# 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?

• I don't understand your last sentence. Could you expand? Explain? If the model is well specified and the error term has high variation, the model will have low $R^2$ even if you have the true model parameters. – Matthew Gunn Jun 30 '17 at 15:22
• The idea is that the fitting of these models is largely unsupervised, to be broadly applied across many different portfolios. I want to specify a model that will not fit a high $\hat{\beta}$ value to two series that are not correlated. I could just use an $R^2$ threshold, but I was curious if there was a more elegant way to do it. – milkmotel Jun 30 '17 at 16:04
• For example, if the fit yields $\hat{\beta} = 5.45$ but $R^2 = 0.02$, I don't want to show that the portfolio response to a 100 bps move would be a 5.45% move when the fitted model is completely spurious. – milkmotel Jun 30 '17 at 16:18