jdlogo

jdlogo

jdlogo

jdlogo

jdlogo

Home

Simple & Compound Interest

Present Value

Ordinary Annuities

Present Value Annuities

General Annuities & Equivalent Rates

Mortgages

Review&Test

 

jdsmathnotes

 

 


 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.

Text Box: ·	The first payment is always at the end of the first interest period of the annuity.  This will be numbered 1 on your time line.
·	The accumulated sum of the future values of each payment at the end of the annuity’s term is called the amount or accumulated value of the annuity. This will be at the last number on your time line.
·	The future value of each payment is evaluated using the formula   A = P(1 + i)n
·	This accumulated amount forms a geometric series (see below).
·	A time line is very helpful in illustrating an annuity.
·	You can use either the geometric series formula OR the amount of an annuity formula (see below) to find the amount of an annuity.
 

 

 

 

 

 

 

 

 

 

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)
Text Box: Formula:
 A = P(1+ i)n
 

 


                                                                                                                       

 

 

 

 

 

 

Text Box: a = the regular payment of the annuity 
n = number of payments or terms
Text Box: Geometric Series Formula:
 

 

 

 

 

 

 

 

 

Text Box: R = the regular payment of the annuity
n  = the number of payments or terms 
i   = interest rate per interest period
A = the accumulated amount of the annuity at the time of the last payment
Text Box: Amount of an Annuity Formula:
 

 

 

 

 

 

 

 

 

 


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

 

 

 

 

 

 

 

Return to top of page

Click here to go back to lesson

Click here to go to homework questions # 4 - 11