Home Simple & Compound Interest Present Value Ordinary Annuities Present Value Annuities General Annuities & Equivalent Rates Mortgages Review&Test

UNIT 11  :  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.

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