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Equivalent Interest Rate Calculator
Convert interest rates between different compounding frequencies. Find the equivalent APR for annual, monthly, daily, or continuous compounding.
Formula:i = q × [(1 + EAR)^(1/q) - 1]
Equivalent Rate
5.1162%
5% Monthly (12/year) = 5.1162% Annually (1/year)
Effective Annual Rate
Original EAR:5.1162%
Target EAR:5.1162%
Proof of Equivalence
Both rates produce the same effective annual rate:
Original:5.116190%
Converted:5.116190%
Difference:0.0000000000%
Rates are mathematically equivalent
Investment Results
Both compounding methods yield the same result:
Monthly (12/year)
$12833.59
Annually (1/year)
$12833.59
Rate Conversion
Enter your interest rate and select compounding frequencies
%
Enter the nominal annual rate
How often interest compounds
Convert to this frequency
Equivalent Rate
5.1162%
annually compounding
Effective Annual Rate
5.1162%
True annual return
Original Rate
5%
monthly compounding
Rate Difference
0.1162%
Nominal rate change
Equivalent Nominal Rates by Compounding Frequency
Formula Reference
Effective Annual Rate (EAR)
EAR = (1 + r/m)^m - 1
Where r is the nominal rate and m is compounding periods per year.
Equivalent Nominal Rate
i = q x [(1 + EAR)^(1/q) - 1]
Where q is the target compounding frequency.
Continuous Compounding
EAR = e^r - 1
For continuous compounding, use the exponential function.
Equivalent Rate
5.1162%
5% Monthly (12/year) = 5.1162% Annually (1/year)
Effective Annual Rate
Original EAR:5.1162%
Target EAR:5.1162%
Proof of Equivalence
Both rates produce the same effective annual rate:
Original:5.116190%
Converted:5.116190%
Difference:0.0000000000%
Rates are mathematically equivalent
Investment Results
Both compounding methods yield the same result:
Monthly (12/year)
$12833.59
Annually (1/year)
$12833.59
Quick Answer: How to Convert Interest Rates
To convert interest rates between compounding frequencies, first calculate the Effective Annual Rate (EAR) using EAR = (1 + r/m)^m - 1, where r is the nominal rate and m is periods per year. Then find the equivalent nominal rate for the target frequency using i = q x [(1 + EAR)^(1/q) - 1]. For example, 5% monthly compounding equals approximately 5.116% annually or 4.889% quarterly.
Key Facts
- Effective Annual Rate (EAR) represents the true annual return accounting for compounding
- More frequent compounding results in higher effective rates for the same nominal rate
- A 5% rate compounded monthly equals approximately 5.116% compounded annually
- Continuous compounding uses the formula EAR = e^r - 1 for the theoretical maximum
- APR (Annual Percentage Rate) is a nominal rate; APY (Annual Percentage Yield) equals EAR
- Daily compounding is very close to continuous compounding in practice
- Converting rates preserves the effective annual rate across different frequencies
What if the interest rate changed?
See how different nominal rates convert across compounding frequencies
0.5%5%25%
Personalized Insights
2 insights based on your inputs
Compounding Boost
Your 5% rate with monthly compounding yields an effective 5.12% annually.
Rate Conversion Matters
The equivalent annually rate is 5.1162% - a 0.12% nominal difference.
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Frequently Asked Questions
APR (Annual Percentage Rate) is the nominal interest rate without accounting for compounding. APY (Annual Percentage Yield) is the effective rate that includes the effect of compounding. APY is always equal to or higher than APR. For example, a 5% APR compounded monthly results in a 5.116% APY.
Banks choose compounding frequencies based on their products. Savings accounts often compound daily to attract depositors (higher effective yield), while mortgages compound monthly by convention. Treasury bonds typically compound semi-annually. The frequency affects the true cost or return of the financial product.
More frequent compounding results in higher effective returns because interest earns interest more often. For example, $10,000 at 5% for 10 years yields: annually = $16,289, monthly = $16,470, daily = $16,487. The difference grows with higher rates and longer periods.
Continuous compounding is the theoretical limit where interest compounds infinitely often. It uses the formula A = Pe^(rt), where e is Euler's number (approximately 2.71828). While no real financial product uses true continuous compounding, it's used in financial modeling and options pricing.
To convert a monthly compounded rate to an annual rate: First calculate EAR = (1 + r/12)^12 - 1, where r is the nominal annual rate. Then the equivalent annual rate equals the EAR. For example, 6% monthly compounding equals 6.168% annual compounding.
Equivalent rates appear different because the same effective annual rate requires different nominal rates depending on compounding frequency. Less frequent compounding needs higher nominal rates to achieve the same effective rate. This is why the nominal rate decreases as compounding becomes more frequent.
The Effective Annual Rate (EAR) is the true annual interest rate that accounts for compounding within the year. It allows fair comparison between products with different compounding frequencies. EAR equals (1 + r/m)^m - 1, where r is the nominal rate and m is the number of compounding periods per year.
Daily compounding (365 times per year) is extremely close to continuous compounding. For a 5% rate, daily compounding yields 5.1267% effective while continuous yields 5.1271%. The difference is only 0.0004 percentage points, making daily compounding practically equivalent to continuous for most purposes.
Joseph OrdunaFounder & Software Engineer
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).
Equivalent Rate
5.1162%
5% Monthly (12/year) = 5.1162% Annually (1/year)
Effective Annual Rate
Original EAR:5.1162%
Target EAR:5.1162%
Proof of Equivalence
Both rates produce the same effective annual rate:
Original:5.116190%
Converted:5.116190%
Difference:0.0000000000%
Rates are mathematically equivalent
Investment Results
Both compounding methods yield the same result:
Monthly (12/year)
$12833.59
Annually (1/year)
$12833.59