Optimal Stake based on edge

In the famous journal by John Kelly, he described an optimal staking plan based on your winning edge. The Kelly criterion is simply a formula for calculating how much you should bet based on the edge you have. The important point about this formula is that it is optimised for growth. Therefore the formula tries to reinvest your profits and subject them to risk, a measure which might be too aggressive for some. However, for others with a large enough bankroll and an aim in increasing the size of the bankroll at a maximum rate possible, this formula is ideal.

Now, what is the main concern of betting here? Since the objective is to increase your bankroll as much as possible, you might want to bet as large a capital as possible. However, if you bet too large an amount, you might subject the capital to too much risk. This means that if you lose, you may be severely handicapped to the point you do not possess enough bankroll to recover from the winning selections. On the contrary, betting too small an amount may be too conservative and limit your profits. So what is the ideal stake to place?

Kelly relates the fraction to bet of your bankroll to bet as follows.

Fraction to bet = [ (Decimal odds – 1)*p – (1 – p) ] / (Decimal Odds – 1)

where p = expected winning probability

Before we move on, let us have a good look at the numerator. This actually represents the winning edge you have. For a simple illustration, consider the actual quoted odds of 1.5. This decimal odds imply the outcome will occur with probability 0.67. (1 / 1.5). Suppose you have reason to believe your expected winning probability for this match is 0.8. If this was true, that means the bookmaker made a mistake in offering you the odds of 1.5. Instead, he should have offered you the odds 1.25! (1 / 0.8) Therefore, you actually have an edge over the bookmaker in predicting this outcome. Since you have an edge, the Kelly formula will of course, naturally make you place a bet. For clarity, suppose now you are a very lousy gambler. Your probability of success for this same match is now 0.5. Plug in all the numbers into the numerator, and you will get a negative number! (Denominator is always positive since decimal odds is always greater than 1) What does it mean to bet a negative fraction? Well, it simply means you should not bet on this outcome occuring. Instead you should cross over and bet on the outcome not occuring!

Now for an actual numerical example. I have a $100 bankroll. The actual odds quoted is 1.8 on a particular horse winning the race. (Implied probability of 0.56) I always bet on horses in this range of odds and I have a 60% strike rate. (Implied probability is 0.6, implied odds is 1.67) Plugging in the numbers according to the formula,

Fraction to bet = [(1.8-1)*0.6 – (1 – 0.6)] / (1.8 – 1) = 0.1

Thus, I should bet a stake of 0.1*100 = $10 on the horse winning the match. One important assumption to note is that in the derivation of the formula, Kelly uses two outcomes, ie either it will happen or not happen. For matches involving three outcomes such as soccer, one may wish to use a modified form of the Kelly’s formula. The reader may wish to read the original journal paper here for a more rigourous discussion.