*Question by Wonhi*: Bookmakers rate of return?

The formula for calculating a bookmakers rate of return is to take the inverse of the sum of all outcomes. Why is this the case? I’m trying to understand this intuitively but can’t seem to wrap my head around it. If anyone can explain I would appreciate it.

For reference, I came across the formula here http://en.wikipedia.org/wiki/Arbitrage_betting. Just search for the section “Arbitrage using bookmakers”.

**Best answer:**

*Answer by Mitch*

The formula is based on the assumption that the bookmaker has set the odds so close to the expected outcome that there will be an amount of money betting on every possible outcome will be inversely proportional to the odds. If the bookmaker has not done this and there is a disproportionate amount of money placed on one side of the bet, this rate of return falls apart and the bookmaker can lose, or win, very big.

So again, this is an ideal case analysis. Let’s keep it simple and assume there are only 2 possible outcomes…like a boxing match. Either fighter A is going to win, or fighter B is going to win.

Now suppose the bookmaker is going to pay 5 to 4 if fighter A (the champ) wins and he is going to pay 15 to 4 if fighter B (the underdog) wins.

Suppose all the bets are placed and fight time starts. If the bookmaker set the odds properly, for every dollar bet on fighter B, would have been bet on fighter A. So lets say at fight time, a total of ,000 was bet on A to win and ,000 was bet on B to win. The bookie took in ,000.

If fighter A wins, the bookmaker must pay out ,500. If fighter B wins, he must pay out 500.

His profit either way will be 00 out of 000 = 6.25%

The “inverse sum of all outcomes” actually computes his profit based on his payout (500), not the money he took in (000), in this case 00/500 = 6.67%. This is a good approximation to his return however:

4/5 + 4/15 = .8 + .2667 = 1.0667 which indicates a 6.67% profit.

Intuitively, the way to think about what’s going on with the inverse sum of all outcomes formula is this. First, realize:

4/5 + 4/15 = 12/15 + 4/15 = 16/15

Now you should see that the bookie is receiving in bets but only paying out . He keeps for every paid out (1/15 = 6.67%) or for every bet. (1/16 = 6.25%)

**What do you think? Answer below!**