Estimating Distribution of Consecutive Losses Based on Winning Percentages Using Monte Carlo Simulation

A large part of trading (whether it is day trading, scalping, swing trading or long term trading) is determining the psychological toll that the system will take on your psyche.  A system may boast a healthy expectancy but have large drawdowns and make you endure a large number of consecutive losing trades before that big winner comes in.  This can cause you to doubt both yourself and the system you are trading.

I have tried to quantify this in the past by projecting the maximum consecutive number of losses, and by calculating the maximum drawdown (peak to peak) and using the Ulcer Index. 

Recall that the maximum consecutive number of losses tells us the maximum number of losing trades on average in a given system.  I simulated random data for our trades based on a set winning percentage (number of trades that reached our profit target) and then simulated how many consecutive number of trades hit our stop loss.  This is a useful number but doesn’t really help us on a trade-to-trade basis to determine if our system is performing.

Maximum drawdown brings in both money management and winning percentage.  Peak to peak drawdown refers to the largest percentage loss you endure when trading.  Say you start with $10,000, run the system up to $15,000 and then have a drawdown of $2,000.  This is 20% of your starting equity but only 13.3% of your current equity.  Recall that for our trading system with a $10,000 starting account, 40% winning percentage and a 2:1 win/loss ratio, our statistics looked like this over 1000 simulations of 500 trades:

Maximum Drawdown
Minimum	-2.93%
Maximum	-23.36%
Average	-7.12%
Median	-6.51%
Mode	-6.25%
Standard Deviation	2.47%
	Low	High
1-SD (68% Confidence)	-4.65%	-9.58%
2-SD (95% Confidence)	-2.19%	-12.05%

We could be confident that our system should never have more than a 12% drawdown.  If it did, then we would need to step back and examine it to determine if the market had changed.

Ulcer Index measures the depth and duration of percentage drawdowns in price from earlier highs. The greater a drawdown in value, and the longer it takes to recover to earlier highs, the higher the UI. Technically, it is the square root of the mean of the squared percentage drawdowns in value. The squaring effect penalizes large drawdowns proportionately more than small drawdowns (the SD calculation also uses squaring).  In effect, UI measures the "severity" of drawdowns.

For our system above, with a $10,000 starting account, 40% winning percentage and a 2:1 win/loss ratio, our statistics looked like this over 1000 simulations of 500 trades:

Ulcer Index
Minimum	0.83%
Maximum	9.80%
Average	2.41%
Median	2.15%
Standard Deviation	1.06%
	Low	High
1-SD (68% Confidence)	1.35%	3.47%
2-SD (95% Confidence)	0.29%	4.53%

What these statistics do not display is the trade-to-trade “grind” that can degrade your ability to effectively execute the next trade.  Should you be worried if you have 5 consecutive losses in a row?  What about 15?  Does it mean that the system has stopped working?  Is it a red flag that you are not executing the trades properly?  Or has something fundamentally changed in the market?

With another Monte Carlo simulation of 500 trades repeated 1000 times, we can begin to get a picture of how many consecutive losses we can expect before a red flag should go up and we should examine whether we are executing the system improperly or something fundamental has changed.

The table below is a summary of this simulation.  Across the top are varying win percentages from 5% to 95%.   On the far left is the average number of consecutive losses that the system had over 500 trades.  A PDF file showing the entire table can be found by going here.

As an example, let’s take a closer look at the system with a 50% winning percentage.

Out of 500 trades, 436 had either “0” or “1”consecutive losses.

Out of 500 trades, we could expect on average that 31 times we would have two losses in a row, 15 times 3 losses in a row, about 8 times 4 losses in a row, 4 times 5 losses in a row, 6 times about 2 losses in a row and 1 time 7 losses in a row.

If you are a “percentages” type person, the table below shows the percent of times you had a given number of losses in a row on average in 1000 simulations.  The full report in PDF format can be found by going here.

These are helpful statistics to have when executing a system.  We know that consecutive losses will occur but now you can tell if they are an “aberration” or in line with what we should expect based on random data.  It may or may not be “okay” to have 5 losses in a row a couple of different times during your trading.  Furthermore, it can help you with money management – if you know that a given number of losses in a row is the max number of losses in a given system, make sure you have enough capital to withstand that type of drawdown.

Keep ya mind right and happy trading.

“To Scale Out or Not to Scale Out?:” Build Your Own Fixed Risk Money Management Monte Carlo Simulator in Excel

What we will attempt to do is build a Monte Carlo Simulation simulating 1000 runs of 500 trades each with multiple contracts, multiple profit targets, multiple stop losses and multiple probabilities that each is hit/not hit.  We will be all in on the initial purchase but pyramid or “scale out” based on specific profit targets being hit.

First, determine how many profit targets you are going to have.  Divide the total number of contracts you are trading by this number.  For this example, assume that you have 3 contracts with 3 separate profit targets.  We will define these as profit targets #1, #2 and #3.

Second, determine your initial “drop dead” stop loss.  This will be the amount of money you will lose per contract assuming that profit target #1 is not hit.  You would be “all out” at this point.

Third, determine your initial risk to reward ratio for each of your profit targets.  For this example, we will assume the following:

Profit Target #1:  Initially risking 10 ticks for a Reward of 20 ticks (1:2 R/R Profile)

Profit Target #2:  Initially risking 10 ticks for a Reward of 30 ticks (1:3 R/R Profile)

Profit Target #3:  Initially risking 10 ticks for a Reward of 40 ticks (1:4 R/R Profile)

Fourth, determine how you will scale out or move your stop loss up if and when Profit Targets #1 and #2 are hit.  For this example, we will do the following:

Scale Out #1:  If Profit Target #1 is hit, we will exit the first contract then we will move the stop loss on the 2 remaining contracts to breakeven plus a tick.

Scale Out #2:  If Profit Target #2 is hit, then we will move the stop loss on the 1 remaining contract to Profit Target #1.

Fifth, determine the Winning Percentage (WP) for each of the individual scenarios that you could have for your scale outs and profit targets.  In this example, we basically have 4 scenarios:

Scenario #1:  Call this the “Worst Case Scenario.”  None of the profit targets are hit and we are stopped out on all 3 contracts.  Our loss is equal to 10 ticks times 3 contracts.

Scenario #2:  Profit Target #1 is hit on 1 contract, but we are stopped out on the remaining 2 contracts at breakeven plus a tick (Scale Out #1).  If Scenario #2 occurs, then we will have 1 contract at a 1:2 R/R and 2 contracts at a breakeven plus a tick for a total profit of 22 ticks.

Scenario #3:  Profit Target #1 is hit on 1 contract and Profit Target #2 is hit on 1 contract, but we are stopped out on the remaining 1 contract at Profit Target #1 (Scale Out #2). If Scenario 3 occurs, then we will have 1 contract at a 20 Ticks, 1 contract at 30 ticks and 1 contract at 20 ticks.

Scenario #4:  Call this the “Best Case Scenario.”  Profit Targets #1, #2 and #3 are all hit.  If Scenario 4 occurs, then we will have 1 contract at a 20 Ticks, 1 contract at 30 ticks and 1 contract at 40 ticks.

Assume for a second that on any given trade, the Stop Loss and Profit Targets happen the following percentage of the time.  Think of the table below as the average that, on a given trade, all 3 contracts would have reached the ultimate profit target or been stopped out:

Scenario	Probability	Total Profit/(Loss)
#1	50%	(30) ticks
#2	25%	+22 Ticks
#3	15%	+70 Ticks
#4	10%	+90 Ticks

Fifth, we will also be using fixed risk money management strategies.  If you recall, this is a type of money management strategies that does 2 things:

1.         It limits the risk on any one trade to a fixed percentage of the account equal to the Account Value divided by your Average Total Loss (across all contracts traded).  For example, if you had an initial account value of $10,000 and never wanted to risk more than 1%, then your fixed risk on any one trade would be $100.  Essentially, you would have 100 capital units to begin with.

2.         It allows the account to compound with the same initial risk on each trade.  If your account grows to $20,000 after 100 trades and you still want to risk 1% of your account, then you should be willing to risk $200 on the next trade. 

One problem that I ran into last time is that what happens when the account value dips below a certain threshold where the individual trade risk is greater than the fixed risk.  To illustrate, take the above example – we have a 1% fixed risk parameter on a $10,000 account.  That means that no more than $100 could be risked on a given trade.  If the first trade in the series is a loss, then our account is now at $9,900.  Our fixed risk is then 1% or a maximum of $99 on the next trade.  If our average loss or individual trade risk is $100, then technically you can’t take the next trade – you would have to risk $0.  This results in no additional trades being taken in the series.  One way to fix this is to delineate between the individual trade risk and the fixed risk and simply reduce the number of contracts.  I will show you how these two work together below.  Another way would be to put a minimum contract specification in your simulation.  This would specify that, no matter the fixed risk on the next trade, you would always trade 1 contract.

Another problem that was encountered last time is the situation where you reach an unrealistic number of contracts traded.  In one instance, the simulation returned that we should trade 2,000 contracts on the next trade.  This is unrealistic for most retail traders.  A simple fix for the simulation is to place a limit on the number of contracts traded. 

Building Our Excel Monte Carlo Simulator

The first step is to create your input page and call it “INPUT” like this:

In this example, we will be simulating 500 trades in the account and repeating this 1000 times.  We will assume that we will trade the NQ futures contract with a tick value of $5 per tick.  We will initially trade 3 contracts with a risk per contract (Scenario #1 above) of 10 ticks.  We never want to risk more than .5% of our account on any given trade.  Finally, we assume that the Scenarios outlined above have the given winning percentages, resulting in a gain/loss per contract outlined above.

Under the 2^nd tab, create a random number generator for each simulation and call it “Rand”.  We want a random number between 0 and 1 that is rounded to 2 decimal places.  You can generate a random number with these parameters by entering the following:

=ROUND(Rand(),2)

The 2^nd Tab then looks like this and is copied over 500 columns (trades are illustrated in each column) and down 1000 rows (simulations are contained in each row):

Under the 3^rd tab, we create another sheet that returns the profit or loss from the random number generation.  Call the 3^rd tab “Result”.  We then apply the following rules based on the winning percentages (“WP”) we entered into the Input tab:

For random numbers less than the WP contained in Scenario 1, return a 1 (this would be a loss). 

For random numbers greater than the WP of Scenario 1 but less than the WP of (Scenario 1 + Scenario 2), return a 2.

For random numbers greater than the WP of (Scenario 1 + Scenario 2) but less than the WP of (Scenario 1 + Scenario 2 + Scenario 3), return a 3.

For random numbers greater than the WP of (Scenario 1 + Scenario 2 + Scenario 3) but less than the WP of (Scenario 1 + Scenario 2 + Scenario 3 + Scenario 4), return a 4.

You can do this by creating a nested “IF” function in Excel.  For cell B2 in the “Results” tab enter the following:

=IF(Rand!B2<Input!$C$17,1,IF(Rand!B2<(Input!$C$17+Input!$C$18),2,IF(Rand!B2<(Input!$C$17+Input!$C$18+Input!$C$19),3,4)))

Our Results tab looks like this:

For instance, in cell b2 under the Rand tab, it returned the random number 0.6 for our first trade in our first simulation.  We judge this to mean that for trade 1 simulation 1, we achieved Scenario 2.  You can see in the picture above, the Result tab returned a 2.

The step will help us determine whether the trade was a win (by achieving either Scenarios 2, 3 or 4) or a loss (by achieving Scenario 1).

For the 4^th Tab called WinLoss, enter the following formula in cell c2:

=IF(Result!B2=1,"L","W")

This tells Excel that if the output in the Results tab is equal to 1, return “L” and otherwise return a “W.”  Your WinLoss tab should look like this:

We then want to calculate the maximum number of losers or “L” that we had in a row.  You can do this by creating a 5^th tab on your workbook and calling it WLStreak.  Three calculations are required.

In cell c2 of WLStreak, enter the following:

=IF(WinLoss!C2="L",1,0)

This tells Excel that if the outcome of the first trade in the first simulation was a loss or an “L”, enter a 1 and otherwise enter a 0.  Copy this formula down Column C of WLStreak.

In cell d2 of WLStreak, enter the following:

=IF(WinLoss!D2="L",C2+1,0)

This tells Excel that if the outcome of the second trade in the first simulation was a loss or an “L”, add 1 to cell C2 and otherwise enter a 0.  This little formula will calculate your losses in a row and reset them if you have a win.  Copy this over to the last trade and down to the last simulation.

Finally, in cell a2 of WLStreak, enter the following:

=MAX(C2:SH2)

This tells Excel that, for the first simulation, what is the maximum number of losses or “L” that you had in a row.  Cell C2 corresponds to the first trade in the first simulation and cell SH2 corresponds to the last trade in the first simulation.  Copy what you have in cell A2 down each row for the 1000 simulations.

WLStreak tab now looks like this:

You can see in the picture above that the 10^th simulation had 17 losses in a row (see cell A11).

Now that we now the outcome of each simulation and trade, we need to construct a 6^th tab and call it AccountValue.  We will apply the fixed risk money rules we talked about and allow us to simulate how the account grows and contracts.

The AccountValue tab looks like this:

For column C, reference the starting account value on the input page.  This is where every simulation will start out.  In cell C2, I entered the following formula and copied it down for each simulation:

=Input!$C$3

Next, create a lookup table called “Scen” using the formula tab.  This will be used to lookup the profit or loss per contract on each trade.  Recall that a “1” in the far left column tells us to return Scenario 1.

Back to the Account Value tab, for the first trade in the first simulation, enter the following:

=VLOOKUP(Result!B2,Scen,6)*Input!$C$7+C2

This tells Excel the following:

1.         Lookup the result of the first trade in the Results tab,

2.         Go to the Scen table, find the corresponding row based on the Result,

3.         Return the 6^th column over which is the profit/loss per contract,

4.         Multiply it by the number of contracts we started with on the Input page

5.         Add it to our starting account value contained in cell c2.

Next, create a second lookup table and call it NoCon or Number of Contracts.  This is where our FRMM comes into play.  Here is a picture of my NoCon:

Back to the Account Value tab, for the second trade in the first simulation, enter the following:

=VLOOKUP(D2*Input!$F$8/Input!$C$6,NoCon,2)*VLOOKUP(Result!C2,Scen,6)+D2

This formula tells Excel the following:

1.         Multiply the previous trade’s account value by your fixed risk percentage on the   Input page.

2.         Divide it by the points per contract.  This result is the maximum number of            contracts you can trade on this trade.

3.         Go to our NoCon table and lookup this value, returning the number in the second column.  For example, if the result of Step 2 is a 5, the maximum number      of contracts you could trade would be 3.  If Step 2 returns a 28, the maximum number of contracts you could trade would be 27.

4.         Multiply the numbers of contracts times the resulting profit/loss per contract and             add it to the previous trades account value.

You have now created a simulation of this system, scaling out at three different profit targets.  You can add tabs for maximum drawdown, maximum losses and the ulcer index by following the steps here.

Interpreting the Results

One simulation of the above gives the following results:

Ending Account Value
Minimum	42,590
Maximum	298,840
Average	127,341
Median	121,585
Mode	132,270
Standard Deviation	43,688
	Low	High
1-SD (68% Confidence)	83,652	171,029
2-SD (95% Confidence)	39,964	214,717
Maximum Drawdown
Minimum	-5.86%
Maximum	-28.56%
Average	-12.26%
Median	-11.68%
Mode	-9.19%
Standard Deviation	3.43%
	Low	High
1-SD (68% Confidence)	-8.83%	-15.69%
2-SD (95% Confidence)	-5.40%	-19.12%
Ulcer Index
Minimum	1.58%
Maximum	10.67%
Average	3.63%
Median	3.41%
Standard Deviation	1.16%
	Low	High
1-SD (68% Confidence)	2.48%	4.79%
2-SD (95% Confidence)	1.32%	5.95%
Max Losing Streak
Minimum	4
Maximum	17
Average	8.22
Median	8.00
Standard Deviation	1.91
	Low	High
1-SD (68% Confidence)	6.30	10.13
2-SD (95% Confidence)	4.39	12.05

This gives a very nice, tradable system with an average ending account value of around 127k, a drawdown of around 12%, UI of 3.63% and max losing streak of 8.  The full report in PDF format can be found here.

The better comparison is whether or not scaling out results in a higher ending account value with a similar risk profile.  If we were to assume that you could get your 1:2 Risk/Reward on all 3 contracts, rather than scaling out, what would the results look like?  Remember, we assumed we hit profit target #1 with all 3 contracts but held on for targets #2 and #3.  Here are the results:

Ending Account Value
Minimum	89,550
Maximum	418,500
Average	240,823
Median	240,150
Mode	285,900
Standard Deviation	50,059
	Low	High
1-SD (68% Confidence)	190,764	290,882
2-SD (95% Confidence)	140,705	340,941
Maximum Drawdown
Minimum	-4.65%
Maximum	-21.55%
Average	-9.74%
Median	-9.28%
Mode	-9.09%
Standard Deviation	2.55%
	Low	High
1-SD (68% Confidence)	-7.20%	-12.29%
2-SD (95% Confidence)	-4.65%	-14.83%
Ulcer Index
Minimum	1.14%
Maximum	5.71%
Average	2.45%
Median	2.33%
Standard Deviation	0.65%
	Low	High
1-SD (68% Confidence)	1.80%	3.09%
2-SD (95% Confidence)	1.16%	3.74%
Max Losing Streak
Minimum	5
Maximum	18
Average	8.19
Median	8.00
Standard Deviation	1.84
	Low	High
1-SD (68% Confidence)	6.36	10.03
2-SD (95% Confidence)	4.52	11.87

This looks a lot more promising, since the average account value is almost double with a lower average drawdown and lower UI.  The full report can be found here.

However, what if we accepted more fixed dollar risk on each trade – instead of .5% of the account, we increased it to 1%?  The table is below and the full report can be found here.

Ending Account Value
Minimum	45,590
Maximum	456,000
Average	247,422
Median	247,015
Mode	147,150
Standard Deviation	53,386
	Low	High
1-SD (68% Confidence)	194,036	300,808
2-SD (95% Confidence)	140,651	354,194
Maximum Drawdown
Minimum	-7.77%
Maximum	-51.69%
Average	-20.07%
Median	-18.78%
Mode	-15.04%
Standard Deviation	6.63%
	Low	High
1-SD (68% Confidence)	-13.44%	-26.70%
2-SD (95% Confidence)	-6.81%	-33.34%
Ulcer Index
Minimum	1.78%
Maximum	25.14%
Average	5.40%
Median	4.77%
Standard Deviation	2.48%
	Low	High
1-SD (68% Confidence)	2.92%	7.89%
2-SD (95% Confidence)	0.43%	10.37%
Max Losing Streak
Minimum	5
Maximum	18
Average	8.27
Median	8.00
Standard Deviation	1.84
	Low	High
1-SD (68% Confidence)	6.43	10.11
2-SD (95% Confidence)	4.60	11.94

Here is the full report if we can reduce our probability of Scenario 1 (all out stop loss) to 40% and increase Scenario #2 to 35%.

Hope this helps.  Keep ya mind right.