

UNIT
9 :
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}