How Does Compound Interest Work?

1.With simple interest, what is each year's interest figured on?

Simple interest: each year's interest = starting amount × rate. With 1,000 at 5 percent, that is 50 a year.

2.1,000 earns 5 percent compound interest a year. After year one, the balance is 1,050. In year two, is the interest still 50?

More than 50: year two pays 52.50, because 5 percent is now taken from 1,050.

3.1,000 earns 5 percent a year, compounded yearly. What is the balance after 2 years?

Year one adds 50 (balance 1,050). Year two adds 5 percent of 1,050, which is 52.50, so the balance is 1,102.50.

4.1,000 is left for 20 years at 5 percent a year, compounded yearly. Which value is the end balance closest to?

About 2,650: 1,000 × 1.05²⁰ = 2,653.30. The interest itself keeps growing.

5.1,000 grows at 5 percent a year for 30 years. How much more does compound interest end with than simple interest?

Simple interest ends at 2,500, compound interest at 4,321.94. The gap of about 1,822 is interest earned on interest.

6.Same example: 1,000 at 5 percent, compounded yearly. In the first ten years the balance grows by about 629. By about how much does it grow in the last ten years (years 21 to 30)?

Same rate, but the last decade grows on a much bigger balance. The early years are the slow part.

7.Which statements about the formula start × (1 + rate)^years are true? Select all that apply.

Several answers can be correct.

Final amount = start × (1 + rate)^years. It needs only the start amount, the rate and the number of years. Real accounts also have fees, taxes and inflation.

8.Over 30 years, 1,000 grows to about 4,322 at 5 percent a year. About how much does it grow to at 8 percent a year?

Rate and time steepen the curve: going from 5 to 8 percent more than doubles the 30-year result (about 10,063 instead of 4,322).

9.Using the rule of 72, about how many years does money need to double at 6 percent a year?

Rule of 72: years to double ≈ 72 ÷ rate in percent. At 6 percent: 72 ÷ 6 = 12 years.

10.True or false: The rule of 72 gives the exact doubling time.

The exact doubling time is ln 2 ÷ ln(1 + rate): 11.90 years at 6 percent. One early written mention of the rule is in Luca Pacioli's math book, printed in Venice in 1494.

11.Start with 1 at an imaginary 100 percent interest a year. What happens as the interest is added more and more often (monthly, daily, and so on)?

Jacob Bernoulli studied this question in the 1680s. The values rise (2, then 2.61, then 2.71 for daily adding) and approach, but never pass, about 2.718, the constant e.

12.True or false: There is solid evidence that Einstein called compound interest 'the eighth wonder of the world'.

The quote is popular on posters, but its link to Einstein is unsupported.

13.Two people each put 1,000 away at 5 percent a year. One starts today, the other ten years later. Thirty years from today, the gap is about 1,669. Which years does the late starter miss?

Starting later does not cut the slow start; it cuts the fast end: 1,000 × 1.05^30 = 4,321.94 against 1,000 × 1.05^20 = 2,653.30.

14.Transfer: 2,000 earns 4 percent a year, compounded yearly. What is the balance after 2 years?

Year one adds 80 (balance 2,080). Year two adds 4 percent of 2,080, which is 83.20, so the balance is 2,163.20.

15.Transfer: with the rule of 72, about how long does money need to double at 9 percent a year?

Years to double ≈ 72 ÷ rate in percent: 72 ÷ 9 = 8.

16.A debt grows at an example rate of 20 percent a year and nothing is paid. About how long until it doubles?

The math is neutral: compound interest makes an unpaid debt grow just like savings. 20 percent is an example rate, not a market rate.

17.Which grows faster over a year: 5 percent added once a year, or 4.9 percent added every month?

How often interest is added can decide when two rates look close. Comparing such rates is covered in a later lesson.