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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

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

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

**Kelly Criterion, Generalized**

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?

Name | Win % | Win Amount | Loss Amount | Bet Size | Gain Factor |
---|---|---|---|---|---|

Game 1 | 19.24% | 500% | 100% | 0.229 | 4.831448744052506e+39 |

Tail Puts | 70% | 50% | 100% | 2.08 | 3.0426425535513843e+141 |

MU call Spreads | 90% | 100% | 100% | 1.79 | 9.430298923255593e+202 |

**Simple Examples**

**Kelly Links**

Use this one for gain factor

Everything We’ve Learned About Modern Economic Theory Is Wrong - BloombergThe Kelly Criterion - Quantitative TradingWall Street University: Betting Markets10K Kelly CriterionThe Kelly criterion: How to size bets