What it means
The number e is a bit like pi, in that it is a fixed value that appears in many places in nature and mathematics. It arises when you ask how much a sum of money would grow if interest were added every instant rather than once a year.
The answer to a 100% annual rate, compounded continuously for one year, is e times the starting amount. In everyday finance, interest is usually added monthly, quarterly or annually.
As you increase the number of compounding periods, the final amount rises, but only up to a limit. Continuous compounding is that limit, and e is what makes the formula work neatly.
You will meet e in several practical settings. Treasury and risk teams use it to convert between rates, quants (mathematicians who build financial models) use it in option pricing, and analysts use it to calculate discount factors and growth rates.
Continuously compounded returns are also convenient because they can be added up over time instead of multiplied. A short word of warning on the name: mathematicians also use "Euler's constant" for a different number, about 0.5772, called the Euler-Mascheroni constant.
In finance and business, the phrase nearly always means e. If you see it in a spreadsheet, the function EXP does the job of raising e to a power.
Most real-world loans and deposits do not compound continuously, so e is a modelling tool rather than a payment term. It gives a clean upper benchmark and makes the maths of growth and decay easier.
Even so, the difference between annual and continuous compounding on a modest rate is small. For a non-finance professional, the practical lesson is that more frequent compounding helps savers and hurts borrowers, but with diminishing effect.
When someone quotes a rate, always ask how often it is compounded before comparing offers.
In practice
Real-world examples.
Example
A bank's risk team in Frankfurt prices a bond using continuously compounded rates. The discount factor for a payment due in 3 years at 4% is e^(-0.04 x 3), or about 0.8869, which makes the model easy to chain together.
Example
A start-up founder in Austin sees that her website visits grew from 20,000 to 30,000 in a quarter. Her analyst uses the natural logarithm, the inverse of e, to calculate a continuously compounded growth rate of about 40.5%.
Example
An engineer-turned-analyst at a manufacturer models how quickly the value of a machine falls over time. A depreciation curve built on e gives a smooth path that is easy to feed into forecasts.
Formula
Calculation
Future value with continuous compounding = P x e^(r x t)
Here P is the starting amount, r is the annual interest rate as a decimal, t is the time in years, and e is approximately 2.71828.
Worked example: you invest $10,000 at 5% a year for 2 years with continuous compounding.
Step 1: Work out the exponent = r x t = 0.05 x 2 = 0.10.
Step 2: Find e^0.10, which is about 1.105171.
Step 3: Future value = $10,000 x 1.105171 = $11,051.71.
For comparison, annual compounding gives $10,000 x 1.05 x 1.05 = $11,025. The continuous method adds just $26.71 more over two years, which shows how small the gain from extra compounding can be.Case study
Seen in the real world.
Northgate Savings is a fictional credit union that wanted to compare deposit products from three partner banks. One quoted an annual rate compounded yearly, one compounded monthly, and one used continuous compounding.
An analyst converted all three to a common basis using e and found that the best headline rate was not the best product once compounding was taken into account. The gap between the best and worst offer on a $50,000 deposit was only a few hundred dollars over five years, but it was enough to change the ranking.
In this illustrative case, the credit union adopted a simple rule: every quoted rate must be converted to an effective annual rate before it goes to the board. The team now has a standard spreadsheet template that does this in one step.
Watch out
Common mistakes.
- Thinking e is a rate or a percentage, when it is a fixed number of about 2.71828 used as a mathematical base.
- Assuming continuous compounding makes a big difference, when at ordinary interest rates the gain over daily or monthly compounding is tiny.
- Mixing up e with the Euler-Mascheroni constant of about 0.5772, which is a different number.
Questions
People also ask.
Why is e used in finance?
It makes continuous growth and discounting easy to calculate, and continuously compounded returns can be added together over time.
Do real banks compound continuously?
Rarely, because most products add interest daily, monthly or annually, so e is mainly a tool for models and comparisons.
How do I calculate e to a power in a spreadsheet?
Use the EXP function, for example =EXP(0.10) returns about 1.105171.
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