PV of Ordinary Annuity
Calculates the present value of an ordinary annuity (discrete annual end-of-period payments) discounted at a continuously-compounded rate k.
When to use: Use when payments arrive annually at year-end but you want to discount them with continuous compounding — the standard convention in continuous-time finance and actuarial models.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PVA | Present Value of Annuity | Total present 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 Discount
$10,000/year for 15 years at 5% continuous discounting.
Given
Step-by-Step
The annuity is worth $102,910.51 in present value terms with continuous discounting.
Example 2: Academic Problem
$2,000/year for 10 years at 8% continuous discounting.
Given
Step-by-Step
The present value is $13,223.45 with continuous discounting.
Frequently Asked Questions
It is used in theoretical finance, academic research, and advanced models where the discount rate is naturally expressed in continuously-compounded form (e.g., from option-pricing or yield-curve models). For practical day-to-day calculations, the m-times version with m matching real-world compounding is more common.
Continuous discounting produces a slightly lower present value than annual discounting at the same nominal rate. This is because continuous compounding implies a higher effective annual rate, meaning future cash flows are worth less today.
The discount factor (1+k)^-t in the discrete formula is replaced by e^(-kt) in the continuous form. The denominator changes from k to e^k - 1 because the annuity factor must reflect the per-year discount factor implied by continuous compounding.