Kelly Criterion question for kut2k2 and anybody who is interested in it

E

espresso

Hi kut2k2 and folks at ET

First, thanks kut2k2 a lot for the great posts on Kelly Criterion (KC) in past years!

I did the calculations based on your posts:
Kelly Approximation:
https://www.elitetrader.com/et/threads/kelly-for-traders.102205/ (Kelly for Traders)
Bad Kelly v.s. LessBad Kelly:
https://www.elitetrader.com/et/threads/bad-kelly.260462/ (Bad Kelly)

In my case, Kelly Approximation (#3 below) produces 2.7177, and LessBad Kelly (#2.1 below) produces 4.6241, which means putting all money in and even borrowing more(??) While traditional KC (#1.2 below) produces 0.4846 (48%), which seems reasonable to me.

I currently use traditional Kelly (#1.2 below) for position sizing.

So, my quesitons are:
a) for LessBad Kelly, #2.1 and #2.2, which one is correct? if #2.1 is correct, how to interpret its output (4.6241)?
b) for Kelly Approximation, #3, is there anything I missed or incorrect in calulation? if it's correct, how to interpret its output (2.7177)?

Thank you!


Trading stats
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Net P&L ____ Loss ____ Profit
295.38 ____ -91.09 ____386.47

_______ Won ____ Lost ____ Total
Trade # ____ 52 ____ 30 ____ 82

Average trade cost $28.977 (per trade)

$ per winning trade: $386.47 / 52 = $7.4321
meaning $7.4321 won given wager $ 28.977

$ per losing trade: $-91.09 / 30 = $-3.0363
meaning $3.0363 loss given wager $ 28.977


>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#1.1 >>>>>>>>> (Original Kelly, with arbitrary $ won/loss ratio)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
K = Kelly criterion fraction of capital to bet
W = arbitrary $ won/loss ratio
Pw = Probability of winning
Pl = Probability of losing

Pw = 52 / 82 = 0.6341
Pl = 1 - Pw = 0.3659
W = 386.47 / 91.09 = 4.2427

K = Pw – (Pl/W)
= 0.6341 - (0.3659 / 4.2427) = 0.6341 - 0.0862 = 0.5479 (~55%)

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#1.2 >>>>>>>>> (Original Kelly, with averaged $ won/loss per trade)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
W' = averaged $ won/loss ratio (per trade)
W' = (386.47 / 52) / (91.09 / 30) = 7.4321 / 3.0363 = 2.4478

K' = Pw – (Pl/W')
= 0.6341 - (0.3659 / 2.4478) = 0.6341 - 0.1495 = 0.4846 (~48%)

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#2 >>>>>>>>> (LessBad Kelly by kut2k2)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
K = p/S - (1-p)/R , where
K is the Kelly ratio,
p is the winrate,
R is the average winning trade return,
S is the absolute value of the average losing trade return.

#2.1 --- based on $ won/loss per $ wagered
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
p = Pw = 0.6341
R = 7.4321 / 28.977 = 0.2565 (avg return per winning trade)
S = - 3.0363 / 28.977 = - 0.1048 (avg return per losing trade)

K = 0.6341 / 0.1048 - 0.3659 / 0.2565 = 6.0506 - 1.4265 = 4.6241 (???)

#2.2 --- based on arbitrary $ won/loss per trade
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
R' = 7.4321 (arbitrary $ per winning trade)
S' = - 3.0363 (arbitrary$ per losing trade)

K' = 0.6341 / 3.0363 - 0.3659 / 7.4321 = 0.2088 - 0.0492 = 0.1596 (~0.16%)


>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#3 >>>>>>>>> (Kelly approximation by kut2k2)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Pw = 52 / 82 = 0.6341
Pl = 1 - Pw = 0.3659
Rw = 7.4321 / 28.977 = 0.2565 (winning return % per trade)
Rl = - 3.0363 / 28.977 = -0.1048 (lose return % per trade)

K = ( Pw * Rw + Pl * Rl) / ( Pw * Rw ^ 2 + Pl * Rl ^ 2 )
= (0.6341 * 0.2565 + 0.3659 * (-0.1048)) / (0.6341 * 0.2565 ^ 2 + 0.3659 * (-0.1048) ^ 2)
= 0.12430033 / (0.6341 * 0.06579225 + 0.3659 * 0.01098304)
= 0.12430033 / 0.045737560061 = 2.7177 (???)
 
I haven't read the threads on Kelly on this forum. The classic Kelly is the only one I have ever heard of.

I was wondering if it is reasonable to apply Kelly criterion on such a small sample of trades, so my first thought was to calculate sigma (standard deviation , considering the implied probability of an average trade) and to see how many standard deviations have you got here. In other words, to try to determine what are the odds of this being just luck/randomness. Does anyone here know how to do that?
 
aex said:
I haven't read the threads on Kelly on this forum. The classic Kelly is the only one I have ever heard of.

I was wondering if it is reasonable to apply Kelly criterion on such a small sample of trades, so my first thought was to calculate sigma (standard deviation , considering the implied probability of an average trade) and to see how many standard deviations have you got here. In other words, to try to determine what are the odds of this being just luck/randomness. Does anyone here know how to do that?
More...

These are true stats for one instrument traded in one month, and these stats remain pretty similar for each month for past two years. However, the point here is that I wanted to know which is the proper way to calc Kelly Criterion. :)
 
(Continued)

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#4 >>>>>>>>> (New Kelly Formula by kut2k2)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
I also tried this one (by kut2k2)
New Kelly Formula
https://www.elitetrader.com/et/threads/a-new-kelly-formula.291307/

It gives output:
S1 0.1243
S2 0.0457
S3 0.0103
S4 0.0028
Nk = 4.6241


>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#5 >>>>>>>>> (Numerical optimization)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Finally, I used Excel solver to find numerically optimal value: K = 4.6241

similar idea is discussed here:
https://www.elitetrader.com/et/threads/kelly-again.102179/


So given my trade stats, the Kelly ratio is 4.6241,
where #5 produces the most accurate output, #4 and #2.1 produes similar value thru approximation.

With this K = 4.6241, does it mean borrow more money or use more leverage, how to interpret it if we use leveraged/margin account?
 
I can't find any reference to another kelly formula besides the original one in probability related sites.
 
espresso said:
Hi kut2k2 and folks at ET

First, thanks kut2k2 a lot for the great posts on Kelly Criterion (KC) in past years!

I did the calculations based on your posts:
Kelly Approximation:
https://www.elitetrader.com/et/threads/kelly-for-traders.102205/ (Kelly for Traders)
Bad Kelly v.s. LessBad Kelly:
https://www.elitetrader.com/et/threads/bad-kelly.260462/ (Bad Kelly)

In my case, Kelly Approximation (#3 below) produces 2.7177, and LessBad Kelly (#2.1 below) produces 4.6241, which means putting all money in and even borrowing more(??) While traditional KC (#1.2 below) produces 0.4846 (48%), which seems reasonable to me.

I currently use traditional Kelly (#1.2 below) for position sizing.

So, my quesitons are:
a) for LessBad Kelly, #2.1 and #2.2, which one is correct? if #2.1 is correct, how to interpret its output (4.6241)?
b) for Kelly Approximation, #3, is there anything I missed or incorrect in calulation? if it's correct, how to interpret its output (2.7177)?

Thank you!


Trading stats
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Net P&L ____ Loss ____ Profit
295.38 ____ -91.09 ____386.47

_______ Won ____ Lost ____ Total
Trade # ____ 52 ____ 30 ____ 82

Average trade cost $28.977 (per trade)

$ per winning trade: $386.47 / 52 = $7.4321
meaning $7.4321 won given wager $ 28.977

$ per losing trade: $-91.09 / 30 = $-3.0363
meaning $3.0363 loss given wager $ 28.977


>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#1.1 >>>>>>>>> (Original Kelly, with arbitrary $ won/loss ratio)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
K = Kelly criterion fraction of capital to bet
W = arbitrary $ won/loss ratio
Pw = Probability of winning
Pl = Probability of losing

Pw = 52 / 82 = 0.6341
Pl = 1 - Pw = 0.3659
W = 386.47 / 91.09 = 4.2427

K = Pw – (Pl/W)
= 0.6341 - (0.3659 / 4.2427) = 0.6341 - 0.0862 = 0.5479 (~55%)

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#1.2 >>>>>>>>> (Original Kelly, with averaged $ won/loss per trade)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
W' = averaged $ won/loss ratio (per trade)
W' = (386.47 / 52) / (91.09 / 30) = 7.4321 / 3.0363 = 2.4478

K' = Pw – (Pl/W')
= 0.6341 - (0.3659 / 2.4478) = 0.6341 - 0.1495 = 0.4846 (~48%)

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#2 >>>>>>>>> (LessBad Kelly by kut2k2)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
K = p/S - (1-p)/R , where
K is the Kelly ratio,
p is the winrate,
R is the average winning trade return,
S is the absolute value of the average losing trade return.

#2.1 --- based on $ won/loss per $ wagered
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
p = Pw = 0.6341
R = 7.4321 / 28.977 = 0.2565 (avg return per winning trade)
S = - 3.0363 / 28.977 = - 0.1048 (avg return per losing trade)

K = 0.6341 / 0.1048 - 0.3659 / 0.2565 = 6.0506 - 1.4265 = 4.6241 (???)

#2.2 --- based on arbitrary $ won/loss per trade
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
R' = 7.4321 (arbitrary $ per winning trade)
S' = - 3.0363 (arbitrary$ per losing trade)

K' = 0.6341 / 3.0363 - 0.3659 / 7.4321 = 0.2088 - 0.0492 = 0.1596 (~0.16%)


>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
#3 >>>>>>>>> (Kelly approximation by kut2k2)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Pw = 52 / 82 = 0.6341
Pl = 1 - Pw = 0.3659
Rw = 7.4321 / 28.977 = 0.2565 (winning return % per trade)
Rl = - 3.0363 / 28.977 = -0.1048 (lose return % per trade)

K = ( Pw * Rw + Pl * Rl) / ( Pw * Rw ^ 2 + Pl * Rl ^ 2 )
= (0.6341 * 0.2565 + 0.3659 * (-0.1048)) / (0.6341 * 0.2565 ^ 2 + 0.3659 * (-0.1048) ^ 2)
= 0.12430033 / (0.6341 * 0.06579225 + 0.3659 * 0.01098304)
= 0.12430033 / 0.045737560061 = 2.7177 (???)
More...

82 trades is not enough to give you much of a statistical idea about what your 'true' win / loss is. So its really a moot point what Kelly formula you use. All the numbers you are quoting for fractional bet size are way too high.

What's your average holding period?

GAT
 
globalarbtrader said:
82 trades is not enough to give you much of a statistical idea about what your 'true' win / loss is. So its really a moot point what Kelly formula you use. All the numbers you are quoting for fractional bet size are way too high.

What's your average holding period?

GAT
More...


Let's simply assume the trading stats given here is reasonable and reliable. :)

In fact, It's a live, leveraged account, mainly for currencies, the $ amount has been proportionally modified, the stats is based on one currency pair for simplicity. However it shows the essence of the performance.

The holding period is a range from a few hours to a few days, some positions could be held for a few weeks. my trading stats shows a winning rate of 55% - 65%, the return could be avg 15% when it won, or avg -10% when it lost. And there is small chance (<10%) that it could lose 30% (due to unexpected events).

In the past, I always allocate a small % of available fund, based on 2/3 of classical Kelly, and further divide it 50/50 for each currency pair (I only trade two pairs). And I have a feeling that the money was somehow under utilized in most time.

That's why I'm looking for opinions on how people calc and use Kelly Criterion in their trading.

With some recent research, now, I'm OK with numerically optimized solution for Kelly Criterion, which means I could always find an optimal ratio using past trading stats.

Now, the problem is that the optimal value of Kelly Criterion given my trading stats is 4.6241, based on numerical optimization.

How could I interpret this ratio and apply it in real trading? I guess kut2k2 may suggest me borrowing more (3.6 times of my current), HAHAHA, but it's already a leveraged account...
 
aex said:
I can't find any reference to another kelly formula besides the original one in probability related sites.
More...

Never mind, aex. I found Kelly approximations (#2.1, #2.2, #3, #4) works most time. But I'm perfectly OK with optimal ratio that's based on numerical optimization. However, as I said above, the optimal value is much larger than 1, how could I apply it practically?
 
I wouldn't consider using those kelly approximations. The basic kelly is the real deal imo and if it offers a larger percentage than the comfort level, I would use fractions of that.
 
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