What Is The Compound Continuously Formula And Why Is It Hypothetical?

Before we learn about the compound continuously formula, which is very popularly called the continuous compounding formula, let us first go through some of the few things that we need to know about compound interest.

We calculate the compound interest on a daily, monthly, weekly, quarterly, annual, or half-yearly basis. In each of the following cases, the total amount of times it is compounding is finite and different.

But have you ever wondered if these calculations were infinite?

This is precisely where the compound continuous formula comes into the picture.

In the continuous compound interest formula, the total number of times which we calculate compounding tends towards infinity.

So, without any further delay, let us now understand the entire concept of the compound continuous formula and look through some of the easy examples that will help you grasp the picture even better.

Key Takeaways

• In most cases, compounding interest is performed monthly, semi-annually, or quarterly.
• Compound continuously formula compounds the interest and adds it back to the balance an infinite number of times.
• The formula that computes the compounding continuous interest considers four variables to do so.
• The idea of the compound continuous formula to calculate interest is very crucial in finances; however, the possibility of practicing the same remains zero.

Compound Calculator

Rather than calculating the interest over finite time periods, like monthly, yearly, semi annually, and so on, the compound continuously formula calculates the interest by assuming a continuous compounding over an infinite time period.

The compound continuously formula considers four variables in order to calculate the interest. These four variables namely are:

PV = which is the present investment value

I = the stated rate of interest

N = the total number of compounding periods

T = the total time in years

The compound continuously formula is a derivation of the future value of an investment bearing investment formula:

Future Value (FV): PV X [1(i/n)] (n x t)

While calculating the limit for this formula, as per the given definition of continuous compounding, we get the compound continuously formula:

FV = PV X e ( i x t)

Where we consider e as the mathematical constant, which is approximated as 2.7183.

What Is Continuous Compounding?

The compounded continuously formula gives us the mathematical limit till which an interest is compounded continuously and reinvested in the balance of an account over a hypothetical infinite number of periods.

While it is not practically possible to practice this, the coconut of continuous compounding is an important subject in finance. It is an extreme compounding case, as the compounding for most of the interests is done on a yearly, semi yearly, quarterly, or monthly basis.

The Compound Continuously Formula 

You will be using the compound continuously formula only when you’re asked to do so within a problem. The formula uses a mathematical constant “e,” the value of which is equal to 2.782818??.

Therefore, the compound continuous formula goes like this:

A = Pen

Where,

P is the initial amount

A is the final amount

R is the interest rate

T is the total time

What Can The Compound Continuously Formula Denote? 

Theoretically speaking, continuous compound interest means that the balance of an account is constantly earning interest. This is not where it ends. The compounding interest is transferred back into the initial balance so that it can earn the interest, too.

The compound continuous formula calculates the interest under the assumption that the interest will compound over an infinite count of periods.

Also, the continuous compounding concept holds a significant position; it is next to impossible in the real world to have infinite periods over which a balance is compounded. Therefore, in practical cases, the compounding interest is performed on a fixed term only.

Examples Of Compound Continuously Formula To Calculate Interest

1. Joe invested $3,000 in a bank that pays an annual rate of interest of 7% that compounds continuously. What amount will Joe receive 5 years later from the bank?

This is a hypothetical situation in which we will solve the question to get an answer. Let us not get to the solution.

Solution

What to find?

The amount that Joe will receive after five years.

The initial amount (P) = $3,000

The rate of interest (r ) = 7% = 7/100 = 0.07

Time (t) = 5 years

Now, we will be substituting these given values within the compound continuously formula.

A = Pe^rt

A = 3000 × e0.07(5) ? 4257

You can calculate the same using a calculator and round it to its nearest integer.

Therefore, the amount that Joe will receive after five years is $4,275.

1. What will be the interest rate for an amount of $5,300 to become within a time span of 8 years if the amount is compounded continuously?

Solution

What to find?

The rate of interest

The principal amount P: $5,300

The final amount A = 2(5,300), which is $10,600.

Time i = 8 years.

Now, we have to substitute all these given values within the compound continuously formula.

A = Pe^rt

10600 = 5300 × er (8)

Divide both sides by 5,300

2 = e8r

Take “In” on both the sides,

In 2 = 8r

Divide both sides by 8

r = (ln 2) / 8 ? 0.087

Therefore, the interest rate = 0.087 × 100 = 8.7

Hence, the initial amount has to incur an interest rate of 8.7% in order to double within 8 years.

Is There Any Reaction Between Compound Continuous Formula And The Annual Percentage Yield?

The annual percentage yield, or the APY, is the real return rate on an investment, which takes the compounding interest into account as well. The APY of a given account that has more frequent or more continuous compounding will have a higher APY compared to the account that has gone through an infrequent compounding, considering that the rate of interest for both accounts is the same.

The Bottom Line

The compound continuous formula to calculate interest may be a hypothetical concept and cannot be achieved in reality, but it has real value for investors and savers. It lets the savers look at the maximum amount that they may earn through interest for a selected period and is also useful when you compare the real-time yield of the account.

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