Problem Set 5solution:
ACTSC 231 Mathematics of Finance, Fall 2010
Q1.
The present value of this perpetuitydue is
1
,
000
v
n
¨
a
∞
= 6
,
561;
where
v
= 9
/
10 i.e.
d
= 1
/
10. We know that ¨
a
∞
= 1
/d
= 10. Thus,
n
=
ln(6
,
561
/
10
,
000)
ln 0
.
9
= 4
.
Q2.
We first need to find the interest rate
i
on the loan, which solves
50
,
000 = 7
,
426
.
39a
8
i.
Using the financial calculator, we find
i
= 4%.
(a) We can then compute OLB
3
= 7
,
426
.
39a
5 4%
= 33
,
060
.
98.
(b) The interest paid in the fourth payment is
i
·
OLB
3
= 1
,
322
.
44.
Q3.
(a) The loan amount
L
is 10,000 and the payment
P
is such that
L
=
P
a
5 6%
. Solving
this equation, we obtain
P
= 2
,
373
.
96.
Consequently, OLB
2
=
P
·
a
3 6%
=
6
,
345
.
63.
(b) The additional payment at time 5 is the accumulated value of the missed payment
at time 3, and hence is given by
P
(1
.
06)
2
= 2
,
667
.
39.
Q4.
This annuity is the sum of the annuity with payment $1,000 from
t
= 0 to
t
= 3 and
the annuity with the first payment $1,000 at
t
= 4 and increasing by 5% per year up
to
t
= 9. The accumulated value at
t
= 4 is
1
,
000¨
s
4 8%
+ 1
,
000
1 + (1
.
05)
v
+ [(1
.
05)
v
]
2
+
· · ·
+ [(1
.
05)
v
]
5
=
1
,
000¨
s
4 8%
+ 1
,
000¨
a
6
i
*
=
4
,
866
.
601 + 5
,
598
.
447
=
10
,
465
.
048
,
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 Spring '10
 WILKIE
 Math, Annual Percentage Rate, Personal Finance, Mortgage loan, Mathematical finance

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