How is the Principal Payment determined on a loan

Question

Hi everyone,

I have a question about how much in excess of the interest is paid in each payment of a fixed rate loan. Suppose you have a 100k loan at 6%. An example from investopedia shows a schedule of

Payment

001: Principal($99.55)____Interest($500.00)___Principal Balance($99,900.45)

180: Principal($243.09)___Interest($356.46)___Principal Balance($71,048.9?)

360: Principal($597.00)___Interest($2.99)_____Principal Balance($0)

I'm having trouble understanding how the initial principal payment is determined. The initial $500 interest payment makes sense: 100k at 6% divided into 12 months. I'm assuming that's how the math works out to have the loan paid in 30 years and keeping an even $600 paid every month. Is that right?

I'm also curious if this sort of paydown schedule yields any tricks for smart investing. For example, the principal paydown is slow at the beginning, but speeds up. Does that mean loan amortization only becomes a significant part of your investment returns after a certain inflection point, does that change for other loan terms or interest rates? Anything you feel is really valuable to know about loan paydowns that may not be obvious?

Chris Mason · Answer

It's just the formula for how annualized interest on monthly payments works. Same exact formula for credit card interest, for a car loan, or anything else, assuming annualized interest on monthly payments.

Rate * balance / 12 = interest due for the month.

How much of an amortized P&I payment is P is just P minus I.

Not to put too fine a point on it, but why do we teach quadratic equations in school, but not this? Most people will not be solving quadratic equations in their entire adult lives, but will be paying interest at some point....

Loan sharks trick people by not having interest annualized. "2% a week" is what they say, and we don't teach this in school, so folks fall for it at the payday loan store, because "2% is a great interest rate."

2% * $10k * 52 / 12 = $866/mo.

Annualized interest rate on "2% a week" is 104%. 2% * 52 weeks. APR would be 104% plus whatever fees they are charging. And people sign up for that all the time, but don't worry! If there's a catastrophe that could be averted by solving a quadratic equation, they're ready!

Chris Mason · Answer

P&I minus (rate * balance / 12) = the amount going to paying down the balance.

Say the P&I is $942 and the current balance is $173,111 and the rate is 5%.

$942 - (5% * $173,111 / 12) = $220.70.

Note that it's not even necessary to know if it's a 30YF or 24YF or 15YF.

In terms of calculating what the total P&I payment will be, there's a giant formula, you do need to know the term in that case, I did it like 3 times when I was first licensed to prove to myself it wasn't witchcraft, after that I just use a financial calculator. HP12-C is the ubiquitous one, like how TI-83s are used by algebra students everywhere, same thing.

Martin Neal · Answer

@Shafi Noss The simple answer is that’s the magic behind an amortization schedule. Long answer is ask a college math teacher or experienced loan officer (lbvs). They are tricks to pay off a loan quickly (search payoff my loan in 5-7 years). Hope this helps.

Matt Devincenzo · Answer

The term you're looking for is a logarithmic function....it's the inverse of an exponential function. I'm an engineer not a mathematician so this may not be the most technical of descriptions, but should convey the general idea....in layman's term: the mortgage payment is mathematically determined based on your interest payment being logarithmic (i.e. it reduces each month) and your principal payment being exponential (it increases each month) such that the total payment stays the same.

The concept that "the bank has chose to front load the interest on a loan" isn't really accurate...as if somehow the bank has concocted a payment that you pay interest up front arbitrarily. The interest you pay each month is specifically, mathematically, exactly based on the balance you have at that point in time. And the overall payment is computed (see again mathematical) so that it will pay off in the correct (previously agreed to) number of months. There is no bank mystery behind the payment such that they are somehow getting one over on the public. The solution to paying less is simple; pay more initially to reduce your balance faster, which shifts your graph to the left reducing the total interest paid....with an ROI which is equal to the interest rate of the note.

If you wanted to pay an identical amount of principal each month and achieve a 30 year payoff that way, you could (again) mathematically determine a function for that....but then your payment amount would change every month and would initially be alot greater, though at the end would be much smaller.

Here is a graph and tables of the two payments...hopefully it helps.

Shafi Noss · Answer

Thanks everyone for the responses, and thank you @Chris Mason for the detailed explanation. Do you know what that formula is called? I'm curious to learn more about it.

Tom S. · Answer

@Shafi Noss  Yes, that payment schedule is correct.  Almost all of your payment goes to interest in the beginning.  $500 in interest for the first payment, $499 in interest for the 2nd payment, etc.

Biggest trick is to use a shorter term loan (I mostly do 15 or 20 year).  It will be a larger payment but you're paying far less interest and build equity quicker.  Or do 30 year loan and pay extra each month. The extra goes straight toward principal.

Last trick is to make sure you have excellent credit.  A 1 point interest rate difference at the start of a 30 year loan can make the difference of thousands in dollars over the life of the loan.

Craig Jeppesen · Answer

The first step is to calculate the payment amount for the loan to pay off over it’s amortization term. Use a pmt calculator; the math is pretty complex.

Next you calculate the interest: balance times rate divided by 360 times 30.

Finally you calculate the principle by subtracting the interest calculated by the total payment.

How is the Principal Payment determined on a loan

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