H
AAS
S
CHOOL
OF
B
USINESS
U
NIVERSITY
OF
C
ALIFORNIA
AT
B
ERKELEY
UGBA 103
A
VINASH
V
ERMA
S
OLUTION
TO
H
OMEWORK
6
1.
Stock A and Stock B are expected to return 6.3% and 3.15% respectively on an annual basis. The variance of
returns on Stock A is 0.64 while the variance of returns on Stock B is 3025%%. Assuming that Stock A and Stock B
have the smallest possible correlation, work out the price of a 1 year zero coupon with face value of 1000.
[10 points]
Given:
E
A
= 6.3%;
E
B
= 3.15%;
σ
A
= SQRT(0.64) = 0.8 =80%; and
B
= SQRT(3025%%)
= 55% = 0.55. Since the smallest possible value of the correlation is –1, we are also
given that
ρ
AB
= –1. We know that the risk-eliminating portfolio of the two stocks has the
following portfolio weights:
;
407407
.
0
55
.
0
8
.
0
55
.
0
=
+
=
+
=
B
A
B
A
x
and
.
59259259
.
0
55
.
0
8
.
0
8
.
0
=
+
=
+
=
B
A
A
B
x
The annual (1-year) return on this risk free portfolio is:
E
p
=
x
A
*
E
A
+
x
B
*
E
B
= 0.407*6.3% + 0.593*3.15% = 4.4333
*
%;
Price of a 1-year zero-coupon bond equals $1000/(1+1-year risk free rate) =
$1000/1.044333
*
= $957.55.
2.
[a]: Ms. Adelgundes Krzyzanowska has $600,000 invested as follows: $120,000 in HANS, $420,000 in CAJ,
and the remainder in the risk free asset. You are given that the standard deviation of annualized returns on HANS, CAJ,
and Ms. Adelgundes Krzyzanowska’s portfolio as a whole, is 25%, 21%, and 18%, respectively. Solve for the correlation
between returns on HANS and returns on CAJ.
[5 points]
Denoting HANS as Security
1
, CAJ as Security
2
, and the risk free asset as Security
3
, we
are given that
x
1
= $120,000/$600,000 = 0.2;
x
2
= $420,000/$600,000 = 0.7; and
x
3
=
$60,000/$600,000 = 0.1. We are also given that
1
= 25%
2
= 21%, and
p
= 18%. The
portfolio variance is (
p
)
2
= (0.18)
2
= 0.0324.
For a three-security portfolio, the portfolio variance, here 0.0324, is given by the sum of
entries in the matrix below:
S
ECURITY1
S
ECURITY2
S
ECURITY
n
S
ECURITY1
11
1
1
x
x
12
2
1
x
x
13
3
1
x
x
S
ECURITY2
21
1
2
x
x
22
2
2
x
x
23
3
2
x
x
S
ECURITY3
31
1
3
x
x
32
2
3
x
x
33
3
3
x
x
However, because Security
3
is risk free, its standard deviation is zero, as is its
covariance with the other two securities. Algebraically,
3
= 0,
31
= 0, and
32
= 0.
Thus, all the entries in the third row and column are zero. Adding the remaining four
terms and equating the total to (
p
)
2
= 0.0324, we get:
0.0324 = (
x
1
1
)
2
+ (
x
2
2
)
2
+ 2
x
1
x
2
12
1
2
= (0.2*0.25)
2
+ (0.7*0.21)
2
+ 2*0.2*0.7*
12
*0.25*0.21
0.0147*
12
= 0.0324 – 0.0025 – 0.021609 = 0.008291
12
= 0.008291/0.0147 = 0.564014.
[b]:
Download into an excel worksheet the historical price data on HANS and CAJ from January 1, 2007 to
date. Compute daily returns on the two stocks for each trading day. Now, using the CORREL function in MS Excel,
compute the correlation between the returns on the two stocks. Also compute the standard deviation of returns on each