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Solutions to Selected Problems at the End of Chapter 5 - Interest Rates Prepping for Exams 1. a. 2. c. 3. b. 4. a. 5. c. 6. c. 7. c. 8. c. 9. c. 10. b. Problems 1. Periodic interest rates. In the following table, fill in the periodic rates and the effective annual rates. Period APR Compounding Per Year Semi-Annual 8% 2 Quarterly 9% 4 Monthly 7.5% 12 Daily 4.25% 365 Periodic Rate Effective Annual Rate ANSWER Period APR Compounding Per Year Periodic Rate Effective Annual Rate Semi-Annual 8% 2 4.0% 8.16% Quarterly 9% 4 2.25% 9.31% Monthly 7.5% 12 0.625% 7.76% Daily 4.25% 365 0.01164% 4.34% Periodic Rate = APR / (C/Y) = 0.08 / 2 = 0.04 = 4.0% Periodic Rate = APR / (C/Y) = 0.09 / 4 = 0.0225= 2.25% Periodic Rate = APR / (C/Y) = 0.075 / 12 = 0.00625= 0.625% Periodic Rate = APR / (C/Y) = 0.0425 / 365 = 0.0001164384 = 0.01164% EAR = (1 + Periodic Rate)C/Y – 1 = 1.042 – 1 = 1.0816 – 1 = 0.0816 = 8.16% EAR = (1 + Periodic Rate)C/Y – 1 = 1.02254 – 1 = 1.0931 – 1 = 0.0931= 9.31% EAR = (1 + Periodic Rate)C/Y – 1 = 1.0062512 – 1 = 1.0776 – 1 = 0.0776 = 7.76% EAR = (1 + Periodic Rate)C/Y – 1 = 1.0001164365 – 1 = 1.0434 – 1 = 0.0434 = 4.34% 2. 3. EAR. What is the effective annual rate of a mortgage rate that is advertised at 7.75% (APR) over the next twenty years and paid with monthly payments? ANSWER Periodic Rate = 0.0775 / 12 = 0.0064583333 EAR = (1 + Periodic Rate)C/Y – 1 = 1.0064583312 – 1 = 1.0803 – 1 = 0.0803 = 8.03% 4. 5. Present value with periodic rates. Let’s follow up with Sam Hinds, the dentist, from Chapter 4 and his remodeling project (Problem 12). The cost of the equipment for the project is $18,000, and the purchase will be financed with a 7.5% loan over six years. Originally, the loan called for annual payments. Redo the payments based on quarterly payments (four per year) and monthly payments (twelve per year). Compare the annual cash outflow of the two payments. Why does the monthly payment plan have less total cash outflow each year? ANSWER Quarterly Payment = $18,000 / (1 – 1/[1 + (0.075/4)]6 x 4 ) / (0.075/4) Quarterly Payment = $18,000 / 19.1845 = $938.26 Monthly Payment = $18,000 / (1 – 1/[1 + (0.075/12)]6 x 12 ) / (0.075/12) Monthly Payment = $18,000 / 57.8365 = $311.22 Annual Cash Outflow Quarterly Payment = $938.26 x 4 = $3,753.04 Annual Cash Outflow Monthly Payment = $311.22 x 12 = $3,734.64 Difference of $18.04 It is lower for the monthly payment because each payment reduces some of the principal and so over the three months between the quarterly payments the average borrowed amount is lower so that the accumulated interest expense is lower. 6. 7. Future value with periodic rates. Matt Johnson delivers newspapers and is putting away $15.00 every month from his paper route collections. Matt is eight years old and will use the money when he goes to college in ten years. What will be the value of Matt’s account in ten years with his monthly payments if he is earning 6% (APR), 8% (APR) or 12% (APR)? ANSWER FV at 6% APR = $15.00 x [(1 + 0.06/12)10 x 12 – 1] / (0.06/12) FV at 6% APR = $15.00 x 163.8793 = $2,458.19 FV at 8% APR = $15.00 x [(1 + 0.08/12)10 x 12 – 1] / (0.08/12) FV at 8% APR = $15.00 x 182.9460 = $2,744.19 FV at 12% APR = $15.00 x [(1 + 0.12/12)10 x 12 – 1] / (0.12/12) FV at 12% APR = $15.00 x 230.0387 = $3,450.58 8. 9. Payments with periodic rates. What payment does Denise (from problem 8) need to make at the end of each month over the coming forty-four years at 6% to reach her retirement goal of $1,000,000? ANSWER Payment = $1,000,000 / [(1 + 0.06/12)44 x 12 -1 ] / (0.06/12) Payment = $1,000,000 / 2,584.2652 = $386.96 10. 11. Amortization schedule with periodic payments. Moulton Motors is advertising the following deal on a new Honda Civic: “Monthly Payments of $400.40 for the next 60 months and this beauty can be yours!” The sticker price of the car is $18,000. If you bought the car, what interest rate would you be paying in both APR and EAR terms? What is the amortization schedule of these sixty payments? ANSWER The periodic or monthly interest rate, r, is the solution to the equation PV = Payment x (1 – 1/(1+r)n) / r $18,000 = $400.40 x (1 – 1/(1+r)60) / r Using an iterative process you will eventually get to a periodic or monthly interest rate of 1%. The annual percentage rate is 12%, periodic rate times 12, 1% x 12 = 12% Or you can solve with TVM keys on a calculator: P/Y = 12; C/Y=12 INPUT 60 Keys N OUTPUT ? I/Y 12.0 -18,000 PV 400.40 PMT 0 FV and the EAR is EAR = 1.0112 – 1 = 12.68%. Amortization Schedule (Can be done effectively on a spread sheet) Cell A1 is Beginning Balance for month 1 Cell B1 is the Monthly Payment Cell C1 is the Monthly Interest Expense and is the periodic or monthly interest rate times the beginning balance: A1 * 0.01 (formula for the cell) Cell D1 is the amount of the monthly payment that is applied to the principal and is the payment minus the interest expense: B1 – C1 (formula for the cell) Cell E1 is the ending balance after the applying of the monthly payment to interest and principal. It is the beginning balance minus the principal reduction: A1 – D1 (formula for the cell). Cell A2 is the ending balance from the previous month or the value in Cell E1. Then for cells B2 through E2 copy the formulas down from the row above. A B C D E 1 $18,000.00 $400.40 $180.00 $220.40 $17,779.60 2 $17,779.60 $400.40 $177.80 $222.60 $17,557.00 3 $17,557.00 $400.40 $175.57 $224.83 $17,332.17 … … 57 $1,562.35 $400.40 $15.62 $384.78 $1,177.57 58 $1,177.57 $400.40 $11.78 $388.62 $788.94 59 $788.94 $400.40 $7.90 $392.50 $396.44 60 $396.44 $400.40 Repeat this for the sixty months… $3.96 $396.44 $0.00 12. 13. Inflation, nominal interest rates, and real rates. Given the information below, estimate the nominal rate with the approximation nominal interest rate equation and the true nominal interest rate equation. Real Rate Inflation Rate 3% 5% 8% 15% 1% 4% 2.5% 3.5% Approximate Nominal Rate True Nominal Rate ANSWER Real Rate Inflation Rate Approximate Nominal Rate True Nominal Rate 3% 5% 8% 8.15% 8% 15% 23% 24.20% 1% 4% 5% 5.04% 2.5% 3.5% 6.0% 6.09% Approximate Nominal Rate = 3% + 5% = 8% Approximate Nominal Rate = 8% + 15% = 23% Approximate Nominal Rate = 1% + 4% = 5% Approximate Nominal Rate = 2.5% + 3.5% = 6% True Nominal Rate = 1.03 x 1.05 – 1 = 1.0815 – 1 = 0.0815 or 8.15% True Nominal Rate = 1.08 x 1.15 – 1 = 1.2420 – 1 = 0.2420 or 24.20% True Nominal Rate = 1.01 x 1.04 – 1 = 1.0504 – 1 = 0.0504 or 5.04% True Nominal Rate = 1.025 x 1.035 – 1 = 1.0609 – 1 = 0.0609 or 6.09% 14. 15. Inflation, nominal interest rates, and real rates. Given the information below estimate the inflation rate with the approximation formula and the true inflation with the Fisher Effect formula. Nominal Rate Real Rate 11% 5% 8% 2% 21% 14% 5.5% 1.25% Approximate Inflation True Inflation ANSWER Nominal Rate Real Rate Approximate Inflation True Inflation 11% 5% 6% 5.71% 8% 2% 6% 5.88% 21% 14% 7% 6.14% 5.5% 1.25% 4.25% 4.20% Approximate Inflation = 11% – 5% = 6% Approximate Inflation = 8% – 2% = 6% Approximate Inflation = 21% – 14% = 7% Approximate Inflation = 5.5% – 1.25% = 4.25% True Inflation = 1.11 /1.05 – 1 = 1.0571 – 1 = 0.0571 or 5.71% True Inflation = 1.08 /1.02 – 1 = 1.0588 – 1 = 0.0588 or 5.88% True Inflation = 1.21 /1.14 – 1 = 1.0614 – 1 = 0.0614 or 6.14% True Inflation = 1.0.55 /1.0125 – 1 = 1.0420 – 1 = 0.0420 or 4.20% 17. Inflation, nominal interest rates, and real rates. The Minister of Finance for the State of Tranquility has just estimated the expected inflation rate for the coming year at 6.75%. If the real rate for the coming year is 3%, what should the nominal interest rates at the central bank of the State of Tranquility be for the coming year? ANSWER Approximate Nominal Rate = 3% + 6.75% = 9.75% True Nominal Rate = 1.03 x 1.0675 – 1 = 1.0995 – 1 = 0.0995 or 9.95% 18. 19. Negative inflation (deflation), nominal interest rates, and real rates. The Republic of Northern Lights, a small stable country in the North Atlantic, is experiencing a negative inflation (deflation) at this time. The annual inflation rate is -4%. If the nominal rate of interest is 6%, what is the real interest rate that the Northern Lightians are getting as a reward for waiting? ANSWER Approximate Real Rate = 6% – (-4%) = 10% True Real Rate = 1.06 / 0.96 – 1 = 1.1042 – 1 = 0.1042 or 10.42% 20. 21. Interest premium. The U.S. government offers two bonds: one selling to yield 6.5% and the other to yield 8.5%. Why would one bond sell for a lower yield if the originator is the same on both bonds? ANSWER If the bonds have different maturity dates the difference in yields is a reflection of the maturity premium where bonds with longer maturities have higher rates. 22. 23. Interest premium. Ben has just purchased a long-term government bond and expects to make a 7% return. Donna has just purchased a stock in a new start-up company but expects to make a 20% return. Why is Donna expecting a higher return? ANSWER If the investments have the same investment horizon, but different issuers then the riskier asset will have a higher expected return to compensate for the additional risk. Ben’s investment in government bonds is probably less risky than Donna’s investment in the new start-up company stock 24. 25. Historical interest rates. Refer to Figure 5.5 in the text. For the risk-free rate, what decade experienced the highest interest rates? The lowest? ANSWER The decade of the 80s had the highest interest rates and the decade of the 50s had the lowest rates. 26. 27. Historical interest rates. Refer to Table 5.6 for the average interest rates for the 2000 to 2010 period and estimate the default premium using the average Treasury Bond rate and the AAA corporate bond rate. ANSWER Default premium (2000 -2010) = AAA Corp. Bond rate – Treasury Bond rate = 5.66% – 4.12% = 1.54% 28. 29. Challenge question I. Michael is shopping for a special automobile. He finds the exact car he wants, a 1966 dark blue Pontiac GTO. This car is currently the property of a neighbor, so in order to buy the car for the agreed-upon price of $35,000; Michael must secure his own financing. Michael visits four different financial institutions and gets the following available loans: Bank 1: 60 monthly payments of $726.54 Bank 2: 48 monthly payments of $870.97 Bank 3: 156 weekly payments of $256.20 Bank 4: 24 quarterly payments of $1,115.81 Which loan should Michael take? (Hint: what loan has the lowest EAR?) ANSWER Bank ONE’s EAR; first find the APR with the TVM keys or a spreadsheet: TVM Keys:Mode P/Y = 12 and C/Y = 12 INPUT 60 ? -35,000 KEYS N I/Y PV OUTPUT 9.00 726.54 PMT 0 FV Periodic Rate = 0.09/12 = 0.0075 EAR = 1.007512 – 1 = 9.38%. Bank TWO’s EAR, first find the APR with the TVM keys or a spreadsheet: TVM Keys:Mode P/Y = 12 and C/Y = 12 INPUT 48 ? -35,000 KEYS N I/Y PV OUTPUT 9.00 870.97 PMT 0 FV 256.20 PMT 0 FV 1,115.81 PMT 0 FV Periodic Rate = 0.09/12 = 0.0075 EAR = 1.007512 – 1 = 9.38%. Bank THREE’s APR is TVM Keys:Mode P/Y = 52 and C/Y = 52 INPUT 156 ? -35,000 KEYS N I/Y PV OUTPUT 9.00 Periodic Rate = 0.09/52 = 0.0017308 EAR = 1.001730852 – 1 = 9.41%. Bank FOUR’s APR is TVM Keys:Mode P/Y = 4 and C/Y = 4 INPUT 24 ? -35,000 KEYS N I/Y PV OUTPUT 8.16 Periodic Rate = 0.0816/4 = 0.0204 EAR = 1.02044 – 1 = 8.41%. Bank Four has the lowest EAR and all else equal Michael should take the quarterly payment choice.