14th August 2013, 11:24 AM
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Member
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Join Date: Jul 2013
Posts: 13
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Quote:
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Originally Posted by Lord Greystoke
How could this be, Ma' Ludi?
LG
PS I call on Demodocus for further input here |
If you have an edge, there is no faster way to attain optimal geometric growth for your assets than the Kelly Criterion. Who should not use the Kelly Criterion to size their bets? Anyone who does not have an edge. If you use the Kelly Criterion and you don’t have an edge, it will evidence itself very quickly. Even for those who do have an edge, the variance and risk of ruin can be gut-wrenching.
note: all examples below use American odds
Example: horse with 5:1 odds and an expected 3% edge
BR = bankroll = $100,000
p = expected win probability
q = expected loss probability
A = net odds (American)
e = (A+1)p-1>0
f* = fraction of BR to invest = e/A
f*BR = amount invested
EV = expected value = ef*BR
p = .1717
A = 5
e = (5+1)*.1717-1 = .03
f* = .03/5 = .006
f*BR = $100,000*.006 = $600
EV = $600*.03 = $18
The short form:
edge = 3%
odds = 5
edge/odds = 3%/5 = .6%
optimal bet = .6% * $100,000 = $600
EV = $600*3% = $18
With a 3% edge and a $100,000 bankroll, your optimal bet on a 5:1 horse is $600. Your expected value is $18.
Constructing an optimal hedge bets on two or more horses is a little more difficult. Let’s say that you have a race with a 5:1 and a 25:1 horse, each with an expected 3% edge:
P(a) = .1717
P(b) = .0396
First you calculate what is called the reserve rate:
r = reserve rate = (1-sum(each p bet))/(1-sum(each 1/(A+1)))
r = (1-(.1717+.0396))/(1-((1/(5+1))+(1/(25+1))))) = .9923
Then you calculate the optimal percentage of your bankroll to invest:
f* = p-r/A
f*(a) = .1717-(.9923/6) = .00629
f*(b) = .0396-(.9923/26) = .00145
The optimal amount that you bet on each horse is:
f*(a)BR = .00629*100,000 = $629
f*(b)BR = .00145*100,000 = $145
Your expected value is:
EV(a) = $629*.03 = $18.87
EV(b) = $145*.03 = $ 4.36
EV(total) = $23.23
Note that EV(a) = $18.87 > EV = $18.00 above. Even though they’re both 5:1 horses with an expected 3% edge, you are able to optimally bet more on a given horse if it is hedged with one or more other horses. Of course, the Kelly Criterion can also be applied to both vertical and horizontal exotics. You should always place all bets in a race for which you have a positive expected value.
With relatively large bankrolls and/or small pools, the calculated optimal Kelly bet may exceed the amount that will return maximum expected value from the parimutuel pool(s). To calculate the optimal bet in these cases involves a delightful foray into the world of non-linear programming, far beyond the scope of this thread.
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