Continuous Compounding Formula

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.

Formula for Continuous Compounding

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,

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

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

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

Solved Examples 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?

Continuous Compounding Formula – FAQs

1. What is Continuous Compounding?

Continuous compounding is a method of calculating interest where compounding occurs an infinite number of times over a given time period. It assumes that interest is added continuously rather than at specific intervals, providing a more precise measure of interest growth.

2. What is the Difference between Continuous Compounding and Regular Compounding?

In regular compounding, interest is added at discrete intervals (e.g., annually, quarterly, monthly). Continuous compounding assumes interest is added constantly, resulting in more accurate interest calculations.

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

The constant “e” (approximately 2.7183) is used to calculate continuous compounding. The formula for continuous compounding is FV = PV x e^{(i x t)}, where “e” represents this mathematical constant.

4. Is Continuous Compounding used in Real-world financial scenarios?

Continuous compounding is a theoretical concept, but it’s used in practice for financial instruments like continuously compounded bonds and certain types of savings accounts. It provides a more accurate representation of interest growth.

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

To calculate the future value (FV) 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.