Ergodicity & Kelly Betting

Arithmetic return is over 1 period. Over long run, geometric return is what matters. Higher vol drags down geometric return. Ex: Heads win 50%, tails lose 40%. You would obviously play this one time (actually more) but if you have to play forever you will mathematically go broke.

He is maximizing the geometric return (not leveraged) — geometric mean frontier and solve for peak

Optimal Kelly = maximum sharpe ratio portoflio (so have to leverage)

Key Takewaways:

• Geometric return is all that matters in long run
• If two assets have same geo return, mix them 50/50, correlations and variance do not matter
• Negative correlation helps, positive hurts. Can include -ve return assets if they hvae -ve correlation.
• Variance is additive, standard deviations are not (for calcing SD of a portfolio)

Geometric Return Formulas:

GR = Geometric Return, AR = Arithmetic Return

$GR = AR -(\frac{1} {2} * \sigma ^2)$

$GR = AR - \lparen\frac{1}{2} * Variance)$

Kelly Criterion, Generalized

https://blogs.cfainstitute.org/investor/2018/06/14/the-kelly-criterion-you-dont-know-the-half-of-it/#:~:text=Created in 1956 by John,investor expects a positive return.

W = win probability

B = loss %

A = gain %

$\LARGE Kelly \% = \frac{W}{B} - \frac{(1-W)}{A}$

Kelly Growth Factor Formula:

$\LARGE GR(f) = (f-\frac{f^2}{2})) * \frac{u^2}{\sigma^2})$

Flip a coin. Heads you win 50%, tails you lose 40%. How much do you bet?

Use this one for gain factor