Finance 100 Problem Set Bonds 1. You have a liability for paying college fees for your children of $20,000 at the end of each of the next 2 years (1998-1999). You can invest your money now (January 1 1998) in 2 different types of investments: 1-year zero-coupon bonds, and 4-year zero-coupon bonds. Both have a yield to maturity of 8%, compounded semi-annually. 1.a How much do you have to put aside today, assuming that interest rates stay unchanged? 1.b Suppose you invest all the money you need to put aside (use the amount from (1)) in the 4-year bonds today, and then sell the appropriate amount for the fee payments. Show that if interest rates remain unchanged, you will meet your obligations exactly. Now consider the case where interest rates change only once, immediately after you made your investment. They can either rise to 9% or drop to 7%. If you continue to meet your payments, then how much are you short (or how much have you left over) immediately after you made the second payment? 1.c Suppose now you invest all the money you decided to put aside into the 1-year zero coupon bonds, then make your fee payment and reinvest the remainder. Show that if interest rates remain unchanged, you will meet your obligations exactly. Now consider the case where interest rates change only 1
once, immediately after you made your investment. They can either rise to 9% or drop to 7%. What is the result now? Compare your results with those under b) and comment. 2. You want to borrow $2 million now. loans: You are being offered two different 1. A short term loan would give you $2 million now at 8% for one year. 2. A longer term loan would give you $2 million today at 7.5% for 2 years. Interest on this loan would accrue and be paid at the end of the second year. You receive a payment one year from now that would be exactly sufficient to repay the short-term loan. If you take the long-term loan, you have to invest this money for one more year. You have an opportunity to lock in the interest rate now and invest this cash flow one year from now at a 1-year forward rate of 7%. Which loan should you take? 3. A 6% corporate bond is due in 12 years. What is the price of the bond if the nominal yield to maturity is 12% p.a.? (note that the bond pays the coupons semiannually and has a face value equal to $100). 4. You have a choice between receiving your salary of $120,000 in equal monthly installments of $10,000 or in a single lump sum at the end of each year. If your required return is 12% p.a. compounding monthly, what year-end salary would you demand? 5. 5.a You have a $1000 par 5% coupon (nominal rate) US Treasury bond with 7 years remaining in its life. Coupons are paid semiannually and the next coupon payment is exactly six months away. The market interest rate is 6% (nominal rate with semiannual compounding). What is the current price of this bond? 2
5.b What is the effective annual rate that corresponds to a nominal interest rate of 6% with semi-annual compounding? 5.c Price a zero coupon bond with a face amount of $1000 maturing in 7 years. Assume that the nominal interest rate is 6% and interest is compounded semiannually. 5.d Assume now that interest rates have instantaneously increased by 1% to 7%. What are the bonds in parts (a) and (c) worth now? 6. 6.a You want to take out a 15 year fixed rate mortgage on a $400,000 house. Current mortgage rates are 9.5% (nominal rate, compounded monthly). Calculate your monthly payment on this mortgage. 6.b Using a spreadsheet, prepare an amortization schedule for this mortgage showing principal outstanding and the portion of each month s payment towards interest and principal. 6.c After exactly one year (i.e., immediately after the 12th payment), you use your $200,000 annual bonus to reduce the principal outstanding on your mortgage. You continue making the same monthly payment that was calculated in part (a). When will the loan be fully paid off? 7. 7.a Suppose that the current one year interest rate is 5% per annum. Also assume that the 1-year forward interest rate, (f 1,1 ), is 6%. This forward rate means that you are able to commit to investing $x one year from now and 3
be certain of receiving $x(1 + f 1,1 ) two years from now. How much money will you have in two years if you invest $100 in the current one-year rate (the spot rate) and commit to investing the proceeds of the one-year investment at the one-year forward rate? Assume that interest is calculated annually. 7.b Assume that investing $100 at the current 2 year interest rate will leave you with the same amount of money that you calculated in part (a) at the end of two years. What is the current 2-year rate? 8. In 1989, a couple purchased a house, financing $155,000 of the purchase with an 11% mortgage (monthly compounded) over 30 years. On the anniversary date of their mortgage in 1999, rates had fallen to 9%. If they refinance their home at this time with a new 20 year loan, they will incur prepayment penalties and closing costs which are equal to 5% on the new mortgage. Assume that the couple can finance both the new mortgage and the prepayment/closing costs at the 9% rate. Assume the couple makes monthly payments. Should they refinance their home? 9. The following table records current prices for zero-coupon bonds of various maturities (all securities have a face value of $100): Bond Maturity (Years) Price ($) A 1 95.24 B 2 89.85 C 3 83.96 Use the prices to value a bond with a coupon rate of 5% per year, $100,000 face value, and three years remaining to maturity (annual coupon payments). 10. The following table records current spot rates for zero-coupon bonds of various maturities: 4
Bond Maturity (Years) Spot Rate (%) A 1 5.00 B 2 5.50 C 3 6.00 Use the spot rates to derive (a) the forward rate between year 1 and 2, (b) the forward rate between year 2 and 3, and (c) the forward rate between year 1 and 3. The forward rates should be expressed on an annual basis. 11. You are interested in purchasing a new car. The list price is $58,000 and the manufacturer provides financing over five years at 5.5% p.a. compounding monthly with repayments to be made monthly. One dealer has offered you a $4,000 discount and has offered to provide financing (over five years) at 6% p.a. compounding monthly with repayments to be made monthly. The question is which terms are more attractive? 12. You own government bonds with a face value of $2 million. The bonds mature 6 years and 3 months from today and have a coupon rate of 12%, paid semiannually. The next coupon will be paid in three months. The current yield on these bonds is 6% p.a. compounded semi-annually. How much are the bonds worth today? The next two questions are more advanced 13. A 3-year coupon bond has payments as follows: Bond Cash Flows ($) by Year Year 1 Year 2 Year 3 8 8 108 This 8% coupon bond is currently trading at par ($100). (a) What is the annually compounded yield of the bond? (b) Compute the Macaulay duration, the modified duration and DV01 of the bond. (c) 5
Using the DV01, how much do you expect this bond s price to rise if the yield on the bond declines by 10 basis points compounded annually? 14. A financial institution has raised $1 million by selling a number of 2-year zerocoupon bonds to individuals. These bonds have a yield-to-maturity of 6%. The institution has used the proceeds to buy a number of long-term coupon bonds. These bonds have a Macaulay duration of 12 years and a yield-tomaturity of 7%. Use the concept of duration to explain how this institution is exposed to changes in interest rates. In particular, what happens to the value of the zero-coupon bonds, the long-term bonds, and the value of the firm as a whole, if the yields on these bonds change by 50 basis points (half of one percent)? [Hint: Use the duration as an approximation for percentage prices changes.] 6