Howto Find the Growth Rate of Interest Compounded Continuously

Knowing the formulas for calculating compound interest, and especially continuous compound interest, can help you evaluate investing opportunities.

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Types of Interest

There are two types of interest: simple and compound.

Simple interest: a set amount that is a percentage of the principal and is paid out once every time period; for example, once every year, semi-annually, or quarterly.
Compound interest: the interest generated is reinvested, thus becoming part of the principal, and interest is then generated on this blended amount. Compound interest is a form of exponential growth, or b ⁿ , where b is the base and n is the exponent.

Everyone is familiar with the case where the base quantity doubles (n=2). The most famous example of this is giving someone the choice between either receiving $1 million today, or else receiving a penny and having it double every day for a month. Most people would take the $1,000,000 but, when the penny grows at 100% per day (doubling) and that interest is reinvested, the penny-investment ends up being worth over $5,000,000 on day 30.

Simple vs. Compound Interest

If you invested $10,000 at 5% simple interest for 10 years, you would receive $500 in interest every year, for a total of $5,000 at the end of year 10. This would make your total of principal plus interest equal to $15,000.

If that same $10,000 were invested at 5% compounded yearly, you would have:

Year	Interest Earned	New Principal
1	$500.00	$10,500
2	$525.00	$11,025
3	$551.25	$11,576.25
4	$578.81	$12,155.06
5	$607.75	$12,762.82
6	$638.14	$13,400.96
7	$670.05	$14,071.00
8	$703.55	$14,774.55
9	$738.73	$15,513.28
10	$775.66	$16,288.95

That is an additional $1,288.95 over what you would receive with simple interest.

What is Continuous Compound Interest?

Continuous compound interest is the amount that can be achieved if interest is calculated continuously, or over the smallest increment of time possible, and reinvested. In reality, interest is compounded annually, semi-annually, quarterly, or monthly.

Compound Interest Formula

The formula for determining compound interest is:

FV = PV * [1 + (r / n)]^{(n * t)}

FV = future value
P = principal
r = interest rate
n = number of compounding periods
t = time in years

Compound Interest Examples

Let's look at the value a $10,000 principal investment placed into an account that pays interest of 20% per year would achieve:

Annual Compounding

$12,000 = $10,000 * [1 + (.20/1)]^{(1 x 1)}, where n = 1

Semi-annual Compounding

$12,100 = $10,000 * [1 + (.20/2)]^{(2 x 1)}, where n = 2

Quarterly Compounding

$12,155 = $10,000 * [1 + (.20/4)]^{( 4 x 1)}, where n = 4

Monthly Compounding

$12,194 = $10,000 * [1 + (.20/12)]^{(12 x 1)}, where n = 12

Weekly Compounding

$12,209 = $10,000 * [1 + (.20/52)]^{(52 x 1)}, where n = 52

Daily Compounding

$12,213 = $10,000 * [1 + (.20/365)]^{(365 x 1)} where n = 365

If we look at the amounts achieved by these different compounding periods, we see that the increases are becoming smaller and smaller as the compounding period shortens:

Annually to semi-annually: $100
Semi-annually to quarterly: $55
Quarterly to monthly: $39
Monthly to weekly: $15
Weekly to daily: $4

Continuous Compounding Formula

What would happen if the interest was compounded every hour, or every minute, every second, or even a time period less than a second? When the number of compounding periods within a given time duration becomes infinitely large, this is known as continuous compounding, and its formula is:

FV = P * e ^rt

P = principal
e = natural log, or 2.7183
r = interest rate
t = time

The number "e" is so named because it is also known as Euler's Number after the Swiss mathematician Leonhard Euler. It is the limit of the formula (1 + 1/n)ⁿ as n approaches infinity. While it often appears as 2.1783, e is actually an irrational number having infinitely many integers following the decimal point.

If we invest $10,000 at an interest rate of 20% compounded continuously, after one year we would have:

$12,214 = $10,000 * 2.7183^{(.20 * 1)}

Notice that this is only $1 more than we would get from daily compounding.

Continuous Compounding Examples

Example 1

If a credit union pays an annual interest rate of 5% compounded continuously, and you invest $10,000, how much will you have in your account after five years?

Here, we're solving for the future value:

FV = $10,000 * 2.7183^{(.05 * 5)} = $12,840

Example 2

If you have a $50,000 investment that you want to increase to $100,000 within eight years, what continuously compounded interest rate will you need to receive?

Here, we're solving for r, our principal is $50,000, our future value is $100,000, and our time period is 8 years. Plugging those values into the formula and solving for r, we get:

$100,000 = $50,000 * 2.7183^{(r * 8)}

Dividing both sides by $50,000, we get

2 = e^8r

Dividing both sides by e, or 2.1783, we get

0.693 = 8r

Dividing both sides by 8, we get

0.087 = r, which when multiplied by 100 = 8.7%

This is the rate that you'd need to receive.

Example 3

After the birth of a child, if a parent wanting to create a college fund invested $5,000 into an account paying 6% compounded continuously, how much would be in the account when the child turned 18?

Here, we're solving for future value:

FV = $5,000 * 2.7183^{(.06 * 18)}= $14,724

This is, sadly, not enough for a four-year education at most colleges or universities.

Bottom Line

Now that you know that the interest achieved over various compounding periods becomes less and less as the periods become smaller and smaller, you're in a better position to evaluate interest offers.

Disclosure: I/we have no stock, option or similar derivative position in any of the companies mentioned, and no plans to initiate any such positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

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Source: https://seekingalpha.com/article/4484912-calculate-continuous-compound-interest