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nominal can be ignored; we’ll discuss what that means in Chapter 3.) On the statement, the
issuer has stated the interest rate charged both as an annual rate and as a monthly rate. The
fact is, even though there may be reasons to use a daily rate, an annual rate actually could
have been used just as well. The rate of 0.05203% per day is equivalent to an annual rate
of 18.99%.Example 1.5.3 Convert the rate 0.05% per day into an equivalent simple interest
rate per year.For the (simplifi ed) exact method, there are 365 days in a year. Thus, the interest for a full
year would be 365 times the interest for a single day. Thus, since 365(0.05%) 18.25%, we
conclude that 0.05% per day is equivalent to 18.25% per year.That 0.05% doesn’t seem so small now!
We can verify this claimed equivalence by calculating interest using the daily rate and
then again using the annual rate.Example 1.5.4 Calculate the interest due for 10 days on a principal of $1,200 using
(a) a simple interest rate of 0.05% per day and (b) a simple interest rate of 18.25%
annually.(a) Since the rate is daily, time should be measured in days, and so T 10.I PRT
I ($1,200)(0.0005)(10)
I $6.00(b) Since the rate is annual, time should be measured in years, and so T 10/365.I PRT
I ($1,200)(0.1825)(10/365)
I $6.00The answers to (a) and (b) agree, as they should. Notice that we had to be careful to make
sure that the time units for the value of T agreed with the time units of the interest rate.Converting from an Annual Simple Interest Rate
In Example 1.5.3 we saw how to convert from a given daily rate to an equivalent annual
simple interest rate. What if we want to go the other direction?Example 1.5.5 Convert an 18% annual simple interest rate to a rate per month.There are 12 months in a year, or, in other words, 1 month is^1 ⁄ 12 of a year. Thus, the inter-
est for a month would be just^1 ⁄ 12 of the interest for a full year. Since 18%/12 1.5% we
conclude that a simple interest rate of 18% per year is equivalent to a simple interest rate of
1½% per month.We can follow the principles of examples 1.5.3 and 1.5.5 whenever we need to change
the time units of an interest rate. If the new time unit is larger (as in Example 1.5.3),
we will end up multiplying; if it is smaller, we end up dividing (as in Example 1.5.5).
Common sense will also help avoid errors here. If you multiply when you should divide
(or vice versa) you will usually wind up with a final answer that is either far too large
(or too small) to make sense. It is always a good idea, in these or any other type of prob-
lems, to give your answers a quick reality check. A nonsensically large or small answer
is an obvious tip off that a mistake has been made and that you should double-check
your work.
Note that when an interest rate is given per day, or even per month, the rounding rule of
two decimal places that we have been using for annual rates will not provide enough preci-
sion. Daily rates should be carried out to at least five decimal places to avoid unacceptably
large rounding errors.1.5 Nonannual Interest Rates (Optional) 45