A New Kelly Formula

[nonlinear5]

Nk = ( s1*s2*s2 - s1*s1*s3 )/( s2*s2*s2 + s1*s1*s4 - 2*s1*s2*s3 ) ,
where
Nk is the estimation of the true Kelly fraction produced by the new formula ;
S1 == sum[ Ri ]_i=1toN ;
S2 == sum[ Ri*Ri ]_i=1toN ;
S3 == sum[ Ri*Ri*Ri ]_i=1toN ;
S4 == sum[ Ri*Ri*Ri*Ri ]_i=1toN.

Well, let's see. Consider a simple case of a trading strategy which always loses, ether 1%, or 2%, with equal probability.

The classically defined Kelly Criterion would prescribe betting 0% of one's capital on this strategy, which makes total sense. Your version would compute Kelly equal to 75, which makes no sense at all.

 

[kut2k2]

Well, let's see. Consider a simple case of a trading strategy which always loses, ether 1%, or 2%, with equal probability.

The classically defined Kelly Criterion would prescribe betting 0% of one's capital on this strategy, which makes total sense. Your version would compute Kelly equal to 75, which makes no sense at all.

Thanks for pointing that out. Fortunately, this is an easy fix:

Nk = max[0, s1]*(s2*s2 - s1*s3)/(s2*s2*s2 + s1*s1*s4 - 2*s1*s2*s3)

 

[nonlinear5]

Thanks for pointing that out. Fortunately, this is an easy fix

That's better. However, something fundamental is still missing. Consider trading strategy TS with returns:
R(TS): {+20%, -5%}

Your formula would compute Kelly equal to 7.5%. That's way too small for this distribution. Intuitively, it should be much higher, somewhere around 95% or so.

 

[kut2k2]

That's better. However, something fundamental is still missing. Consider trading strategy TS with returns:
R(TS): {+20%, -5%}

Your formula would compute Kelly equal to 7.5%. That's way too small for this distribution. Intuitively, it should be much higher, somewhere around 95% or so.

The formula computes 7.5, not 7.5%

 

[nonlinear5]

The formula computes 7.5, not 7.5%

Ok, so what is the meaning of that 7.5? Borrow 650% of your stake, and bet everything on every trade?

 

[kut2k2]

Ok, so what is the meaning of that 7.5? Borrow 650% of your stake, and bet everything on every trade?

Of course. The most you could lose on any single trade is 5%, which means you'd have to trade 20 times your account to risk losing it all at once. You're only trading 7.5 times your account. When you lose, you lose 37.5%, but when you win, you gain 150%. These are equally likely outcomes.

 

[nonlinear5]

The most you could lose on any single trade is 5%, which means you'd have to trade 20 times your account to risk losing it all at once.

That's not how I would calculate it, as a trader. If I borrow 6.5 times of my equity, and bet the entire amount on every trade, it would take only 2 losing trades in a row to experience a 73% drawdown on my equity, with this particular strategy. And after only 3 losing trades in a row, you would in all likelihood get a margin call, at which point you would be totally wiped out.

 

[kut2k2]

That's not how I would calculate it, as a trader. If I borrow 6.5 times of my equity, and bet the entire amount on every trade, it would take only 2 losing trades in a row to experience a 73% drawdown on my equity, with this particular strategy. And after 3 or 4 losing trades in a row, you would in all likelihood get a margin call, at which point you would be totally wiped out.

Two loses in a row will give you a 61% drawdown on your equity. And if your luck is so bad that you experience a string of losses in the beginning (with no compensating wins) enough to wipe you out, that's prolly a sign that you shouldn't be trading.

 

[nonlinear5]

Two loses in a row will give you a 61% drawdown on your equity.

Let's see. Suppose my equity is $100K. Then I borrow $650K, and bet the entire available amount on every trade.

-- after losing trade #1, my equity is 750,000 * 0.95 - 650,000 = $62,500
-- after losing trade #2, my equity is 712,500 * 0.95 - 650,000 = $26,875

So, I had 100K, and ended up with $26,875. That's a 73% drawdown on equity after 2 trades. And after 1 more losing trade, my equity would go negative, which would generate a margin call. At that point, I would be forced to return the entire borrowed capital, and my equity would be entirely wiped out.

Now compare this to a well known, "standard" solution. The continuous Kelly (developed by Edward Thorp) would be about 2.4. Taking the conventional "quarter Kelly" would evaluate to 2.4 * 0.25 = 0.6. That is, with this this particular strategy, you should bet 60% of your account size. That's quite reasonable, given the distribution. Your formula gives the Kelly 12.5 times larger than that.

 

[kut2k2]

Let's see. Suppose my equity is $100K. Then I borrow $650K, and bet the entire available amount on every trade.

-- after losing trade #1, my equity is 750,000 * 0.95 - 650,000 = $62,500
-- after losing trade #2, my equity is 712,500 * 0.95 - 650,000 = $26,875

So, I had 100K, and ended up with $26,875. That's a 73% drawdown on equity after 2 trades. And after 1 more losing trade, my equity would go negative, which would generate a margin call. At that point, I would be forced to return the entire borrowed capital, and my equity would be entirely wiped out.

Now compare this to a well known, "standard" solution. The continuous Kelly (developed by Edward Thorp) would be about 2.4. Taking the conventional "quarter Kelly" would evaluate to 2.4 * 0.25 = 0.6. That is, with this this particular strategy, you should bet 60% of your account size. That's quite reasonable, given the distribution. Your formula gives the Kelly 12.5 times larger than that.

OK, this is my last post on this sub-topic .

After one loss, your drawdown is 1-(1-.05*7.5) = 37.5%
After 2 losses, your drawdown is 1-(1-.05*7.5)^2 = 61%
After 3 losses, your drawdown is 1-(1-.05*7.5)^3 = 76%
After 4 losses, your drawdown is 1-(1-.05*7.5)^4 = 85%
After 5 losses, your drawdown is 1-(1-.05*7.5)^5 = 90.5%

More importantly, your expectation is √((1+.20*7.5)*(1-.05*7.5)) = 1.25 : when you bet 7.5 times your equity in [ +.20, -.05 ], you can expect to gain 25 cents on average for every equity dollar wagered.

Thorp's ratio = mu/sigma^2.
Mu = .5*.2 - .5*.05 = .075 ; sigma^2 = 0.015625
Thorp's ratio = 4.8

Thorp's expectation = √((1+.2*4.8)*(1-.05*4.8)) = 1.2204917 :
When you bet 4.8 times your equity, you can expect to gain 22 cents on average for every equity dollar wagered.

That is all.