91Ó°ÊÓ

To find the outstanding balance at the end of the first 9 years, we first need to find the annual mortgage payments at an 8% yield: \(A = P \frac{i(1+i)^n}{(1+i)^n - 1}\) Where \(A\) is the annual payment, \(P\) is the mortgage amount, \(i\) is the annual interest rate, and \(n\) is the number of years. Using the given data, \(P = 80{,}000\), \(i = 0.08\), and \(n = 20\), so we can solve for \(A\): \(A = 80{,}000 \frac{0.08(1+0.08)^{20}}{(1+0.08)^{20} - 1} \approx 8{,}247.50\) Now, we can calculate the outstanding balance \(B\) after 9 years: \(B = A \frac{(1+i)^n - (1+i)^k}{i}\) Where \(k\) is the number of years that have passed. Plugging in the values: \(A = 8{,}247.50\), \(i = 0.08\), \(n = 20\), and \(k = 9\), we calculate the outstanding balance as: \(B \approx 56{,}853.09\)

Now, let's consider the lump-sum payment of \(\$ 5,000\). We will subtract this payment from the outstanding balance: \(B' = B - 5{,}000 \approx 51{,}853.09\)

Now that we know the outstanding balance and the lender's required yield for the remaining years, we can calculate the revised annual payment \(A'\) for the next 9 years at a 9% yield. Using the same formula as in Step 1, with the new outstanding balance, new interest rate, and remaining term: \(A' = 51{,}853.09 \frac{0.09(1+0.09)^9}{(1+0.09)^9 - 1} \approx 8{,}372.83\) Thus, the revised annual payment for Scenario A is approximately $8,372.83. #Scenario B: Lender insists on 9% yield during entire life of the mortgage#

The mortgage loan consists of the original mortgage amount (\(80,000) minus the lump-sum payment (\)5,000) made nine years later. \(P' = P - 5{,}000 = 80{,}000 - 5{,}000 = 75{,}000\) Now, we need to calculate the annual mortgage payment (\(A''\)) with a 9% yield for 18 years (9 years past and 9 years to come): \(A'' = 75{,}000 \frac{0.09(1+0.09)^{18}}{(1+0.09)^{18} - 1} \approx 8{,}595.15\) The revised annual payment for scenario B is approximately $8,595.15. In conclusion, we have calculated the revised annual payments \(A'\) for both scenarios: a) If the lender is satisfied with an 8% yield for the past 9 years and insists on a 9% yield for the next 9 years, the revised annual payment = \(\$ 8,372.83\) b) If the lender insists on a 9% yield during the entire life of the mortgage, the revised annual payment = \(\$ 8,595.15\)