Deriving the Kelly Criterion to maximise Profits - Niklas Oberhuber
In a fictional casino which offers even odds on a fair coin toss game, how much of your money should you invest? If you said anything other than 0, you’re leaving broke at the end of the night.
If you want to know how much to invest every flip, you should apply the Kelly Criterion. It’s a way of calculating the optimal fraction of your capital to invest in order to maximise growth over a long series of bets.
The expected value of a bet E[X]E[X] E [ X ] doesn’t make sense in the real world. Betting all your money on a razor thin edge might be worth it in theory, but in reality you’re as likely to lose and what are you going to invest with then?
Let’s take the example of a simple even odds game in which you invest half your money every turn. If you win, you get one and a half times your wealth, if you lose, you still have half your wealth. The expected value is 1, and thus in theory, playing an infinite number of times should net you zero losses.
For these odds and a fair coin, according to Kelly, the growth rate of your wealth is negative. In a typical, median world, the coin will show heads as many times as tails and after three turns, you lose most of your capital.
