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UNIT
9 :
MATHEMATICS OF INVESTMENT
LESSON 3:
ORDINARY ANNUITY HOMEWORK QUESTIONS PAGE 1
Quick Review:
Definition: A sequence of payments made at regular
intervals is called an annuity.
Interest Period 0
1 2 3 18
19 20
Payment 200 200 200 200 200 200
An ordinary annuity has the following properties.
![Text Box: P = principal [amount borrowed
or invested]
n = number of interest periods
i = interest rate per interest period as
as a decimal
A = accumulated amount (due or payable)](./ordannhw_files/image011.gif)
Homework Questions: (Solutions below)
1. Find the amount of each of the following
annuities. Include a time line diagram
for #c, d.
a) 150 + 150(1.06) + 150(1.06)2 + .
. . + 150(1.06)18 + 150(1.06)19
b) 2500 +
2500(1.04) + 2500(1.04)2 + .
. . + 2500(1.04)49 +
2500(1.04)50
c) 700 + 700(1.045) + 700(1.045)2 +
. .
. + 700(1.045)14
d) 2000 + 2000(1.054) + 2000(1.054)2
+ .
. . + 2000(1.054)24
2. Find the amount of each of the following
annuities. Include a time line diagram
and the terms of the series for each.
a) $500 at the end of each year for 8 years
with interest at 5.5%/a, compounded annually.
b) $150 at the end of each month for 6 years
with interest at 6.4%/a, compounded monthly.
c) $1000 at the end of every ½ year for 10
years with interest at 4.25%/a, compounded semi-annually.
d) $500 at the end of every 3 months for 7
years with interest at 5.9%/a, compounded quarterly.
3. Find the payment for each of the following
annuities.
a) Half–yearly payments for 10 years at 7%/a,
compounded semi-annually. The
accumulated amount of the annuity is to be
$20 000.
b) Quarterly payments for 8 years at 6.6%/a,
compounded quarterly. The accumulated
amount of the annuity is to be $30 000.
c) Monthly payments for 5 years at 5.6%/a,
compounded monthly. The accumulated
amount of the annuity is to be $150 000.
4. For the past 5 years Amane has been
depositing $100 every month into an investment account . If the interest rate is 5.4%/a, compounded
monthly, how much has she accumulated at the time of her last deposit? Include a time line diagram in your
solution.
5. Find the annual payment for an annuity of 10
years duration at a rate of 5.6%/a, compounded annually, that will amount to
$10 000 at the time of the last payment.
6. Somaiah is saving for her college
education. She wishes to have $12 000
available for her first year’s tuition in 4 years. How much should she deposit each month in an account that pays
5.4%/a, compounded monthly to achieve her goal?
7. Stuart plans to buy a new trenching machine
in 8 years to replace the current one.
He will need $40 000 at this time.
How much should he set aside each month to achieve this goal if interest
is 8%/a, compounded monthly?
8. Calculate the amount of an annuity of $200 per month for 12 years if interest is
6%/a, compounded monthly for the first 4 years and 8%/a, compounded monthly for
the last 8 years.
9. On the birth of their son Patrick, Marie and
Jim started an RESP. They will deposit
$100 per month in an educational savings account for Patrick. For each $110 they deposit the government
will contribute $20. They plan to
contribute until his 15th birthday.
a) How much will they have in the account at
this time if interest is 8.4%/a, compounded monthly?
b) If they leave the money to accumulate until
his eighteenth birthday without any further monthly deposits, how much will it
accumulate to? Assume the same interest
rate.
10. At the end of each year for 10 years the
Frieda deposits $2000 into an investment account which pays 6%/a, compounded
annually. If she then leaves this
amount to accumulate for another 10 years without any further yearly deposits
at the same rate, how much will she have accumulated at this time?
11. Michael invests $500 into an account every
month. The account pays 5.4%/a,
compounded monthly. How many months
will he have to pay in order to accumulate $20 000?
Solutions:
1. Find the amount of each of the following
annuities. Include a time line diagram
for #c, d.
Solution:
Interest period0 1
2 . . . 13 14
15 Accumulated
value
Payment 700 700 700 700 700
700
700(1.045)1
700(1.045)2
.
.
.
700(1.045)13
700(1.045)14
Solution:
Interest period0 1
2 . . .
23 24
25 Accumulated
value
Payment 2000 2000 2000 2000
2000
2000
2000(1.054)1
2000(1.054)2
.
.
.
2000(1.054)23
2000(1.054)24
2. Find the amount of each of the following
annuities. Include a time line diagram
and the terms of the series for each.
a)
$500
at the end of each year for 8 years with interest at 5.5%/a, compounded
annually.
Solution:
Interest period 0 1 2
6 7 8 Accumulated value
Payment 500 500 500 500 500
500
500(1.055)1
500(1.055)2
.
.
.
500(1.055)6
500(1.055)7
b)
$150
at the end of each month for 6 years with interest at 6.4%/a, compounded
monthly.
Solution:
Interest period 0 1 2
70 71 72 Accumulated value
Payment 150 150 150 150 150
150
150(1.005333333)1
150(1.005333333)2
.
.
.
150(1.005333333)70
150(1.005333333)71
c)
$1000
at the end of every ½ year for 10 years with interest at 4.25%/a, compounded
semi-annually.
Solution:
Interest period0 1
2 . . . 13 14
15 Accumulated
value
Payment 1000 1000 1000 1000
1000
1000
1000(1.02125)1
1000(1.02125)2
.
.
.
1000(1.02125)18
1000(1.02125)19
d) $500 at the end of every 3 months for 7
years with interest at 5.9%/a, compounded quarterly.
Solution:
Interest period0 1
2 . . . 26 27
28 Accumulated
value
Payment 500 500 500 500 500
500
500(1.01475)1
500(1.01475)2
.
.
.
500(1.01475)26
500(1.01475)27
3. Find the payment for each of the following
annuities.
a) Half–yearly payments for 10 years at 7%/a,
compounded semi-annually. The
accumulated amount of the annuity is to be
$20 000.
Solution:
Let the
yearly payment be $R, with the first payment at the end of the first year.
Interest period0 1
2 . . . 18 19
20 Accumulated
value
Payment R R
R R R
R
R(1.035)1
R(1.035)2
.
.
.
R(1.035)18
R(1.035)19
b) Quarterly payments for 8 years at 6.6%/a,
compounded quarterly. The accumulated
amount of the annuity is to be $30 000.
Solution:
Let the
yearly payment be $R, with the first payment at the end of the first year.
Interest period0 1
2 . . .
30 31
32 Accumulated
value
Payment R R
R R R
R
R(1.0165)1
R(1.0165)2
.
.
.
R(1.0165)30
R(1.0165)31
c) Monthly payments for 5 years at 5.6%/a,
compounded monthly. The accumulated
amount of the annuity is to be $150 000.
Solution:
Let the
yearly payment be $R, with the first payment at the end of the first year.
Interest period0 1
2 . . . 58 59
60 Accumulated
value
Payment R R
R R R
R
R(1.004666666)1
R(1.004666666)2
.
.
.
R(1.004666666)58
R(1.004666666)59
