## Thursday 4 March 2010

### Applying the Kelly criterion

David Pennock at Oddhead notes the Kelly criterion for maximizing long-term wealth: invest a fraction of your wealth equal to edge/odds.
The Kelly criterion dictates exactly what fraction of wealth to wager on any available gamble. First consider a binary gamble that, if correct, pays \$x for every \$1 risked. You estimate that the probability of winning is p. As Poundstone states it, the Kelly rule says to invest a fraction of your wealth equal to edge/odds, where edge is the expected return per \$1 and odds is the payoff per \$1. Substituting, edge/odds = (x*p – 1*(1-p))/x. If the expected return is zero or negative, Kelly sensibly advises to stay away: don’t invest at all. If the expected return is positive, Kelly says to invest some fraction of your wealth proportional to how advantageous the bet is. To generalize beyond a single binary bet, we can use the fact that, as it happens, the Kelly criterion is entirely equivalent to (1) maximizing the logarithm of wealth, and (2) maximizing the geometric mean of gambles.
Let's translate this into iPredict language.

The current trading price of a stock we'll call s. Buying a contract for \$s will get you \$1 if you win. Investing a dollar in that contract will give you 1/s contracts, and consequently \$1/s if you win. So the \$x is \$1/s.

The probability of winning is your best guess as to the true price of the stock, which we'll keep at p.

The Kelly formula then says ((1/s)*p - 1*(1-p))/(1/s).

So for the National to win government in 2011 contract, where I very loosely estimate a true probability of 0.9 and the contract's trading at \$0.75, we'd have:
[(1/.75)*.9 - 1*(1-.9)]/(1/.75) = [1.2 - .1]/1.33 = 0.825.
I should invest 82.5% of my wealth in this contract. On the climate contracts, where I had far more precise estimates of true probability and where the price difference was larger, I'd have needed to have staked more.

Even if I apply fractional Kelly and shade my estimate of the true probability, the iPredict deposit limits of \$1000/6 month period clearly are binding....

Do read Pennock's caveats before following the Kelly rule.