to use financial calculators or Excel to ensure maximum accuracy.
** Hereafter in this book, the terms payment and receipt will be used interchangeably. A payment by one party in a transaction becomes a receipt to the other and vice versa,
How Do You Calculate Future Values? - How Money Grows
Simple Interest
Simple interest is the interest calculated on the amount of the principal only. It is the return on (or growth of) the principal for one time period. The following equation expresses simple interest.
Interest = p x i x n
Where p = principal
i = rate of interest for a single period n = number of periods
Example 1
Barstow Electric Inc. borrows $10,000 for 3 years with a simple interest rate of 8% per year. It computes the total interest it will pay as follows.
Interest = p x i x n = $10,000 x .08 x 3 = $2,400
Compound Interest
Compounding interest means that interest earns interest. The future value of a dollar is its value at a time in the future given its present sum. The future value of a dollar is affected both by the interest rate and the time at which the dollar is received. For the discussion of the concepts of compounding and time value, let us define:
Then,
The future value of an investment compounded annually at rate i for n years is
where FVF(i,n)=T1(i,n) is the future value (compound amount) of $1 and can be found in Table 1.
Example 2
To illustrate the difference between simple and compound interest, assume that Nolan Company deposits $1,000 in the First Bank, where it will earn simple interest of 8% per year. It deposits another $1,000 in the Second Bank, where it will earn compound interest of 8% per year compounded annually. In both cases, Nolan will not withdraw any interest until 3 years from the date of deposit.
Simple interest:
$1,000 × .08 × 3 years = $240; the future value = $1,240
Compound interest:
Note: Simple interest uses the initial principal of $1,000 to compute the interest in all 3 years. Compound interest uses the accumulated balance (principal plus interest to date) at each year-end to compute interest in the succeeding year. This explains the larger balance in the compound interest account. Obviously, any rational investor would choose compound interest, if available, over simple interest. In the example above, compounding provides $20 of additional interest revenue. Simple interest usually applies only to short-term investments and debts that involve a time span of one year or less.
Example 3
You place $1,000 in a savings account earning 8 percent interest compounded annually. How much money will you have in the account at the end of 4 years?
From Table 1, the T1 for 4 years at 8 percent is 1.360. Therefore,
An excerpt from Table 1 is given over the page.
Table 1: The Future Value of $1.00 (Compound Amount of $1.00) (1 + i)n = FVF (i,n) = T1 (i,n)
Example 4
You invested a large sum of money in the stock of TLC Corporation. The company paid a $3 dividend per share. The dividend is expected to increase by 14 percent per year for the next 3 years. You wish to project the dividends for years 1 through 3.
Intrayear Compounding
Interest is often compounded more frequently than once a year. Banks, for example, compound interest quarterly, daily, or even continuously. If interest is compounded m times a year, then the general formula for solving the future value becomes
The formula reflects more frequent compounding (n x m) at a smaller interest rate per period (i/m). For example, in the case of semiannual compounding (m = 2), the above formula becomes
Example 5
You deposit $10,000 in an account offering an annual interest rate of 16 percent. You will keep the money on deposit for five years. The interest rate is compounded quarterly. The accumulated amount at the end of the fifth year is calculated as follows:
Where P = $10,000
Therefore,
Example 6
As the example shows, the more frequently interest is compounded, the greater the amount accumulated. This is true for any interest for any period of time. How often interest is compounded can substantially affect the rate of return. For example, an 8% annual interest compounded daily provides an 8.33% yield, or a difference of 0.33%. The 8.33% is the effective yield, frequently called annual percentage rate (APR). The annual interest rate (8%) is the nominal, stated, coupon, or face rate. When the compounding frequency is greater than once a year, the effective interest rate will always exceed the nominal rate.
The formula for calculating the effective interest rate or annual percentage rate (APR), in situations where the compounding frequency (n) is greater than once a year, is as follows.
APR
