Sizing a position to a risk budget
Size is not conviction. Each idea gets a slice of risk, and a stop-loss rule sits beside it. The smaller answer wins.
- Size a position to a contribution-to-risk budget
- Size it to a stop-loss limit, and say which rule binds
- Back out the correlation a risk system is assuming
- Resize a position after a jump in volatility
The intuition
An insurer does not decide how many homes to cover in a town by how much it likes the town. It asks how much a bad storm could cost, and how many of those homes the same storm would hit. Homes spread across the country get more cover than homes on one flood plain. And it caps any single policy at the loss it can afford.
A hedge fund sizes positions the same way. Each position may add only so much risk to the fund: its weight × its volatility × its correlation with the rest of the book. A second rule caps the loss if the stock falls to the stop, the price at which the thesis is wrong.
Weight under the risk rule = risk budget ÷ (volatility × correlation). Weight under the stop rule = maximum loss ÷ distance to the stop. The position is the smaller of the two.
Why it works
- The conventions here: a position's contribution to risk ≈ weight × annual volatility × correlation with the rest of the book, all as shares of NAV. The stop rule: weight × distance to stop ≤ maximum loss. The position takes the smaller weight.
- Correlation shrinks a position because a stock that moves with the book adds to risk already there. At a correlation of 0.5 only half its volatility counts against the budget.
- Volatile names get small positions. Double the volatility and the risk rule halves the weight.
- A distant stop means a small position. If the thesis is only proved wrong 25% lower, the stop rule allows less than if it is wrong 10% lower.
- Whichever rule binds tells the analyst what to change: a lower correlation or a bigger budget when the risk rule binds, a tighter stop when the stop rule binds.
- Volatility-sized books sell together. When volatility jumps, the risk rule forces a trim, often at the same moment as everyone else's.
| Risk rule: 0.5% ÷ (40% × 0.5) | 2.50% |
| Stop rule: 1% ÷ 15% | 6.67% |
| Allowed: the smaller | 2.50%, or $25M |
| Check the stop: 2.50% × 15% | 0.375% of NAV, inside 1% |
| Volatility doubles to 80%: 0.5% ÷ (80% × 0.5) | 1.25%, so sell $12.5M |
| Correlation if 2.50% adds 0.5%: 0.5% ÷ (2.50% × 40%) | 0.50 |
With a stop 30% away the stop rule would allow 3.33%, still above the risk rule's 2.50%.
The formulas
How much of the book's volatility this position adds.
Solve the contribution for the weight.
What the book loses if the stop is hit.
Solve that for the weight.
Both rules must hold.
Run it backwards.
Worked example
Size under each rule, take the smaller, then check that the other rule still holds.
See it move
Same fund. Change the stock's volatility and correlation with the book, the risk budget, the distance to the stop and the loss cap.
- Raise the volatility. The risk-rule weight falls, the stop-rule weight does not move, and the position allowed never grows.
- Raise the correlation. The risk-rule weight falls.
- Move the stop further away. The stop-rule weight falls.
- Raise the loss cap. The stop-rule weight rises.
Run it backwards
Same stock, reversed: the risk system reports a position's weight and the risk it adds. What correlation is it assuming?
The position's risk on its own is weight × volatility. The reported contribution is that times the correlation, so divide one by the other.
A high implied correlation means the stock mostly adds to bets the book already has. A low one means it is close to an independent idea, which earns a bigger size for the same budget.
Traps
Say it in the interview
“How do you decide how big a position should be?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Risk rule: weight = budget ÷ (volatility × correlation).
- Stop rule: weight = maximum loss ÷ distance to stop.
- Take the smaller; it names the constraint.
- A volatility jump forces a trim, often at the worst time.