Quote:
Originally Posted by sports_quant99
I know what you're saying, in that the shape of the bell curve will not be even/symmetrical, but how can that be represented by one number?
..
. The only thing that changes is your volatility, but your long term bankroll will grow.
|
If you're more accustomed to thinking in terms of standard deviation, I'd point out that standard Markowitz utility (the utility function implicitly assumed by financial economists when considering the Markowitz efficient frontier of the traditional CAPM model) is actually a small variable approximation of Kelly.
To wit, if X is our random variable representing return, and μ represents the mean of X, and σ the standard deviation of X, then recalling that the Taylor expansion of ln(X) = X - X
2/2 + X
3/3 - X
4/4 + ..., for small values of X ≈ 0, expected Kelly utility as a function of X is given by:
K(X) = E(ln(1+X))
≈ E(X) - E(X2)/2
≈ μ - σ2/2
which is simply Markowitz utility with a coefficient of risk aversion of 1.
In fact, Kelly is really a generalization of Markowitz that includes all higher-order moments.