How to calculate unsystematic risk?

Question

We know that there are 2 types of risk which are systematic and unsystematic risk. Systematic risk can be estimate through the calculation of β in CAPM formula. But how can we estimate the unsystematic risk quantitatively? is there any formula or calculation that can be related to the measurement of unsystematic risk?

Answer 1

I would use the identity that Total Variance = Systematic Variance + Unsystematic Variance. Therefore, you can calculate systematic variance via: Systematic Risk = beta * std(x), so , Systematic Variance = (Systematic Risk)^2, and then you can rearrange the identity above to get:

Unsystematic Variance = Total Variance - Systematic Variance, or if you want the number as "risk" (i.e. standard deviation), then:

Unsystematic Risk = sqrt(Total Variance - Systematic Variance).

NOTE: You're making assumptions here that that the Covariance of Unsystematic and Systematic is 0 (which in my experience holds up a good bit of the time).

Answer 2

I'm not sure about the "CAPM formula" that you are referring to.

I assume you are referring to the estimated coefficient of a regression of a security on a market portfolio. That is to say

\begin{equation} \beta_{security,market} = \frac{\sigma_{security,market}}{\sigma^2_{market}} \end{equation}

The idiosyncratic risk is the portion of risk unexplained by the market factor. The value of $1 - R^2$ of the regression will tell you this proportion.

Empirically, the idiosyncratic risk in a single-factor contemporaneous CAPM model with US equities is around 60-70%.