# The price of a home is $240,000. The bank requires 15% down payment. The buyer is offered two mortgage options: 15 year fixed at 6.5% or 30 year fixed at 6.5%. Calculate the amount of interest paid...

The price of a home is $240,000. The bank requires 15% down payment. The buyer is offered two mortgage options: 15 year fixed at 6.5% or 30 year fixed at 6.5%. Calculate the amount of interest paid of each option? How much interest is saved with the 15 year option?

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An amount of $36 000 (`15/100 times 240 000 = 36 000` )is requried as the down payment. We must borrow $204 000. As we know the amount NOW we must use the present value formula:

`P_(v) = (x[1-(1+i)^(-n)])/i`

We know that `P_(v) = 204 000`

x is unknown

`i = 6.5% = 6.5/100 = 0.065`

which we divide by 12 (compounded monthly as not specified in the question but it is usual for mortgage bonds) `i=0.065/12`

`n= 15` years or `n=30` years which are x12 (due to compounding monthly.

now substitute into the formula:

15 years: `204 000 = (x[ 1 - (1 + 0.065/12)^(-15 times 12)])/ (0.065/ 12)`

30 years: `204 000 = (x[1-(1+0.065/12)^(-30 times 12)])/(0.065/12)`

CARE not to round off too soon. Plug into the calculator and do calculations. I will have to do some measure of rounding off here for the purposees of the example:

15 years:`204 000 times (0.065/12) = (x[1-(1.0054166667) ^(-180)])`

30 years:`204 000 times (0.065/12) = x[1-(1.0054166667)^(-360)]`

15 years:`1 105 = x(1- 0.378186146)`

30 years: `1105 = x(1- 0.143024761)`

15 years: `1105/ 0.621813854= x`

`x= 1777.06`

30 years: `1105/ 0.856975239`

`x= 1289.42`

`therefore` repayment over 15 years = 1777.06 x 180 months

= $319 870.80

repayment over 30 years= 1289.42 x 360 months

= $ 464 191.20

Hence the interest will be the total paid less the original capital and you can calculate the amount of interest saved from these results.

**15 years: 319 870.80 - 204 000 = $1150870.80 interest**

**30 years: 464 191.20 - 204 000 = $ 260 191.20 interest**

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