Problem Set 4:
ACTSC 231 Mathematics of Finance, Winter 2011
Q1.
A deferred perpetuitydue begins payments at time
n
with annual payments of $1,000
per year. If the present value of this perpetuitydue is equal to $6,561 and the effective
rate of interest
i
= 1
/
9, find
n
.
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.
A loan of $50,000 is made at time 0 and is repaid by making regular annual payments
of $7426.39 for 8 years in total. (a) What is the outstanding loan balance at the end of
three years? (b) What is the interest payment included in the fourth regular payment?
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.
Julia borrows $10,000 and repays the loan by making annual (end of the year) payments
of
X
for 5 years. The annual effective rate on the loan is 6%. (a) Determine
P
and the
outstanding loan balance at the end of two years. (b) Julia skips the third payment.
What additional payment does she need to make at time 5 in order to repay the loan
within 5 years, as originally planned?
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.
Annual deposits are made into a fund at the beginning of each year for 10 years. The
first 5 deposits are $1,000 each and deposits increase by 5% per year thereafter. If the
fund earns 8% effective per year, find the accumulated value at the end of 10 years.
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 Winter '09
 Chisholm
 Annual Percentage Rate, Interest, Payment, Mortgage loan

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