B2. Duration hedging A company agreed many years ago to pay pensions to its workers. Many of these workers are now reaching retirement. The company estimates that it will need to make aggregate pension payments amounting to exactly 1.5m GBP at the end of this year and at the end of next year. Currently, the spot rates at all maturities are equal to 4% (with continuous compounding).
a. Calculate the present value of the pension liabilities.
b. The company has access to a 3-year bond that pays a coupon of 5.5% at the end of every year. i. Define the concept of (modified) duration. ii. Calculate the (modified) duration of the bond (with respect to the continuous-compounded yield to maturity). [11 marks]
c. The company would like to buy an amount of the bond, such that the combined present value of the liability from the financial product and the position in the bond does not change for small parallel shifts in the spot curve. Calculate the position that the company would have to take in the bond. Explain any approximations that you need to make. In your opinion, how well is this hedge likely to perform in practice? [12 marks]
d. Check how the portfolio that you have set up in part c) performs in two scenarios:
i. In a scenario in which the spot rates with maturities of 1, 2, and 3 years decrease by 10bp.
ii. In a scenario in which spot rates with maturities of 1 and 2 years decrease by 10bp, but the spot rate with a maturity of 3 years increases by 10bp. [12 marks]