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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?

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You should specify a lot more details in your question. As it is I bet yuo get much answers. Regards –  TheBridge Oct 20 '11 at 15:59
    
You mean the calculation of beta in the Capital Asset Pricing Model, no? As @Quant Guy mentioned, I was also puzzled by the "CAPM formula" in the question. –  Feral Oink Oct 21 '11 at 1:30
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Hi Norlyda, welcome to quant.SE. Systematic and idiosyncratic (unsystematic) risk are estimated simultaneously in a CAPM-type regression equation. I believe you are misunderstanding CAPM. Please read the relevant wikipedia pages and come back if you still have a question. As it is, this question may be off topic or not a real question. –  Tal Fishman Oct 22 '11 at 23:37
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5 Answers 5

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%.

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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).

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do a regression where stock returns is dependent and market return is independent variable. Value of R^2 is Systematic risk and value of 1-R^2 is unsystematic risk...

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Actually, the value of R2 is the percent of total risk explained by systematic risk..so you need to compute total risk, which is the sd of your stock returns...and then annualize it (i.e. if your data is monthly, just multiply the sd you computed by sqrt of 12) and then multiply it with R2 to obtain your systematic risk. The rest is unsystematic. –  user6157 Sep 24 '13 at 16:52
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If Y is the excess returns of your asset and X is that of the market, then CAPM tells you $Y = \beta X + \epsilon$ Taking the variance of both sides yields $$ \\ \sigma^2_{Y} = \beta^2 \sigma^2_{X} + \sigma^2_{\epsilon} \\ $$ We know that $$\beta = \frac{\sigma_{X,Y}}{\sigma^2_{X}} = \rho_{X,Y}\frac{\sigma_{Y}}{\sigma_{X}}$$ Where $\sigma_{X,Y}$ is the covariance and $\rho_{X,Y}$ the correlation. Hence, substituting for $\beta$ and solving for $\sigma^2_{\epsilon}$ we get: $$\sigma^2_{\epsilon}= \sigma^2_{Y}(1-\rho^2_{X,Y}) $$

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In your top example, are you setting the intercept(alpha) to zero? Why? Ties into my question...quant.stackexchange.com/questions/9118/… –  Bob Hopez Oct 10 '13 at 21:56
    
Is \sigma^2_{\epsilon}= \sigma^2_{Y}(1-\rho^2_{X,Y}) unsystematic in your example? –  Bob Hopez Oct 10 '13 at 21:58
    
The variance of a constant is zero so it doesn't matter. –  SpeedBoots Oct 14 '13 at 12:58
    
True again. But when the RFR is time-varying? Moreover, the forced zero-intercept Beta alters the correlation used in the variance of the error. –  Bob Hopez Oct 14 '13 at 15:35
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Actually, the value of R2 is the percent of total risk explained by systematic risk..so you need to compute total risk, which is the sd of your stock returns...and then annualize it (i.e. if your data is monthly, just multiply the sd you computed by sqrt of 12) and then multiply it with R2 to obtain your systematic risk. The rest is unsystematic.

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