110 MORE INTERESTFORMULAS
per year and paid a nominal interest rate per year,r,theinterest rate per compounding
subperiodwould ber/m,and the total in the account at the end of one year would be.$t(1+:)m or simply (1 +:)m
If we deduct the $1 principal sum, the expression would be
Therefore,
Effective interest rate per year (4-32)
where r=nominal interest rate per year
m=numberof compoundingsubperiodsper yearOr, substitutingthe effectiveinterest rate per compounding subperiod,i=(r/m),
Effective interest rate per year (4-33)
where i=effectiveinterest rate per compounding subperiod
m=number of compounding subperiods per yearEither Equation 4-32 or 4-33 may be used to compute an effectiveinterest rate per year.
One should note thatiwas describedin Chapter 3 simply as the interestrate per interest
period. We were describing the effective interest rate without maldng any fuss about it.
A more precise definition, we now know, is thatiis theeffectiveinterest rate per interest
period. Although it seems more complicated, we are describing the same exact situation,
but with more care.
The nominalinterest rateris often givenfor a one-yearperiod (but it could be given for
either a shorter or a longer time period). In the special case of a nominal interest rate that is
given per compounding subperiod, the effective interest rate per compounding subperiod,
i,equals the nominal interest rate per subperiod,r.
In thetypicaleffectiveinterestcomputation,there aremultiplecompoundingsubperiods
(m>1). The resulting effective interest rate is either the solution to the problem .or an
intermediatesolution,which allowsus to use standardcompound interest factors to proceed
to solve the problem.
For continuous compounding (which is described in the next section),Effective interest rate per year (4-34)