# What is Continuous Compounding

From incentivising investment to promoting saving, interest rates are an integral role of the functions of an economy. In many ways, it is the very notion of interest that fuels credit, and in turn has allowed our world to finance itself.  There exist a number of ways interest is calculated, from simple interest, compounding interest to more complex concepts such as the real rate of return etc. In this article however, we will have a look at the concept of continuous compounding, the way it is calculated including the continuous compounding formula along with scenarios for where it can come in handy.

## Simple interest v/s compound interest

In order to understand what continuous compounding is and how the continuous compounding formula works, we must first understand the basics.

Simple interest is, as the term suggests, simply an interest earned on the principal amount term after term. With simple interest, the interest earned is not added to the principal amount and interest is paid year after year on the original principal amount. Evidently, this interest payment method is not sustainable as it does not account for the time value of money.

Compound interest on the other hand, does. In compound interest, the principal amount changes to accommodate the interest earned as well. Therefore, if you are getting 10% interest yearly, you would get 10% of a 1000 (your principal amount for the sake of this example), or 100 rupees at the end of year 1. At the end of year 2 however, you would now get interest in 1100, or 110, as the previous interest payment was then added to the principal amount.

## What is continuous compounding?

Continuous compounding can be best understood when compared with other forms of interest accumulation. For example, let us assume that there is a principal amount of 1 rupee that is being compounded twice a year, or bi annually. The formula would look something like this :

(1 + ½)^2 = 2.25

Similarly, if the amount is being compounded on quarterly basis, the continuous compounding formula for this scenario would be :

1 + ¼) ^ 4 = 2.44

Now, following a similar continuous compounding formula and a similar conceptual approach, we will eventually arrive at the amount being compounded daily. This would result in the following equation :

(1 + 1/365 ) ^ 365 = 2.7145.

We can now conclude that continuous compounding is the compounding of interest that takes place every hour, minute, second and so on. However for practical purposes, most of us will stop at a daily compounding rate, as the difference then is seen merely in the decimal points and is of negligible importance.

Continuous compounding remains to be a theoretical concept as it sees no real world application (mostly due to its questionable practicality) and yet, it is an important tenet of business and finance.

## The continuous compounding formula

The continuous compounding formula, or continuous compounding interest formula is derived from the formula applied to calculate the future value of an investment that is interest bearing, and is as follows.

Future Value (FV) = PV x [1 + (i / n)](n x t)

This concept is then applied in order to arrive at the continuous compounding interest formula. As the formula rinses and repeats and the the value of “n”, or the compounding time period nears the value of infinity (since compounding interest is calculated even at the smallest theoretical intervals of time, which then also renders it a theoretical concept), the continuous compounding formula is arrived at, which looks like :

FV = PV x e (i x t)

FV stands for future value while PV stands for present value and i and t stand for interest rate and time respectively. The e is assumed to be a constant of 2.7183.

## Importance of continuous compounding

Contrary to what the vastly different time periods in the continuous compounding interest formula would have you believe, continuous compounding does not offer significantly higher yields than years, bi annually or quarterly interest payments. For example, while you would receive 1500 rupees as yearly interest on your initial investment of 10,000 rupees at 15% interest rate, using continuous compounding interest formula would give you about 1618 rupees. A mere 118 rupees extra.

Conclusion

While continuous compounding appears to be a concept that would offer significantly higher yield, it does not do so. Additionally, in most cases, continuous compounding is limited to the theoretical sphere as it barely materialises itself in real world transactions. Even if it did, the interest rate would be limited to per day, as going any lower proposes negligible additions to the interest earned.