Introduction

There are two kinds of interest: simple and compound. Both are expressed as percentages. We will illustrate the difference with an interest-earning deposit.

A person makes a deposit with a financial institution, which promises a certain rate of interest per year, paid after specified intervals of time. With simple interest, the amount of the deposit remains the same, and the amount of interest is paid to the depositor at the end of each interval of time. With compound interest, the amount of the deposit rises because the interest is added to the deposit at the end of each interval of time.

For example, suppose the deposit is $1000, the rate of interest is 6 percent per year, and the payment intervals are quarterly.

• If this is simple interest, the financial institution will pay the depositor $15 at the end of each quarter, for a total of $60 interest earned for the year (6 percent of $1000). The total assets of the depositor after one year will be $1060.
  
  

• If this is compound interest, the payment will still be $15 at the end of the first quarter, but the interest will be added to the deposit, making the deposit now $1015.
  
  At the end of the second quarter, the interest will be calculated using this larger amount and will come to $15.225, which will be added to the deposit, making the new total $1030.225.
  
  The interest paid at the end of the third quarter will be calculated using the second-quarter total, and will come to $15.453375, which will again be added to the deposit, for a total of $1045.678375.
  
  At the end of the last quarter, the interest will be calculated based on the third-quarter amount, and will come to $15.685175625. Thus the total assets of the depositor at the end of the year will be $1061.363550625.
  
  Of course, financial institutions do not keep track of these fractions of cents, and interest payments are rounded to the nearest cent. This means that the actual amounts paid are not the numbers shown above. Instead, the second interest payment will be $15.23, the third will be $15.45, the fourth will be $15.69, and the total assets will be $1061.37.

Notice that with compound interest, the depositor's assets at the end of the year are $1.37 more than with simple interest. This is because during the last three quarters of the year, the depositor is earning interest on the interest previously earned.

Note: All the formulas below assume that interest earned is computed exactly, and not rounded at all. The effect of rounding is usually an extremely small amount over the course of many payments.

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Simple Interest

Let the annual rate of interest be i (as a fraction, that is 100i percent), the amount of the principal be P, the number of years be n, and the amount after n years be A. Then

A = P(1+ni).

If you want to know what principal to deposit in order to have an amount A after n years at interest rate i, that principal is called the present value, and is given by

P = A/(1+ni).

To find the interest rate i, use

To find the interest rate i, use

i = ([A/P]-1)/n.

To determine how many compounding periods are needed to reach a given amount,

n = ([A/P]-1)/i.

Example: Suppose you deposit $6000 in a bank and receive 4% per year simple interest for 7 years. Then the parameters will be principal P = $6000, interest rate per period i = 0.04, and number of periods n = 7. The amount of interest you will have received by the end of 7 years will be Pni = ($6000)(7)(0.04) = $1680, so you will have A = $7680.

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Compound Interest

Let the annual rate of interest be i (as a fraction, that is 100i percent), the amount of the principal be P, the number of years be n, the number of times per year that the interest is compounded be q, and the amount after n years be A. Then

A = P(1+[i/q])nq.

Then the present value is given by

P = A(1+[i/q])-nq.

To find the interest rate i, use

i = q([A/P]1/nq - 1).

To determine how many years are needed to reach a given amount,

n = log(A/P)/(q log[1+(i/q)]).

Interest may be compounded quarterly, monthly, weekly, daily, or even more frequently. As the frequency of compounding increases, the amount A increases, but ever more slowly -- in fact it approaches a limit with continuous compounding. The formulas for this situation are found by taking the limit of the formulas above as q increases without bound. They take the form

A = Pein,
P = Ae-in,
i = log(A/P)/(n log[e]),
n = log(A/P)/(i log[e]).

Example: Suppose you deposit $6000 in a bank and receive 4% per year compound interest for 7 years, compounded monthly. The parameters will be principal P = $6000, annual interest rate i = 0.04, number of years n = 7, and number of periods per year q = 12. Over these 7 years the principal will grow over to the amount A, where

A = P(1+[i/q])nq,
  = ($6000)(1+[0.04/12])(7)(12),
  = ($6000)(1.003333333...)84,
  = $7935.08.

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Annual Percentage Rate (APR)

The annual percentage rate (APR) is the rate r of simple interest for 1 year, which will be equal to the actual amount of compound interest for that year at rate i compounded q times per year. It is given by the formula

r = (1+[i/q])q-1.

To find the interest rate i given the APR r, use

i = q[(1+r)1/q-1].

The APR is mainly used to compare loans with different interest rates and payment intervals. The lower the APR, the lower the cost of the loan to the borrower.

Example: Suppose your credit card charges 18% interest per year, but you have to pay the interest due monthly. What is the annual percentage rate? Here the parameters are rate i = 0.18 and the number of compounding periods q = 12. Then the annual percentage rate (APR) r is given by

r = (1+[i/q])q-1,
  = (1+[0.18/12])12-1,
  = (1.015)12-1,
  = 0.195618...,

or an APR of 19.5618%

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