313.7
7. You've found a $200,000 home that you want to buy. A mortgage broker gets you a 7% APR on a 30 year loan with monthly payments. Your first payment is due in one month.
This is a straight-up annuity. PV=$200,000, r=.07/12, n=360, CF=?
\( \LARGE CF = \frac{200{,}000} { \left [ \frac{1}{\frac{.07}{12}} - \frac{1}{\frac{.07}{12} \left (1+\frac{.07}{12} \right )^{360}} \right ] } = 1330.60 \)
If you are only planning to be in the house for 6 years, is this option a good deal? (6)
The paydown will get you lower payments, but the question is: will your savings on your payments add-up to enough so that paying $2000 for the savings is a good deal? Well, your new lower payments (using r = .0675/12 and still figured on a 30-year mortgage) will be:
\( \LARGE CF = \frac{200{,}000} { \left [ \frac{1}{\frac{.0675}{12}} - \frac{1}{\frac{.0675}{12} \left (1+\frac{.0675}{12} \right )^{360}} \right ] } = 1297.20 \)
but you will only plan on making 6 years worth (72) of them. Will the PV of your savings over 6 years be worth at least $2000? The difference between what your original payments are and your new payments are is $33.40 per month. The PV of 72 months of that savings is:
\( \LARGE PV = 33.40 \left [ \frac{1}{\frac{.07}{12}} - \frac{1}{\frac{.07}{12} \left (1+\frac{.07}{12} \right )^{72} } \right ] = 1959.06 \lt 2000 \)
so no, it is not a good deal.
In order to pay-off your mortgage, you must give the bank an amount equal to the present value of all of the payments you still owe it. After 6 years, you still owe 360-72 = 288 payments of $1330.60. The PV (in year 6) of that stream is:
\( \LARGE PV = 1330.60 \left [ \frac{1}{\frac{.07}{12}} - \frac{1}{\frac{.07}{12} \left(1+\frac{.07}{12} \right )^{288} } \right ] = 185{,}382.41 \)
Leaving you with $225,000 - $185,382.41 = $39,617.59 from the sale.