194 Chapter 4 Annuities
Wayne’s future account value is less of a challenge. Once his payments start, they continue
on to the end, and so Wayne’s account is an annuity:FV PMT s _n (^) | (^) i
FV $2,500 s 30 | (^) .084
FV $2,500(121.9393221)
FV $304,848.30
So even though he makes the payments for 30 years instead of Wanda’s 10, Wayne winds
up with less money in his account in the end!
The results of this example may be a little surprising. Wanda’s account ran for 40 years
versus Wayne’s 30—that doesn’t seem that much longer. But Wayne made contributions
for 30 years versus Wanda’s 10, so he contributed 3 times as much as she did—that does
seem like a big difference. Yet Wanda came out ahead because of the power of com-
pound interest over time. Another view of those future values may help explain Wanda’s
victory:
Example 4.6.3 How much of Wanda’s future value came from interest? How much
of Wayne’s did?
Wanda contributed $2,500 a year for 10 years, so her total deposits were ($2,500/year)
(10 years) $25,000. Since her future value totaled $414,993.99, her total interest earn-
ings were $414,993.99 $25,000 $389,993.99.
Wayne contributed $2,500 a year for 30 years, so his total deposits were ($2,500/year)
(30 years) $75,000. His total interest earnings, then, were $304,848.30 $75,000
$229,848.30.
Those extra 10 years that Wanda had compound interest working for her made all the
difference, even making up for her much smaller total contributions. This is interesting
mathematically, but it also interesting from a practical point of view: if you want to have
an account grow into a large future value, how soon you start putting money to work can
be far more important than how much money you put to work.
“Sinking Funds” Whose Payments Stop
What if, instead of letting the payments determine the future value, we let a target future value
determine the payments? This is basically the same idea as a sinking fund, except that now
we are open to the possibility that the payments might stop at some point along the way.
Example 4.6.4 Dario wants to have $1,000,000 in his retirement account when he
reaches age 70, 45 years from now. Dario thinks that his account can earn 9.6%, and
he’d like to reach his goal by making monthly deposits for just the next 10 years. How
much should each deposit be?
Here we need to work backwards. For the last 35 years, the account simply earns interest,
and so at the start of the period its value must have been enough to grow to the target
$1,000,000. So:
FV PV(1 i)n
$1,000,000 PV (^) 1 ____0.096 12
420
PV $35,202.74
So the payments for the fi rst 10 years must accumulate to this amount:FV PMT s _n (^) | (^) i
$35,202.74 PMT s ___ 120 | (^). 096
⁄ 12
PMT $175.82