Exercise Sheet 7
Exercise 1
Assume there are two stocks, A and B, with A = 1:4 and B = 0:8. Assume
also that the CAPM model applies.
(i) If the mean return on the market portfolio is 10% and the risk-free rate
of return is 5%, calculate the mean return of the portfolios consisting of:
a. 75% of stock A and 25% of stock B,
b. 50% of stock A and 50% of stock B,
c. 25% of stock A and 75% of stock B.
(ii) If the idiosyncratic variations of the stocks are A = 4; B = 2 and
the variance of the market portfolio is 2M = 12, calculate the variance of the
portfolios in (a), (b), (c).
(iii) What are the mean return and variance of the portfolios if they are 50%
…nanced by borrowing?
Solution 1
(i) The security market line can be used to write
rA
= rf +
A
(rM
=
5 + 1:4 (10
=
12;
rf )
5)
and
rB
=
5 + 0:8 (10
=
9:
5)
For the portfolios
a: rp
b: rp
c: rp
=
XA rA + XB rB
=
0:75
=
11:25;
12 + 0:25
=
0:5
=
10:5;
=
0:25
=
9:75:
12 + 0:5
12 + 0:75
9
9
9
(ii) The beta of a portfolio is found using
p
= XA
A
+ XB
B;
and the variance
2
p
=
2 2
p M
2
+ XA
1
2
A
2
+ XB
2
B
:
Applying these results
a:
p
2
p
0:75
1:4 + 0:25
=
1:25:
0:8
=
1:252 12 + 0:752 16 + 0:252 4
=
28:
b:
p
2
p
c:
=
0:5
=
1:1;
1:4 + 0:5
0:8
=
1:12 12 + 0:52 16 + 0:52 4
=
19:52:
p
2
p
=
=
0:25
1:4 + 0:75
=
0:95;
0:8
=
0:952 12 + 0:252 16 + 0:752 4
=
14:08:
(iii) If 50% …nanced by borrowing the portfolio proportions are Xp = 2 and
Xf = 1. So the expected return and variance are
2
r = Xp rp + Xf rf ;
= Xp2
2
p:
Evaluating for the individual portfolios
a: r = 2
b: r = 2
c: r = 2
11:25
10:5
9:75
1
1
1
2
5 = 17:5;
5 = 16;
5 = 14:5;
2
2
= 22
= 22
= 22
28 = 112:
19:52 = 78:08:
14:08 = 56:32:
Exercise 2
Assume there are just two risky securities in the market portfolio. Security
A, which constitutes 40% of this portfolio, has an expected return of 10% and
a standard deviation of 20%. Security B has an expected return of 15% and a
standard deviation of 28%. If the correlation between the assets is 0.3 and the
risk free rate 5%, calculate the capital market line.
Solution 2
The expected return on the market is
rM
= XA rA + XB rB
=
0:4
10 + 0:6
2
15 = 13;
and the variance of the market is
2
M
2
= XA
2
A
2
2
+ XB
2
2
B
+ 2XA XB
2
=
0:4 20 + 0:6 282 + 2
=
426:88:
AB A B
0:4
0:6
0:3
The standard deviation of the market portfolio follows as
20:661: Hence the capital market line is
rp
rM
= rf +
20
M
28
=
p
426:88 =
rf
p
M
=
=
13 5
20:661
8
5+
20:661
5+
p
p:
Exercise 3
The market portfolio is composed of four securities. Given the following
data, calculate the market portfolio’s standard deviation.
Security Covariance with market Proportion
A
242
0.2
B
360
0.3
C
155
0.2
D
210
0.3
Solution 3
The market beta must satisfy
1
= XA
=
1
=
A+
AM
XA 2
M
0:2
242
2
M
XB
+ XC
B
BM
2
M
+ XB
+ 0:3
360
2
M
C
+ XD
+ XC
+ 0:2
155
2
M
CM
2
M
+ 0:3
D
+ XD
DM
2
M
210
2
M
so
M
= 15:824
Exercise 4
Given the following data, calculate the security market line and the betas of
the two securities.
Security 1
Security 2
Market portfolio
Risk free asset
Solution 4
Expected return
Correlation with market portfolio
Standard deviation
15.5
9.2
12
5
0.9
0.8
1
0
2
9
12
0
3
The security market line is
ri
= rf +
=
5+
i
(rM
i (12
rf )
5) :
For security 1
15:5 = 5 +
(12
5) ;
(12
5) ;
1
= 1:5:
1
For security 2
9:2 = 5 +
2
2
= 0:6:
Exercise 5
Consider an economy with just two assets. The details of these are given
below.
Number of Shares Price Expected Return Standard Deviation
A 100
1.5
15
15
B 150
2
12
9
The correlation coe¢ cient between the returns on the two assets is 1=3 and
there is also a risk free asset. Assume the CAPM model is satis…ed.
(i) What is the expected rate of return on the market portfolio?
(ii) What is the standard deviation of the market portfolio?
(iii) What is the beta of stock A?
(iv) What is the risk free rate of return?
(vi) Construct the capital market line and the security market line.
Solution 5
(a) Value of A: 100 1:5 = 150
Value of B: 150 2 = 300
Total market value = 450
So
150
1
300
2
XA =
= ; XB =
=
450
3
450
3
Since
rM = XA rA + XB rB
this gives
rM =
2
1
15 + 12 = 13
3
3
(b)
2
M
2
M
2
= XA
=
1
3
2
A
2
+ XB
2
152 +
2
B
+2
XA
2
2
3
92 + 2
So
M
(c) By de…nition,
A
=
AM
2
M
=9
:
4
XB
2
3
1
3
A
AB
1
3
15
B
9 = 81
To …nd
AM
:
rA ) (rM
= E [(rA
AM
rM )]
but using the de…nition of the return on the market
AM
= E [(rA
rA ) (XA rA + XB rB
(XA rA + XB rB ))]
Collecting terms
AM
= E [(rA
rA ) (XA (rA
rA ) + XB (rB
rB ))]
Hence
AM
= XA E [(rA
rA ) (rA
AM
AM
AM
=
rA )] + XB E [(rA
= XA
= XA
2
A
2
A
+ XB
+ XB
1
21
225 +
3
33
rA ) (rB
rB )]
AB
AB A B
15
9 = 105
Therefore
105
= 1:2963
81
(iv) The risk-free return is derived from the either the capital market line or
the security market line.
The security market line gives
A
=
rA = rf +
A
[rM
rf ]
So
rf =
rA
1
A rM
=
15
1
A
1:2963 13
= 6:25
1:2963
(v) Capital market line
rp = rf +
rM
rf
p
M
rp = 6:25 + 0:75
p
Security market line
rp = rf +
p
5
[rM
rf ]
rp = 6:25 + 6:75
p
Exercise 6
Consider an economy with three risky assets. The details of these are given
below.
No. of Shares Price Expected Return Standard Deviation
A 100
4
8
10
B 300
6
12
14
C 100
5
10
12
The correlation coe¢ cient between the returns on any pair of assets is 1/2
and there is also a risk free asset. Assume the CAPM model is satis…ed.
(i) Calculate the expected rate of return and standard deviation of the market portfolio.
(ii) Calculate the betas of the three assets.
(iii) Use solution to (ii) to …nd the beta of the market portfolio.
(iv) What is the risk-free rate of return implied by these returns?
(v) Describe how this model could be used to price a new asset, D.
Solution 6
(i) Total value of risk assets is
W = 100 4 + 300 6 + 100 5 = 2700
The proportions of the assets are
18
5
4
; XB =
; XC =
27
27
27
XA =
The expected return on the market portfolio is
rp =
4
27
8+
18
27
12 +
5
27
10 = 11:037
The standard deviation of the market portfolio is
2
p
=
4
27
+2
=
2
2
2
(10) +
4
27
5
27
2
18
5
4
18
2
2
(14) +
(12) + 2
27
27
27
27
1
18
5
1
10 12 + 2
14 12
2
27
27
2
132:1
Hence
p
=
p
132:1 = 11:493
6
1
2
10 14
(ii) This follow from
AM
= Cov (rA ; rM )
= E ((rA
rA ) (rM
= E ((rA
rA ) (XA (rA
XA 2A
=
+ XB
=
rA ) + XB (rB
+ XC AC
18
1
10 14 +
27
2
rB ) + XC (rC
rC )))
AB
4
2
(10) +
27
72:593
=
rM ))
So
A
=
5
27
1
2
10 12
72:593
= 0:54953
132:1
For the other two
=
BM
=
XA
+ XB
2
B
+ XC
BC
156:59
156:59
= 1:1854
B =
132:1
= XA
CM
AB
AC
+ XB
BC
+ XC
2
C
=
91:556
91:556
= 0:69308
C =
132:1
(iii) The beta of the market portfolio is
M
= XA A + XB B + XC C
4
18
=
0:54953 +
1:1854 +
27
27
5
27
0:69308
(iv) The risk-free rate can be found from the security market line
ri = rf +
So
rf =
i
(rM
i rM
i
rf )
ri
1
Using asset A
rf
=
=
0:54953 11:037
0:54953 1
4:2952
8
(v) In practice, the value of D would be computed and the security market
line employed to give the predicted return. In this case there are only three
existing assets. If the proportion of the new asset were a signi…cant part of
the market portfolio then the computations would have to begin again from the
beginning since all the data will change.
7