In Example 5.10, we showed you how to work out mortgage payments. Log on to the
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Question
In Example 5.10, we showed you how to work out mortgage payments. Log on to the personal finance page of www.bankrate.com and find the mortgage payment calculator. Assume a 20- year mortgage loan of $100,000 and an interest rate (APR) of 12%, what is the amount of the monthly payment? Check that you get the same answer when using the annuity formula. Now look at how much of the first month's payment goes to reduce the size of the mortgage. How much of the payment by the 10th year? Can you explain why the figure changes? If the interest rate doubles, would you expect the mortgage payment to double? Check whether you are right Example 5.10 Home Mortgages Sometimes you may need to find the series of cash payments that would provide a given value today. For example, home purchasers typically borrow the bulk of the house price from a lender. The most common loan arrangement is a 30-year loan that is repaid in equal monthly installments. Suppose that a house costs $125,000 and that the buyer puts down 20% of the purchase price, or $25,000, in cash, borrowing the remaining $100,000 from a mortgage lender such as the local savings bank. What is the appropriate monthly mortgage payment? The borrower repays the loan by making monthly payments over the next 30 years (360 months). The savings bank needs to set these monthly payments so that they have a present value of $100,000. Thus Present value- mortgage payment x 360-month annuity factor $100,000 $100,000 Mortgage payment 360-month annuity factor Suppose that the interest rate is 1% a month. Then $100,000 $100,000 Mortgage payment- 97.218-$1,028.61 01 .01(1.01)360Explanation / Answer
Mortgage Borrowing = $ 100000, APR = 12 % per annum or 1 % per month, Tenure = 20 years or 240 months
Present Value of Mortgage = Monthly Payments (assumed to be $ K) x 240-month annuity factor
100000 = K x (1/0.01) x [1-{1/(1.01)^(240)}]
K = $ 1101.086 per month
Out of the monthly repayments a part goes towards paying off the interest on the outstanding mortgage balance and remaining is used to reduce the mortgage balance.
At the end of month 1, Mortgage Balance = $ 100000
Interest on Mortgage Balance = Monthly Interest Rate x Mortgage Balance = 0.01 x 100000 = $ 1000
Reduction in Mortgage Balance = 1101.086 - 1000 = $ 101.086
Mortgage Balance at the end of Year 8 and 11 months = 1101.086 x (1/0.01) x [1-{1 / (1.01)^(133)}] = $ 80794.37 approximately.
Interest Expense on Mortgage Balance at the end of Year 9 (8 years 12 months or by the 10th Year) = 80794.37 x 0.01 = $ 807.9437
Principal Reduction = 1101.086 - 807.9437 = $ 293.1423
The amount of principal reduction increases with each monthly repayment because amortizing mortgages use initial periodic repayments majorly to pay off the interest expense. In other words, only a small portion of the initial periodic repayments go towards reducing the outstanding mortgage balance.
The periodic repayments are inversely proportional to the annuity factor which in turn is inversely related to a combination of the interest rate and interest rate raised to the power of the mortgage duration. Hence, the repayments will not be exactly linearly proportional to the interest rates. The doubling of interest rate will result in repayments approximately double of the previous value.
New Interest Rate = 2 % per month
Let repayments be $ M
Therefore, 100000 = M x (1/0.02) x [1-{1/(1.02)^(240)}]
M = $ 2017.408 which is approximately but not exactly equal to double of the previous value of $ 1101.086