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ECO 362
December 6, 2007
PROBLEM SET 7
1
FINANCIAL INVESTMENTS
Professor: Burt Malkiel
Exercise 1
The approximate percentage price change due to a change
∆
y
is
∆
P
P
=
−
D
∗
×
∆
y
=
−
D
×
1
1+
y
×
∆
y
.
where
D
and
D
∗
denote Macaulay and modi
f
ed duration, respectively. Given the information:
y
= 10%
,
D
=7
.
194
,and
∆
y
=0
.
5%
,wehave
∆
P
P
=
−
7
.
194
×
1
1+0
.
10
×
0
.
005 =
−
0
.
0327
.
Then, the price would decline by
3
.
27%
.
Obs
: Since the wording did not specify what duration
7
.
194
was, I will consider right those
answers that used
D
∗
=7
.
194
instead, and hence obtained a decrease of
3
.
6%
. In any case, you have
to be sure of being using the right number in the Final!
Exercise 2
A
6%
coupon bond making annual coupon payments has a coupon of
60 = 6%
×
F
=6%
×
1
,
000
(where
F
=1
,
000
as usual). Furthermore, we know that
y
=6%
,sow
ehav
ea
l
lth
en
e
ed
ed
information to compute the Macaulay’s duration. As last week, we can construct the following table:
(1)
(2)
(3)
(2)
×
(3)
(4)
(1)
×
(4)
Semester
Payment
Discount rate
Disc. Payment (6%)
Weight
Contrib. to
D
1
60
0.9434
56.60
0.0566
0.0566
2
60
0.8900
53.40
0.0534
0.1068
3
1060
0.8396
890.00
0.8900
2.6700
So that the Macaulay duration is
D
=2
.
833
years. Consequently, the modi
f
ed duration is
D
∗
=
D
1+
y
=
2
.
833
1+0
.
06
=2
.
673
years.
If instead, one assumes
y
= 10%
, doing the same as above:
(1)
(2)
(3)
(2)
×
(3)
(4)
(1)
×
(4)
Semester
Payment
Discount rate
Disc. Payment (10%)
Weight
Contrib. to
D
1
60
0.9091
54.55
0.0606
0.0606
2
60
0.8264
49.59
0.0551
0.1102
3
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View Full DocumentECO 362
December 6, 2007
So that the Macaulay duration is
D
=2
.
824
years —which is less than the duration at
y
=6%
—
and the modi
f
ed duration is
D
∗
=2
.
567
years.
Intuition
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 Fall '08
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