ࡱ> 5@ bjbj22 "XX(.z9z9z9z9,9DK&; < < < <>>>JJJJJJJ$LRN)J>>^>>>)J < <y:K???>" < <J?>J?0??? <; \Xz9?d?@ PK0K?Oi?XO?O?>>?>>>>>)J)J6z9? z9Quantitative Problems Chapter 12 1. Compute the required monthly payment on a $80,000 30-year, fixed-rate mortgage with a nominal interest rate of 5.80%. How much of the payment goes toward principal and interest during the first year? Solution: The monthly mortgage payment is computed as: N = 360; I = 5.8/12; PV = 80,000; FV = 0 Compute PMT; PMT = $469.40 The amortization schedule is as follows: MonthBeginning BalancePaymentInterest PaidPrincipal PaidEnding Balance1$80,000$469.40$386.67$82.74$79,917.262$79,917.26$469.40$386.27$83.14$79,834.133$79,834.13$469.40$385.86$83.54$79,750.594$79,750.59$469.40$385.46$83.94$79,666.655$79,666.65$469.40$385.06$84.35$79,582.306$79,582.30$469.40$384.65$84.75$79,497.557$79,497.55$469.40$384.24$85.16$79,412.388$79,412.38$469.40$383.83$85.58$79,326.819$79,326.81$469.40$383.41$85.99$79,240.8210$79,240.82$469.40$383.00$86.41$79,154.4111$79,154.41$469.40$382.58$86.82$79,067.5912$79,067.59$469.40$382.16$87.24$78,980.35Total$5,632.83$4,613.18$1,019.652. Compute the face value of a 30-year, fixed-rate mortgage with a monthly payment of $1,100, assuming a nominal interest rate of 9%. If the mortgage requires 5% down, what is maximum house price? Solution: The PV of the payments is: N = 360; I = 9/12; PV = 1100; FV = 0 Compute PV; PV = 136,710 The maximum house price is 136,710/0.95 = $143,905 3. Consider a 30-year, fixed-rate mortgage for $100,000 at a nominal rate of 9%. If the borrower wants to payoff the remaining balance on the mortgage after making the 12th payment, what is the remaining balance on the mortgage? Solution: The monthly mortgage payment is computed as: N = 360; I = 9/12; PV = 100,000; FV = 0 Compute PMT; PMT = $804.62 The amortization schedule is as follows: MonthBeginning BalancePaymentInterest PaidPrincipal PaidEnding Balance1$100,000$804.62$750.00$54.62$99,945.382$99,945.38$804.62$749.59$55.03$99,890.353$99,890.35$804.62$749.18$55.44$99,834.914$99,834.91$804.62$748.76$55.86$99,779.055$99,779.05$804.62$748.34$56.28$99,722.776$99,722.77$804.62$747.92$56.70$99,666.077$99,666.07$804.62$747.50$57.12$99,608.958$99,608.95$804.62$747.07$57.55$99,551.409$99,551.40$804.62$746.64$57.98$99,493.4110$99,493.41$804.62$746.20$58.42$99,434.9911$99,434.99$804.62$745.76$58.86$99,376.1312$99,376.13$804.62$745.32$59.30$99,316.84 Just after making the 12th payment, the borrower must pay $99,317 to payoff the loan. 4. Consider a 30-year, fixed-rate mortgage for $100,000 at a nominal rate of 9%. If the borrower pays an additional $100 with each payment, how fast with the mortgage payoff? Solution: The monthly mortgage payment is computed as: N = 360; I = 9/12; PV = 100,000; FV = 0 Compute PMT; PMT = $804.62 The borrower is sending in $904.62 each month. To determine when the loan will be retired: PMT = 904.62; I = 9/12; PV = 100,000; FV = 0 Compute N; N = 237, or after 19.75 years. 5. Consider a 30-year, fixed-rate mortgage for $100,000 at a nominal rate of 9%. A S&L issues this mortgage on April 1 and retains the mortgage in its portfolio. However, by April 2, mortgage rates have increased to a 9.5% nominal rate. By how much has the value of the mortgage fallen? Solution: The monthly mortgage payment is computed as: N = 360; I = 9/12; PV = 100,000; FV = 0 Compute PMT; PMT = $804.62 In a 9.5% market, the mortgage is worth: N = 360; I = 9.5/12; PMT = $804.62; FV = 0 Compute PV; PV = $95,691.10 The value of the mortgage has fallen by about $4,300, or 4.3% 6. Consider a 30-year, fixed-rate mortgage for $100,000 at a nominal rate of 9%. What is the duration of the loan? If interest rates increase to 9.5% immediately after the mortgage is made, how much is the loan worth to the lender? Solution: The monthly mortgage payment is computed as: N = 360; I = 9/12; PV = 100,000; FV = 0 Compute PMT; PMT = $804.62 The duration calculation is exactly the same as those done in previous chapters. However, there are 360 payments to consider. Using a spreadsheet package, the duration can be calculated as 108 months, or roughly 9 years.  EMBED Equation.DSMT4  From the interest rate change, the value of the mortgage has dropped by over 4.1%. 7. Consider a 5-year balloon loan for $100,000. The bank requires a monthly payment equal to that of a 30-year fixed-rate loan with a nominal annual rate of 5.5%. How much will the borrower owe when the balloon payment is due? Solution: The required payment is computed as: N = 360; I = 5.5/12; PV = 100,000; FV = 0 Compute PMT; PMT = $567.79 The amortization schedule is as follows: MonthBeginning BalancePaymentInterest PaidPrincipal PaidEnding Balance1$100,000$567.79$458.33$109.46$99,890.542$99,890.54$567.79$457.83$109.96$99,780.583$99,780.58$567.79$457.33$110.46$99,670.124$99,670.12$567.79$456.82$110.97$99,559.155$99,559.15$567.79$456.31$111.48$99,447.686$99,447.68$567.79$455.80$111.99$99,335.697$99,335.69$567.79$455.29$112.50$99,223.19(56$93,170.80$567.79$427.03$140.76$93,030.0457$93,030.04$567.79$426.39$141.40$92,888.6458$92,888.64$567.79$425.74$142.05$92,746.5959$92,746.59$567.79$425.09$142.70$92,603.8960$92,603.89$567.79$424.43$143.36$92,460.53 Just after making the 60th payment, the borrower must make a balloon payment of $92,461. 8. A 30-year, variable-rate mortgage offers a first-year teaser rate of 2%. After that, the rate starts at 4.5%, adjusted based on actual interest states. The maximum rate over the life of the loan is 10.5%, and the rate can increase by no more than 200 basis points a year. If the mortgage is for $250,000, what is the monthly payment during the first year? Second year? What is the maximum payment during the 4th year? What is the maximum payment ever? Solution: The required payment for the 1st year is computed as: N = 360; I = 2/12; PV = 250,000; FV = 0 Compute PMT; PMT = $924.05 The required payment for the 2nd year is computed as: N = 348; I = 4.5/12; PV = $243,855.29; FV = 0 Compute PMT; PMT = $1,255.84 The maximum required payment for the 4th year is computed as: N = 324; I = 8.5/12; PV = $236,551.31; FV = 0 Compute PMT; PMT = $1,865.02 The maximum possible payment would occur in the 5th year if the 10.5% rate is required. The payment would be: N = 312; I = 10.5/12; PV = $234,187.24; FV = 0 Compute PMT; PMT = $2,193.93 9. Consider a 30-year, fixed-rate mortgage for $500,000 at a nominal rate of 6%. What is the difference in required payments between a monthly payment and a bi-monthly payment (payments made twice a month)? Solution: The required payment for monthly payments is computed as: N = 360; I = 6/12; PV = 500,000; FV = 0 Compute PMT; PMT = $2,997.75 The required payment for bi-monthly payments is computed as: N = 720; I = 6/24; PV = 500,000; FV = 0 Compute PMT; PMT = $1,498.21 Notice that this save about $1.33/month. Often times, mortgages with bi-monthly payments (automatically debited from your checking account) will offer a lower rate as well. 10. Consider the following options available to a mortgage borrower: Loan AmountInterest RateType of MortgageDiscount PointOption 1$100,0006.75%30-yr fixednoneOption 2$150,0006.25%30-yr fixed1Option 3$125,0006.0%30-yr fixed2 What is the effective annual rate for each option? Solution: Option 1: (1 + 0.0675/12)12 - 1 = 0.069628 Option 2: First, compute the effective monthly rate based on the points as follows: N = 360, I/Y = 6.25/12, PV = 150,000, compute PMT = 923.58 PMT = -923.58, N = 360, PV = 148,500, compute I/Y = 0.528789 Based on this, (1 + !    , . 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Two mortgage options are available: a 15-year fixed-rate loan at 6% with no discount points, and a 15-year fixed-rate loan at 5.75% with 1 discount point. Assuming you will not pay off the loan early, which alternative is best for you? Assume a $100,000 mortgage. Solution: Determine the effective annual rate for each alternative. 15-year fixed-rate loan at 6% with no discount points (1 + 0.06/12)12 -1 = 0.061678 15-year fixed-rate loan at 5.75% with 1 discount point N = 180; I = 5.75/12; PV = $100,000; FV = 0 Compute PMT; PMT = $830.41 PMT = 830.41; N = 180; PV = 99,000; FV = 0 Compute I; I = 0.4921841 (1 + 0.004921841)12 -1 = 0.060687 Based on these, you will pay a lower effective rate by paying points now. 12. Two mortgage options are available: a 30-year fixed-rate loan at 6% with no discount points, and a 30-year fixed-rate loan at 5.75% with 1 discount point. How long do you have to stay in the house for the mortgage with points to be a better option? Assume a $100,000 mortgage. Solution: The two loans have the same effective rate at the point of indifference. 30-year fixed-rate loan at 6% with no discount points This option has an effective monthly rate of 0.5%. Use this to back into N, as follows: N = 360; PV = 99,000; FV = 0; I = 6/12 Compute PMT; PMT = 593.55 I = 5.75/12; PV = $100,000; FV = 0; PMT = 593.55 Compute N; N = 345 You will have to live in the house for more than 345 months (28.75 years) for the mortgage with points to be a cheaper option. 13. Two mortgage options are available: a 30-year fixed-rate loan at 6% with no discount points, and a 30-year fixed-rate loan at 5.75% with points. If you are planning on living in the house for 12 years, what is the most you are willing to pay in points for the 5.75% mortgage? Assume a $100,000 mortgage. Solution: 30-year fixed-rate loan at 6% with no discount points This option has an effective monthly rate of 0.5%. I = 6.0/12; PV = $100,000; FV = 0; N = 360 Compute PMT; PMT = 599.55 Use this to back into points, as follows: I = 5.75/12; PV = $100,000; FV = 0; N = 360 Compute PMT; PMT = 583.57 The difference over 12 years is worth: N = 244; FV = 0; I = 6/12; PMT = 599.55 - 583.57 Compute PV; PV = 2,249.65 If the points on the 5.75% loan are less than 2.249, the 5.75% mortgage is a cheaper option over the life of the loan. 14. A mortgage on a house worth $350,000 requires what down payment to avoid PMI insurance? Solution: $350,000 ( 20% = $70,000. With this down payment, home owners are usually allowed to make their own property tax payments, instead of including it with their monthly mortgage payment. 15. Consider a shared-appreciation mortgage (SAM) on a $250,000 mortgage with yearly payments. Current market mortgage rates are high, running at 13%, 10% of which is annual inflation. Under the terms of the SAM, a 15-year mortgage is offered at 5%. After 15 years, the house must be sold, and the bank retains $400,000 of the sale price. If inflation remains at 10%, what are the cash flows to the bank? To the owner? Solution: The discounted payment is calculated as: I = 5; PV = $250,000; FV = 0; N = 15 Compute PMT; PMT = 24,085.57 The full payment is calculated as: I = 13; PV = $250,000; FV = 0; N = 15 Compute PMT; PMT = 38,685.45 So, the bank is accepting a lower payment of $14,599.87 per year. In terms of dollars today, this is worth: I = 13%; PMT = 14,599.87; N = 15; FV = 0 Compute PV; PV = 94,349.92 The expected house price is $250,000 ( (1.10)15 = 1,044,312 The owner will retain $644,312. The bank will retain $400,000. 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