Equivalent Rate Calculator
Convert a nominal annual rate from one compounding frequency to another's equivalent rate, and compute the AER.
Input Data
Results
At a glance:The Equivalent Rate converts the same real annual return from one compounding frequency's nominal rate to another's: i = q × ((1 + r/m)^(m/q) − 1), with r nominal, m original, q target; it also gives AER = (1 + r/m)^m − 1. The AER is unchanged by conversion — only the compounding frequency used to express the same real return differs.
Formula
Equivalent nominal rate i = q × ((1 + r/m)^(m/q) − 1).
AER = (1 + r/m)^m − 1 (discrete) or e^r − 1 (continuous original).
When m = q, the equivalent rate equals the nominal rate.
$$$i = q\\left(\\left(1+\\dfrac{r}{m}\\right)^{m/q}-1\\right)$$$$$$AER = \\left(1+\\dfrac{r}{m}\\right)^{m}-1$$$$$(5%→)$i\\approx5.0209\\%$$AER\\approx5.1162\\%$$$How to Use
- Enter the nominal annual rate.
- Select its original compounding frequency.
- Select the target frequency to convert to, and get the equivalent nominal rate and AER.
Nominal 5% (monthly) converted to various target frequencies
| Original freq | Target freq | Equivalent nominal | AER |
|---|---|---|---|
| Monthly (12) | Quarterly (4) | 5.0209% | 5.1162% |
| Monthly (12) | Annual (1) | 5.1162% | 5.1162% |
| Semi-annual (2) @6% | Monthly (12) | 5.9263% | 6.0900% |
Case Studies
Case 1: Convert monthly to quarterly
Product nominal 5% compounded monthly (m=12). Convert to quarterly (q=4) equivalent nominal rate.
i = 4 × ((1 + 0.05/12)^(12/4) − 1) = 4 × ((1.0041667)^3 − 1) ≈ 5.0209%; AER = (1 + 0.05/12)^12 − 1 ≈ 5.1162%.
Interpretation: monthly 5% expressed quarterly is about 5.0209% — the same real return (AER 5.1162%), just a different frequency and slightly different nominal number. Conversion puts different-frequency products on the same basis for fair comparison.
Case 2: AER stays unchanged after conversion
Same example: nominal 5% monthly converted to quarterly (5.0209%) and annual (5.1162%). Quarterly AER = (1 + 0.050209/4)^4 − 1 ≈ 5.1162%; annual AER = 5.1162% (annual compounding: nominal = AER).
Three different nominals (5% monthly, 5.0209% quarterly, 5.1162% annual) all give the same AER 5.1162%.
Interpretation: conversion only changes the frequency expressing the same real return; AER never changes. Facing quotes like '5% monthly' vs '5.1% quarterly', convert them (or compare AER) to see the real return and avoid nominal-number traps. Converting to annual makes the equivalent nominal exactly equal AER. Continuous original uses the e-based formula (AER = e^r − 1).
FAQ
Why convert equivalent rates?
Products compound at different frequencies (monthly vs quarterly). Converting them to the same frequency's equivalent rate lets you compare the real return or cost fairly.
What is the Annual Effective Rate (AER)?
AER is the real annual return after compounding at the given frequency — it reflects the compounding effect. The higher the frequency, the more AER exceeds the nominal rate.
Does the AER change after conversion?
No. Equivalent-rate conversion keeps the AER unchanged — it only expresses the same real annual return at a different compounding frequency, so AER is identical before and after.
Why keep AER unchanged, and what is it for?
The core principle of equivalent-rate conversion is 'keep the AER unchanged' — because AER is the product's true annual return; conversion only changes the frequency used to express that same return, not the return itself. Think of one sum of money with one real annual growth: you can call it '5% monthly', '5.0209% quarterly' or '5.1162% annual' — three descriptions of the exact same thing, like one distance in km, m or miles. If the AER changed after conversion, it would not be 'equivalent' but a different return, defeating the purpose. Uses: (1) most importantly, fairly compare products at different frequencies — product A '5% monthly' vs B '5.05% quarterly' cannot be compared by nominal numbers alone; convert to the same frequency (or compare AER) to see which is really higher. (2) Provide the frequency a specific calculation needs (cash-flow discounting) without changing the real return. (3) Understand the true return behind a quote — the same AER can be expressed by countless nominal rates; always return to AER as the real benchmark. So equivalent rate is a practical conversion tool, and 'AER unchanged' is its foundation.
How is continuous-compounding equivalent rate computed, and how different is it?
Continuous compounding is the limit as frequency → infinity (compounding every instant), using the natural constant e. If the original rate is continuous nominal r, the equivalent nominal rate at q times per year is i = q × (e^(r/q) − 1); its AER = e^r − 1 (not (1+r/m)^m − 1). Example: nominal 5% continuous, convert to quarterly (q=4): i = 4 × (e^(0.05/4) − 1) = 4 × (e^0.0125 − 1) ≈ 5.0314%; AER = e^0.05 − 1 ≈ 5.1271%. Difference from normal compounding? Minimal. At nominal 5%: continuous AER ≈ 5.1271%, daily (m=365) ≈ 5.1267%, monthly ≈ 5.1162% — daily already nearly reaches the continuous limit, differing only by a few ten-thousandths. Compounding has a mathematical ceiling; no frequency exceeds continuous by much, and the gain shrinks as frequency rises. Continuous compounding is mainly for mathematical finance, option pricing (Black-Scholes), rates derivatives — consumers rarely see beyond daily compounding, which is already near continuous. This calculator's 'original frequency' offers a 'continuous' option that switches to the e-based formula.
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References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.