Derive formula for amount of monthly mortgage payment

leyou

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Hello !

So, I'm coming here because I don't even know how to start my answer. Moreover, I'm not a native english speaker so it's really hard to understand everything it this problem.. The vocabulary is quiet complicated for me.
The problem is the number 12 of this file : http://www.cs.uiowa.edu/~rus/Courses/Theory/assign1.pdf

An idea :/ ?

Thanks in advance
 
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Please list the terms that you do not understand.

Thank you.
 

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Hello Leyou:

Do you know how to solve an equation for one of its symbols?

You need to be able to solve the equation in exercise 12 for Y.

Please let us know if you understand enough about algebra to be able to write the following answer from the given information.

Solve the equation for D

\(\displaystyle A\;=\;B \cdot C\;-\;D\cdot\left(\frac{C\;-\;1}{E\;-\;1}\right)\)

The answer is:

\(\displaystyle D\;=\;(B \cdot C\;-\;A)\cdot\left(\frac{E\;-\;1}{C\;-\;1}\right)\)

Also - please tell us about any words in exercise 12 that you do not understand.

Thank you.


My edit: deleted redundant phrase
 
Re: Problem about loan and interest rate

Well Leyou, I sure don't see why they gave you that formula...
Also, since the problem states m = 1 + i/12, then the rate which is stated as "5% annual"
should be stated as "5% annual compounded monthly".

I don't like the variables used in the equation they gave you; like, P suggests Payment,
but is used to represent balance owing, which also becomes amount borrowed at time 0.

n = number of months (360)
i = interest rate monthly (.05/12)
a = amount borrowed (100000)
p = payment monthly (?)

Formula: p = ai / [1 - 1/(1+i)^n]

For your problem:
p = 100000(.05/12) / [1 - 1/(1 + .05/12)^360]

As far as I can tell, formula they show in problem is to calculate balance owing after a given number of months.
This is really same as (future value of amount borrowed) - (future value of monthly payments).
If t is used as the given months:
balance owing after t payments = a(1+i)^t - [p(1+i)^t - 1] / i

Hope that helps you...
 
Re: Problem about loan and interest rate

Denis said:
... I sure don't see why they gave you that formula...

Denis, I believe that #12 is an exercise to refresh memory for solving a given equation for one of its parameters, followed by plugging in given numbers to determine that parameter's value. If you check out the other exercises at that link, then you'll see that the class is probably not a business math course! The exercise sheet also suggests that part of exercise #12 is for comprehension of notation (eg: subscripts) and concepts (eg: loan, interest, monthly payments, bank schemes :twisted: )

It took a few minutes, but I understand what the given theorem represents; for me, once I realized that P is P[sub:3satg0er]1[/sub:3satg0er] and that t = 360 after the last payment, then the solution steps became clear on how to determine Y.

I hope the original poster lets us know whether or not they can solve the equation for Y. Then we can begin the process of building the formula for Y in terms of P, I, and t (that will be messy).

Cheers,

~ Mark :)
 
Re: Problem about loan and interest rate

That formula I'm "complaining(!)" about:
P(subscript t) = PM^t - Y(M^t - 1)/(M - 1)
is effectively the SAME as the one I posted:
balance owing after t payments = a(1+i)^t - [p(1+i)^t - 1] / i

I don't have a problem with understanding it: I'm simply surprised it's so horribly "non-standard":
using P for amount borrowed, M for interest factor and Y for payment amount

I gave the poster the standard formula: up to him to translate it back into his P/M/Y/t variables.
He should come up with:
P = amount borrowed (100000)
t = number of months (360)
M= 1 + interest (1 + .05/12)
Y = monthly payment (?)

Y = P(M - 1) / (1 - 1/M^t)
 
Re: Problem about loan and interest rate

Thanks for your replies !

I don't have enough time to look at everything now, but I'll study it tonight and tell you if I progress.
 
Re: Problem about loan and interest rate

Denis said:
... I'm simply surprised it's so horribly "non-standard":
using P for amount borrowed, M for interest factor and Y for payment amount

I agree when I'm wearing my applied mathematics hat; otherwise it doesn't matter to me.

I see your perspective, since I know of your commerce and finance background! I see that I botched my previous remark, also. I intended to say that I understand why this "financial" exercise was used on the worksheet.

Denis, your method is certainly valid; I hope that I did not belittle it.

If you would like to see another example of horribly non-standard formulas, then look at my version!

\(\displaystyle Y\;=\;\frac{1}{12} \cdot P \cdot i \cdot \frac{(12 + i)^t}{(12 + i)^t - 12^t}\)

:)


My edit: directed comment to Denis
 
Re: Problem about loan and interest rate

Thanks again for your replies.

So, this is what I finally understood :
I need to resolve the equation to calculate Y. (I misunderstood the word "derive", I thought it was about derivation...)
So I get :
\(\displaystyle Y = \frac{(PM^t-P_t)(M-1)}{M^t-1}\)
\(\displaystyle Y = \frac{(PM^t-P_t)(M-1)}{M^t-1}\)

We know the values of :
M=1+0,5/12
P=100000
t=360
\(\displaystyle P_t=0\)

So, \(\displaystyle Y = \frac{(100000*(1+\frac{0,5}{12})^{360}-0)((1+\frac{0,5}{12})-1)}{(1+\frac{0,5}{12})^{360}-1}\)
\(\displaystyle Y = \frac{(100000*(1+\frac{0,5}{12})^{360})(\frac{0,5}{12})}{(1+\frac{0,5}{12})^{360}-1}\)
(ouch)
So, Y=4 167

It seems to be a lot... 4 167 $ every months, during 360 months ? Total = 360*4 167 = 1 500 120 $ (??)

Else, I had some trouble to understand :
loan outstanding (i know what is a loan, but not a loan outstanding)
derive (I thought it was getting a derivation...)
mortgage (??)
 
Check Your Arithmetic

leyou said:
So I get :

\(\displaystyle Y = \frac{(PM^t-P_t)(M-1)}{M^t-1}\)

So, \(\displaystyle Y = \frac{(100000*(1+\frac{0,5}{12})^{360}-0)((1+\frac{0,5}{12})-1)}{(1+\frac{0,5}{12})^{360}-1}\)
\(\displaystyle Y = \frac{(100000*(1+\frac{0,5}{12})^{360})(\frac{0,5}{12})}{(1+\frac{0,5}{12})^{360}-1}\)

So, Y=4 167

It seems to be a lot... 4 167 $ every month ... ? I'm glad to see you thinking ... yes, this is too much

loan outstanding (i know what is a loan, but not a loan outstanding) "outstanding" refers to the amount still owed

derive (I thought it was getting a derivation...) verb "to derive" also means "to get by reasoning", "to build or form by calculations" (nothing to do with differentiation in calculus)

mortgage (??) when a loan is for buying land and/or buildings, the load is called a mortgage

Hi Leyou:

Good job on resolving the equation for Y.

Please check your arithmetic; I get 536,82 $ monthly payment.

Cheers,

~ Mark :)
 
Re: Problem about loan and interest rate

Nice :)
I was using 0,5 instead of 0,05 for 5%.
So now I get 536,821623
Finally it was not really complicated, but with all this financial vocabulary, I was a little bit lost.

However I haven't understood how they get the formula of \(\displaystyle P_t\)...

Thanks for your help !
 
Re: Problem about loan and interest rate

leyou said:
... I haven't understood how they get the formula of \(\displaystyle P_t\)...

Starting from nothing? Neither do I!

Here another interesting question.

The theorem gives the monthly payment as $536.82162301456 ...

The bank only gets $536.82 per month.

What happens to the $0.00162301456... ?

I seem to remember a case a few years ago where somebody working at one of the major financial institutions in the US (maybe CitiCorp?) programmed the bank's computers to take small amounts such as these from thousands of loans and deposit them into a personal account. They guy got caught after he started driving Jaguars to work wearing $800 suits!

~ Mark :)
 
Re: Problem about loan and interest rate

mmm4444bot said:
Denis, your method is certainly valid; I hope that I did not belittle it.
No problems...nothing wrong with a little "belittling" anyway!

"My" method is the standard (so not mine) and usually shown this cute simple way:
payment = i / (1 - v) where v = 1 / (1 + i)^n

That gives results for $1 loan; so becomes a factor applicable to any borrowed amount.

In Leyou's problem: i = .05/12 and n = 360
 
Re: Problem about loan and interest rate

mmm4444bot said:
If you would like to see another example of horribly non-standard formulas, then look at my version!

\(\displaystyle Y\;=\;\frac{1}{12} \cdot P \cdot i \cdot \frac{(12 + i)^t}{(12 + i)^t - 12^t}\)
Maybe I'm just sleepy but I get the impression that this formula is a bit off.
 
jonah said:
Maybe I'm just sleepy but I get the impression that this formula is a bit off.
KIndly provide clarification, including your corrected version and your reasoning, so that the original poster knows what you mean. Thank you! :D

Eliz.
 
Re:

stapel said:
KIndly provide clarification, including your corrected version and your reasoning, so that the original poster knows what you mean. Thank you! :D

Eliz.
Applying this formula to leyou’s particular problem gives result nowhere near the expected monthly payment of $536.82.

A counter example: if
P = $100,000
i = .05/12
t = 360
and Y is the monthly payment, then
\(\displaystyle Y = \tfrac{1}{{12}} \times 100,000 \times \tfrac{{.05}}{{12}} \times \frac{{\left( {12 + \tfrac{{.05}}{{12}}} \right)^{360} }}{{\left( {12 + \tfrac{{.05}}{{12}}} \right)^{360} - 12^{360} }} \approx {\text{295}}{\text{.548422}}\)
Unless one has access to a powerful CAS like Maple or Mathematica, one would be lucky indeed if his/her calculator gets past the (12+0.05/12)^360 calculator limitation. TI-89 probably could. Perhaps Mark himself could provide the corrected version and the reasoning that went behind this (in his own words “horribly non-standard”) formula since he came up with it. Life, being a constant learning process, I for one would very much like to see the derivation process that went behind it.
 
MARK RESPONDS ...

Hello Jonah:

It looks like you think that i = 0.05/12

The interest rate is 5%. (That much is standard.)

~ Mark :wink:


PS: I would have liked to have retrieved the scrap paper from the recycle bin to be able to upload an image of my work for you (I had already pulled it out once to type my version for Denis; I would gladly have dug through the bin a second time), but, alas, the recycling truck came earlier today.
 
Jonah, Mark's monstrosity does work:

100000(.05)/12 * 12.05^360 / (12.05^360 - 12^360) = 536.821623...

The division by 12 in 1st term effectively makes the interest .05/12
 
mmm4444bot said:
Hello Jonah:
It looks like you think that i = 0.05/12
Exactly, that’s what my conditional “if P = $100,000; i = .05/12…” and my (actual) sleepy comment was for. My bad. I guess temporarily quitting the bottle has its undesirable effect of making me miss the obvious.
mmm4444bot said:
The interest rate is 5%. (That much is standard.)
I’d have to disagree with you on that one. i is almost always defined as nominal rate divided by the interest period in finance math jargon.
mmm4444bot said:
PS: I would have liked to have retrieved the scrap paper from the recycle bin to be able to upload an image of my work for you (I had already pulled it out once to type my version for Denis; I would gladly have dug through the bin a second time), but, alas, the recycling truck came earlier today.
Can you perhaps recreate the process of your work that led to this formula? Impractical as it seems (my opinion, as in 12^360) and monstrous as it was (as Denis called it), I’m almost certain I’m not the only one curious to see how you managed to derive it.
Denis said:
Jonah, Mark's monstrosity does work:

100000(.05)/12 * 12.05^360 / (12.05^360 - 12^360) = 536.821623...

The division by 12 in 1st term effectively makes the interest .05/12
So it does.
By the way, I’d like to thank you for that insightful formula that you cited at:
http://www.freemathhelp.com/forum/viewtopic.php?f=17&t=26743
It is essentially a generalization or clever manipulation (depending on how you look at it) of the so-called retrospective method for determining the outstanding liability just after any particular payment.
Had I not gotten used to obtaining the same results from spreadsheet amortization schedules, I might have come up with it in like a few years after retiring. If I might inquire with you: Did you perhaps learned it from a textbook? Or did you perhaps “discovered” it while you were still active in the workplace? Either way, I am definitely green with envy. To refresh your memory:
Denis said:
So as example: a $3000 loan at rate 12% cpd monthly is being repaid at $100 per month;
how much interest is paid between payment number x and payment number y (y > x) ?

100(y - x) - {3000(1 + .01)^x - 100[(1 + .01)^x - 1] / .01 - 3000(1 + .01)^y - 100[(1 + .01)^y - 1] / .01}
A slight correction would be:
100(y - x) - {3000(1 + .01)^x - 100[(1 + .01)^x - 1] / .01 - 3000(1 + .01)^y + 100[(1 + .01)^y - 1] / .01}
which then simplifies to
100(y - x) + (100/.01 – 3,000)[ (1 + .01)^x – (1 + .01)^y]

Let's see, how many "perhaps" did I use?
 
jonah said:
mmm4444bot said:
It looks like you think that i = 0.05/12

Exactly, that’s what my conditional “if ... i = .05/12 ...


I'm using the definition for i given by the original post. The exercise requires a formula for Y in terms of P, i and t.

jonah said:
mmm4444bot said:
The interest rate is 5%. (That much is standard.)

I’d have to disagree with you on that one. i is almost always defined as nominal rate divided by the interest period in finance math jargon.

We can disagree because my experiences are different; I learned formulas that employ distinct variables for interest rate (i) and compounding period (n).

When a financial institution claims an interest rate (eg: 5%), then I treat i as the interest rate verbatim. For me, the value of i is not a function of how it is used in a majority of formulas.

There are plenty of formulas, and even more schemes, to suit personal preference; I stress definitions over interpretation. The original post defines i = 0.05. If everybody is on the same page with the definitions, then there should be no ambiguity.

At the end of the day, it's a free world, brother. You can define i to take any form you want. :wink:

jonah said:
Can you perhaps recreate the process of your work that led to this formula?

You betcha, although there's not much of interest to be discovered ...

I started by solving the original equation for Y in the normal way. Then, since the exercise calls for a formula in terms of P, i and t, I started mucking around without using M to see if I could simplify the initial result for Y.

Basically, I amused myself by factoring out the fraction 1/12 everywhere I could, while resimplifying, to the extent possible, at each step.

It was more of an exercise in symbolic manipulation with exponents than anything else. I had no intention of presenting it as a solution to the exercise.

~ Mark :)
 
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