Portfolio Variance and Mahalanobis Distance for Correlated Returns

Summary

The document asks how to measure risk for a portfolio of correlated assets under a multivariate return model. It contrasts the familiar portfolio standard deviation, formed from weights and the covariance matrix, with a proposed Mahalanobis distance that uses the inverse covariance matrix and a difference between weights and expected returns. It also asks how the distribution of portfolio values depends on weights, means, and covariance.

The text presents these as questions and does not include an answer or empirical evidence. It therefore does not establish that Mahalanobis distance can replace portfolio volatility or provide a distribution formula. The two expressions describe different mathematical objects: portfolio variance aggregates return covariance using portfolio weights, whereas Mahalanobis distance measures a standardized displacement in a vector space. The proposed expression also places weights and expected returns in the same difference, an assumption whose meaning and units would need clarification. Any risk calculation would depend on the modeled random variable, return horizon, and distributional assumptions, which the prompt leaves unresolved.

Key ideas

• The standard portfolio volatility expression uses portfolio weights and the return covariance matrix.
• The document proposes Mahalanobis distance as an alternative but does not provide an answer validating that substitution.
• Mahalanobis distance measures a covariance-scaled displacement, while portfolio variance aggregates return risk for a weighted portfolio.
• The proposed difference between weights and expected returns requires a clear interpretation and compatible units.
• The prompt does not derive the distribution of portfolio values or provide evidence for a risk methodology.

Tags

# Portfolio risk of correlated assets using Mahalanobis distance


# Portfolio risk of correlated assets using Mahalanobis distance












I am trying to understand if there is an agreed methodology to measure the total risk in a portfolio of correlated assets.

I am taking a simple model of stock prices following geometric Brownian motion. As I understand it, we may describe the correlation between stock prices via the covariance matrix $\Sigma \in \mathbb{R}^n$.

Assume:

- We hold a portfolio of $n$ assets with weights $\textbf{w}$ and drifts $\mu$

- Asset prices $S_i$ evolve following the SDE:

$$ \frac{dS_i}{S_i} = \mu_i \cdot dt + \sigma_i \cdot dW{_t}{_i} $$

- Log-returns for each asset follow $\mathcal{N}(\mu,\sigma^2_i)$

- The portfolio log-returns hence follow a multivariate Normal distribution whose shape depends on $\Sigma$

Now, I have commonly seen the portfolio risk described as:

$$ \sigma_p = \sqrt{\textbf{w}^{\top} \Sigma \textbf{w}} $$

It seems to me that the correct metric to use in this case is actually the Mahalanobis distance:

$$ \sigma_p^M = \sqrt{(\textbf{w} - \mu)^{\top} \Sigma^{-1} (\textbf{w} - \mu)} $$

which generalises the Z-score, and which is (for example) relevant to the VaR approach to modelling risk. In this way one can describe the probability distribution of all portfolio values as some function of ($\textbf{w}$, $\mu$ and $\Sigma$)

My questions:

- Is the above reasoning valid, or have I made a mistake?

- If the above is valid, it seems that one method can be substituted for another. What are the implications of doing this?

Bonus question:

- What is the shape of the function which describes the probability distribution of portfolio values in terms of ($\textbf{w}$, $\mu$ and $\Sigma$)?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.