ACF329 Interest Theory
Fall 2010
University of Texas at Austin
HW 3  Solutions
Instructor: Milica
ˇ
Cudina
3.7.2. We split the annuitydue from the problem into two annuitiesdue: one with payments
of 100 and the other with payments of 300. The latter annuity is one halfyear deferred.
With
i
= 0
.
06, the present value of Suzanne’s annuitydue can then be calculated as
100¨
a
20
i
+ (1 +
i
)

1
/
2
·
300¨
a
20
i
= 100¨
a
20
i
(1 + 3(1 +
i
)

1
/
2
)
≈
100
·
12
.
158
·
3
.
914
≈
4
,
758
.
51
.
3.7.4 Let
P
= 100,
i
1
= 0
.
15 and
i
2
= 0
.
06. Then, the accumulated value of Bill’s deposits at
the end of 13 years equals
P
(
s
5
i
1
·
(1 +
i
2
)
8
+
s
8
i
2
) =
P
1
.
15
5

1
0
.
15
·
(1
.
06)
8
+
1
.
06
8

1
0
.
06
= 20
.
4638
P.
On the other hand, Seth’s deposits must satisfy
20
.
4638
P
=
P
·
s
13
i
=
P
·
(1 +
i
)
13

1
i
.
Using our calculator, we solve for
i
, we get
i
≈
0
.
0738.
3.7.6 The present value of the annuity
X
is
PV
(
X
) = 1000¨
a
12
+ 2000
v
12
¨
a
18
with
v
= 1

d
= 0
.
9. So,
PV
(
X
) = 1
.
1111(6458
.
475 + 4320
.
98553) = 1
.
1111
·
10779
.
4605 = 11977
.
0586
..
The present value of the perpetuity
Y
can be expressed as
PV
(
Y
) =
Qa
20
+ 3
Qv
20
a
∞
=
Q
(
a
20
+ 3
v
20
a
∞
)
=
Q
(7
.
9064 + 3
·
0
.
9
20
/
0
.
1111 = 11
.
1893
Q.
So,
Q
= 11977
.
0586
/
11
.
1893
≈
1070
.
40285
.
Note: The answer in the back of the textbook is different due to rounding errors.
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 Spring '08
 HIETMANN
 Time Value Of Money, Perpetuity, Mathematical finance

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