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.

• Hint: can you rewrite the total portfolio variance as a sum of the hedge portfolio variance, the covariance of hedge and existing portfolio, and the variance of the original portfolio? Once you have that, you introduce any new constraints; eg sum of hedge = 0. Then you can use the usual machinery. Lerne know if that works for you. Dec 23 '20 at 6:50

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 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

$$\begin{pmatrix}\mathbf{h}^*\\ \lambda^*\end{pmatrix}=\begin{pmatrix}\Sigma_3&\mathbf{1}\\\mathbf{1}^T&0\end{pmatrix}^{-1}\begin{pmatrix}-\Sigma_2^T \mathbf{w}\\0\end{pmatrix}$$

• what source is this from? Dec 23 '20 at 12:33