Effective Annual Rate

Banks like to quote a rate and a rhythm separately: 5% “compounded monthly,” 4.9% “compounded daily.” Those phrases are not yet comparable. The effective annual rate (EAR, sometimes APY in US deposit advertising) folds the rhythm into one number: how much a balance grows over a year if you add nothing and remove nothing.

The conversion

EAR = (1 + r/n)^n − 1

where r is the nominal annual rate as a decimal and n is the number of compounding periods per year.

5% compounded monthly. (1 + 0.05/12)^12 − 1 ≈ 5.116%.

5% compounded daily, using 365. (1 + 0.05/365)^365 − 1 ≈ 5.127%.

5% compounded annually. EAR = 5% exactly. Nominal and effective match when interest is credited once a year.

The monthly-versus-daily gap on a 5% rate is about one hundredth of a percentage point in effective terms. On €10,000 that is about a euro a year. Advertising copy that treats “daily” as a different class of return, at the same nominal rate, is selling the euro.

The bottom rows are why the conversion stops being a rounding topic. Credit-card-like nominal rates, compounded often, hide several extra points inside the frequency. A loan or card disclosure may already be required to show a single comparable charge; deposits may show an APY. When you only have the nominal and the word “monthly,” compute the EAR yourself before you rank two offers.

What EAR does not include

It does not include fees. A 5.12% effective savings rate with a €40 annual fee is a bad description of a €1,000 balance: the fee is 4% of the balance, eating most of the interest. Fold large fees into a balance comparison, not into a slogan about daily compounding.

It does not include tax. If interest is taxed at 20% and the tax is paid from the account, your spendable growth is lower. A rough adjustment is to compare after-tax rates only when both products are taxed the same way. A tax-exempt account versus a taxable one is not an EAR contest until you have applied the tax.

It does not describe a year in which you contribute every month. Contributions make the internal rate of the whole cash-flow stream a different statistic. EAR is specifically the empty-handed year: principal in, nothing added, interest left alone, see the percent. For a contribution plan, project the balance at your horizon instead of anointing the EAR as the growth of every euro. Early contributions grow for more than a year; late ones grow for less. EAR is the rate each euro experiences per year it is present, which is still the right building block, but the blended result will not equal “principal × (1 + EAR)^years” once deposits keep arriving.

Nominal rates in loans

Borrowing disclosures often use APR in a regulated way that may or may not match this EAR formula, because fee treatment and day count are specified by law. Do not assume “APR” on a mortgage and “effective annual rate” on a savings account are the same algebra. APR versus interest rate separates the borrowing vocabulary. Use the formula on this page when you are holding a nominal rate and a compounding frequency and nobody has handed you a single comparable percent yet.

A ranking example

Offer A: 4.80% compounded daily. EAR ≈ (1 + 0.048/365)^365 − 1 ≈ 4.92%.

Offer B: 5.00% compounded annually. EAR = 5.00%.

B wins the rate comparison. A wins only if B has a catch — a minimum balance you will not keep, a teaser that expires, a tax difference. The effective rate breaks the tie between frequencies. It does not break ties between contracts. Once the EARs are within a few basis points, go read the fees and the lockup. The exponent has finished its job.

Continuous compounding, only so you can ignore it

The mathematical limit as n grows without bound is e^r − 1. At 5%, e^0.05 − 1 ≈ 5.127%. Daily compounding already landed on about 5.127%. “Compounded continuously” on a consumer savings account is a rounding story next to daily, not a new digit in your balance. I mention it so a brochure that says continuous does not sound like a different asset class. At 20% the continuous effective rate is about 22.14%, and daily was about 22.13%. The drama, again, is the 20, not the adverb.

One year of a real deposit, checked

€4,000 at 3.6% nominal, compounded monthly. Period rate = 0.036/12 = 0.003. After 12 months the factor is 1.003^12 ≈ 1.03660. EAR ≈ 3.660%. Interest earned ≈ €146.40. If the bank posts €146.40, you are looking at this convention. If the bank posts €144.00, that is simple 3.6% of 4,000, and the account did not do what “compounded monthly” led you to expect — or a fee took the difference. Two euros is small. The habit of checking one year on a round principal is not.

When to calculate it

Calculate EAR when two savings offers quote different compounding rhythms, or when a nominal rate looks moderate but compounds monthly at a high level (the 18% row). You can also get it indirectly: divide a one-year compound projection from the compound interest calculator by the principal, subtract 1, and you have reconstructed the EAR. If that reconstruction disagrees with a bank’s advertised APY, someone is using a different day count (360 versus 365) or including a bonus that is not interest. Ask which. The formula is too short to argue with; the inputs are where offers get slippery. Do not use EAR to rank a loan’s APR against a deposit’s APY without reading what those labels include.