Estimating Unsystematic Variance from Beta and Total Risk

Summary

The discussion applies a single-factor decomposition of asset variance into systematic variance linked to market exposure and residual, or unsystematic, variance. In the example, the asset has a beta of zero and a stated standard deviation of six percent. Under the decomposition, the beta-squared market variance term is zero, so the residual variance equals total asset variance, calculated by squaring the standard deviation.

This result depends on the model assumptions and on beta being measured against the relevant market factor. The expected return figure in the question does not enter this variance calculation. The response calls the model CAPM, though the variance decomposition is more precisely associated with a single-factor return model. It does not provide market variance or estimate beta; it only explains what follows from the stated zero-beta assumption.

Key ideas

• A single-factor model splits total asset variance into systematic and residual components.
• The systematic variance term is beta squared multiplied by market variance.
• With beta equal to zero, the example assigns all total variance to the unsystematic component.
• Variance is the square of standard deviation, so the stated six percent volatility is squared.
• The expected return does not determine the variance decomposition.

Tags

# variance unsystematic component


# variance unsystematic component












I was wondering how to calculate the variance of the unsystematic component in an asset.

For example, if an asset's expected return is 10% with standard deviation of 6% and a beta of zero. What would the variance of the systematic component be? Can I use the CAMP model to find it?

## Answer by Question Anxiety (score 2)

https://quant.stackexchange.com/a/43210

What do you want to calculate now the systematic or unsystematic component?

The equation is total risk = systematic + unsystematic risk. Or more formally: o_i^2 = B_i^2 * o_m^2 + o_e^2 So if you have a beta of zero and want to calculate the unsystematic component it would be equal to the total risk of the asset in your case 0.06^2.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.