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Future Values and Compound Interest

You have $100 invested in a bank account. Suppose banks are currently paying an interest rate of 6 percent per year on deposits. So after a year, your account will earn interest of $6:

Interest = interest rate initial investment = .06 $100 = $6

You start the year with $100 and you earn interest of $6, so the value of your investment will grow to $106 by the end of the year:

Value of investment after 1 year = $100 + $6 = $106

Notice that the $100 invested grows by the factor (1 + .06) = 1.06. In general, for any interest rate r, the value of the investment at the end of 1 year is (1 + r) times the initialinvestment:

Value after 1 year = initial investment (1 + r) = $100 (1.06) = $106

What if you leave this money in the bank for a second year? Your balance, now $106, will continue to earn interest of 6 percent. So

Interest in Year 2 = .06 $106 = $6.36

You start the second year with $106 on which you earn interest of $6.36. So by the end of the year the value of your account will grow to $106 + $6.36 = $112.36. In the first year your investment of $100 increases by a factor of 1.06 to $106; in the second year the $106 again increases by a factor of 1.06 to $112.36. Thus the initial $100 investment grows twice by a factor 1.06:

Value of account after 2 years = $100 1.06 1.06 = $100 (1.06)2 = $112.36

If you keep your money invested for a third year, your investment multiplies by 1.06 each year for 3 years. By the end of the third year it will total $100 (1.06)3 = $119.10, scarcely enough to put you in the millionaire class, but even millionaires have to start somewhere.

Clearly for an investment horizon of t years, the original $100 investment will grow to $100 (1.06)t. For an interest rate of r and a horizon of t years, the future value of your investment will be

Future value of $100 = $100 _ (1 + r)t

Notice in our example that your interest income in the first year is $6 (6 percent of $100), and in the second year it is $6.36 (6 percent of $106). Your income in the second year is higher because you now earn interest on both the original $100 investment and the $6 of interest earned in the previous year. Earning interest on interest is called compounding or compound interest. In contrast, if the bank calculated the interest only on

your original investment, you would be paid simple interest.

Table 1.5 and Figure 1.3 illustrate the mechanics of compound interest. Table 1.5 shows that in each year, you start with a greater balance in your account your savings have been increased by the previous year`s interest. As a result, your interest income also is higher.

Obviously, the higher the rate of interest, the faster your savings will grow. Figure 1.4 shows that a few percentage points added to the (compound) interest rate can dramatically affect the future balance of your savings account. For example, after 10 years $1,000 invested at 10 percent will grow to $1,000 (1.10)10 = $2,594. If invested at 5 percent, it will grow to only $1,000 (1.05)10 = $1,629.

Calculating future values is easy using almost any calculator. If you have the patience, you can multiply your initial investment by 1 + r (1.06 in our example) once for each year of your investment. A simpler procedure is to use the power key (the yx key) on your calculator. For example, to compute (1.06)10, enter 1.06, press the yx key, enter 10, press = and discover that the answer is 1.791. (Try this!)

If you don`t have a calculator, you can use a table of future values such as Table 1.6. Check that you can use it to work out the future value of a 10-year investment at 6 percent.First find the row corresponding to 10 years. Now work along that row until you reach the column for a 6 percent interest rate. The entry shows that $1 invested for 10 years at 6 percent grows to $1.791.

Now try one more example. If you invest $1 for 20 years at 10 percent and do not withdraw any money, what will you have at the end? Your answer should be $6.727. Table 1.6 gives futures values for only a small selection of years and interest rates. Table A.1 at the end of the material is a bigger version of Table 1.6. It presents the future value of a $1 investment for a wide range of time periods and interest rates. Future value tables are tedious, and as Table 1.6 demonstrates, they show future values only for a limited set of interest rates and time periods. For example, suppose that you want to calculate future values using an interest rate of 7.835 percent. The power key on your calculator will be faster and easier than future value tables. A third alternative is to use a financial calculator. These are discussed in two boxes later.

FUTURE VALUE

Amount to which an investment will grow after earning interest.

COMPOUND INTEREST

Interest earned on interest.

SIMPLE INTEREST

Interest earned only on the original investment; no interest is earned on interest.