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The EIR - Effective Interest Rate
The use of the effective rate is enshrined in legislation in many countries but both the law and the terminology used varies. For example in the USA the term APR is used to refer to the nominal annual rate i.e. without taking compounding into account but in the UK the same term, APR, refers to the Annual Percentage Rate which does take compounding (and other one off payments) into account. You can see that confusion is not only possible but likely in reading any financial document. Your only hope of getting things right is to understand what is being calculated.
In general the term Effective Interest Rate (EIR) is used to refer to the general concept of including compounding in the quoted rate but this is sometimes called by other names, for example APR (Annual Percentage Rate) in the UK.
As we already know the formula to convert nominal to effective rate you might think that there is very little more to say about EIR but in fact there is a great deal. The reason is simply that there are many different forms of loan involving different repayment schemes - regular payments, single payments and the effect of additional charges to consider. However in one very simple situation - where the interest is simply calculated on the balance at a periodic interval - we do indeed know how to calculate the EIR.
More complex loans will be covered in a later chapter.
Credit card EIR/APR
Although the EIR/APR has to be quoted for a great many types of loan, the most commonly encountered is the credit card.
In this case the nominal interest rate is quoted as a monthly rate and conversion has to be made to an effective annual rate.
For example, if an interest rate of 1% per month is quoted, the nominal annual rate (the APR in the USA) is 12% while the effective annual rate (the APR in many countries and the EIR in the USA) is 12.6825%.
Similarly a 2% per month nominal rate is equal to an EIR/APR of 26.8%. It is very important to notice that this simple calculation assumes that there are no charges other than the monthly percentage quoted.
Cards that charge fixed annual fees and cash advances which attract a single surcharge have an EIR/APR that is calculated by slightly more complicated methods.
A spreadsheet to calculate the EIR/APR on the monthly rate is easy enough to construct. First enter the text into column A as shown in Figure 4.
Next enter the formula:
into B5. This calculates the effective annual rate from the monthly nominal rate.
In the UK the effective rate has to be truncated to one decimal place to be called the APR and this is worth calculating because it demonstrates how to truncate a percentage in general.
To do this we can use the TRUNC(val,n) function which simpley chops off the decimal places in val to leave just n digits after the decimal point. You might think that you need TRUNC(val,1) but if you remember that percentages are actually stored as decimal fractions you should be able to see that the correct formula if TRUNC(val,3) e.g. 52.13% is in fact stored as 0.5213 and truncating this to three decimal places gives 0.521 which displays as 52.1. So truncating a percentage to one decimal place corresponds to truncating the equivalent decimal fraction to three decimal places. This makes the correct formula:
which should be entered into B6 to complete the spreadsheet.
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:
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:
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:
The effective interest rate E% is just the simple interest rate that gives the same future value. That is:
which if you compare the two gives
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 10-year period is:
E=(1+0.1)^10-1 = 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:
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.
- 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.