Minimum Returns, Concave And Convex Bettors

Minimum Return Ratio

Returning once again to our prototypical horse race - Haigh (1999), Appendix V - we will, on this occasion, calculate a Minimum Return Ratio (using our three selections - Alpha, Bravo, and Charlie), which is the following metric:

MinimumRatio = MininumReturn / TotalStakes.

The simple calculation is demonstrated in this Excel image:

and can be calculated in a PowerShell CLI using this one-liner:

@(@(2.625,30),@(3.25,53),@(5.50,31)) | ForEach-Object{$TotalStakes=0; $MinReturn=[Double]::MaxValue} {If ($_.GetType().IsArray) {$TotalStakes+=$_[1]; $MinReturn=[Math]::Min($MinReturn,$_[0]*$_[1])}} {New-Object PSObject -Property @{ 'MinimumReturn' = ('{0:N2}' -f $MinReturn); 'TotalStakes' = ('{0:N2}' -f $TotalStakes); 'MinimumRatio' = ('{0:N2}' -f (($MinReturn/$TotalStakes)))}} | Format-Table -Auto

with the output, as follows:

TotalStakes MinimumReturn MinimumRatio
----------- ------------- ------------
114.00      78.75         0.69

Simple Example

The Minimum Return Ratio for a bet is calculated by dividing the minimum return by the total stakes. Here's a step-by-step guide on how to calculate it:

1. Identify the selections: Alpha, Bravo, and Charlie.
2. Calculate the total stakes: Add up the total amount you have staked on all selections.
3. Calculate the minimum return: The minimum return is the smallest amount you could win from any of your selections.
4. Calculate the Minimum Return Ratio: Divide the minimum return by the total stakes.

Say we have bet on the three selections with the following odds and stakes:

• Alpha: Odds 2.625, Stake £30
• Bravo: Odds 3.25, Stake £53
• Charlie: Odds 5.50, Stake £31

The total stakes would be £30 + £53 + £31 = £114.

The returns for each horse would be:

• Alpha: 2.625 * £30 = £78.75
• Bravo: 3.25 * £53 = £171.25
• Charlie: 5.50 * £31 = £170.50

The minimum return is the smallest of these, which is £78.75. So, the Minimum Return Ratio would be £78.75 / £114 = 0.69.

Therefore, the smallest amount you could win (if Alpha wins) is only 69% of the total amount you have staked.

Concave And Convex Bettors

Our Minimum Return Ratio (MinimumRatio) allows us to operationally distinguish between concave and convex bettors:

• Concave Bettor: [MinimumRatio > 0.00]. They are willing to accept bets where the potential return may be less than the total stakes, indicating a higher risk tolerance.
• Convex Bettor: [MinimumRatio > 1.00]. They only accept bets where the potential return is greater than the total stakes, indicating a lower risk tolerance.

In other words, even when both bettors agree on what is a value bet and calculate the recommended Kelly or Bayes stake, the Convex Bettor has the additional requirement that the minimum return from all three selections is greater than the total stakes invested! Put this difference in approach down to their contrasting risk profiles.

These terms are derived from the mathematical concepts of concave and convex functions. In a concave function, the slope decreases as you move along the x-axis, similar to the risk tolerance of a concave bettor. Conversely, in a convex function, the slope increases as you move along the x-axis, similar to the risk tolerance of a convex bettor.

It's important to note that these strategies reflect different risk profiles and neither is inherently better than the other. The choice between being a concave or convex bettor depends on an individual's risk tolerance and trading objectives.