University of Pennsylvania
The Wharton School
FNCE 100
A. Craig MacKinlay
PROBLEM SET #3
Fall Term 2005
Diversification, Risk and Return
1. We have three securities with the following possible payoffs.
Probability
Return on
Return on
Return on
State
of Outcome
Security #1
Security #2
Security #3
1
.
10
.
25
.
25
.
10
2
.
40
.
20
.
15
.
15
3
.
40
.
15
.
20
.
20
4
.
10
.
10
.
10
.
25
(a) What is the expected return and standard deviation on each security?
(b) What is Cov(
R
1
, R
2
), Cov(
R
1
, R
3
), and Cov(
R
2
, R
3
)?
What is Corr(
R
1
, R
2
), Corr(
R
1
, R
3
), and Corr(
R
2
, R
3
)?
(Cov(
·
,
·
) denotes covariance and Corr(
·
,
·
) denotes correlation).
(c) What is the expected return,
E
(
R
p
), and standard deviation,
σ
(
R
p
), of a portfolio
which has half of its funds invested in Security #1 and half in Security #2?
(d) What is
E
(
R
p
) and
σ
(
R
p
) of a portfolio which has half of its funds invested in
Security #1 and half in Security #3?
(e) What is
E
(
R
p
) and
σ
(
R
p
) of a portfolio which has half of its funds in Security #2
and half in #3?
(f) Compare your answers in Parts (a), (c), (d), and (e). Comment on the effects of
diversification.
2. Assume there are
n
securities, each having:
E
(
R
i
)
=
.
01
i
=
1
,
2
, . . . , n
σ
2
(
R
i
)
=
.
01
i
=
1
,
2
, . . . , n
cov(
R
i
, R
j
)
=
.
005
i
=
1
,
2
, . . . , n
;
j
=
1
,
2
, . . . , n
;
j
6
=
i
31