The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

4.1 What Is an Annuity? 143

payments are all equal, and are made at regular intervals, they constitute an

annuity. Was the amount she borrowed this annuity’s present value or future value?

The $160,000 was received at the start of the annuity payments. Therefore it would be the

present value of the annuity.

Just as with compound interest, the terms “present” and “future” here do not necessarily

correspond to the actual present or future. When Dylan reaches retirement, the value of his

account will still be considered a future value, even though his retirement will no longer be

in the future. Likewise, regardless of how long ago Thalissa took out her loan and bought

her house, we still call the $160,000 the annuity’s present value.

The Timing of the Payments

Our definition of an annuity demands that payments be made at regular intervals (annu-

ally, monthly, weekly, daily, etc.), but it is silent about the timing of the payments within

each interval. If I agree to pay $245 monthly for 60 months for a car loan, this clearly

fits the definition of an annuity. But do my payments begin right away (the payment due

at the start of each month) or can I wait 30 days before making the first payment (the

payment is due at the end of each month)? Our annuity definition says nothing about

this detail.

An annuity actually can be set up either way; either timing of the payments would

meet the definition. However, since the timing of the payments does matter, we do want to

distinguish annuities whose payments come at the start of each period from those whose

payments come at the end.

Definitions 4.1.3

An ordinary annuity is an annuity whose payments are made at the end of each time

period.

An annuity due is an annuity whose payment are made at the beginning of each time

period.

As the name ordinary suggests, annuities are assumed to be ordinary unless otherwise

specified.

Example 4.1.2 Distinguishing between ordinary annuities and annuities due.

(a) John took out a car loan on May 7. Payments will be made monthly. The fi rst

payment is not due until June 7 (the second will be due on July 7, etc.). Is this an

ordinary annuity or an annuity due?

Because his fi rst payment is not made until the end of the fi rst month, the second at the end

of the second month, and so on, his car payments are an ordinary annuity.

(b) Jenna won a lottery jackpot which will pay her $35,000 per year for

the next 26 years. She does not have to wait an entire year to get her fi rst

check—she will be paid the fi rst $35,000 right away. Her second payment will

come a year from now, at the start of the second year, and so on. Is this an

ordinary annuity or an annuity due?

Since the payments come at the start of each year, her prize payout is an annuity due.

Don’t misunderstand the terms start and end here. In John’s example, we said the pay-

ments were made at the end of the month, even though they were in fact due on the seventh

day—nowhere near the time to turn the page of the calendar. End of the month refers to

a month counted from the date the annuity begins. It does not mean end of the calendar

month. Likewise, Jenna does not necessarily receive her lottery payment at the start of each

calendar year (on January 1). Rather, the years in question are measured from the date her

payout annuity begins. She would only receive her payments on January 1 if by coinci-

dence that happened to be the starting date of the annuity.