Compounded Interest

How money or value or perhaps even The Blob grows. In the simplest of all settings, interest can be as straight forward as, I’ll loan you $10 for a hamburger today, and you pay me back $11 on Tuesday. In this case, Wimpy is taking out a loan of $10 at 10% simple interest.

That’s great for a cartoon, but it’s not the case in most real-life examples of interest. In real life, a loan you take today might accrue interest every year, month, day, or even continuously. Although the last case is one of the most interesting ones as it deals with a special number ‘e’, I just want to address the more intuitive cases.

Let’s say Wimpy is keen on that hamburger today, but he won’t have any cash until Tuesday again. This time, his usual rubes are all cash strapped as well, so poor Wimpy has to go to a formal lender. This lender is OK with the loan, but as insurance against Wimpy’s ability to pay back the loan on time, he insists on compounding interest every week.

“Let’s be clear about this Wimpy,” his loan officer says, as he walks him through the conditions. “You can have your $10 today at 10% interest. The loan is due on Tuesday, and that will come to a total of $11. If you can pay it off then, great. But if you need more time, we will be compounding the interest – that means that you will essentially be getting a new loan of $11, at the same 10% rate.”

“OK,” says wimpy and leaves his mark on the loan document.

 

This is exactly the right way to think about it.

1. initial loan is made: $10 at 10%, due in one week.
2. If the loan continues, another 10% is charged on the new total.
3. Week after week, this goes on until Wimpy can pay up or he’s referred to collections and they repossess his barbershop.

 

Mathematically, this takes the Principal (loan amount ) and multiplies it by 10% every week.

1. week 0: $10
2. week 1:$10 + 10% = Principal x 1.1                                                          -> $11.00
3. week 2: ($10 + 10%) x 110% = (Principal x 1.1)²                                              -> $12.10
4. week 3: (($10 + 10%) x 110%) x 110% = (Principal x 1.1)³       -> $13.31

 

This can be generalized by the formula:

Amount owed at time      t = P (1 + R)^t

Where P = principal

R = rate (expressed as a decimal)

T = the number of times interest is compounded

(whether its days, years, months, whatever)

(10)(1+.1)³               -> $13.31

 

This goes for any compounded growth.

The Blob arrived in Downingtown, PA in 1958. At first it was just something riding into town on a meteorite. But soon after, an old man touched it and got it stuck to himself. Steve McQueen comes to the rescue and gets the old fellow into town to see a doctor. Meanwhile, it becomes evident that the blob is not letting go, and is hurting terribly. Dr. Hallen decides to amputate, but before he can, the blob grows large enough to eat the old man, then a nurse, and then the doctor.

From then on, the thing just keeps growing. Let’s say it grows at a rate of about 50% an hour and use the same formula…

 

1. hour 0: 100g
2. hour 1:100g + 50% = Principal x 1.5                                             -> 150g
3. hour 2: (100 + 50%) x 150% = (Principal x 1.5)²                                               -> 225g
4. hour 3: ((100 + 50%) x 150%) x 150% = (Principal x 1.5)³        -> 338g
5. hour 24:         ——-à                                                                       ->1,683,411g

 

You can really see how this thing gets huge fast (or at least massive, we never talked about the density of this thing).

Graphically, the blob’s growth looks like this:

One troubling thing is that this could also represent the balance on a credit card that isn’t attended to.

“It crawls… It creeps… It eats you alive!”

-Tagline, The Blob 1958

 

For a good explanation of interest, compound interest, and ‘e’ – check out Khan Academy’s lectures on this or this site that does a great job illustrating the difference between several types of interest.