**The formula**

When you have >50% chance to win in a game, how much money would you bet each time? If you bet 100% of your money, you may lose it all just because of bad luck; if you bet 1% every time, your capital could grow so slow that the game is not worth playing. According to Kelly Optimization Model (the original Kelly paper), the relationship between the probability of winning *p* and the optimal bet size *f* is:

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*Demonstration: *

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**A Quick Example**

In this game, you put your bet and blindly pick one card out of a deck. If you got a spade, you would lose 100% of your bet; if you got anything else, you would gain 100%. So no matter how much you bet for each game, you either double it or lose it all. Theoretically, your expected return, (1 / 4) * (-100%) + (3 / 4) * 100% = 50%, is in your favour. But in real world, if you lose all your money in just one game, you wouldn’t have any equity left to enjoy the infinite 50%s in the future.

To find out the optimal allocation in R:

f <- seq(0, 1, 1/1000) test <- rep(1, 1000) test[which(rbinom(1000, 1, 1/4) == 1)] <- -1 gain <- function(f) prod(1 + f * test) rets <- sapply(f, gain) plot(f, rets, type = 'l')

The test shows the optimal allocation that maximizes your total return in the long-run is 50%, which equals to the result by using Kelly: 3 /4 * 2 – 1 = 50%.

**An Implementation**

To find out if Kelly can help improve investment strategies, I plugged it into a RSI(14)50/50 model. The RSI(14) indicator, measures oversold and overbought with a percentage number, makes itself a good proxy for the chance to win (p in Kelly formula)*.*

Note: Reminded by friend Mark Leeds, I noticed this implementation actually doesn’t meet all pre-conditions required by Kelly Criterion. Kelly Criterion is only applicable with a binomial outcome between 100% and -100% like in the example above. This implementation ignores that the gain/loss in trading is a variable that is proportional to initial investments. Although this test seems effective (probably because in this test the investor assumes he’s in a much more risky environment than he really is), the Kelly formula should be substituted by Optimal f (found in this journal) here.

And to better illustrate the effects brought by Kelly.

As shown above, Kelly strategy has the same winning rate as RSI(14) but much less volatile returns. By implementing this bet sizing strategy, we effectively reduced our exposure to the fat-tail risk of RSI(14).

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