Interpreting Asset Contributions to Mahalanobis Distance
Summary
The document asks how to decompose an asset’s marginal contribution to portfolio Mahalanobis distance into volatility and correlation parts. It presents a contribution formula based on each asset’s return and the corresponding row of the inverse covariance matrix, then separates the terms into a same-asset component and cross-asset components. The author questions whether these labels are valid because entries in the precision matrix themselves reflect both variances and dependence among assets.
No answer or worked derivation is included, so the proposed split is not confirmed. The useful methodological point is that a term involving only one asset’s return need not represent volatility alone when its coefficient comes from the full inverse covariance matrix. Likewise, cross terms should not automatically be read as pure correlation effects. A valid economic decomposition would need to define the desired components and account for how covariance structure enters the precision matrix; the document provides no such method or empirical evidence.
Key ideas
- Mahalanobis distance measures returns relative to a covariance structure through the inverse covariance matrix.
- The proposed marginal contribution formula separates same-asset and cross-asset return terms.
- A same-asset return term is not necessarily a pure volatility contribution when its coefficient comes from the precision matrix.
- The document raises the decomposition question but supplies no answer or validated volatility and correlation split.
Tags
Full text
# Contribution to Mahalanobis Distance
# Contribution to Mahalanobis Distance
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?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.