WEB APPENDIX 5B
GROWING ANNUITIES
Normally, an annuity is defined as a series of
constant
payments to be received
over a specified number of periods. However, the term
growing annuity
is used to
describe a series of payments that are growing at a constant rate. For example, a
payment of $100 that is growing at a rate of 5% per year for 20 years is a growing
annuity. As we shall see, growing annuities are quite important in the real world.
Example 1: Finding a Constant Real Income, First
Withdrawal Made Immediately
Growing annuities are often used in the area of financial planning. Suppose a
prospective retiree wants to determine the constant
real,
or
inflation-adjusted,
withdrawals he or she can make from a given amount of money over a specified
number of years. For example, suppose your uncle, who is 65 years old, is con-
templating retirement. He expects to live for another 20 years, has a $1 million nest
egg, expects to earn 8% on his or her investments, expects inflation to average 3%
per year, and wants to withdraw a constant
real
amount annually over the next
20 years. If the first withdrawal is to be made
today,
what is the amount of the
initial withdrawal?
This problem can be solved two ways:
1
(1) Use a financial calculator, where
we first calculate the real rate of return, which is the nominal rate adjusted for
inflation, and then use it for
I
to find the initial withdrawal. (2) Set up a spread-
sheet model that is similar to an amortization table, where the account earns 8%
per year and withdrawals rise at the 3% inflation rate. We then use Excel
’
s Goal
Seek function to find the initial inflation-adjusted withdrawal that produces a zero
balance at the end of the 20th year. We illustrate these procedures in the Growing
Annuity tab of the chapter spreadsheet model. The calculator approach is easier to
use, but the spreadsheet model shows the value of the retirement portfolio,
earnings, and each withdrawal over the 20-year planning horizon. Also, the