# 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. 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.

## 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 Formula

The continuous 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 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} = Pe^{rt}

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

Thus, the continuous compound interest formula is,

A = Pe^{rt}

We can see the applications of the continuous compounding formula in the section below.

## Examples Using 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 = 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 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 = Pe^{rt }

10600 = 5300 × e^{r (8)}

Dividing both sides by 5300,

2 = e^{8r}

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 = Pe^{rt}

A = 5000 × e^{0.09(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 Formula

### What 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 e^{rt}.

### 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} = 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 in Continuous Compounding Formula?

The continuous compounding formula says A = Pe^{rt} 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.

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