Effective Annual Rate Calculator
Solve EAR = (1 + r/n)ⁿ − 1 for any of its three quantities — effective annual rate, nominal annual rate, or compounding frequency. Compares discrete compounding to the continuous limit, shows the compounding staircase, and includes a full step-by-step solution plus a scenario library.
EAR equals nominal rate only for annual compounding
When n = 1, the formula reduces to EAR = r — annual compounding credits interest only once at year-end, so the nominal rate is the effective rate. For any n > 1, EAR > r. This is why regulators force lenders to disclose the effective rate (APR, AER, EAY) rather than just the nominal rate: a "5 % loan compounded daily" sounds like 5 % but is really 5.127 %. Without the EAR disclosure, borrowers would systematically underestimate the true cost of credit.
Compounding more often increases EAR but with diminishing returns
Going from annual to monthly compounding at 6 % adds about 0.17 pp to the effective rate. Going from monthly to daily adds only 0.006 pp more. Going from daily to continuous adds another 0.0005 pp — negligible. The gains from more frequent compounding shrink rapidly, and the series converges to a hard ceiling: e^r − 1. At 6 %, the continuous limit is 6.1837 %, only 0.0159 pp above daily compounding. In practice, daily and continuous compounding are practically indistinguishable for typical rates.
EAR is the only correct way to compare rates
Two loans can have the same nominal rate but different EARs if they compound differently. A 5 % nominal rate compounded annually is 5.000 % effective; compounded monthly it is 5.116 %; compounded daily it is 5.127 %. To compare which loan is cheaper, you must use EAR — or equivalently APR with the same fee treatment. The Truth in Lending Act and its international equivalents require lenders to disclose the APR (which for a fee-free loan equals the EAR) precisely so that consumers can compare apples to apples.
The gap grows with the rate
The spread EAR − r is roughly r²/(2n) for small r — so it grows quadratically with the nominal rate and inversely with the compounding frequency. At 3 % monthly, the spread is about 0.046 pp; at 6 % it is 0.168 pp; at 12 % it is 0.618 pp; at 24 % it is 2.36 pp. This is why the difference matters most for high-rate products like credit cards and payday loans, where the nominal-to-effective gap can add real money to the cost of borrowing — a 24 % nominal credit card compounded daily is effectively 27.1 %, nearly 3 pp more than the advertised rate.
How to Use This Calculator
The effective annual rate is the true annual cost (or yield) of money after compounding is accounted for. Give the calculator two of the three quantities — nominal rate, compounding frequency, or effective rate — and it will solve for the third.
1. Pick a compounding frequency
Click any preset — annual, semi-annual, quarterly, monthly, weekly, daily, or continuous — to set n in one click. The frequency you pick determines the shape of the compounding staircase and how large the gap is between nominal and effective rates.
2. Enter the nominal rate
Type or drag the stated annual rate r. The preset row gives quick jumps from 1 % through 25 %; the slider spans the full 0.001 %–100 % range for exotic cases like credit cards or hyperinflation scenarios.
3. Choose what to solve for
Pick EAR to compute the effective rate from nominal rate and frequency; r to reverse-engineer the nominal rate from a quoted EAR; or n to find the compounding frequency that produces a given EAR — useful for identifying whether a rate quote is really quarterly, monthly, or something in between.
4. Read the result and the staircase diagram
The result card shows the solved value plus a colour-coded verdict based on the spread. The staircase diagram below draws the growth of $1 over one year — with each compounding event shown as a step and the continuous limit overlaid as a dashed curve for comparison.
5. Compare frequencies visually
The main chart plots EAR against the nominal rate for several frequencies, so you can see how the annual-compounding line (EAR = r) sits below the continuous limit and where the monthly, quarterly and semi-annual lines fall between them.
6. Switch to Advanced for depth
Advanced mode adds a frequency-sweep chart showing EAR vs n, a +1 % sensitivity check on each input, and a companion metrics panel with the EAR/r ratio, the spread in percentage points, the continuous-equivalent rate, and the doubling time at the effective rate.
What This Tool Offers
Built around the classic relation EAR = (1 + r/n)ⁿ − 1, the correct way to compare rates across different compounding conventions.
Three-way solving
Solve for the effective annual rate, the nominal annual rate, or the compounding frequency — including a numerical solver for n when it is not an integer.
Compounding staircase
A live chart of one year of growth showing each compounding event as a discrete step, with the continuous limit overlaid as a dashed curve for direct visual comparison.
Rate comparison chart
All frequencies on the same axes — annual, quarterly, monthly, and continuous — showing how EAR grows with the nominal rate and how the curves diverge at high rates.
Frequency sweep
Advanced mode plots EAR against compounding frequency n, revealing the diminishing-returns curve that asymptotes to er − 1.
Full conversion suite
Instantly converts between EAR, nominal rate, continuous rate, and the spread in percentage points, with monthly, quarterly, semi-annual and daily equivalents.
Reference tables
A frequency comparison at 5 %, an EAR lookup grid across rates and frequencies, frequency definitions, and the full variable reference.
Reference Tables & Conversions
A frequency comparison table, an EAR lookup grid across rates, and the reference definitions.
Rate converter — quick reference
Effective Annual Rate by Compounding Frequency — at 6 % Nominal
The same 6 % nominal rate produces different effective rates depending on how often interest is credited.
| Frequency | n (periods/yr) | Periodic rate | EAR (%) | Spread vs nominal |
|---|
EAR Lookup Grid
Effective annual rate for a given nominal rate and compounding frequency. Values in % per year.
| Nominal \ Freq | Annual | Semi | Quarterly | Monthly | Daily | Continuous |
|---|
Variable Reference
| Symbol | Meaning | Formula | Notes |
|---|---|---|---|
| EAR | Effective annual rate | (1 + r/n)ⁿ − 1 | The true annual rate after compounding |
| r | Nominal annual rate | n·((1+EAR)^(1/n) − 1) | The stated rate; lower than EAR when n > 1 |
| n | Compounding periods per year | (solved numerically) | 1, 2, 4, 12, 52, 365, or ∞ |
| r/n | Periodic rate | — | The rate applied each compounding period |
| er − 1 | Continuous limit | as n → ∞ | The hard ceiling on EAR for any nominal rate |
| APR | Annual percentage rate | ≈ EAR | In the US, includes some fees; regulator-defined |
| ln(1+EAR) | Continuous-equivalent rate | rcc | The continuous rate that produces the same EAR |
What Does EAR = (1 + r/n)ⁿ − 1 Measure?
The effective annual rate is the true annual rate of interest after compounding is taken into account. When interest is credited more than once per year, each interest payment itself starts earning interest before the year is out, so the total year-end balance is higher than the nominal rate would suggest. The EAR converts any compounding schedule into a single number that represents "what percentage would the year-end balance grow by if the nominal rate were credited once at year-end at the same final total?" It is the apples-to-apples comparison rate that consumers, regulators, and financial analysts use to compare different financial products fairly.
The Effective Annual Rate Formula
The defining relationship, for discrete compounding n times per year, is:
As the compounding frequency n grows toward infinity, the expression converges to the continuous-compounding limit:
Rearranged to solve for the nominal rate given the effective rate and the frequency:
For continuous compounding, the inversion is direct:
The compounding frequency that produces a given EAR from a given nominal rate is found by solving:
Because this equation has no closed-form solution, it is solved numerically by bisection. There is no real solution when the target EAR exceeds the continuous limit er − 1 — meaning no finite compounding frequency can produce that EAR at the given nominal rate.
Effective Annual Rate Variables
| Symbol | Meaning | Notes |
|---|---|---|
| EAR | Effective annual rate | Always ≥ r; always < er − 1 for finite n |
| r | Nominal (stated) rate | What is advertised; the periodic rate times n |
| n | Periods per year | Positive real number; typically 1, 2, 4, 12, 52, 365 |
| r/n | Periodic rate | The actual rate applied each period |
| EAR − r | Compounding spread | Grows with r and shrinks with n |
| rcc | Continuous-equivalent rate | ln(1 + EAR); the exact force of interest |
How Each Input Affects the Result
The nominal rate r is the base; the effective rate is always at least as large as r, and grows monotonically with r. Doubling r does not double the spread — it roughly quadruples it, since the spread is approximately r²/(2n) for small r.
The compounding frequency n compresses the gap between r and EAR. Going from n = 1 to n = 2 (annual to semi-annual) closes about half the remaining gap to the continuous limit. Going from n = 2 to n = 4 closes another quarter of the original gap. Going from n = 4 to n = 12 closes another eighth. The pattern continues: each doubling of n halves the remaining distance to the continuous limit. This is why daily compounding and continuous compounding are almost indistinguishable in practice — the residual gap is less than a thousandth of a percentage point at typical rates.
The relative effect of n is largest at low n and high r. At 3 % nominal, the spread from annual to continuous is only 0.046 pp. At 24 % nominal, the same shift is 2.36 pp. This is why credit cards — which often quote 18–30 % nominal rates compounded daily — can end up with effective rates several points above their advertised rate, and why regulators insist on the APR or EAR disclosure for consumer credit.
Effective Annual Rate Calculation Examples
Periodic rate = 6 % / 12 = 0.5 % per month
EAR = (1.005)¹² − 1 = 1.061678 − 1 = 6.1678 % / yr
Spread = 0.1678 pp above the nominal rate
EAR = (1 + 0.06/365)³⁶⁵ − 1 = 6.1831 % / yr
Spread = 0.1831 pp; only 0.0153 pp more than monthly
EAR = e0.06 − 1 = 6.1837 % / yr
This is the theoretical maximum for a 6 % nominal rate
r = 4 · ((1.07)1/4 − 1) = 4 · 0.017059 = 6.8236 % / yr
A quarterly-compounded loan quoted at 6.82 % nominal has the same cost as a 7 % annual one.
Frequently Asked Questions
In the United States, APR (Annual Percentage Rate) is a legally defined term under the Truth in Lending Act and includes not just the interest rate but also certain fees and charges — origination fees, points, mortgage insurance, and the like. It is meant to be a comprehensive cost-of-credit disclosure. EAR (Effective Annual Rate) is a purely mathematical construct that converts a nominal rate and a compounding schedule into a single effective annual rate, without regard to fees. For a loan with no fees, APR and EAR are the same number when the APR is quoted on an effective annual basis. For loans with fees, APR is higher because it amortises the fees over the loan. Outside the US, regulators often use "AER" (Annual Equivalent Rate) or "EAY" (Effective Annual Yield) for the purely mathematical version and reserve "APR" for the fee-inclusive version, though the terminology varies by country. Always check what definition a quoted rate uses before comparing products.
Because compounding credits interest periodically, and each interest payment starts earning interest itself for the remainder of the year. The compounding effect means the effective annual growth is larger than what a simple r × 1 year calculation would suggest. The exact size of the difference depends on both the rate and the frequency: it is small at low rates (0.17 pp at 6 % monthly) and grows quadratically with the rate (2.36 pp at 24 % daily). Only when n = 1 (annual compounding) does the periodic rate apply exactly once per year, so that EAR = r exactly. For every other finite n > 1, EAR > r. And for any finite n, EAR < er − 1, the continuous-compounding ceiling.
Because the function f(n) = (1 + r/n)ⁿ is monotonically increasing in n and converges to er as n → ∞. Adding more compounding events always helps (each one credits interest a little sooner), but the improvements shrink rapidly — each doubling of n closes only half the remaining gap to the limit. The mathematical identity behind this is that (1 + r/n)ⁿ → er as n → ∞, which is one of the standard definitions of the number e. So er − 1 is the theoretical maximum effective annual rate achievable from a nominal rate r under any compounding scheme, discrete or continuous.
Every time you compare two financial products. A savings account paying 5 % compounded daily yields more than one paying 5.05 % compounded annually, even though the second has a higher nominal rate. A mortgage quoted at 5.5 % with monthly compounding costs more than one at 5.5 % with annual compounding. Credit cards are especially important: a card quoted at "18 % APR" compounded daily is really 19.72 % effective — nearly 1.7 pp higher. The general rule: use EAR to compare any two rates that differ in compounding frequency, and use EAR when forecasting growth over multiple years with a periodic reinvestment assumption. The nominal rate alone is not enough to know the true cost or yield.
Yes — this is precisely what EAR is designed for. If you know the effective annual rate, you can compute the balance after any number of years as FV = PV × (1 + EAR)ⁿ, where n is the number of years. This treats each year as a single compounding period at the effective annual rate, which correctly reproduces the same final balance as the original monthly (or daily) compounding schedule over full years. It does not work for fractional years if the original schedule compounds more frequently than annually — for those you must use the original periodic rate and the exact number of periods. Over whole years, though, EAR is the correct rate to use for compound growth calculations, which is why it appears in NPV, IRR, and loan-amortisation formulas throughout finance.
The continuous-equivalent rate (rcc) is the continuous-compounding rate that produces the same EAR as your actual compounding schedule: rcc = ln(1 + EAR). It is always smaller than the nominal rate — for example, 6 % nominal compounded monthly produces an EAR of 6.1678 %, whose continuous-equivalent is ln(1.061678) = 5.9855 %. The continuous rate is the natural "force of interest" and is used extensively in derivatives pricing (the Black–Scholes model), in academic finance, and in mathematical treatments of compound growth because it converts multiplicative compounding into additive linear arithmetic: over time t, a continuous rate rcc multiplies wealth by er_cc · t, and log-returns become linearly additive. Convert your discrete EAR to the continuous-equivalent rate whenever you need the cleanest analytical form.
At low rates the spread is small, but the direction is always the same — more frequent compounding is always better for savings and worse for debt. At 3 % nominal, the spread from annual to continuous is about 0.046 pp, which on a $10,000 balance is $4.60 per year — trivial. At 6 % it is $16.78 per year — noticeable. At 18 % (typical credit card) it is $1.56 per year on a $1,000 balance but $156 per year on a $10,000 balance, and over a decade the compounding difference compounds further. At 30 % (payday-lending rates) the spread between annual and daily compounding is nearly 4 pp — a massive difference. So the general principle is: at low rates the frequency choice barely matters, and at high rates it matters a great deal. Always compare effective rates when rates are high or compounding differences are large.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| EAR | Effective annual rate | |
| r | Nominal annual interest rate | |
| n | Number of compounding periods per year |
What it means
The Effective Annual Rate (EAR) reflects the actual annual interest rate when compounding occurs more than once per year. It is calculated from the nominal rate r and the number of compounding periods n per year. EAR is higher than the nominal rate if n>1. It is used to compare financial products with different compounding frequencies, such as savings accounts, loans, and credit cards. It ensures that investors can make apples‑to‑apples comparisons. Understanding EAR is crucial for making informed financial decisions and for understanding the true cost of borrowing or return on saving.
Worked example
Effective Annual Rate – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| Nominal Rate | 18% |
| Compounding (n/year) | 12 |
| Parameter | Value |
|---|---|
| Nominal Rate | 5% |
| Compounding | 365 |
Common mistakes
- Stated rate r: The nominal annual interest rate (APR) – in decimal form.
- Compounding periods n: The number of compounding periods per year.
- EAR: The effective annual rate – accounts for compounding within the year.
- EAR is always ≥ APR (for n>1) – equal only if n=1.
- Continuous compounding: For continuous, EAR = e^r − 1.
Applications
The Effective Annual Rate (EAR) adjusts the nominal annual rate for compounding within the year, giving the true annual cost of borrowing or return on investment. EAR is crucial for comparing financial products with different compounding periods. Banks and consumers use it to evaluate credit cards, loans, and savings accounts. By converting all rates to EAR, one can make an apples‑to‑apples comparison. This formula is also used in corporate finance to evaluate financing options. Understanding EAR helps avoid underestimating the cost of debt and ensures that investment returns are accurately measured. It is a fundamental concept in personal and corporate finance, ensuring transparency and fair comparisons.
- Comparison of credit card APRs with different compounding
- Selection of savings accounts and certificates of deposit
- Corporate borrowing – choosing among loan offers
- Investment performance reporting and benchmarking
- Regulatory compliance – truth in lending disclosures
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