Continuous Compounding Formula

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. Incontinuous 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.

What IsContinuous Compounding Formula?

Thecontinuous 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 thecontinuous compounding formula.

Thecontinuous compounding formula is,

A =Pe^rt

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 formula is,

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 = lim$_{n \rightarrow \infty}$P (1 + r/n)^nt= Pe^rt

The final step is by using one of the limit formulaswhich says, lim$_{n \rightarrow \infty}$(1 + r/n)ⁿ= e^r.

Thus, the continuous compound interest formula is,

A =Pe^rt

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Examples UsingContinuous Compounding Formula

Example 1:Tina invested $3000 in a bankthat 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 =Pe^rt

A = 3000× e^0.07(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 yearsif the amount is compounding continuously? Round your answer to the nearest tenths.

FAQs onContinuous Compounding Formula

What IsContinuous 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 e^rt.

How To DeriveContinuous Compounding Formula?

Let us recall the compound interest formula which says,A = P (1 + r/n)^nt, wheren isthe 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 asn→∞.

A = lim$_{n \rightarrow \infty}$P (1 + r/n)^nt= Pe^rt (∵ lim$_{n \rightarrow \infty}$(1 + r/n)ⁿ= e^r)

Thus, the continuous compound interest formula is,

A =Pe^rt

What Is r inContinuous Compounding Formula?

The continuous compounding formula saysA =Pe^rtwhere '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 einContinuous Compounding Formula?

'e'in thecontinuous 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 number2.7183.