Chapter 5 Time Value of Money 135annuity. If the payments are made at the beginning of each year, the annuity is an
annuity due. Ordinary annuities are more common in! nance; so when we use the
term annuity in this book, assume that the payments occur at the ends of the peri-
ods unless otherwise noted.
Here are the time lines for a $100, 3-year, 5% ordinary annuity and for the same
annuity on an annuity due basis. With the annuity due, each payment is shifted to
the left by one year. A $100 deposit will be made each year, so we show the pay-
ments with minus signs:
Ordinary Annuity:
PeriodsPayments !$100 !$100 !$100(^0) 5% 1 2 3
Annuity Due:
Periods
Payments !$100 !$100 !$100
(^0) 5% 1 2 3
As we demonstrate in the following sections, we can! nd an annuity’s future
and present values, the interest rate built into annuity contracts, and the length of
time it takes to reach a! nancial goal using an annuity. Keep in mind that annuities
must have constant payments and a " xed number of periods. If these conditions don’t
hold, we don’t have an annuity.
Ordinary (Deferred)
Annuity
An annuity whose
payments occur at the end
of each period.
Annuity Due
An annuity whose
payments occur at the
beginning of each period.
Ordinary (Deferred)
Annuity
An annuity whose
payments occur at the end
of each period.
Annuity Due
An annuity whose
payments occur at the
beginning of each period.
SEL
F^ TEST What’s the di# erence between an ordinary annuity and an annuity due?
Why would you prefer to receive an annuity due for $10,000 per year for
10 years than an otherwise similar ordinary annuity?
5-7 FUTURE VALUE OF AN ORDINARY ANNUIT Y
The future value of an annuity can be found using the step-by-step approach or
using a formula, a! nancial calculator, or a spreadsheet. As an illustration, con-
sider the ordinary annuity diagrammed earlier, where you deposit $100 at the end
of each year for 3 years and earn 5% per year. How much will you have at the end
of the third year? The answer, $315.25, is de! ned as the future value of the annuity,
FVAN ; it is shown in Table 5-3.
As shown in the step-by-step section of the table, we compound each payment
out to Time 3, then sum those compounded values to! nd the annuity’s FV, FVA 3!
$315.25. The! rst payment earns interest for two periods, the second payment
earns interest for one period, and the third payment earns no interest at all because
it is made at the end of the annuity’s life. This approach is straightforward; but
if the annuity extends out for many years, the approach is cumbersome and
time-consuming.
As you can see from the time line diagram, with the step-by-step approach, we
apply the following equation, with N! 3 and I! 5%:
FVAN! PMT(1 " I)N#^1 " PMT(1 " I)N#^2 " PMT(1 " I)N#^3
! $100(1.05)^2 " $100(1.05)^1 " $100(1.05)^0
! $315.25
FVAN
The future value of an
annuity over N periods.
FVAN
The future value of an
annuity over N periods.