Unsystematic Risk as Regression Residual Variance

Summary

The document asks how to calculate a firm's unsystematic risk from a market regression, given its R-squared and the regression's standard deviation. It distinguishes total return variability from the part left unexplained by the market. The answer points to the regression decomposition: a firm's return consists of an intercept, a market-linked component, and a residual. Under the stated assumption that the market factor and residual are independent, the total return variance separates into explained market variance and residual variance; the residual variance represents unsystematic risk.

The response is a conceptual hint rather than a worked calculation. It does not show how to convert the supplied standard deviation into variance or derive a numerical result, and it cautions against treating one minus R-squared itself as risk. The example's market variance is not supplied, so the residual risk cannot be calculated from the listed figures alone using the decomposition.

Key ideas

• Unsystematic risk is represented by the variance of the regression residual.
• Total return variance decomposes into market-explained variance and residual variance when the components are independent.
• R-squared describes the share of variance explained by the regression, but one minus R-squared is not itself a risk measure.
• A numerical residual-risk estimate requires the relevant total and market variance information.

Tags

# Calculate unsystematic-risk of a firm in a regression with SD or R2?


# Calculate unsystematic-risk of a firm in a regression with SD or R2?












i'm studying for a finance exam and i can't answer this question. It is asking me to calculate the unsystematic risk of firm A and i have the following information:

Return firm A: 0,1% + 1,2Returnstockexchange

R² = 0,15

SD of regression = 0,38

Risk free rate = 5%

Market rate = 8%

I know that the SD is the total risk of the firm A, but how can i find the unsystematic risk from this?

Some post in the internet says that the Unsystematic risk is (1-R²) but i coudn't find anything solid about this subject, does someone here can answer this?

## Answer by mbison (score 0, accepted)

https://quant.stackexchange.com/a/42697

I'm not going to give you the answer directly (since this is your homework), but as a hint: Y = a + bX + epsilon

variance_Y = b^2 variance_X + variance_epsilon, under the assumption that X and epsilon are independent. The variance of epsilon is what you need.

Hope this helps.

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