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Quiz 2 solution sketches 1:00 Lecture, Version A Note for multiple-choice questions: Choose the closest answer Growing Annuity Aubrey will receive $50 today. She will receive 6% more each subsequent year. The last payment she will receive will be 30 years from today. What is the total present value of all payments if the effective annual discount rate is 18%? PV = 50 + 53/(.18-.06) * [1 – (1.06/1.18)30] PV = 50 + 53 * 7.99950 PV = $473.97 Interest-Only Loan Carrie borrows $5,000 today from the Isla Vista Monster Bank. She makes monthly payments of $100 for 48 months, starting one month from today. She makes one additional payment 50 months from today to completely pay off the loan. How much will the payment be if the stated annual interest rate is 24%, compounded monthly? Interest-Only Loan Interest-only loan for first 48 months (1st month’s interest = 5000 * .02 = $100) Balance in 48 months = $5000 Payment in 50 months: 5000 * (1.02)2 = $5,202 Time to Double The stated annual interest rate is 0.9%, compounded continuously. How many years will it take to double an initial deposit made today? Assume that any interest gets re-invested. e(.009T) = 2 .009 * T = ln 2 T = (ln 2)/.009 = 77.016 years Profitability Index Seamus invests $8,000 today. He is set to receive $2,000 per year, forever, starting six years from now. What is the profitability index for this investment if the effective annual interest rate is 20%? Profitability Index PVbenefits = 2000/.2 * 1/(1.2)5 PVbenefits = $4,018.78 P.I. = PVbenefits / Today’s cost P.I. = 4018.78 / 8000 = 0.50235 Internal Rate of Return Maxton invests $5,000 today and will receive $500 per year forever, starting later today. What is the internal rate of return for this investment? -5000 + 500 + 500/IRR = 0 500/IRR = 4500 IRR = 11.11% Payback Period Method Use the undiscounted payback period method, with the cutoff date 8 years, 4 months from now. (In other words, the payback period is 8 years, 4 months.) The effective annual discount rate is 14%. Which of the following offers should be picked if someone uses this method? Payback Period Method A: $1,000 per year forever, starting 4 years from now B: $510 per year forever, starting today C: $4,800 every 8 years forever, starting 8 years from now D: $10,000 every 10 years forever, starting 10 years from now E: A one-time payment of $4,500 today Payback Period Method Undiscounted future value at 8y 4m: A: $1,000 * 5 = $5,000 B: $510 * 9 = $4,590 C: $4,800 (one payment at year 8) D: $0 (first payment not until year 10) E: $4,500 Using the undiscounted payback period method, one should choose option A PV of Perpetuity Bayleigh will receive $25,000 per year, forever, starting 18 months from now. What is the present value of this perpetuity if her effective annual discount rate is 12.36%? PV = 25,000/.1236 * 1/(1.1236)1/2 PV = $190,816 PV of a Single Payment Orlando will receive $30,000 from his grandfather in 10 years. What is the present value of this payment if his stated annual discount rate is 12%, compounded every 30 minutes? PV of a Single Payment 2 ways to solve Approximate with continuous compounding: PV = 30,000/e(.12*10) = $9,035.83 Compound 48x/day, or 17,520x/year EAR = (1 + .12/17520)17520 – 1 = 12.7496% PV = 30,000/(1.127496)10 = $9,035.86 Long-Answer: Credit Card Balance Transfers Paloma has just charged $60,000 of construction equipment on her Vampire Express credit card, which charges 15% stated annual interest, compounded monthly. After getting home, she finds she has just been approved for a new Mummy Express credit card, which charges 12% stated annual interest, compounded monthly. Long-Answer: Credit Card Balance Transfers The Mummy Express charges a 2% fee for any balances that are transferred. If Paloma makes monthly payments of $2,000 to pay off the equipment, should she pay off her Vampire Express card, or transfer her balance to the Mummy Express card to pay off the higher balance? Assume the first $2,000 will be one month from today. Long-Answer: Credit Card Balance Transfers Vampire: monthly rate = .15/12 = .0125 60,000 = 2,000/.0125 * [1 – 1/(1.0125)T] 60,000 = 160,000 – 160,000/(1.0125)T 160,000/100,000 = (1.0125)T T = (ln 1.6)/(ln 1.0125) = 37.8348 months Long-Answer: Credit Card Balance Transfers Mummy: monthly rate = .12/12 = .01 60,000 * 1.02 = 2,000/.01 * [1 – 1/(1.01)T] 61,200 = 200,000 – 200,000/(1.01)T 200,000/138,800 = (1.01)T T = (ln 1.4409)/(ln 1.01) = 36.7107 months Choose the Mummy Express card because the number of months to pay off is lower.