## SemiannuaI and Other Compounding Periods

Future (or Compound) Vatue. Up to now, we have assumed that interest is paid annually. It is easiest to get a basic understanding of the time value of money with this assumption. Now, however, it is time to consider the relationship between future value and interest rates for different compounding periods. To begin, suppose that interest is paid semiannually. If you then deposit \$100 in a savings account at a nominal, or stated,8 percent annual interest rate, the future value at the end of six months would be

FV0.5 = \$100(1 + [0.08/2])= \$104

In other words, at the end ofone half-year you would receive 4 percent in interest, not 8 percent. At the end of a year the future value of the deposit would be

FV1 = \$100(1 + [0.08/2])2 = \$108.16

This amount compares with \$108 if interest is paid only once a year. The \$0.16 difference is caused by interest being earned in the second six months on the \$4 in interest paid at the end of the first six months. The more times during the year that interest is paid, the greater the future value at the end ofa given year.

The general formula for solving for the future value at the end of n years where interest is paid m times a year is

FVn= PV0(1 + [i/m])mn (3.18)

To illustrate, suppose that now interest is paid quarterly and that you wish to know the future value of\$100 at the end ofone yearwhere the stated annual rate is 8 percent. The future value would be

FV1 = \$100(1 + [0.08/4])(4)(1) = \$100(1 + 0.02)4 = \$108.24

which, of course, is higher than it would be with either semiannual or annual compounding. The future value at the end of three years for the example with quarterly compounding is

FV3 = \$100(t + [0.08/4])(4)(3) = \$100(1 + 0.02]12 = \$12682

compared with a future value with semiannual compounding of

FV3 = \$100(1 + [0.08/2])(2)(3) = \$100(1 + 0.04)6 = \$126.53

and with annual compounding of

FV3 = \$100(1 + [0.08/1])(1)(3) = \$100(1 + 0.08)3 = \$125.97

Thus, the more frequently interest is paid each year, the greater the future value. When m in Eq. (3.17) approaches infinity, we achieve continuous compounding. Shortly, we will take a special look at continuous compounding and discounting.

Present lor Discounted) Vatue. When interest is compounded more than once a year, the formula for calcuiating present value must be revised along the same lines as for the calculation of future value. Instead of dividing the future cash flow by ( 1 + i)" as we do when annual compounding is involved, we determine the present value by

PVo= FVn/(1+[i/m])mn (3.18)

where, as before, Fli is the future cash flow to be received at the end of year n, m is the number of times a year interest is compounded, and I is the discount rate. We can use Eq. (3.18), for example, to calculate the present value of \$100 to be received at the end of year 3 for a nominal discount rate of 8 percent compounded quarterly:

PV0 = \$100/(1 + [0.08/4])(4)(3) = \$100/(1 + 0.Q2)12 = \$78.85

If the discount rate is compounded only annually, we have

P% = \$100/(1 + 0.08)3 = \$79.38

Thus, the fewer times a year that the nominal discount rate is compounded, the greater the present value. This relationship is just the opposite of that for future values.