Monthly rate to annual rate: APR vs EAR
When a lender or credit card quotes a monthly interest rate, there are two ways to express it as an annual figure. The choice between them is not cosmetic — it determines whether you are looking at the nominal cost or the true economic cost of the money, and the difference can be significant at high monthly rates.
The APR (Annual Percentage Rate) is the simple proportional annual rate: the monthly rate multiplied by 12. It is the rate most commonly disclosed in consumer credit agreements because regulators require a standardised nominal rate for comparison purposes. A 1% monthly rate gives a 12% APR. The EAR (Effective Annual Rate), also called AER on savings products, accounts for the compounding effect: each month's interest is added to the balance and itself earns interest in subsequent months. A 1% monthly rate gives a 12.683% EAR. The 0.683 percentage point gap is the cost of compounding that APR does not capture.
The formulas
APR (simple annual) = monthly rate × 12
EAR (compound annual) = (1 + monthly rate)^12 − 1
Quarterly rate (compound) = (1 + monthly rate)^3 − 1
Semi-annual rate (compound) = (1 + monthly rate)^6 − 1
Daily rate (simple) = APR ÷ 365
Daily rate (compound) = (1 + monthly rate)^(1/30.4375) − 1
All rates are in decimal form inside the formula. For a 1% monthly rate r = 0.01. EAR = (1.01)^12 − 1 = 0.12683 = 12.683%. The daily compound formula uses 30.4375 days per month (365 ÷ 12).
Reference table: monthly rates to annual equivalents
| Monthly rate | APR (simple) | EAR (compound) | Uplift | Quarterly (compound) | Daily (simple) |
|---|
| 0.25% | 3.00% | 3.042% | 0.042 pp | 0.752% | 0.00822% |
| 0.50% | 6.00% | 6.168% | 0.168 pp | 1.508% | 0.01644% |
| 0.75% | 9.00% | 9.381% | 0.381 pp | 2.267% | 0.02466% |
| 1.00% | 12.00% | 12.683% | 0.683 pp | 3.030% | 0.03288% |
| 1.50% | 18.00% | 19.562% | 1.562 pp | 4.568% | 0.04932% |
| 2.00% | 24.00% | 26.824% | 2.824 pp | 6.121% | 0.06575% |
| 3.00% | 36.00% | 42.576% | 6.576 pp | 9.273% | 0.09863% |
Why the uplift grows non-linearly
At 0.5% monthly the APR-to-EAR gap is just 0.168 percentage points, easily dismissed as negligible. At 2% monthly the gap is 2.824 points. At 3% monthly it reaches 6.576 points. This exponential growth happens because the compounding formula raises (1 + r) to the 12th power, making the result increasingly sensitive to the value of r at higher rates. It is precisely at the high rates — credit cards, payday loans, store finance — where the APR figure is most misleading and the EAR most important to know.
Using the monthly rate in loan calculations
For loan amortisation — calculating monthly payments using the standard formula M = P × r × (1+r)^n / ((1+r)^n − 1) — the correct rate to use is the simple monthly rate, which is APR ÷ 12. This is because the formula already builds in the compounding effect through the (1+r)^n terms. Using the compound monthly equivalent (derived from EAR) instead would double-count the compounding and produce incorrect payments. The simple monthly rate and the compound monthly rate converge at low rates but diverge significantly at high rates: at 2% monthly compound equivalent from a 26.824% EAR, the compound monthly is 1.989%, not 2.000%.
Worked examples
Example 1: Credit card at 1.5% per month
A credit card charges 1.5% per month. APR = 1.5% × 12 = 18.00%. EAR = (1.015)^12 − 1 = 19.562%. On a $3,500 balance carried for a full year without payment, simple interest would total $630 (18% × $3,500) while compound interest totals $684.67 (EAR × $3,500). The $54.67 gap is the cost of monthly compounding that the APR figure does not show. Over multiple years the gap widens further.
Example 2: Payday loan at 3% per month
A short-term loan charges 3% per month. APR = 36%. EAR = (1.03)^12 − 1 = 42.576%. The 6.576 percentage point gap between APR and EAR is the largest of any common lending product. On $1,000 over one year: simple (APR) = $360 interest; compound (EAR) = $425.76 interest. A borrower rolling over this loan monthly pays the EAR, not the APR, regardless of what the headline figure states.
Example 3: Savings account paying 0.5% per month
A savings account credits 0.5% interest monthly. APR = 6.00%. AER (equivalent to EAR) = (1.005)^12 − 1 = 6.168%. The account statement will show 6.168% AER, not 6.00%, because AER is the regulatory disclosure for UK savings products and reflects the full compounding benefit. A saver depositing $10,000 earns $616.78 after one year (6.168% × $10,000), not $600 as a simple 6% would suggest.
Example 4: Reverse conversion — finding the monthly rate from an annual target
To find the monthly rate that produces a given annual EAR: monthly rate = (1 + EAR)^(1/12) − 1. For a 12.683% EAR: monthly = (1.12683)^(1/12) − 1 = 1.000% exactly. For a 6.168% EAR: monthly = (1.06168)^(1/12) − 1 = 0.500%. This confirms the round-trip consistency of the formulas. Use the APR vs Nominal Rate Calculator for this reverse direction.
| Scenario | Monthly rate | APR | EAR | Interest on $10,000 / year (compound) |
|---|
| Savings account | 0.50% | 6.00% | 6.168% | $616.78 |
| Personal loan | 1.00% | 12.00% | 12.683% | $1,268.25 |
| Credit card | 1.50% | 18.00% | 19.562% | $1,956.18 |
| High-rate card | 2.00% | 24.00% | 26.824% | $2,682.42 |
| Payday loan | 3.00% | 36.00% | 42.576% | $4,257.61 |
Frequently Asked Questions
Why is EAR always higher than APR for the same monthly rate?+
APR is calculated by simply multiplying the monthly rate by 12, ignoring the fact that each month's interest is added to the balance and itself earns interest in subsequent months. EAR captures this snowball effect. At 1% monthly, the first month earns 1% on the original principal. In month two the balance is 1.01 times larger, so it earns 1% on a higher base. After 12 months the total growth is 12.683% of the original, not 12%. The higher the monthly rate, the larger the gap between APR and EAR because the compounding formula (1 + r)^12 is increasingly sensitive to r at higher values. At 0.5% monthly the gap is 0.168 pp; at 3% monthly it is 6.576 pp.
Which rate do lenders and credit cards actually use?+
In the EU and UK, consumer credit lenders are required by regulation to disclose an APR, which is the simple nominal annual rate (monthly rate × 12). This is the lower of the two figures and makes the product look cheaper than the true compound cost. Savings accounts in the UK must quote the AER (Annual Equivalent Rate), which is identical to EAR and is the higher of the two. The result is a deliberate asymmetry: borrowers see the lower figure (APR) and savers see the higher figure (AER). When you see a credit card advertising an 18% APR, the true annual cost if you carry a balance month-to-month is closer to 19.56% EAR.
Should I use APR or EAR to compare two credit cards?+
If both cards compound monthly (which is standard), comparing APR figures is valid because the compounding frequency is the same. APR A versus APR B is a fair comparison when both use monthly compounding. EAR becomes essential when comparing products with different compounding frequencies, or when comparing a loan APR against a savings AER, since these use different calculation conventions. For a quick everyday comparison between two credit cards both compounding monthly, APR is fine. For understanding the true annual cost of a specific card independently, use EAR.
What monthly rate do I need to achieve a target annual rate?+
To find the compound-equivalent monthly rate for a given EAR: monthly rate = (1 + EAR)^(1/12) − 1. For example, to achieve 12% EAR: monthly = (1.12)^(1/12) − 1 = 0.9489%, not 1.000%. To find the simple monthly rate for a given APR: monthly = APR ÷ 12. For 12% APR: monthly = 1.000%. The compound method gives a slightly lower monthly rate because the compounding of 12 periods naturally amplifies it to hit the annual target. Use the APR vs Nominal Rate Calculator for a full bidirectional conversion with all compounding frequencies.
How does this calculator differ from the Interest Rate Converter?+
This calculator is specifically focused on converting a monthly rate to all other common periods (daily, quarterly, semi-annual, annual) and produces a month-by-month compounding table showing exactly how interest accumulates. The Interest Rate Converter is more general: it accepts any input period (daily, weekly, bi-weekly, monthly, quarterly, semi-annual, annual) and converts to all other periods in a single table, with both simple and compound modes. Use this calculator when you have a monthly rate and want quick answers. Use the Interest Rate Converter when you need to convert between any two non-monthly periods.
Why does the compounding table show different interest amounts each month?+
Each month's interest is calculated on the closing balance of the previous month, not on the original principal. Because the previous month's interest is added to the balance, the base for the next month's calculation is slightly larger. At 1% monthly on $10,000: month 1 earns $100 (1% of $10,000). Month 2 earns $101 (1% of $10,100). Month 3 earns $102.01 (1% of $10,201). After 12 months the cumulative interest is $1,268.25, not $1,200. This is precisely the difference between the EAR of 12.683% and the APR of 12.000%, made visible row by row.