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Test 1 solution sketches

Test 1 solution sketches. Note for multiple-choice questions: Choose the closest answer. Loan calculations.

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Test 1 solution sketches

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  1. Test 1 solution sketches Note for multiple-choice questions: Choose the closest answer

  2. Loan calculations • Billy’s Pianos receives a loan of $180,000 today. The stated annual interest rate is 8.4%, compounded monthly. Payments are monthly, starting one month from today. The loan is amortized over 30 years.

  3. Loan calculations • If Billy pays an equal amount of principal each month, how much will the first payment be? • Monthly rate = .0084 / 12 = 0.7% • Amount of principal paid each month = $180,000 / 360 = $500 • Amount of interest accrued in first month = $180,000 * .007 = $1,260 • First payment = 500 + 1,260 = $1,760

  4. Loan calculations • If Billy makes equal month payments each month, how much will the first payment be? • 180,000 = C / .007 * [1 – 1 / (1.007)360] • 180,000 = 131.262 * C • C = $1,371.31

  5. Loan calculations • If Billy pays an equal amount of principal each month, how much will the last payment be? • Principal owed in 359 months = 180,000 / 360 = 500 • Interest owed = 500 * .007 = 3.50 • Last payment = 500 + 3.50 = $503.50

  6. Loan calculations • If Billy makes equal month payments each month, how much will the last payment be? • Note: equal payments means first = last (so same answer as #2) • 180,000 = C / .007 * [1 – 1 / (1.007)360] • 180,000 = 131.262 * C • C = $1,371.31

  7. Profitability Index • Carly Rae pays $50,000 to open her dating service. She receives $2,700 per year in cash flow, starting in two years. Annual discount rate is 5%. What is the profitability index? • PV of benefits = 2700 / .05 * 1 / 1.05 = 51,429 • PV of costs = 50,000 • PI = 51,429 / 50,000 = 1.029

  8. Effective Discount Rates • If the effective annual discount rate is 15%, then what is the effective discount rate for 8 months? • (1.15)8/12 – 1 = 9.76534%

  9. PV of Annuity • Wolfgang will receive royalty payments of $500 every year, starting 5 years from today and ending 25 years from today. What is the present value of these payments if the effective annual discount rate is 15%? • Annuity formula for 21 payments, discounted by 4 years due to 1st payment in year 5 • 500/.15 * [1 – 1 / 1.1521] * 1 / 1.154 = $1,804.59

  10. Real payments • If the inflation rate this year is 5% and the nominal interest rate is 15%, then what is the real interest rate? • (1 + real)(1 + inflation) = (1 + nominal) • (1 + real)(1.05) = 1.15 • 1 + real = 1.15 / 1.05 = 1.0952381 • Real = 9.52381%

  11. Discounted vs. undiscounted payback periods • Reba’s Rabbits invests $50,000 today, and will earn $10,000 each year starting one year from today. The effective annual discount rate is 9%. • If Reba uses discounted cash flows, how many years is the payback period for this investment? • 50000 = 10000/.09 (1 – 1/1.09T) • 61111 = (10000/.09)/(1.09T) • 1.09T = (10000/.09)/61111 = 1.81818 • T = ln(1.81818)/ln(1.09) = 6.93726 ≈ 7

  12. Discounted vs. undiscounted payback periods • If Reba uses undiscounted cash flows, how many years is the payback period for this investment? • 50000 / 10000 = 5

  13. Pyotr’s Beauty Products • Pyotr’s Beauty Products is considering buying a new device. This machine would cost $8,000 today, and require maintenance costs of $600 every three years, starting in 2 years and ending in 11 years. The machine lasts 12 years, and the effective annual discount rate is 14%.

  14. Part (a) • What is the present value of all costs of the machine over its life? • Purchase cost today and maintenance costs in years 2, 5, 8, and 11 • 8000 + 600/(1.142) + 600/(1.145) + 600/(1.148) + 600/(1.1411) = $9,125.61

  15. Part (b) • Pyotr pays $X per year for five years, starting today. These payments will have the same present value as the answer you got from part (a). Find X. • X + X/1.14 + X/(1.142) + X/(1.143) + X/(1.144) = 9125.61 • 3.91371 * X = 9125.61 • X = $2,331.70

  16. Yield to Maturity • A bond has a face value of $750. It pays a coupon of 10% today, one year from today, and two years from today. Two years from today, the bond matures. If the current selling price of the bond is $800, what is the yield to maturity (expressed as an effective annual discount rate)?

  17. Yield to Maturity • 800 = 75 + 75/(1+r) + 825/(1+r)2 • 725(1+r)2 – 75(1+r) – 825 = 0 • 725r2 + 1375r – 175 = 0 • 29r2 + 55r – 7 = 0 Ignore negative root. r = 0.119716 so r = 11.97%. Or…

  18. Yield to Maturity • 800 = 75 + 75/(1+r) + 825/(1+r)2 • 725(1+r)2 – 75(1+r) – 825 = 0 • Let x = 1+r • 29x2 – 3x – 33 = 0 Ignore negative root. x = 1.1197 so r = 11.97%

  19. Balloon Payment • Michael is taking out a loan of $1,000,000 today and he will pay $22,000 per month for the next 10 years (120 payments, starting one month from today). The stated annual interest rate is 24%, compounded monthly. 13 years from today, Michael will make one additional payment to pay off the loan. How much will this payment be?

  20. Balloon Payment • PV of monthly payments: • 22000/.02 * [1 – 1/(1.02120)] = 997,818.55 • PV of payment made in 13 years: • 1,000,000 – 997,818.55 = 2,181.45 • FV of payment made in 13 years: • 2,181.45 (1.02)12*13 = $47,904.10

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