Consider three risk-free securities: a zero-coupon bond with a face value of $1,000 and 3 years to maturity, tradingat $711.78, an annuity that pays $1,000 in year 1 and year 2. The annuity costs $1,809.72, a bond that pays off $1,000 in one year, $2,000 in two years, and $3,000 in threeyears. This bond trades at $4,802.40.(a) What is the term structure of spot interest rates implied by these instruments?What is the term structure of forward rates?(b) What are the YTMs (yields to maturity) of these three instruments? Write downthe equation that you need to solve to compute YTMs. Solve the equations for thethree-year zero-coupon bond and the annuity. Solve the third equation numerically,that is, compute the YTM using the function IRR (or using Goal Seek or Solver Tool)in Excel.Suppose now that there is also a fourth instrument: a zero-coupon bond with the facevalue of $1,000 and two years to maturity. This bond trades at $857.34.(c) Create a portfolio of the original three instruments that replicates the cash flowsof the two-year zero-coupon bond.(d) What is the YTM of the two-year zero-coupon bond? What is the YTM of thereplicating portfolio you found in part (c)? (Try to answer this question without doingany calculations. However, if you wish, use whichever method you like to compute theYTM.)(e) Suppose now that the two-year zero-coupon bond trades at $900.00 (all other pricesare unchanged). Is there a free lunch available here? Describe a trading strategy thatwill allow you to get it.1

UNIVERSITY OF TORONTO Joseph L. Rotman School of Management RSM332 PROBLEM SET #1 1. Suppose you own a farm that, if run efficiently, can produce corn according to the following \"transformation\" formula: 2 W1 = 90 I03 where I0 is the number of bushels of corn planted in date 0, and W1 is the number of bushels turned over to you in date 1, net after all payments to labor and other hired inputs. Your utility function of consumption at date 0 and consumption at date 1 is: 1 3 U (C0 , C1 ) = C04 C14 (a) If 64,000 bushels of corn are planted, what will be the net output of corn at date 1? (b) If you set a target for output of 110,250 bushels, what is the minimum number of bushels that must be planted? (c) Suppose capital markets do not exist, and you can neither borrow, lend, or store any corn at date 0. If you have 40,500 bushels of corn at date 0, what will your production plan be? Your consumption plan? What will be the average rate of return on investment of corn? What will be the rate of return on the marginal investment? (d) If 64,000 bushels are planted, what will be the average rate of return on the investment? What will be the rate of return on the marginal investment? (e) If a capital market exists, and the rate of interest is 50%, what will be your optimal investment? (f) If you have no corn at date 0, a capital market exists (50% rate of interest), and you invest optimally, what is your equity in the venture? What will be your optimal consumption plan? Outline your sources and uses of funds for date 0 and date 1. (g) Will you loan your farm for a period for 36,000 bushels of corn? Why? If you have no corn at date 0, and you decide to loan the farm, what will be your consumption plan? 2. Your brother works as an engineer. Today is his 24th birthday. At his birthday party, he asks for your advice on saving for his retirement. He plans to retire at 65 years old and he expects to live for another 20 years afterwards. He wants an income of $30,000 1 per year during his retirement years, to be paid annually on his birthday (starting from his 65th birthday). He plans to save some amount at each birthday from the age 25 to 64. He thinks about saving a constant amount for the first 10 years and then increases his saving at 3% each year until the last one before his retirement. The bank provides two types of accounts. One account pays 6.9%/year compounded quarterly. The other account pays 7%/year compounded annually? (a) Which account would you recommend? Why? (b) After choosing the proper account, how much should your brother save each year for the first 10 years? (c) What is the balance of your brother's account right after he makes his deposit in his saving account on his 50th birthday? (d) In fact, your brother is not entirely sure how long he will live. Although he expects to live until 85 years old, there is actually an equal probability that he will die at the age of 75, 85, or 95. If this is the case, would you change your answer to part (b)? 3. You obtain a $250,000 mortgage loan from Bank of Montreal to buy a house. The mortgage has a 5-year fixed rate of 4%/year (using Canadian mortgage convention), and the amortization period of the mortgage is 20 years. (a) What is the monthly mortgage payment? (b) How much do you owe the bank after the 36th monthly payment? (c) For the 45th monthly payment, how much of it is for interest, and how much of it is for principal repayment? (d) What is the present value of the principal repayment portion of the first 60 payments? 4. Management Fee You are about to invest some money in a bond fund. The management fee of the fund is quite low, it only charges a fee of 1%/year on the assets managed. However, you do not believe the bond fund manager has superior ability to beat the market and you expect him to earn a return of 5%/year (before management fee) on the assets of the fund. This is the same return that you (and everyone else) will be able to get but you just do not want the hassle of managing your own money. (a) Suppose you plan to leave the money in the bond fund for 20 years. For every dollar that you invest in the bond fund today, how much are you effectively giving to the fund manager for his service over the next 20 years? (Hint: How much are you willing to give to the manager today if he is willing to waive the management fee in the future.) (b) How would your answer in part (a) change if you plan to leave the money in the bond fund for a very long period of time (say for T years, where T is a very large number). 2 5. The duration of a bond is defined as the weighted average maturity of its cashflows, with the weights proportional to the present values of the cashflows at different maturities. If a bond has cashflows C1 , C2 , . . . , CT with maturities 1 year, 2 years, to T years, then the duration of the bond is defined as D= T X t=1 t P V (C1 ) P V (C2 ) P V (CT ) P V (Ct ) =1 +2 + + T , P0 P0 P0 P0 where P V (Ct ) stands for the present value of the cashflow at time t, and P0 = PT t=1 P V (Ct ) is the current price of the bond. For example, a 2-year 5% annual coupon bond with a face value of $1000 has $50 cashflow matures in one year and $1050 matures in two years. Assuming the term structure of interest rates is flat and the annual interest rate is r, then the duration of the bond is equal to D =1 where P0 = 50 1+r P0 +2 1050 (1+r)2 P0 , 1050 50 + . 1 + r (1 + r)2 For this question, we assume the term structure of interest rates is flat and the market interest rate is r = 10% per year. (a) What is the duration of an annual coupon bond with a coupon rate of 5%/year, and a maturity of 10 years. (b) What is the duration of a perpetuity that pays $10/year? 6. Consider three risk-free securities: a zero-coupon bond with a face value of $1,000 and 3 years to maturity, trading at $711.78, an annuity that pays $1,000 in year 1 and year 2. The annuity costs $1,809.72, a bond that pays off $1,000 in one year, $2,000 in two years, and $3,000 in three years. This bond trades at $4,802.40. (a) What is the term structure of spot interest rates implied by these instruments? What is the term structure of forward rates? (b) What are the YTMs (yields to maturity) of these three instruments? Write down the equation that you need to solve to compute YTMs. Solve the equations for the three-year zero-coupon bond and the annuity. Solve the third equation numerically, that is, compute the YTM using the function IRR (or using Goal Seek or Solver Tool) in Excel. 3 Suppose now that there is also a fourth instrument: a zero-coupon bond with the face value of $1,000 and two years to maturity. This bond trades at $857.34. (c) Create a portfolio of the original three instruments that replicates the cash flows of the two-year zero-coupon bond. (d) What is the YTM of the two-year zero-coupon bond? What is the YTM of the replicating portfolio you found in part (c)? (Try to answer this question without doing any calculations. However, if you wish, use whichever method you like to compute the YTM.) (e) Suppose now that the two-year zero-coupon bond trades at $900.00 (all other prices are unchanged). Is there a free lunch available here? Describe a trading strategy that will allow you to get it. 4