Calculating the most effective trade size

Hey,

Let's assume we have a strategy which has the following odds:

 

1. 70% chance to win 100 pips
2. 30% chance to lose 100 pips

So let's keep it that simple, you either lose or win 100 pips, 1 trade at a time.

-----


So here are 3 questions waiting to get answered:

Question 1:
What is the optimal % of our account-equity we should risk each trade to gain as much as possible?



Question 2:
Our goal is to get 100% return (example: turn $100 to $200).
We always use the "updated" % of our current full account balance/equity before each trade (compounding returns).
How much % should we risk each trade, to get the 100% return?
What would the average drawdown be?
Of course, we want to have that 100% return with as few trades as possible (fastest).



Question 3:
If we risk 100% of our equity each trade (Starting with $100, 70% chance to double account to $200, 30% chance to lose it all. if we win, we go on and try to double that $200 to $400 and so on. until we fail), how high would our account on average be before losing it all?



Any answers of math wizards or statisticians appreciated . It would be nice if you could tell how you calculated your (best) value or optimal values (if it's not too complicated )

not a realistic scenario....your system shows no standard deviations or normal distributions?

but i will get back to you with your answers in a bit.

hey redlion,

of course, this is just a theoretical environment, but let's assume we trade in this perfect scenario with no deviations (or maybe 5-10%) for reasons of simplicity

I am sure there are smarter people here in Forex Factory with a more robust background in trading and mathematics as well as finance but here is my attempt at it.


your system of 70% win 30% loss exactly. Without adjusting lot size till you double your account.

assuming we are taking 100 trades. (Not even doing a binomial probability distribution, as we are 100% certain of the 70/30).

in those 100 trades you will lose 30 of them, so if you risk 1% of account each trade, then worst case scenario you are down 30% before your winning streak of 70 trades kicks in.....but to maximize returns then 2% as you will be down 60% before your winning streak kicks in.
your equity is now 40%, a 3% risk is equal to 90% of your equity gone in a 30 trade losing streak out of 100.

so my answer would be with 100% certainty that you have a 70/30 then risk 2%.

you do not need average draw down since you know with 100% certainty that 30% is will be your draw down.

if you risk 100% of equity then you can expect to lose it the first trade that goes south. This could be the first trade you take.

90% equity gone with a risk of just 3% per trade in 100 trades.... whew, that better not happen that often but it's not that probable anyway is it

the bad thing is also that, after losing 30 trades a row, you are earning less the next trades because your equity is smaller and you always bet a % of your equity.

somewhere i read that risking more is good and makes your way faster/shorter, but at some point risking more actually hurts your overall expectancy. because once lost, you need more margin to recover from that loss. Now where is this point, the "sweetspot"?^^ just a thought

it all depends what you mean by %

I assumed a fix % of initial equity, and only upping it once account doubles.

however % return by simple returns yes a 50% loss requires 100% gain to get back to b/e

but if you are measuring by Log returns then no. By the way with your system of 70/30 it is expected to happen, you will lose 30 out of 100.
if your system is more realistic then it gets worst because no system actually has exactly a certain ratio....it is all sort of statistical performance with standard deviations, heavy tails, possible market change, and you will have to deal with probabilities not certainty.

in 100 trades you will lose 30*(1000*.02)= 600 your remaining equity is 400 now 70*(1000*.02)=1400 so on I expect roughly you make
80% in 100 trades as first 30 trades loser = 400 remaining + 70 next trades winner = 1400.............so 1400+remaining equity= (1800-initial account balance/ initial account balance = 80%


you would need 1250 trades to make 1000% return

Maybe this is what I've searched for. Kenny Formula it's called.
(this may be informative on that topic, just took a glimpse over it: https://www.tradestation.com/en/educ...ize-formulas-2)


It gives me a Kelly value of 40%.
(70% chance of win, average win same size as average loss)

at 80% win it is 60% kelly value
at 90% win it is 80% kelly value
at 50% win it is 0% kelly value
at 51% win it is 2% kelly value



So, for my values of 70% win, i could risk up to 40% each trade? that should be the optimal value for the long run as i understood. ofcourse, risking 40% of your equity each trade is heavy. i gotta think about that a bit and read into it, maybe someone got some insight, cya

Dagobart, You are on the right path looking for a way to determine position size. May I suggest "Trade your way to Financial Freedom" by Van K. Tharp. Don't be put off by the title, it's a great read for dealing with the question of how much to risk per trade.

As to the Kelly Formula I'll pass along one of my favorite Yogi Berra quotes. I have no idea the context he made this observation, put I have found it applies to many trading topics (back test vs foward test, demo vs real money etc); "In theory there is no difference between theory and practice. In practice there is". Good trading - Lou

Here is another way to calculate risk of ruin http://www.futuresmag.com/2009/08/01...of-ruin?page=2 :

For example, suppose a trader has a mean return of 6% per month, with a standard deviation of 13% (see “Risk of loss,” above). The risk of losing one standard deviation, or 13% of the account, is given by this formula:
R = e^(-2*a/d)
Where:
R =
risk of losing one standard
deviation
e =
2.71828, the base of the
natural logarithm
a =
average, or mean return,
in this case 6%, or 0.06. a must be a positive number, or else R is defined as 100%.
d =
standard deviation of returns,
in this case 13%, or 0.13. d must be a positive number between 0 and 1.
Plugging in the numbers, we find that
R = e^(-(2*0.06)/0.13)
R = e^-0.923
R = 0.397
There is a 39.7% chance of losing 13% of the account from any point on the equity curve. This trader will spend 39.7% of his time at least 13% below a prior maximum equity high.
Now suppose this same trader wants to calculate the probability of losing half of his account. From Cox and Miller, the equation is:
R = e^(-(2*a*z)/(d*d))
Where:
R =
risk of losing z fraction
of the account
e =
2.71828, the base of the
natural logarithm
z =
The fraction of the account that might be lost, in this case 50%,
or 0.50
a =
average, or mean return,
in this case 6%, or 0.06
d =
standard deviation of returns,
in this case 13%, or 0.13
Plugging in the numbers, we find that
R = e^(-(2*0.06*0.50)/(0.13*0.13))
R = e^-3.550
R = 0.029
There is a 2.9% chance of losing 50% of the account from any point on the equity curve. This trader will spend 2.9% of his time at least 50% below a prior maximum equity high.
This equation assumes that the trader cannot reduce the trade size as the account value falls. The monthly standard deviation remains at 13% of the original account size and the average return remains at 6% of the original account size.

Hi Dagobert,

If I understand correctly the questions are about gamble-trading à la James Dean, "Grow fast, lose quick and leave a good looking blown account..."

Question 1:
Don't confuse optimal and maximal. Optimal means the best according to some tradeoff based on some constraints. Here you give no constraints. To "gain as much as possible" is the maximum gain and the answer is 100%, go all-in hit or miss.

You cannot make more since you cannot bet more. It's of course also the highest risk because you cannot lose more...

Question 2:
Here you give implicitely a tradeoff between the probability (we want to have) and the speed (as few trades as possible). Kelly is indeed the answer.

With 40% risk and RR 1:1 you almost double (x1.96) with 2 winners (win, win=49%) and more than double (x2.74) with 3 winners (win, win, win=34%).

You lose 92% of the account if 5 losers come in sequence. You need 8 winners to recover this drawdown.

In average in 100 trades you have 70 winners and 30 losers. A winner multiplies the account by 1.4, a loser by 0.6. The order is unimportant. You multiply your account by 1.4^70*0.6^30=3745.

$374500 from $100 after 100 trades sounds promising BUT if your system is actually 65% winning you multiply by 54 (69 times less). If your actual win/loss is 60% or less your system loses money! To use Kelly you must absolutely be certain of your win/loss ratio. And you just cannot.

If you use 2% you only double after 100 trades but if your win/loss ratio drops to 55% you are still in the money (+20%). Drawdown is 19% for 10 losers.

But you want grow fast, don't you?
10% is the optimal tradeoff. For 100 trades, if your win/loss is 70% you multiply by 33.5 while you still make +65% if the win/loss drops to 55%. The DD is 65% for 10 losers. That's clearly huge but we target $3K from $100. That's a risk of $65. Come on!

Question 3:
You have 30% chance of not making it one single time. Probability of not doubling N times is 1-0.7^N

Probability of not doubling N times:
1 => 30%
2 => 51%
3 => 66%
4 => 76%
5 => 83%
6 => 88%
7 => 92%
8 => 94%
9 => 96%
10 => 97%

So in average $400 or $800 no more. Desappointing isn't it? Only 2.8% chance hitting the million this way. Okay that's much better than lotery

any person who risks more than 2.5% is a gunslinger.... how many can accept 50-70%drawdowns? this are the draw Downs needed to produce really outstanding results.

Sure but put things in perspective. We are talking 80% DD on... a hundred bucks. Nothing harmful. Like people scratching a lotery card every week... --At least that's the way I understood the OP's question--

This said, if he really has got a 70% win/loss with 1:1 RR, burning $100 per month just to try doubling 10 times or lose them all... There is 47.9% chance is becomes a millionnaire in less than 9 years, after burning less than $10k.

Now I agree with you 100% for a real amount. Seriously, let's imagine he made it 13 times. He's now sitting on half a million. Would he really be ready to pull the trigger the 14th time?

ERRATUM: I was wrong by one zero... You have 47.9% chance becoming a millionnaire after 108 months. You need doubling 14 times... That's still a million for 10K flipping a coin for ten years.

Pemully, I think you can't say that. Ofcourse, if you don't have a serious edge, it's not wise to risk too much.^^
But if you know your expectations, risking too less is just lost opportunity.


Thank you very much for your replies.
PipMeUp especially. Calculations like yours are what i was looking for .

It's helping to get your initial investment out when you reached higher amounts. I was actually looking for a 80%+ calculation, but said 70% to be on the safe side.

That 10% equity risk for each trade calculation you did sounds nice. Also what happens when your win% drops from 70% win to 55% win.

Can you tell what expectations there are for 80% win chance, which may drop to 70% or upto 65% win? More drawdown risk but faster "doubling" time? Much more gain than the +3350% equity with 70% win?


Cheers & good luck to your trading,
D

Hi Dagobert,

Keep in mind all of this is highly theoritical because I didn't take spread, minimum tradable lot size or margin requirement into account. As Redlion said you will have variations in the performances of your system.

As long as you only have one trade open at a given time, here are the explanations:
p = your win/loss for 100 trades
r = percentage of risk
w = the multiplication factor for a winner
l = the multiplication factor for a loser

For R:R = 1:1 you have
w = 1+(r/100)
l = 1-(r/100)
Example with a 5% risk, a winner multiplies the account by 1.05 (+5%) and a loser multiplies the account by 0.95 (-5%).

Say you have this sequence: win-win-loss-win-loss your account is multiplied by w*w*l*w*l. The multiplication is commutative (AxB=BxA) so the order of the win-loss has no importance. If you take 100 trades, in average, p win and 100-p lose. The account will be multiplied by w^p x l^(100-p).

p = 80%
r = 10%
=> 1.1^80 x 0.9^(100-80) = 249.04 (rather irrealistic)

p = 70%
r = 10%
=> 1.1^70 x 0.9^(100-70) = 33.478 (a super high performance)

p = 55%
r = 10%
=> 1.1^55 x 0.9^(100-55) = 1.65

p = 40%
r = 10%
=> 1.1^40 x 0.9^(100-60) = 0.08 (expectancy is now negative the system cannot win)

Do you have by any chance a long enough history of trades (>100)?

Have fun but only trade this kind of MM with pocket money or demo (pleeease!).

Have you tried the fixed ratio?

Thanks for the reply PipMeUp. That's the right advice for anyone reading here, first do a set of trades. All trades must striclty follow your ruleset. Try atleast 20 trades to get an idea about the performance. I know that there was/is much more upside potential when doing my trades so that's why I'm curious to tweak it even more in terms of growth. Ofcourse this tweaking is kind of theoretical and leaves no room for errors if you maximize your reward by increasing your risk based on your past performance (for example using the kelly formula). But these "maximum risk values without hurting yourself" give a good estimate, for example if you have a good strategy but risk far too less than the optimum/maximum you calculated - then you should consider (maybe slowly) approaching this value.

These for example are 25+ trades (pic). Risk taken i varied each trade, but maximum risk was never greater than 8%. Also the RR actually is not 1:1 but greater. Win at around ~80% mostly, 84% at this one. So there is still room up for risk. I also tried with 80% risk, which is obviously far too much, reached 10+ times equity, then crashed (insane but funny ).
So for anyone reading, I think the Kelly value is right, it's the maximum you should ever risk after perfectly knowing your system and forwardtesting it.

@alterbest same size everytime you mean? That's probably a good choice for most average profitable traders. The downside of it is that you can't grow small to big that fast and the more you win, the more opportunity is lost if you don't increase your size (which is good for stable, risk decreasing returns ofcourse).

Hi,

Thumb up for the trades

Kelly's formula doesn't work if the outcome isn't "win X or lose Y". The generalized version is a little bit more complicated. It becomes a high degree polynomial to solve. If you have the values in pips I may try. Also interresting is that you have got a cluster of 3 losers but the sample is too small to know wether it is meaningful or not.