3: The Time Value of MoneyNot surprisingly, the maximum effective annual rate for a given stated annual rate occurs when
interest compounds continuously. The effective annual rate for this extreme case can be found by using
the following equation:
Eq. 3.14a EAR (continuous compounding) = er – 1
For the 8% stated annual rate (r = 0.08), substitution into Equation 3.14a results in an effective
annual rate of 8.33%, as follows:
e0.08 – 1 = 1.0833 – 1 = 0.0833 = 8.33%
In Australia, credit card suppliers are required to advise holders of their cards of the average annual
percentage rate (AAPR). This also applies to those with loans from lenders covered by the ‘truth-in-lending’
requirements of the National Credit Code. The AAPR is based on the stated nominal annual rate charged
on the credit card or loan, and is found by multiplying the periodic rate by the number of periods in one
year; but it is also required to include fees such as upfront, ongoing and exit fees.
The AAPR, however, understates the actual cost of a credit card account. The actual cost is
determined by calculating the annual percentage yield (APY). The APY is the same as the effective annual rate
(sometimes called the effective APR), which (as discussed earlier) reflects the impact of compounding
frequency. For a credit card charging 1.5% per month interest, the effective annual rate is [(1.015)^11 – 1]
= 0.1956, or 19.56%. This means that paying interest at 1.5% per month is the same as paying 19.56%
if interest were charged annually.
3 -7c CALCULATING DEPOSITS NEEDED TO ACCUMULATE A
FUTURE SUM
Suppose that someone wishes to determine the annual deposit necessary to accumulate a certain amount of
money at some point in the future. Assume that you want to buy a house five years from now and estimate
that an initial down payment of $20,000 will be required. You want to make equal end-of-year deposits
into an account paying annual interest of 6%, so you must determine what size annuity results in a lump
sum equal to $20,000 at the end of year 5. The solution can be derived from the equation for finding the
future value of an ordinary annuity.
Earlier in this chapter, we found the future value of an n-year ordinary annuity (FV) by applying
Equation 3.4. Solving that equation for PMT, in this case the annual deposit, we get Equation 3.15:
Eq. 3.15 PMT
FV
r
rn
=
[(1+1) − ]
Once this is done, we substitute the known values of FV, r and n into the right-hand side of the
equation to find the annual deposit required.
11 The Excel function for continuous compounding is ‘=exp(argument)’. For example, suppose you want to calculate the future value of $100
compounded continuously for five years at 8%. To find this value in Excel, first calculate the value of e 0.08 × 5, by entering ‘=EXP(0.08*5), and
then multiply the result, 1.492, by $100 to obtain a future value of $149.20.
annual percentage yield
(APY)
The annual rate of interest
actually paid or earned,
reflecting the impact of
compounding frequency. The
same as the effective annual
rate (sometimes called the
effective APR)average annual
percentage rate (AAPR)
The stated annual rate
calculated by multiplying the
periodic rate by the number of
periods in one year