# 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?

## 1 Answer

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.

• Hey man, thanks for the answer. This isn't actually a homework, but a question that is likely to appear in my admission test. That said, you mean that the unsystematic-risk (the portion unexplained from the regression) is the variance of the error? Since the question give me only the SD of the regression, does that mean that this is the SD of the error and the variance of the error then would be (0,38²)? Do you have any thoughts in the (1-R²) for the unsystematic error too? Sorry to bother, this questions is extremely important to me! – Renato Chavez Nov 19 '18 at 12:31
• regarding (1-R^2): please note that $R^2 = cov(R_m, R_i)/(\sigma_m \sigma_i)$. Thus the relation between $R^2$ and B is given by: $B = cov(R_m, R_i) / \sigma_m ^2 = R^2 *\frac{\sigma_i}{\sigma_m}$. So (1-R^2) = $1 - \frac{\sigma_m^2 B^2}{\sigma_i ^2}$. simplifying results into $(1-R^2) = \frac{(\sigma_\epsilon)^2}{(\sigma_m)^2}$. Where $\sigma_\epsilon$ is the residual risk. So, asuming i did not mess up my maths it looks like (1-R^2) gives you the ratio between residual risk and market risk (squared) – mbison Nov 19 '18 at 19:05
• Assuming that you are given the value of R2, and assuming you know the variance of the market portfolio, you can isolate the residual risk $\sigma_\epsilon$. (The next link gives you the same result, but derives it in a different way win-vector.com/blog/2011/11/correlation-and-r-squared) – mbison Nov 19 '18 at 19:15
• You are helping me a lot, you seen to be very good at econometrics. So it seens i can't find it with R² since i don't have the variance of the market. About you first answer,i have the variance of X (0,38²), but how can i go about finding the variance of Y if i don't have access to the data used to estimate this regression? I only have the information given in the exercise. My unsystematic would be => Var epsilon = variance_Y/(1.2² x 0,38²) going about your formula, can you give me another tip if its possible to get to the Variance_Y in this exercise? – Renato Chavez Nov 19 '18 at 20:32
• Hey man, its me again, i'm reading a lot of different stuff here since i coudn't find the answer with your explanation (i only have the information in this post, no data to find the variance of Y) I have 2 theories now to find the unsystematic risk: 1 - Since this is a 1 stock portfolio, the SD of Beta (0,38) gives me the total risk, so the unsystematic risk would be 0,38 - market risk. (But i don't know how to find market risk) 2 - If the Beta (1.0) is the market risk, and the beta of firm is 1.2, i have 20% more risk than the market, and this 20% is the Unsyst.risk. What do you think? – Renato Chavez Nov 19 '18 at 20:59