In compound interest, we would have heard about the terms like "compounded annually" or "compounded semi annually" or "compounded quarterly" or compounded monthly".

But, always we have question about compounded continuously.

To understand "compounded continuously", let us consider the example given below.

When we invest some money in a bank, it will grow continuously. That is, at any instant the balance is changing at a rate that equals "r" (rate of interest per year) times the current balance.

Continuous compounding formula

Many real world phenomena are being modeled by functions which describe how things grow continuously at any instance.

Formula :

The formula given below is related to compound interest formula and represents the case where interest is being compounded continuously.

That is, at any instant the balance is changing at a rate that equals "r" times the current balance.

We use this formula, when it is given "compounded continuously"

A ---> Ending amount

P ---> Beginning amount

r ---> Growth rate

t ---> Time

Continuous compounding formula - Example

Example 1 :

You invest $2500 in bank which pays 10% interest per year compounded continuously. What will be the value of the investment after 10 years ?

Solution :

We have to use the formula given below to know the value of the investment after 3 years.

Here,

A = Final value of the deposit

P = 2500, r = 10% or 0.1, t = 10, e = 2.71828 and also

rt = 0.1x10 = 1

A = 2500(2.71828)¹

A = 6795.70

Hence, the value of the investment after 10 years is $6795.70

Example 2 :

If David invests $500 at annual rate of 20% compounded continuously, calculate the final amount that David will have after 5 years.

Solution :

We have to use the formula given below to know the final amount that David will have after 5 years .

Here,

A = Final value of the deposit

P = 500, r = 20% or 0.2, t = 5, e = 2.71828 and also

rt = 0.2x5 = 1

A = 500(2.71828)¹

A = 1359.14

Hence, the final amount that David will have after 5 years is $1359.14

After having gone through the stuff given above, we hope that the students would have understood "Compounded continuously formula".

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