Section 3.4
Kyle Matthews
”A penny saved is a penny earned.”
When saving money we generally deposit relatively small quantities over
long periods of time. An account in which a fixed amount of money deposited
over regular intervals is called a systematic savings plan.
Our first goal is to come up with an easy to use formula to calculate the
amount of money that accumulates in a systematic savings plan. To do this let
us first consider an example to help guide us:
Suppose $100 is deposited every month for 5 months into a savings account
earning 6% interest compunded monthly. The first deposit made at the end of
the first month will be in the account earning interest for 4 months. Thus, using
4
12· 12
the compound interest formula this deposit will be worth 100(1 + .06
=
12 )
100(1.005)4 The next deposit will be in the account for 3 months so it will be
3
12· 12
worth 100(1 + .06
= 100(1.005)3 We keep doing this until we get to the
12 )
fifth deposit which is made at the end of the fifth month so it earns no interest
is worth just $100.
Thus, the total amount in the account will be the sum:
100(1.005)4 + 100(1.005)3 + 100(1.005)2 + 100(1.005) + 100
Finding this is not too bad for 5 months, but imagine trying to calculate 5
years which would be a sum of 60 terms! Let us work to come up with a more
computationally friendly formula.
We first notice that our reasoning works for any number of k months. That
is, the amount in the account after k months will be:
100(1.005)k−1 + 100(1.005)k−2 + · · · + 100(1.005) + 100
= 100((1.005)k−1 + (1.005)k−2 + · · · + 1.005 + 1)
A sum of the form ak−1 + ak−2 + · · · + a + 1 is called a geometric series. For
a 6= 1 there is a nice formula:
ak − 1
a−1
Applying this to our situation above we see that after k months the amount
of money in the account will be
(1.005)k − 1
100
0.005
ak−1 + ak−2 + · · · + a + 1 =
1
Notice how much easier a little thinking made this formula.
For the general formula notice that $100 is just the amount of each deposit,
0.005 is just the periodic interest nr where r is the annual interest and n is the
number of periods in a year. Note that k is the total number of periods which
means k = nt where t is the number of years the account is held. Hence, putting
this together and using our above derivation we have the following:
Theorem 1 (Systematic Savings Plan). The future value F of a systematic
savings plan is given by the formula
(1 + nr )nt − 1
F =D
r
n
where D is amount of each deposit, r is the annual interest rate, n is the number
of periods in a year, t is time in years. This formula uses the convention that
deposits are made at the end of each period.
Lets do some example using our newly derived formula.
Example 1.
Ken is an evil genius and is saving up to buy a tank of sharks
with freaking lazers attached to their freaking heads. If he saves $300 a month
in a savings account earning 4.9% interest compounded monthly for 20 years
how much will be in the account and how much of this will be interest earned?
Solution. Using our formula we have that D = 300, t = 20, r = 0.049, n = 12
which means that the future value will be
20·12
(1 + .049
−1
12 )
F = 300
≈ 121, 896.36
.049
12
To calculate the interest we first find how much was deposited which is just
300 · 12 · 20 = 72, 000. Hence, the amount of interest earned is
121, 896.36 − 72, 000 = 49, 896.36.
Example 2.
The evil genius Ken has just had little evil genius babies. He
wants to save money to send them to a good(evil?) university. They will enter
college in 17 years and he estimates their education will cost about $250, 000.
He decides to put money into some stocks which are estimated to earn 7%
compounded quarterly over the next 20 years. How much should Ken invest
every quarter to ensure he will reach $250, 000 in 17 years?
Solution. We have F = 250, 000, r = 7%, n = 4, t = 17 and we want to find D.
That is, we want to solve the equation
4·17
−1
(1 + .07
4 )
250, 000 = D
0.07
4
2
for D.
Calculating the numbers this is:
250, 000 ≈ D · 128.7669
and hence D ≈ 1941.49.
Thus, Ken should put about $1941.49 every three months into the stock.
Example 3.
You want to go on a dream vacay to some remote islands.
The total cost of the trip is about $3, 000. If you put $50 into an account every
month which earns 3% interest compounded monthly, how long will it take you
to have enough to go on the trip?
Solution. We seek to find the time. We have that F = 3, 000, D = 50, n =
12, r = .03. Thus we want to solve the equation
(1.0025)12t − 1
3, 000 = 50
0.0025
for t.
We isolate (1.0025)12t since this is the term with t. We get that
3, 000 · 0.0025 + 50
= (1.0025)12t
50
Thus we want to solve an equation with the variable in the exponent so we
take the log of both sides:
log(1.15) = log((1.0025)12t )
Using log properties we have that
log(1.15) = 12 · t · log(1.0025)
Solving for t we get that t =
log(1.15)
12 log(1.0025
3
≈ 4.66 years