Last updated: 2026-05-25

Effective annual interest rate

The effective annual interest rate (EAR), also called the annual percentage yield (APY), is the actual return earned — or the actual cost paid — on a loan or investment after accounting for compounding within the year. It is always greater than or equal to the annual percentage rate (APR), which is the nominal rate that lenders and issuers are required to disclose.

The two diverge whenever interest compounds more than once a year. Under monthly compounding, each month's interest is added to the balance and earns interest in the following months, so the amount accumulated over a full year exceeds the stated nominal rate. A nominal "6% APR, compounded monthly" therefore corresponds to a true annual cost of 6.168%, not 6%.

How it's calculated

The effective annual rate follows directly from the nominal rate and the number of compounding periods:

where:

• $r$ = nominal annual rate (APR) as a decimal (e.g. 0.06 for 6%)
• $n$ = number of compounding periods per year

The periodic rate — the rate applied to the balance in each period — is the nominal rate divided by the number of periods:

As the number of periods grows without bound ($n \to \infty$), the expression approaches the continuous-compounding limit:

Worked example

A savings account advertises 5.5% APR, compounded monthly (n = 12).

• Periodic rate: 5.5% ÷ 12 ≈ 0.4583%/month
• Effective rate: (1 + 0.055/12)^12 − 1 ≈ 5.6408% APY

The effective rate adds 0.14 extra percentage points over the stated APR. On a $50,000 deposit that is about $70 extra per year, and the gap accumulates over time as the higher effective rate applies to a growing balance.

How compounding frequency affects the effective rate

The more frequently interest compounds, the higher the effective rate, but the increments diminish quickly:

Moving from annual to monthly compounding at 6% adds about 0.17 percentage points, while moving from monthly to daily adds only another 0.015 pp. Above daily, the effective rate is essentially at the continuous-compounding ceiling and further subdivision adds nothing meaningful.

The reason the increments shrink is that the limit $e^r$ is approached asymptotically. The jump from annual to monthly is large because the period count rises from 1 to 12; each subsequent increase in frequency (12→52→365) captures a progressively smaller slice of the remaining gap between the nominal and continuous rates. At 6% APR the difference between monthly and continuous compounding is only 0.0159 percentage points (1.59 basis points) — on a $100,000 balance, $15.90 per year.

Continuous compounding

Continuous compounding is the mathematical limit where interest accrues at every instant rather than at discrete intervals. The effective rate is $e^r - 1$, using the number e (Euler's number, ≈ 2.71828). It is the absolute upper bound on how much any discrete compounding frequency can earn for a given APR. No retail financial product compounds continuously, but the formula appears in options pricing (Black-Scholes), bond mathematics, and theoretical finance.

APR and APY in practice

The distinction between the nominal and effective rate determines which number to compare when evaluating products:

• Borrowing: Credit card issuers and lenders advertise APR (the lower number), while the APY reflects the true cost. At 24% APR compounded daily the effective cost is 27.11% APY — a meaningful difference on a large balance.
• Saving and investing: High-yield savings accounts and certificates of deposit advertise APY (the higher number). Comparing APYs directly shows which account pays more, regardless of how often each compounds.
• Regulation: In the US, the Truth in Lending Act requires APR disclosure for loans, and the Truth in Savings Act requires APY disclosure for deposit accounts. Neither law requires both.