We described an interest rate (*r*) is the rate of return that equates the value of different cash flows on different dates.

If you could pay $975 today to receive $1,000 in 1 year, you can consider the value of waiting 1 year to be $25 ($1,000 – $975.) It can be expressed as an interest rate, *r *= 25/975 = 0.02564 or 2.56%.

In many cases, interest is paid more frequently than annually. In such cases the future value formula can be expressed as:

FV_{N} = PV(1+*r*_{s}/m)^^{mN}

where m is the number of compounding periods and N is the number of years.

Interest can also be compounded continuously, in which case FV_{N} = PV_{e}^{rsN}where e is the transcendental number e = 2.718… Most financial calculators have a function for e^{x}.

The frequency of compounding can have a considerable effect on the ending value for an investor. Consider a $1,000 investment being made for 5 years at a stated annual interest rate of 6%. If the interest is paid annually, the investor would receive $60 at the end of each of the first four years, and 1,060 (the original investment plus the final interest payment) at the end of the fifth year. The total amount received would be 1,300.

If instead the interim interest payments could be reinvested at the original rate, the outcome would be different. The $60 payment at the end of year 1 would increase the total size of the investment to $1,060. This entire amount would earn 6% in the second year, for an interest payment of $63.60. The value at the end of five years would be 1,000(1.06)^{5} = $1,338.23.

With semiannual compounding, we would get 1,000(1.03)^{10} = $1,343.92.

Finally, with continuous compounding the result would be 1,000e^{.04} = $1,349.86.