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UNIT 11  :  MATHEMATICS OF INVESTMENT

LESSON 4: PRESENT VALUE ANNUITY

Definition 1: A sequence of payments made at regular intervals is called an annuity.

Definition 2: When we calculate the present values of the sequence of payments made at regular intervals this is called the Present Value of the annuity.

A Present Value annuity has the following properties.

Example 1: Finding the Present Value of an annuity.

Heywood recently won \$5 000 000 in the lottery.  He plans to purchase an annuity that will pay him \$50 000 every year for 25 years and spend the rest.  How much of his winnings would he need to pay today for that annuity  if  interest is 7.6%/a, compounded annually?  Include the series and a time line diagram in the solution.

Solution:

We calculate the present values of the 25 future payments of \$50 000 each.  Notice the arrows go to the left for a present value annuity.

Interest Period   0          1         2          3                                                                                                                                         23   24    25

Payment( 1000’s)           50       50                                                                                                                                                   50    50    50

50000(1.076)-1

50000(1.076)-2

.

.

50000(1.076)-23

50000(1.076)-24

50000(1.076)-25

This forms the following geometric series:  Note – write the last term first.

50 000(1.076)-25 + 50000(1.076)-24 +  . . . + 50000(1.076)-2 + 50000(1.076)-1

Hence Michael needs \$552 492.55 of his winnings to purchase the annuity. He would have \$5 000 000 - \$ 552 492.55 = \$4 447 507.45 left to spend.

Alternate Solution:

Notice this present value amount is much less than the total amount paid out over 25 years which would be 25 x \$50 000=\$1 250 000.

Example 2: Finding the Payment for a Present Value annuity.

Richard Rummy recently retired from Rofasco Inc.  He received a retirement gratuity from profit sharing of \$300 000.  He wishes to purchase an annuity that will pay him a fixed income every month for 25 years.  If interest is 8.4%/a, compounded monthly, determine his monthly income

Solution:

Let the monthly payment be \$R, with the first payment at the end of the first month.

Interest Period   0          1         2          3                                                                                                                                         298   299   300

Payment( 1000’s)           R         R        R                                                                                                                                          R      R      R

R(1.007)-1

R(1.007)-2

.

.

R(1.007)-298

R(1.007)-299

R(1.007)-300

This forms the following geometric series:

R(1.007)-300 + R(1.007)-299 + . . . + R(1.007)-2 + R(1.007)-1

Hence  Richard will receive a monthly income of \$2395.50 for 25 years from his gratuity by purchasing an annuity.

Example 3: Borrowing money – finding the monthly payment.

Tanya plans to borrow \$30 000 to use as a down payment on the purchase of her dream home.   She will repay the loan with monthly payments over a 10 year period.  How much will the monthly payment be if  interest is 4.2%/a, compounded monthly?  Include a time line diagram in the solution.

Solution:

As in the previous question, the money in question is borrowed now – at point 0 on the time line.  Hence this is a PV annuity question

Interest Period   0          1         2          3                                                                                                                                         118   119  120

Payment( 1000’s)           R         R        R                                                                                                                                          R      R      R

R(1.0035)-1

R(1.0035)-2

.

.

R(1.0035)-118

R(1.0035)-119

R(1.0035)-120

This forms the following geometric series:

R(1.0035)-120 + R(1.0035)-119 + . . . + R(1.0035)-2 + R(1.0035)-1