Payment from PV
Beginning-of-period annual payment that amortizes a present value when the rate is continuously compounded.
When to use: Use to size annual annuity-due 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 (paid at start of period) | $ |
| PV | Present Value | Current lump sum value | $ |
| k | Interest Rate | Nominal annual interest rate | % |
| n | Number of Years | Time period in years | years |
Real-Life Examples
Example 1: Continuous Due Payment
$80,000 obligation at 6% continuous compounding, 10-year term, payments at start.
Given
Step-by-Step
Beginning-of-period payments are $10,325.70 with continuous compounding.
Example 2: Academic Example
$25,000 at 5% continuous, 8 years, payments at start of period.
Given
Step-by-Step
The beginning-of-period payment is $3,698.33.
Frequently Asked Questions
It is the inverse of the PVA due continuous formula. Divide the ordinary continuous PMT by e^k to adjust for beginning-of-period timing, which reduces the required payment since each deposit earns interest longer.
Because each payment is made at the beginning of the period, it earns interest for one extra period. This extra earning means you need less per payment to achieve the same result.
This formula is primarily theoretical, used in academic finance and actuarial science. Practical lease and loan calculations use the m-times formula with the appropriate payment frequency.