In practice, interest is sometimes compounded continuously. Therefore it is useful to consider how this works. Recall that the general formula for solving for the future value at the end of year n, Eq. (3.17), is

*FV _{n}* =

*PV*

_{0}(1 + [

*i|m*])

^{mn}As zn, the number of times a year that interest is compounded, approaches infinity (-), we get continuous compounding, and the term (1 + liln) )-' approaches e"', where e is approximately 2.71828. Therefore the future value at the end of n years of an initial deposit of PVo where interest is compounded continuously at a rate of I percent is

*FV _{n}* =

*PV*

_{0}(e)

*(3.19)*

^{in}For our earlier example probiem, the future value of a $ 100 deposit at the end of three years with continuous compounding at 8 percent would be

*FV*_{3}= $100(e)^{(0.0813)} = $100(2.718281)^{(0 24)} = $127.12

This compares with a future value with annual compounding of

*FV*_{3} = $100(1 + 0.08)^{3} = $125.97

Continuous compounding results in the maximum possible future value at the end of n periods for a given nominal rate of interest.

By the same token, when interest is compounded continuously, the formula for the present value ofa cash flow received at the end ofyear n is

*FV*_{0}= *FV _{n}* / (

*e*)

*(3.20)*

^{in}Thus the present value of $1,000 to be received at the end of 10 years with a discount rate of 20 percent, compounded continuously, is

*FV*_{0} = $1,000/(*e*)^{(0.0813)} = $1,000/(2.71828)^{2} = $135.34

We see then that present value calculations involving continuous compounding are merely the reciprocals of future value calculations. AIso, although continuous compounding results in the maximum possible future value, it results in the minimum possible present value.

Based on a congressional act, the Federal Reserve requires that banks and thrifts adopt a standardized method of calculating the effective interest rates they pay on consumer accounts. It is called lhe annual percentage yield (APY). The APY is meant to eliminate -confusion caused when savings institutions apply different methods of compounding and use various terms, such as effective yield, annual yield, and effective rate. The APY is similar lo the ffictive annual interest rate. The APY calculation, however, is based on the actual number of days for which the money is deposited in an account in a 365-day year (366 days in a leap year).

In a similar vein, the Truth-in-Lending Act mandates that all financial institutions report the effective interest rate on any loan. This rate is called the annual percentage rate (APR). However, the financial institutions are not required to report the "true" effective annual interest rate as the APR. Instead, they may report a noncompounded version of the effective annual interest rate. For example, assume that a bank makes a loan for less than a year, or interest is to be compounded more frequently than annually. The bank would determin e an effective periodic interest rate -based on usable funds (i.e., the amount of funds the borrower can actually use) * and then simply multiply this rate by the number of such periods in a year. The result is the APR.