Question

You estimated a regression model using annual returns of ExxonMobil (as a dependent variable) and of the market (as an independent variable). The R-squared of this regression is 0.2, and the total variance of ExxonMobil's returns in the estimation window is 0.0625. In this case, the variance of the unsystematic (or idiosyncratic) component of ExxonMobil's returns is:

Answer 1

R-squared = 0.2

Total Variance of ExxonMobil = 0.0625

As Formula of R-squared = Variance explained by Market / Total Variance

We have R-squared and Total Variance, hence, we can find Explained Variance

Putting values in the formula

=> 0.2 = Explained Variance / 0.0625

The Explained Variance is also known as Systematic Variance, since, it is the variance explained by Market index.

=> Systematic variance = 0.2 * 0.0625 = 0.0125

As Total Variance = Systematic Variance + Unsystematic Variance

Therefore, Total Variance = 0.0125 + Unsystematic Variance

We already have Total variance

Hence, 0.0625 = 0.0125 + Unsystematic Variance

Unsystematic Variance = 0.0625 - 0.0125 = 0.05

Hence, Unsystematic Variance is 0.05