APR to APY Calculator

Why APR and APY differ

An annual percentage rate states a yearly rate without saying anything about how often interest is applied. An annual percentage yield states what a year actually comes to once interest has been applied at whatever interval the product uses, so it includes interest charged or earned on interest already added during the year. The two are the same number when interest is applied once a year and diverge as the interval shortens. At 5% applied monthly the effective figure is 5.1162%; applied daily it is 5.1267%; and the limit as the interval shrinks towards zero is 5.1271%. The gap widens as the rate rises: at 20% applied monthly it is 193.91 basis points, where a basis point is one hundredth of a percentage point.

Which figure applies to which product

Convention rather than arithmetic decides which of the two a product quotes. Deposit products (savings accounts, term deposits, money-market accounts) are commonly quoted as a yield, which is the compounded figure. Credit products (most cards and consumer loans) are commonly quoted as a rate, which is not, and in many markets that rate also carries a defined fee treatment that this conversion does not touch. The practical consequence is that a quoted deposit figure and a quoted credit figure are usually stated on different bases, so putting them side by side compares two things that are not the same measurement. Converting the credit figure is what makes them comparable, and that is the whole of what this calculator does.

Reading the result rows

The headline is the effective annual figure. Below it, the difference row states the gap in percentage points at four decimals, because at ordinary rates the gap lives in the third and fourth places and rounding it to two would show 0.12 for a wide band of inputs. It is also the route between the two precisions on the panel: the headline is rounded to two decimals, so the four-decimal effective figure is the stated rate plus the difference row, and it is that figure the ceiling should be measured against. Rate per Compounding Period is the figure actually applied at each interval, the rate divided by the frequency, which is what a statement line shows. The last two rows bound the result: the continuous ceiling is the limit the effective figure approaches as the interval shrinks towards zero, and the distance row states how far the entered frequency sits below it, in basis points. At monthly compounding on a 5% rate that distance is about 1.09 basis points; at daily it is under a tenth of one.

The formula

The effective figure is one plus the period rate, raised to the number of periods, minus one. The period rate is the annual percentage divided by 100 and then by the number of periods, which is the step most often skipped when the arithmetic is done by hand, since substituting the percentage figure itself rather than its decimal form inflates the answer by orders of magnitude. The continuous limit replaces the power with the exponential of the decimal rate. Both are standard effective-annual-rate identities and neither depends on a balance, a term or a currency.

Frequency, and where it stops mattering

Each increase in frequency adds less than the one before, because the interest accrued within each interval is smaller as the interval shortens, so there is less of it left to earn further interest before the year ends. On a 5% rate, moving from annual to monthly adds 11.62 basis points, monthly to daily adds a further 1.06, and daily to the continuous limit adds 0.04. That is why the choice between daily and continuous is immaterial in practice while the choice between annual and monthly is not. The input takes periods per year as a number, so annual, semiannual, quarterly, monthly, weekly and daily are all expressible; continuous is not a frequency that can be entered, which is why it appears as the ceiling row rather than as an option.

Where the gap becomes material

Two things drive it: the rate, which sets how much interest there is to compound, and the frequency, which sets how often. Neither alone settles the size. At a 10% rate the gap is nothing at all applied annually, 25.00 basis points semiannually, 38.13 quarterly, 47.13 monthly, 50.65 weekly and 51.56 daily, so a claim that a given rate reaches a given gap only holds once the frequency is named, and the same rate can sit either side of a threshold depending on it. Below a 3% rate the gap stays under 5 basis points at every frequency including the continuous limit, which on a balance of 10,000 held for a year is under 5 units of currency.

Disclosure conventions

Which figure has to be disclosed, and how it must be calculated, is set by each jurisdiction rather than by any common standard. Some require a rate on consumer credit and a yield on deposits; some use different names for the compounded figure, and some define the fee treatment inside the disclosed rate so that it is not a pure interest measure at all. Within one jurisdiction and one product category, disclosed figures are usually directly comparable because they are calculated the same way. Across categories or across borders they may not be, and converting to a common basis is what makes the comparison meaningful.

What this calculator does not model

The conversion re-expresses one rate as another. It does not add interest to anything: an effective figure is the same interest described on a different basis, not extra interest on top of the stated one. There is no balance and no term in the model, so it cannot show the result as an amount; a gap of one basis point is one unit per 10,000 of balance held for a year, which is the conversion to money if it is needed. The model also assumes the rate holds for the whole year and that interest is applied at even intervals with nothing added or withdrawn, and it excludes fees, taxes, promotional periods, and any minimum-balance condition attached to the quoted figure.