About Continuous Compounding Formula
Let us review a few points regarding compound interest before learning the continuous compounding formula. Compound interest is often calculated every day, week, month, quarter, half-year, or year. The number of times it compounds is distinct and finite in each of these circumstances. What if, however, this number is infinite? This leads to the formula for continuous compounding. The number of times compounding occurs in continuous compounding is approaching infinity. Let's look at the continuous compounding formula and several examples with solutions.
What Is Continuous Compounding Formula?
The continuous compounding formula should be used when they mention specifically that the amount is "compounded continuously" in a problem. This formula involves the mathematical constant "e" whose value is approximately equal to2.7182818.... Here is the continuous compounding formula.
Continuous Compounding Formula
The continuous compounding formula is,
A =Pert
where,
- P = the initial amount
- A = the final amount
- r = the rate of interest
- t = time
- e is a mathematical constant where e≈ 2.7183.
Continuous Compounding Formula Derivation
We will derive the continuous compounding formula from the usual formula of compound interest.
The compound interest formulais,
A = P (1 + r/n)nt
Here, n = the number of terms the initial amount (P)is compounding in the time t andA is the final amount (or) future value.For the continuous compound interest, n→∞. So we will take the limit of the above formula asn→∞.
A = limn→∞n→∞P (1 + r/n)nt= Pert
The final step is by using one of the limit formulaswhich says, limn→∞n→∞(1 + r/n)n= er.
Thus, the continuous compound interest formula is,
A =Pert
Solved example of Continuous Compounding Formula
Example 1:Tina invested $3000 in a bank that pays an annual interest rate of 7% compounded continuously. What is the amount she can get after 5 years from the bank? Round your answer to the nearest integer.
Solution:
To find: The amount after 5 years.
The initial amount is P = $3000.
The interest rate is, r = 7% = 7/100 = 0.07.
Time is, t = 5 years.
Substitute these values in the continuous compounding formula,
A =Pert
A = 3000× e0.07(5)≈4257
The answer is calculated using the calculator and is rounded to the nearest integer. To get all the Maths formulas check out the main page.
Pdf of Continuous Compounding Formula
Continuous Compounding Formula
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