Future value and compounding

Time value of money
How much will my money grow to?

Learning goals

After this page, you can:

  • Explain why a dollar today is worth more than a dollar next year.
  • Calculate the future value of one amount with the formula and with Excel.
  • Compare compound interest with simple interest, and say why the gap grows over time.

The idea

Your friend asks to borrow $100. They say, “I will pay you back $100 next year.” Is that a good deal for you? Not really. If you keep the $100, you can put it in a bank and earn interest. Next year you will have more than $100.

So money has a time value. A dollar today is worth more than a dollar later, because a dollar today can earn interest.

The money you have today is the present value (PV). The money it grows to is the future value (FV). The interest rate (r) is the percent you earn each year. The number of periods (n) is how many years the money grows.

After one year, $100 at 5% becomes $100 × 1.05 = $105. In the second year, you earn 5% on $105, not on $100. So you get $105 × 1.05 = $110.25. The extra 25 cents is interest on interest. This is called compounding.

With simple interest, you earn interest only on the first $100: $5 every year. With compound interest, the base gets bigger every year. Over a few years the gap is small. Over 30 or 40 years the gap is very large.

Key formula

\[ FV = PV \times (1 + r)^n \] PV = money today · r = interest rate per period (as a decimal) · n = number of periods

Try it

Move the sliders. Try these:

  1. Set the rate to 6% and the years to 12. The money is about double. (Rule of 72: 72 ÷ 6 = 12.)
  2. Keep the rate. Move the years from 20 to 40. Does the money double, or more than double?
  3. Look at the gap between the two lines. When does it get big?

Worked example

Problem. Maria saves $2,500 from her summer job. She puts it in an account that pays 4% per year, compounded once a year. How much will she have in 8 years?

Step 1. Write what you know. PV = $2,500, r = 4% = 0.04, n = 8.

Step 2. Put the numbers in the formula. FV = 2,500 × (1.04)8.

Step 3. Calculate. (1.04)8 = 1.36857. So FV = 2,500 × 1.36857 = $3,421.42.

Check with Excel. Type =FV(0.04, 8, 0, -2500). Excel shows 3,421.42. The PV is negative in Excel because the money leaves your pocket today.

Answer. Maria will have $3,421.42. Of that, $921.42 is interest.

Check yourself

Check 1.

You put $100 in a bank at 10% per year for 2 years, compounded yearly. How much do you have at the end?

Year 1: $100 × 1.10 = $110. Year 2: $110 × 1.10 = $121. The extra $1 (compared with $120) is interest on interest.

Check 2.

Which change makes the future value bigger? (Everything else stays the same.)

More years means more compounding periods, so the money grows more. A lower rate, fewer years, or a smaller start all make FV smaller.

Check 3.

In Excel, which formula gives the future value of $5,000 at 3% for 10 years?

The rate goes in as a decimal (0.03), then the number of periods (10), then the payment (0 here), then the present value as a negative number.

Video

Video coming soon.