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Effective rate  a broader view
There is a tendency to think of the effective rate of interest as something that relates only to the way compounding increases the effect of an annual rate of interest applied monthly  but this is just the one manifestation of the way compounding can affect a simple interest rate.
In all transactions the fundamental quantity is the simple interest rate  the percentage per time period. The fundamental calculation is always the trivial one of simple interest per time period.
For example, if an investment earns I% per month then at the end of one month the amount has grown to:
FV=PV*(1+I)
where PV is the amount on deposit for the month concerned.
If the same amount is on deposit for n months the final sum could be calculated by using the simple interest formula n times or the shortcut formula:
FV=PV*(1+I)^n
could be used instead.
In this sense compound interest calculations have to be seen as a shortcut to obtaining the result that the repeated application of simple interest would give.
Notice that if the amount on deposit each month varies then the compound interest calculation cannot be used and there is no alternative but to use the simple interest calculation using the amount on deposit at the end of each month.
When the amount on deposit changes in a regular and predictable way, e.g. in an annuity where a fixed sum is withdrawn each time period  then there are shortcut methods of calculation similar to compound interest and these are described in later chapters.
The effective rate is also a shortcut way of working out the effect that compounding has on the nominal or simple interest rate. If you deposit PV for n periods at a nominal/simple interest rate of I% then the future value is given by:
FV=PV*(1+I)^n
The effective interest rate E% is just the simple interest rate that gives the same future value. That is:
FV=PV*(1+E)=PV*(1+I)^n
which if you compare the two gives
E=(1+I)^n1
which is the fundamental relationship between nominal and effective rates derived earlier for an annual rate applied monthly, i.e. with n equal to 12.
This more general form of effective interest rate allows the effect of compounding over any period to be summarised as a simple interest rate.
For example, if you deposit $100 at 10% per annum paid yearly the annual nominal and effective interest rates are the same because there is no compounding. However if you leave, or plan to leave, the deposit untouched for 10 years then annual compounding produces a future value of:
FV=100*(1+0.1)^10 = $259.37
and the effective interest rate for a 10year period is:
E=(1+0.1)^101 = 1.5937
or 159.37% per 10 years.
If you would like to check these figures try calculating the future value as simple interest using the effective rate, that is:
FV=100*(1+1.5937) =$259.37
As you can see, the effective rate gives the same future value using a simple interest calculation as compound interest calculated using the nominal rate.
In general:
 the effective rate is the simple interest equivalent of a rate that is compounded over a given number of periods
 to find the effective rate write down the formula for the compounded rate and a simple interest formula for the same time period. Solve for the simple interest rate that gives the same future value.
The same ideas hold for a loan as for an investment.
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^{i}1
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.
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