# Compound Interest

Interest is an amount charged for the use of money. There are different methods for calculating this charge. A common method is "compound interest"; this method adds the interest from prior periods to the amount invested or borrowed (the "principal") before the current period interest is calculated. In other words, prior periods' interest is considered to be reinvested, and current period interest is calculated on the principal plus prior periods' interest.

## Using Tables to Learn Compound Interest

Although compound interest solutions can be quickly and precisely obtained by using hand-held calculators, software applications, and Internet websites, compound interest tables are also still widely used.

Tables are especially helpful for learning how to solve various compound interest problems such as determining future values, present values, payment amounts, interest rates, and loan balances. There are several reasons for this:

- Table values can be interpreted as multiples or ratios (to present and future single amounts, and, payments) which enhances understanding of the results of calculations.
- The effects of changes in an interest rate or number of periods are easily viewed by looking at the changes in the multiples - sometimes these are quite dramatic.
- By looking across rows and down columns, the direction of change in the multiples provides an intuitive understanding of the effects of compounding and discounting.

## Compound Interest Tables

- Single Sum of $1 Future Value Table: How much $1 today will be worth compounded at
**i**% interest per period for**n**periods. - Single Sum of $1 Present Value Table: How much $1 in the future is worth today, discounted at
**i**% interest per period for**n**periods. - Ordinary Annuity of $1 Future Value Table: How much
**n**payments of $1 each received at the end of each period, compounded at**i**% interest between each payment, will be worth when the last payment is made. (Note: "annuity" means a series of equal payments at equal intervals compounded at the same interest rate between payments.) - Ordinary Annuity of $1 Present Value Table: How much
**n**payments of $1 each received at the end of each period, discounted at**i**% interest between each payment, is worth today.

## Other Common Methods

**Simple Interest**: Simple interest calculates interest only on the principal. Prior periods' interest is not assumed to be reinvested; therefore, current interest is not earned on prior interest. The formula for simple interest is: P x i x n, where P is the principal amount, i is the interest rate and n is the number of periods. For example, if $10,000 were invested for 9 months at 10% annual interest, the total interest would be $10,000 x .10 x 9/12 = $750. (The interest rate is usually annual, so time is expressed in years or parts of a year.)

**Discount Method**: Interest is calculated on a principal amount and subtracted from the principal at the time of a loan. The borrower receives the difference. For example, suppose that you borrowed the same $10,000 as in the above example. The $750 would be subtracted from the principal and you would have received $9,250 and be required to repay $10,000. Notice that the true interest rate is higher than the stated 10% because you have the use of only $9,250, not $10,000. $750 / $9,250 x 12/9 = 10.81% annual rate.

**Rule of 78**: The rule of 78 is an older method used for relatively short-term consumer loans. A pre-computed finance charge is calculated and added to the principal, and then paid off in equal monthly installments. For example assume that you buy an appliance for $3,000 and a $300 finance charge is calculated. The loan term is for one year, so the equal payments would be $3,300 / 12 = $275.

The amount of the finance charge earned by the lender is calculated by using a declining fraction. The numerator of the fraction is the most recent month and the denominator of the fraction is the sum of the months in the loan term. The denominator can be determined by the formula (n + n^{2})/2 where n is the number of months. In the twelve-month example above, (12 + 12^{2})/2 = 78 (That’s the source of the name). The lender earns $300 x 12/78 in the first month = $46.15. In the second month, the lender earns $300 x 11/78 = $42.31, and so on, until the last month, when the lender earns 1/78 of the finance charge. Making early or additional payments does not save interest. Only a full prepayment will save interest (sometimes called a ‘rebate’). Even with a full loan payoff, the lender earns a disproportionate amount of interest in the early months, to the disadvantage of the borrower.