Percentage Contributions to Portfolio Risk and Loss

Summary

The document asks how to show that a portfolio’s percentage contribution to loss equals its percentage contribution to risk, in the setting of a quantitative equity portfolio management exercise. It defines percentage contribution to risk for asset i as that asset’s weight multiplied by the sensitivity of portfolio volatility to its weight, divided by total portfolio volatility. The questioner says they have already obtained the optimal portfolio weights and variance from a mean-variance optimization problem.

The optimization maximizes expected portfolio return less a risk penalty proportional to portfolio variance, subject to a fully invested constraint. However, the document does not provide the definition of percentage contribution to loss, the remaining conditions from the exercise, or a derivation of the requested equality. It therefore frames a portfolio attribution problem rather than resolving it. Any proof would need the precise loss measure and assumptions used in the source exercise; the provided risk contribution formula and optimization setup alone do not establish the claimed identity.

Key ideas

• Percentage contribution to risk is defined using portfolio volatility sensitivity to each asset weight.
• The contribution formula scales each asset’s marginal volatility effect by its weight and total portfolio volatility.
• The stated portfolio allocation comes from mean-variance optimization with a full-investment constraint.
• The document asks whether percentage contributions to loss equal percentage contributions to risk but gives no derivation.
• A proof depends on how loss is defined and on assumptions beyond the information provided.

Tags

# Show that portfolio's percentage contribution to loss (PCL) equals PCR (risk)


# Show that portfolio's percentage contribution to loss (PCL) equals PCR (risk)












I came across this question during self study on a quantitative book (Question 3.6 on Page 75 of Quantitative Equity Portfolio Management: Modern Techniques and Applications By Edward E. Qian, Ronald H. Hua, Eric H. Sorensen ), can someone help me out? I got stuck on part(c). FYI, PCR = Percentage Contribution to Risk, PCL = Percentage Contribution to Loss, the definition for PCR is: $$PCR_i=\frac{\omega_i\frac{\partial\sigma}{\partial w_i}}{\sigma}$$ and I have already found the optimal $\displaystyle \omega^*$ and optimal $\displaystyle (\sigma^*)^2$ from mean-variance optimization problem formed as follows. $$Maximize \ \ \ \ \ \ \omega^T\cdot f-\frac{1}{2}\lambda(\omega^T\Sigma\omega )\\ subject \ \ to \ \ \ \ \ \ \omega ^T i=1$$

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