Equivalent rate
Periodic rate that, compounded over the reference period, gives exactly the same effective annual rate as a reference annual rate.
An annual rate of 5% does not convert to 0.4167% a month: the equivalent monthly rate is (1.05)^(1/12) − 1 = 0.4074%, and it is the only one that gives back 5% after twelve months. A periodic rate is said to be equivalent to a reference annual rate when it is linked to it by compound capitalization, that is when the monthly rate raised to the twelfth power returns exactly one plus the annual rate. It stands against the proportional rate, obtained by simply dividing the annual rate by the number of periods, which is a linear approximation of the same conversion. The gap between the two is negligible at low rates and becomes significant at high rates, because it grows as the square of the rate: it therefore became visible again after the 2022 rate rise, having been of no practical consequence for a decade. Three uses in insurance depend on it. Converting a portfolio yield between periodicities, where using the proportional rate introduces a systematic bias always in the same direction. Discounting monthly benefit cash flows from an annual curve. And comparing rates quoted on different periodicities, where confusing the two notions distorts the ranking of offers before any calculation.
Textbook case: annual rate 8%: equivalent monthly rate = (1.08)^(1/12) − 1 ≈ 0.643%; proportional monthly rate = 8/12 = 0.667%. Compounded over twelve months, the proportional rate gives back 8.30% instead of 8%: 0.3 point too much a year, from a 0.024 point gap a month.
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