Rate from Lump Sum
Find the continuous compounding rate.
When to use: Use for continuous-time finance calculations.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| k | Continuous Rate | Continuously compounded rate | % |
| FV | Future Value | Ending value | $ |
| PV | Present Value | Starting value | $ |
| n | Number of Years | Time period | years |
Real-Life Examples
Example 1: Continuous Return
An investment went from $10,000 to $25,000 in 15 years. Continuous rate?
Given
Step-by-Step
The continuously compounded return was about 6.11%.
Example 2: Bond Yield
A zero-coupon bond priced at $600 matures at $1,000 in 10 years. Continuous yield?
Given
Step-by-Step
The continuous yield is about 5.11%.
Frequently Asked Questions
A continuously compounded return assumes growth happens at every instant rather than at discrete intervals. It equals ln(FV/PV)/n, using the natural logarithm. It is commonly used in quantitative finance and options pricing.
Continuous returns are mathematically convenient because they are additive across time periods (unlike discrete returns, which must be compounded). A 5% continuous return for 2 years is simply 10%, making multi-period analysis simpler.
To convert a continuous rate to an effective annual rate: EAR = e^k - 1. To go the other way: k_continuous = ln(1 + EAR). For example, a 6% continuous rate equals an effective rate of e^0.06 - 1 = 6.18%.