Question

The monthly payment for a home loan is given by a functionf(P, r, N),where P is the principal (the initial size of the loan), r the interest rate expressed as a decimal (a 6% interest rate is denoted by r = 0.06), and N the length of the loan in months. If P = $100,000, r = 0.06, and N = 240 (a 20-year loan), then the monthly payment is f(100,000, 0.06, 240) = 716.43. Furthermore, with these values, we have?f?P= 0.0072,?f?r= 5,769,?f?N= ?1.5467.Estimate the following values.(a) The change in monthly payment per $1,000 increase in loan principal.?f = $_____________(b) The change in monthly payment if the interest rate increases to r = 6.5%.?f = $_____________(c) The change in monthly payment if the length of the loan increases to 24 years.?f = $_____________

Answer 1

Answer:

a) a change in the principal of 100 dollar will increase the monthly payment for 7.164 dollars

b) and increase inthe rate to 6.5% from 6% will increase the monthly payment for $   29.14

c) an increase inthe lenght of the loan to 24 years will decrease the payment per month by 60.45 dollars

Explanation:

We calcualte the annuity of 1,000 dollar under the same conditions:

$PV \div \frac{1-(1+r)^{-time} }{rate} = C$

PV  $1,000.00

time 240

rate 0.005 (6% annual divide into 12 months per year)

$1000 \div \frac{1-(1+0.005)^{-240} }{0.005} = C$

C  $ 7.164

Then, the change if rate increase to 6.5%

$100000 \div \frac{1-(1+0.00541667)^{-240} }{0.00541667} = C$

C of 6.50%  $ 745.573

C of 6.00%       $ 716.43

Difference:      $   29.14

last, if n = 24 years = 24 x 12 = 288

$100000 \div \frac{1-(1+0.005)^{-288} }{0.005} = C$

C of 24 years  $ 655.98

C of 20 years  $  716.43

Differnece       $  (60.45)