PV of Lump Sum
Calculates the present value when discounting continuously.
When to use: Use for continuous-time finance models and derivative pricing.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PV | Present Value | The current worth of a future sum | $ |
| FV | Future Value | The value of the investment at a future date | $ |
| k | Interest Rate | Nominal annual interest rate as a decimal | % |
| n | Number of Years | Investment time horizon in years | years |
Real-Life Examples
Example 1: Continuous Discounting
What is $50,000 receivable in 10 years worth today at 8% continuously compounded?
Given
Step-by-Step
$50,000 in 10 years is worth $22,466.48 today with continuous discounting.
Example 2: Option Pricing Input
A payoff of $75,000 in 3 years at a 5% continuous rate. What is the PV?
Given
Step-by-Step
The present value is $64,552.20 using continuous discounting.
Frequently Asked Questions
Continuous discounting is used primarily in academic finance, options pricing models like Black-Scholes, and theoretical analyses where time is treated as a continuous variable rather than discrete periods.
Continuous discounting produces a slightly lower present value than periodic discounting at the same nominal rate, because continuous compounding is the fastest possible rate of growth. The difference is small for typical rates and time periods.
Euler's number (e ≈ 2.71828) is a mathematical constant that naturally arises in continuous growth and decay processes. It is the base of the natural logarithm and appears whenever compounding happens infinitely often, making it the foundation of continuous compounding formulas.