Minimum-Variance Hedging of an Existing Portfolio
Summary
The document explains how to choose a self-financing hedge portfolio when an investor already holds positions in part of an asset universe. It partitions the full return covariance matrix into blocks for the existing holdings and the assets available for hedging. Total variance includes the existing portfolio’s variance, the covariance between the existing positions and hedge assets, and the hedge portfolio’s own variance.
The proposed method minimizes this combined variance subject to a zero-budget constraint on hedge weights. A Lagrange multiplier yields a linear system whose solution gives the hedge weights from the covariance blocks and the existing portfolio weights. The formulation assumes multivariate normal returns and a fixed asset universe, and it restricts the hedge to assets not already held. The document does not discuss transaction costs, estimation error, bounds on positions, or other practical constraints, so those would require extensions.
Key ideas
- The objective is the variance of the combined existing and hedge portfolios.
- Cross-covariances between current holdings and hedge assets affect the optimal hedge.
- The hedge weights are constrained to have zero net budget.
- A block covariance matrix and Lagrange multiplier produce a linear system for the hedge.
- The setup assumes a fixed asset universe and does not model trading costs or position limits.
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Full text
# how do we use portfolio optimization to hedge an existing portfolio?
# how do we use portfolio optimization to hedge an existing portfolio?
I am working on a risk management project and want to create a custom hedge portfolio to add on to an existing portfolio. I am wondering how do we treat the existing portfolio in the optimization problem?
Eg. I want to minimize the variance of both portfolios instead of only minimizing the variance of the hedge portfolio. The MVP I studied in school only deals with finding optimal weights for reducing current portfolio's variance but what do I do if I want to use it as a hedge? Should I include the existing portfolio as a single asset in the problem, or set its factors as constraints. ect.
## Answer by Kermittfrog (score 3)
https://quant.stackexchange.com/a/60125
Let us fix the asset universe with $N$ assets whose returns are multivariate normally distributed with covariance matrix $\Sigma$. You are already invested in $K<N$ assets (your portfolio) and you wish to add other assets from that universe to your portfolio to form a hedge(d) portfolio. Let us assume that the hedge should be self-financing.
Let us reorganize and partition the covariance matrix into four blocks:
$$ \Sigma \equiv \begin{pmatrix}\Sigma_{1} &\Sigma_{2} \\ \Sigma_{2}^T & \Sigma_{3}\end{pmatrix} $$ where the dimensions of the four submatrices from top left to bottom right are $(K,K), (K,N-K), (N-K,K)$ and $(N-K,N-K)$. Note that due to the symmetry of the covariance matrix, the lower left submatrix is the transpose of the upper right submatrix.
Your existing portfolio vector $\mathbf{w}$ is invested into the first $K$ assets and has a portfolio variance amounting to $\sigma_p^2=\mathbf{w}^T\Sigma_{1}\mathbf{w}$. You now add a hedge portfolio $\mathbf{h}$, which may only invest in the remaining $N-K$ assets, and your total portfolio variance is then
$$ \sigma_{total}^2=f(\mathbf{h})=\sigma_p^2+2\mathbf{h}\Sigma_{2}^T\mathbf{w}+\mathbf{h}^T\Sigma_3\mathbf{h} $$
We can now try to minimize this expression taking the usual route, assuming zero budget for the hedge weights
$$ L(h,\lambda)=\frac{1}{2}\left(\sigma_p^2+2h\Sigma_{2}^Tw+h^T\Sigma_3h\right)+\lambda(h^T1) $$ With FOC $$ \begin{pmatrix}\Sigma_3&\mathbf{1}\\\mathbf{1}^T&0\end{pmatrix}\begin{pmatrix}\mathbf{h}\\ \lambda\end{pmatrix}=\begin{pmatrix}-\Sigma_2^T \mathbf{w}\\0\end{pmatrix} $$
You can then solve for $\mathbf{h},\lambda$ as
$$ \begin{pmatrix}\mathbf{h}^*\\ \lambda^*\end{pmatrix}=\begin{pmatrix}\Sigma_3&\mathbf{1}\\\mathbf{1}^T&0\end{pmatrix}^{-1}\begin{pmatrix}-\Sigma_2^T \mathbf{w}\\0\end{pmatrix} $$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.