PV of Perpetuity Due
Present value of infinite beginning-of-period payments with continuous compounding.
When to use: Use in academic models when perpetual payments are made at the start of each period with continuous discounting.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PV | Present Value | Value of the perpetuity due | $ |
| PMT | Payment | Periodic payment | $ |
| k | Interest Rate | Nominal annual rate | % |
Real-Life Examples
Example 1: Continuous Perpetuity Due
$8,000/year paid at start of year forever, 5% continuous compounding.
Given
Step-by-Step
The perpetuity due is worth $163,967.04 with continuous compounding.
Example 2: Academic Example
$2,000/year at start forever, 8% continuous compounding.
Given
Step-by-Step
The perpetuity due is worth $26,011.38.
Frequently Asked Questions
Multiply the continuous ordinary perpetuity value by e^k to account for each payment being received one period earlier. Equivalently, PV = PMT × e^k / (e^k - 1).
The perpetuity due is worth e^k times the ordinary perpetuity. At 5% continuous, this is about 5.13% more. The first payment is received immediately, which adds significant value over an infinite horizon.
The annual version uses PV = PMT + PMT/k = PMT(1+k)/k, while the continuous version uses PMT × e^k / (e^k - 1). At typical rates, the continuous version gives a slightly lower value because continuous compounding produces a higher effective discount rate.