PV of Perpetuity
Present value of an infinite stream of equal payments with continuous compounding.
When to use: Use in academic continuous-time models for perpetuity valuation.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PV | Present Value | Value of the perpetuity today | $ |
| PMT | Payment | Periodic payment amount | $ |
| k | Interest Rate | Nominal annual rate | % |
Real-Life Examples
Example 1: Continuous Perpetuity
An investment pays $5,000/year forever. Rate is 6% with continuous compounding. What is it worth?
Given
Step-by-Step
The perpetuity is worth $80,849.08 with continuous compounding.
Example 2: Academic Example
$1,000/year forever at 10% continuous compounding.
Given
Step-by-Step
The perpetuity is worth $9,508.42 with continuous compounding.
Frequently Asked Questions
With continuous compounding, the effective rate per period is e^k - 1 instead of k. Since e^k - 1 > k, the discount rate is higher, making the perpetuity worth slightly less than the annual compounding version. For example, at 6%, the continuous value is about PMT/0.0618 vs. PMT/0.06 for annual.
The term e^k - 1 is the effective annual rate under continuous compounding. For discrete annual payments discounted with continuous compounding, each payment's present value uses e^(-kt), and the infinite geometric sum yields PMT/(e^k - 1).
In academic finance, theoretical modeling, and as a mathematical reference point. In practice, the annual formula PV = PMT/k is more commonly used, and the difference is small at typical interest rates.