What does it mean for interest to be continuously compounded? Part one of this series of posts explains the meaning and how continuously compounded interest is calculated. Part 2 will discuss how an understanding of the principles behind interest calculation can help both savers and borrowers navigate the confusing subject of annual interest rates to determine what they are really earning or paying.
The best way to describe how continuous interest is calculated is by using an example. Let’s say that a bank comes to you and offers to give you 100% interest if you will deposit $1,000 into a savings account with them for 1 year. (Lucky you!)
$1,000 1 Year 100% Interest $2,000 one year
This is a very simple calculation as $1,000 (1+1.00), 100% interest = $2,000 after one year. $1,000 of interest and $1,000 of the original principal.
Another bank comes to you and offers to pay you 100% interest but will pay 50% of the 100% six months after the deposit and then the other 50% at the end of one year.
$1,000 6 Months, 50% Interest Paid $2,250 one year
$1,000(1+.50) = $1,500 $1,500(1+.50) =$2,250
WOW!! It looks like we have a money-making machine!! By compounding every six months we increase the interest payment by 25%!!
A third bank come to you and offers 100% interest for your $10,000 but they offer to pay out or compound the interest monthly for one year! Let see what happens in this case.
$1,083 $1,174 $1,271 $1,377 $1,492 $1,616 $1,751 $1,897 $2,055 $2,226 $2,411 $2,613
BOOM!! Compounding every month increases the interest payment by 29% bi-annual (6 months) compounding. This really might be a money-making machine!! We will see later that it is true that the more often interest is compounded the more interest is earned, but, as it is with most things, there is a limit.
Calculating compounded interest uses an algebraic formula that makes it easier to calculate the interest earned so that we don’t have to do each calculation for every compounding period as I’ve done for you in the illustration above. The formula is:
$1,000(1+(1.00 (100% interest))/12 )^12=$2,613
This formula yields the same result as my illustration where each month was calculated individually. Algebra is a faster way to determine the result!
Let’s now go a little crazy and find the limit to compounded interest. Let’s see what the interest amount is if we compound every day of the year. To avoid calculating every day for 365 days the algebra formula will be used for ease of calculation. The formula is:
$1,000(1+(1.00 (100% interest))/365 )^365=$2,714.57
Now the increase in interest is slowing way down. The increase in interest paid from compounding monthly to compounding daily is only 3.89%, from $1,613 to $1,714.
Now let’s try compounding every minute of each day for 365 days. Every minute is 365 X 24 hours X 60 minutes = 525,600 minutes in a year.
$1,000(1+(1.00 (100% interest))/525,600 )^525,600=$2,718.28
We are really moving to the limit now as the difference in the interest from compounding daily to compounding every second is very small at $3.21 or 1/5 of one percent.
Let’s go one step further and compound every second. There are 525,600 minutes *60 seconds in a minute = 31,536,000 seconds in a year. Let’s see if my Casio Scientific calculator can handle this equation:
$1000(1+(1.00 (100% interest))/31,536,000 )^31,536,000=$2,718.28
See what happens? We have found the limit to two digits to the right of the decimal place for continuously compounded interest. The limit is compounding every minute for one year. This illustrates that compounding interest more often is advantageous up to the point of compounding every minute for savers. Unfortunately, it works as a disadvantage to borrowers. Stay tuned for my next post, which will tackle that subject.
Practically, banks would never pay 100% interest, but this serves as a good example. The equation we used for the calculations above was discovered by a 17th century Swiss mathematician named Leonhard Euler. While Euler discovered the equation by trying to prove the mathematics for compound interest, his equation and work have applications in all kinds of scientific fields like physics, chemistry, biology and any field where growth is measured. It has become so famous that Euler has his own number called ℯ or Euler’s number. Euler’s number is 2.718281828459045 to 15 digits. Euler’s number, like the more popular number , is an irrational number that never repeats and goes on into infinity. If we multiply the $1,000 in our savings account for one year, by Euler’s number we get:
This is the same result we got using the algebra formula used above to calculate daily and every second compounding. Euler’s number can be used to calculate compound interest for any interest rate. Let’s say we want to know what the compounded interest rate is for 6%. The simple interest rate is $1,000*(1+.06) = $1,060 or $60 of interest for one year. The continuously compounded rate for 6% is:
$1,000*e^.06=$1,061.84 or $61.84 of interest annually.
Increasing interest earned through compounding interest more often ultimately hits a wall. The maximum increase is reached at hourly compounding. Leonhard Euler discovered the limit by using a mathematical formula which provides a convenient shortcut for figuring out the total earnings on savings in a compounded interest account. The result of his work is boiled down to the mathematical constant, ℯ. Thanks, Euler!