Compound Interest
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.
Suppose that we deposit $1000 in a bank account offering 3% annual interest rate (APR) and that the interest is 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%) per month
- 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, math is good at giving us ways to take shortcuts. First, let’s review an important Law of Exponents.
Recall: Multiplying terms containing exponents
In the example below, you’ll need to use the rule for multiplying “like” bases containing exponents
[latex]a^{m}a^{n}=a^{m+n}[/latex].
That is, when multiplying like bases, we add the exponents.
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.
Now we are ready to find our formula to calculate the total amount of our $1000 investment after m months.
Example
Build a formula for the growth of $1000 deposited in a bank account offering 3% annual percentage rate, compounded monthly.
View this video to see this example worked out. Note: the problem statement in the video should state that 3% is the annual rate.
This formula works just fine. However, knowing that investments are usually left to grow over years rather than over a number of compounding periods, we’ll adjust the formula slightly. If N is the number of years, then [latex]m = N\cdot k[/latex]. Making this change gives us the standard formula for compound interest.
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.
- k is the number of compounding periods in one year.
Reminders:
If the compounding is done annually (once a year), k = 1. If the compounding is done monthly, k = 12.
If the compounding is done quarterly, k = 4. If the compounding is done daily, k = 365.
[latex]m=N\cdot k[/latex]
How did we get [latex]m = N\cdot k[/latex]?
Recall that [latex]m[/latex] represents the number of compounding periods that an investment remains in the account, and [latex]k[/latex] represents the number of times per year that your interest is compounded. If your deposit earns interest compounded monthly, then [latex]k = 12[/latex]. If you leave the deposit in for [latex]3[/latex] years, then there are [latex]3\cdot 12[/latex] compounding periods for your investment. That is, [latex]m = 3\cdot 12[/latex].
In general, we have that the total number of compounding periods equals the number of years times the number of times the interest is compounded per year: [latex]m = N\cdot k[/latex].
Ex. An investment of $1000 earning annual interest of 4%, compounded quarterly (4 times per year) is left in the account for [latex]3[/latex] years. Find the total number of compounding periods for the investment.
Solution: We have [latex]4[/latex] compounding periods per year, so [latex]k = 4[/latex]. Thus, if we leave our money in for [latex]3[/latex] years, [latex]m = 3\cdot 4=12[/latex].
Remember 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.
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.
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% annual interest, compounded monthly. How much will you have in the account after 20 years?
A video walk through of this example problem is available below.
In the last example, we invested $3000 in a CD earning 6% annual interest (0.5% per month) compounded monthly. Let us compare this investment opportunity with one in which the simple interest (rather than monthly compounding) is used. The data for these two investment opportunities are found in the chart and graph below.
| 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 |
As you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth.
Summary: Linear growth vs exponential growth
Recall that linear growth increases at a constant rate. A graph of linear growth will describe a straight line between any two points on the graph. The output changes by the same additive amount per unit of input.
For example, a bank account that grows at $15 per month experiences linear growth.
Exponential growth describes a quantity growing at a rate proportional to itself for each unit of input. The output changes by a multiple of its current value per unit of input. The graph will depict a quickly rising curve.
For example, an account that grows at 0.5% per per month experiences exponential growth.
Some Words About 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 5 significant digits (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 may give you a “close enough” answer for your own finances, but keeping more digits is always better. In the online assessments, if you are getting an answer “almost right,” look for a rounding error.
The video for the next example includes a great demonstration of how to avoid rounding issues when evaluating the compound interest formula using your calculator. There is also a written description of this calculator demonstration after the example.
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 |
Notice how large errors can occur when we round even to 3 or 4 decimal places. We certainly would not want to lose $259.15 because our bank rounded incorrectly!
Be sure to view the following to see this example worked and for a demonstration of how to avoid rounding issues when evaluating the compound interest formula using your calculator. There is also a written description of this calculator demonstration below.
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 |
| [latex]y^x\ 360 =[/latex] . | 4.46774431400613 |
| × 1000 = . | 4467.74431400613 |
Important Note: 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:
[latex]1000\times(1+0.05÷12)y^x360=[/latex]
Try It
Example
You determine that you would like to save $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
Solving For Time using the Compound Interest Formula
Often we are interested in how long it will take to accumulate money or how long we’d need to extend a loan to bring payments down to a reasonable level. When interest is being compounded, solving this type of problem requires algebra that we will not cover in the course. However, these interesting problems can be solved using spreadsheets which we will investigate in class activities and the project for this module.
