The ‘over-round’

In a two-horse race, if both horses have an equal chance of winning (objectively), and both are offered at evens, then the expected profit of the market-maker (and of the bettor) is zero, ignoring operating, information and transactions costs.

In a two-horse race, if both are offered at evens (regardless of the respective probabilities of victory of the two horses), then it would require a stake of £x (split equally between the two horses) to be sure of being returned that £x (a net profit of zero) whichever horse wins.  In this circumstance, the over-round of the bookmaker is said to be 100%, i.e. a notional profit margin of zero.

In practice, even if the notional profit margin is zero, the bookmaker is at a disadvantage if the horses are not equally matched, as a sophisticated bettor can take advantage by staking more than half on the horse with the greater chance of winning.

More generally, the over-round does not yield an accurate indicator of the bookmaker’s profit margin if bettors do not stake across all options in such a way as to ensure that their total stake of £x yields a certain return of £x, factored by the over-round.

For example, if the over-round is 120%, the notional margin to the bookmaker is 20%, and put simply bettors would have to stake £120 to ensure a return of £100.  Say, for instance, that both horses in a 2-horse race are being offered at 4 to 6.  Then the bettor would need to stake £60 on each (£120 in total) to be guaranteed a return of £100 (£40 plus the £60 stake returned) whichever horse won.  In such circumstances, the bookmaker is guaranteed at 20% profit, regardless of the outcome.

If one horse is offered at 4 to 6 and the other at 6 to 4, the bettor can guarantee a zero profit (and loss) by staking £60 at 4 to 6 and £40 at 6 to 4.  That way, a £100 return is guaranteed for a total stake of £100, regardless of the outcome.  Again, if the horse offered at 4 to 6 is actually a 4 to 7 chance, and bettors stake exclusively on this horse, their expected return is positive (although there is now a risk of losing the entire stake), and the expected return of the bookmaker is negative (though the actual return may be positive).

To summarize, the notional margin, as implied in the over-round, formally equates to the actual margin only if bettors stake proportionately more on the outcome offered at shorter odds.

Creating an over-round

Take as an example the following odds offered about a binary proposition to players, where the odds-maker believes that the objective probability of X winning is 1 in 5 (0.2) and of Y winning is 4 in 5 (0.8).

Assuming an over-round of 100% (i.e. margin of zero), the odds-setter (taken here to be a bookmaker) would set the following odds:

Odds about X = 5.0 (4 to 1): Odds about Y = 1.25 (1 to 4).

Assume now that the odds-maker wishes to create an over-round of 108%.

In each case the odds offered should be cut, by 8 per cent in each case. So 8% of 5.0 = 0.4. Deducting 0.4 from 5.0 gives 4.6. 8% of 1.25 = 0.1. Deducting 0.1 from 1.25 gives 1.15.

So in the particular example, the odds offered would be as follows:

Assuming an equal amount bet (say £1,000) bet on both sides of the proposition (i.e. a total of £2,000, consisting of perhaps 200 people betting £10 each), the profit (loss) to the bookmaker would vary depending on the outcome.

If horse X wins, the bookmaker will pay out:

4.6x £1,000 = £4,600

Total amount staked (on X and Y) = £2,000.

Net profit to bookmaker if horse X wins = £2,000 – £4,600 = – £2,600

So if horse X wins, bookmaker loses £2,600.

If horse Y wins, the bookmaker will pay out:

1.15 x £1,000 = £1,150

Total amount staked (on X and Y) = £2,000

Net profit to bookmaker if horse Y wins = £2,000 – £1,150 = £850

Expected value of profit = expected value of profit from X + expected value of profit from Y = (-£2,600) x 0.2 + (£850) x 0.8 = -£520 + £680 = £160.

This is assuming that the implied probabilities in the odds are the correct probabilities, i.e. odds of 4/1 = probability of 1/5 (0.2); odds of 1/4 = probability of 4/5 (0.8).

Note also that £160 = 8% of total stake on X and Y (£2,000).

This all assumes, as observed, that the objective probabilities are correctly observed and that the amount staked on both sides of the proposition are equal.

Even if we assume that the objective probabilities are correctly observed then there is still substantial volatility of outcome (i.e. risk) for the bookmaker. If the objective probability is incorrectly observed, however, the outcome for the bookmaker may be worse, i.e. a systematic loss.

For example, assume the probability of horse X winning is actually 25%; assume probability of horse Y winning is 75%.

At the given odds levels, and assuming equal stakes across both propositions, we derive the following.

As above, if horse X wins, the bookmaker will pay out, as before:

4.6 x £1,000 = £4,600

Total amount staked (on X and Y) = £2,000.

Net profit to bookmaker if horse X wins = £2,000 – £4,600 = – £2,600

So if horse X wins, bookmaker loses £2,600.

If horse Y wins, the bookmaker will pay out, as before:

1.15 x £1,000 = £1,150

Total amount staked (on X and Y) = £2,000

Net profit to bookmaker if horse Y wins = £2,000 – £1,150 = £850

Expected value of profit = expected value of profit from X + expected value of profit from Y = (-£2,600) x 0.25 + (£850) x 0.75 = -£650 + £637.50 = -£12.50, i.e. a loss of £12.50.

Insofar as the objective probability of horse X winning is greater than 20%, the expected profit to the bookmaker will decline. At 24.65%, the profit (rounded to the nearest pound) can be shown to be equal to zero, and above that to turn negative.

Assume objective probability of horse X winning = 0.2465; objective probability of horse Y winning = 0.753.

Then, expected value of profit = expected value of profit from X + expected value of profit from Y = (-£2,600) x 0.2465 + (£850) x 0.7535 = -£640 + £640 = 0

To the extent that the objective probabilities are inaccurately estimated, therefore there is significant potential from the bookmaker’s point of view for a negative expected (as well as actual) profit.

Using the probabilities from the original example, the staking pattern from the bettor’s point of view that will lead to a unique expected loss (8% in this case) across both betting propositions is to bet more on the favourite and less on the longshot, in this case £1,600 and £400 respectively.

This leads to the following outcomes:

Profit to a £400 bet on horse X (if it wins) at 4.60 = £1,840

Profit to a £1,600 on horse Y (if it wins) at 1.15 = £1,840

Guaranteed profit by staking these sums on each horse from the bettor’s point of view = – £160, i.e. a net loss of 8% of total stake.

Insofar as bettors can be induced to bet in these proportions, the operator is guaranteed a profit regardless of the outcome. If the average bet size is the same for bets made on either side, then we need four times as many bettors on the favourite as the longshot to achieve this. Otherwise, the same outcome can be achieved if those who are backing the favourite bet four times as much in total as those backing the longshot.

Another way to manage risk in the face of unbalanced staking patterns is to move the odds so as to limit the maximum loss.

In order to reduce the maximum downside (i.e. when X wins) the bookmaker may move the odds in such a way as to attract money on one horse and away from the other horse. To do this, the odds about one horse may be lengthened and those about the other horse shortened before a negative downside is occurred to ether outcome. While such a strategy may reduce the exposure of the operator, the price may be paid in reduced profits.

Ultimately, line management from the operator’s point of view is about balancing risk and return, while maintaining an edge in favour of the ‘house’. From the bettor’s point of view, it is about exploiting opportunities which might arise where one (or more) of the odds making up that over-round are mispriced in the bettor’s favour, a possibility which can arise even when the over-round favours the ‘house.’