The monthly payment for a home loan is given by a function f (P, r, N), where P
ID: 2893773 • Letter: T
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 = 0.072, partial differential f/partial differential r = 5, 769, partial differential f/partial differential N = -1.5467. Estimate the following values. (a) The change in monthly payment per $1,000 increase in loan principal. Delta f = $ _________________ (b) The change in monthly payment if the interest rate increases to r = 6.5%. Delta f = $ _________________ (c) The change in monthly payment if the length of the loan increases to 24 years. Delta f = $ _________________Explanation / Answer
f/P=0.0072,f/r=5729 ,f/N=-1.5467
(a)
P=1000
f=(f/P)*P
f=(0.0072)*1000
f=7.2 $
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(b)
r=0.065-0.06=0.005
f=(f/r)*r
f=(5769)*0.005
f=28.845 $
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(c)
N=(24*12)-(20*12)=288-240=48
f=(f/N)*N
f=(-1.5467)*48
f=-74.2416 $