This is the capstone. Five lessons built one relentless idea: what you actually keep is what compounds, and what compounds is the geometric mean — always less than the arithmetic average, and dragged lower by every bit of volatility. You learned why a +50% then −50% year leaves you down 25%, why a positive average return can still compound to ruin, and how the Kelly criterion turns that math into a bet size by maximizing expected log wealth. You met continuous Kelly (f* = mu/sigma squared), fractional Kelly and its growth-versus-volatility trade, the multi-asset vector solution, and the cliff where over-betting flips growth negative. No formula sheet, no hints, no take-backs: every answer locks the instant you submit, the wrong options are the exact traps that fool real bettors and traders, and your score stays hidden until the end.
How this exam works
This is a graded exam. Questions arrive one at a time. Once you submit an answer it is final — there is no going back, no second try, and a wrong answer simply fails that question. Your score stays hidden until the very end, where you need 70% to pass. Read every option before you commit.
You start with 100 dollars, gain 50% one year, then lose 50% the next. Where do you end up?
Select an answer to continue.
Passed? Here's what you now own
You can look at any return stream and know that the geometric mean — not the seductive arithmetic average — is what compounds, and that volatility drag quietly taxes every point of variance. You can size a bet with Kelly from edge and odds, choose a sensible fraction to survive your own estimation errors, extend it to a correlated multi-asset book, and recognize the over-betting cliff before you walk off it. Most of all, you carry the instinct that a great average return means nothing if the path runs through zero.
Big picture
Kelly & Geometric Growth — the whole toolkit
- Kelly & Geometric Growth
- Two means
- +50% then −50% leaves you down 25%
- Geometric mean is what compounds
- Losses need bigger gains to recover
- CAGR & volatility drag
- CAGR is the annualized geometric mean
- g ≈ mu − sigma squared / 2
- Positive average can still compound negative
- The Kelly criterion
- Maximize expected LOG wealth
- f* = p − q/b = edge/odds
- Growth zero at 2·f*, negative beyond
- Continuous & fractional Kelly
- f* = mu / sigma squared
- Max growth = half the Sharpe squared
- Half-Kelly: 75% growth, half the vol
- Multi-asset Kelly & ruin
- f* = Sigma inverse times mu (tangency direction)
- Positive correlation shrinks the bet
- Err LOW; ruin is path-dependent
- Two means
That’s geometric growth and Kelly betting, end to end. You now own the math that separates the return you brag about from the return you keep — and the bet-sizing discipline that lets a real edge compound for a lifetime instead of detonating on a single bad streak.