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 ANNUITIES

 

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:
 

 

 

 

 

 

 

 

 


Example 1: Finding the accumulated amount of an annuity.

Laura plans to save for her retirement by investing $200 every month.  How much will she have after 20 years if interest is 6%/a, compounded monthly?  Include a time line diagram in the solution.

Solution:

Interest period0           1         2                            .  .  .                                                                        238       239     240   Accumulated value

                               

Payment                    200       200                                                                                                          200       200       200

 


                                                                                                                                                                                                     200

 

                                                                                                                                                                                                     200(1.005)1

 

                                                                                                                                                                                                     200(1.005)2

 

                                                                                                                                                                                                                .

 

                                                                                                                                                                                                                .

 

                                                                                                                                                                                                                .

 

                                                                                                                                                                                                     200(1.005)238

 

                                                                                                                                                                                                     200(1.005)239

                                                                                                                                                                       

 

 

There are 12 x 20 = 240 payments over the term of the annuity. The first payment occurs at the end of the first month as in the diagram.  The last payment occurs on the last day of the 240th month and collects no interest as we evaluate the series at that point. We evaluate each term of the series separately

 using the formula  A = P(1 + i)n

 

Hence Laura will have accumulated $92 408.18 after 20 years.

 

Example 2: Finding the regular payment of an annuity.

Tanya plans to save for her retirement by investing a fixed amount once a year.  How much should she invest each year if she wishes to have $150 000 available after 35 years and  interest is 8%/a, compounded annually?  Include a time line diagram in the solution.

Solution:

Let the yearly payment Be $R, with the first payment at the end of the first year.

Interest period0       1       2                     .  .  .                                                    33      34     35    Accumulated value

                               

Payment                       R         R                                                                                                            R           R          R

 


                                                                                                                                                                                                     R

 

                                                                                                                                                                                                     R(1.08)1

 

                                                                                                                                                                                                     R(1.08)2

 

                                                                                                                                                                                                                .

 

                                                                                                                                                                                                                .

 

                                                                                                                                                                                                                .

 

                                                                                                                                                                                                     R(1.08)33

 

                                                                                                                                                                                                     R(1.08)34

 

 

 

 

There are 35 payments over the term of the annuity. The first payment occurs at the end of the first year as in the diagram.  The last payment occurs on the last day of the 35th year and collects no interest as we evaluate the series at that point. We evaluate each term of the series separately using

 the formula  A = P(1 + i)n

 

 

 

 

Return to top of page

Click here to go to homework questions