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More than one year
Given that by now you understand the idea of effective rate fully, you might like to consider the following question before you read on.
A deposit attracts 10% per annum compounded monthly, giving an effective annual rate of 10.47%. If the deposit is left undisturbed for five years what is the future value?
The future value at the end of one year can be calculated using simple interest and the effective annual rate of 10.47%, that is:
FV=100*(1+0.1047) =$110.47
The question is can the simple interest calculation be used to give the Future Value after five years using five times the effective rate?
The answer is obviously no because there is now another compounding effect to take into account. The interest at the end of each year itself earns interest and this is not taken into account in the effective rate.
The point is that the effective rate is an annual rate and only summarises the influence of a single year’s compounding.
You can still use the annual effective rate to calculate the Future Value after five years but only by using the compound interest formula:
FV=100*(1+0.1047)^5 = $164.52
which of course gives the same result as using the monthly interest rate for the entire 10 years:
FV=100*(1+0.1/12)^(5*12) = $164.53
The difference of 1 cent is within the accuracy of the calculation. If the effective annual rate had been used to more decimal places then the results would have tallied more closely.
The point is that the effective interest rate summarises the effect that compounding has over a specified period.
In the example above, the effective rate summarises the monthly compounding to give a yearly rate but if you want to calculate what happens in subsequent years you must remember to use the compound interest formula.
The compounding period
If you were offered an investment rate of 10% per annum compounded monthly, weekly or daily which would you choose?
The answer should be obvious after only a few moments’ thought. The more frequent the compounding the higher the effective rate. For example the effective rate of 10% compounded monthly is:
(1+0.1/12)^121 = 10.47131%
but 10% compounded daily is:
(1+0.1/365)^3651 = 10.515%
Of course daily isn’t the upper limit and higher effective rates could be reach by compounding more often than once a day. If you allow the compounding to be performed so often that it can be considered to be continuous then the form of the equation giving the effective rate of interest changes to:
effective rate = EXP(I)1
Most spreadsheets have an EXP or exponential function so working out the effective rate due to continuous compounding is relatively easy. In maths books you will also see the same formula written as:
e^i1
where e is the exponential number roughly equal to 2.71828. The spreadsheet function =EXP(I) is identical to e^I.
Using this formula for continuous compounding gives the effective annual interest rate for 10% as:
=EXP(0.1)1=10.517%
which you can see is bigger than all of the effective rates that we have calculated so far. Of course this is no accident because by continuously compounding the nominal rate we reach the largest effective rate possible.
If you want to calculate the Future Value of an investment at any time t then the appropriate formula is;
FV=PV*EXP(t*I)
Notice that if I is an annual rate then t has to be in years and fractions of years, e.g. t=1.5 is 1 year 6 months.
