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Test 1 solution sketches Note for multiple-choice questions: Choose the closest answer 1) Interest rate conversions If the effective annual discount rate is 12%, what is the stated annual discount rate if compounded monthly? (1 + r/12)12 = 1.12 1 + r/12 = 1.0094888 r/12 = .0094888 r = .1138656 2) PV of future payments Mauricio will receive $900 three months from now and $1500 nine months from now. The effective annual discount rate is 16%. Find the total of the present value of the two payments. PV = 900/(1.16)1/4 + 1500/(1.16)3/4 PV = $2209.20 3) Profitability Index Shane invests $6,000 in a new invention. He will receive a positive cash flow of $1,000 in 9 months, followed by a positive cash flow of $1,000 every 12 months thereafter. What is the profitability index of this investment if the effective annual discount rate is 15%? 3) Profitability Index PV of future payments = 1000/.15 * (1.15)1/4 = $6903.72 P.I. = 6903.72/6000 = 1.1506 4 & 5) Future Values For the next 2 questions, assume that today is February 4, 2014. You invest $3,000 today. Find the future values on the following dates, given the stated annual interest rates and frequency of compounding. 4) Future Value May 4, 2015, 20% interest rate, compounded every three months Rate every 3 months = .20/4 = 5% FV = 3000 * (1.05)5 = $3828.84 5) Future Value February 4, 2054, 3% interest rate, compounded continuously FV = 3000 * e.03*40 = $9960.35 6) Equivalent Annual Cost Cheyenne buys a machine that will produce $5,000 worth of hair dye each year. The machine must be purchased for $20,000 today, and a maintenance cost of $6,000 must be paid 4 years from today. If the machine lasts for 7 years, the equivalent annual cost of the machine is _____ if the effective annual discount rate is 8%. 6) Equivalent Annual Cost PV of costs = 20,000 + 6000/(1.08)4 PV of costs = $24,410 EAC calculation 24,410 = C/.08 * [1 – 1/(1.08)7] 24,410 = 5.2064 * C C = $4,688 7) Annuity factor What is the annuity factor if Frank receives $5,000 per year each year for 20 years, starting one year from today? Assume the effective annual interest rate is 10%. A.F. = 1/.1 * [1 – 1/(1.1)20] A.F. = 8.5136 8) Real interest rate If the nominal interest rate is 500% and the inflation rate is 400%, the exact real interest rate is ______. (1 + real)(1 + inflation) = (1 + nominal) (1 + real)(1 + 4) = 1 + 5 1 + real = 6/5 = 1.2 Real = 20% 9) FV of annuity Leona puts $1,000 per year into a Roth IRA, starting today. The account pays 8% effective annual interest. How much will Leona have in this account immediately after the deposit is made 3 years from today? FV = 1000 * (1.08)3 + 1000 * (1.08)2 + 1000 * 1.08 + 1000 FV = $4,506.11 10) Formula assumptions Donte will receive $500 today, $550 one year from now, $600 two years from now, $650 three years from now, etc. Donte will receive these payments forever. If we knew Donte’s effective annual discount rate, which of the following formulas could you use to calculate the present value of this stream of payments? 10) Formula assumptions A) annuity B) growing annuity C) perpetuity D) growing perpetuity E) none of the above Because the growth rate is not constant, we cannot use any formula 1) Loan amortization Jaelyn borrows $45,000 today from the Party Polka National Bank. She is currently negotiating how to pay back the loan. (a) If she pays the loan back in 12 equal monthly installments, starting seven months from today, how much will each payment be if the stated annual interest rate is 12%, compounded monthly? 1) Loan amortization (a) Note monthly rate = .12/12 = 1% If paid in months 1-12: 45,000 = C/.01 * [1 – 1/(1.01)12] 45,000 = 11.255 * C C = 3998.20 But since each payment is 6 months later (months 7-18): Payment = 3998.20 * (1.01)6 = $4,244.17 1) Loan amortization (b) Suppose instead that Jaelyn pays back the loan with 3 years payments such that the principal is reduced by the same amount each year. The payments would be made 1 year, 2 years, and 3 years, from today. If the effective annual interest rate is 16%, how much total interest will Jaelyn pay? 1) Loan amortization (b) Principal per year= 45000/3= 15000 1st year: 15000 + 45000(.16)= $22,200 2nd year: 15000 + 30000(.16)= $19,800 3rd year: 15000 + 15000(.16)= $17,400 Total payments = $59,400 Total interest = 59,400 – 45,000 Total interest = $14,400 2) Internal Rates of Return Lucia invests in a company that has the following guaranteed cash flows: She must pay $3,000 today, she will receive $6,910 one year from today, and she must pay $3,930 two years from today. (a) Find all internal rates of return for this investment. 2) Internal Rates of Return (a) Let x = 1 + r -3000 + 6910/x – 3930/x2 = 0 -3000x2 + 6910x – 3930 = 0 = −6910± 69102 −4(−3000)(−3930) 2(−3000) x = 1.1516675 ± .127813 x = 1.27948, 1.023854 IRR = 27.948%, 2.3854% 2) Internal Rates of Return (b) What is the net present value of this investment fi the effective annual discount rate is 25%? PV = -3000 + 6910/1.25 – 3930/(1.25)2 PV = $12.80 2) Internal Rates of Return (c) From your answers in parts a and b, explain why the following statement is true or false: “All effective annual discount rates of 20-30% for this project lead to a positive net present value for this investment.” 2) Internal Rates of Return (c) False, above 27.948%, NPV is negative since NPV is 0 at IRRs Possible explanations: At r=29%: PV = -3000 + 6910/1.29 – 3930/(1.29)2 PV = -$5.05 The NPV equation is a downward-opening parabola because the coefficient on the squared term (-3000) is negative. So the range between IRRs must give positive NPV and outside of that range must have negative NPV.