324 Chapter 7 Retirement Plans
The inflation rate we used in this example is slightly higher than the long-term average, so
this $416.20 predicted price assumes inflation over the next 20 years that runs a bit above
the long-term rate. However, 4% is close enough to the long-term average that we can say
that, while this prediction may prove inaccurate, it is not completely unreasonable either.
Let’s take another look at the example mentioned at the start of this section: how much
does someone need to set aside each week in order to really have $1,000,000 in 40 years?Example 7.3.2 Suppose you want to have $1,000,000 in your 401(k) account in
40 years. How much do you need to deposit into this account each week to achieve
your goal? For the types of investments you plan to make, you expect to earn 9% on
your investments. Also assume that your goal is not $1,000,000 in actual dollars, but
instead is $1,000,000 in today’s dollars.Our fi rst step is to fi gure out what the actual goal is. Of course, since we can’t know what
infl ation will be over the next 40 years, there is no way to know this for sure. However, just
as with predictions about investment returns, we can reasonably approach the question by
using an infl ation rate based on what has happened in the past. So we will assume a 3.5%
infl ation rate in this problem, and use it to predict the cost of $1,000,000 (at today’s prices)
worth of goods and services in 40 years.FV PV (1 i)n
FV $1,000,000(1 0.035)^40
FV $3,959,260So your goal is actually $3,959,260.Now, using this as the future value of a sinking fund, we can calculate the required weekly
deposits:FV PMT s −n| (^) i
$3,959,260 PMT s 2080 .09⁄ 52
$3,959,260 PMT(20502.17)
PMT $193.11
So it turns out you would need to deposit $193.11 each week to reach this million dollar goal.
This is a deeply disappointing result. In Chapter 4, we saw that fairly modest deposits could
grow into enormous future values, and the goal of becoming a millionaire seemed attain-
able, even easy. If we had used a $1,000,000 future value without adjusting for inflation,
we would calculate that the weekly deposits need to be only $48.78. Weekly deposits of
$48.78 amount to a bit over $2,500 a year—not pocket change, but not completely unat-
tainable either. But $193.11 a week amounts to over $10,000 a year, an investment that
most people would find challenging to say the least.
There is more to this story, though. We overlooked something in the prior example. We
adjusted the targeted future value for inflation, but we did not make any adjustments to the
payments. Our solution to the problem assumed that you will deposit $193.11 in the first
week, and every week in between, so over the course of 40 years your weekly deposits never
increase. If that is the goal—to find the fixed and never-changing weekly payment to reach
the target—then our solution is correct. But as prices rise over the course of the next 40 years,
hopefully your income will too. Probably a more realistic scenario would be to assume that
you start making deposits at a certain rate today, but that over time as prices and your income
rise, you will increase your deposits at a similar rate. This assumption changes things.
Projections in Today’s Dollars
First of all, we have to make some assumption about the rate at which you will increase
your payments. The simplest approach is to assume that you want to maintain a contribu-
tion rate that stays the same in today’s dollars, and so assume that your deposits increase