Portfolio and Risk, lesson 9
Sizing bets
Kelly, fractional Kelly, and the cost of betting too much.
Volatility drag from Risk and return (typical growth is about the average return minus half the variance), and the Sharpe ratio.
Part 1. How big?
A simulated strategy earns 5% a year over cash with 10% volatility. You choose how big to run it, borrowing at the cash rate. Find the size that makes the typical path grow fastest.
Bigger meant faster growth at first. Past a point, the swings took away more than the extra return added, and at twice that size typical growth falls to about nothing.
Example 1.1
Growth against size
- g(L)=L × μ − L² × σ² / 2
Volatility drag again, with every return scaled by L, the size: dollars of position for each dollar you own. Return grows like L, and drag like L².
- best L=μ / σ²=0.05 / 0.10²=5
Where g stops rising: its slope, μ − Lσ², is zero.
- g(5)=0.25 − 0.125=12.5% a year
The peak growth rate is the Sharpe ratio squared over 2: 0.5² / 2. Compounded, that's about 13% a year, as the lab found.
- g(10)=0.5 − 0.5=0
Twice the best size: all the growth is gone, and the book swings 100% a year.
The Kelly criterion
The size that makes wealth grow fastest over the long run is the Kelly size. For a strategy with roughly normal returns it's μ / σ², excess return over variance. For a single bet it's the share of your money to stake. Here it means $5 of positions for every $1 you own.
Your turn
You think a strategy's excess return is 10% a year, but it's really 5%, with 10% volatility. You size at what you believe is Kelly. What happens?
Part 2. Half the size
Keep going with Pro
You've done the first part. Pro opens the rest of this lesson, and every other one.
- Half the size
- A worked problem
- When the risk itself moves
- The same thing, for real
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