This article deals with continuous compound interest formula and its derivation. Continuous compounding most certainly refers to the mathematical limit that compound interest is capable of reaching if it is calculated and reinvested into an account’s balance over an infinite number of periods. This usually may not be possible in practice. However, the concept of continuously compounded interest is of paramount importance in finance. It is certainly an extreme case of compounding. This is because, compounding of most interest takes place on a monthly, quarterly or semi-annual basis. In theory, continuously compounded interest means that a particular account balance is earning interest constantly and also refeeding that interest back into the balance so that it earns interest too.

**What is Continuous Compound Interest?**

Continuous compound interest refers to the mathematical limit of the general compound interest such that its compounding takes place infinitely many times each year. Furthermore, in continuous compound interest, one receives payment with every possible time increment.

Instead of calculating interest on a finite number of periods, continuous compounding calculates interest by assuming constant compounding which takes place over an infinite number of periods.

Examples of a finite number of periods can be yearly or monthly basis. Even when we are talking of very large investment amounts, the difference in the total interest earned through continuous compounding is certainly not significantly high when its comparison takes place to traditional compounding periods.

Continuous compound return is what takes place when the interest earned on a particular investment is calculated and reinvested back for a number of periods which is infinite. The calculation of the interest takes place on the principal amount. Furthermore, the accumulation of the interest takes place over the given periods and their reinvestment takes place into the cash balance.

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**Continuous Compound Interest Formula**

The continuous compounded interest formula is below:

Continuous compounded interest = \(\lim_{N\rightarrow /\infty }\)\(\left [ \left ( 1+\frac{annual interest rate}{N} \right )^{N time}-1 \right ]\)

Where,

N is the number of times interest is compounded in a particular year

Furthermore,

The formula can also be as follows:

A = \(Pe^{rt}\)

Here,

A = amount

P = principal

e = mathematical constant e

r = rate of interest

t = time in years

**Continuous Compound Interest Formula Derivation**

A common definition of constant e is below:

e = \(\lim_{m\rightarrow \infty }\)\(\left ( 1+\frac{1}{m} \right )\)

Therefore, we certainly have, with a change of variables which gives n = mr, that

\(\lim_{n\rightarrow \infty }P\left ( 1+\frac{r}{n} \right )^{nt}\)

= \(\lim_{m\rightarrow \infty }P\left ( 1+\frac{1}{m} \right )^{mrt}\)

= \(p\left ( \lim_{m\rightarrow \infty }\left ( 1+\frac{1}{m} \right )^{m} \right )^{rt}\)

= \(Pe^{rt}\)

Hence, this is the derivation of the continuous interest formula.

**Solved Examples**

Q1 An individual invests $1,000 at an annual interest rate of 5% compounded continuously. Find out the final amount you will have in the account after five years?

A1 The formula for finding the amount in case of continuous compounding is as follows:

A = \(Pe^{rt}\)

A = 1000e(0.05.5)

so, A = 1284.02

Hence, the amount in this case is dollars 1284.02.

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