Hello and HELP!!! U Mathematical GURUS!...

[SittingBull]

U secured a 35-YR loan to buy a home costing 150K. U make a 15% down payment.. The bank gives U a 7.3% annual rate compounded monthly.
What's your monthly payment and your equity after 18 years?
Thanks,
SittingBull

[BruceZ]

This was a useful exercise. It was the first time I've tried to derive the mortgage equations. Your niece and nephew should be having this fun instead of me having fun doing their homework. [img]/images/graemlins/smile.gif[/img] They may have been given some different formulas or methods to use other than what I've done here, I don't know. I'm depending only on the geometric series.

I checked my results below with Excel which has built in functions for doing this kind of thing, and it gives exactly the same answers I derived.

You have a loan of $127,500 at 7.3% for 35 years. If you just paid (7.3%/12)*127,500 = $775.63 each month, that would only cover the interest. If we pay more than this each month by an amount p, then the balance owed after the first payment will decrease by p, and after the second payment it will decrease by p + the interest we save on p, or p + (7.3%/12)*p, etc. In 35 years there will be 35*12 = 420 payments, and the total balance paid down will be:

sum{n = 0 to 419} p*(1 + 0.073/12)^n = 127,500

From the formula for the geometric series, this is:

p*[1 - (1 + 0.073/12)^420] / [1 - (1 + 0.073/12) ] = 127,500

p = $65.88694

So our total monthly payment is: $775.63 + $65.89 = $841.51/mo.

Excel gives exactly the same answer by using the function =PMT(7.3%/12,35*12,127500).

After 18 years, our equity will be, from the last equation evaluated at 18*12 = 216 payments:

equity = 65.88694*[1 - (1 + 0.073/12)^216 / (-.073/12) ] = $29,310.62.

Excel gives exactly the same answer by using the function =CUMPRINC(7.3%/12,35*12,127500,1,18*12,0). I had to load a special analysis tool pack to get this function as it explains in the help. It took me longer to find this function than to derive it.

Generalizing the above, we have derived this formula for the monthly payment:

monthly payment = P*i/n*{1 + 1/[ (1 + i/n)^(nT) - 1 ]}

where P = principal amount borrowed
i = interest rate per year
n = number of times compounded per year
T = length of the loan in years


The second term is p = (P*i/n) / [ (1 + i/n)^(nt) - 1 ]. If we substitute this into the expression for equity above, we get:

equity = P*[ (1 + i/n)^(nt) - 1 ] / [ (1 + i/n)^(nT) - 1 ]

Where t is the time we have been making payments in years,
and T is the total length of the loan. Again, P is the amount of the loan, and n is the number of times it is compounded per year.

[BruceZ]

The equity I computed was from paying off the loan, but you also have equity from the down payment of $22,500. So after 18 years you actually have equity of $22,500 + $29,310.62 = $51,810.62.

equity = P*[ (1 + i/n)^(nt) - 1 ] / [ (1 + i/n)^(nT) - 1 ] + down payment

[SittingBull]

I certainly will give U credit when I forward this info to my nephew and Niece. Do NOT want them to think that these were my ideas!
Thanks again! [img]/images/graemlins/smile.gif[/img]
SittingBull

[SittingBull]