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Ergodicity & Kelly Betting

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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

Kelly Criterion, Generalized

W = win probability

B = loss %

A = gain %

Kelly Growth Factor Formula:

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

Kelly Calculator

NameWin %Win AmountLoss AmountBet SizeGain Factor
Game 1
10%
2,000%
100%
0
1
Tail Puts
70%
50%
100%
NaN
NaN
MU call Spreads
90%
100%
100%
0
1
Simple Examples
Kelly Links

Use this one for gain factor

Everything We’ve Learned About Modern Economic Theory Is Wrong - Bloomberg
The Kelly Criterion - Quantitative Trading
Wall Street University: Betting Markets
10K Kelly Criterion
The Kelly criterion: How to size bets