### How to calculate unsystematic risk?

• 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?

You should specify a lot more details in your question. As it is I bet yuo get much answers. Regards

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.

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.

just a point, your 'unsystematic' risk is usually termed 'idiosyncratic' risk.

• 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).

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?

@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.

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?

@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!

• 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

$$\beta_{security,market} = \frac{\sigma_{security,market}}{\sigma^2_{market}}$$

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%.

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?

• 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: $$\sigma^2_{\epsilon}= \sigma^2_{Y}(1-\rho^2_{X,Y})$$

In your top example, are you setting the intercept(alpha) to zero? Why? Ties into my question...http://quant.stackexchange.com/questions/9118/should-the-standard-deviation-points-of-cml-x-axis-be-calculated-with-excess-ret

Is \sigma^2_{\epsilon}= \sigma^2_{Y}(1-\rho^2_{X,Y}) unsystematic in your example?

The variance of a constant is zero so it doesn't matter.

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.

• 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...

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.

• 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.

• 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:

• 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

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

Hi Akhtar Rasheed, welcome to Quant.SE! What about the _un_systematic risk?

• 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

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

• 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

• 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 $$\beta_{PM}$$.