FINANCE 261 Introduction to Investments Assignment 2 ALL Answers CORRECT,100% SCORE

Department of Accounting and Finance FINANCE 261 Introduction to Investments Assignment 2 Suggested Answers 1. Suppose that a two-factor model, where the factors are the market return (Factor 1) and the growth rate o ...

Department of Accounting and Finance FINANCE 261 Introduction to Investments Assignment 2 Suggested Answers 1. Suppose that a two-factor model, where the factors are the market return (Factor 1) and the growth rate of industrial production (Factor 2), correctly describes the return generating processes of all assets and the corresponding two-factor APT correctly prices three well-diversified portfolios, A, B, and C. Portfolio Expected Return Sensitivity to Factor 1 Sensitivity to Factor 2 A 15% 1 0 B 20% 1 1 C 5% 0 0 a. What are i) the risk premiums of the two factors and ii) the risk-free rate? (4 marks) Equating the expected returns to their APT counterparts: A: 15 = λ0 + λ1 B: 20 = λ0 + λ1 + λ2 C: 5 = λ0 Solving: λ0 = 5%, λ1 = 10%, λ2 = 5% b. Another well-diversified portfolio D has sensitivities 0 and 1 to factor 1 and factor 2, respectively. What is the APT-consistent expected return on Portfolio D? (2 marks) 5 + 5 = 10% c. Suppose that Portfolio D’s expected return is 8%. Given your answers above, design an arbitrage strategy involving Portfolios A, B, C, and D. (Hint: an arbitrage strategy requires no initial investment, has no risk and yet generates a positive return.) (4 marks) D is overpriced. Short-sell D or weight on D is -1 (for every dollar shorted): From 0 net investment: -1 + wA + wB + wC = 0 From 0 factor 1 exposure: wA + wB = 0 From 0 factor 2 exposure: -1 + wB = 0 Solving: wA = -1 wB = 1 1wC = 1 (Total for Question: 10 marks) 2. You just bought a newly issued bond which has a face value of $1,000 and pays its coupon once annually. Its coupon rate is 5%, maturity is 20 years and the yield to maturity for the bond is currently 8%. a. Do you expect the bond price to change in the future when the yield stays at 8%? Why or why not? Explain. (No calculation is necessary.) (2 marks) Since the coupon rate is lower than the ytm, the bond is selling at a discount. As investors get a part of ytm in the form of capital gains, the price is expected to appreciate. b. Calculate what the bond price would be in one year if its yield to maturity stays at 8%. (2 marks) PMT: 50, N: 19, I/YR: 8, FV: 1000 → PV18% = $711.89 c. Find the before-tax holding-period return for a one-year investment period if the bond sells at a yield to maturity of 7% by the end of the year (year 1). (2 marks) PMT: 50, N: 20, I/YR: 8, FV: 1000 → PV0 = 705.46 PMT: 50, N: 19, I/YR: 7, FV: 1000 → PV1 = 793.29 Before-tax holding period return = (793 – 705.46 + 50) / 705.46 = 19.54% d. When the ordinary income tax rate is higher than the capital gains tax rate, tax authorities typically tax anticipated price appreciations from bonds at the ordinary income rate in order to prevent tax aversion with discount bonds. Suppose that from the total dollar return in part c), the coupon payment and the difference between the hypothetical prices in part b) and the purchase price are taxed at the ordinary income tax rate, 40%. The rest of the dollar return is considered capital gains (due to unanticipated change in yield-to-maturity from 8% to 7%) taxed at 30%. In other words, coupon payments and the anticipated price appreciation are taxed at the ordinary income tax rate and the rest at the lower capital gains rate. Using your answers in part b) and c), calculate the after-tax holding period return over one year if the yield to maturity is 7% at the end of the year. 2(5 marks) Interest income to be taxed at 40% = 50 + (711.89 – 705.46) = 56.44 Capital gains to be taxed at 30% = 793.29 – 711.89 = 81.40 After-tax dollar return = 0.6×56.44 + 0.7×81.40 = 90.84 After-tax holding period return = 90.84 / 705.46 = 12.88% e. Find the realized compound yield before taxes for a two-year holding period, assuming that 1) investor who bought the newly issued bond now will sell the bond in two years, ii) bond’s yield-to-maturity will be 7% at the end of the second year, and iii) the coupon in year 1 will be reinvested for one year at a 3% interest rate. Ignore taxes. (3 marks) PMT: 50, N: 18, I/YR: 7, FV: 1000 → PV2 = 798.82 Total income from the coupon payments including reinvestment income: (50 ×1.03) + 50 = 101.50 Total funds after two years: 798.82 + 101.50 = 900.32 RCY=(900.32 705.46)1/2−1=12.97 % (Total for Question: 14 marks) 3. Consider a pension plan that will pay $10,000 once a year for a 5-year period (5 annual payments). The first payment will come in exactly 5 years (at the end of year 5) and the last payment in 9 years (at the end of year 9). a. What is the duration of the pension obligation? The current interest rate is 10% per year for all maturities. (3 marks) Year CF PV y*PV/Pric e 5 10000 6209.21 3 1.199079 5 6 10000 5644.73 9 1.308086 7 7 10000 5131.58 1 1.387364 7 8 10000 4665.07 4 1.441417 8 9 10000 4240.97 6 1.474177 3 25891.5 8 6.810126 3b. To generate the scheduled pension payments, the pension fund wants to invest the present value of the future payouts in bonds and match the duration of its obligation in part a). If the fund uses 5-year and 10-year zero-coupon bonds to construct its investment position, how much money (dollar amount) ought to be placed in each bond now? What should be the total face value (not current market value) of each zero-coupon bond held? (3 marks) The pension fund needs to invest $25891.58 now to meet its obligation. In addition, it wants to match the duration: w5×5+(1−w5)×10=6.81→ w5=¿ 0.6380. So the amount to be invested in 5-year zero is 0.6380 ×25891.58=16518.18 Amount to be invested in 10-year zero is 9373.40. Face values of the investments are: 5-year: 16518.18 ×1.15=26602.69 10-year: 9373.40 ×1.110=24312.20 c. Right after the fund made its investment outlined in part b), market interest rates for all maturities dropped from 10% p.a.to 9% p.a. Show that the investment position constructed in part b) can still fund (approximately) the future payments by showing that the fund’s net investment is close to 0 at the end of year 9 after making all the scheduled payments. Assume that interest rates will remain at 9% p.a. Any excess cash from the 5-year investment will be reinvested at 9% and any fraction of the 10- year bonds held can be sold at the going market price at any time to fund the annual payments. (6 marks