Rate from Annuity
Find the continuously-compounded rate implied by an annuity of discrete annual payments. Solved numerically via Newton-Raphson.
When to use: Use in continuous-time finance contexts when payments are still annual but the rate is quoted in continuously-compounded form. Mostly academic; for practical loan APRs use the m-times variant.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| k | Continuous Rate | Continuously-compounded annual rate (found iteratively) | % |
| PV | Present Value | Loan amount or annuity price | $ |
| PMT | Payment | Annual payment | $ |
| n | Number of Years | Number of years | years |
Real-Life Examples
Example 1: Continuous-Discount Annuity
A $50,000 obligation is repaid with $8,000 annually for 10 years under a continuously-compounded rate. What rate is implied?
Given
Step-by-Step
The implied continuously-compounded rate is 9.18%. The equivalent EAR is e^0.0918 − 1 ≈ 9.61% — matching the annual-compounding result on the same cash flows, as expected.
Example 2: Academic Loan Comparison
$20,000 borrowed, repaid $4,000/year for 6 years under continuous discounting.
Given
Step-by-Step
The continuously-compounded rate is 5.33%, slightly below the 5.47% annual-compounding rate for the same cash flows (since continuous compounding implies a higher EAR per unit of k).
Frequently Asked Questions
The continuous rate is always slightly below the annual rate. Reason: at the same effective annual return, continuous compounding implies a smaller k because more compounding "work" happens per unit of k. The two are linked by k_continuous = ln(1 + k_annual).
Mostly in academic finance, actuarial models, and quantitative work where rates are naturally expressed continuously. Black-Scholes-style settings sometimes need a continuously-compounded rate inferred from discrete cash flows.
For sensible inputs (PMT × n > PV, positive rates, reasonable bounds) it converges in under 20 iterations. Pathological inputs (e.g. PMT × n ≤ PV with no positive solution) may diverge — the safety guard at k ≤ −1 catches these and returns NaN.