Of Sizing Bets – Kelly’s Formula

“To me, it’s obvious that the winner has to bet very selectively. It’s been obvious to me since very early in life. I don’t know why it’s not obvious to very many other people.” – Charlie Munger

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Q: How much to invest in a given opportunity?
A: One way is to use “Kelly’s Formula”. A scientific way of determining an optimal bet size given the Edge and Odds. Edge is also referred to as the “Expected Value”.

Kelly’s Formula is defined as:
K= (p* +veOdds – q * -veOdds)/Sum of all +veOdds
K: is the fraction of the current bankroll to wager

  • p: = the probability of winning
  • q: = the probability of incurring a 100% loss = 1 – p
  • +ve Odds = the net winning odds received on the wager.
    Usually quoted as +veOdds against 1 eg. 3 to 1, 10 to 1,…. If the odds are 7 to 2, then the +veOdds = 3.5 to 1
  • -ve Odds = the net losing odds incurred on the wager.
    Usually quoted as +veOdds against 1 eg. 3 to 1, 10 to 1,…. If the odds are 7 to 2, then the +veOdds = 3.5 to 1
Example 1:
Say the probability of winning an event is 70% and the winning Odds are 4:1. The Odds of losing would be 30% and loss size would be 100%. Then:
p=0.7, +veOdds=4,
q=(1-0.7)=0.3, -veOdds=1
then, K=(0.7*4-0.3*1)/4 = 0.625
So the optimal bet size would be 62.5% of the bankroll.
Use Kelly’s Extended Formula, for multiple probabilities and odds:
K= (p1*+veOdds1 + p2*+veOdds2 + p3*+veOdds3 – q*-veOdds)/(+veOdds1+ +veOdds2+ +veOdds3)

Example 2:

Event 1, p1(60%) and the +veOdds1=2:1, or 200%
Event 2, p2(20%) and the +veOdds2=4:1 or 400%
Event 3, p3(10%) and the +veOdds3=6:1 or 600%
Event 4, q, the probability of loss (10%), and –veOdds=1:1 or 100%

K = (0.6*2+0.2*4+0.1*6 – 0.1*1) / (2+4+6) = 0.2 or 20% of the bankroll.

Husain Kothari
January 13th, 2013

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