# Continuous Compounding

## Did you know?

Continuous compounding is the mathematical limit reached by compound interest when it’s calculated and reinvested to an account balance over a theoretically endless number of periods. Put simply, the account balance continually earns interest, and that interest gets added to the balance, which then also earns interest and it continues to grow.

The concept of continuously compounding is important in finance though it is not possible in practice. The majority of the interest is compounded on a monthly, quarterly, or semiannual basis, so it’s is an extreme case of compounding.

## Continuous Compounding Formula

$FV = PV \times e^{it}$
• FV = the future value of the investment
• PV = the present value of the investment, or principle
• e = Euler’s number, the mathematical constant 2.71828
• i = the interest rate
• t = the time in years

This formula basically determines the interest that is earned on an amount through constantly compounding it to a theoretically infinite amount of compounding periods. Put simply, it takes compounding to the furthest theoretical limit.

Interest will be calculated assuming that there is a constant compounding over an infinite number of periods instead of being calculated on a finite period such as yearly or monthly.

The constant compounding formula is derived from the future value of an interest-bearing investment formula, which is more commonly referred to as the compound interest formula:

$FV = PV \times \bigg( 1 + \dfrac{i}{n} \bigg)^{nt}$
• FV = the future value of the investment
• PV = the present value of the investment, or principle
• i = the interest rate
• n = the number of compounding periods
• t = the time in years

Without getting too mathematical, the continuous compounding formula is much simpler because with finite compounding you have to raise the value to a large exponent, which is messier than the cleaner continuous formula.

## Continuous Compounding Example

Jeremy is looking at investment opportunities and has \$15,000 to invest, with an expected interest of 14% over the next year. We can use the finite and continuous compounding formulas above to find the ending value of the investment:

$Annual\: Compounding = \15{,}000 \times \bigg( 1 + \dfrac{14\%}{1} \bigg)^{1 \times 1} = \17{,}100$ $Monthly\: Compounding = \15{,}000 \times \bigg( 1 + \dfrac{14\%}{12} \bigg)^{12 \times 1} = \17{,}240.13$ $Daily\: Compounding = \15{,}000 \times \bigg( 1 + \dfrac{14\%}{365} \bigg)^{365 \times 1} = \17{,}253.64$ $Continuous\: Compounding = \15{,}000 \times 2.71828^{15\% \times 1} = \17{,}254.11$

As you can see in these examples, continuous compounding is only marginally more than daily compounding. Even though we’re using a theoretically infinite number of compounding, the final amount is not much more because the effect of each compound becomes smaller each time.

## Continuous Compounding Conclusion

When calculating continuous compounding, the below points are worth bearing in mind as a quick recap of what it is, why it’s used, and how to use it:

• Continuous compounding is the mathematical limit reached by compound interest when it’s calculated and reinvested over unlimited periods
• The balance continually earns interest which is added to the balance, which then earns more interest
• It’s an important financial concept, but not possible in practice to have an infinite number of periods
• Continuous compounding takes compounding to the furthest theoretical limit
• The formula uses Euler’s number, which is the mathematical constant 2.71828
• The formula is also derived from the future value of an interest-bearing investment formula (most commonly known as the compound interest formula)
• The continuous compounding formula is simpler than the compound interest formula from a math perspective, which is why you might use it

## Continuous Compounding Calculator

You can use the continuous compounding calculator below to work out your own future value and compare it with finite compounding periods.