Payment from PV
Calculates the discrete annual end-of-period payment needed to amortize a present value when the rate is continuously compounded.
When to use: Use to size annual payments on a loan or withdrawal stream when the rate is quoted in continuously-compounded form.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PMT | Payment | Periodic payment amount | $ |
| PV | Present Value | Current lump sum value | $ |
| k | Interest Rate | Nominal annual interest rate as a decimal | % |
| n | Number of Years | Time period in years | years |
Real-Life Examples
Example 1: Continuous Loan Payment
A $100,000 obligation at 6% continuous compounding over 20 years.
Given
Step-by-Step
Annual end-of-year payments of $8,848.89 amortize the $100,000 obligation.
Example 2: Academic Example
$50,000 at 8% continuous compounding over 10 years.
Given
Step-by-Step
Annual payments of $7,562.33 amortize the $50,000 over 10 years.
Frequently Asked Questions
The discrete-rate annual formula uses k directly in the annuity factor. This continuous-compounding form replaces (1+k)^-t with e^-(kt) and the annuity factor with [1-e^(-kn)]/(e^k-1). For typical rates (5-10%), the difference is small but real.
For the same nominal rate, continuously-compounded discounting implies a slightly higher effective annual rate, so the required payment is slightly higher. The gap is a few basis points for rates under 10%.
Primarily in academic finance, actuarial science, and theoretical models, or when the rate is naturally quoted in continuously-compounded form (some yield-curve models). For practical loan calculations, the m-times formula with m=12 is the standard.