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Estimating Multi-Asset Portfolio Sensitivity to an Equity Shock

Article Quant Q&A · Author: milkmotel
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Summary

The document asks how to estimate the change in a diversified portfolio’s value after a specified upward shock to the S&P 500. It proposes fitting an ordinary least squares regression of portfolio value on the index and multiplying the estimated beta by the assumed index move. The author points out a practical concern: portfolio returns may vary considerably for reasons unrelated to the index, leaving the regression with low explanatory power and an unstable sensitivity estimate.

The text poses, but does not answer, whether a dummy variable or another technique could reduce beta estimation error in this setting. It offers no data, empirical results, or comparison of alternatives. The question therefore serves as a starting point for discussing exposure estimation under noisy portfolio returns; it does not establish that low R-squared alone biases beta downward or identify a preferred model.

Key ideas

  • The proposed sensitivity estimate is the estimated regression beta multiplied by the assumed S&P 500 move.
  • The regression uses portfolio value as the dependent variable and the index as the explanatory variable.
  • The author is concerned that high residual variation and low explanatory power may make the estimate unreliable.
  • The document asks about dummy variables or alternative methods but does not provide a solution.

Tags

Multi-assetStatisticsPortfolio construction
Full text
# Sensitivity analysis


# Sensitivity analysis












If I want to test the sensitivity of a diversified, multi-asset class portfolio to say a 100 bps shock upwards in the S&P 500, the simplest solution would be to run an OLS defined as:

$\hat{Y}_{portfolio} = \alpha +\hat{\beta_x}x_{sp500} + \epsilon$

then define the portfolio sensitivity as:

$ \Delta P = (\Delta X_{sp500})(\hat{\beta_x})$

The obvious flaw with this method is that your fitted model would most likely have a very high variation and low $R^2$. Is there a standard method for using a dummy variable or some other technique to minimize $\hat{\beta}$ if the error term has high variation?

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