Copyright © 2008, The McGraw-Hill Companies, Inc.
at the same rate as inflation. If you contribute $50.00 per week this year, and inflation runs
at 4%, you will increase your deposit next year by 4% to $52.00 a week. But the $52 in
actual dollars next year is really the same as $50 in today’s dollars, since our 4% inflation
rate assumes that goods and services that could be bought for $50 today would cost $52
next year.
Of course, once we assume that your deposits change, we no longer have an annu-
ity, and so our annuity formulas won’t work. We could deal with this by forgetting about
using annuity formulas and instead switching over to the spreadsheet methods developed
in Chapter 5. That will work, but there is another way.
Let’s take a different approach than what we did in Example 7.3.2. In that example,
we changed our future value to be the actual dollars you would need, and the payment we
calculated was the actual dollars that you would deposit. Since annuities require equal pay-
ments, our solution had to require the same actual dollar deposits.
We can, though, work out the problem assuming that all dollar amounts are understood
as being in today’s dollars. Our equal deposits, then, would be equal in today’s dollars, but
not in actual dollars. When we calculate the payment in today’s dollars, we understand that
in reality the actual dollar deposits are increasing at the assumed inflation rate. Taking this
approach, the future value would be $1,000,000, understood as 1 million today’s dollars,
which we understand will probably mean a much larger number of actual dollars 40 years
from now.
We will also need to adjust the rate of return that we assume. If you earn 9.1% on your
account balance, $100 will turn into $109.10 in one year. But, if prices rise by 3.5%, you
haven’t really gained a full 9.1%, because in that year prices rose as well. What $100 buys
today will cost $103.50 in a year. So in fact, you’ve actually only earned $109.10 – $103.50
$5.60, or 5.6% on your money. In other words, of the 9.1% we assume you earn, 3.5%
is lost to inflation, so your real rate of return is actually 9.1% – 3.5% 5.6%.^6 If we are
going to work the problem out in today’s dollars, we need to express our rate of return as
the return on today’s dollars, using 5.6% instead of 9.1%.Example 7.3.3 Working entirely in today’s dollars, what amount would you need to
deposit each week to reach $1,000,000 in 40 years, assuming that your account earns
9.1% and infl ation averages 3.5%?Since we are working in today’s dollars, the future value remains $1,000,000. As explained
above, we will use a rate of 9.1% – 3.5% for the sinking fund to fi nd the required payment:FV PMT s −n| (^) i
$1,000,000 PMT s 2080 0.056⁄ 52
$1,000,000 PMT(7,783.300973)
PMT $128.48
So the required payment is $128.48 in today’s dollars.
Don’t forget that this must be understood as today’s dollars. This year, $128.48 would
be the payment, but that amount needs to be adjusted upward each year to keep up
with inflation. It also must be understood that if all works according to plan, the value
of your account will be $1,000,000 in today’s dollars; the actual dollar amount in the
account 40 years from now would actually be much larger, though whatever the amount
is, we are projecting that it will have the same purchasing power as $1,000,000 does
today.
7.3 Assessing the Effect of Inflation 325
(^6) Mathematically, this isn’t entirely clear cut. Subtracting treats the gain of $5.60 as a percent of the original $100,
when in fact you don’t have that $5.60 until the prices have risen to $103.50. Your buying power after 1 year is
better stated as $109.10/$103.50 $105.41, and it might be more accurate to say that your real rate of return is
5.41%. That said, it is conventional to fi nd real rates using subtraction. Unless the rates of return and/or infl ation
are very high, the difference is minimal, and the rates we are using are approximations anyway. Because of this,
though, if you work the problem out in actual dollars with increasing payments using a spreadsheet, the results
will not exactly agree with the answer to Example 7.3.3.