Minimum-Variance Hedge Ratio

Long one share of stock A and hedge with h shares of stock B. Choose h to minimize Var(RA − hRB).
1.00
1.00
0.70
Portfolio variance as h changes
The variance is a convex quadratic, so its lowest point gives the optimal hedge.
hedge ratio h variance − 0 +
1. Expand the variance
Var(RA − hRB) = Var(RA) + h²Var(RB) − 2hCov(RA,RB) = σA² + h²σB² − 2hρσAσB
2. Differentiate with respect to h
dVar/dh = 2hσB² − 2ρσAσB Set equal to 0: 2hσB² − 2ρσAσB = 0
Optimal hedge ratio
h* = ρσA / σB
= 0.700
3. Minimum variance
Var_min = σA²(1 − ρ²)
0.510
How to interpret the hedge ratio
ρ
Stronger positive correlation means B is a better hedge, so you short more B.
σA
If A is more volatile, there is more risk to hedge, so h increases.
σB
If B is more volatile, each share provides more offsetting movement, so fewer shares are needed.