Effective Annual Rate

The construction has three steps. First, strip the stated (nominal) annual rate down to its true periodic rate by dividing by the number of compounding periods per year: a rate quoted at 12% compounded quarterly is really 3% per quarter. Second, grow one unit of currency forward across every period of the year by raising one plus that periodic rate to the power of the number of periods. Third, subtract the original unit to isolate the growth. What remains is the proportional increase in wealth over twelve months when interest earns interest.

The same logic runs in reverse. Given an effective annual rate, the periodic rate is one plus the EAR raised to the power of one divided by the number of periods, minus one; multiplying that periodic rate by the number of periods recovers the stated annual rate. As frequency rises without limit, the expression converges to e raised to the stated rate, minus one, which is the continuous-compounding case.

Deposit products in the United States are advertised as an annual percentage yield, which is already an effective rate, while consumer credit is disclosed as an annual percentage rate, which is nominal and typically excludes intra-year compounding on revolving balances. A bank can therefore quote a lower number on its loan than on its deposit and still earn a spread, purely through presentation conventions.

Fixed income carries the same asymmetry. US Treasury and most corporate yields are quoted on a semiannual bond basis, whereas many European issues pay annually. Comparing a semiannual-pay yield directly against an annual-pay yield overstates the annual-pay instrument's relative attractiveness. The correction is to convert the semiannual yield to an effective annual figure, or to restate the annual-pay yield on a semiannual basis, before ranking the two. Money market instruments add a further wrinkle: discount-basis and add-on-basis quotes with 360-day conventions must first be turned into a genuine holding period return.

Three tasks recur. Converting a stated rate at any periodicity into an effective annual rate; converting an effective annual rate back into a stated rate at a specified periodicity, which is the step candidates most often skip; and choosing between two instruments quoted on different bases by placing both on a common footing.

Discipline in the discounting also matters. When cash flows arrive quarterly, discount them at the quarterly rate over quarterly periods rather than mixing an annual rate with fractional exponents. Financial calculators handle this through the payments-per-year and compounding-periods-per-year settings, but a mismatch between those two settings silently corrupts every answer that follows. Verifying that the periodic rate, the number of periods and the payment frequency describe the same calendar unit is the single highest-value habit in this part of the curriculum.

The effective annual rate assumes that every intra-year cash flow is reinvested at the same periodic rate for the remainder of the year. For a savings account left untouched, that assumption is close to descriptive. For a bond whose coupons must be reinvested in an uncertain future market, the effective annual yield is a convention rather than a forecast, and realised return will differ whenever reinvestment rates move.

The measure also says nothing about fees, points, insurance or timing of drawdowns, which is why an all-in cost of borrowing can exceed even the correctly compounded rate. Two further boundary cases are worth noting: at a stated rate of zero the effective annual rate is zero regardless of frequency, and for negative stated rates more frequent compounding makes the effective rate less negative rather than more.