Resolving a dispute over the Kelly formula

G

Ghost of Cutten

In another thread (http://elitetrader.com/vb/showthread.php?s=&threadid=251271) I am in dispute with another poster (kut2k2) over the correct calculation of the Kelly formula. Because of this, we are disagreeing over which trade is the more attractive and should be bet on with bigger size. Since kut2k2 responded to my disputation of his calculations by putting me on ignore, I am unable to see how he would respond to my post clarifying why I disagreed. I would therefore like to get input from other people familiar with the Kelly criterion to double check which approach they think is correct, so I can be more sure which one is right.

Here is a summary:

Initial post: "Consider two hypothetical setups. One setup has a winning rate of 55% with a potential profit of $433 and a potential loss of $407. The other setup has a winning rate of 10%, a potential profit of $1000 and a potential loss of $50. Both have the same expectancy at $55.

Which setup do you find more attractive and why?"

My Kelly calculation: "System A: Kelly = (.55*(433/407+1)-1)/(433/407) = 12.7%

System B: Kelly = (0.1*(1000/50+1)-1)/(1000/50) = 5.5%"

kut2k2's calculation:

"Finally we look at the Kelly fractions:

k(A) = .55/(407/500) - .45/(433/500) = 0.156

k(B) = .1/(50/500) - .9/(1000/500) = 0.55

k(C) = .001/(45.05/500) - .999/(100,000/500) = 0.0061

• Setup B wins this comparison by a wide margin."

It appears to me as though kut2k2 is mistakenly adjusting the win rate by the *loss* payout, rather than by the win payout, and therefore getting an incorrect result.

Here is the Kelly formula:

http://www.investopedia.com/article...g/04/091504.asp

'The Basics
There are two basic components to the Kelly Criterion:
• Win probability - The probability that any given trade you make will return a positive amount.
• Win/loss ratio - The total positive trade amounts divided by the total negative trade amounts.

These two factors are then put into Kelly's equation:
Kelly % = W – [(1 – W) / R]

Where:
W = Winning probability
R = Win/loss ratio

The output is the Kelly percentage,'

Using the examples above:

Setup A: W (winning probability) = 55%. R (win/loss ratio) = 433/407. Therefore Kelly % = .55 - [(1-.55)/(433/407)] = 0.127 = 12.7%

Setup B: W = 10%. R = 1000/50. Therefore Kelly % = .1 - [(1-.1)/(1000/50)] = 0.055 = 5.5%

Setup A is therefore superior according to Kelly.

I would appreciate clarification, double-checking, and input from other posters.
 
Quote from Ghost of Cutten:

In another thread (http://elitetrader.com/vb/showthread.php?s=&threadid=251271) I am in dispute with another poster (kut2k2) over the correct calculation of the Kelly formula. Because of this, we are disagreeing over which trade is the more attractive and should be bet on with bigger size. Since kut2k2 responded to my disputation of his calculations by putting me on ignore, I am unable to see how he would respond to my post clarifying why I disagreed. I would therefore like to get input from other people familiar with the Kelly criterion to double check which approach they think is correct, so I can be more sure which one is right.

Here is a summary:

Initial post: "Consider two hypothetical setups. One setup has a winning rate of 55% with a potential profit of $433 and a potential loss of $407. The other setup has a winning rate of 10%, a potential profit of $1000 and a potential loss of $50. Both have the same expectancy at $55.

Which setup do you find more attractive and why?"

My Kelly calculation: "System A: Kelly = (.55*(433/407+1)-1)/(433/407) = 12.7%

System B: Kelly = (0.1*(1000/50+1)-1)/(1000/50) = 5.5%"

kut2k2's calculation:

"Finally we look at the Kelly fractions:

k(A) = .55/(407/500) - .45/(433/500) = 0.156

k(B) = .1/(50/500) - .9/(1000/500) = 0.55

k(C) = .001/(45.05/500) - .999/(100,000/500) = 0.0061

• Setup B wins this comparison by a wide margin."

It appears to me as though kut2k2 is mistakenly adjusting the win rate by the *loss* payout, rather than by the win payout, and therefore getting an incorrect result.

Here is the Kelly formula:

http://www.investopedia.com/article...g/04/091504.asp

'The Basics
There are two basic components to the Kelly Criterion:
• Win probability - The probability that any given trade you make will return a positive amount.
• Win/loss ratio - The total positive trade amounts divided by the total negative trade amounts.

These two factors are then put into Kelly's equation:
Kelly % = W – [(1 – W) / R]

Where:
W = Winning probability
R = Win/loss ratio

The output is the Kelly percentage,'

Using the examples above:

Setup A: W (winning probability) = 55%. R (win/loss ratio) = 433/407. Therefore Kelly % = .55 - [(1-.55)/(433/407)] = 0.127 = 12.7%

Setup B: W = 10%. R = 1000/50. Therefore Kelly % = .1 - [(1-.1)/(1000/50)] = 0.055 = 5.5%

Setup A is therefore superior according to Kelly.

I would appreciate clarification, double-checking, and input from other posters.
More...

I don't think there is enough information to decide what "Kelly fraction" to bet. The figures given are absolute dollar amounts, and nothing is given about your current bankroll that produces those dollar returns. Are we getting 433/407 or 1000/50 based on traded size of $1000 or $100000? we do not know, so the Kelly fraction is to be applied to what?

Normally they would say you get $x for each dollar bet, meaning if it does not work out, the loss is $1 (100%). Even if the percentage returns are varied we still need to know the bet size.

The purpose of the fraction is to show what will get you the best compounded return in the long run (assuming the probabilities work out!).
 
Quote from neke:

I don't think there is enough information to decide what "Kelly fraction" to bet. The figures given are absolute dollar amounts, and nothing is given about your current bankroll that produces those dollar returns. Are we getting 433/407 or 1000/50 based on traded size of $1000 or $100000? we do not know, so the Kelly fraction is to be applied to what?

Normally they would say you get $x for each dollar bet, meaning if it does not work out, the loss is $1 (100%). Even if the percentage returns are varied we still need to know the bet size.

The purpose of the fraction is to show what will get you the best compounded return in the long run (assuming the probabilities work out!).
More...

Ok revisiting this. If the amount of loss ($407 and $50) are assumed to be your bet size, yes 12.7% and 5.5% are the Kelly fractions. Which means you need smaller bankroll ($909) for(B) than $3204 for (A) (divide bet size by Kelly fraction). Both generate the same $ expectancy, but as a fraction of the bank roll (55 / 909, 55 / 3204), you get more percentage (6%) for (B) as opposed to (1.7%) for (A). So in the long run, (B) will grossly outperform (A). Of course Kelly cares only about performance, not about reliability or smoothness of curve.
 
Quote from neke:

I don't think there is enough information to decide what "Kelly fraction" to bet. The figures given are absolute dollar amounts, and nothing is given about your current bankroll that produces those dollar returns. Are we getting 433/407 or 1000/50 based on traded size of $1000 or $100000? we do not know, so the Kelly fraction is to be applied to what?

Normally they would say you get $x for each dollar bet, meaning if it does not work out, the loss is $1 (100%). Even if the percentage returns are varied we still need to know the bet size.

The purpose of the fraction is to show what will get you the best compounded return in the long run (assuming the probabilities work out!).
More...

It's actually not necessary to know the bankroll to arrive at a Kelly fraction, since the Kelly fraction is a % of bankroll. If your bankroll is 100k then 12.7% is 12,700, if it's 1 mill then it's 127k etc. All you need to know is the win % rate and the win/loss ratio. Your account may not be big enough to then bet the desired Kelly fraction (e.g. a 5k account with a contract that has a 3k margin requirement can only bet a 1 lot, even if Kelly says bet 1.5 contracts), but that is a problem with your account size, not the ability to calculate the Kelly fraction.

Anyway, my main curiosity here was to decide if I or kut2k2 had made the error in calculating the Kelly fraction, so thanks for your input. Any other takers?
 
Your calculations, ghost, are correct. Note that both of you had the same answer, 5.5%, for setup B.

However, your conclusion that set up A is better because the Kelly fraction is higher than that of setup B is incorrect. All the Kelly fraction says is how much you should bet, on each trade, of your existing bankroll, in order to maximise the long run return.
 
With Kelly betting you will never go broke but you will get close to it very fast.:)

These are correct according to my understanding of Kelly:


My Kelly calculation: "System A: Kelly = (.55*(433/407+1)-1)/(433/407) = 12.7%

System B: Kelly = (0.1*(1000/50+1)-1)/(1000/50) = 5.5%"

All you have to do is divide $55 in each case by average winner size.
 
Quote from Ghost of Cutten:

System A: Kelly = (.55*(433/407+1)-1)/(433/407) = 12.7%
System B: Kelly = (0.1*(1000/50+1)-1)/(1000/50) = 5.5%
More...

I use a simpler formula which gives me the same result as yours:
Kelly = probabilityOfWin - (1 - probabilityOfWin) / (aveProfit / aveLoss)

Kelly(A) = 0.55 - (1 - 0.55) / (433 / 407) = 12.7%
Kelly(B) = 0.1 - (1 - 0.1) / (1000 / 50) = 5.5%
 
Quote from Ghost of Cutten:

Kelly % = W – [(1 – W) / R]

Where:
W = Winning probability
R = Win/loss ratio
More...

Correct.

Quote from nonlinear5:

I use a simpler formula which gives me the same result as yours:
Kelly = probabilityOfWin - (1 - probabilityOfWin) / (aveProfit / aveLoss)
More...

Not simpler. And also not correct, if "aveProfit" stands for average profit. This formula only applies if there are only two possible outcomes. If there are more than two possible outcomes, you can't just average them and apply this equation.
 
Quote from euclid:
This formula only applies if there are only two possible outcomes. If there are more than two possible outcomes, you can't just average them and apply this equation.
More...

There is either a gain or a loss. What's the other "possible outcome"? A break-even trade?
 
Quote from nonlinear5:

There is either a gain or a loss. What's the other "possible outcome"? A break-even trade?
More...


I've never understood that either. What, one of your OTHER systems placed a trade and you don't know which system will hit it's target first?

I've never needed to fully understand Kelly so I never fully investigated it but I suspect that Kelly's formula is just too elegant and too simple to be accepted by some... I noticed something a very long time ago: really smart people, I mean IQ >175 maybe, they come up with very simple solutions to complex problems. I love to work with people like that, they are rare but I found that trying to emulate the way they approach things can be very helpful. I'm assuming that Kelly's thingy is one of those very simple solutions. The simplifying assumptions are that all the trades have about the same win/lose probability and the same win/loss sizes. It's all normalized to percent of account size so it does indeed appear elegant.

Investing pros argue against it because of the potential for account volatility. They know their clients won't like it at some point. Those arguments mean nothing in the context of an individual trader seeking the steepest account balance curve.
 
You must log in or register to reply here.
Back
Top