Effective Annual Rates when Loan Interest is Paid

The term “effective annual rate” (EAR) is very often explained with reference to the compounding of a nominal or stated rate using the formula:

For example, if the stated rate is 12% p.a., compounded quarterly, then the EAR is (1.03)^4 -1 i.e. around 12.6% p.a.

These explanations are potentially ambiguous, since they do not make clear how any repayments of the loan interest should be treated, and hence whether the EAR shown above is a general definition, or one that applies only in specific circumstances.

For example, does the above formula apply if compounding of interest were done quarterly and payments of accumulated interest were paid out every six months? Note that, once interest is paid out, then this interest is no longer compounded, so interest on that interest is not earned (nor charged).

It is possible to derive more general formula(s) that reflects the payouts as well as the compounding. For example, if all accumulated interest is paid out at least annually (e.g. at the end of the year, or every six months, or quarterly etc.), then one can derive the total paid interest in the year per \$1 initial loan (total payout rate or TPR) as:

In this formula, r is the per period interest rate (e.g. monthly), “delta-c” is the number of period in between compounding points (e.g. 3 for quarterly compounding based on monthly periods), and “delta-p” is the number of periods between payout times.

(The formula applies to the cases where payout frequency is less than or equal to compounding frequency, so that payout dates are a sub-set of compounding dates, and where the periods of of each length and with a year being made up of a whole number of periods.)

The derivation of the formula is covered in our Level I course “I.3 Core Calculations for Economic Modelling”.

The formula can be simplified in some common and special cases:

First, if there is only one payout per year (delta-p=12), and with monthly compounding (delta-c=1), then it gives:

(for the derivation of TPR from r, or vice versa)

Second, if payment is made at the end of every compounding period (i.e. the payment and compounding time points are identical), so that delta-p is equal to delta-c:

The first of these is essentially the same as pure compounding, and the second is rather like simple interest (in this second case, the payment of interest occurs at the same time as the interest would have otherwise been compounded, so the compounding process is essentially non existent). That is, as soon as the accrued interest is paid out, then it is no longer compounded.

One should therefore be particularly careful to ensure that calculation relating to loans (such as finding periodic rates from effective rates, or vice versa) use clear terminology and are done in a consistent way. The use of two separate terms (EAR and TPR) can help in this context.

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