PV of Growing Annuity
Present value of a growing payment stream with continuous compounding.
When to use: Use in academic continuous-time models for valuing growing cash flow streams.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PVA | Present Value of Annuity | Present value of all growing payments | $ |
| PMT | First Payment | The first payment amount (or continuous payment rate) | $ |
| k | Interest Rate | Nominal annual discount rate | % |
| g | Growth Rate | Continuous growth rate of payments | % |
| n | Number of Years | Time horizon in years | years |
Real-Life Examples
Example 1: Continuous Growing Cash Flow
A project generates $50,000/year growing at 3%, discounted at 8% continuously for 20 years.
Given
Step-by-Step
The growing cash flow stream is worth $632,120.56 with continuous compounding.
Example 2: Academic Example
$10,000/year growing at 2%, 6% continuous discounting for 15 years.
Given
Step-by-Step
The present value is $112,795.22 with continuous compounding.
Frequently Asked Questions
It values a stream of payments growing at rate g and discounted at rate k, both applied continuously. The formula PMT × (1 - e^(-(k-g)n)) / (k-g) is the continuous-time analog of the discrete growing annuity formula.
When k = g, the formula simplifies to PMT × n. The growth and discounting exactly offset, so the present value is simply the payment rate times the number of years.
The continuous version uses exponential functions instead of power functions. At the same nominal rates, continuous compounding produces a slightly lower present value because the effective discount rate is higher.