The Calculus of Uncertainty: the Problem of Random Inputs Insofar As It Relates to a Rapidly-Diminishing Curve. Louis Shalako.

"Come on, Louis. You can do it." Miss Purviss, my Grade One teacher.

Louis Shalako

I need to talk to a mathematician.

(Either that, or invent The Calculus of Uncertainty. – ed.)

***

Okay. A couple of weeks ago on Facebook, I said that if you could put only two extra payments on a $10,000.00 loan over four years, it is better to do it sooner rather than later.

I also said I couldn't really show the math. I will attempt to do so now. Just to keep it simple, we'll call it a 12 % Annual Percentage Rate, or one percent per month. Every month, they’re going to whack you with one percent on the remaining balance.

Assuming roughly a $250.00 monthly payment, let's say you throw an extra fifty bucks on that very first payment. Calculating on the balance, instead of owing $9,750.00, now you only owe $9,700.00. Either way, the bank tacks one percent onto the balance, and this is what you now owe. One percent of $9,750.00 equals $97.50, so now you owe $9,847.50. One percent of $9,700.00 is $97.00. Now you owe $9,797.00. Subtract one from the other. You have saved exactly fifty cents in interest, and there’s your fifty bucks too.* But after your next $250.00 payment, having put in an extra fifty only once, you owe $9,547.00. Add in one percent, i.e., $95.47. Not having put down any extra this time, your new balance would be $9,642.47. This is where it gets interesting. Subtract one from the other and you get a figure of $95.47. You have done something very strange there...your fifty bucks is there, and the other $45.47 had to come from somewhere.

So let's make that second payment of an extra fifty bucks on a balance of $9547.00.

That’s right, a measly fifty bucks.

That's now $9,497.00. One percent of that is $95.00 rounding off. The interest on your first, unmodified payment was $97.50, now, with only an extra pair of fifties on there, it is down to only $95.00. What happened there? Your interest is calculated on the balance, which is going down. By throwing in extra money, you are reducing the balance, as well as the interest payment, as quickly as possible. (It's still a one-percent rate for our purposes, and truth is, this math don't quite work out as my own rate is 8.99 %. I've also rounded up the payment a little, just to keep it simple.)

At this point, you still have forty-six payments. So, it’s $2.50 times forty-six…right? So, all other things being equal, you have saved roughly $115.00 TOTAL interest, over the course of the four-year term, and your last payment will obviously be a hundred less because you've already paid it off. If you did that about halfway through the loan period, you would arguably only save about $57.50 in TOTAL interest. You will also save any interest on the last hundred bucks, because you’ve already paid that, this is where my own lack of math skills makes the logic a bit muddy. (Or maybe you’ve already said it, three different ways. – ed.) Now, if your loan rate is higher, you save more money with everything extra you can put on it.

If I’m paying ten percent a year in interest, and you’re paying twenty percent, that is all calculated on the balance. The story is not really about rates per se. Lower is obviously better, no matter what you do with the payments.

However, you can look up stuff like this on any number of websites. These guys tend to think of making ‘extra’ payments a regular and monthly occurrence, for the full term of the loan, so their online calculator reflects that. In my current model, I cheated a bit and calculated ONE extra (full) payment of $250.00 as an ‘extra’ $5.20 per month in order to use the calculator. (It works out, more or less, although it must clearly have some curve of inaccuracy.) In any case, the sources agree as to the ‘mathematical certainty’ of saving a lot of money in interest and not incidentally, paying off the loan before term. So far, I have only made one extra payment on the loan, about three or four months in. My big goal is to put something extra on there in October, if I can possibly do it—it doesn’t have to be a full payment. A hundred would be an achievement in itself. The hours at work are slowing down and money will be tight over the winter. In January, I will get an HST rebate, and that might represent another opportunity. The opportunity to make a short-term sacrifice in order to receive a long-term benefit.

If you make one full, extra payment, $250.00, that very first month, you knock one month off the term of the loan, and that’s two-point-five times what we have done with our pair of fifties. So, that works out to $287.50 (or something on that order. – ed.) in TOTAL interest savings over the term of the loan, based on the simplistic reasoning of our first model. It’s actually more, and I would invite the reader to do the math on their own, using their own figures and their own loan or loans for a model. It bothers me that the bankers can do this math, (obviously), but I can’t. I have no way to check their figures, this is especially true in terms of throwing down random extra payments. I know it works, I just don’t quite know how…or by how much. Hence the Calculus of Uncertainty: the Problem of Random Inputs, Insofar As It Relates to a Rapidly-Diminishing Curve. I say that because conventional wisdom, as imparted by bankers to consumers, is that much of your first payment is taken up by interest, (which must, in effect, diminish over the course of the loan), while in my own interpretation, the interest is all on the back end of the loan—an example of what I call ‘inversion’, a kind of way of looking at bullshit and making sense of what they are Really Telling You.

What I'm telling you, is that if the cost of a four-year, $10,000.00 loan is $1,500.00, then the last six payments are essentially the cost of the loan. You just have six months of debt slavery to work off, and then you're done.

Basically, the less you know, the more they can take you for—but my own math shows that it’s a hundred out of the first two-fifty.

Other reasons for murkiness in the math include the fact that I might have gotten the loan halfway through the month. My first payment would, therefore, only be two weeks away.

The loan will be theoretically paid off, halfway through a month.

I might have put the extra payment down partway through the month as well.

Poor people study the sports pages, rich people study the interest rates. If my financial literacy was better, I could explain it better.

Here’s a nice, simple model. If you go in exactly two months before term, and put down exactly two payments of $250.00, or $500.00, at a twelve percent APR, (or one percent in our model), you should save exactly one percent, or five dollars. In other words, you give them $495.00. Your balance is zero, there is no compounding on zero. The real question, is what happens when you do it at the front end of the loan as opposed to the back end. 

(Editor: he was saying ten bucks, then he got confused and said five bucks. But it's two months at five each, better yet, five the second last month and two-fifty for the last one.)

As you can imagine, however intuitively, it should look like a pretty predictable curve. It starts high on one end, and ends up low on the other.

Any input, positive or negative, along that time-line will affect the resulting radius…simple, really. The real question is how much, in dollars and cents.

Two hundred and ninety bucks, for that one extra payment in total interest savings, is the best answer I can come up with. We save money, and shorten the term of the loan, which are two separate and distinct ends. Interestingly, the longer I wait, the less benefit I will receive…kind of an allegory there.

Speaking of the end, we have reached the point where I am only confusing myself further.

*Just as the interest compounds on the balance, your savings will also compound, so it’s actually more than fifty cents.

#math #work #money #business #superdough #Louis

Thank you for reading.

END

(Editor’s Note. Your first interest charge would be calculated on $10,000.00, or about a hundred bucks, but other than that, the story really is quite all right.) He also has some books and stories on Google Play.