## ARB202: Fundamental mathematics of arbitrage

Don’t be scared! This is just a guide to help you to appreciate the mathematics of sports arbitrage. You do not need to be a maths genius to arb, far from it, but it is worthwhile appreciating the basic principles – **all alert services do the grunt work for you and tell you what bets to place and how much on each side**.

**The underlying principle**

An arbitrage opportunity exists when *the sum of the inverse of the odds for all bets on an event are less than 1*.

As always examples explain better than words, so using the same assumptions as our example in ARB102, being:

- Australian Open final: Murray versus Djokovic
- Bet365 offers 2.15 for Murray to win
- William Hill offers 2.05 for Djokovic to win

The calculation would be:

Arbitrage calculation = ( 1 / 2.15 ) + ( 1 / 2.05 )

= 0.953

Because the result is less than 1 this is an opportunity from which you can profit. The profit you can generate on your funds is roughly how much below 1 the result is – in this case 4.7% (1 – 0.953).

If the result is greater than 1 then the odds offered do not give rise to a profitable arb and you should move on.

**Expressing arbitrage percentages**

All arbs are expressed as a percentage, however there are two different ways commonly used to display arbs:

*Approach 1:* The most widespread way to display an arb percentage is something like “4.7%”. From the example above, this would define the arb as profitable and the return on investment to be roughly 4.7%. You’ll notice that we say ‘roughly’ because 4.7% is not the true return on investment. The true return on investment in the above example is actually higher, at 4.9% of the total amount wagered.

If an arb is not profitable then the percentage shown would be negative, such as “-1.8%”.

This approach for displaying arbs is by far the easiest to understand and is the method used by our preferred alert service, BetOnValue.

*Approach 2:* A less prevalent way of displaying an arb percentage is by expressing the mathematical result of the above calculation, such as “0.953” or “95.3%”. This would define the arb as profitable because the amount is less than 1.

If an arb is not profitable then the percentage shown would be greater than 1, such as “1.018” or “101.8%”.

**Calculators do the hard work for you**

As noted earlier, you won’t ever need to manually calculate arbs yourself but it is beneficial to appreciate the mathematics involved in sports arbitrage. All alert services do the grunt work for you, tell you what bets to place, how much to bet on each side, and what profit you will be making on the arb.

In any case, speed is a major factor of successful arbitrage and manual calculation of arbs would be too slow to be effective – that is why subscribing to a good quality alert service is critical.