FV of Growing Annuity
Future value of a growing payment stream with continuous compounding.
When to use: Use in academic continuous-time models for growing savings accumulation.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| FVA | Future Value of Annuity | Accumulated value of growing payments | $ |
| PMT | First Payment | The first payment amount (or continuous payment rate) | $ |
| k | Interest Rate | Nominal annual rate of return | % |
| g | Growth Rate | Continuous growth rate of payments | % |
| n | Number of Years | Time horizon in years | years |
Real-Life Examples
Example 1: Continuous Growing Savings
$8,000/year growing at 3%, 7% continuous compounding for 25 years.
Given
Step-by-Step
Growing contributions accumulate to $727,519.32 with continuous compounding.
Example 2: Academic Example
$5,000/year growing at 2%, 6% continuous for 15 years.
Given
Step-by-Step
Growing payments accumulate to $138,715.86.
Frequently Asked Questions
The formula PMT × (e^(kn) - e^(gn)) / (k-g) is the future value analog of the continuous growing PVA. It can also be derived by multiplying the continuous growing PVA by e^(kn).
When k = g, the formula simplifies to PMT × n × e^(kn). Both the growth and the interest rate are the same, so the future value is the payment rate times years, compounded to the future.
Continuous compounding produces a slightly higher future value than annual compounding at the same nominal rate, because interest is being reinvested at every instant rather than once per year.