How to calculate for mortgage? 30 year fixed rate loan w/ 4.25% interest rate

[Kayh]

How do I manually calculate the unknown mortgage for a 30 year fixed rate loan with annual interest rate of 4.25%, if only given the following information?
Monthly Principal & interest payment = $1420 at 4.25% (FHA Loan)
I've been given an answer of $285,400- but i don't know how that answer was derived.

 

[JeffM]

How do I manually calculate the unknown mortgage for a 30 year fixed rate loan with annual interest rate of 4.25%, if only given the following information?
Monthly Principal & interest payment = $1420 at 4.25% (FHA Loan)
I've been given an answer of $285,400- but i don't know how that answer was derived.

There is a formula, but I doubt giving you the formula would answer your question. It is ugly to calculate without a computer or a good hand calculator. Excel will compute the formula for you.

Explaining the formula is hard.

Demonstrating in Excel that the formula is correct is fairly easy.

Please clarify your question.

 

[JeffM]

WHO gave you that answer?

I get ~288,653 or ~291,442,
depending on how the rate is treated.

Agree, Jeff?

The problem here is that we are dealing with federal regulation, not pure math. There is a formula required for private lenders, worked out but not defined in a so-called Schumer Box, but the FHA is a federal agency that may not be covered by the same law (the government likes to do that: pass one set of rules for itself and another for the rest of the world).

Rates on consumer loans from regulated private entities like banks are quoted in annual rates (not yields, which is for bank deposits). So, assuming we are not dealing with some legal weirdness, I get:

\(\displaystyle 1,420 * \dfrac{1 - \dfrac{1}{\left (1 + \dfrac{0.0425}{12} \right )^{30 * 12}}}{\dfrac{0.0425}{12}} \approx 288,653.15.\)

Frequently, however, the last payment is slightly less than the other payments due to rounding issues, but that cannot be enough to account for this discrepancy. What is likely to be happening is deferred fees. I forget how they are to be handled under Federal Reserve Regulation Z, and perhaps the FHA is not covered by Federal Reserve Rules. But if there are deferred fees, the standard annuity formula must be adjusted. As I say, this is not pure math, but math determined by legal rules.

EDIT: One of my first jobs when I started working in banking was to translate legal rules into mathematical algorithms, but that was 40 years ago. I have not picked up the Federal Register in the six years since I retired and do not plan to resume now. If the OP gives us the details in the Schumer Box, we can figure it all out for him. But my bet is that it is deferred fees or points.

SECOND EDIT: The OP asked for the formula. There it is above. I doubt it makes him or her feel any better. It makes me feel slightly nauseated every time I look at it.

 

[JeffM]

My point is that stated is: "annual interest rate of 4.25%".

~288,653 is the result IF 4.25 APR (i = .0035416...).

But if 4.25 EPR, interest is lower (i = .0034744...).
Results in mortgage of ~291,442

I was trying to indicate to OP that this needs clarification.

Sorry. I got that, but my response was not clear.

My point is that when we get into the minutiae of mortgage calculations in the US, we descend into a legal quagmire where pure math will not suffice. The legal standard in the US is to give an annual percentage rate, and you find the monthly rate simply by dividing by 12. You are correct that that is a mathematically dumb way to do it, but it is the legally approved default on consumer loans in the US. So I only bothered to confirm your result on that assumption. There is then a way to deal with fees, costs, and "points." Many of those get "financed." I suspect that is what is happening here.

So, yes, you are correct. We need a lot of detailed and technical information to be able to explain why that payment at that rate and term will support a loan of that amount.

 

[JeffM]

Mortgage rates up here compound semi-annually.
In my days, we had to calculate EPR from that,
then use that to effect the interest charge,
plus use number of days in a given month...
we all hoped no borrower would ask for an
explanation of how arrived at

That sounds close to what is done on bank deposits in the US, where what is quoted is an annual percentage yield.

Back before Y2K and all the frenzy that induced in the US, I told the computer people to prepare a program that would print out interest computations day by day for up to 366 consecutive days at the teller terminals in the branches so we could show people how interest was computed. It terrified me that some poor branch person was going to have to do the computation by hand. There were remarkably few people who ever asked.