FV of Growing Annuity Due
Future value of growing beginning-of-period payments with continuous compounding.
When to use: Use in academic models for growing beginning-of-period savings under continuous compounding.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| FVA | Future Value of Annuity | Accumulated value | $ |
| PMT | First Payment | The first payment amount | $ |
| k | Interest Rate | Nominal annual rate | % |
| g | Growth Rate | Continuous growth rate | % |
| n | Number of Years | Time horizon | years |
Real-Life Examples
Example 1: Continuous Growing Due FV
$6,000/year growing at 3%, deposited at start, 8% continuous for 20 years.
Given
Step-by-Step
Growing beginning-of-period deposits accumulate to $407,045.06.
Example 2: Academic Example
$3,000/year growing at 2%, start of period, 5% continuous for 10 years.
Given
Step-by-Step
Growing due deposits reach $44,936.41 with continuous compounding.
Frequently Asked Questions
Multiply the continuous ordinary growing annuity FV by e^k. The e^k factor accounts for each payment being made one period earlier, giving it an extra period of continuous compounding.
The due adjustment multiplies the result by e^k. At 8% continuous, this is about 8.3% more than the ordinary version. Over long time horizons with growing payments, this compounds to a meaningful difference.
When k = g, the formula simplifies to PMT × n × e^(kn) × e^k. The growth and compounding rates cancel in the annuity factor, leaving the payment rate times years, compounded forward and adjusted for due timing.