Kelly Sizing, Semivariance, and Empirical Return Distributions

# Why not calculate Kelly using semivariance? As w Sortino


# Why not calculate Kelly using semivariance? As w Sortino












Kelly is calculated as mu / sigma^2. If we remove our highest performing returns from our calculations this actually increases our Kelly leverage, which does not make sense to me. A less profitable return history means we should be lowering our Kelly factor, not increasing it

I've seen Kelly derivations that account for higher moments (skew, kurtosis) but never this. Can anyone help?

## Answer by Mild_Thornberry (score 2)

https://quant.stackexchange.com/a/64093

The underlying assumption to your mu/simga^2 formula is that the pricing process follows geometric Brownian motion, so your returns are therefore symmetric and normal. The existence of a very high positive return implies the possibility of a very low negative return, even if you haven’t realized it yet in the time series you used to calibrate your sigma. If the very low negative return wasn’t as likely as the very high positive return, your return distribution wouldn’t be normal (because it’s not symmetric) and you shouldn’t be using that formula anyway. Then, the log utility function you are maximizing punishes losses more than it does gains, like most utility functions. Depending on your time series, the chance of the big loss isn’t worth the chance of another big gain, so your total allocation decreases.

## Answer by André Bittencourt (score 0)

https://quant.stackexchange.com/a/73834

The Kelly's derivation is the maximization of the expectation of a function (log) of a random variable, which is usually approximate by the Taylor expansion of this function. For small $E[X]$, the variance is a good proxy of $E[X^2]$ which appears in the expectation of the Taylor expansion.

You could substitute the variance by the semivariance, for sure, but it wouldn't maximize the expected log-return. Maybe maximize some type of utility function, though.

## Answer by Jordi Muñoz (score 0)

https://quant.stackexchange.com/a/81749

Why not simply compute Kelly numerically?

You don't even need a model for the random process distribution. You can take empirical distribution, and numerically compute the fraction that maximizes log1p expected returns.

This is extremely practical for many reasons. One important one: you can manually introduce 1 single -1 sample, and assign it any probability you wish, outside of any specific fat tail model. You can have empirical plus single sample black swan, and calculate exact Kelly optimal fraction for it.

A lot more flexible and robust than working with models with estimated parameters.