Problem 2.6. Portfolio A consists of a two-year zero-coupon bond with a face value of $3,000 and a 10-year zero-coupon bond with a face value of $7,000. Portfolio B consists of a 6.88-year zero-coupon bond with a face value of $10,000. The current yield on all bonds is 5% per annum. (a) Show that both portfolios have the same duration. (6) Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same to 0.01%. (c) What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields? Problem 2.7. A portfolio manager plans to use a Treasury bond futures contract to hedge a bond portfolio over the next three months. The portfolio is worth $54 million and will have a duration of 6 years in three months. The futures price is 120, and each futures contract is on $100,000 of bonds. The bond that is expected to be cheapest to deliver will have a duration of 11 years at the maturity of the futures contract. (a) What position in futures contracts is required? (6) Suppose that all rates increase over the three months, but long-term rates increase less than short-term and medium-term rates. What is the effect of this on the performance of the hedge? Problem 2.6. Portfolio A consists of a two-year zero-coupon bond with a face value of $3,000 and a 10-year zero-coupon bond with a face value of $7,000. Portfolio B consists of a 6.88-year zero-coupon bond with a face value of $10,000. The current yield on all bonds is 5% per annum. (a) Show that both portfolios have the same duration. (6) Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same to 0.01%. (c) What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields? Problem 2.7. A portfolio manager plans to use a Treasury bond futures contract to hedge a bond portfolio over the next three months. The portfolio is worth $54 million and will have a duration of 6 years in three months. The futures price is 120, and each futures contract is on $100,000 of bonds. The bond that is expected to be cheapest to deliver will have a duration of 11 years at the maturity of the futures contract. (a) What position in futures contracts is required? (6) Suppose that all rates increase over the three months, but long-term rates increase less than short-term and medium-term rates. What is the effect of this on the performance of the hedge