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UNIT 9  :  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.

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