March 6, 2010

Is there free money crossing Betfair and Betdaq?

Tags: Gambling

It was recently suggested to me that there could be risk-free money to be made crossing Betfair and Betdaq. Betfair and Betdaq are betting exchanges - they allow their users to bet on various events, from the traditional horse races to more unusual fare like netball or the UK general election. Unlike a normal bookmaker, the exchanges don’t provide the odds and take the opposing side of the bet. Instead the odds are determined by other users and the other side is another user. So while there may be someone offering 1.15 odds that Labour will win (a back bet) the next UK election, someone else may offer 1.2 that they will lose (a lay bet). When the back and lay odds match then a bet is made. Note, all the odds quoted in this article are decimal odds as this is the type of odds used on the exchanges.

While Betdaq and Betfair are two different companies and run their own separate exchanges, there is a great deal of overlap in the events on which they allow betting. Thus it is possible that the lay odds on one exchange are less than the back odds on the other. If this is the case, then by placing a lay bet on one and a back bet on the other, a profit will be made regardless of the actual result. For example, in theory, if I back Labour to win the next election for £10 at 1.2 odds then I profit $2 if Labour wins and lose $10 if they lose. If I lay Labour to win the next election for £10 at 1.1 odds, then I lose £1 if they win and profit £10 if they win. If I can make both bets at the same time, then if Labour wins I profit £1 (£2 from the back bet minus £1 from the lay) and if they lose I’m even (£10 loss on the back and £10 gain on the lay). Risk free arbitrage!

A friend watched the two exchanges manually and saw that occasionally the odds on the same event crossed. Does this mean there is money to be made? Let’s see.

In the general case, a back bet of size £X at P odds will cost £X, and in the event of a win return £XP, resulting in a profit of £X (P-1). For a lay bet of size £Y at Q odds, the cost is £Y (Q-1), if the bet wins then it returns £YQ for a profit of £Y. To hedge a back bet the cost of the back has to be matched by the profit on a lay, which according to the above means that X = Y (i.e. the lay bet has to be the same size as the back bet). If this hedge is to be profitable then the cost of the lay has to be less than the profit on the back. That is:

       Y(Q-1) < X(P-1) , but X=Y
    => X(Q-1) < X(P-1) 
    =>    Q-1 < P-1 
    =>      Q < P

Thus, as long as the lay odds are less than back odds it is possible to make a risk-free bet by hedging the back with an equivalent sized lay and the profit will the difference between the back & lay odds multipled by the bet size.

So far so good, but the exchanges charge a fee on any winning bet (losing bets are free). For both exhanges this fee is expressed as a percentage of the profit on the bet and starts at 5% (although this can drop based on activity). How does this fee affect the equations above? Let Z be the fraction of the profit left after fees, that is if the fee is 5% of profit, Z=0.95. Then a back bet of size £X at P odds will cost £X and will result in a profit of £XZ (P-1) for a win. For a lay bet of size £Y at Q odds, the cost is £Y (Q-1), for a profit of £YZ. Thus to hedge a back, the back cost must equal the lay profit, X = YZ. The size of the lay has to be X/Z (which since Z has to be less or equal to 1, means the lay has to larger than the back). The hedge is profitable when:

      cost of lay < profit on back
           Y(Q-1) < XZ(P-1) , but Y=X/Z
    => (X/Z)(Q-1) < XZ(P-1) 
    =>     X(Q-1) < XZ^2(P-1) 
    =>        Q-1 < Z^2(P-1)
    =>          Q < Z^2(P-1)+1

This means that if the back odds are 1.2 then the lay odds have to be 1.18 (to 2 decimal places) at 5% fees and 1.19 at 2% fees. The difference between back and lay odds only diverges further from there. As shown in the graph below.

Armed with this extra information I manually checked whether Betfair and Betdaq markets cross to this degree. While there are regular crosses, I never saw the exchanges cross enough to make a profit.