(a) Find a recurrence relation for the balance B(k) owed at the end of k months
ID: 3167815 • Letter: #
Question
(a) Find a recurrence relation for the balance B(k) owed at the end of k months on a loan at a rate r if a payment P is made on the loan each month ths. e) Suppose you take out a fixed-rate mortgage for SIM at the current (historically low rate 3% and want to pay it off in 20 years. What monthly payment should you make? (d) Now suppose the same mortgage of $1M but you have qualified only for the rate 5% and the maximum monthly payment you can afford is S5K. How many years will it take you to pay off that mortgage?Explanation / Answer
a) Let Loan Amount (Principal)= A
rate of interest = r
Monthly payment = P
Interest per year = Amount borrowed * rate of interest / 100 = A * r/100
Total interest per year = r/100 * A
Interest paid per month = 1/12 * r/100 * A
Total interest to be paid for k months = k * 1/12 * r/100 * A
Total Amount to be paid after k months = Loan Amount + interest of k months
so, Total Amount to be paid after k months = A + (k * 1/12 * r/100 * A)
Payment made for k months if each month payment is P = k * P
So the balance owed B(k) = Total Amount for k months - payment made for k months
B(k) = A + (k * 1/12 * r/100 * A) - (k * P)
b) Total loan Amount borrowed = A
interest per month = Amount borrowed * (rate of interest / 100) * (1/12) = A * r/100 * 1/12
Interest for T months = Interest per month * T = A * r/100 * 1/12 * T
Total Amount to be paid after T months = Loan amount borrowed + Interest for T months
= A + (A * r/100 * 1/12 * T)
Payment to be paid per month to clear the total amount after T months = 1/12 * (A + (A * r/100 * 1/12 * T))
c) Amount borrowed = $ 1 M
rate of Interest = 3% per year
Interest for t years = Amount borrowed * (rate of interest / 100) * t
Interest for 20 years = Amount borrowed * (rate of interest /100) * 20
= 1,000,000 * (3 / 100) * 20
= $ 600,000
Total Amount to be repaid after 20 years = Amount borrowed + Interest for 20 years
= $ (1,000,000 + 600,000)
= $ 1,600,000
Therefore to repay the total amount in 20 years , monthly payment to be made = 1/12 * 1,600,000 = $133,333.33
d) Mortgage Amount (Principal)= $1,000,000
Rate of Interest = 5% p.a
Interest to be paid after t years = Principal * (rate of interest /100) * t
= 1,000,000 * (5/100) * t = 50000 * t
Total amount to be paid after t years = Principal + Interest
= 1,000,000 + 50,000*t
Monthly payment made = $5,000
Yearly Payment made = $ 5,000 * 12 = $60,000
Payment made in t years = $ 60,000 * t
borrowed amount plus interest = Total Payment made in t years
1,000,000 + 50,000 t = 60,000 * t
1,000,000 = 10,000 * t
t = 100 years
Comment = Loan Amount Borrowed(A) should be provided in a and b parts of the question, along with the information of the period for the given rate of interest . Rate of interest r should be considered monthly, quaterly,half yearly or per annum must be mentioned. By default I considered it per annum.
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