The effective interest rate formula converts a stated annual rate into the rate a year of that arrangement actually produces once interest compounds inside the year. It is EAR = (1 + i / n)^n - 1, where i is the stated annual rate and n is the number of times interest compounds in a year. The answer, called the effective annual rate, is what makes two quotes carrying different compounding frequencies comparable.
The Formula, and What Each Term Means
EAR = (1 + i / n)^n - 1
i the stated (nominal) annual rate, written as a decimal
n the number of compounding periods in one year
EAR the effective annual rate, what a full year of that rate producesThe quantity i divided by n is the periodic rate, the slice of the annual rate charged in a single compounding period, and raising that periodic factor to the power of n is what lets each period’s interest earn interest for the rest of the year. Principal never enters, because the formula measures a rate rather than an amount, so the same conversion holds on a balance of any size. A stated rate says nothing on its own about how often interest compounds, and what an annual quote leaves open is set out in per annum.
A Worked Example
Say a lender quotes one stated annual rate and the only thing that changes across these four lines is how often interest is added to the balance.
Stated annual rate i = 6%
Compounding n Formula EAR
annually 1 (1 + 6%/1)^1 - 1 6.0000%
quarterly 4 (1 + 6%/4)^4 - 1 6.1364%
monthly 12 (1 + 6%/12)^12 - 1 6.1678%
daily 365 (1 + 6%/365)^365 - 1 6.1831%The stated rate never moves. The distance between the first line and the last is what the extra compounding is worth, and it widens as the compounding period gets shorter. Two related calculations sit outside it: how much interest has built up between payment dates (accrued interest), and how an annual figure is split across a part-year (prorated).
Where a US Firm Meets the Effective Rate
Financial reporting asks for a rate under the same name, and that rate is arrived at differently. The description of a note has to include the effective interest rate (FASB Accounting Standards Update 2015-03, Subtopic 835-30, paragraph 45-2), and the rate meant there is the one that amortizes discount or premium over the life of the note so as to result in a constant rate of interest applied to the amount outstanding at the beginning of any given period (APB Opinion No. 21, paragraph 15). It comes out of the note’s own present value against its face amount rather than out of a compounding-frequency conversion, and the schedule behind it is worked through in bonds payable.
That disclosed rate is not always the rate printed on the note. A note issued solely for cash equal to its face amount is presumed to earn the stated rate, and where the parties exchange unstated or stated rights or privileges as well, a discount or premium arises and the effective interest rate differs from the stated rate (APB Opinion No. 21, paragraph 7).
The ordinary route is simpler. The stated or coupon interest rate frequently differs from the prevailing rate applicable to similar notes, so the proceeds of the note differ from its face amount, and that difference is shown as discount or premium and amortized over the life of the note (APB Opinion No. 21, paragraph 6).
Consumer credit uses a similar word for a different figure. Under Regulation Z the annual percentage rate is a measure of the cost of credit, expressed as a yearly rate, determined by either the actuarial method or the United States Rule method (12 CFR 1026.22(a)(1)). The general rule in appendix J, which covers closed-end credit transactions, is that the annual percentage rate is the nominal annual percentage rate, found by multiplying the unit-period rate by the number of unit-periods in a year, the unit-period normally being the common period, not to exceed 1 year, that occurs most frequently in the transaction (Regulation Z, appendix J to part 1026, paragraphs (b)(1) and (b)(4)(i)).
A closed-end APR is therefore a nominal figure. Where an open-end plan’s finance charge is computed with one or more periodic rates, the disclosed annual percentage rate is reached the same way, by multiplying each periodic rate by the number of periods in a year (12 CFR 1026.14(b)). Either becomes an effective annual rate only once that conversion is applied.
What the Effective Interest Rate Is Not
An effective tax rate shares the adjective and measures something unrelated: total tax for a period divided by a stated income measure, worked through in how to calculate effective tax rate.
Common Questions
The nominal rate is the annual figure a contract quotes, and it says nothing about how often interest compounds. The effective rate is what a year of that nominal rate produces after compounding, so the two match only when interest compounds once a year, and for a positive rate more frequent compounding always leaves the effective figure the larger.
Twelve compounding periods, so n is 12 and the periodic rate is a twelfth of the stated rate.
(1 + 6%/12)^12 - 1 = 6.1678%That is the figure to hold a competing quote against, whatever compounding the other quote carries.
Put the disclosed APR in as i and the number of unit-periods in a year as n, then run the same formula, because a Regulation Z closed-end annual percentage rate is the nominal annual rate reached by multiplying the unit-period rate by the number of unit-periods in a year (Regulation Z, appendix J to part 1026, paragraph (b)(1)). Where the unit-period is a month there are 12 unit-periods in a year, so n is 12.
What comes out is an effective annual cost of credit rather than a note’s effective interest rate, because a Regulation Z annual percentage rate is built on the finance charge, which takes in any charge imposed directly or indirectly by the creditor as an incident to or a condition of the extension of credit and not interest alone (12 CFR 1026.4(a)).
Accountably places trained offshore accountants and tax preparers inside US CPA and EA firms, and a firm can put a block of its own amortization schedules and interest computations through a trial task before anything client facing depends on it.
