I am given R-squared = 0.25 How do I find the standard deviation of the residuals ? Give me only the formula, how R-squared is related to the residual st. dev; I couldn’t find it. Thanx!

Y = alpha + beta*X+residuals R^2 = 1-Var(residuals)/Var(Y) -> StDev(residuals) = StDev(Y)*(1-R^2). Example, StDev(Y) = 12%, R^2 = 0.25 -> StDev(residuals)=12%*(1-0.25)=9%.

I still don’t see how I can apply this in the question below: The index model for stock A has been estimated with the following result: RA = 0.01 + 0.9RM + e(A) If st. dev. of the market = 0.25 and the R-square R2A = 0.25, the standard deviation of return of stock A is A) 0.2025 B) 0.2500 C) 0.4500 D) 0.8100 Answer: C I tried the following: Variance of Stock A = Beta-squared X Market Variance + Variance of residuals. Then I take the sq. root of the result to obtain the st. dev. but I don’ know how can I use the R-squared to derive the st. dev. of the residuals.

2x2equals4 Wrote: ------------------------------------------------------- > I still don’t see how I can apply this in the > question below: > > The index model for stock A has been estimated > with the following result: > RA = 0.01 + 0.9RM + e(A) > If st. dev. of the market = 0.25 and the R-square > R2A = 0.25, the standard deviation of return of > stock A > is > A) 0.2025 > B) 0.2500 > C) 0.4500 > D) 0.8100 > Answer: C > > I tried the following: Variance of Stock A = > Beta-squared X Market Variance + Variance of > residuals. Then I take the sq. root of the result > to obtain the st. dev. but I don’ know how can I > use the R-squared to derive the st. dev. of the > residuals. Actually I forgot to take square root of (1-R^2) in my formula above. Let’s look at your derivation. You are correct that Var(A)=beta^2*Var(M)+Var(Res) Var(Res)=0.75*Var(A) -> beta^2*Var(M)=0.25*Var(A) Therefore, Var(A)=4*beta^*Var(M)->StDev(A)=2*beta*StDev(M)=2*0.9*25%=45% -> C

Thanx a lot!