Confusions over OLS

[Liming]

Dear David,
I’ve had some confusion, misunderstanding and doubts when doing 09 Level I Annotated Boot Camp. Appreciate your kind help on this!

My questions related to question 3 are(question 3 is appended below):
1) Given an OLS relationship such as A = -1.6% +1.8B, do we mean this relation holds for the level of A and level of B, OR the mean return of A and mean return of B ((change/return in A) = -1.6% + 1.8(change/return in B) )?

2) In the subquestion 3, considering that the OLS for A and B is A = -1.6% +1.8 B as calculated from subquestion 2. I don’t understand your answer to sub question 3, where you calculated that the expected annual return of stock A: E(A|B) = average A + 1.8 *(b – average B). I think maybe the intercept -1.6% was omitted by you here. Since we’ve got the OLS: A = -1.6% +1.8 B, why don’t we substitute B in this OLS with (b – average B), therefore getting (A - average A) = -1.6% +1.8 (b – average B). Or should we directly apply A = -1.6% +1.8 B and get that A = -1.6% +1.8*3% ?

Question 3:
Consider two stocks A and B. Assume their annual returns are jointly normally distributed and the marginal distribution of each stock has a mean of 2%. Stock A has standard deviation of 20% and stock B has standard deviation of 10%. Their correlation is 0.9. [source: repurposed 2009 L1.03]
1. What is the beta of Stock A with respect to Stock B? of Stock B with respect to A? (this distinction gives rise to the error)
2. Now that we have slope (i.e., beta from above) and both means, can we specify the OLS line?
Answer: 1. Beta of A with respect to B = Covariance (A,B)/Variance(B) = 1.8. Beta of B with respect to A = 0.45
2. Yes, please keep in mind the OLS passes through the mean of X and mean Y: average Y = intercept + slope(average X). Therefore, intercept = average Y - (slope)(average X) = 2%-(1.8)(2%) = -1.6%. OLS line: Y = -1.6% + (1.8)X
3. E(A|B) = average A + beta * (b - average B) = 2% + (1.8)(3%-2%) = 3.8%

Thank you for your enlightenment and correction!
Cheers
Liming
15/11/09

 

[David Harper CFA FRM]

Hi Liming,

FYI, this question appears here: http://forum.bionicturtle.com/viewthread/1141/

1. The units (A & B) are returns; i.e., both stocks have a mean return of 2% and stock A has a standard deviation (of returns) of 20%. So, the OLS gives the A's return conditional on B'd return.

2. I just showed this way because I wanted to remind of the alternative.
as you suggest, it is more natural to use:
intercept + slope*B = 3% * 1.8 - 1.6% = 3.8%

it is the same result given by using the alternative:
E(A|B) = average A + slope * (B - average B) = 2% + (1.8)(3%-2%) = 3.8%

same result, i only used the latter b/c it's easily forgotten!

David