APR versus effective rate: what a quoted rate leaves out

Why it matters

The U.S. Treasury’s 4-week bill is about as short as lending gets, and Kitalpha records its quoted rate on 4-Week Treasury Bill (coupon-equivalent) as 3.85% per year as of 2026-09-15. Every rate on the site is quoted per year, and so is the rate on a credit card, a savings account or a mortgage. But a quote per year is silent on one thing: how often interest is added to the balance during that year. Ask what the quoted figure actually costs, or actually accrues, over a full year, and the answer is a different number. How different, and why, is this lesson.

The concept

A quoted annual rate is a convention. Lenders and markets state a rate per year even when interest is charged or credited more often, and the convention for getting from the period to the year is multiplication: the rate for one period times the number of periods in a year. A card that charges 1.5% a month is quoted at 18% a year; a deposit that credits 0.5% a month is quoted at 6%. The label varies, annual percentage rate or APR on consumer credit, stated or nominal annual rate in a textbook, but the arithmetic is the same; a consumer-credit APR may also fold in fees, which this lesson leaves aside.

The effective annual rate is the growth actually accrued over one year once each period’s interest is added to the balance and earns interest itself. Lesson 2 showed compounding across years; the same thing happens inside a year whenever there is more than one period. Take the 6% quote credited monthly. Month one adds 0.5% of the balance. Month two adds 0.5% of a balance that is already 0.5% larger, and so on for twelve months. Multiplying 1.005 by itself twelve times gives about 1.0617, so the balance is 6.17% larger after a year, not 6.00%. The extra 0.17 percentage points, 17 basis points, is interest earned on interest inside the year. Nobody hid it; the quote simply leaves it out.

The formula follows directly. Write the quoted annual rate as a decimal, y, and the number of compounding periods in a year as m. The rate per period is y divided by m. Growth over one period multiplies the balance by (1 + y ÷ m); growth over the year multiplies by that factor m times, which is (1 + y ÷ m) raised to the power m. Subtract one and the effective annual rate is what remains: (1 + y ÷ m) to the power m, minus 1. With m equal to one the formula collapses to y, so the quoted and effective rates coincide only when interest compounds once a year. For any larger m the effective rate is higher, and the same quote implies a different effective rate at every frequency, which is why a quote on its own leaves open what is actually paid or earned.

Two features are worth carrying away. First, the effect of frequency shrinks as frequency rises. Moving from once a year to monthly captures most of the gap; weekly adds a fraction of a basis point more; daily adds less still, and the figures crowd toward a ceiling. Second, the gap grows with the rate: at a low rate the interest on interest is tiny, at a high one it is material, which is why the same monthly frequency matters more on a card than on a bill. Two quotes can only be compared once they are converted to the same footing, and the effective annual rate is that footing.

The one visual for this lesson is a small table. It takes today’s quoted rate on the 4-week bill and converts it to the effective annual rate at monthly, weekly and daily compounding. Look for the shape of the increases down the table: one large step, then two small ones.

Worked example

Take the record from the opening: the 4-week bill on 4-Week Treasury Bill (coupon-equivalent), quoted at 3.85% per year as of 2026-09-15. A bill pays no interest along the way; it is issued below its face value and repaid at face value four weeks later, so the record itself does not compound monthly. The steps treat its quoted figure as a nominal annual rate and ask what that figure implies at two compounding frequencies. Every number below comes from that record on the date shown.

Read the steps in order. The quote becomes a decimal, then a rate per month, one-twelfth of the annual figure. Compounding that monthly rate twelve times gives the effective annual rate at monthly compounding, and fifty-two weekly periods give the weekly figure. The last two lines measure the gaps in basis points. The first gap, effective minus quoted, is what the quote leaves out: the interest on interest within the year, and the number the misconception sets to zero. The second gap, weekly minus monthly, is smaller by an order of magnitude, because most of the frequency effect has already appeared by the first step.

Faded example

Now a second record: the 13-week bill on 13-Week Treasury Bill (coupon-equivalent), quoted at 4.07% per year as of 2026-09-15. The decimal rate and the rate per month are given below. Complete the last step, the effective annual rate at monthly compounding, then reveal the answer and compare it with the quote.

Retrieval check

Mark your confidence before each answer. Every option carries an explanation; read the ones you rejected as well as the one you chose.

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