Correlation Attribution is Exposure Times Risk Times Correlation Divided by Portfolio Risk

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TL;DR

Two portfolios share common holdings. You are curious which holdings are largest contributors to correlation of the two returns. The result below, assisted by Deepseek R1, tells you exactly how to decompose correlation in an additive and non-overlapping way. You may be surprised by the fact that even non-common holdings can contribute significantly to correlation through indirect channels.

Final Simplified Formula:
\(\text{Contribution}_{\rho}(k) = \frac{\sigma_k}{2} \left( \frac{w_{A_k}}{\sigma_A} \rho_{r_B, r_k} + \frac{w_{B_k}}{\sigma_B} \rho_{r_A, r_k} \right)\)

Symbol Definitions:

• $\sigma_k$: Volatility of asset $k$’s returns ($r_k$)
• $w_{A_k}$, $w_{B_k}$: Weight of asset $k$ in portfolios $A$ and $B$
• $\sigma_A$, $\sigma_B$: Volatilities of portfolio $A$’s returns ($r_A$) and portfolio $B$’s returns ($r_B$)
• $\rho_{r_B, r_k}$: Correlation between portfolio $B$’s returns ($r_B$) and asset $k$’s returns ($r_k$)
• $\rho_{r_A, r_k}$: Correlation between portfolio $A$’s returns ($r_A$) and asset $k$’s returns ($r_k$)

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Derivation Walkthrough:

1. Euler Theorem for Bilinear Covariance:
   Covariance $\text{Cov}(r_A, r_B)$ is a bilinear function:
   \(\text{Cov}(t r_A, t r_B) = t^2 \cdot \text{Cov}(r_A, r_B)\)
   By Euler’s theorem:
   \(\sum_{k=1}^N \left( w_{A_k} \cdot \frac{\partial \text{Cov}(r_A, r_B)}{\partial w_{A_k}} + w_{B_k} \cdot \frac{\partial \text{Cov}(r_A, r_B)}{\partial w_{B_k}} \right) = 2 \cdot \text{Cov}(r_A, r_B)\)
   Intuition: The factor (1/2) fairly splits covariance contributions between portfolios (A) and (B).

2. Express Partial Derivatives:
   \(\text{Contribution}_{\text{Cov}}(k) = \frac{w_{A_k} \cdot \text{Cov}(r_B, r_k) + w_{B_k} \cdot \text{Cov}(r_A, r_k)}{2}\)

3. Convert to Correlation:
   \(\text{Contribution}_{\rho}(k) = \frac{\text{Contribution}_{\text{Cov}}(k)}{\sigma_A \sigma_B}\)

4. Substitute Covariance with Correlation:
   \(\text{Contribution}_{\rho}(k) = \frac{\sigma_k}{2} \left( \frac{w_{A_k} \rho_{r_B, r_k}}{\sigma_A} + \frac{w_{B_k} \rho_{r_A, r_k}}{\sigma_B} \right)\)

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Key Takeaways:

• Dual Dependence: Contributions depend on correlations between both portfolios and asset $k$.
• Volatility Scaling: Higher asset volatility ($\sigma_k$) amplifies impact.
• Normalized Weights: Weights are scaled by portfolio volatilities ($\sigma_A, \sigma_B$), penalizing riskier allocations.

This formula quantifies how shared holdings drive portfolio correlation, enabling precise adjustments for diversification or systemic risk management.