Time from Lump Sum
Time for a lump sum to reach a target with continuous compounding.
When to use: Use for continuous-time models.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| n | Number of Years | Time to reach goal | years |
| FV | Future Value | Target value | $ |
| PV | Present Value | Starting amount | $ |
| k | Interest Rate | Continuous rate | % |
Real-Life Examples
Example 1: Continuous Doubling
How long to double at 7% continuous compounding?
Given
Step-by-Step
Continuous compounding at 7% doubles money in 9.90 years.
Example 2: Growth Target
How long for $5,000 to reach $15,000 at 10% continuous?
Given
Step-by-Step
It takes about 11 years to triple with continuous compounding at 10%.
Frequently Asked Questions
With continuous compounding, the time formula simplifies to n = ln(FV/PV) ÷ k. This is the simplest form of the time-to-goal formula because the natural logarithm and exponential function are mathematical inverses.
Yes, continuous compounding gives the shortest possible doubling time for a given nominal rate. However, the difference from daily compounding is negligible — typically less than a day for most practical interest rates.
At 7% continuous: n = ln(2)/0.07 = 0.6931/0.07 = 9.90 years. Compare to annual compounding: n = ln(2)/ln(1.07) = 10.24 years. Continuous compounding shaves about 4 months off the doubling time.