Trade Size Optimisation

I am thinking to make an optimisation for my trade sizes, leverage optimisation.

There are the Trade Explorers and lets select a period of time. Hopefully it will be losing trades to not only winning, because there are all wining trades, than the max allowed leverage by broker is the optimum and is done.

Let be a winning ration of 70% and lets select a period, where are 7 consecutive wins and 3 consecutive lose trades and R:R ratio 1:1.

If we take the same return from the trade of the 1% than around 7-3 = 4% will be, not exactly, but around that: +4.025863%

If we take 10 % return ( loss ) than the winning will generate 194.78% and a drawdown to 72,9% => +41.99% and there is an answer for When to broke the 2% rule.

If 20%, than+83.45%

if 40% , than +129.85%

if 80%, than -51.022%! Negative return ( drawdown)!

If we take a 100% return ( loss ) than no matter how much we made with 7 consecutive win, because we are losing all in the first loser trade.

So the optimum is less than 100% and it seems it is bigger than 1%, but how much is it? How is the formula containing the consecutive lose trade numbers?!

I believe it is a math for school, and I forgot the optimum search for a "geometrical progression"? Anybody can help?

Lets look at trade explorer back to 1 month / 1 year, so we know the win and lose rates.
Let be the case of the R:R 1:1 for calculations, (maybe it was different at trade explorer)

For eg if we lose, than we lose1% and gain 1%, if 10% is gained, than 10% lost.

The question it is how much should be the risk percent, trade size.

In a winning trade:
Balance2 = Balance1 + Balance1 * X = Balance1*(1+X)

In a losing trade:
Balance2 = Balance1 - Balance1 * X = Balance1*(1-X)

For a series of random trades the balance looks like:
Balance2 = Balance1*(1+X)*(1+X)*(1+X)*(1+X)*(1-X)

4 win and 1 lose ( 80:20 win lose ratio)

X can be positive number only and less than 1, so it is 0...1 range ( 0% and 100%)

For a longer period it seems it doesn't matter how much consecutive loses are inside, because there are multiplications and they are commutative! - if we don't care the max drawn down, just the maximum of Balance2 , at end of the period ( month )


Balance2 = Balance1*(1+X)*(1+X)*(1+X)*(1+X)*(1-X)*(1+X)*(1+X)*(1+X)*(1+X)*(1-X)*(1+X)*(1+X)*(1+X)*(1+X)*(1-X)*(1+X)*(1+X)*(1+X)*(1+X)*(1-X)*(1+X)*(1+X)*(1+X)*(1+X)*(1-X)....*(1+X)*(1+X)*(1+X)*(1+X)*(1-X)

Balance2 = Balance1 * POW( (1+X), a) * POW( ( 1-X),b)

* can't write html source code, a and b should be "superscript" and I have introduced a function from programming languages: POW, which takes 2 parameters ( maybe in reverse order) Here POW ( x, y) is " X at power of y".

In this formula is the number of winning trades and is the number of loosing trades in the examined period. We want to maximize Balance2 and searching for the X value in the 0...1 range.

Balance1 isn't important in optimisation, because it is a constant.

derivative( POW( (1+X), a) * POW( ( 1-X),b) ) = 0 will have the maximum value of Balance2.

[f(x)*g(x)]’= [f(x)]’*g(x) + f(x)*[g(x)]’
[X^a]’ = a * X^(a-1) - the power of deviated


f(x) = (1+X)^a
g(x) = ( 1-X)^b
[f(x)]’ = (1+X)’ * a * (1+X)^(a-1) = 1 * a * (1+X)^(a-1) = a * (1+X)^(a-1)
[g(x)]’ = (1-X)’ * b * (1-X)^(b-1) = -1* b * (1-X)^(b-1) = -b * (1-X)^(b-1)


[a*(1+X)^(a-1) * ( 1-X)^b ] + [ (1+X)^a * -b * (1-X)^(b-1) ] = 0


[a*(1+X)^(a-1) * ( 1-X)^b ] - [ (1+X)^a * b * (1-X)^(b-1) ] = 0

take the negative part to the other part of the equation and do a ln for the equation:

(a-1)*ln(1+x) + b*ln(1-x) = a*ln(1+x) + (b-1)*ln(1-x)



a2 = (a-1)/a
b2 = b / (b-1)

a2 * ln(1+x) + b2 * ln (1-x) = 2

ln(((1 + x)^a2) * ((1 - x) ^ b2)) = 2

foo(x) = ((1 + x)^a2)*((1 - x)^b2) - e^2 = 0, x in (0,1) range

How to solve this?

Hi!

Kelly's formula (http://en.wikipedia.org/wiki/Kelly_criterion) is the solution of your equation.

You may be interrested in these two threads as well:

Kelly Criterion - Optimizing risk
http://www.forexfactory.com/showthread.php?t=322455

Trading Game System Discussion
http://www.forexfactory.com/showthread.php?t=384691

well first of all i did not understand the initial statements of the math problem you're triyng to solve but i did get what you want. you want to know how much to risk per trade. first of all you have to have the system parameters for your system. second do the monte carlo simulations for the system that you have . if the simulations are ok then move to the next step. find the maximum historical or simulated drawdown for the system. then set a system stopped working scenario. if the maximum drawdown of historical or simulated are 10 points then you can set a 20-30 point drawdown as the sistem stoped working scenario. then set a risk amount of the starting capital. how much are you willing to risk . i set mine at 30 % drawdown. if you have 30 point drawdown and a 30% drawdown then the risk per trade becomes 1%.
so basically there are 2 limits that you have to set. 1 sistem stopped working scenario in terms of point of risk. 2 sistem stopped working scenario in terms of %of equity.
this is my solution for the problem http://www.forexfactory.com/showthread.php?p=5128495.

In the second post I have realized the draw down doesn't matter. I don't care if I had 1 billion usd and it was a draw down to 1 million. From 1000 usd starting balance. I will take the risk and I will survive psychical with that instead of taking smaler risks and have 1100 usd at the same trades


It seems Kelly criterion is my solution, checking. Or at least something like that.

Thanks PipMeUp!

Let be the Win 70%, Loose 30%. In this case in Kelly formula:
W = 0.7
R = 2.33333

And the result is 87%, well that is not exactly the true, because I have calculated above with this numbers the optimum should be beween 40 and 80%

I wrote a little software to get the result:

100 trades
70 win
30 lose

The risk went from 1% to 99% by 1 in each step.

The maximum it is at 40% , the capital will be increased with 374,541.56%

The 2% will result: 218.17%
The 25% will result: 108,527.26%
The 70% will result: 278.65%
The 71% will result: 151.94%

And you won't believe when is the optimum of 2% risk!
At 51:49 win:lose ratio !!!

The 2% rule...

you should do a monte carlo analysis on that. "market system analyzer" is a software that can accomplish that and tell you the trade size you should take.