APR from EAR
Backs out the continuously-compounded nominal rate equivalent of a given effective annual rate.
When to use: Use when you have an effective annual yield and need the equivalent continuously-compounded rate — common in options pricing, fixed-income analytics, and academic research.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| APR | Continuous Rate | Continuously-compounded nominal rate | % |
| EAR | Effective Annual Rate | True annual return | % |
Real-Life Examples
Example 1: Effective to Continuous
An asset returns 6.18% effective annually. What is the equivalent continuously-compounded rate?
Given
Step-by-Step
6.18% effective annual is equivalent to 6.00% continuously compounded — the inverse of the EAR-from-continuous example.
Example 2: High-Volatility Asset
A volatile asset realizes 16.18% effective annually. Translate to a continuously-compounded rate.
Given
Step-by-Step
16.18% EAR is equivalent to 15% continuously compounded.
Frequently Asked Questions
Continuous rates are mathematically convenient — they add across time periods (a 5% continuous rate over 2 years compounds to e^(0.05×2) = 10.52%, not (1.05)^2). Options pricing, yield-curve modeling, and most quantitative finance use continuous rates internally even when inputs are quoted otherwise.
Yes. For a given EAR, continuous compounding produces the lowest nominal rate because each marginal unit of time is doing maximum compounding work. As m increases from 1 (annual) to ∞ (continuous), the implied APR decreases monotonically.
It is the inverse of e^x. Since EAR = e^k - 1 for continuous compounding, solving for k gives k = ln(1 + EAR). The natural log is the discrete counterpart of continuous-time growth.