Continuous Compounding Formula - Derivation and Examples (2024)

Continuous Compounding Formula is a financial concept where interest is continuously computed and added to an account’s balance over an infinite number of time intervals.

In this article, we will discuss about Continuous compounding formula in detail starting with the continuous compounding formula understanding followed by solved examples and practice problems on the continuous compounding formula.

What is Continuous Compounding Formula?

Continuous compounding Formula in practical applications is an infinite process of idealization and serves as a fundamental principle in finance. Typically, interest is compounded at regular intervals, such as monthly, quarterly, or semiannually, which differs from the theoretical continuous approach.

Continuous compounding formula denotes the investment calculation where interest is continuously computed and added to the investment account’s balance over the mentioned time interval.

Continuous Compounding Formula

The formula for continuous compounding is derived from the concept of calculating limit as the number of compounding periods (n) approaches infinity. The Formula for continuous compounding is given as:

FV = PV x e^{(i x t)}

Where,

• FV (Future Value): The amount of money after a certain time period.
• PV (Present Value): The initial investment amount.
• i (Interest Rate): The stated annual interest rate (expressed as a decimal).
• t (Time): The duration in years.
• e: The mathematical constant

In this formula, “e” denotes the mathematical constant, which is roughly equivalent to 2.7183. This equation offers a precise estimation of interest growth under the assumption of continuous compounding.

Continuous Compounding Definition

Continuous Compounding formula is a method for determining interest, assuming compounding takes place over an unending series of intervals, offering a more accurate assessment of interest accrual.

Continuous Compounding Formula Proof

The formula for continuous compounding is derived from the compound interest formula, and it involves using the mathematical constant ‘e.’

• PV (Present Value): The initial investment amount.
• i (Interest Rate): The stated annual interest rate.
• t (Time): The duration in years.

Here’s a concise proof:

Start with the compound interest formula:

A = PV(1+ i/n)^nt

Now, let’s consider the limit as ‘n’ approaches infinity to achieve continuous compounding:

A = PV lim _n→∞ (1+ i/n)^nt

As ‘n’ approaches infinity, the expression inside the limit simplifies:

lim _n→∞ (1+ i/n) ^nt = e ^it

So, the formula for continuous compounding becomes:

A = PV e^it

^Where

^{A = Amount of money after a certain amount of time}

^{P = Principle or the amount of money you start with}

^{e = Napier’s number, which is approximately 2.7183}

^{i = Interest rate and is always represented as a decimal}

^{t = Amount of time in years}

This formula represents the future value of an investment when interest is compounded continuously.

Basic Math’s Formulas for CBSE

• Basic Math Formulas
• Compound Interest Formula
• Simple Interest
• CBSE Class 11 Maths Formulas
• Class 12 Maths Formulas
• CBSE Class 9 Maths Formulas
• CBSE Class 10th Maths Formulas
• CBSE Class 8 Maths Formulas

Calculation on Continuous Compounding Formula

Example 1: Suppose you invest Rs 1,000 at an annual interest rate of 5% compounded continuously. What will be the investment after one year?

Solution:

Given we want to invest Rs 1,000 at an annual interest rate of 5% compounded continuously, the future value (FV) can be calculated as follows:

FV = PV x e^{(i x t)}

After one year, the future value (FV) can be calculated as follows:

FV = Rs 1,000 x e^{(0.05 x 1)} ≈ Rs 1,051.27

After one year, your investment would be worth approximately Rs 1,051.27.

Example 2: Suppose you deposit Rs. 5,000 into a savings account with a stated annual interest rate of 4.5% that compounds continuously. How much will you have in the account after 3 years?

Solution:

Given we want to invest Rs. 5,000 into a savings account with a stated annual interest rate of 4.5% that compounds continuously, the future value (FV) can be calculated as follows for three years:

FV = PV x e^{(i x t)}

FV = Rs 5,000 x e^{(0.045 x 3)} ≈ Rs 5,659.47

After 3 years, your savings account would hold approximately Rs 5,659.47.

Example 3: You decide to invest Rs. 12,000 in a savings account with a stated annual interest rate of 4.75% that compounds continuously. How much will your investment be worth after 3 years?

Solution:

Given we want to invest Rs. 12,000 into a savings account with a stated annual interest rate of 4.75% that compounds continuously, the future value (FV) can be calculated as follows for three years:

FV = PV x e^{(i x t)}

FV = Rs 12,000 x e^{(0.0475 x 3)} ≈ Rs 13,764.11

After 3 years, your savings account would hold approximately Rs 13,764.11.

Example 4: You have Rs. 9,500 to invest in a certificate of deposit (CD) with a stated annual interest rate of 5.5% that compounds continuously. How much will you have in the CD after 4 years?

Solution:

Given we want to invest Rs.9,500 into a certificate of deposit (CD) with a stated annual interest rate of 5.5% that compounds continuously, the future value (FV) can be calculated as follows for four years:

FV = PV x e^{(i x t)}

FV = Rs 9,500 x e^{(0.055 x 4)} ≈ Rs 11,048.46

After 4 years, your savings account would hold approximately Rs 11,048.46.

Example 5: You decide to invest Rs. 16,500 in a bond with a stated annual interest rate of 4.25% that compounds continuously. Calculate the future value of your investment after 5 years.

Solution:

Given we want to invest Rs.16,500 with a stated annual interest rate of 4.25% that compounds continuously, the future value (FV) can be calculated as follows for five years:

FV = PV x e^{(i x t)}

FV = Rs 16,500 x e^{(0.0425 x 5)} ≈ Rs 19,438.24

After 4 years, your savings account would hold approximately Rs 19,438.24.

Practice Problems on Continuous Compounding Formula

Q1: Calculate the future value of a Rs 2,500 investment at a continuous annual interest rate of 6% after 4 years?

Q2: If you invest Rs 10,000 at a continuous interest rate of 3.5%, how long will it take for your investment to double in value?

Q3: You open a continuous compounding savings account with an initial deposit of Rs1,200. After 2 years, the account balance is Rs 1,500. What was the annual interest rate?

Q4: Determine the present value (PV) of an investment if you want it to grow to Rs 8,000 after 5 years with continuous compounding at an annual rate of 4.2%.

Q5: Suppose you invest Rs 18,000 at a continuous interest rate of 5%. How long will it take for your investment to triple in value?

Q6: Calculate the future value of a Rs 15,000 investment after 6 years with a continuous compounding rate of 4%.

Q7: Find the continuous compounding interest earned on a Rs 3,000 investment after 2 years at a 7% annual interest rate.

Q8: Determine the time required for Rs 2,500 to grow to Rs 5,000 with continuous compounding at a 4% interest rate.

Q9: Calculate the present value needed to achieve Rs 10,000 after 5 years with a continuous compounding rate of 6%.

Q10: If an investment triples in 10 years with continuous compounding, what is the annual interest rate?

Conclusion

The continuous compounding formula provides a sophisticated method to calculate the growth of investments by assuming interest is compounded continuously. This approach offers a more accurate representation of interest accumulation compared to traditional methods. By understanding and applying this formula, investors can make informed financial decisions and evaluate different investment opportunities more effectively

Continuous Compounding Formula- FAQs

What is Continuous Compounding?

Continuous compounding is a method where interest is compounded an infinite number of times over a given period, providing a more precise measure of interest growth.

What is the Difference between Continuous Compounding and Regular Compounding?

Regular compounding adds interest at specific intervals (e.g., annually, quarterly), whereas continuous compounding assumes interest is added constantly, leading to more accurate calculations.

How is the Mathematical Constant “e” used in Continuous Compounding?

The constant “e” (approximately 2.7183) is used in the continuous compounding formula to calculate the future value of investments. It represents continuous growth.

Is Continuous Compounding used in Real-world financial scenarios?

Yes, continuous compounding is applied in certain financial instruments and accounts, providing a more precise representation of interest growth over time.

How can I Calculate the Future Value of an Investment with Continuous Compounding?

Use the formula: FV = PV x e(i x t), where PV is the present value, i is the interest rate, t is the time in years, and e is the mathematical constant.

Is the continuous compounding formula applicable to all types of investments?

It is mainly used for investments that accrue interest over time, like bonds and savings accounts.

Can continuous compounding be used for short-term investments?

Yes, continuous compounding can be applied to both short-term and long-term investments, but its benefits are more significant over longer periods.