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Continuously Compounded Interest Explained


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Continuously Compounded Interest Explained
Continuously compounded interest is yet another financial concept that theoretically brings great returns. This financial theory operates on the premise that your interest can be compounded over and over, and your investment will just keep on growing. It really is a magical idea. However, you do need to understand it.

Here we will explore continuously compounded interest. We will compare it to regular compound interest or discrete compounding as it is sometimes called. We will also compare the formulas and look at some examples. We have come to understand that examples often help the figures come to life and really allow you to compare and understand the concepts. So without further ado, let’s get into it!

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What Is Continuously Compounded Interest?

Continuously compounded interest can be defined as compound interest being taking to the extreme.

It hinges on the concept that interest can be added to the account balance (principal and all the interest that is previously accrued) and reinvested for an infinite or never-ending number of periods.

It is the idea that an investment can go on in perpetuity. It is an addictive concept. This, of course, is impossible in real life. It is simply a theory. However, it has its place in the financial world and must be understood.

As an investor you will enjoy the perpetual growth of your investment. Continually compounded interest still yields the most interest compared to interest earned monthly, quarterly, bi-annually, or annually with discrete compounding.

Continuously Compounded Interest

This is important!!!

As an investor you will enjoy the perpetual growth of your investment. Continually compounded interest still yields the most interest compared to interest earned monthly, quarterly, bi-annually, or annually with discrete compounding.

What Is The Formula For Continuously Compounded Interest?

The formula for continuously compounded interest has its origins in a formula that you may not have come across before. This formula is for the future value of an interest-bearing investment. When speaking about continuously compounded interest, you will see we talk in terms of future value and present value. Future value is the value of your current investment at a future date. The present value is the value of your money today. Now that you know these terms you can see the formula here:

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

In this equation the values represent the following:

PV =Present value

i =the expressed interest rate

n = The number of compounded periods

t= the time or term in years

This formula is modified to create the continuously compounded interest as follows:

FV = PV x e ^ (i x t),

As you may have noticed that “e” was added to this equation. This stands for the mathematical constant calculated as 2.7183.

I should also mention that the upside-down V in the equation stands for exponent which means ‘raised to power’ or “to the power of”.

As always, we provide an example so that you can see continuously compounded interest in action.

FV = PV x e ^ (i x t)

FV = £5,000 x 2.7183 ^ (15% x 1) = £5,809.17

What is compound interest?

The easiest compound interest definition to understand is:


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When you use compounding, interest is calculated on the initial amount or principal and includes all of the accrued interest from previous periods on loan or in the case of an investment, the deposit.

When compounding interest, you must always consider the frequency or the number of compounding periods, since this is a pivotal part of the equation.

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Compound Interest Accounts

Continuously Compounded Interest

The Compound Interest Formula

What is the difference between compound interest and continuously compounded interest?

The difference between compound interest and continuously compounded interest lies in the approach of the calculation. Compound interest or discretely compounded interest as it is sometimes called adds interest at various intervals i.e., annually, bi-annually, monthly, weekly, and even daily.

Continuously compounded interest on the other hand uses the formula created from the future value formula and adds interest at minuscule intervals.

In order to truly compare continuously compounded interest and discretely compounded, we should look at the numbers together:

Annual or yearly compounding: FV = £5,000 x (1 + (15% / 1)^(1 x 1) = £5,750.00

Semi-Annual or bi-annual compounding: FV = £5,000 x (1 + (15% / 2)^(2 x 1) = £5,778.13

Quarterly Compounding: FV = £5,000 x (1 + (15% / 4)^ (4 x 1) = £5,793.25

Monthly Compounding: FV = £5,000 x (1 + (15% / 12)^ (12 x 1) = £5,803.77

Daily Compounding: FV = £5,000 x (1 + (15% / 365)^ (365 x 1) = £5,808.99

Continuous Compounding: FV = £5,000 x 2.7183 ^(15% x 1) = £5,809.17

As you can tell from the numbers, the daily compounding and the continuous compounding are very close. So as for as investments go, there is not much difference between the two amounts. Although, continuously compounded interest is marginally more.

What Are The Benefits Of Continuously Compounded Interest?

What Are The Benefits Of Continuously Compounded Interest?

Reinvestment will continue forever.

We know there is no “forever” in finance. However, one of the advantages of continuous compounding is that your interest is reinvested into your portfolio over and over again for a theoretically infinite number of periods. So as an investor you will enjoy the perpetual growth of your investment. Continually compounded interest still yields the most interest compared to interest earned monthly, quarterly, bi-annually, or annually with discrete compounding.

Interest just keeps on growing.

When your investment is being continuously compounded, both the principal and the loan keep on increasing. It makes it easier to grow the returns in the long term. The other types of compounding that we discussed only earn interest on your initial investment or principal. The interest is paid out as it is accrued against your account. This reinvestment of interest allows you to earn at a rapid rate for an immeasurable number of periods.

Do any banks offer continuously compounded interest?

Banks often offer daily compounding which as we can tell from the figures above are really close to what would be achieved with continuously compounded interest. Sometimes financial institutions may claim that they compound continuously but it is paid monthly. Always remember that continuously compounded interest is an academic concept.

As you can tell continuously compounded interest is the holy grail of compounding investing in theory. However, we simply recommend that you choose an investment that offers compounded interest. You should ask if interest is being compounded daily since this offers the best rates.

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