Payment from FV
Calculates the discrete annual end-of-period payment needed to accumulate a target future value when the rate is continuously compounded.
When to use: Use to size annual deposits toward a savings goal when the rate is quoted in continuously-compounded form.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PMT | Payment | Periodic payment amount | $ |
| FV | Future Value | Future 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 Savings Goal
Accumulate $500,000 in 25 years at 7% continuous compounding.
Given
Step-by-Step
Annual deposits of $7,625.05 reach $500,000 over 25 years.
Example 2: Academic Example
Reach $100,000 in 10 years at 5% continuous compounding.
Given
Step-by-Step
Annual deposits of $7,903.41 reach the $100,000 target.
Frequently Asked Questions
A discrete annual end-of-year deposit, sized so the deposits compounded at a continuously-compounded rate accumulate to the target future value.
For the same nominal rate, continuous compounding implies a higher effective annual rate, so each deposit grows faster — the required deposit is slightly LOWER than the discrete-annual version.
This formula is the algebraic inverse of the continuous FVA formula. If FVA = PMT × (e^(kn) - 1) / (e^k - 1), then PMT = FV × (e^k - 1) / (e^(kn) - 1).