FV of Ordinary Annuity
Calculates the future value of an ordinary annuity (discrete annual end-of-period payments) when interest is compounded continuously at the rate k.
When to use: Use when payments are made annually at year-end but interest accrues with continuous compounding — the limit case of the m-times formula as m approaches infinity for the discounting side, while payments remain discrete.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| FVA | Future Value of Annuity | Total future value of all payments | $ |
| PMT | Payment | Periodic payment amount | $ |
| k | Interest Rate | Nominal annual interest rate as a decimal | % |
| n | Number of Years | Time period in years | years |
Real-Life Examples
Example 1: Continuous Growth Fund
$5,000/year into a fund at 6% continuous compounding for 20 years.
Given
Step-by-Step
The fund reaches $187,600.78 with discrete annual deposits compounded continuously.
Example 2: Academic Example
$1,000/year at 10% continuously compounded for 5 years.
Given
Step-by-Step
The annuity grows to $6,168.26 with discrete annual payments and continuous compounding.
Frequently Asked Questions
The m-times formula treats both payments AND compounding as occurring m times per year. This continuous-compounding form keeps payments discrete (one per year) while letting the rate compound continuously. As m → ∞ in the m-times formula with PMT/m as the per-period payment, the continuous limit is identical; with PMT as the per-period payment the m-times sum diverges.
Use it in academic coursework, theoretical finance research, or when modeling annual cash flows in a continuous-time framework (Black-Scholes-style settings, actuarial models). In day-to-day work, the m-times formula with m matching the actual compounding frequency is more common.
Continuous compounding produces a slightly higher future value than any discrete compounding frequency at the same nominal rate, since interest is being added at every possible instant. The difference from daily compounding is typically very small (a few basis points per year).