Impact of Outliers in Algorithmic Trading Strategies

Author: Marcello Chiuminatto
Python version: 3.10.4

Overview

In the context of FOREX trading, in the commitment for finding profitable trading strategies, you will find in books, articles, papers, web sites and maybe from your own inspiration, strategies that shows positive profits, based on cumulative profit over a period. Well, that is not enough unfortunately, because does not necessarily indicate consistency in the profits.

Would be profit average (expected profit) then be a good indicator of profits consistency? The answer is not straight forward: not necessarily. Why? and the answer to this question is the topic of this analysis: Outliers.

Outliers will distort the reading of cumulative profits and expected profits in such a way that a losing strategy, can look like a profitable one.

Consider the following example: Let's define a strategy S, from which we get a sample of 50 trades whose profits are distributed as shown in the next table:

	Trades	Profit/trade[pips]	Total Profit[pips]
	25	-5.0	-125.0
	24	2.0	48.0
	1	200.0	200.0
--------	-------	--------------------	-------------------
Total	50		123.0
mean			2.5
median			-1.5

Total profit and average profit are positive, but clearly this is strongly influenced by the 200 pip trade. Pay close attention to median though, is telling something about this case.

Based on a real but simple trading strategy, it will be shown how outliers impacts mean and total profits, however median signals that something is hidden behind the scenes.

Throughout this article, it will be shown with a real, but simple trading strategy, the danger of not paying attention to outliers and in the process of showing so, it is presented the trading strategy research life cycle: design / back-testing, analysis, and conclusions.

Strategy Design

Definitions and Concepts

This strategy is one of the simplest trend-following or momentum FOREX trading strategy.

For simplicity, this analysis focuses only on buying (long) strategy.

The rationale behind the strategy is the following: “When price “accelerates” enough to the upside it is likely that will remain that way for a while, so is a good signal for opening a buying position and keep it open until market decelerate.”

To have an actionable and testable strategy is necessary to define in mathematical terms the concepts involved in the above statement: acceleration and deceleration. It necessary to remark that these concepts are defined in the context of this trading strategy analysis only and under no way are general concepts that can necessarily be applied to ALL trading strategies or financial markets.

Being that said and reducing the abstraction level a bit, market acceleration is defined as: when prices close above a fast-moving average, which is above a medium-moving average, which at the same time is above a slow-moving average.

Market deceleration is defined as: when the fast-moving average cross below the medium-moving average.

What is a Moving Average (MA) of $n$ periods?

It is the average of the last $n$ available price periods. Generally, is calculated on close price (which is the case for this research), but can also be calculated on open, high, low, typical price etc.)

Mathematically:

$MA(close;n)=\frac{1}{n}\sum_{i=t-n+1}^{i=0} close_i$

Simple moving average on close parametrized with n (periods)

Where:

• $n$ Number of periods
• $close_i$: close price at i-th period

What are fast, medium and slow moving averages?

The concept of speed for a MA's is given by the number of periods the MA is calculated on; less number periods will make the MA to react faster to price action whilst greater number of periods will make it slower.

In this context the periods for the three different MA are defined as follows:

• Fast MA ($MA_f$): 10 periods
• Medium MA ($MA_m$): 25 periods
• Slow MA ($MA_s$): 50 periods

Long entry/exit rules

Now it is time to concretely define the strategy Entry/Exit/Management rules.

Entry

Enter long when close cross above $SMA_f$ and $SMA_f$ is above $SMA_m$ and $SMA_m$ is above $SMA_s$

Mathematically:

Enter long when the following two conditions are true

1. $close_{t-1}<SMA_f \land close_t > SMA_f$
2. $SMA_s < SMA_m < SMA_f$

Exit

Exit a long position when $close_i$ cross below $SMA_m$

Mathematically

Exit when

1. $close_i < SMA_m < close_{i-1}$

Strategy Back-Testing

Strategy back-testing is in essence simulating the strategy logic on historic data, expecting that past performance will repeat in the feature. But this is not necessarily the case and depends a lot on how the back-testing process is designed and executed, among other conditions. Detailed back-testing discussion is out of the scope of this analysis. As a starter you can find more here: https://www.investopedia.com/terms/b/backtesting.asp.

What is relevant for this analysis is that back-testing will produce a set of trades that if considered right out of the process they seem to be produced by a profitable strategy, but hypothetically this is not true and the real losing results are hidden by outliers.

In this section are presented some steps to implement a simple vectorized back-testing process. The term vectrorized means that instead of looping throughout the elements of an array or a matrix performing scalar operations, the operations are performed on entire arrays or matrices (vectors). Why to do so? because the reduction of processing time is significant, specially when working with millions or billions of records. You can find more here: https://www.youtube.com/watch?v=BR3Qx9AVHZE

Data

The data for this study is for the instrument (currency cross pair) EURUSD, format OHLC (Open/High/Low/Close) aggregated at a fix frequency of one hour (H1) in the period from 2015-01-01 22:00:00 until 2021-11-24 23:00:00

Sample:

Time (UTC)	Open	High	Low	Close	Volume
2021-11-24 19:00:00	1.11925	1.12034	1.11883	1.11985	3332.690
2021-11-24 20:00:00	1.11984	1.12053	1.11976	1.12050	1500.710
2021-11-24 21:00:00	1.12049	1.12054	1.11962	1.11962	1051.825
2021-11-24 22:00:00	1.11957	1.12020	1.11944	1.12001	662.320
2021-11-24 23:00:00	1.11997	1.12042	1.11997	1.12025	465.710

Feature Calculation

Besides price, the three moving averages need to be calculated: $SMA_f$ (fast), $SMA_m$ (medium) and $SMA_s$ (slow).

The image belo shows the three SMA plotting.

Strategy Vectorization

For this strategy, the required steps for vectorization are the following:

• Mark entry and exit signal
• Mark trading periods
• Reduce trading periods to trades
• Calculate profits

These steps that are followed here are not mandatory and can and probably will change from strategy to strategy, from trader to trader and for many other reasons.

Worth to mention that key tools for vectorization are python libraries: Numpy and Pandas.

Marking Entry and Exit Signal

A signal is a flag a light or anything indicating that some condition has been reached and someone or something must execute an action accordingly.

In the case of this strategy and depending on the required action that needs to be triggered, there are two kind of signals: entry signal, triggering a buy action to open a position, and exit signal, triggering a sell action to close a position.

In the context of this strategy also, the signals are calculated and risen once a period is closed, when period's close price is available and therefore it is possible to calculate all moving averages based on it. This is not the only approach though, some strategies could trigger signals based on tick data arriving each second for example, when a period is not yet closed.

Remember that this is a buy only (long) strategy, so the signal space is limited. Broader strategies could have wider signal spaces, containing for example entry, adjust, increase size, decrease size, close signals, to name a few.

Entry Signal: Cross over condition

The entry signal is risen when the last available close price crosses above the fast SMA, along with fast SMA being above med SMA and med SMA being above the slow SMA.

$signal_i = (open_{i} < SMA_{fast, i} < close_{i}) \land (SMA_{slow, i} < SMA_{med, i} < SMA_{fast, i})$

Where:

• $Close_{i}$: Last closed bar close price.
• $Open_{i}$: Last closed bar open price.
• $SMA_{fast, i}, SMA_{med, i}, SMA_{slow, i}$: SMA fast, med and slow respectively, calculated at the last bar close.

Entry period

The entry period is marked right at the opening of current bar, if signal was risen for the last closed bar.

$entry_i=\begin{cases} 1 \text{ if }signal_{i-1} = True\ \emptyset \text{ otherwise} \end{cases} $

Where:

• $entry_i$: Indicates if a trade needs to be opened at current period.
• $signal_{i-1}:$ Signal at last closed bar.

Exit Signal

The exit signal is risen when the close price crosses below the med SMA, therefore, following the entry.

Exit Signal

$x_signal_i = close_{i} < SMA_{med, i} < close_{i-1}$

Exit Signal

$exit_i = \begin{cases} -1 \text{ if }x_signal_{i-1} = True \ \emptyset \text{ otherwise} \end{cases} $

In the example below:

t	Entry/Exit Signal	Entry/Exit Period	Trade
1
2	1
3		1	1
4			1
5			1
6	-1		1
7		-1
8

• $t=2$: entry signal risen at the close of 2nd bar.
• $t=3$: entry condition is met, so a long trade is open right at the opening of the 3rd bar.
• $t=6$: exit signal is risen at the close of 6th bar.
• $t=7$: exit condition is met, so the trade is closed right at the opening of the 7th bar.

The following image shows entry/exit period example

The following image shows trading periods marking.

Trade Reduction

This process is also part of the strategy vectorization and consists of transforming from period-wise trade markings to trade-wise records. The attributes that define a trade could be open date/time, close date/time, open price, close price, number of trading periods, to name a few. What attributes to include is a decision that depends on the problem's context. In this case open date/time, close date/time and gross profit are enough.

trade_id	gross_profit	date_open	date_close
1	-40.1	1/12/2015 4:00	1/12/2015 8:00
2	-38.7	1/22/2015 9:00	1/22/2015 13:00
3	-47.1	1/30/2015 11:00	1/30/2015 12:00
4	-18.3	2/3/2015 1:00	2/3/2015 3:00
5	94	2/3/2015 11:00	2/4/2015 9:00
...	...	...	...
620	-34.3	10/29/2021 1:00	10/29/2021 8:00
621	-14.7	11/2/2021 11:00	11/2/2021 13:00
622	13.4	11/8/2021 11:00	11/9/2021 11:00
623	-10.9	11/9/2021 14:00	11/9/2021 15:00
624	-8.7	11/10/2021 0:00	11/10/2021 2:00

Profit Calculations

With trades in place, now it is possible to perform profits calculations along with some profits related metrics.

In the context of this study, one profit related calculation is required: net profit, which is the result of subtracting trade cost from gross profit:

$NP = GP - Cost$

Where

$NP$: Net Profit
$GP$: Gross Profit
$Cost$: Trading Cost

Trading Cost

Usually overlooked, Trading Cost is a critical variable that can make the difference between a profitable strategy and a losing one. Maybe you have read books, papers or publications that shows very profitable strategies not mentioning trading cost though. Most probably is they didn't consider it, because if they did, the strategy wouldn’t be as profitable as they presented it.

More detail on trading cost calculation is out of the scope of this study and deserve an analysis by itself, enough is to say that its calculation is a function of spread and commission. The value used in this analysis is 0.5 pips per trade

The following image presents the gross and net profit for the strategy under analysis.

trade_id	gross_profit	date_open	date_close	net_profit	cum_profit
1	-40.1	1/12/2015 4:00	1/12/2015 8:00	-40.6	-40.6
2	-38.7	1/22/2015 9:00	1/22/2015 13:00	-39.2	-79.8
3	-47.1	1/30/2015 11:00	1/30/2015 12:00	-47.6	-127.4
4	-18.3	2/3/2015 1:00	2/3/2015 3:00	-18.8	-146.2
5	94	2/3/2015 11:00	2/4/2015 9:00	93.5	-52.7

Profit Metrics

Metric	Value
total_trades	624
winning_trades	184
losing_trades	440
avg_winning	46.12
avg_losing	-17.34
total_pl	857.3
total_profit	8486.3
total_loss	-7629.0
stdev_pl	43.93
stdev_cum_pl	372.23
avg_profit	1.37
median_profit	-8.25
skew_pl	3.36

Observations

1. The strategy is profitable, after cost removal: 857.3 pips.

2. Winning trades proportion: 184 out of 624 trades are winning ones. A 29.5%, or roughly a third.

3. The average profit is 1.37 pips, the median is -8.25 though.

4. The profit distribution chart shows that the distribution is right tailed, which is confirmed by a positive skewness. Also, it is possible to observe that the profits are concentrated around zero.

5. The average profit of winning trades is 46.12 pips, while the losing trade average profit is -17.34

Results Analysis

Considering that proportion of winning trades (observation 2) is low and that e the strategy is profitable (observation 1) it is possible to hypothesize that the winning trades average should be high and losing ones low, which is confirmed by observation 4.

There are two possible explanations for high winning trades average: a) Concentration around the average, which is signal that the strategy is consistent and b) The presence of outliers, which signals the opposite than a). Clearly a) is not the case because winning trades average profit is not concentrated around 46.12 and it is evident the presence of positive outliers (observation 3).

It is almost evident that the strategy is heavily dependent on outliers, so next step is comparing the strategy performance with and without outliers through an statistical tool: Confidence Intervals.

Outliers Removal

Previous Considerations

Before removing outliers two aspects, related to data and the confidence intervals tool, need to be taken into consideration.

a) Profit Distribution. Profit distribution is not normal, and confidence intervals relies on normality which invalidates raw profit for the analysis. To overcome this issue, it is at hand the Central Limit Theorem. It is necessary then to define a statistic and do sampling to calculate it. For this analysis the statistic of choice is profit mean, so the sampling process will produce profit mean sampling, which is assumed as normal distributed. This means that instead of analyzing raw profit, the analysis now shifts to profit mean.

b) Sampling on time series. With cross sectional data (not time related) it is possible to perform sampling techniques like bootstrapping for example, but profits variable is a time series, and there could be correlations (auto-correlation) between consecutive values. For example, opening time for trade $t+1$ was affected because trade $t$ was open at the time it should have been opened. Even tough autocorrelation detection is out of the scope of this analysis and for simplicity auto-correlation will be assumed. You can go deeper in some auto-correlation detection techniques like Auto Correlation Function (ACF) and Partial Auto Correlation Function (PACF)

For time series bootstrapping there are some techniques, like block bootstrapping, with and without overlapping, among others. To maximize the number of samples, simple block bootstrapping with overlapping is used in this analysis. The following image describe in essence what this technique consists of.

Now it is possible to calculate the confidence interval for profit mean.

Confidence Interval Parameters

To calculate confidence interval the following parameter sare used:

• Confidence level: 95 (alpha = 0.05)
  
• Sample size: 60

Confidence Interval with outliers

• sampling mean: 1.17

• sampling median: 0.25

• sampling std: 6.42

• sampling min: -13.02

• sampling max: 16.64

• confidence interval: [0.64, 1.7], mean: 1.17, standard error: 0.27

Observations

With a 95% of probability, the profit mean is within the range 0.64 and 1.7 pips. For the purpose of this analysis, this is a benchmark only.

Removing Outliers

There are many approaches to remove outliers like trimmed mean, values out of a certain number of standard deviation range, out of interquartile ranges, to name a few. For this analysis, two approaches are being taken into consideration: The trading approach and a mathematical approach.

The trading approach to remove outliers uses trading's risk management tools: Stop Loss and Take Profit. In general terms, Stop Loss controls the loss risk (under a normal market context of volatility not like this context:https://www.dailyfx.com/forex/market_alert/2015/01/15/Swiss-Franc-Skyrockets-as-SNB-Announce-End-of-Currency-Floor.html) and Take Profit, limit the profits. The advantage of this approach is that no data need to be removed from the sample and makes sense from the trading perspective because stop loss and take profit are simple to apply (implemented in most trading platforms) and will work as expected most of the time.

In the other hand, using mathematical approach to remove outliers like trimming the data (removing data above 90 and below 10 percentiles) for example, is actually difficult to implement because it requires forecasting tools that can predict ahead in time when an outlier will occur, which is complex.

Being that said, the most feasible approach is the trading one, both will be applied though for research purposes.

Trading Approach

This approach considers applying risk management tools to reduce outliers: Stop Loss (SL) and Take Profit (TP)

SL levels and TP levels are set using standard deviation of net profits. SL will be set at one standard deviation and TP will be set at two standard deviations (there are as many criteria to define take profit and stop loss levels as traders exists)

This way, the profit range is:

$PR = [-s, 2s]$

Where:

$s$: profit standard deviation

Before Removing Outliers:

Min profit -114.1 max profit 406.8

After Removing Outliers:

Min profit -43.93 max profit 87.85

• sampling mean: -0.45,
• sampling median: -0.98
• sampling std: 4.77,
• sampling min: -11.8,
• sampling max: 11.94
• confidence interval: [-0.84, -0.06], mean: -0.45, standard error: 0.2

Observations

Removing outliers using stop loss and take profit, moved the profit sampling mean towards the left to a negative confident interval, which supports the hypothesis that the strategy relayed only in outliers.

Mathematical Approach

As mentioned before, this approach is considered only for research purposes, but is not applicable in practical trading, unless you can apply a very accurate outlier prediction model.

The criteria that are applied here is to remove all trades for which their profits are either below the 10th percentile and above the 90th percentile.

Once trades are removed the sampling process is run again abd the same confidence interval is used: alpha = 0.05 (confidence level of 95%)

• sampling mean: -5.54,

• sampling median: -5.32

• sampling std: 2.17,

• sampling min: -10.47,

• sampling max: -0.33

• confidence interval: [-5.74, -5.34], mean: -5.54, standard error: 0.1

Observarions

Removing outlier trades clearly moves the sampling profit mean towards the negetive side, confirming what was found with the trading approach.

Conclusions

Using relatively basic statistics tools, like central limit theorem, confidence intervals and bootstrap sampling for time series (or autocorrelated data) it has been shown that a profitable strategy was not consistently profitable and relayed only on outliers.

Consistent strategies are those where profits are produced by a high proportion of positive trades, presenting positive median and equal or very approximated profit mean.

What is the difference with profit produced by outliers? Outliers are rare situations and there is not certainty they will occur. Instead, consistent strategies capture more frequent events producing positive profits, probably smaller profits (or losses) produced by outliers.

It is possible to also appreciate the importance of performing this basic analysis before calculating more specialized metrics to measure the potential of a trading strategy like maximum drawdown, Sharpe ratio to name a few. If profit median is negative and profit mean is not, something is not right and possibly outliers are the culprit.

Besides the analysis conclusions, throughout this work is presented an oversimplified process for algorithmic trading strategy back testing; trading idea, model design, vectorized back-testing implementation, back testing running, metrics collecting, analysis, and conclusions and even you can see that the process is not linear, because when a difference between mean and median was spotted, the process had to add additional information to support hypothesis testing.

And last and not least there is a technology dimension not visible in this analysis because the volume of data was relatively low, but if that wouldn’t have been the case, a more robust platform than a notebook and a Jupyter notebook should have been evaluated and used.