# math

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I am answering questions concerning equity, the problem is a story problem, and it concerns a couple who bought a 129000 home, and then borrowed 107,000 at 7.25% for 30 years. A) How much equity does the couple have in their home before they make their first mortgage payment? B) How much equity would they have 10 years later if they could sell the home for 165,000? Also assume that the couple must pay 6.75% closing costs.
Then I have to modify question B under the assumption that they DID pay an extra \$75 in principal each month during the same 10 years but still have 6.75 closing costs. How much equity does the couple have then?

• math -

Before they made their first payment, the house is presumably still worth 129,000 and they borrowed 107,000 so their initial equity is what they paid in cash:
129,000 - 107,000 = 22,000 initial equity

Now after ten years they will owe less principal on their loan. I will assume that you have mortgage tables that give the payment, interest amount, principal amount, and remaining principal for each month of the thirty years.
From the tables you find that after ten years your remaining principal debt will be 92,352
So now, after ten years you owe 92,352 to the bank.
But you can sell the house for (1-.0675)*165,000 = 153,863 in cash after closing costs (I am rounding to the nearest dollar)
So now their equity is 153,863 - 92,352 = 61,511
(almost triple their initial 22,000 initial investment) This is the power of leveraging as long as prices go up (as they no longer are).

Now if they pay an additional 75 per month, that will increase their equity by decreasing the principal they owe the bank. One way to do this is to figure out how much \$75 monthly payments add up to at 7.25% for ten years by tables or with annuity computation:
monthly interest rate,
r= .0725/12=.00604167
number of payments in ten years,
n = 12*10 = 120
accumulation = \$75* [ (1+r)^n -1]/r
accumulation = 75* [2.060232-1]/r
= 75 * 175.49
= 13,161 principal accumulated by depositing 75/month for ten years
so at the end of ten years you really owe: 92,352 - 13,161 = 79,191
You got the same cash, 153,863 after closing costa but now owe the bank only 79,190
153, 863 - 79,190 = 74,672

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