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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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  • $\begingroup$ You should specify a lot more details in your question. As it is I bet yuo get much answers. Regards $\endgroup$ – TheBridge Oct 20 '11 at 15:59
  • $\begingroup$ 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. $\endgroup$ – Ellie Kesselman Oct 21 '11 at 1:30
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    $\begingroup$ 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. $\endgroup$ – Tal Fishman Oct 22 '11 at 23:37
  • $\begingroup$ just a point, your 'unsystematic' risk is usually termed 'idiosyncratic' risk. $\endgroup$ – lehalle Nov 11 '15 at 9:33

12 Answers 12

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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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  • $\begingroup$ The above definition is useful to get the idiosyncratic market risk and has to do with shares. As equity is even subordinated debt with respect to bond holders, the idiosyncratic credit risk (not reflected in market price varaitions) should not be included? $\endgroup$ – user7056 Mar 10 '15 at 14:27
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I would use the identity and three step process that:

$$\textrm{Total Variance} = \textrm{Systematic Variance} + \textrm{Unsystematic Variance}$$

You can calculate systematic variance via:

$$\textrm{Systematic Risk} = \beta \cdot \sigma_\textrm{market} \Rightarrow \; \textrm{Systematic Variance} = (\textrm{Systematic Risk})^2$$

then you can rearrange the identity above to get:

$$\textrm{Unsystematic Variance} = \textrm{Total Variance} - \textrm{Systematic Variance}$$

Or if you want the number as "risk" (i.e. standard deviation), then:

$$\textrm{Unsystematic Risk} = \sqrt{(\textrm{Total Variance} - \textrm{Systematic Variance})}$$

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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  • $\begingroup$ To clarify. $B$ is Beta of the asset to the market. Since Beta is equal to the covariance of the asset to the market divided by the variance of the market, Systematic Risk, when simplified, is just the covariance correct? $\endgroup$ – Jack Armstrong Feb 18 '18 at 18:35
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    $\begingroup$ @Jack, the OP asked about Idiosyncratic / unsystematic risk. If you're looking to quantify Systematic Risk, then you'd get \beta \cdot \sigma_{\textrm\market} or covariance divided by the volatility (not variance) of the market. $\endgroup$ – benjaminmgross Feb 19 '18 at 1:53
  • $\begingroup$ Yes you are right. Another question came up though. Isn't BETA the same as Systematic Risk or at least how we think of it? Also, I don't think the Total Variance equation is correct. I see it logically, but not mathematically. Maybe my steps are wrong, but Total Risk=Sys Risk + Unsys Risk. When we square Total Risk to get Total Var, we have to square (Sys Risk + Unsys Risk) together, not separately. If Total Risk is the standard deviation and Sys Risk is Beta, then Unsys Risk is simply their difference. Could that be another way? $\endgroup$ – Jack Armstrong Feb 24 '18 at 21:49
  • $\begingroup$ @Jack, the equation I used was total variance (not total risk). Systemic risk = \beta * \sigma_M. See some of the other answers here and you'll see they (and the Math I used in my problem) all line up! $\endgroup$ – benjaminmgross Feb 25 '18 at 2:28
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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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  • $\begingroup$ 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. $\endgroup$ – 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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  • $\begingroup$ In your top example, are you setting the intercept(alpha) to zero? Why? Ties into my question...quant.stackexchange.com/questions/9118/… $\endgroup$ – Bob Hopez Oct 10 '13 at 21:56
  • $\begingroup$ Is \sigma^2_{\epsilon}= \sigma^2_{Y}(1-\rho^2_{X,Y}) unsystematic in your example? $\endgroup$ – Bob Hopez Oct 10 '13 at 21:58
  • $\begingroup$ The variance of a constant is zero so it doesn't matter. $\endgroup$ – SpeedBoots Oct 14 '13 at 12:58
  • $\begingroup$ 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. $\endgroup$ – Bob Hopez Oct 14 '13 at 15:35
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I have studied unsystematic risk [USR] for more than two decades. In fact, I wrote a book (which is here) whose central focus is how to deal with USR in the valuation of non-public companies. It is a multifaceted, complex, and difficult issue. Modern Portfolio Theory did professionals in my line of work no favors when it assumed away the existence of USR because few small-business owners hold diversified investment portfolios.

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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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For calculating systematic risk(beta) for a company which is registered on stock exchange can be calculated in excel through following steps. 1. co variance of both will be multiplied 2. Divided by the variance of stock exchange index A common expression for beta is

for further see link http://en.wikipedia.org/wiki/Beta_(finance)

by Akhtar rasheed international islamic university islamabad BBA 24(A)

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    $\begingroup$ Hi Akhtar Rasheed, welcome to Quant.SE! What about the _un_systematic risk? $\endgroup$ – Bob Jansen Oct 1 '14 at 7:46
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I guess one can figure out the unsystematic risk by using the following formula:

$ Unsystematic Risk = [R_A - E(R_A)] - [R_M - E(R_M)] * \beta $

Where:

$R_A$ is the actual return on the asset

$E(R_A)$ is the expected return on the asset

$R_M$ is the actual return on the market

$E(R_M)$ is the expected return on the market

You can think of the ACTUAL - EXPECTED as how far the actual returns deviate from the expected returns i.e. the residuals

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  • $\begingroup$ Can you provide me the sources of your formula: UnsystematicRisk=[R A −E(R A )]−[R M −E(R M )]∗β ? Thank you in advance $\endgroup$ – user18314 Nov 19 '15 at 8:02
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the simple answer is to make an adjustment to the beta of company. let me give you an example say, beta is 1.0 & correlation of the company with market is 0.5 (which is 50% of the movement in the prices is explained by the market and rest is because of some other reason). so, now one thing is clear that if we some how make this correlation equals to 1 (i.e 100% of the movement is explained by market it self) we can get the total risk.

so, total beta=total risk=Beta/Correlation(r) =1/.5 = 2 total beta = 2.

thanks

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Unsystematic risk of a single stock can be calculated as follows:

$$\sigma_\lambda-\rho_{\lambda,m}\sigma_\lambda=\sigma_\lambda(1-\rho_{\lambda,m})$$

where $\sigma_\lambda$ is the volatility of the stock $\lambda$ and $\rho_{\lambda,m}$ is the correlation between this stock and the market.

Written differently this is the same as:

$$\sigma_\lambda-\beta_\lambda\sigma_m$$

which means that the unsystematic risk of a single stock is its volatility minus its beta scaled by the market volatility.

Sources:

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I assume here you're trying to calculate appraisal ratio, the measure of systematic risk-adjusted excess return relative to idiosyncratic risk. I also agree with a previous comment that the current trend is to call unsystematic risk either specific or idiosyncratic risk.

Specific risk equals the standard deviation of alpha, or alpha plus an error term. You can't really ex ante use any result with an error term because you can't predict when a factory will blow up and such.

I think I saw a correct description of alpha earlier, but it is: $$r_P - [r_F + \beta_{PB}(r_B - r_F)]$$ where $rP$ is portfolio return, $rF$ is the risk-free rate, $\beta_{PB}$ is beta for the portfolio against the benchmark, and $rB$ is the benchmark return. You can use $\beta_{PM}$ and $r_M$ (market measures rather than benchmark measures), but a portfolio manager should be able to beat his benchmark rather than a market index... unless he's an index manager.

I'm not sure how deep your desire to know this goes, but the benchmark should include all the securities from which the manager could select to implement his strategy in the weights appropriate to implement it. If he's just trying to beat the S&P500, use $r_M$ and $\beta_{PM}$.

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Systematic risk and unsystematic risk

1) when total risk assume to be equal to standard deviation of portfolio

Systematic risk= B × standard deviation of market portfolio

And unsystematic risk = standard deviation of portfolio - syetamatic risk ( i.e total risk - systamatic risk)

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  • $\begingroup$ Hi mohit, welcome to Quant.SE! While not wrong this answer is not the quality we hope for here. Due to spelling errors it's hard to read and it also reiterates information already stated. So, unfortunately, this answer has little value for others and I've downvoted it. $\endgroup$ – Bob Jansen Nov 22 '15 at 18:16

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