Amortization tables... just another fun thing to do with finance

So, one of the most exciting parts of what I do is work with amortization tables. Amortization tables are essentially a way of keeping track of what portion of your loan payment goes toward reducing your principal, and which portion goes towards interest. In general, loan payments that we want to amortize are done on a month-to-month basis.

To figure out an amortization table for a loan is relatively easy; the first step, as usual, is to define your financial criteria: the loan amount, the amount of years/periods, and interest rate.

So, lets say we have loan amount = 100,000$, with a 30 year time frame, paid monthly, and an annual interest rate of 5%.

One of the most important things to understand about finance is that your "periods" are not how many years that you have, but how many payments you are making. In this case, you are going to be making 12 x 30 payments = 360 payments. This means that your true interest rate is not 5%, but (1.05)^(1/12)-1. The difference isn't HUGE on a small scale, but if your principal is big enough, your time length long enough, or your compounding periods small enough, the difference can add up... for example

FV= 100(1.05)^1 = $105.00

FV= 100(1.004)^1 + 100.40(1.004)^2 + 101.20(1.004)^3... k(1.004)^12= $105.13 and you can see where this is going. In other words, the more frequent the compounding, the more "effective" the interest.

In the case of a loan, sadly, this works against you, but you can think about the implications of that yourself: just remember, when looking at a car payment APY is written yearly for a reason!

So, now that we know our monthly interest rate (AKA the effective interest rate) we can calculate the periodic payment using the annuity function:

Payment= [interest*principal(1+interest)^# of payments]/[(1+interest)^# of payments -1]

So for our example, that's essentially: [.004*100,000(1.004)^360]/[(1.004)^360-1]

in other words, we're looking at a monthly payment of $536.82.

But this is where the fun part comes in... AMORTIZATION... you see, because this is a loan, you are really having to pay both the amount of money you borrowed and the accumulated interest on that money. So, by the end of this lovely investment, you have pulled not $100,000 from your pockets, but $193,255.20! (which, conveniently, is $536.82 x 360) Why? Because finance is fun. Now, we can obviously deduce from the above equation why the monthly payment is greater than the "divide loan by number of years payment," but an amortization table lets us see how much of that loan payment goes towards the principal and how much goes towards the interest; it is, in a sense, the mechanism by which the formula is explained.

Our amortization table for the above would appear like this:

MONTH PAYMENT TO INTEREST TO PRINCIPAL BALANCE

1 $536.82 416.66 120.15 99879.14

2 $536.82 416.16 120.65 99759.18

....

The underlying math is based on the concept that the loan balance is only reduced by the money paid to the principal, and not the money paid to the interest. Oh, those sneaky financial folk!

First month: loan balance= $100,000, interest = .0041666, loan x interest = 416.66 <-- the amount paid to interest

$536.82 - $416.66 = $120.15<-- the money paid to principal

In the second month, the balance is reduced by only $120.15, leaving a balance that is larger than what we would "expect."

As you can guess, as time goes on, the amount of money paid to interest lessens, whereas the amount of money paid to principal increases; both always add up to $536.82, however.

I was thinking earlier today that there is probably a good way to express this process with matrices. Now I am not so sure that is the case, but I am also not fully confident in matrices. Part of the glory of the amortization table is that it is so laborious. In fact, if you haven't already guessed, the word "amortization" is related to the latin prefix "mort" -- which means death. No, not that it's deathly boring to run an amortization table, but that you can really dig yourself a ditch with how much interest you're paying.

Why is this important? In some accounting and property situations, your interest can be written off as deductible. Knowing how much interest you can deduct is important; this ability to deduct will obviously decrease over time. Additionally, your net worth is obviously affected by how much you actually "owe" (liabilities)... not how much you paid off in interest. In the long run, it is always valuable to look at an amortization table before getting into any kind of loan...

In short, amortization is the tedious way of stating the mantra of financial folk world wide: YOU BETTER BE KNOWIN' WHERE YOUR MONEY'S GOIN'.