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UNIT
9 :
MATHEMATICS OF INVESTMENT
LESSON 3:
ORDINARY ANNUITY HOMEWORK QUESTIONS CONT’D
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.
Homework Questions cont’d:
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.
Solution:
$100 at
the end of every month for 5 years with interest at 4.25%/a, compounded
monthly.
Interest period0 1
2 . . . 238 239
240 Accumulated value
Payment 100 100 100 100 100
100
100(1.0045)1
100(1.0045)2
.
.
.
100(1.0045)59
100(1.0045)59
Solution:
Let the
yearly payment be $R, with the first payment at the end of the first year; i
= 0.056 and 1 + i = 1.056
Interest period0 1
2 . . . 8
9 10 Accumulated value
Payment R R
R R R
R
R(1.056)1
R(1.056)2
.
.
.
R(1.056)8
R(1.056)9
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?
Solution:
Let the
yearly payment be $R, with the first payment at the end of the first year.
Interest period0 1
2 . . . 46 47
48 Accumulated
value
Payment R R
R R R
R
R(1.0045)1
R(1.0045)2
.
.
.
R(1.0045)46
R(1.0045)47
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?
Solution:
Let the
yearly payment be $R, with the first payment at the end of the first year.
Interest period0 1
2 . . . 94 95
96 Accumulated
value
Payment R R
R R R
R
R(1.006666666)1
R(1.006666666)2
.
.
.
R(1.006666666)94
R(1.006666666)95
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.
Solution:
STEP1: Calculate
the accumulated amount after 4 years,
Interest period0 1
2 . . . 46 47
48 Accumulated value
Payment 200 200 200 200 200
200
200(1.005)1
200(1.005)2
.
.
.
200(1.005)46
200(1.005)47
STEP2:a) Now calculate what
this sum will accumulate to over the next 8 years using A = P(1 +i)n
STEP2:b) Find the amount of the
annuity of $200 at 8%/a, compounded monthly for 8 years.
Interest period0 1
2 . . . 94
95 96 Accumulated value
Payment 200 200 200 200 200
200
200(1.006666666)1
200(1.006666666)2
.
.
.
200(1.006666666)94
200(1.006666666)95
STEP2:c) Add a) + b)
Therefore
the total value of the annuity is $20 475.57 + $56 773.72 = $77 249.29
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 $100 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.
Solution:
Interest period0 1
2 . . . 178 179
180 Accumulated value
Payment 120 120 120 120 120
120
120(1.007)1
120(1.007)2
.
.
.
120(1.007)179
120(1.007)180
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?
Solution:
Interest period0 1
2 . . . 8
9 10 Accumulated value
Payment 2000 2000 2000 2000
2000
2000
2000(1.06)1
2000(1.06)2
.
.
.
2000(1.06)8
2000(1.06)9
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?
Solution:
Let the
number of months be n.
Interest period0 1
2 . . . n-2
n-1 n Accumulated value
Payment 500 500 500 500 500
500
500(1.0045)1
500(1.0045)2
.
.
.
500(1.0045)n - 2
500(1.0045)n - 1