What you will learn
- Calculate simple interest and show why it adds the same amount each year
- Calculate compound interest and show why its yearly growth speeds up
- Explain how time and compounding frequency affect the result
- Estimate doubling time with the Rule of 72
The previous lesson showed how money can grow to a future value. The details depend on the kind of interest you earn. Simple interest adds a fixed amount, while compound interest builds on earlier interest. At first, their results look close. Give them enough time and the gap becomes large. Interest itself is the price of using money over time.
Simple video explaing some key concepts of simple and compound interest. Watch before moving on!
Simple interest adds the same amount each year
Simple interest is based only on the original amount, called the principal. It never includes interest you earned earlier. The formula is interest = principal x rate x time. Since the principal stays the same, the interest added each year stays the same too. On a graph, the balance follows a straight line.
Earn simple interest on 1,000 dollars
You lend 1,000 dollars for 3 years at 5 percent simple interest. How much total interest do you earn?
- Find one year's interest. 5 percent of 1,000 dollars is 50 dollars. That amount does not change from year to year.
- Count all three years. 50 dollars x 3 years = 150 dollars.
Why it matters: Simple interest adds the same 50 dollars every year because it is always calculated from the original 1,000 dollars.
Try a simple interest calculation
You put 2,000 dollars into an account that pays 4 percent simple interest for 5 years. How much total interest will you earn?
Compound interest builds on itself
Compound interest uses the principal plus all the interest already earned. You earn interest on your interest. The balance used for the calculation gets a little larger each year, so the dollar amount of interest also grows. On a graph, that produces an upward curve. It is the future value calculation from the last lesson: multiply the balance by one plus the rate in every period.
Key terms
- Principal
- The amount originally invested or borrowed, before interest is added.
- Simple interest
- Interest calculated only from the original principal. It adds the same amount each year.
- Compound interest
- Interest calculated from the principal and earlier interest, so the growth speeds up over time.
- Rule of 72
- A shortcut for doubling time: divide 72 by the annual percentage rate.
| When | Balance with simple interest | Balance with compound interest |
|---|---|---|
| Year 1 | $1,100 | $1,100 |
| Year 5 | $1,500 | $1,611 |
| Year 10 | $2,000 | $2,594 |
| Year 30 | $4,000 | $17,449 |
By year 30, simple interest has turned 1,000 dollars into 4,000 dollars, or four times the starting amount. Compound interest at the same 10 percent rate has produced 17,449 dollars, more than seventeen times the starting amount. The compound balance kept earning a return on previous interest, and that difference grew with time.
Simple interest adds. Compound interest multiplies, so the difference grows over time.
Time matters most
- Time has the largest effect. Money that starts compounding earlier gets more chances to grow.
- Compounding frequency also helps. Monthly compounding produces a little more than yearly compounding, although each extra increase in frequency adds less benefit.
- The math works on debt too. A credit card balance can compound just as quickly as savings, only now the growth works against you.
The Rule of 72
The Rule of 72 gives you a quick estimate of doubling time. Divide 72 by the annual percentage rate. At 8 percent per year, the estimate is 72 divided by 8, or about 9 years. It is an approximation, but it is useful when you want a fast answer without a calculator.
The Rule of 72 for compound interest (Khan Academy)
The video explains where the Rule of 72 comes from and how to use it to estimate doubling time in your head.
Estimate a doubling time
An investment compounds at about 9 percent per year. According to the Rule of 72, about how many years will the money take to double?
Why can an early start matter so much?
Alex and Blake invest the same amount at the same compound rate. Alex invests from age 25 through 35. Blake starts at 35 and continues through 65. Alex often finishes with more money despite investing for fewer years. Why?
Money invested earlier has more years to earn returns on earlier returns. That extra time can outweigh investing more money later.When compounding works against you
In a couple of sentences, explain how compound interest can make an unpaid credit card balance grow.
Write an answer before comparing it with the model response.
Model answer
Unpaid interest gets added to the credit card balance. The next interest charge is then based on that larger balance, so the debt grows along a curve instead of a straight line. At a high rate, an unpaid balance can grow quickly.
Compounding can grow the number of dollars you have. Inflation pushes the other way by reducing what those dollars can buy, which is the topic of the next lesson.