6.1 Simple Interest


A generic line chart.

5.4

Compound Interest Discretely

Learning Objectives

Upon completion of this section, you should be able to

  • Calculate the value of an account for which interest is compounded n times per year (discretely).
  • Solve various problems involving compounding interest discretely.

Compound Interest

In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on
that interest. This reinvestment of interest is called compounding.

Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly.  How will our
money grow?

The 3% interest is an annual percentage rate (APR) – the total interest to be paid during the year. 
Since interest is being paid monthly, each month, we will earn
3 % 12 = 0.25 %
per month. 

In the first month,
P = $1000
r = 0.0025 (0.25%)
I = $1000 (0.0025) = $2.50
FV = $1000 + $2.50 = $1002.50

In the first month, we will earn $2.50 in interest, raising our account balance to $1002.50. 
In the second month the balance is now higher, so we calculate the interest on that new balance

P = $1002.50
I = $1002.50 (0.0025) = $2.51 (rounded)
FV = $1002.50 + $2.51 = $1005.01

Notice that in the second month we earned more interest than we did in the first month.  This is because
we earned interest not only on the original $1000 we deposited, but we also earned interest on the $2.50 of
interest we earned the first month.  This is the key advantage that compounding of
interest gives us.

Calculating out a few more months:

Month Starting balance Interest earned Ending Balance

1

1000.00

2.50

1002.50

2

1002.50

2.51

1005.01

3

1005.01

2.51

1007.52

4

1007.52

2.52

1010.04

5

1010.04

2.53

1012.57

6

1012.57

2.53

1015.10

7

1015.10

2.54

1017.64

8

1017.64

2.54

1020.18

9

1020.18

2.55

1022.73

10

1022.73

2.56

1025.29

11

1025.29

2.56

1027.85

12

1027.85

2.57

1030.42

To find an equation to represent this, let us show the calculations at each step for the first three months to
see a pattern.

Month Starting balance Interest earned Ending Balance

1

 1000.00 1000 ( 0.025 ) 1000 + 1000 ( 0.025 ) = ( 1 + .025 )

2

 1000 ( 1 + 0.025 ) 1000 ( 1 + .025 ) ( 0.025 ) 1000 ( 1 + 0.025 ) + 1000 ( 1 + 0.025 ) ( 0.025 ) = 1000 ( 1 + .025 ) ( 1 + 0.025 ) = 1000 ( 1 + .025 ) 2

3

 1000 ( 1 + 0.025 ) 2 1000 ( 1 + 0.025 ) 2 ( 0.025 ) 1000 ( 1 + 0.025 ) 2 + 1000 ( 1 + 0.025 ) 2 ( 0.025 ) = 1000 ( 1 + 0.025 ) 2 ( 1 + 0.025 ) = 1000 ( 1 + 0.025 ) 3

Observing a pattern, we could conclude that after m months the future value (or balance) of the account would
be

F V = 1000 ( 1 + 0.025 ) m

Notice that the $1000 in the equation was P\, the starting amount.  We found 0.025 by taking the
annual rate and dividing by how often the interest was being reinvested (compouneded), so rate r
divided by 12, since we were compounding 12 times per year.

Generalizing our result, we could write

F V = P ( 1 + r n ) m

In this formula:
m is the number of compounding periods (months in our example)
r is the annual interest rate
n is the number of compounds per year.

While this formula works fine, it is more common to use a formula that involves the number of years, rather
than the number of compounding periods.  If t is the number of years, then
m = t ⋅ n
.  Making this change gives us the standard formula for compound interest.

Compound Interest (discretely)

F V = P ( 1 + r n ) t ⋅ n

FV  is the balance in the account after t years.
P\ is the starting balance of the account (also called initial deposit, or principal)
r is the annual interest rate in decimal form
n is the number of compounding periods in one year. 

  • If the compounding is done annually (once a year), n = 1. 
  • If the compounding is done quarterly, n = 4.
  • If the compounding is done monthly, n = 12.
  • If the compounding is done weekly, n = 52.
  • If the compounding is done daily, n = 365.

The most important thing to remember about using this formula is that it assumes that we put money in the
account once and let it sit there earning interest.

Example 1

A certificate of deposit (CD) is savings instrument that many banks offer.  It usually gives a higher
interest rate, but you cannot access your investment for a specified length of time.  Suppose you deposit
$3000 in a CD paying 6% interest, compounded monthly.  How much will you have in the account after 20
years?

Solution

In this example,

P = $ 3000   the initial deposit r = 0.06 6 %  annual rate n = 12 12 months in 1 year t = 20 since we ’ re looking for how much we ’ ll have after 2 0  years

So
FV = 3000 ( 1 + 0.06 12 ) 20 × 12 = $ 9930.61
 (round your answer to the nearest penny)

Try it Now 1

Peter deposits $200 that was given to him upon graduation into an account with an annual rate of 7%
compounded weekly. What is the value of that deposit after 4 years? After 8 years?

Hint 1

Since we are compounding the interest you will need to use the compouind interest formula for each
question. The compound interest formula is given by
F V = P ( 1 + r n ) t ⋅ n
. Since we are compounding weekly you will use
n = 52

Answer

The only thing we need to change is the time, t, in the compounding interest formula.

For  t = 4 P = 200 r = 0.07 n = 53 t = 4 F V = 200 ( 1 + 0.07 52 ) 4 ⋅ 52 ≈ $ 264.58 For  t = 8 P = 200 r = 0.07 n = 53 t = 4 F V = 200 ( 1 + 0.07 52 ) 8 ⋅ 52 ≈ $ 350

Let us compare the amount of money earned from compounding against the amount you would earn from an account
that earned simple interest.

Years Simple Interest ($15 per month) 6% compounded monthly = 0.5% each month.

5

$3900

$4046.55

10

$4800

$5458.19

15

$5700

$7362.28

20

$6600

$9930.61

25

$7500

$13394.91

30

$8400

$18067.73

35

$9300

$24370.65
Graph displaying output from table showing simple interest growing linearly and compound interest growing exponentitally. Both graphs are increasing, but after ten years a big difference is noted between the output of compound interest compared to the output of simple interest.

As you can see, over a long period of time, compounding makes a large difference in the account balance. You
may recognize the graphs as the difference between linear growth and exponential growth.

Example 2

You know that you will need $40,000 for your child’s education in 18 years.  If your account earns
4% compounded quarterly, how much would you need to deposit now to reach your goal?

Solution

In this example, we’re looking for P0.

r = 0.0 4 4 % n = 4 4 quarters in 1 year t = 18 Since we know the balance in 18 years FV = $ 40 , 000 The amount we have in 18 years

In this case, we’re going to have to set up the equation, and solve for P.

40000 = P ( 1 + 0.04 4 ) 18 ⋅ 4 40000 ( 1 + 0.04 4 ) 18 ⋅ 4 = P P ≈ $ 19 , 539.84

So you would need to deposit $19,539.84 now to have $40,000 in 18 years. 

Try it Now 2

How much should be deposited today if the goal is to have $2000 five years from now when it is known that the
account the money is to be deposited has an annual rate of 8.2% compounded daily?

Hint 1

The question is looking for the initial deposit,
P
. Use the compound interest formula to find
P
when we now how much money is in the account when
t = 5
.

Answer

Use the compounding interest formula to find P0.

F V = $ 2000 r = 0.082 n = 365 t = 5 F V = P ( 1 + r n ) t ⋅ n 2000 = P ( 1 + 0.082 365 ) ( 5 ⋅ 365 ) 2000 ( 1 + 0.082 365 ) ( 5 ⋅ 365 ) = P P ≈ $ 1 , 327.36

Rounding

It is important to be very careful about rounding when calculating expressions with exponents.  In
general, you want to keep as many decimals during calculations as you can if you are not able to evaluate the
entire epxression for a solution at once.  Be sure to keep at least 3 significant digits
(numbers after any leading zeros).  Rounding 0.00012345 to 0.000123 will usually give you a “close
enough” answer, but keeping more digits is always better.

Example 3

To see why not over-rounding is so important, suppose you were investing $1000 at 5% interest compounded
monthly for 30 years.

Solution

P = $ 1000     the initial deposit r = 0.05 5 % n = 12 12 months in 1 year t = 30 since we ’ re looking for the amount after 3 0  years

If we first compute
r n
, we find
0.05 12 = 0 .00416666666667

Here is the effect of rounding this to different values:

r n
rounded to:		 Gives FV to be: Error

0.004

$4208.59

$259.15

0.0042

$4521.45

$53.71

0.00417

$4473.09

$5.35

0.004167

$4468.28

$0.54

0.0041667

$4467.80

$0.06

no rounding

$4467.74

 

If you’re working in a bank, of course you wouldn’t round at all.  For our purposes, the
answer we got by rounding to 0.00417, three significant digits, is close enough – $5 off of $4500 isn’t
too bad.  Certainly keeping that fourth decimal place wouldn’t have hurt.

Exercises

  1. You deposit $300 in an account with an annual interest rate of 5% compounded annually.  How much will
    you have in the account in 10 years?
  2. How much will $1000 deposited in an account with an annual interest rate of 7% compounded annually be worth
    in 20 years?
  3. You deposit $2000 in an account with an annual interest rate of 3% compounded monthly. 
  4. How much will you have in the account in 20 years?
  5. How much interest will you earn?

You deposit $10,000 in an account with an annual interest rate of 4% compounded monthly.

  1. How much will you have in the account in 25 years?
  2. How much interest will you earn?

How much would you need to deposit in an account now in order to have $6,000 in the account in 8
years?  Assume the account has an annual interest rate of 6% compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the account in 4
years?  Assume the account has an annual interest rate of 5% interest compounded monthly.
If $800 is invested in an account for 5 years compounded monthly with an annual rate of 4%, then would
doubling the annual interest rate double the amount of interest earned? Explain.
If $500 is invested in an account for 2 years compounded monthly with an annual rate of 3%, then would
doubling the principal double the amount of interest earned? Explain.
Sam invests $1500 into a savings account in 2022. Ten years later anther $1500 is invested into the
account. If we assume the interest is compounded monthly and has the same rate of the entire time of 4.5%,
then how much is in the account 20 years after the original investment?

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