Multiple Selections and Sharpe Ratio

Full or fractional Kelly staking has been in the foremost of betting staking plans. Using the available odds, and our estimate of the "fair" probability of an event happening, we can calculate what proportion of our bank should be staked in each bet. However, what happens if there are more than one, say n, concurrent selections? How much should one stake in total on the n selections, and how much on each of the selections?

What if we looked at the concept of expected reward-to-risk ratio? In finance, this is also known as Sharpe Ratio and it measures the excess return of an investment per unit of risk. Let's denote a bet with a letter i. The estimated probability of the bet winning is p_i while the available odds are o_i.

The bet would represent value if p_i > 1/o_i. Now, if we invest one unit in this bet the expected profit becomes:

E[Profit_i] = (p_i * o_i - 1)

The standard deviation (a measure of the risk) of the bet's profit can be shown to be:

SD[Profit_i] = o_i * Sqrt[ p_i * (1 - p_i) ]

Denoting the expected profit and its standard deviation as E_i and SD_i respectively, the Sharpe ratio for a unit stake in a single bet is given by:

SR_i = E_i / SD_i

When we have more than one selections, i = 1, 2, ..., n, and we invest x_i units in bet i, the total expected profit is given by:

ExpTotProf = x₁ * E₁ + x₂ * E₂ + ... + x_n * E_n = Sum(x_i * E_i)

whereas its standard deviation (because of independence between bets) now becomes:

SDTotProf = Sqrt ( (x₁ * SD₁)² + (x₂ * SD₂)² + ... + (x_n * SD_n)² )

The Sharpe Ratio is given by:

SR = ExpTotProf / SDTotProf

This means that SR is now simply a function of the available odds, the estimated probabilities and the stake in each bet. We can therefore select those stakes (x₁, x₂, ..., x_n) which maximize SR with respect to some constraint e.g Sum(x_i) = 1.

It turns out that the solution to this optimization problem is simply selecting stakes x_i proportional to E_i / SD_i² or more specifically:

x_{i =} (E_i / SD_i²) / Sum[ E_i / SD_i² ]

Although this may sound heavy, it's quite simple in practical terms. Consider 4 bets, available at odds of 2.05, 1.50, 5.00 and 2.50. Let the estimated probabilities for these bets be 60%, 68%, 22% and 45% respectively. If you follow the steps above you should get the following figures:

Note that with the calculated allocation we have achieved a reward-to-risk ratio of 0.256, which is higher than the ratio that we would have achieved had we invested all of our stake on Bet 1 (which had a Sharpe ratio of 0.229).