Estimating Unsystematic Risk from Market-Model Residuals

# How to calculate unsystematic risk?


# How to calculate unsystematic risk?












Systematic risk can be estimate through the calculation of β in CAPM formula. 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 by benjaminmgross (score 11)

https://quant.stackexchange.com/a/3042

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).

## Answer by Ram Ahluwalia (score 10)

https://quant.stackexchange.com/a/2206

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%.

## Answer by SpeedBoots (score 4)

https://quant.stackexchange.com/a/8329

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

## Answer by paracha (score 3)

https://quant.stackexchange.com/a/8324

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...

## Answer by Warren Miller (score 2)

https://quant.stackexchange.com/a/27938

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.

## Answer by vonjd (score 2)

https://quant.stackexchange.com/a/31497

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:

- Cara M. Marshall (2015) Isolating the systematic and unsystematic components of a single stock’s (or portfolio’s) standard deviation, Applied Economics, 47:1, 1-11, DOI: 10.1080/00036846.2014.959652

- Fabio Pizzutilo (2015): Isolating the systematic and unsystematic components of a single stock’s (or portfolio’s) standard deviation: a comment, Applied Economics, DOI: 10.1080/00036846.2015.1068925

## Answer by user6157 (score 1)

https://quant.stackexchange.com/a/8992

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.

## Answer by AKHTAR (score 1)

https://quant.stackexchange.com/a/14914

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)

## Answer by Rime (score 1)

https://quant.stackexchange.com/a/15368

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

## Answer by Imran (score 1)

https://quant.stackexchange.com/a/21661

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

## Answer by Ben Pierce (score 1)

https://quant.stackexchange.com/a/41898

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 $\beta_{PM}$.

## Answer by mohit (score 0)

https://quant.stackexchange.com/a/21857

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)