

Exc. RR: it's the difference between the 'reward / risk' of the anomaly and the 'other trades' one.

Exc. Winning %: it is the difference between the 'positive percentage' of the anomaly and the 'other trades' one.

Note that:

• Excess metrics are computed on 'other trades'.
• The 'other trades' are trades done in the same instrument but in a different periods than the anomaly one.
• For example, the other trades of "Apple TDW 1" are "Apple TDW 2,3,4,5".
• The 'other trades' are duration-adjusted according to the average trade duration of the anomaly. E.g. in the example, the "Apple TDW 2,3,4,5" returns are divided by 4, to become comparable to "Apple TDW 1" returns.

Excess Metrics (portfolios)

Exc. Avg [R] 1y on Bench.: it's the average yearly return of the portfolio minus the benchmark one.

Exc. Avg [R] 1y on RF: it is the average yearly return of the portfolio minus the Risk Free one.

Exc. RR 1y on Bench.: the yearly Reward / Risk of the portfolio minus the benchmark one.

Winning % 1y on Bench.: the percentage of years in which the return of the portfolio has been higher than the benchmark one.

Winning % 1y on RF: it's the percentage of years in which the return of the portfolio has been higher than the Risk Free one.

Note that:

• Excess metrics are computed 'on benchmark' or 'on Risk-free rate'
• The 'benchmark' is the S&P 500.
• The 'Risk-free rate' is the annualized US 3 months T-Bill rate.

Score and Rating

Score: The 'Rating' is derived from the 'Score'. The p-value 'Score' is the inverse of the p-value (score = 1 / p-value) get from a statistical test, different according to the distribution of returns.

There are two types of scores:

• 'Score on zero': (for 'anomalies' and 'portfolios') measures how much a set of returns is significantly different from a set of returns with zero-mean.
• 'Score on others': (for 'anomalies' only) measures how much a set of returns is significantly different with another set of returns.

Rating: the 'Rating' goes from 0 to 5. The higher the rating, the more the anomaly returns are statistically significantly different from zero. It is derived from the Score, and the Rating is given according to 'Score' clusters.

Rating FC: it's a variation of the Rating, and it is computed as the weighted average of three different 'Ratings' over the same period (IS, OS or ALL); its formula is: 0.4 * 'Rating Net' + 0.4 * 'Rating on Others' + 0.2 * 'Rating Gross'.

Rating FC summary: it is a rating derived from the 'Rating FC - all period' and, in addition, gives a bonus to the anomalies that were able to keep the good In-Sample performance also in the Out-of-Sample period; while vice-versa it gives a malus.

The list of the pages about Anomalies:

1. Anomalies ranking: contains the list of the anomalies and their statistical significativity, both in net and gross returns.

2. Anomaly details: contains the backtest of the anomaly (net and gross equity lines, table with statistics) in the In-sample, out-of-sample and entire period, with distinct statistics for gross and net returns.

3. My Anomalies: contains the anomaly you saved.

4. Customize an Anomaly: starting from an anomaly, you can create and save a variation, by changing SL, TP and/or trading costs.

5. Optimize an anomaly, starting from an anomaly, you can search for the best combination of Stop Loss and Take Profit among different modalities (in percentage, in price distance etc.).

6. Portfolios: it is possible to create portfolios, which are containers of anomalies. In this page you will able to filter for many metrics (Avg Return, Reward / Risk, Stability of equity line etc.) to find the best portfolios of all the web-site users.

7. Portfolio details: to see the backtest of the portfolio, with net equity lines and statistics.

8. Compare Portfolios: to compare the performance and the backtest metrics of two or more portfolios.

9. Portfolio Trading Calendar: to create the trading calendar of one of your portfolio, pair your telegram account to FC bot, in order to receive notifications when your portfolio will open and close a position.

Anomalies

1. Anomalies ranking: contains the list of the anomalies. It is possible to:
   • search for specific instruments, e.g., to find just 'Apple' and 'Amazon' anomalies.
   • filter by instrument info, e.g. by asset class (stock indices, bonds, etc.), sector (financials, energy, etc.), size (large cap, small cap), etc.
   • filter by anomalies' statistics, e.g., net avg annualized return >= 10% and stability of equity line >= 90%.
   • filter for gross and/or net statistic parameters, e.g., avg gross return and/or avg net return. To compute net returns, we used average trading cost for each instrument, considering commission, bid-ask spread and overnight fee.
   • Save template, to save the set of filters you have created, to load it the next time.
   • Save an anomaly in 'My Anomalies', to access them quickly.
   • See anomaly details, to see the backtest of the anomaly, in the 'Anomaly details' page.

2. Anomaly details: This page contains the backtest of the anomaly. it is possible, also from here, to save the anomaly in 'My Anomalies'. The page contains:
   • Equity line chart: with the gross and the net equity lines
   • Table with statistics: a table with many statistics, such as avg return, stdev, Reward/Risk ratio, Profit factor, Sharpe ratio, Sortino ratio, scores of statistical significativity, etc. it is also possible to see distinct statistics for:
     • Gross and/or net: statistics computed using gross and/or net returns.
     • IS, OS, ALL: distinct statistics for the in-sample, out-of-sample and entire periods.
     • Score to others the statistical significativity of the anomaly's returns versus the 'other periods' returns of the same instrument. For example, how much significant is a positive Friday of Gold, with respect to the other weekdays (Mon, Tue, Wed, Thu).
     • Score to zero it says how much we can be confident that the anomaly's returns are statistically different from zero. The 'scores' are computed using statistical test, such as the t-test or non-parametric ones, chosing the most appropriate according to returns distributions.

3. My Anomalies: this page contains the anomaly you saved. It is possible to:
   • Delete an anomaly you previously added.
   • Customize an anomaly, starting from an anomaly, you can set these parameters (SL mode, SL value, TP mode, TP value, commission %, bid-ask spread %, overnight fee %) to see the results and save it, creating a new anomaly, that you can add to one of your portfolio. We applied average trading costs, but you can use a broker which has such different costs for a particular instrument.
   • Optimize an anomaly, finding the best stop loss and take profit, setting the modality of them (in price distance, in percentage, multiple of past average range).
   • Add an anomaly to 'My Portfolio': Add an anomaly to one of your portfolio you created in 'My Portfolios'

4. Customize Anomaly: starting from an anomaly, you can create a variation of that anomaly by changing one or more parameters among these: SL mode, SL value, TP mode, TP value, commission %, bid-ask spread %, overnight fee %) to see the results and decide whether to save it into 'My Anomalies'. To explain the parameters:
   • Stop Loss and Take Profit
     • SL_mode: you can chose among different type of Stop Loss.
       • No Stop Loss
       • In percentage: for example, in a long trade, if the entry price is 100$ and the SL_value is 1%, the stop loss will be set to price 99$.
       • In price distance: for example, in a long trade, if the entry price is 100$ and the SL_value is 1, the stop loss will be set to price 99$.
       • In average daily range multiple: for example, in a long trade, if the entry price is 100$, the SL_value is 1, and the average daily range of that instrument in the last 50 days has been 1$, the stop loss will be set to price 99$. Otherwise, if the SL_value is set to 2, the stop loss will be 98$.
     • SL_value: is the value of the stop loss, according to its mode. Read the examples in the 'SL mode' descriptions to understand its use.
     • TP_mode and TP_value: the same description written in the SL_mode and SL_value description.
   • Trading costs We applied, for each instrument, average trading costs, computed by looking at different tradable instruments in different brokers, and by applying to non-tradable instrument trading costs equal to the ones of similar tradable-instruments. But you may be using a broker that offers an instrument with different trading costs, so you can change the default ones we used to compute 'net returns' from 'gross returns':
     • Commission percentage: it is a commission paid each time a trade is opened. This commission is not paid when the trade is closed.
     • Bid-Ask spread percentage: it a cost given by the level of liquidity of that instrument, if the instrument has an high liquidity, bid-ask spread will be low. For example, an ETF on S&P 500 has a such lower Bid-Ask spread than a small-cap stock. The Bid-Ask spread is deducted from each trade performance.
     • Overnight fee percentage (in annual terms):
       • Spot account: If you operate using a spot account, set the overnight daily fee percentage to zero, because you don't have to pay nothing to keep the position open overnight.
       • Explanation: it is a fee deducted each day that the position is kept open. We added this fee since many traders operate with leverage account using CFDs, that have this fee. For example, in Metatrader 4, you can find this fee, but expressed in intrument's points and not in percentage, in the 'contract specification', it's the voice 'Swap for long position' or 'Swap for short position'.
       • Why are there different ovn_fees depending on the instrument: the overnight fee is different depending on the instrument you trade. Without entering too much into details and to give a simplistic hint to understand the point, the factor that impacts the most in the overnight fee is the currency in which the instrument is quoted. For example, if you buy S&P 500 which is expressed in USD you will have an annual overnight cost of about 3%/4%, because the annual US interest rates are about 3%. While, if you would buy an 'Argentina stock index', and Argentine peso interest rates are around 20%, you will have to pay an annual overnight cost of about 20%/30%. Our input paramenter is the 'annual ovn_fee percentage'. The formula to get the daily_ovn_fee_percentage from the annual one, is: Daily_ovn_fee_perc = (1 + annual_ovn_fee_perc) ^(1/365) - 1

5. Optimize an anomaly, this section is useful to find the best stop loss and take profit for an anomaly. The stop loss and take profit are computed using gross returns. You can't save an anomaly from year, but after you have found the best combination of SL and TP, you can go to 'Customize Anomaly' page, inserting these SL and TP and then save the anomaly. In these page you can:
   • Optimize SL or TP: you can optimize SL and TP one by one, choosing a maximum of 10 values of optimization. For example, you optimize SL, you choose SL_mode = 'in percentage' and as SL_value (start = 1%, step = 1%, stop = 10%), meaning that it will try 10 combinations of SL (1%, 2%, 3%, ...10%). To know more about 'SL and TP_mode and SL and TP_value', read the description of 'Customize anomaly'.
   • Chose the optimization period: it is possible to choose the years of optimization, instead of the entire history of the instrument e.g. from 2012 to 2020.
   • Lauch the optimization: by clicking a button to launch the computation, it may require a minute, especially if there are many years of history.
   • See the results: when the computations are completed you can see the results by looking at two charts and one table:
     • 1st chart: equity lines to see, one by one, if the optimized anomaly is better than the base anomaly (without SL and TP). You can switch the optimization value and see its equity line, e.g. from the equity line with SL 1%, to the equity line with SL 2%.
     • 2nd chart: statistical metrics, you can choose one metric among a set of ones (Tot[R], Avg[R], Reward/Risk, Profit factor, etc.) to see which is the best optimized value of SL or TP. E.g., you find that the best TP is 5% because it has the highest average return and the nearest values (4% and 6%) are the second and third best TPs.
     • Statistical table: each row of this table refers to an optimized value and its backtest metrics (Avg[R], RR, Pos%, PF, etc.). E.g., you can see that TP of 5% is a very good one because it is the best under most of the metrics (RR, pos%, Avg[R], PF) etc.

NB: We use daily historical data to backtest. We suggest you to choose not a too tight SL and/or TP (especially SL, also for 'stop hunting' phenomena), because if in the same day the instrument hit the SL and the TP (it should be rare), the SL will have the priority.

Portfolios

6. Portfolios this page contains all the portfolios previoulsy created by you and other users of the web-site. It is possible to search and filter them, by then seeing some of their synthetic statistics. You can:
   • Create Ptf, create a new portfolio by choosing its name
   • Rename Ptf, rename a portfolio you previously created
   • Delete Ptf, delete a portfolio you previously created
   • Filter and sort Ptfs, choose the set of filters and sort the portfolios in the page, by one of their key metric (avg 1y return, reward/risk, Sharpe ratio, equity line stability, max drawdown etc)
   • Save the template, save the template of filters. to load it the next time.
   • Go to 'Portfolio details', by clicking in a portfolio, you'll be directed to the 'Portfolio details' page, in which you can change its components (if the portfolio is yours) and lanuch the backtest computations (equity lines and statistics, with respect to benchmark).
   • Go to 'Compare Portfolios': in this page it is possible to compare the performance and the backtest metrics of two or more portfolios, including the benchmark.
   • Go to 'Portfolio Trading Calendar': in this page you can create the trading calendar of one of your portfolio, pair your telegram account to FC bot, in order to receive notifications when your portfolio will open and close a position.

7. Portfolio details: this page contains the components and the backtest of one of your portfolios.
   • Change the components:: If the portfolio is yours you can change the components (a component is an anomaly you added to it), by removing them or changing their 'weight'. For example, if a weight is set to 1, it means that the portfolio will open the position with the entire capital; so, if the performance of the instrument during the trading timespan is 5%, the portfolio will record performance of 5%.
   • Save portfolio and components: if the portfolio is the one of another user, with a click and by choosing a new portfolio name, you can add the portfolio to 'My Portfolios' and all the anomalies that compose it to 'My Anomalies'.
   • Compute portfolio results: launch the computations of the ptf backtest, updating data and creating the equity line chart and the table with statistics.

8. Compare Portfolios: in this page it is possible to compare the performance and the backtest metrics of two or more portfolios, including the benchmark. You can:
   • choose the portfolios you want to compare, by choosing them from the yours and the ones of other users.
   • Insert the starting and ending date of the comparison (e.g, from 2005 to 2018)
   • Click the button to launch the computations to create the comparison.

10. Portfolio Trading Calendar: in this page you can:
    • create the trading calendar of one of your portfolio
    • pair your telegram account to FC bot, in order to receive notifications when your portfolio will open and close a position. One user can pair one telegram account.
    • activate and disactivate the notifications
    • delete the pairing between your telegram account and FC bot
    • delete the trading calendar of your portfolio.

As options do not expire on the same day each month, regular seasonal charts cannot measure this effect, as they do not consider the different days in which occur a holiday (for example Thanksgiving) which occur at different day each year.

In ForecastCycles there two types of OE strategies:

• OPT_WE: Options, Week of Expiration. In the week of the options expiration, it opens the position at Monday opening price and close it at the next Friday closing price.
• OPT_WAE: Options, Week After Expiration. In the week following the options expiration, it opens the position at Monday opening price and close it at the next Friday closing price.

Each stratetegy has two tactics:

• US_Stocks: US stocks options, that expire the 3rd Friday of each month.
• US_Stock_indices: US stock indices options expire the 3rd Friday of each quarter (Mar, Jun, Sep, Dec).

While a lunar year lasts about 354 days, a calendar year lasts 365 days: there are 9 days of difference every year. After just 10 years, the difference would be worth 3 months. So, it is an error to say that periods of MOY and WTM include the lunar cycle ones. More detailed, the moon takes 27.3 days to orbit Earth, but the lunar phase cycle (from new Moon to new Moon) is 29.5 days. The Moon spends the extra 2.2 days "catching up" because Earth travels about 45 million miles around the Sun during the time the Moon completes one orbit around Earth.

The moon cycle can be divided in different phases:

• 4 phases: there are four principal lunar phases:
  • 1/4 - New Moon
  • 2/4 - First Quarter
  • 3/4 - Full Moon
  • 4/4 - Last Quarter (also known as a third or final quarter)
• 8 phases: these eight phases are, in order, new Moon, waxing crescent, first quarter, waxing gibbous, full Moon, waning gibbous, third quarter and waning crescent.
  • 1/8 - New Moon
  • 2/8 - Waxing Crescent
  • 3/8 - First Quarter
  • 4/8 - Waxing Gibbous
  • 5/8 - Full Moon
  • 6/8 - Waning Gibbous
  • 7/8 - Last Quarter
  • 8/8 - Waning Crescent

Countries in the different hemispheres see the Moon from a completely different vantage point from each other:

• in the northern hemisphere the first quarter looks like a growing "D", while in the southern hemisphere it looks like a "C".
• In the northern hemisphere the last quarter looks like a "C", while in the southern hemisphere looks like a "D".

Qadan er al. in their paper “Seasonal patterns and calendar anomalies in the commodity market for natural resources”, starting from a study in which it was found a correlation between lunar cycle, particularly with the full moon periods, and stock returns, they checked the correlation also in commodity market, but without noticing statistically significant results about the full moon effect:

“Yuan et al. (2006) provide evidence that market index returns are correlated with the lunar cycle, such that stock returns are lower on the days near full moons (FM) than on other days.
Floros and Tan (2013) report that this anomaly exists in emerging markets but is weak in the US. Lucey (2010) addresses this effect in precious metals (gold, silver, and platinum) for 1998 to September 2007, and reports that the lunar cycle effect is more pronounced in silver than gold, with very little evidence for an effect in platinum.

To test if such an anomaly also exists in other commodities, we define FM using three different windows of time. The first full moon dummy variable (FM 1) captures the full moon day itself; the second dummy captures a window of (−3, +3) days around full moon days (FM 3), and the last one (FM 7) captures (−7, +7) days around them. The results in show that the precious metals market is efficient regardless of the sample period considered. In fact, in PanelsA (FM1) and C (FM3) the FM i coefficients are generally negative.

Even though they are statistically insignificant, the negative tendency is noticeable. The only exception is copper, which exhibits positive returns for FM 3 and FM 7. There is a clear tendency for less volatility on FM days as evident in the coefficients of FM > 04 and FM < 04 for silver, oil, heating oil and natural gas. Thus, the overall findings suggest that such an anomaly does not exist in precious metals.

Finally, the FM contribution to conditional variance differs between commodities but seems to be associated with less influence for the period before 2004. Anyway, in most cases, FM is not statistically significant in the variance part.”

Another test that could be made is the check whether the distance of the moon to the Earth, which is like an elastic, and its tendency to move closer or away to the Earth are correlated to financial asset returns.

Literature review

In his research of 2014 “Why don’t you trade only four days a year? An empirical study into the abnormal returns of quarters first trading day”, Gil Cohen found a very brief period in which returns of S&P 500 tend to concentrate, the first trading day of each quarter:

“In this research I have called attention to a calendar anomaly that occurs at the beginning of each quarter. By calculating daily and annual returns for the S&P500 over 34 years, I have shown the existence of a First Day-of-Quarter (FDQ) effect. This anomaly becomes statistically significant beginning in the year 2000.

By trading only four days a year from the beginning of 2000 until the end of 2013, an investor could have gained 113.1% return, while exposing himself to stock risk only for 56 days. Moreover, for 11 of those 14 years of trading, the FDQ contributed more than 10% to the annual returns. Only for two years (2001 and 2005) the FDQ contributed negatively to the annual returns.

The biggest gainer of the FDQ is the financial sector, which for the last 13 years of investing has been non-productive, showing a −6.12% total return. Investing only at the beginning of each quarter for a total of 52 days would have yielded a return of 40.17%. The next beneficiary of the FDQ is the technological sector. The 82.5% of total return gained in this sector over the last 13 years could have been gained in only 52 days of trading.”

Summarizing, he measured that, from 2000 to 2013, an investor would have earned the 113.1% of return in his portfolio, by investing in S&P 500 just 56 trading days (the FDQ days) out of 3500 trading days (14 years).

In their paper “Seasonal patterns and calendar anomalies in the commodity market for natural resources”, T.Levy and J.Yagil wrote:

“According to Cooper et al. (2006), the other January effect, sometimes known as the “January barometer,” is related to the documented observation that stock market performance in January predicts market returns over the following 11 months. This effect predicts that if the January returns are positive, the returns for the next 11 months will be positive as well, but if January concludes with negative returns, the returns in the following 11 months will be negative as well.

For commodities, the returns for the 11 months following a positive January tend to be positive, but they are statistically insignificant. In addition, the returns for the 11 months following a negative January also are no different from zero. The only significant result is the relatively high number of positive January months for silver (20 vs. only 12 negative Januarys), platinum (23 vs. 9) and palladium (25 positive Januarys vs. only 7 negative Januarys).”

In ForecastCycles we analyze this anomaly through the "Max_Min" strategy. The strategy is built by taking the most important maximum and minimum from the seasonal line, computed over the entire history of the financial instrument.

For example, starting from the detrended seasonality, used to identify better the top and bottoms, our algos identify the most important Max and Min of the seasonality, in this case of Apple Inc.

In this case the indications are:

• Long: from 09.Mar to 09.May
• Short: from 09.May to 03.Jul
• Long: from 03.Jul to 03.Sep
• Short: from 03.Sep to 03.Oct
• Long: from 03.Jul to 05.Nov

Literature review

In their paper, T. Levy and J. Yagil, “The week-of-the-year effect: Evidence from around the globe” they present 2 weeks characterized by an anomaly in return in most of world stock indices, the weeks 43 and 44.

“Using the weekly returns on the stock market indexes of 20 countries worldwide, for a period that ends in December 2010, found that returns in Week 44, which starts on October 29 and ends on November 4, are positive in 19 of the 20 countries, and in 18 of them, it is also statistically significant. In contrast, the returns for Week 43, which starts on October 22 and ends on October 28, are negative in 19 of the 20 countries, and statistically significant for most of the countries.

For Week 44, the mean return across all 20 countries (2.27%) is more than 10 times larger than the corresponding mean for all other weeks (0.21%). In Week 43 the mean return across all countries (-1.34%) is far below the corresponding mean return of all other weeks (0.21%).

We also find that as the distance of the country from the equator increases, the significance level of both Week 43’s negative performance and Week 44’s positive performance increases (in other words, the P-value is lower). This finding seems consistent with both the MTO and SAD anomalies.”

They say that results seem to be consistent with MTO (May to October), TOW (Turn of the Winter) and SAD (Seasonal Affective Disorder) anomalies. Anomalies of weeks 43 and 44 should be explained by the daylight change, which should increase seasonal disorder. It occurs in US every year in a slightly different date, between October 28^th and November 6^th.

The incoherent thing about this explanation is that every year the daylight shift date is different, while the timespan of the weeks n° 43 and n° 44 are the same every year. About SAD disorder, a later sunset could be the cause of the improvement in mood. It should be verified if, taking the exact daylight shift date instead of fixed dates, results may be even more precise and explanatory.

To trade automatically this anomaly, you can purchase the "WOY Expert Advisor". The Expert Advisor needs the "Main EA and libraries" to work. If you have an active ForecastCycles membership, contact us to get 20% discount.

For example:

• MOY 1: opens the position at the first trading day of January and closes it at the last trading day of January.
• MOY_multi 1_3: opens the position at the first trading day of January and closes it at the last trading day of March.

Literature review

In literature, research show anomalies in months of the year in multiple markets. Keloharju et al. , to evaluate MOY effect, did a regression on monthly return data from January 1963 to December 2016 of stocks listed on the NYSE, Amex and Nasdaq. By looking that the same-month coefficients they state that the same-month stock return is significantly predictive of future stock returns up to 20 years into the future.

The January effect
The January effect is a calendar anomaly reported in the financial markets wherein the returns in the month of January are higher than the returns during any other month of the year. Keim observed that in January tended to concentrate practically the entire excess return generated in the medium term by small caps compared to large stocks. And that the entire stock market, on average in January, has generated performances above the other months of the year.

Numerous studies done in the following years have validated the persistence of the January Effect over time, not only with reference to the US stock market, but also on other world stock markets . Although the presence of the January Effect is unanimously proven, there is no accepted interpretation on its causes.

Siegel, 2010 highlighted how the January Effect has significantly attenuated during the 90s, although it has not disappeared, only to reappear over the last decade.

September effect linked to the school calendar
September is one of the most evidenced months in literature, because it is the unique month in which mostly of the stock market indices have a negative average performance, on average 1.62% lower than other months returns. L.Fang et.al. in their paper “School Holidays and Stock Market Seasonality, linked the September effect to the post-school holidays, having found a strong link with negative equities performance. They have found that for some regions, in which schools start the new year in August, the September effect is shifted to August.

To demonstrate that they firstly found a local bias among investors: institutional investors use to own more local firms because they have, or believe to have, an information advantage on them. There are some States, countries where school does not start on September but starts in August, such as US southern states. So, for stocks headquartered in that southern states not September’s, but August returns are negative. To summarize their findings:

“The September effect is true for all “Western” and developed economies in the Northern Hemisphere and the differences are almost always significant. Averaged across all countries, September returns are 1.62% lower than other months’ returns. We note that August is also usually a low return month. As a comparison, January returns are, on average, 1.45% higher than other months.

Of the 47 countries analyzed, 37 are in the North Hemisphere September column is dominated by negative average returns. In fact, of the 37 Northern Hemisphere countries, 28 have average negative September returns. Looking at the global average, September is also the lowest return month with a negative return of − 0.03%, while all other months have positive averages.

But in US southern states (Georgia, Indiana, Nevada, Oklahoma, Tennessee, and Hawaii), the school holiday ends in early August. For stocks headquartered in these states, we find that it is August, rather than September, that exhibits particularly low returns. Returns during the months after major school holidays are 1.1% lower than other months with a t-stat of 9.61. This effect can be explained by investor inattention during school holidays because traders are more away from the market, either on vacation or in summer homes so they pay less attention to the market and are less likely to trade.

We noticed an investor local bias leaded by the fact that local institutional investors have an information advantage and, as such, their trading plays an important role in incorporating the information of local firms. In addition, we evidenced that low postschool holiday returns are driven by stocks that release negative news, and especially those that also experience light trading, during school holidays, due to investor inattention during school holidays resulting in news, and especially negative news, being incorporated more slowly into prices.

Prior research indicates that prices generally respond more slowly to negative news than positive news. And another discovery is that the postschool holiday effect is stronger among larger stocks and stocks with greater institutional holdings. Finally, we found that high-yield bond returns are 1.83% lower after school holidays than at other times.”

The US dollar pattern

US dollar tends to show weakness at the end of the year and strength at the beginning of the new year. The fact that the trend in the US dollar reverses precisely at the turn of the year already provides a strong hint regarding the reason for this seasonal pattern: it must be related to the end of the calendar year.

The question is: what happens at that point in time? It is the balance sheet date. The weakness in the US dollar at the end of the year is driven by US tax legislation. Many US-based companies save on their tax liabilities by reporting small amounts of cash at the balance sheet date at the end of the year. It can be worthwhile to shift money to the accounts of overseas subsidiaries.

The additional demand for foreign currencies affects the exchange rate, therefore the US dollar typically declines at the end of the year. After the turn of the year, the tide immediately turns because companies transfer sizable amounts back to the US.

In the tables below, there are the most relevant MOY statistics I found in forex crosses, computed respectively in December and in January. I evidenced in the table the rows involving USD: in red the bearish indications for USD and in green the bullish ones.

There are two strong negative evidence of a weak dollar in December with a p-value lower than 0.05 (so a score greater than 20).

In the January month table, there are three evidence of a strong USD. Below I inserted two charts showing the backtest of a speculation in EurUsd every December (left) and in every January (right) from 1971 to 2020.

Equity lines are steadily increasing since 70’s with a slowdown in the more recent years in the December’s equity. But in general, both equity lines have had a great historical stability, meaning that this phenomenon has been continuous during the last 50 years.

Precious metals demand in January

BullionVault is one of the world largest online investment service for physical precious metals. It reported that in this decade it has seen in January, 7 times out of 10 more new bullion investors than the monthly average, and that January was the very best month of the year 5 times.

Precious metals prices used to rise in the New Year: gold, and silver in the last 20 Januarys have both risen 14 times and Platinum 17 times. Gold consumers demand is important for the price dynamic and it use to peak during major countries holidays. China has become the 1^stworld's gold consumer nation and demand in China peaks during the Chinese New Year, that occurs every year in a different date but always between January and February. In the next chart there are the seasonalities of precious metals from 1987 to 2020. I evidenced with a black circle the January’s average upward movement of all the four metals.

All metals have been strong, on average, in the last 34 years from the days before Christmas to mid-February, in correspondence to the Chinese New Year and St. Valentine Day. After that periods metal prices tended to significantly slow down the upward trend decrease.

January, as seen above, has been one of the most bullish months for precious metal prices, and also a bullish month for US dollar. Gold and silver are two assets negatively correlated to the Dollar Index.

The deduction is that precious metal priced in other currencies than USD, like Euro or Chf should have had a greater strength in January. In fact, the charts of the back-tests of XauUsd (left) and XauChf (right) in January confirms that: XauChf has been more bullish and stable than XauUsd.

In the following bullets, there are additional explanation given by some authors to explain the positive January for precious metals, but these explanations have not been confirmed with data:

• In January, investors use to review their portfolio and they rebalance their holdings of bullion, equities, and bonds; in particular, wealth managers and private savers give more focus on potential risks to their money, so they choose to buy a little investment insurance for protection.
• About both silver and platinum, they find most of their demand today from industrial use. Some automakers, chemical plants, pharmaceutical companies, and other manufacturers book their supplies for the coming year in January.

To trade automatically this anomaly, you can purchase the "MOY Expert Advisor". The Expert Advisor needs the "Main EA and libraries" to work. If you have an active ForecastCycles membership, contact us to get 20% discount.

In ForecastCycles potential TDW anomalies are investigated through three kind of strategies:

• TDW: a strategy of this family enters at the opening price of a weekday and closes the position at the closing price of the same weekday. For example, if the TDW tactic is "1", the asset is bought at the opening price of Monday and sold at the closing price of the same Monday.
• TDW_ovn: a strategy of this family enters at the opening price of a weekday, keeps the position overnight, and closes the position at the opening price of the following trading day (Tuesday, if the market is open). For example, if the tactic is "1", the asset is bought at the opening price of Monday and sold at the opening price of Tuesday.
• TDW_multi: a strategy of this family enters at the opening price of a weekday, keeps the position unitl the closing price of another weekday. For example, if the tactic is "123", the asset is bought at the opening price of Monday and sold at the closing price of Wednesday.

In short term trading, commission and spread have a heavy impact, causing in most cases positive gross returns but negative net returns. The unique instruments that can be used for short term trading in my opinion are the ones with very low spread and commissions, two of them are:

• Futures, largely traded by banks with a commission of a half of a basis point; retail traders usually trade the mini-contract version of these futures or trade the futures through CFD, instruments that replicates futures.
• CFD, they are derivatives built over single stocks, equity indices, forex pairs and commodities. Many retail brokers offer zero commission costs, the unique costs are the Bid-Ask spread and the overnight fee, making them very useful for TDW strategies.

Literature review

In literature the day of the week effect is called “DOW” and less often TDW (“Trading Day of the Week). Personally, to avoid the confusion with “the Dow”, the Dow Jones Industrial Average I prefer to call them “TDW”. Citing Birru from its paper "Day of the week and the cross-section of returns“, the week is a source of temporal organization and strongly influences the organization and structure of our activities:

The week it is not associated with environmental factors in the same way as the month of the year is. For instance, weekends aren’t associated with more sunshine than weekdays. Rather, mood fluctuations across days of the week result from lifestyle and sociocultural factors.

The week is the source of much temporal organization and strongly influences the organization and structure of our activities. Consistent with this, day-of-the-week variation in mood is more strongly exhibited among people who are not retired (Stone, Schneider, and Harter, 2012), is stronger among full-time workers than part-time workers (Helliwell and Wang, 2015), and is stronger among employed than unemployed (Young and Lim, 2014).

The most documented effect in literature about TDW anomalies is the “Weekend Effect”, a phenomenon in financial markets in which stock returns on Mondays are often significantly lower than those of the immediately preceding Friday.

This fact is anomalous for some authors because for example, in a historical appreciating instrument like an US stock index, Monday should have greater return because the closing price of Monday should incorporate the returns of 3 days: Saturday, Sunday, and Monday itself. The logic is that even if the exchanges are closed, there should be an increased demand of the asset in these days that should be incorporated in Monday’s return.

The other two effects about TDW anomalies are Monday effect and Friday effect. Literature, until 90’s, documented US stock indices Monday’s average return significantly negative and Friday average return significantly positive with respect to the other TDW returns. The explanation can be linked to the mood of the investor, before the weekend they tend to feel a better mood, so they are more inclined to risk. On Monday, when they come back to work, they feel a bit more depressed and less aware on what happened during the weekend, especially if some negative news has been announced, so they are more averse to risk.

After 90’s for Monday and 2000 for Friday returns, the day of week anomaly in S&P 500 vanished. Other studies on mood fluctuation during the week have been made. In a 2011 study, Golder and Macy, assess variation in mood by using a sample of 2.4 million individuals making over 500 million tweets from February 2008 through January 2010. Their analysis again confirms that mood is higher on Friday than it is on Monday through Thursday. Their analysis has many advantages over previous studies.

Golder and Macy (“Diurnal and Seasonal Mood Vary with Work, Sleep, and Daylength Across Diverse Cultures”, 2011) used textual analysis of Twitter data to identify average mood across each hour of the day for each day of the week, measuring mood through Linguistic Inquiry and Word Count. The average Twitter user appear to be representative of the typical stock market participant since Twitter users are more adult people than average. Focusing on the average mood, measured from 3pm to 4pm of NY, to capture the mood at the daily close of the market, and consistent with past findings, the level of positive mood is the lowest on Monday and highest on Friday.

Chinese Monday returns
Alia and Ülküb (“Another Look at Calendar Anomalies”, 2019) found that Monday effect is different in China, instead of being negative it is positive, especially regarding small-cap stocks:

“In contrast to the well-known pattern of negative/low Monday returns, Chinese stocks earn positive/higher returns early in the week, which are more pronounced in small-cap, less-profitable (speculative) stocks, and stocks with lack of institutional investor domination. As in US there is on Monday a higher volatility.”

Speculative stocks
Birru (“Day of the week and the cross-section of returns“, 2018), using US stocks data from July of 1963 to December of 2013, found that speculative stocks earn low return on Mondays and high return on Fridays. These findings are attributable to the variation in mood across days of the week, with a decreased mood on Mondays, that consequently leads to a relatively low returns for speculative stocks, and an increased mood on Fridays that leads to relatively high returns for speculative stocks.

He defines a speculative portfolio as “the subset of stocks predicted to be most strongly affected by investor sentiment: small, young, high volatility, unprofitable, non- dividend paying, lottery-like, distressed, extreme growth, low-priced, lottery-like".

Intraday vs Overnight returns
Birru (“Day of the week and the cross-section of returns“, 2018), using data of US stocks (NYSE, AMEX, Nasdaq), from Jul.1992 to Dec.2013, computed the intraday and overnight returns: intraday returns are calculated using the opent and closet prices, while overnight returns are calculated using the closet and the opent+1.

He found that there is a significant anomaly regarding Monday and Friday returns attributable to mood variation. Instead, he showed that there is not a significant variation in the overnight returns, although most of the firm-specific news is released outside of trading hours. In addition, he found that the anomalies tend to be strongest in speculative stocks, recognizable also by low institutional ownership. He states that all the variation on Monday and Friday anomaly returns occurs intraday.

“The intraday analysis shows that during the day, the difference in strategy returns between Friday and Monday is statistically significant at the 1% level. There is no day-of-the-week variation in anomaly returns for the overnight period, as the difference in anomaly returns between Friday and Monday is small and typically in the opposite direction as the pattern observed for the close-to-close returns. [..]

The cross-sectional variation is strongest for firms with low institutional ownership. The evidence is consistent with an explanation in which speculative stocks experience increases in stock price concurrent with increases in sentiment (Fridays) and decreases in stock price concurrent with decreases in sentiment (Mondays).

Borowski and Lukasik (“Analysis of Selected Seasonality Effects in the Following Metal Markets: Gold, Silver, Platinum, Palladium and Copper”, 2017) found statistical significance on gold and copper in the overnight return between the closing price of Friday and the opening price of Monday. They analyzed daily data from 1995 to 2015, spot prices in USD. The first value in the cell represents statistic test for z and the second is the p-value.

Copper is statistically significant on 0.05 confidence level (p-value of 0.033), while gold on 0.002 level (p-value of 0.00138). That means that the chances for their return to be equal to zero in this timespan are respectively equal to 1/30 (copper) and 1/878 (gold).

Backtest of TDW anomalies

This is a backtest I have made by choosing a set of TDW strategies on futures, having a good in-sample backtest. The criterion to choose these strategy is the in-sample period is the p-value score greater than 10, meaning meaning a p-value lower than 0.10.

The in-sample period goes from 1963 to 2010, while the out-of sample period goes from 2011 to 2021. The vertical line in the chart delineates the start of the out-of-sample period.

The gold line is the portfolio composed by all the TDW and TDW_ovn strategies (9 strategies), while the red line is the equitty composed by the best strategy for each weekday (5 strategies). The black line is the S&P 500, used as benchmark.

Even if the log-scale hide them, TDW strategies have had a large max drawdown (-69% and -64%). It is also true that they are not far to the max drawdown of the S&P (-60%). Almost every other statistic of the two TDW portfolio is greater than the benchmark’s: Sharpe ratio is around 3 times bigger, Sortino is almost 10 times bigger, due to very few years having a negative return. It is important to remark that these strategies can only work if trading costs are low, otherwise they would overwhelm all the positive performances.

To trade automatically this anomaly, you can purchase the "TDW Expert Advisor". The Expert Advisor needs the "Main EA and libraries" to work. If you have an active ForecastCycles membership, contact us to get 20% discount.

In ForecastCycles, holiday's effect is studied through 3 different families of strategies:

• HOL "A": (After H) a strategy of this family enters N days after the holiday and exit M days after. For example, HOL_A_1_3, enters at the opening price of the 1st trading day after the holiday and closes the position at the closing price of the 3rd trading day after the holiday.
• HOL "B": (Before H) a strategy of this family enters N days before the holiday and close M days before the holiday. For example, HOL_B_3_1, enters at the opening price of the 3rd trading day before an holiday and closes the trade at the closing price of the trading day before the holiday.
• HOL "C": (Cross H) a strategy of this family operates across holidays, it enters N days before the holiday and exits M days after the holiday. For example, HOL_C_3_3, enters at the opening price of the 3rd trading day before the holiday and closes the position at the closing price of the 3rd trading day after the holiday.

Each finiancial instrument, in general, has its own set of holidays according to its country. There are particular rules for forex and commodities.

• Stocks, country stock indices, government bonds and ETFs have holidays of the country to which they belong, for example S&P 500 has the United States holidays.
• European stock indices have the most common european holidays.
• Forex pairs have the holidays of two countries, for example AudJpy has the holidays of Australia and Japan.
• commodities have the holidays of United States.

Literature review

Qadan et al. studied in their paper “Seasonal patterns and calendar anomalies in the commodity market for natural resources” the holidays effect in commodities, and they found significant results for several commodities in different holidays. Around Christmas days, they found a significant increase in commodity prices, more prominent after 2004:

“One of the most compelling pieces of evidence for the existence of market inefficiency is the seasonality of returns around Christmas. As results indicates, commodities follow this trend and in fact confirm the hypothesis that the positive sentiment associated with Christmas will be accompanied by an increase in commodity prices. Note that the stock market is closed on Christmas itself, so the response in returns is captured on the first trading day after Christmas.

In fact, during the full sample period, each of the metals studied here (excluding natural gas) is accompanied by positive returns on the first trading day after Christmas. Looking at each sub-period, one can see that the positive sentiment that prevails around holidays (Qadan and Aharon, 2018) is more prominent in the period of the financialization of commodities.

The sign test results reported provide strong support for our story as evidenced in the large ratios reported on the days before and after Christmas. These findings suggest that the precious metals market is generally inefficient because it consistently yields significant positive returns around this holiday period.”

They get significant results also around New Year's Day:

“The findings also demonstrate positive returns around New Year's Day. Investors' anticipation of a fresh start to the year may be associated with optimism reflected in the positive returns. In the total sample period (P1 +P2), all commodities, but the natural gas, are associated with positive returns. Gold, palladium, platinum, and oil are statistically significant.

As the significance level of the abnormal returns indicates, these findings are more evident in the financialization period (P2) than the prior period (P1). Though we might expect that the financialization period would be characterized by more efficiency (i.e., insignificant returns), the empirical evidence here shows the opposite and contradicts the efficient market hypothesis.

They indicate with P1 the “pre-financialization period”, which starts in 1896 and ends in 2003. While they call P2 the “financialization period”, which starts in 2004 and ends in 2018, the current year when they did the research. They argue that contrary to literature and logic expectations, the financialization period has generally increased anomalies in the commodity markets and reduced their efficiency. They attribute this “paradox” to the increased liquidity that permits more volatility, which sometimes assume a defined direction:

“During the financialization of commodities era many international portfolio and hedge fund managers, as well as retail investors, increased their exposure to commodities. […] Our results are robust for different estimation procedures and subsamples. Though we might expect that the financialization period would be characterized by more efficiency (i.e., insignificant returns), the empirical evidence here shows the opposite and contradicts the efficient market hypothesis.

The literature argues that the more liquid the market the more efficient it should be. However, one possible “dark side” of liquidity is that it permits inefficiency, which is evident in the returns on metals regarding calendar anomalies. In the same way, sometimes illiquid securities might not appear to be volatile because there are no transactions that can express the true degree of their volatility. In contrast, liquidity does reveal the true volatility of a stock. Similarly, the financialization period may act as an “alignment mechanism” (Aharon et al., 2017) that allows the appearance of abnormal returns around New Year's Day.”

Fang et al., in their paper “School Holidays and Stock Market Seasonality", investigated US stock market indices around labor market day, noticing that the days after this holiday (the first Monday in September), coincide for most of the people with the end of the summer and “the start of a new season”: the return to the normal working and schooling life. After this holiday, it has started most of the times a period of significantly negative returns on US stock market indices:

“In the United States, Labor Day (the first Monday in September) is typically associated with the formal conclusion of the summer season, especially in the northeastern part of the country. Many public schools and universities start the new academic year after this school holiday. If the weak market-level return we find is related to school holidays, we should see that the so-called September effect in the United States is concentrated in the days after Labor Day.

To test this, for each year from 1962 to 2012, we identified the exact date of Labor Day. We split the September dates into before and after Labor Day, and test that the weak September return is concentrated after Labor Day. Univariate results confirm our conjecture. In fact, the average daily returns for the days before Labor Day are positive. The average daily returns for the days after Labor Day are mostly negative and the differences are significant. Thus, not all September days are equal when it comes to returns. The weak returns are concentrated after Labor Day.”

To trade automatically this anomaly, you can purchase the "HOL Expert Advisor". The Expert Advisor needs the "Main EA and libraries" to work. If you have an active ForecastCycles membership, contact us to get 20% discount.

A month can be divided in multiple parts:

2 parts
The month is divided in 2 parts:

• the 1st part goes from the 1st day of month to the 15th.
• the 2nd goes from the 16th day to the end of the month.

The Ariel's 1987 study covers the period from 1963 to 1981 of Dow Jones Industrial Average and consists of dividing the months into two parts: the results showed that in the second half of the month returns were negative, and all monthly profit was derived from the first 15 days. He noticed that returns of the Dow Jones Industrial Average of the first half of the month were, on average, nine times higher than the returns produced in the second half of the month.

3 parts
The month is divided in 3 parts:

• the 1st part goes from the 1st day of month to the 10th.
• the 2nd goes from the 11th day to the 20th.
• the 3rd goes from the 21th day to the end of the month.

In the paper by M.Qadan et.al “Seasonal patterns and calendar anomalies in the commodity market for natural resources”, they present studies on the commodity markets and interesting results:

“…dividing the whole month into 3 parts results in abnormal returns in the first part, but a decreasing trend in the second, and in third part returns are either very low or negative. Results demonstrating that the time-of-the-month effect does exist mainly in copper, gold, silver, zinc and oil. According to these findings, the coefficients of the returns in the second half of each month are positive and significant mainly for the period after 2004.”

4 parts or Weeks of the Month
In this case the month is divided in 4 parts:

• the 1st part goes from 1st day of the month to the 7th,
• the 2nd goes from 8th to 15th day,
• the 3rd from 16thto 23rd,
• and finally, the 4th from 24th day to the end of the month.

30 parts (DOM) and 23 parts (TDOM)
The month can be divided in units, according to calendar days (DOM), or trading days (TDOM).

Such anomalies seem to be explained, at least partially, with the irregularity of private capital inflows on the stock market.

WTM returns of S&P 500 and Treasury Note future

In the following table there are the returns in the different monthly parts of the S&P 500 (from 1928 to 2020) and Treasury Note future (from 1982 to 2020).

About S&P 500 returns, they have been a way better in the 1st half of the month: 0.49% vs 0.14% of the second half, but the most significant indication can be noticed by dividing the month in 4 parts: more than two third of the monthly return is concentrated in the 1st quarter of the month: 0.44% vs a mean quarter return of 0.15%.

While, about TNote returns, the average return of the 2nd half of the month has been the double (0.24%) than the one of the 1st half (0.12%); and the best quarters of TNote have been the 2nd (0.14%) and the 4th (0.18%), in which returns have been more than four times greater than the quarters n° 1 and 3.

To trade automatically this anomaly, you can purchase the "WTM Expert Advisor". The Expert Advisor needs the "Main EA and libraries" to work. If you have an active ForecastCycles membership, contact us to get 20% discount.

In the first version of Forecastcycles, it was just the investigation of one calendar anomaly: the Month Of the Year (MOY). In the new version there will be investigated over 10 calendar anomalies (MOY, Weekday effect, Halloween effect, Holidays effect etc.).

But what is an anomaly? An anomaly is a behavior of a financial market that contradicts the efficient market hypotesis (EMH). According to EMH:

• The operators are perfectly rational, and their behavior can be represented by the model of maximization of expected utility.
• Historical price data is already priced in and cannot be used to make profits. Technical Analysis (TA) and Time Series Analysis (TSA) do not work.
• It should not be possible to find elements of regularity in the behavior of financial assets, and financial assets follow paths of pure randomness.
• If elements of statistical regularity are found, the action of the arbitrageurs should lead in a short time to eliminate them, making it impossible to benefit in terms of profits.

So, according to the prevailing theory, it would not possible to identify and profit from a specific period of a financial market because every period should be equal to all other one.

From 1990's there has been a rise in the importance of the Behavioral Finance theories, that oppose to the EMH and random walk theories. According to BF theories:

• The model of maximization of expected utility, fails to correctly portray the real behavior of economic operators. Operators are not perfectly rational, because are conditioned or rather limited in making perfectly rational decisions by the interaction of several disturbing factors (limited time, too complex problems, etc.)
• The 2002 Nobel Prize for economics Daniel Kahneman is the definitive affirmation of the discipline and the acceptance of the presence of "anomalies“ in the behavior of financial markets, attributable to the absence of perfect rationality in making investment decisions and the inability of market prices to react instantly to the dissemination of new information.

In literature, there are different families of market anomalies:

• Anomalies related to fundamentals (PE, DY, PBV).
• Anomalies linked to past-price performance or ‘technical anomalies’ (momentum/trend following)
• Anomalies attributable to seasonality:
  • Calendar anomalies: anomalies linked to specific calendar days or periods
  • Weather related anomalies: anomalies linked to climate.

Additional concepts linked to anomalies:

• Not all the anomalies are profitable: once the existence of an anomaly has been verified, it is necessary to analyze whether it can be exploited to set arbitrage strategies, aimed at profiting from the presence of such irregularities, by deducting trading costs.
• An anomaly is better if there is a reason behind it. And a trader / investor who trade an anomaly should understand the underlying causes and monitor them, to adapt to any changes.
• It would be better to integrate calendar anomalies to other kinds of elements, for example with the expansions / recession's phases of economy and the phase of the long-term Kondratieff cycle.

In literature, two of the main explanation about the causes of the anomalies are:

• Change in human mood and then in propension to risk due to event such as a holiday.
• The irregularity of the capital inflows and outflows in a market, for example due to the timing of monthly cash flows received by pension funds, which are reinvested in the stock market.
  There are contradictions in literature about the existence of anomalies, even of the same one in the same market, and this could be one of the cause of their persistence.

In ForecastCycles there are the following strategies to investigate a potential TOM anomaly in a market:

• TOM FNM: it opens the position at the opening price of the 1st trading day of the new month.
• TOM LPM: it opens the position at the opening price of the last trading day of the ending month.
• TOM L4PM: it opens the position at the opening price of the fourth to last trading day of the ending month.

All the strategies close the position at the closing price of the 3rd trading day of the new month.

Literature review

Starting from 1987 Ariel’s work, in which he showed that Dow Jones returns in the 1^stpart of the month are significantly higher than the ones of the 2^ndpart of the month, Lakonishok and Smidt (1987) noted that the performance in the 4 days around the end of the month was 0.473%, while the average of returns over a period of 4 days was 0.0612%. Results can be subsequently filtered to investigate for the “Turn of the Quarter” (or “End of the Quarter”) effect. The "end of quarter effect" refers to the end of each quarter: in March, June, September, and December.

Although the turn of the month is a simple anomaly, it is a big challenge for the academic world to explain the potential reasons for the functionality. The turn-of-the month effect is not concentrated at calendar-year quarter-ends. The reason for functionality also is not a risk-based; the paper has explored whether higher “risk” at the turn-of-the-month can explain this pattern. Using the standard deviation of returns as a measure of risk it was found that risk is not higher during the four turn-of-the-month days than over the other 16 trading days of the month. This implicates that higher risk does not explain the turn-of-the-month effect. Moreover, also a systematic monthly shift in interest rates does not appear to explain the turn-of-the-month pattern in equity returns. Interestingly, the turn-of-the-month effect occurs in 30 different markets, so they concluded that the effect is not due to a factor unique to the U.S. market structure.

On the other hand, Ogden (1990) proposed that the turn-of-the-month effect is due to a “regularity in payment” dates, at least in the U.S., since investors receive a preponderance of compensation from employment, dividends, and interest at month-ends. Consequently, as investors seek to invest these funds, equity prices are pushed up. Unfortunately, the paper provided tests that reject this hypothesis. The overall problem of finding some reason to functionality is also supported by the work of McConnell and Xu: “Equity Returns at the Turn of the Month“ in which they affirm that:

“This persistent peculiarity in returns remains a puzzle in search of an answer.”

However, the majority of the research ascribe this effect to the timing of monthly cash flows received by pension funds, which are reinvested in the stock market. The end of the month is also a natural point for portfolio/trading models rebalancing both for retail and professional investors. The authors cited above, recommend taking caution to implements this strategy in a portfolio, since calendar effects tend to vanish or rotate to different days in a month.

The "end of quarter" periods are considered very important for investors because different data are published regarding various market indicators. It is now widespread the belief that many investment funds and financial companies review the composition of their portfolio at the end of each quarter to change their strategy or to set new goals to adapt their portfolio to the events of the quarter, or to try to get better performance.

A traditional rebalancing that many managers do, is to sell assets that have given good returns during the quarter and buy assets with low performance (losers), with the aim of taking profits on the winning positions and allocating part of these to the loosing stocks, with the hope (supported by mean-reversion studies) that they can turn bullish.

TOM strategy applied on S&P 500

Following, the average return of Forecastcycles TOM strategy on S&P 500. The strategy is the "TOM LPM", it buys at the opening price of the last trading day of the ending month, and sell at the closing price of the third trading day of the new month. It keeps the position open 4 trading days each month.

It can be noticed that the average gross return of the strategy from 1928 to 2021 (0.48%), has been 4.36 times greater than the average return of S&P 500 in 4 trading days (0.11%).

To trade automatically this anomaly, you can purchase the "TOM Expert Advisor". The Expert Advisor needs the "Main EA and libraries" to work. If you have an active ForecastCycles membership, contact us to get 20% discount.

The list of the anomalies available on Forecastcycles:

• Halloween effect (HAL and HAL_others): also known as “Sell in May and go away” is a market-timing strategy based on the hypothesis that stocks perform better between Nov. 1st (Halloween) and Apr. 30th, than from May to October.
• Turn of the Month effect (TOM): Lakonishok and Smidt (1987) noted that the S&P performance in the 4 days around the end of the month was 0.473%, which is 7.7 times bigger than the average of returns over a period of 4 days of S&P.
• Within the Month (WTM): A behavior of a financial instrument is also studied ‘within the month’, by splitting of the month in multiple parts and measuring the returns in each one. A possible explanation of this anomaly (and not only) is the irregularity of capital inflows in a market which affect asset returns.
• Holiday’s effect (HOL): Interesting periods to be studied are the days that precede and follow public holidays, such as Christmas and New Year. In fact, days preceding holidays are generally associated to a positive mood, which cause a greater propension to risk. Instead, the days following holidays are generally associated to a negative mood and so more aversion to take risks.
• Day of the Week effect (TDW, TDW_ovn and TDW_multi): The discretion of whether buy or sell an instrument will be purely given by the day of the week (Monday = 1, Sunday = 7).
• Month of the Year effect (MOY and MOY_multi): these strategies enters in the market at the beginning of a month (1: January, 12: December) and close the position at the end of the same month (MOY) or at the end of a following month (MOY_multi).
• Week of the Year effect (WOY): the calendar year is splitted into 52 weeks and it is analyzed the return in each week.
• Within the Year effect (WTY or Max-Min): there can be a positive period of a year that goes beyond a period of one single month. Every financial instrument can have a different timespan in which its returns are significantly positive or negative. For example, Nikkei can be strong from half of March to the end of June while Copper can be weak from the start of October to the half of December.
• January Barometer (JB): Some research argues, regarding S&P 500, that a positive January is usually associated to a positive year, instead a negative January is a predictor of a negative year.
• First Day of the Quarter effect (FDQ): these strategies enters in the market at the beginning of each quarter: the 1st of Jan, Mar, Jul, Sep and keep the position open for N days.
• Lunar Cycle effect (Moon): As moon influences natural events, it can be studied whether moon phases influence human mood and financial markets.
• Option Expiration effect (OE): The options expiration day and the preceding days are important dates for each market as option sellers and buyers strive to push the price in their favor when options expire, which often causes specific price moves.