PV of Growing Annuity Due
Present value of a growing beginning-of-period payment stream with continuous compounding.
When to use: Use in academic models for beginning-of-period growing cash flows under continuous compounding.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PVA | Present Value of Annuity | Present value of growing payments | $ |
| PMT | First Payment | The first payment amount | $ |
| k | Interest Rate | Nominal annual discount rate | % |
| g | Growth Rate | Continuous growth rate | % |
| n | Number of Years | Time horizon in years | years |
Real-Life Examples
Example 1: Continuous Growing Due
$20,000/year growing at 3%, paid at start, 7% continuous discounting for 15 years.
Given
Step-by-Step
The growing annuity due is worth $241,923.07 with continuous compounding.
Example 2: Academic Example
$5,000/year growing at 4%, paid at start, 10% continuous for 10 years.
Given
Step-by-Step
The present value of the growing annuity due is $41,553.34.
Frequently Asked Questions
Multiply the continuous ordinary growing annuity PV by e^k. This accounts for beginning-of-period timing, where each payment earns one extra period of continuous interest.
Under continuous compounding, one period of growth equals e^k rather than (1+k). The factor e^k is always slightly larger, reflecting the higher effective rate of continuous compounding.
When k = g, the formula simplifies to PMT × n × e^k. The growth and discounting offset in the annuity factor, leaving just the payment rate times years, adjusted for beginning-of-period timing.