Minimum-Variance Peer Hedges with Fixed Long Exposure

# Hedging with peer companies and optimize the weights


# Hedging with peer companies and optimize the weights












I am trying to long a security that is expected to outperform its peers after certain corporate actions, but want to hedge using the same group of peers (so short ~5 names). So the goal here is to hedge out any risk from the sector and left with the alpha from the event. What's the best way to figure out the hedge ratio and the weights to put on for each peer? Should I consider just use beta to weight it? Or there is a better way to optimize long/short portfolio? Thanks!

## Answer by Pontus Hultkrantz (score 0, accepted)

https://quant.stackexchange.com/a/70890

One method is to calculate the minimum variance (hedged) portfolio (MVP), given a fixed holding in one security. That is, given that you invest one dollar in one security, how much should you invest in the other five securities in order to minimize your total portfolio variance.

Let there be $n$ assets (1+5=6 in your case) with log return covariance matrix $\Sigma$. Let $\vec{w}$ be a $n\times 1$ weight vector with elements representing your dollar holdings in each asset. For simplicity, we assume that the first security is the one you want to hedge, and that you hold one dollar in it, i.e. $w_1=1$.

The variance of your portfolio is then $\vec{w}^T \Sigma \vec{w}$. Using Lagrange multiplier, we can solve the convex minimization problem $$ \min_w w^T \Sigma w \quad \text{subject to} \quad \vec{w}^T\vec\alpha=1, $$ where $\vec\alpha=[1, 0, 0,...]^T$, such that $\vec{w}^T\vec\alpha=w_1=1$. This ensures that the first asset is held fixed. The solution is then given by $$ \vec{w}^* = \frac{\Sigma^{-1}\vec\alpha}{\vec\alpha^T\Sigma^{-1}\vec\alpha}. $$

That is, for every dollar invested in the first security, you invest $w_k$ in security $k=2, 3, 4, 5, 6$.

Note that

- while most of these hedging weights will be negative, some might not be. This is only a problem if you are not allowed to go long in the hedging securities, which is rarely the case.

- the total net dollar investment (cash delta) might not equal zero. That is, the cash made from going short might not equal the funding needed for the long positions. If this is important, additional constraints need to be made in order to ensure this.

If you want a dollar neutral portfolio you will have to minimize $$ L(\vec{w}, \lambda_1, \lambda_2) = \vec{w}^T \Sigma \vec{w} - \lambda_1(\vec{w}^T\vec{\alpha}-1) - \lambda_2(\vec{w}^T\vec{\psi}-0), $$ where $\vec{\alpha}$ is defined as before, and $\vec{\psi}=[1,1,1,...]^T$ is a vector of ones to ensure that sum of weights adds up to zero. Then you solve the three equations, $\frac{\partial L}{\partial \vec{w}}=0$, $\frac{\partial L}{\partial \lambda_1}=0$, $\frac{\partial L}{\partial \lambda_2}=0$. Since we enforce dollar neutrality, variance might not be lower than only holding the single long security, so check whether the variance is reduced.