Contribution of an asset's variance to portfolio variance

Question

How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$?

I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\lambda)}{\partial \sigma_i}$) of the Lagrangean formulation of the minimum-variance portfolio's objective function, for example, but not sure if this is the right approach since the idea is to speculate before optimization, how an asset's variance will contribute to portfolio variance, based on that asset's stand-alone variance (or volatility) level. Besides, asset variances don't appear in the portfolio variance's Lagrangean.

A demonstrated derivation of the $\frac{\partial L_{\sigma_p^2}(w,\lambda)}{\partial \sigma_i}$ first order conditions might be marked best answer, but also open to alternative suggestions.

Answer 1

In this answer, I am assuming that you want to keep correlations constant.

To begin with, note that the $N\times N$ covariance matrix $\Sigma$ with element $\Sigma_{i,j}=Cov(x_i,x_j)$ can be written as

$$ \Sigma = \mathbf{SRS} $$

where $\mathbf{S}$ is a diagonal matrix of the simple volatilties $\sigma_i$, and $\mathbf{R}$ is the correlation matrix. Thus in a matrix sense,

$$ \frac{\partial \mathbf{\Sigma}}{\partial\sigma_i}=\frac{\partial\mathbf{SRS}}{\partial\sigma_i}=\frac{\partial\mathbf{S}}{\partial\sigma_i}\mathbf{RS}+\mathbf{SR}\frac{\partial\mathbf{S}}{\partial\sigma_i} $$

The derivative of $\mathbf{S}$ with respect to $\sigma_i$ is a diagonal matrix of zeros, whose $i$th element is $1$. It is a single-entry matrix or a selector matrix, which we shall denote $\mathbf{E}_i$. For example, for $\mathbf{E}_2$ is

$$ \mathbf{E}_2=\begin{pmatrix}0&0&0&\ldots &0\\ 0&1&0&\ldots &0\\ \ldots &\ldots &\ldots &\ldots &\ldots \\ 0&0&0&\ldots &0\\\end{pmatrix} $$

Hence,

$$ \frac{\partial\mathbf{\Sigma}}{\partial\sigma_i}=\mathbf{E}_i\mathbf{RS}+\mathbf{RS}\mathbf{E}_i $$

Thus, the marginal impact of a change in (any) of the volatilities on portfolio variance $v=\mathbf{w}^T\mathbf{\Sigma} \mathbf{w}$ can be computed as (after some algebra)

$$ \frac{\partial v}{\partial \mathrm{diag(S)}}=\mathbf{w}^T\frac{\partial \mathbf{\Sigma}}{\partial \mathrm{diag(S)}}\mathbf{w}=2\mathbf{w}\otimes \left(\mathbf{RSw} \right)$$ where $\otimes$ denotes element wise multiplication, i.e. $x\otimes y = \mathrm{diag}(xy^T)$. Conveniently, this formulation returns all derivatives at once.

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Example: With

$$ \mathbf{S}=\mathrm{diag}\begin{pmatrix}0.1&0.2&0.3\end{pmatrix} $$ and $$ \mathbf{R}=\begin{pmatrix}1 & 0.5 & 0.25 \\ 0.5 & 1 & 0.1 \\ 0.25 & 0.1 & 1\end{pmatrix} $$ and a weight vector

$$ \mathbf{w}=\begin{pmatrix}0.2 & 0.3 & 0.5\end{pmatrix}^T $$

we find

$$ \frac{\partial v}{\partial \mathrm{diag(S)}}=2\mathbf{w}\otimes \left(\mathbf{RSw} \right)=2\begin{pmatrix}.2\\.3\\.5\end{pmatrix}\otimes\begin{pmatrix}0.0875\\0.085\\0.161\end{pmatrix}=\begin{pmatrix}0.035\\0.051\\0.161\end{pmatrix} $$

Thus, for example, the sensitivity of the portfolio variance with respect to the first volatility is 0.035.

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With a bit more of algebra, you can find the impact of standalone vols on any portfolio optimisation solution, e.g. the MVP. Using the results from above and the fact that $\sigma_{MVP}=\frac{1}{\mathbf{1}^T{\Sigma^{-1} 1}}$ and the knowledge that $\mathbf{\Sigma}^{-1}=\mathbf{S}^{-1}\mathbf{R}^{-1}\mathbf{S}^{-1}$.

Answer 2

The Lagrangian 'solution' can yield negative contributions to portfolio risk, which is a bad look. An alternative definition is via the symmetric square root of the covariance, $\Sigma^{1/2}$. For portfolio $\vec{w}$ define $$ \vec{r} = \Sigma^{1/2}\vec{w}. $$ The norm of $\vec{r}$ is the volatility of the portfolio. Moreover, this definition is equivariant with respect to rotations of the asset space (using the Cholesky square root would not yield this property), and thus elementwise $\vec{r}$ can be identified with the individual assets. See also no parity like risk parity.