PV of Annuity Due
Present value of an annuity due (discrete annual beginning-of-period payments) discounted at a continuously-compounded rate.
When to use: Use when payments arrive at the start of each year and the rate is quoted in continuously-compounded form.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PVA | Present Value of Annuity Due | Total present value of beginning-of-period payments | $ |
| PMT | Payment | Periodic payment amount (paid at start of period) | $ |
| k | Interest Rate | Nominal annual interest rate | % |
| n | Number of Years | Time period in years | years |
Real-Life Examples
Example 1: Continuous Due Valuation
$10,000/year paid at start of year for 15 years, 5% continuous discounting.
Given
Step-by-Step
The annuity due is worth $108,186.84 with continuous discounting.
Example 2: Academic Example
$3,000/year at start for 10 years, 7% continuous discounting.
Given
Step-by-Step
The present value of the annuity due is $22,338.85.
Frequently Asked Questions
Take the continuous ordinary PVA formula and multiply by e^k. This accounts for each payment being received one period sooner, making it worth more in present value terms.
Because each payment arrives at the start of the period rather than the end, it is discounted one fewer period. The e^k multiplier reflects this timing advantage under continuous compounding.
The continuous version produces a slightly different result because continuous discounting is marginally stronger. The annual version uses (1+k) as the due factor while continuous uses e^k.