Why Minimum-Variance Weights Are Not Hedge Ratios
Summary
The document examines whether a two-asset minimum-variance portfolio can determine how much of one stock to buy or short against a holding in another. It describes calculating the portfolio weight from the assets’ return volatilities and correlation, then solving for the dollar amount of the second asset that produces that weight under a fully invested, weights-sum-to-one constraint.
The answer cautions that this calculation completes a constrained portfolio; it does not produce an optimal hedge position. Mean-variance optimization seeks weights that minimize portfolio variance while satisfying its portfolio constraints, with diversification as its purpose. Negative or offsetting weights in larger portfolios may arise from collinearity and estimation error, and should not automatically be interpreted as hedges. The discussion does not derive a dedicated hedge ratio or account for practical constraints such as transaction costs and changing estimates.
Key ideas
- A two-asset minimum-variance model selects portfolio weights subject to its weight constraints.
- Solving for the second asset’s dollar amount from a portfolio weight completes the portfolio under the stated constraint.
- Minimum-variance weights are not, by themselves, an optimal hedge ratio.
- Large offsetting positions in optimized portfolios can reflect asset collinearity and estimation error.
- The method is framed as diversification, not as a dedicated hedging procedure.
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# Deriving investment amount for one asset of a two asset minimum-variance portfolio
# Deriving investment amount for one asset of a two asset minimum-variance portfolio
Suppose I bought $100 worth of stock A and I want to hedge it by shorting stock B, they have correlation of rho and respective standard deviations. How do I know how much of Stock B to sell? that's the problem I am trying to solve.
More generally, suppose we own $\$X$ of Asset A and we wish to hedge this by buying $\$Y$ of Asset B, we know that the standard deviations of the Returns of A, and B are $\sigma_A$ and $\sigma_B$ respectively, and that the returns have a correlation of $\rho$. Using portfolio theory we want to minimize $VAR[w_aX+(1-w_A)Y]$ and we get the optimal weight of asset A in the portfolio as $w_A = \frac{\sigma_B^2-\rho\sigma_A\sigma_B}{\sigma_A+\sigma_B^2-2\rho\sigma_A\sigma_B}$
Furthermore, our total portfolio in dollar terms is $X+Y$, and the percentage of our portfolio in asset A is $\frac{X}{X+Y}$, and since we want to this to equal the weight neccessary to be the minimum variance portfolio we set $w_A=\frac{X}{X+Y}$ and solve for Y to find how much of asset B in dollar terms we need to short/purchase in order to acheive our desired portfolio.
Is this method valid for finding the optimal hedging strategy in terms of asset B?
## Answer by develarist (score 1, accepted)
https://quant.stackexchange.com/a/50061
No, this approach to the mean-variance model should not be interpreted as an optimal hedging strategy in terms of asset $B$. Given $w_A$, solving for $Y$ simply provides the solution for the total dollar amount of the $B$ side of the portfolio that completes and satisfies the $w_A + w_B = 1$ constraint, if that constraint is in fact kept intact somewhere in the equations you provided.
The purpose of the mean-variance model is to compute optimal weights that sum to 1 that minimize portfolio variance, it's not meant for finding offseting hedge positions, especially not for a simple two-asset portfolio.
Moving up to $N>2$ asset portfolios, you might see optimized weights being very aggressive and completely offset by negative weights of the same magnitude in other assets, but this is due to those particular assets' collinearity and well-known misestimation error that the model is known for, and should not be interpreted as hedging since hedging is not an intended mechanism of mean-variance optimization, diversification is.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.