Compound Interest

Learning Outcomes

  • Calculate compound interest given an interest scenario
  • Calculate the initial balance given an interest scenario

Compounding

With simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding.

a row of gold coin stacks. From left to right, they grown from one coin, to two, to four, ending with a stack of 32 coins

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 [latex]\frac{3%}{12}[/latex]= 0.25% per month.

In the first month,

  • P0 = $1000
  • r = 0.0025 (0.25%)
  • I = $1000 (0.0025) = $2.50
  • A = $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,

  • P0 = $1002.50
  • I = $1002.50 (0.0025) = $2.51 (rounded)
  • A = $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 interest gives us.

Calculating out a few more months gives the following:

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

We want to simplify the process for calculating compounding, because creating a table like the one above is time consuming. Luckily, there is a formula for compound interest (found below) that we can use!

Compound Interest

[latex]P_{N}=P_{0}\left(1+\frac{r}{k}\right)^{Nk}[/latex]

  • PN is the balance in the account after N years.
  • P0 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 years
  • k is the number of compounding periods in one year
    • If the compounding is done annually (once a year), k = 1.
    • If the compounding is done quarterly, k = 4.
    • If the compounding is done monthly, k = 12.
    • If the compounding is done daily, k = 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. 

In the next example, we show how to use the compound interest formula to find the balance on a certificate of deposit after 20 years.

don’t forget to convert percent to a decimal

Usually, in order to perform calculations on a number expressed in percent form, you’ll need to convert it to decimal form. The rate [latex]r[/latex] in interest formulas must be converted from percent to decimal form before you use the formula.

Example

A certificate of deposit (CD) is a 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?

A video walkthrough of this example problem is available below.

Let us compare the amount of money earned from compounding against the amount you would earn from simple interest.  Suppose that we deposit $3000 into two accounts (one earning 6% simple interest annually and the other earning 6% compounded monthly).  Notice in both the table and the graph how the account earning compound interest grows much more rapidly than the one earning simple interest.

Years Simple Interest ($15 per month) 6% compounded monthly = 0.5% each month.
0 $3000 $3000
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 $24370.65
Line graph. Vertical axis: Account Balance ($), in increments of 5000 from 5000 to 25000. Horizontal axis: years, in increments of five, from 0 to 25. A blue dotted line shows a gradual increase over time, from roughly $2500 at year 0 to roughly $10000 at year 35. A pink dotted line shows a more dramatic increase, from roughly $2500 at year 0 to $25000 at year 35.

As you can see, over a long period of time, compounding makes a large difference in the account balance.

Try It

APY – Annual Percentage Yield

  • APY is the actual rate of return that will be earned in one year if the interest is compounded.
  • Compound interest is added periodically to the total invested, increasing the balance. That means each interest payment will be larger, based on the higher balance.
  • The more often interest is compounded, the higher the APY will be.
  • APY has a similar concept as annual percentage rate (APR), but APR is used for loans.

To calculate the Annual Percentage Yield (APY), here is the formula:

[latex]APY=\left(1+\frac{r}{k}\right)^{k} - 1[/latex]

This will give you APY as a decimal.  You will then need to multiply by 100 (or move the decimal point two places to the right) to convert the APY to a percentage.

Example

 

Evaluating exponents on the calculator

When we need to calculate something like [latex]5^3[/latex] it is easy enough to just multiply [latex]5\cdot{5}\cdot{5}=125[/latex].  But when we need to calculate something like [latex]1.005^{240}[/latex], it would be very tedious to calculate this by multiplying [latex]1.005[/latex] by itself [latex]240[/latex] times!  So to make things easier, we can harness the power of our scientific calculators.

Most scientific calculators have a button for exponents.  It is typically either labeled like:

^ ,   [latex]y^x[/latex] ,   or [latex]x^y[/latex] .

To evaluate [latex]1.005^{240}[/latex] we’d type [latex]1.005[/latex] ^ [latex]240[/latex], or [latex]1.005 \space{y^{x}}\space 240[/latex].  Try it out – you should get something around 3.3102044758.

Example

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?

Try It

Rounding

It is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. 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

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

P0 = $1000 the initial deposit
r = 0.05 5%
k = 12 12 months in 1 year
N = 30 since we’re looking for the amount after 30 years

If we first compute r/k, we find 0.05/12 = 0.00416666666667

Here is the effect of rounding this to different values:

 r/k rounded to: Gives P­30­ 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.

View the following for a demonstration of this example.

 

Using your calculator

In many cases, you can avoid rounding completely by how you enter things in your calculator. For example, in the example above, we needed to calculate [latex]{{P}_{30}}=1000{{\left(1+\frac{0.05}{12}\right)}^{12\times30}}[/latex]

We can quickly calculate 12×30 = 360, giving [latex]{{P}_{30}}=1000{{\left(1+\frac{0.05}{12}\right)}^{360}}[/latex].

Now we can use the calculator.

Type this Calculator shows
0.05 ÷ 12 = . 0.00416666666667
+ 1 = . 1.00416666666667
yx 360 = . 4.46774431400613
× 1000 = . 4467.74431400613

Using your calculator continued

The previous steps were assuming you have a “one operation at a time” calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter:

1000 ×  ( 1 + 0.05 ÷ 12 ) yx 360 =