Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use **compound interest**. The term *compounding* refers to interest earned not only on the original value, but on the accumulated value of the account.

The **annual percentage rate (APR)** of an account, also called the **nominal rate**, is the yearly interest rate earned by an investment account. The term *nominal* is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being *greater* than the nominal rate! This is a powerful tool for investing.

We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time *t*, principal *P*, APR *r*, and number of compounding periods in a year *n*:

For example, observe the table below, which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.

Frequency | Value after 1 year |
---|---|

Annually | $1100 |

Semiannually | $1102.50 |

Quarterly | $1103.81 |

Monthly | $1104.71 |

Daily | $1105.16 |

### A General Note: The Compound Interest Formula

**Compound interest** can be calculated using the formula

where

*A*(*t*) is the account value,*t*is measured in years,*P*is the starting amount of the account, often called the principal, or more generally present value,*r*is the annual percentage rate (APR) expressed as a decimal, and*n*is the number of compounding periods in one year.

### Example 7: Calculating Compound Interest

If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?

### Solution

Because we are starting with $3,000, *P *= 3000. Our interest rate is 3%, so *r* = 0.03. Because we are compounding quarterly, we are compounding 4 times per year, so *n *= 4. We want to know the value of the account in 10 years, so we are looking for *A*(10), the value when *t *= 10.

The account will be worth about $4,045.05 in 10 years.

### Try It 7

An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?

### Example 8: Using the Compound Interest Formula to Solve for the Principal

A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?

### Solution

The nominal interest rate is 6%, so *r *= 0.06. Interest is compounded twice a year, so *k *= 2.

We want to find the initial investment, *P*, needed so that the value of the account will be worth $40,000 in 18 years. Substitute the given values into the compound interest formula, and solve for *P*.

Lily will need to invest $13,801 to have $40,000 in 18 years.

### Try It 8

Refer to Example 8. To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?