Deriving Residual Risk from a Benchmark Regression

Summary

Residual risk is the standard deviation of the part of portfolio returns left unexplained by a linear regression on benchmark returns. The document writes portfolio returns as an intercept, a benchmark exposure measured by beta, and a residual; it then derives the residual variance from the variance of that regression error.

Using beta as covariance divided by benchmark variance makes the covariance term simplify, yielding residual variance equal to portfolio variance minus beta squared times benchmark variance. This interpretation assumes the stated linear regression setup and consistent return and variance estimates. The derivation explains the relationship to benchmark exposure but does not address estimation error, changing betas, or whether a single benchmark adequately captures portfolio risk.

Key ideas

• Residual risk is the standard deviation of returns unexplained by a linear benchmark regression.
• Beta measures the portfolio’s covariance with the benchmark relative to benchmark variance.
• The residual variance follows by expanding the variance of portfolio returns minus beta-scaled benchmark returns.
• The simplified formula relies on beta being estimated as covariance divided by benchmark variance.

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# Mathematical Derivation of Residual Risk


# Mathematical Derivation of Residual Risk












I understand the difference between Excess, Residual and Active Returns.

I also understand what Active Risk; defined as: $\sigma_{r_P-r_B}$ (i.e. standard deviation of the difference in returns between our portfolio and benchmark).

Now, what exactly is Residual Risk? I often see it defined (e.g. here) as:

$\omega_p = \sqrt{\sigma^ 2_p-\beta^2_p\sigma^2_B}$

with $\beta_P = \frac{\text{Cov}(r_p, r_B)}{Var(r_B)}$

Where does this derivation come from? What is residual risk exactly?

## Answer by Gordon (score 7, accepted)

https://quant.stackexchange.com/a/26343

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \alpha+\beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. The standard deviation of the residual is called the residual risk. Specifically, \begin{align*} std(\varepsilon) &= \sqrt{var(r_P-\beta r_B-\alpha)}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta \rho(r_P, r_B) \sigma_P\sigma_B}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta\, \beta\, var(r_B)}\\ &= \sqrt{\sigma_P^2 - \beta^2 \sigma_B^2}, \end{align*} since $cov(r_P, r_B) = \rho(r_P, r_B)\sigma_P \sigma_B =\beta\, var(r_B)$.

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