Equivalent Rate Calculator – AER

Convert a nominal interest rate compounded at one frequency into its Annual Equivalent Rate (AER), or into an equivalent rate at any other compounding frequency, using AER = (1 + r/n)^n − 1.

Quick Facts

Formula
AER = (1 + r/n)^n − 1
r is the nominal annual rate as a decimal and n is the number of compounding periods per year.
Equivalent rate
r₂ = n₂ × [(1 + AER)^(1/n₂) − 1]
Restates the AER as a rate compounded n₂ times per year with identical annual growth.

Your Results

Calculated
Equivalent rate
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Annualized, at your target compounding frequency
AER (effective annual rate)
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True annual growth rate (compounded once a year)
Rate per period
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Periodic rate at the target compounding frequency
Value after 1 year
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Principal grown at the AER

Ready

Enter the nominal rate, its compounding frequency, and a target frequency, then press Calculate.

How the Equivalent Rate Calculator works

Lenders and savings providers advertise a "nominal" interest rate together with how often it compounds — monthly, quarterly, daily, and so on. Because compounding lets interest already earned start earning interest of its own before the year is out, two accounts with the same nominal rate but different compounding frequencies grow by different amounts over a year. The Annual Equivalent Rate (AER) restates any nominal rate as the single annual-compounding rate that produces the same yearly growth, so rates quoted on different compounding schedules can be compared on equal terms.

The formula

For a nominal annual rate r (as a decimal) compounded n times per year, the Annual Equivalent Rate is:

AER = (1 + r/n)n − 1

To express that same growth as a rate compounded at a different frequency n2 instead of once a year, convert the AER back down with:

r2 = n2 × [(1 + AER)1/n2 − 1]

Setting n2 = 1 in this second formula simply returns the AER — annual compounding is the special case where the "equivalent rate" and the AER are the same number.

Worked example

A savings account advertises 6% compounded monthly (n = 12). The periodic monthly rate is 6% / 12 = 0.5%, and the AER is (1 + 0.005)12 − 1 ≈ 6.1678%. A $10,000 deposit grows to $10,000 × 1.061678 ≈ $10,616.78 after one year — whether interest is credited monthly at 0.5% each time or, equivalently, once a year at 6.1678%.

What moves the AER most

  • Compounding frequency: the more often interest compounds, the higher the AER climbs above the nominal rate — daily compounding produces a slightly higher AER than monthly compounding at the same nominal rate, which in turn produces a higher AER than annual compounding.
  • Rate size: the gap between the nominal rate and the AER widens faster than the rate itself grows, because compounding applies to a larger periodic amount.
  • Target frequency: converting the AER to a higher target frequency (e.g., monthly instead of annually) lowers the quoted periodic-equivalent annual figure slightly, even though the underlying annual growth is unchanged.

Comparing quoted rates fairly

Because two products can quote the same nominal rate with different compounding frequencies, comparing headline nominal figures directly can be misleading. Converting each to its AER — or to a common target frequency — strips out the compounding-frequency effect and leaves only the actual annual growth rate for comparison. This is why AER (or an equivalent effective-rate figure) is commonly disclosed alongside the nominal rate on savings and lending products.

Frequently Asked Questions

What is the Annual Equivalent Rate (AER)?
AER is the annual interest rate you would need if interest compounded only once a year to produce the same growth as a nominal rate that compounds more frequently. It is calculated as AER = (1 + r/n)^n − 1, where r is the nominal annual rate as a decimal and n is the number of compounding periods per year.
How do I convert a rate to a different compounding frequency?
First convert the nominal rate to its AER using (1 + r/n)^n − 1. Then convert the AER to the target frequency n2 using r2 = n2 × [(1 + AER)^(1/n2) − 1]. The two rates compound on different schedules but produce identical growth over a year.
Why is the AER higher than the nominal rate?
Whenever compounding happens more than once a year (n greater than 1), interest earned in early periods itself earns interest before the year ends, so the effective annual growth exceeds the simple nominal rate. The gap widens as compounding frequency increases and shrinks to zero when n = 1.
Does this calculator account for fees or taxes?
No. The AER and equivalent-rate formulas here are pure interest-rate mathematics based on compounding frequency alone. Account fees, early-withdrawal penalties, and taxes are not included and can change the return you actually receive.

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