The monthly payment for a home loan is given by a function f(P, r, N), where P i

Question

The monthly payment for a home loan is given by a function f(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 partial differential f/partial differential p = 0.0072, partial differential f/partial differential r = 5, 769, partial differential f/partial differential N = -1.5467. Estimate the following values. The change in monthly payment per $1,000 increase in loan principal. The change in monthly payment if the length of the loan increases to r = 6.5 %. The change in monthly payment if the length of the long increases to 24 years.

Explanation / Answer

given f/P=0.0072,f/r=5769,f/N=-1.5467

a)P=1000

f=(f/P)*P

f=(0.0072)*1000

f=7.2 dollars

b)r increases from 0.06 to 6.5%

6.5%=0.065

r=0.065-0.06=0.005

f=(f/r)*r

f=(5769)*0.005

f=28.845 dollars

c)N increases 24 years =24*12 =288 months

N=288-240=48

f=(f/N)*N

f=(-1.5467)*48

f=-74.2416 dollars

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