Calculating Asset and Correlation-Weighted Portfolio Risk Contributions

Summary

The document presents two matrix expressions for portfolio risk attribution: one labeled asset risk contribution and another labeled correlation-weighted asset risk contribution. It defines the identity matrix, diagonal weight and volatility matrices, asset correlation matrix, and weight vector, then asks how to modify the expressions so each contribution is a share of total portfolio volatility and the shares sum to 100%.

The text offers no derivation or worked example, so it does not establish whether the displayed formulas are correctly oriented or normalized. Its useful contribution is to frame the distinction between attributing risk by asset and incorporating correlations in the allocation. Any practical calculation would need a consistent covariance-based definition of portfolio volatility and an explicit normalization step; these are not supplied in the document.

Key ideas

• The document distinguishes asset risk contributions from correlation-weighted contributions.
• It represents portfolio weights and volatilities as diagonal matrices and correlations as a matrix.
• It asks how to normalize contributions so they sum to total portfolio volatility.
• No derivation or numerical example is provided.

Tags

# Portfolio Risk Contribution


# Portfolio Risk Contribution












I came across a paper that shows calculations for two types of portfolio risk contribution. The first shows "Asset risk contributions" and the second shows "Correlation-weighted asset risk contributions" as so:

$I \times (W \times (V \times C \times V \times w))$

and

$C \times (W \times (V \times C \times V \times w)$, respectively where:

```
x denotes matrix multiplication
I is the identity matrix
W is the diagonal matrix of asset weights
V is the diagonal matrix of asset volatilities
C is the asset correlation matrix
w is the vector of asset weights
```

Unfortunately, after playing around with these two formulas with dummy data, I was not able to see anything that looked like a "risk contribution". A risk contribution should represent a percentage of total portfolio volatility and all risk contributions should add to 100%. Can anyone provide guidance on how these formulas can be modified to achieve their intention?

Thank you.

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