Mind solving should show you got your answers

31. As a mid-size company, you have a pension plan which pays out $10 million a year forever. The first payment is exactly one year from now. The term structure is currently flat at 5%. (a) Compute the present value of your pension liabilities. (b) Suppose that the interest rate goes down by 0.1%. How does the value of your liability change? (c) Given your answer to (b), what is the modified duration of your pension liability? (d) Suppose that the pension plan is fully funded (i.e., the value of your assets equal the value of pension liabilities). You want to invest all your assets in bonds to avoid any interest rate risk. What should the duration of your bond portfolio be? (e) Suppose that this portfolio is a single zero-coupon bond. What should its maturity and total par value be? 32. On a job interview, you were handed the following quotes on U.S. Treasuries: Bond | Maturity (years) | Coupon Rate | Yield to Maturity 5% 4.5% 5% 5.0% 0% 5.5% Assume that the par value is $100 and coupons are paid annually, with the first coupon payment coming in exactly one year from now. The yield to maturity is also quoted as an annual rate. You are then asked the following questions: (a) What should be the price of a bond with a maturity of 3 years and coupon rate of 5%, given the above information? (b) What should be the 1-year forward rate between years 2 and 3? (c) What is the modified duration of a bond portfolio with 30% invested in bond 1 and 70% invested in bond 3? (d) How much would the value of the portfolio in (c) change if the yields of all bonds increased by 0.15%? 33. You have the following data on Treasury bonds. Assume that there are no taxes, only annual coupon payments are made and the first coupon payment occurs a year from now Bond Year of Maturity Coupon Face Value at T Price Today 25 100 100.00 50 500 422.61 50 N0 - 20 300 232.28 1000 192.31 Fall 2008 Page 17 of 66 (a) Calculate the following four annualized forward rates: f from 0 to 1, f from 1 to 2, f from 2 to 3, and f from 3 to 10. (b) Is it a good investment if it costs $21 million now and yields the following risk-free cash inflows? Year 2 3 Cash Flow (in million dollars) 9 10 11 34. Which of the following investments is most affected by changes in the level of interest rates? Suppose interest rates go up or down by 50 basis points (+ 0.5%). Rank the investments from most affected (largest change in value) to least affected (smallest change in value). (a) $1 million invested in short-term Treasury bills. (b) $1 million invested in Treasury strips (zero coupons) maturing in December 2016. (c) $1 million invested in a Treasury note maturing in December 2016. The note pays a 5.5% coupon. (d) $1 million invested in a Treasury bond maturing in January 2017. The bond pays a 9.25% coupon.35. Valerie Smith is attempting to construct a bond portfolio with a duration of 9 years. She has $5012},m0 to invest and is considering allocating it between two zero coupon bonds. The first new coupon bond matures in exactly 6 years, and the second zero coupon bond matures in exactly 16 years. Both of these bonds are currently selling for a market price of $100. Suppose that the yield curve is flat at 7.5%. Is it possible for Valerie to construct a bond portfolio having a duration of 9 years using these two types of zero coupon bonds? If so. how? (Describe the actual portfolio.) If not, why not? 38. Given the bond prices in the question above, you plan to borrow $15 million one year from now (end ofyear 1}. It will be a two-year loan [from year 1 to year 3} with intemst paid at the ends of year 2 and 3. The cash ow is as follows: Year 1 Year '2 Year 3 Borrow $15M Pay interest Pay interest plus principal of 15M Explain how you could arrange this loan today and \"lock in\" the interest rate on the loan. What transactions today would be required? What would the interest rate be? You can buy or sell any of the bonds listed above (in the previous question]. 37. You purchased a 3 year coupon bond one year ago. Its par value is $1,000 and coupon rate is 6%. paid annually. At the tinie you purchased the bond, its yield to maturity was 6.5%. Suppose you sell the bond after receiving the rst interest payment. Fall 2008 Page 13 of 66 [a] What is the total rate of return from holding the bond for the year if the yield to maturity remains at 6.5% when you sell it? (b) What if the yield to maturity becomes 6.0% when you sell it? 38. You manage a pension fund1 which provides retired workers with lifetime annuities. The fund must pay out $1 million per year to cover these annuities. Assume for simplicity that these payments continue for '20 years and then cease. The interest rate is 4% (at term structure]. You plan to cover this obligation by investing in 5- and 'Z-year maturity 'Il-easury strips. {a} What is the duration of the funds 20-year payout obligation? (b) You decide to minimize the funds exposure to changes in interest rates. How much should you invest in the 5- and 21'.)- year strips? What will be the par value of your holdings of each strip? (c) After three months. you reexamine the pension funds investment strategy. Interest rates have increased. You still want to minimize exposure to interest rate risk. Will you invest more in 20-year strips and less in 5-year strips? Explain briey. 39. Duration and Convexity. Consider a 10 year bond with a face value of $100 that pays an annual coupon of 3%. Assume spot rate; are at at 5%. [a] Find the hond's price and duration. [b] Suppose that lyr yields increase by lbps. Calculate the change in the hond's price using your band pricing formula and then using the duratiou apprmrimatiOn. How big is the difference? (c) Suppose now that lilyr yields increase by ZUprs. Repeat your calculations for [mm (d) Given that the bond has a convexity of 33.8. use the convexity adjustment and repeat parts (h) and (c). I-las anything changed? 40. The yield to maturity of a 10-year zero-coupon bond is 4%. [a] Suppose that you buy the bond today and hold it for 10 years. What is your return? (Express this retnm as an annual rate.) (bl Given only the information provided, can you compute the return on the bond if you hold the bond only for 5 years? If you answered yes. compute the return. If you answered no1 ear-plain why. 41. Refer to Table 2. {a} What was the quoted ask price [in dollars] for the 8.?5s of 2020'? Assume par value = 310.000. You can ignore accrued interest. {b} What cash ows would you receive if you bought this bond on August 13, 2006 and held it to maturity? Specify amounts and timing [by month]. {c} Suppose you buy $10 million (par value] of the 4.125s of 2008 and sell short $10 million [par value] of the 3.253 of m. You hold each trade until the bond matures. What cash flows would you pay or receive? Specify amounts and timing. You can ignore any fees or margin requirements for the short sale. Table 2: Treasury Prices and Yields, August 3. 2005 assayed Bonds and Notes: 3.25 . 5.09% 4.125 . : 4.39 6.0 . 4.92 5.25 . 4.86 4.375 . 4.88 12.5 . 4.86 3.?5 . 5.11 3.125 . 5.11 Strips: 5.08% 4.93 4.91 4.80 4.8? 4.92 5.05 42. Refer again to Table 2. {a} What were the l. 2, 3. 4. 6 and 10-year spot interest rates? {b} What was the forward interest rate from August 2007 to August 2003? From August am to August 2010? {c} The 8.75s of August 2020 will pay a coupon in August 2010. What was the PV of this payment in August 2\"? (d) 'What did the slope of the term structure imply about future interest rates? Ex- plain briey. Fall 2003 Page 20 of 36 Express your answers to (a) and (h) as effective annual interest rates. 43. Refer again to Table 2. Use the quoted yield on the August 2012 note to calculate the preseut value for the cash payments on the August 2012 note. Assume that the rst note coupon comes exactly six months after August 13. 2006. Note: The quoted yields are rounded. Your PV may not match the Asked Price exactly. 44. Lu August 2006 you learn that you will receive a $10 million inheritance in August 2007. You have committed to invest it in Treasury securities at that time._ but worry that interest rates may fall over the next year. Assume that you can buy or sell short any of the Treasuries in Table 2 at the prices listed in the table. {a} How would you lock in a one-year interest rate from August 2007 to August sues? What transactions \"mild you make in August 2008? Show how the transactions that look in the rate. (h) Suppose you wanted to lock in a 5-year interest rate from August 2007 to August 2012. How does your answer to part a change? 45. Assume the yield curve is flat at 4%. There are a 3-year zero coupon bond and a 3-year coupon bond that pays a 5% coupon annually. (a) What are the YTMs of these two bonds? (b) Suppose the yield curve does not change in the future. You invest $100 in each of the two bonds. You re-invest all coupons in zero coupon bonds that mature in year 3. How much would you have at the end of year 3? 46. The attached chart shows the fixed obligations of the Edison Mills pension plan, which is also managed by the Renssalear Advisors. Year Benefits ($MM) 2000 $10.60 2001 $11.24 2002 $11.91 2003 $12.62 Total $46.37 Using an assumed interest rate of 6%, the present value of this stream of fixed cash outflows is $40 million. You are given $40 million to invest in U.S. Treasury bonds for the pension plan. Your boss insists that you only invest in 1 year and 10 year STRIPS. Your task is to minimize the exposure of the Edison Mills pension fund to unexpected changes in the level of interest rates. Your performance will be evaluated after one year. Answer the following questions. Use the backs of this page and the next page if needed to complete your answer. Fall 2008 Page 21 of 66 (a) What is duration of your obligation? (b) Describe - step by step - how you would choose and manage the portfolio of 1-year and 10-year STRIPS to minimize the exposure to interest rate risk. 47. Bond underwriting. Bond underwriters agree to purchase a corporate client's new bonds at a specific price. usually near 100% of face value, and then attempt to resell the bonds to the public. The act of reselling takes some time. Underwriting fees increase with the maturity of the bonds. Provide an explanation for this pattern of fees. 48. You have just been given the following bond portfolio: Bond Maturity (yrs) Coupon rate (%) Holdings ($ million) 7.00 10 7.25 20 7.50 20 20 8.00 10 Coupons are paid semi-annually. The current yield curve is flat at 6%. (a) What is duration for each of the bonds in your portfolio? (b) What is the duration of your total portfolio? (c) What is the percentage change in the value of your portfolio if the yield moves up by 20 basis points? 19. A U.S. Treasury bond makes semi-annual payments of $300 for 10 years. (The investor receives 20 $300 payments at 6-month intervals.) At the end of 10 years, the bonds principal amount of $10,000 is paid to the investor. (a) What is the present value of the bond if the annual interest rate is 5%? (b) Suppose the bond is observed trading at $11,240. What discount rate are investors using to value the bonds cash flows? (This discount rate is called the bonds "yield to maturity.")50. A savings bank has the following balance sheet ($ millions, market values). Assets Liabilities Treasuries:$200 Deposits:$900 Floating rate mortgage loans:$300 Equity:$100 Fixed rate mortgage loans:$500 Total: $1,000 Total: $1,000 Fall 2008 Page 22 of 66 Durations are as follows: Treasuries 6 months Floating rate mortgage loans 1 year Fixed rate mortgage loans 5 years Deposits 1 year (a) What is the duration of the banks equity? Briefly explain what this duration means for the banks stockholders. (b) Suppose interest rates move from 3% to 4% (flat term structure). Use duration to calculate the change in the value of the banks equity. Will the actual change be more or less than your calculated value? Explain briefly. 51. Fixed Income Management: A pension fund has the following liability: A 20-yr annuity, that will pay coupons of 7% at the end of each year. (t=1...t=20). The pension fund's liability has a face value of 100. The yield curve is flat at 5%. (a) Calculate the PV and duration of this liability. (b) The same pension fund has the following assets: a 1-yr discount bond with face value 100, and a 20-yr discount bond which also has a face value of 100. Calculate the PV and duration of the portfolio of assets. (c) How would you change the portfolio composition of assets (keeping the PV of assets the same), so that the NPV of the firm, defined as PVA - PVz, that is Present Value of assets minus the Present Value of liabilities, is unaffected by interest rate changes? (d) After making the change above in (c), what is the change in the NPV of the firm if interest rates increase by 10 basis points. 52. Three bonds trade in London and pay annual coupons Bond Coupon Maturity Price 5% 100.96% 6.5% 106.29% 2% 93.84% Prices are in decimals, not 32nds. (a) What is each bond's yield to maturity? (b) What are the 1, 2 and 3-year spot rates? What are the forward rates?53. Assume the spot rates for year 1, year 2 and year 3 are 3.5%, 4% and 4.5%, respectively. There are a 3-year zero coupon bond and a 3-year coupon bond that pays a 5% coupon annually. (a) What are the YTMs of the bonds? (b) Calculate all 1-year forward rates. (c) Calculate the realized returns of the two bonds over the next year if the yield curve does not change. (In year 1 the 1-year spot rate is 3.5%, the 2-year spot rate is 4% and the 3-year spot rate is 4.5%.) 54. A pension plan is obligated to make disbursements of $1 million, $2 million and $1 million at the end of each of the next three years, respectively. Find the durations of the plan's obligations if the interest rate is 10% annually. 55. A local bank has the following balance sheet: Asset Liability Loans $100 million | Deposits $90 million Equity $10 million The duration of the loans is 4 years and the duration of the deposits is 2 years. (a) What is the duration of the bank's equity? How would you interpret the duration of the equity? (b) Suppose that the yield curve moves from 6% to 6.5%. What is the change in the bank's equity value? The term structure is flat at 6%. A bond has 10 years to maturity, face value $100, and annual coupon rate 5%. Interest rates are expressed as EARs. 56. (a) Compute the bond price. (b) Compute the bond's duration and modified duration. (c) Suppose the term structure moves up to 7% (still staying flat). What is the bond's new price' (d) Compute the approximate price change using duration, and compare it to the actual price change. 57. Suppose Microsoft, which has billions invested in short-term debt securities, undertakes the following two-step transaction on Dec. 30, 2009. (1) Sell $1 billion market value of 6-month U.S. Treasury bills yielding 4% (6 month spot rate). (2) Buy $1 billion of 10- year Treasury notes. The notes have a 5.5% coupon and are trading at par. Microsoft does not need the $1 billion for its operations and will hold the notes to maturity. Fall 2008 Page 24 of 66 (a) What is the impact of this two-step transaction on Microsoft's earnings for the first 6 months of 2010? (b) What is the transaction's NPV? Briefly explain your answers. 58. The Treasury bond maturing on August 15, 2017 traded at a closing ask price of 133:16 (i.e., $133 16/32) on August 31, 2007. The coupon rate is 8.875%, paid semi-annually. The yield to maturity was 4.63% (with semi-annual compounding). (a) Explain in detail how this yield to maturity was calculated. (b) Discount the bond's cash flows, using the yield to maturity. Can you replicate the ask price? (The replication should be close but won't be exact.) Show your calculations