PSCC - Continuously Compounding Interest Lesson

Continuously Compounding Interest

Calculating Interest Using Pe^rt

We've learned that you can earn more money by using compounding interest. You also earn the most depending on how many times a year that interest is compounded. So, what's the most you can earn? If you compound the interest daily, or 365 times a year? Can you compound the interest more than that? How about every hour or even every minute? 

The answer to this is to continuously compound the interest. Although it's hard to picture how you could compound interest an infinite amount of times, this is mathematically possible using the formula below.

 

$A=Pe^{rt}$

 

A = Total Amount

P = Principal Amount

r = interest rate

t = time in years

e = the exponential constant (approximately 2.718)

 

What is this "e" in the formula? Well, when interest is compounded more frequently, the amount of interest earned each time actually becomes smaller, but the total amount grows faster. Since the amount grows smaller each time, it will eventually get so small, that the difference is insignificant. This is known as a "limit". The exponential constant "e" signifies this phenomenon. In fact, this value (approximately 2.718) occurs so frequently in physics and economics that it has been given the symbol of "e" in a similar way that 3.14 is represented by the symbol "π".

 

Let's look at an example to compare simple interest, compound interest, and continuously compounding interest. 

 

Example: Anayo deposits $800 in her savings account. The account pays an annual interest rate of 6.00%. If she makes no more deposits or withdrawals for a year, find the total amount in her savings account after a year. Let's start with simple interest. How much would Anayo have in her account if the interest paid out just once a year?

Simple Interest: $I = PRT$

$I=800\cdot0.06\cdot1$

$I=48$

The total amount in savings would be 800 + 48 = $848.00.

 

Now, let's say Anayo's account uses monthly compounding interest instead.

Compound Interest: $A=P(1+\frac{r}{n})^{nt}$

$A=800(1+\frac{0.06}{12})^{12\cdot1}$

The total amount in savings would be $849.34.

 

Lastly, let's see what Anayo's account would earn using continuously compounding interest.

Continuously Compounding Interest: $A=Pe^{rt}$

$A=800e^{0.06\cdot1}$

The total amount in savings would be $849.47.

 

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Graphing Interest Growth 

There doesn't seem to be much of a difference in interest earned after only a year. But this would change greatly the longer the money was left in the savings account. Take a look at the graphs below. How are they different? The graphs shown represent a simple interest account, an annual compounding account, and a monthly compounding account. Answer the questions shown. Click on the checkmark under the activity to check your answers.

IMAGES CREATED BY GAVS