I am using Mahalanobis Distance to measure abnormal behavior within a portfolio consisting of a handful of general asset types, and am trying to figure out how to decompose this measurement into marginal contributors on a granular level. My variables are as follows:
- $r_i$ is scalar, representing the return from asset $i$
- $\Sigma_{i,j}^{-1}$ is the $(i,j)$ element of the inverse covariance matrix ($\Sigma^{-1}$)
- $MD$ represents the Mahalanobis Distance value
- $MD_i$ represents the marginal contribution to Mahalanobis Distnace from asset $i$
I've already split it up into marginal contribution from each asset (done similarly as marginal contribution to portfolio volatility): $$ MD_i = r_i\frac{\partial MD}{\partial r_i} = \frac{r_i\sum_j(r_j\Sigma^{-1}_{i,j})}{MD} $$ But would like to split it up even further into individual volatility and correlation components. Below is my attempt at that: \begin{align} MD_i &= \frac{r_i\sum_j(r_j\Sigma^{-1}_{i,j})}{MD} \\ &= \frac{r_i^2\Sigma^{-1}_{i,i}}{MD} + \frac{r_i\sum_{j\neq i}r_j\Sigma^{-1}_{i,j}}{MD} \\ &= \underbrace{\frac{r_i^2\Sigma^{-1}_{i,i}}{MD}}_{\text{Volatility Component}_i} + \underbrace{\frac{r_ir_j\Sigma^{-1}_{i,j}}{MD} + \frac{r_ir_k\Sigma^{-1}_{i,k}}{MD} + \dots + \frac{r_ir_l\Sigma^{-1}_{i,l}}{MD}}_{\text{Correlation Component}_i} \end{align} My logic is that if I split up all terms into those that:
- Contain return data just from asset $i$ ($r_i^2$)
- Contain return data from both assets $i$ and $j\neq i$ ($r_ir_j$)
Then I will have split up this marginal contribution into terms that pertain to volatility (1) and correlation (2). However, I've read a bit about how the inverse covariance matrix (precision matrix) and learned that each element contains data regarding volatilities and correlation, which makes me think that my attempt isn't valid.
Can anybody shed some light on whether my logic is indeed off, and perhaps how to go about accomplishing my goal?
