# Model Testing – Measuring Profit & Loss

When developing a betting model it is important to properly measure its performance. Having a formal measure of performance is important because it provides a benchmark with which to test alternative models.

Obviously, when using a model for betting a key measure of performance is the profit or loss obtained when using it, but it’s nice to have more than one metric to identify differences between similarly performing models.

This article is the first of two parts. This part looks at various ways to measure profit. The second article, Model Testing – Measuring Forecast Accuracy, outlines various measures of forecast accuracy in the context of betting models.

If you have not done so already, be sure to check out our article on Post-Sample Evaluation – the Importance of Creating a Holdout Set before Calibrating a Betting Strategy.

### Understanding Sigma (Σ) Notation

Many of the formulas below using sigma notation. For those who can’t recall how to use Σ from their school days, click here to brush up on sigma notation.

# Profit / Loss

It may seem a trivial matter to measure betting profit, however there are numerous ways to go about it. For example, are you accounting for the re-use of funds from previous winning wagers? Below are a four options for measuring the profit – be it hypothetical or real – for a betting model.

### Average Profit (AP)

Presuming the betting model determines your stake amount as well as your bet selection, you can calculate your Average Profit (AP) as the net wager payouts divided by the total number of units wagered:

Where n is the total number of wagers, i denotes the bet number, μi denotes the number of units wagered on bet i and πi denotes the net profit per unit for bet i.

For example, suppose your model makes the following three wagers (we are using a small number to illustrate the calculation):
1. 10 units at 1.92 odds -> winning bet
2. 5 units at 1.90 odds -> losing bet
3. 7 units at 1.94 odds -> winning bet

Our Average Profit is: (1/(10+5+7)) * [ 10*(1.92 – 1) + 5*(-1) + 7*(1.94 – 1) ] = 0.49 = 49%

This figure reflects the average rather than total profit, so it is not impacted by the frequency of betting.

When using this figure it’s important to understand what it represents. The AP figure doesn’t take into account the re-use of betting funds from previous winning wagers. Instead it reflects the average profit per wager. Anything over 0% is excellent, because if you were to factor in the re-use of funds for a betting strategy, the result would be a greater realised profit than the AP figure. Conversely, for anything below 0%, a greater frequency of wagering would result in a larger loss of funds.

### Unit Profit (UP)

The downside of Average Profit is if you have two models that provide the same average – say 1.5%, but one bets at a higher frequency than the other, the figures don’t reflect the improved value from wagering more frequently.

To illustrate, suppose you have two models that achieve the same Average Profit of 1.5% over the course of a season, but the first model made 100 wagers of 10 units each while the second model made only 50 wagers of 10 units. AP ranks the two models the same but there’s better value in a profitable model that bets more frequently. Conversely, if the two models had a -1.5% AP, the one that wagered less frequently would have lost less money.

To factor in bet frequency, we can simply avoid dividing our total profit by the number of units wagered. The result is the Unit Profit (UP):

Where n is the total number of wagers, i denotes the bet number, μi denotes the number of units wagered on bet i and πi denotes the net profit per unit for bet i.

To illustrate, using the same three wagers:
1. 10 units at 1.92 odds -> winning bet
2. 5 units at 1.90 odds -> losing bet
3. 7 units at 1.94 odds -> winning bet

Our Unit Profit is: [ 10*(1.92 – 1) + 5*(-1) + 7*(1.94 – 1) ] = 10.78 units

To see how AP and UP compare, suppose we have an alternative model that makes the following wagers:

1. 10 units at 1.92 odds -> winning bet
2. 10 units at 1.92 odds -> winning bet
3. 5 units at 1.90 odds -> losing bet
4. 5 units at 1.90 odds -> losing bet
5. 7 units at 1.94 odds -> winning bet
6. 7 units at 1.94 odds -> winning bet

The AP and UP are as follows:

Average Profit (AP) = (1/(10+10+5+5+7+7)) * [ 10*(1.92 – 1) + 10*(1.92 – 1) + 5*(-1) + 5*(-1) + 7*(1.94 – 1) + 7*(1.94 – 1) ] = 0.49 = 49%

Unit Profit (UP) = [ 10*(1.92 – 1) + 10*(1.92 – 1) + 5*(-1) + 5*(-1) + 7*(1.94 – 1)+ 7*(1.94 – 1) ] = 21.56 units

The AP is the same for the two models, however the UP is higher for the second model due to it’s higher frequency of wagering for a positive average profit.

### Currency Profit (CP)

If you prefer to measure your profit in actual currency you can multiply the unit profit by the currency amount that each unit represents. For example if 1 unit equals $10.00 AUD, your currency profits for the above examples would be$107.80 and $215.60, respectively. The formula for Currency Profit is: Where n is the total number of wagers, i denotes the bet number, μi denotes the number of units wagered on bet i, πi denotes the net profit per unit for bet i and β denotes the currency amount that each unit represents. Using the following example: 1. 10 units at 1.92 odds -> winning bet 2. 5 units at 1.90 odds -> losing bet 3. 7 units at 1.94 odds -> winning bet 1 unit =$10 (β = $10) we calculate the currency profit as: Currency Profit (CP) =$10.00 * [ 10*(1.92 – 1) + 5*(-1) + 7*(1.94 – 1) ] = \$107.80

### Percentage Profit (PP)

If you have a betting model that calculates stakes as a percentage of funds available rather than by using set range of units, then you can calculate the Percentage Profit (PP) for the model. This is calculated as a percentage of your starting balance.

Note that because we will be multiplying bet results rather than summing them, the following notation uses Π rather than Σ. For those who are rusty on their high school maths, the illustrative example that follows will show how the Π notation works.

Where n is the total number of wagers, i denotes the bet number, σi denotes the percentage of the account balance wagered on bet i and πi denotes the net profit per unit for bet i.

Consider a model that makes the following wagers:
1. 10% of balance at 1.92 odds -> winning bet
2. 5% of balance at 1.90 odds -> losing bet
3. 7% of balance at 1.94 odds -> winning bet

The Percentage Profit is calculated as:

PP = (1 + 0.10*(1.92 – 1))*(1 + 0.05*(-1))*(1 + 0.07*(1.94 – 1)) – 1 = 10.57%.

Hence, using the wagers above we would have improved our betting balance by 10.57%.

So what happens if the betting frequency is doubled? Consider the following wagers:
1. 10% of balance at 1.92 odds -> winning bet
2. 10% of balance at 1.92 odds -> winning bet
3. 5% of balance at 1.90 odds -> losing bet
4. 5% of balance at 1.90 odds -> losing bet
5. 7% of balance at 1.94 odds -> winning bet
6. 7% of balance at 1.94 odds -> winning bet

The Percentage Profit becomes:

PP = (1 + 0.10*(1.92 – 1))*(1 + 0.10*(1.92 – 1))*(1 + 0.05*(-1))*(1 + 0.05*(-1))*(1 + 0.07*(1.94 – 1))*(1 + 0.07*(1.94 – 1)) – 1 = 22.25%.

Like the Unit Profit measure, Percentage Profit rewards models that make a higher frequency of wagers for a positive net gain.

### Coming up Next

This article outlines four measures of betting profit: average profit, unit profit, currency profit and percentage profit. The next article in this series will outline various measures of forecast accuracy in the context of betting models.