Payment from FV
Beginning-of-period annual deposit needed to reach a future value when the rate is continuously compounded.
When to use: Use to size annual annuity-due deposits toward a savings goal when the rate is quoted in continuously-compounded form.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PMT | Payment | Periodic payment amount (paid at start of period) | $ |
| FV | Future Value | Future lump sum value | $ |
| k | Interest Rate | Nominal annual interest rate | % |
| n | Number of Years | Time period in years | years |
Real-Life Examples
Example 1: Continuous Due Savings
Reach $300,000 in 20 years at 7% continuous, deposits at start of period.
Given
Step-by-Step
Deposit $6,638.47 at the start of each period to reach $300,000.
Example 2: Academic Example
$50,000 goal in 5 years, 6% continuous, deposits at start.
Given
Step-by-Step
The beginning-of-period deposit is $8,322.71.
Frequently Asked Questions
Divide the ordinary continuous PMT from FV by e^k. Beginning-of-period timing means each deposit has one extra period of continuous growth, so less is needed per deposit to reach the same goal.
The annuity due payment is lower by a factor of 1/e^k compared to the ordinary version. At 7% continuous, this is about 6.8% less per deposit, since each deposit earns interest for one additional period.
In academic finance and theoretical modeling. For practical savings calculations, use the m-times annuity due formula with the appropriate deposit frequency (m=12 for monthly, m=4 for quarterly).