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Benjamin Dixon
21230
Benjamin Dixon 2015.02.08 09:04 
Martingale: The gambler doubles his bet after every loss...

There are a few complicated Martingale math papers floating around and even those are mostly about roulette but I thought of a way to make a really simple math based presentation.

First of all I am assuming the goal is to make as much profit as possible from a given risk.

Lets take a very small run and calculate all possibilities. We will start with 4 units to risk over 2 trades with 4 possible total outcomes. The system has a 50% chance of winning and wins 1.1x risk when it wins. Lets define Risk of Ruin as losing 75% of the 4 unit start account. Of course we don't normally only make 2 trades and risk such a high proportion of our risk capitol but I will show later how the conclusions hold true when we add trades or increase our edge or reduce risk or reduce the martingale multiplier. 

Standard risk 1 unit on each trade non martingale:
sum “L” “L”
-2 -1 -1
sum “W” “L”
0.1 1.1 -1
sum “L” “W”
0.1 -1 1.1
sum “W” “W”
2.2 1.1 1.1
EV: 0.4


OK now lets double the bet after each loser:
sum “L” “L”
-3 -1 -2
sum “W” “L”
0.1 1.1 -1
sum “L” “W”
1.2 -1 2.2
sum “W” “W”
2.2 1.1 1.1
EV: 0.5
Hey the Martingale really is more profitable! Well, of course it is, we have taken on more risk so it's not a fair comparison. notice in the martingale we have a 25% chance of RoR. 

We can run the system NON martingale and risk 1.5 units on each trade and wind up with an identical RoR of 25%:
sum “L” “L”
-3 -1.5 -1.5
sum “W” “L”
0.15 1.65 -1.5
sum “L” “W”
0.15 -1.5 1.65
sum “W” “W”
3.3 1.65 1.65
EV: 0.6
Winner! 

What if we make the martingale multiplier less than double? We can use a 1.5x martingale multiplier and a start lot of 1.2 to again come up with a 25% RoR strategy:
sum “L” “L”
-3 -1.2 -1.8
sum “W” “L”
0.12 1.32 -1.2
sum “L” “W”
0.78 -1.2 1.98
sum “W” “W”
2.64 1.32 1.32
EV: 0.54

Still no dice for Marty. The highest EV for a 25% RoR strategy is 0.6 achieved by the non martingale. OK well, we have a winner. Lets see if it holds true when we increase the run size to 3 trades and reduce the risk. Lets try some 12.5% RoR strategies:
Martingale 2.0 multiplier:
sum “L” “L” “L”
-3.01 -0.43 -0.86 -1.72
sum “W” “L” “L”
-0.82 0.47 -0.43 -0.86
sum “L” “W” “L”
0.09 -0.43 0.95 -0.43
sum “W” “W” “L”
0.52 0.47 0.47 -0.43
sum “L” “L” “W”
0.60 -0.43 -0.86 1.89
sum “W” “L” “W”
0.99 0.47 -0.43 0.95
sum “L” “W” “W”
0.99 -0.43 0.95 0.47
sum “W” “W” “W”
1.42 0.47 0.47 0.47
EV of Martingale (2.0x): 0.77

Martingale 1.5 multiplier:
sum “L” “L” “L”
-2.99 -0.63 -0.95 -1.42
sum “W” “L” “L”
-0.88 0.69 -0.63 -0.95
sum “L” “W” “L”
-0.22 -0.63 1.04 -0.63
sum “W” “W” “L”
0.76 0.69 0.69 -0.63
sum “L” “L” “W”
-0.02 -0.63 -0.95 1.56
sum “W” “L” “W”
1.10 0.69 -0.63 1.04
sum “L” “W” “W”
1.10 -0.63 1.04 0.69
sum “W” “W” “W”
2.08 0.69 0.69 0.69
EV of Martingale (1.5x): 0.774

Standard non Martingale:
sum “L” “L” “L”
-3.00 -1.00 -1.00 -1.00
sum “W” “L” “L”
-0.90 1.10 -1.00 -1.00
sum “L” “W” “L”
-0.90 -1.00 1.10 -1.00
sum “W” “W” “L”
1.20 1.10 1.10 -1.00
sum “L” “L” “W”
-0.90 -1.00 -1.00 1.10
sum “W” “L” “W”
1.20 1.10 -1.00 1.10
sum “L” “W” “W”
1.20 -1.00 1.10 1.10
sum “W” “W” “W”
3.30 1.10 1.10 1.10
EV of non martingale: 1.20
Non Martingale still best, what if our edge is better?
Lets try more 3 trade samples with an edge of 1.5 instead of 1.1 again all runs have 3 trades and 12.5% RoR
Martingale2.0
sum “L” “L” “L”
-3.01 -0.43 -0.86 -1.72
sum “W” “L” “L”
-0.65 0.65 -0.43 -0.86
sum “L” “W” “L”
0.43 -0.43 1.29 -0.43
sum “W” “W” “L”
0.86 0.65 0.65 -0.43
sum “L” “L” “W”
1.29 -0.43 -0.86 2.58
sum “W” “L” “W”
1.51 0.65 -0.43 1.29
sum “L” “W” “W”
1.51 -0.43 1.29 0.65
sum “W” “W” “W”
1.94 0.65 0.65 0.65
ev 3.87

Martingale1.5
sum “L” “L” “L”
-2.99 -0.63 -0.95 -1.42
sum “W” “L” “L”
-0.63 0.95 -0.63 -0.95
sum “L” “W” “L”
0.16 -0.63 1.42 -0.63
sum “W” “W” “L”
1.26 0.95 0.95 -0.63
sum “L” “L” “W”
0.55 -0.63 -0.95 2.13
sum “W” “L” “W”
1.73 0.95 -0.63 1.42
sum “L” “W” “W”
1.73 -0.63 1.42 0.95
sum “W” “W” “W”
2.84 0.95 0.95 0.95
ev 4.65

Standard
sum “L” “L” “L”
-3.00 -1.00 -1.00 -1.00
sum “W” “L” “L”
-0.50 1.50 -1.00 -1.00
sum “L” “W” “L”
-0.50 -1.00 1.50 -1.00
sum “W” “W” “L”
2.00 1.50 1.50 -1.00
sum “L” “L” “W”
-0.50 -1.00 -1.00 1.50
sum “W” “L” “W”
2.00 1.50 -1.00 1.50
sum “L” “W” “W”
2.00 -1.00 1.50 1.50
sum “W” “W” “W”
4.50 1.50 1.50 1.50
ev 6.00

It just doesn't matter, add more trades, reduce risk and lot size or increase the edge we are still better off staying away from martingale when it comes to maximizing profits for a given RoR. If we want to take on more risk to increase profits, it's always better to simply increase base lot size instead of going martingale.

But wait, what if we reduce risk after loss, not increase it! lets try and risk a % of current equity on each trade instead...
Here we find that a 3 trade run with 1.5 edge risking 37% of equity on each trade gives a 12.5% RoR Run to compare to the last three runs:
sum “L” first trade bal “L” 2nd bal “L” fin bal
-3.00 -1.48 2.52              -0.93 1.59     -0.59 1.00
sum “W” first trade bal “L” 2nd bal      “L” fin bal
-1.53 2.22 6.22              -2.30 3.92       1.45 2.47
sum “L” first trade bal "W” 2nd bal     “L” fin bal
-1.53 -1.48 2.52               1.40 3.92       -1.45 2.47
sum “W” first trade bal “W”  2nd bal         “L” fin bal
2.09 2.22 6.22                3.45 9.67       -3.58 6.09
sum “L” first trade bal “L” 2nd bal “W” fin bal
-1.53 -1.48 2.52                -0.93 1.59        0.88 2.47
sum “W” first trade bal “L” 2nd bal "W” fin bal
2.09 2.22 6.22          -2.30 3.92        2.17 6.09
sum “L” first trade bal “W” 2nd bal “W” fin bal
2.09 -1.48 2.52                 1.40 3.92         2.17 6.09
sum “W” first trade bal  “W”   2nd bal     “W” fin bal
11.04 2.22 6.22 3.45 9.67 5.37 15.04
ev 9.73
Nearly triple the EV of the martingale 2.0!!!! I can assure you will find the risk percent of equity strategy the best ratio of profits to RoR no matter what. Always always always risk a % of your account and stay away from Martingale when it comes to going for the best profit to RoR ratio.

Sure if you add enough starting balance you can make the ev differences more subtle, but now the true danger of Martingale becomes apparent! It's possible to make an unprofitable system look profitable or hide its draw-down risk for many trades by using Martingale with a larger starting account vs risk. If you check out the martingale runs above you will notice the number of runs  that were profitable  were disproportionate to the number of losing runs. This is how the illusion is produced, the double down effect makes it much less likely to end on any given trade in a loss (even if the system is a loser). This is not a good thing, aside from the reduced ev  proven above, it masks true performance and risks of the system (It takes a much much larger sample size to converge on performance if a tiny start lot/huge start account/martingale is used). Go ahead and checkout any martingale EA, turn Martingale off, make up some unprofitable settings, make a tiny lot size compared to you starting balance turn martingale back on and see you can (usually) make an unprofitable system look profitable for some time. In fact this is the concept behind that scam Roulette Bot going around.

If you want maximize profits vs RoR and you want to clearly asses the risks of the system then demand your EA provider/signal provider not use martingale. Demand from the market that no risk masking money management be allowed. Watch out for EA's with built in but hidden dangerous risk hiding MM. If you want to gamble it up, reduce your ev, increase your RoR, hide the true risks of your systems than by all means increase your lot size after each loss.

Ingvar Engelbrecht
4412
Ingvar Engelbrecht 2015.02.08 18:33  

bendex77!

Excellent exercise!  Thank you. Have already  adopted the percentage of account or free margin as the method to use. This just reinforces my belief  :-)

Benjamin Dixon
21230
Benjamin Dixon 2015.02.11 07:47  

And here is an example of an illusion.

Here is a losing strategy:

 

And with a simple Martingale MM applied,
we can usually make it look profitable for some time:

 

As you can see from this example, we can (usually) delay our downswings with martingale but in return it makes them bigger. We can even disguise a losing strategy as a winning one for some time. I'm not sure why the market allows this.

Sergey Golubev
Moderator
52147
Sergey Golubev 2016.11.30 06:58  

Forum on trading, automated trading systems and testing trading strategies

Martingale

Royal-Flush, 2016.11.30 02:38

Many of the MT4 top rated signals are martingale strategies especially those over 1,000% return and are well subscribed. Yet most of them are holding large positions which would be a huge draw down if covered. Many of these loosing trades are from Brexit.  

There are 2 excellent articles on Martingale;

https://www.mql5.com/en/articles/1446

https://www.mql5.com/en/articles/1481

which describe martingale as being meaningless.

So in conclusion are these highly rated martingale strategies deserving of the rating and the subscriptions or are the articles correct and the strategy is meaningless or is there some other way of resolving this contradiction?


Marco vd Heijden
Moderator
4067
Marco vd Heijden 2016.11.30 11:02  

I noticed that the time element is forgotten or excluded most of the time in these examples people come up with.

They only go about the position sizing and number of winning/losing streaks.

But you can also add the time element in which all losing blocks are forced to close at for example the end of the day, week or month...

This will prevent the risk of accumulation of those toxic position blocks.

Like they always call "no more bet's please" and the run of the game has to be done in a known time value it can not last forever.

Carmine Pinto
4233
Carmine Pinto 2016.11.30 11:41  

Personally i don't like martingale and I NEVER use this management type in my manual trading but I must admit that martingale, coupled with a strategy that make sense according to the high probabilities of the specific strategy that is implemented, will outperform most of the strategy out there if we are talking about algorithmic trading; yes it will be risky because you never know when extreme adverse market conditions can materialize in front of you and blow up your account, but surely in the right periods (even years) can give exceptional returns than most of the mechanical strategies that don't implement martingale.

As for manual trading I think that martingale is still effective when used wisely but if you are an experienced manual trader would be better to trade without martingale just because you don't need it, in fact your edge is extremely robust in this case and you don't need them.

So, here the decision to use or not martingale all depends on the type of capital you would trade on.

Is it a pure speculative trading capital so that you can even lose it without fear? In this case martingale can be a good choice.

Do you have a decent amount of capital that is vital for your existence and you can't afford to charge too risk on it? In this case the best option (in my opinion) would be to trade manual WITHOUT martingale and don't be greedy with the lotsize: you can live on that capital for the rest of your life if you learn how to trade manually with realistics expectations.

Actually Martingale is a way to virtually improve an edge and it can be effective but would be wise to base your choice on the meaning of your trading capital.


Happy trading community! :)

splendorfx
110
splendorfx 2016.11.30 21:54  
Carmine Pinto:

Personally i don't like martingale and I NEVER use this management type in my manual trading but I must admit that martingale, coupled with a strategy that make sense according to the high probabilities of the specific strategy that is implemented, will outperform most of the strategy out there if we are talking about algorithmic trading; yes it will be risky because you never know when extreme adverse market conditions can materialize in front of you and blow up your account, but surely in the right periods (even years) can give exceptional returns than most of the mechanical strategies that don't implement martingale.

As for manual trading I think that martingale is still effective when used wisely but if you are an experienced manual trader would be better to trade without martingale just because you don't need it, in fact your edge is extremely robust in this case and you don't need them.

So, here the decision to use or not martingale all depends on the type of capital you would trade on.

Is it a pure speculative trading capital so that you can even lose it without fear? In this case martingale can be a good choice.

Do you have a decent amount of capital that is vital for your existence and you can't afford to charge too risk on it? In this case the best option (in my opinion) would be to trade manual WITHOUT martingale and don't be greedy with the lotsize: you can live on that capital for the rest of your life if you learn how to trade manually with realistics expectations.

Actually Martingale is a way to virtually improve an edge and it can be effective but would be wise to base your choice on the meaning of your trading capital.


Happy trading community! :)

I agree with you, martingale can be profitable under certain market circumstances in extraordinary manner.

However great trading edges don't need martingale.

You're a wise man :-)

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