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annuity
A stream of equal periodic cash
flows, over a specified time
period. These cash flows can be
inflows of returns earned on
investments or outflows of funds
invested to earn future returns.
ordinary annuity
An annuity for which the cash
flow occurs at the end of each
period.
annuity due
An annuity for which the cash
flow occurs at the beginning of
each period.
4–5 What is meant by “the present value of a future amount”? What is the
general equation for present value?
4–6 What effect does increasing the required return have on the present value
of a future amount? Why?
4–7 How are present value and future value calculations related?
Annuities
How much will you have at the end of 5 years if your employer withholds and
invests $1,000 of your year-end bonus at the end of each of the next 5 years, guaranteeing
you a 9 percent annual rate of return? How much would you pay today,
given that you can earn 7 percent on low-risk investments, to receive a guaranteed
$3,000 at the end of each of the next 20 years? To answer these questions, you
need to understand the application of the time value of money to annuities.
An annuity is a stream of equal periodic cash flows, over a specified time
period. These cash flows are usually annual but can occur at other intervals, such
as monthly (rent, car payments). The cash flows in an annuity can be inflows (the
$3,000 received at the end of each of the next 20 years) or outflows (the $1,000
invested at the end of each of the next 5 years).
Types of Annuities
CHAPTER 4 Time Value of Money 143
There are two basic types of annuities. For an ordinary annuity, the cash flow
occurs at the end of each period. For an annuity due, the cash flow occurs at the
beginning of each period.
EXAMPLE Fran Abrams is choosing which of two annuities to receive. Both are 5-year,
$1,000 annuities; annuity A is an ordinary annuity, and annuity B is an annuity
due. To better understand the difference between these annuities, she has listed
their cash flows in Table 4.1. Note that the amount of each annuity totals
$5,000. The two annuities differ in the timing of their cash flows: The cash flows
are received sooner with the annuity due than with the ordinary annuity.
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Although the cash flows of both annuities in Table 4.1 total $5,000, the
annuity due would have a higher future value than the ordinary annuity, because
each of its five annual cash flows can earn interest for one year more than each
of the ordinary annuity’s cash flows. Similarly, the present value of the annuity
due would be greater than that of the ordinary annuity, because each annuity due
cash flow is discounted back one less year than for the ordinary annuity. In general,
both the future value and the present value of an annuity due are always
greater than the future value and the present value, respectively, of an otherwise
identical ordinary annuity.
Because ordinary annuities are more frequently used in finance, unless otherwise
specified, the term annuity is used throughout this book to refer to ordinary
annuities. In addition, discussions of annuities in this book concentrate on ordinary
annuities. For discussion and computations of annuities due, see the book’s
Web site at www.aw.com/gitman.