## Compounding and the Time Value of Money

The concepts of compound growth and the time value of money are widely used in all aspects of business and economics. Compounding is the principle that underlies growth, whether it is growth in value, growth in sales, or growth in assets. The time value of money— the fact that a dollar received in the future is worth less than a dollar in hand today—also plays an important role in managerial economics. Cash flows occurring in different periods must be adjusted to their value at a common point in time to be analyzed and compared. Because of the importance of these concepts in economic analysis, thorough understanding of the material on future (compound) and present values in the appendix is important for the study of managerial economics.

FUTURE VALUE (OR COMPOUND VALUE)

Suppose that you deposit \$100 in a bank savings account that pays 5% interest compounded annually. How much will you have at the end of one year? Let us define terms as follows:

PV = Present value of your account, or the beginning amount, \$100

i = Interest rate the bank pays you = 5% per year, or, expressed in decimal terms, 0.05

I = Dollars of interest earned during the year

FVn = Future value, or ending amount, of your account at the end of n years. Whereas PV is the value now, at the present time, FVn is the value n years into the future, after compound interest has been earned. Note also that FV0 is the future value zero years into the future, which is the present, so FV0 = PV.

In our example, n = 1, so FVn = FV1, and it is calculated as follows:

We can now use Equation A.1 to find how much the account is worth at the end of 1 year:

Your account earned \$5 of interest (I = \$5), so you have \$105 at the end of the year.

Now suppose that you leave your funds on deposit for 5 years; how much will you have at the end of the fifth year? The answer is \$127.63; this value is worked out in Table A.1.

Notice that the Table A.1 value for FV2, the value of the account at the end of year 2, is equal to

FV3, the balance after 3 years, is

In general, FVn, the future value at the end of n years, is found as

Applying Equation A.2 in the case of a 5-year account that earns 5% per year gives

which is the same as the value in Table A.1.

If an electronic calculator is handy, it is easy enough to calculate (1 + i)n directly.1 However, tables have been constructed for values of (1 + i)n for wide ranges of i and n, as Table A.2 illustrates. Table B.1 in Appendix B contains a more complete set of compound value interest factors. Interest compounding can occur over periods of time different from 1 year. Thus, although compounding is often on an annual basis, it can be quarterly, semian-nually, monthly, or for any other period. 