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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 15^th 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}

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