Continuous Compound Interest Formula and Examples

continuous compound interest formulaCompound interest is different from simple interest, because, as you might have gathered, it compounds. This mean that it builds on the principal of a money amount, so that the added interest also accrues interest as well. To calculate compounded interest and determine how much money will be owed after a specific amount of time, we must use the continuous compound interest formula:

V = P(1+r/n) nt

Read on for a more detailed explanation of compound interest, and learn how its related formula works. Or, check out this course on simple interest and this course on compound interest for a more in-depth guide to both.

What is Compound Interest?

Assuming that no payments have been made on it, a theoretical loan of $1000 with an annual simple interest rate of 2% will accrue $20 of interest after the first year, increasing to $1020.  After the second year, it will have accrued $20 more dollars of interesting, increasing to $1040. This is because the principal loan amount does not compound interest itself. It remains $1000, with interest calculated from this principal amount and added separately.

This is important to know if we want to understand what compound interest it. Compound interest, unlike simple interest, is actually added to the principal amount, so that every time interest accrues, it’s calculated from the total amount with the added interest. What does this mean in the case of our theoretical $1000 loan that has had no payments made on it?

Well, after the first year with a 2% annual compound interest rate, this loan would be at $1020 exactly. For the second year, the compounded interest rate would base itself on the new $1020 amount instead of the principal $1000. 2% of 1020 is 20.4, so our loan amount at the end of the second year would be $1040.40, which is $20.40 added to $1020. Learn the ins and outs of financial math in this course.

Continuous Compound Interest Formula

It’s easy to calculate compound interest in our head with an easy number and interest rate like the one in the example above. When the numbers get bigger, and the years more numerous, though, there’s that handy continuous compound interest formula we can use to calculate the impending value of a debt, loan, or deposit after a certain amount of time.

V = P(1+r/n) nt

If that looks scary to you, maybe you should check out this course on finance for some background on financial mathematics and concepts like interest rate.

If you’ve got the math part down, the value of each of the variables in our equation are as follows:

  • P = principal amount
  • r = annual interest rate
  • n = the number of times per year that interest is compounded
  • t = number of years
  • = calculated amount after t time has passed

Remember, principal amount (P) is the starting point, or the initial investment, of the money amount we’re dealing with, whether it’s a loan or a deposit or something else. This is the starting point before the compounded interest.

The annual interest rate (r) is the percentage of the principal amount that is accrued in interest every year.

The number of years (t) is how many years we want to let the interest in our formula accrue. If you want to test how much the money amount will be after five years, you’d plug in the number 5 here.

The calculated amount (V) is the money amount after the compounded interest has accrued following a certain amount of time, provided by t. Of course, our end goal doesn’t have to be finding V if we already know V. Maybe you already know the value of everything but the number of years passed. In that case you’d plug everything in and solve for t.

If you’re feeling lost, check out this course on the foundations of algebra!

Compound Interest Example

So let’s say we have a loan of $28,532. The annual interest rate is 3.5%. We want to know what the loan amount will be after five years.

This means:

  • P = 28,532
  • r = 0.035
  • n = 1
  • t = 5
  • V = ?

V = P(1+r/n) nt

V = (28,532)(1+0.035/1)(1)(5)

V = 33,887.07

With a compounded interest rate of 3.5% and an initial investment amount of $28,532, our new balance at the end of five years will be $33,887.07. Confused about how we got to this number just by plugging in values? Check out this course on beginner calculus for some extra help.

Compound Interest Example Quiz

Try to solve one for yourself! Let’s say you have a principal investment amount of $43,613. If you pay nothing on it for 10 years (life tip: don’t do this), with a compound annual interest rate of 12%, how much will the new balance be?

Answer

  • P = 43,613
  • r = 0.12
  • n = 1
  • t = 10
  • V = $135,455.36

Check out this course on understanding loans for more information about concepts like interest rate.