Before going to learn the continuous compounding formula, let us recall few things about the compound interest. Compound interest is usually calculated on a daily, weekly, monthly, quarterly, half-yearly, or annual basis. In each of these cases, the number of times it is compounding is different and is finite. But what if this number is infinite? This leads to the continuous compounding formula. In continuous compounding number of times by which compounding occurs is tending to infinity. Let us learn the continuous compounding formula along with a few solved examples. Show
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 to 2.7182818.... Here is the continuous compounding formula.  Continuous Compounding FormulaThe continuous compounding formula is, A = Pert where,
Continuous Compounding Formula DerivationWe will derive the continuous compounding formula from the usual formula of compound interest. The compound interest formula is, A = P (1 + r/n)nt Here, n = the number of terms the initial amount (P) is compounding in the time t and A is the final amount (or) future value. For the continuous compound interest, n → ∞. So we will take the limit of the above formula as n → ∞. A = lim\(_{n \rightarrow \infty}\) P (1 + r/n)nt = Pert The final step is by using one of the limit formulas which says, lim\(_{n \rightarrow \infty}\) (1 + r/n)n = er. Thus, the continuous compound interest formula is, A = Pert Have questions on basic mathematical concepts? Become a problem-solving champ using logic, not rules. Learn the why behind math with our certified experts Book a Free Trial Class We can see the applications of the continuous compounding formula in the section below. Examples Using Continuous Compounding FormulaExample 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. Answer: The amount after 5 years = $4,257. Example 2: What should be the rate of interest for the amount of $5,300 to become double in 8 years if the amount is compounding continuously? Round your answer to the nearest tenths. Solution: To find: The rate of interest, r. The initial amount is, P = $5,300. The final amount is, A = 2(5300) = $10,600. Time is, t = 8 years. Substitute all these values in the continuous compound interest formula, A = Pert 10600 = 5300 × er (8) Dividing both sides by 5300, 2 = e8r Taking "ln" on both sides, ln 2 = 8r Dividing both sides by 8, r = (ln 2) / 8 ≈ 0.087 (using calculator) So the rate of interest = 0.087 × 100 = 8.7 Answer: The rate of interest = 8.7%. Example 3: Jim invested $5000 in a bank that pays an annual interest rate of 9% compounded continuously. What is the amount he can get after 15 years from the bank? Round your answer to the nearest integer. Solution: To find: The amount after 15 years. The initial amount is P = $5000. The interest rate is, r = 9% = 9/100 = 0.09. Time is, t = 15 years. Substitute these values in the continuous compounding formula, A = Pert A = 5000 × e0.09(15) ≈ 19287 The answer is calculated using the calculator and is rounded to the nearest integer. Answer: The amount after 15 years = $19,287. FAQs on Continuous Compounding FormulaWhat Is Continuous Compounding Formula?The continuous compounding formula is nothing but the compound interest formula when the number of terms is infinite. This formula says, when an amount P is invested for the time 't' with the interest rate is r% compounded continuously, then the final amount is, A = P ert. How To Derive Continuous Compounding Formula?Let us recall the compound interest formula which says, A = P (1 + r/n)nt, where n is the number of terms the initial amount (P) is compounding in the time t. Here, A is the final amount. For the continuous compound interest, the number of terms is infinite, i.e., n → ∞. So we will take the limit of the above formula as n → ∞. A = lim\(_{n \rightarrow \infty}\) P (1 + r/n)nt = Pert (∵ lim\(_{n \rightarrow \infty}\) (1 + r/n)n = er) Thus, the continuous compound interest formula is, A = Pert What Is r in Continuous Compounding Formula?The continuous compounding formula says A = Pert where 'r' is the rate of interest. For example, if the rate of interest is given to be 10% then we take r = 10/100 = 0.1. What Is e in Continuous Compounding Formula?'e' in the continuous compounding formula is a mathematical constant and its value is approximately equal to 2.7183. We can use the button 'e' on the calculator for more accurate calculations instead of using the number 2.7183. How long will it take money to double at 3% compound interest?To use the rule, divide 72 by the investment return (the interest rate your money will earn). The answer will tell you the number of years it will take to double your money. For example: If your money is in a savings account earning 3% a year, it will take 24 years to double your money (72 / 3 = 24).
How long will it take money to double if compounded continuously?The basic rule of 72 says the initial investment will double in 3.27 years.
How many years does it take to double your money if the continuously compounded interest rate is 6 %?To use the Rule of 72 in order to determine the approximate length of time it will take for your money to double, simply divide 72 by the annual interest rate. For example, if the interest rate earned is 6%, it will take 12 years (72 divided by 6) for your money to double.
How many years does it take to double your money if the continuously compounded interest rate is 10 %?How the Rule of 72 Works. For example, the Rule of 72 states that $1 invested at an annual fixed interest rate of 10% would take 7.2 years ((72/10) = 7.2) to grow to $2. In reality, a 10% investment will take 7.3 years to double ((1.107.3 = 2).
How long will it take money to double if it is compounded annually at a rate of 10 percent per year?Given a 10% annual rate of return, how long will it take for your money to double? Take 72 and divide it by 10 and you get 7.2. This means, at a 10% fixed annual rate of return, your money doubles every 7 years.
How do you calculate how long it will take to double something?The Rule of 70 is a simplified way of determining the doubling time using the equation, doubling time = 70 / r , where r is the rate of growth for a population in percent.
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How long will it take for an amount to double when it is compounded continuously at an annual rate of 3%?
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