Original mathematical analysis
How Bookmaker Margin Compounds in Accumulators
Accumulator odds get bigger as more selections are added. But the pricing disadvantage built into each leg can compound too. This worked model shows how quickly that effect can grow.
The question
Suppose an outcome has a true probability of 50%. Its fair decimal price is 2.00. If the available price is 1.90 instead, the bettor is receiving 95% of the fair decimal return for that one selection.
What happens if five or eight identically priced selections are multiplied into an accumulator?
Single-leg starting point
| Measure | Fair | Offered |
|---|---|---|
| Probability | 50% | 52.63% implied by 1.90 |
| Decimal odds | 2.00 | 1.90 |
| Return ratio vs fair | 100% | 95% |
The 1.90 price does not mean the outcome's true chance suddenly became 52.63%. In this model we deliberately hold true probability at 50% and use 1.90 as the quoted price.
Compounding the prices
If the selections are independent, the fair accumulator price is found by multiplying the fair prices. The quoted accumulator price is found by multiplying the quoted prices.
Fair acca odds after n legs = 2.00n
Quoted acca odds after n legs = 1.90n
Return ratio vs fair = 1.90n ÷ 2.00n = 0.95n
| Legs | Fair odds | Quoted odds | Quoted return vs fair | Modelled shortfall |
|---|---|---|---|---|
| 1 | 2.00 | 1.90 | 95.00% | 5.00% |
| 2 | 4.00 | 3.61 | 90.25% | 9.75% |
| 3 | 8.00 | 6.86 | 85.74% | 14.26% |
| 5 | 32.00 | 24.76 | 77.38% | 22.62% |
| 8 | 256.00 | 169.84 | 66.34% | 33.66% |
| 10 | 1024.00 | 613.11 | 59.87% | 40.13% |
A £10 five-leg example
Five independent 50/50 selections all winning has a true model probability of 3.125%, or fair odds of 32.00. At quoted odds of 1.90 per leg, the accumulator pays 24.76099.
Fair return on £10 at 32.00 = £320.00
Quoted return on £10 at 24.76099 = £247.61
Expected quoted return = 3.125% × £247.61 = about £7.74
Modelled expected loss from a £10 stake = about £2.26
The important point is not that every five-leg accumulator has a 22.62% disadvantage. It does not. The point is that repeating a small disadvantage on every leg can magnify it substantially when prices are multiplied.
Why this is different from adding the margins
It would be tempting to say five 5% disadvantages simply make 25%. The exact modelled effect here is 22.62% because the return ratios multiply: 0.95 × 0.95 × 0.95 × 0.95 × 0.95 = 0.7738.
This distinction matters because accumulator maths is multiplicative, not additive.
Limitations
- Real selections do not all have 50% true probability.
- Real bookmaker margins are not necessarily distributed evenly between outcomes.
- Accumulator legs can be correlated, which changes the fair combined probability.
- Bookmakers may apply special same-game or related-contingency pricing.
- This model isolates price compounding; it does not estimate any specific bookmaker's actual hold.
What the analysis tells us
A small gap between fair price and quoted price can look modest on a single selection. When that same relative gap is repeated across many independent legs, the combined offered return can move much further away from the fair model price.
That is one mathematical reason the headline potential return of an accumulator does not tell the whole story.
