HOMEWORK 1 (32 points)
1. .
a. (2 points) Compute expected return and standard deviation of the market return
b. (2 points) Compute expected return and standard deviation of stock XYZ
c. (2 points) Compute the covariance of the return of stock XYZ and the the market
return
d. (2 points) Are the market return and the stock XYX return independent random
variables? Why or why not?
2. (4 points) Consider the simple linear regression model
yi = β0 + β1 xi + ui
Derive the OLS estimators:
β̂0 = ȳ − β̂1 x̄
β̂1 =
Pn
i=1
Pn(yi −ȳ)(x12−x̄)
i=1 (xi −x̄)
3. (2 points) Prove Property V3 (see Lecture 1):
For any constant a and b:
V ar(aX + bY ) = a2 V ar(X) + b2 V ar(Y ) + 2abCov(X, Y )
1
4. (2 points) Let Ȳ =
1
n
Pn
i=1 Yi
be the mean estimator of a random sample with E(Yi ) = µ
and V ar(Yi ) = σ 2 . Prove:
V ar(Ȳ ) =
σ2
n
5. (2 points) Let W be an estimator of parameter θ (See Lecture 1). Prove that the mean
square error M SE(W ) of W satisfies:
M SE(W ) = V ar(W ) + [Bias(W )]2
6. Matrix Algebra questions
Please detail all computations to arrive at your final answer. No credit (0
points) will be given to answers stating just the result (obtained with software).
Let
A=
h 1 −9
0
7 i
;B =
1 −5
" 1 0
0 #
1 2 −1
3 1
0
a. (2 points) Does A − B exist? Why?
b. (2 points) Compute the product AB
c. (2 points) Does BA exist? Why?
d. (4 points) Verify that (AB)0 = B 0 A0
e. (2 points) Can you compute tr(AB)? Why?
e. (4 points) Verify that tr(CD) = tr(DC)
2
;C =
h 3
h 1 7 i
6 i
;D =
8 −1
−4 2