6. (A bond project ) You are the manager of XYZ Pension Fund. On November 5, 2021, XYZ must purchase a portfolio of U.S. Treasury bonds to meet the fund's projected liabili- ties in the future. The bonds available at that time are those of Exercise 4 in Chapter 4. Short selling is not allowed. Following the procedure of the earlier exercise, a 4th-order polyno- mial estimate of the term structure is constructed as r(t) =Q0+at+a2t2 +azt3 +2414. The liabilities of XYZ are as listed in Table 5.8. (a) (Simple cash matching) Construct a minimum-cost liability-matching portfolio by buying Treasury bonds assuming that excess periodic cash flows may be held only at zero interest to meet future liabilities. TABLE 5.8 LIABILITIES OF XYZ PENSION FUND Liabilities Occur on 15th Feb 2022 Aug 2022 Feb 2023 Aug 2023 Feb 2024 Aug 2024 Feb 2025 Aug 2025 Feb 2026 $2,000 $20,000 $0 $25,000 $1,000 $0 $20,000 $1,000 $15,000 (b) (Complex cash matching Construct a minimum-cost liability-matching portfolio by buying Treasury bonds assuming that all excess periodic cash flows may be rein- vested at the expected interest rates (implied by the current term structure) to meet future liabilities. No borrowing is allowed. (c) (Duration matching) Construct a minimum-cost portfolio with present value equal to that of the liability stream. Immunize against a change in the term structure param- eters. Do this for five cases. Case 1 is to guard against a change in Q1, case 2 to guard against changes in Qy and a2, and so on. TABLE 4.6 BOND QUOTES Ask price Coupon 6.625 9.125 7.875 8.250 8.250 8.375 8.000 8.750 6.875 8.875 6.875 8.625 7.750 11.250 8.500 10.500 7.875 8.875 Maturity 15-Feb-22 15-Feb-22 15-Aug-22 15-Aug-22 15-Feb-23 15-Feb-23 15-Aug-23 15-Aug-23 14-Feb-24 14-Feb-24 15-Aug-24 15-Aug-24 15-Feb-25 15-Feb-25 15-Aug-25 15-Aug-25 15-Feb-26 15-Feb-26 100.00 100.67 100.69 100.03 100.22 100.38 100.81 102.03 98.16 102.28 97.41 101.72 99.16 109.13 101.41 107.84 99.41 103.00 6. (A bond project ) You are the manager of XYZ Pension Fund. On November 5, 2021, XYZ must purchase a portfolio of U.S. Treasury bonds to meet the fund's projected liabili- ties in the future. The bonds available at that time are those of Exercise 4 in Chapter 4. Short selling is not allowed. Following the procedure of the earlier exercise, a 4th-order polyno- mial estimate of the term structure is constructed as r(t) =Q0+at+a2t2 +azt3 +2414. The liabilities of XYZ are as listed in Table 5.8. (a) (Simple cash matching) Construct a minimum-cost liability-matching portfolio by buying Treasury bonds assuming that excess periodic cash flows may be held only at zero interest to meet future liabilities. TABLE 5.8 LIABILITIES OF XYZ PENSION FUND Liabilities Occur on 15th Feb 2022 Aug 2022 Feb 2023 Aug 2023 Feb 2024 Aug 2024 Feb 2025 Aug 2025 Feb 2026 $2,000 $20,000 $0 $25,000 $1,000 $0 $20,000 $1,000 $15,000 (b) (Complex cash matching Construct a minimum-cost liability-matching portfolio by buying Treasury bonds assuming that all excess periodic cash flows may be rein- vested at the expected interest rates (implied by the current term structure) to meet future liabilities. No borrowing is allowed. (c) (Duration matching) Construct a minimum-cost portfolio with present value equal to that of the liability stream. Immunize against a change in the term structure param- eters. Do this for five cases. Case 1 is to guard against a change in Q1, case 2 to guard against changes in Qy and a2, and so on. TABLE 4.6 BOND QUOTES Ask price Coupon 6.625 9.125 7.875 8.250 8.250 8.375 8.000 8.750 6.875 8.875 6.875 8.625 7.750 11.250 8.500 10.500 7.875 8.875 Maturity 15-Feb-22 15-Feb-22 15-Aug-22 15-Aug-22 15-Feb-23 15-Feb-23 15-Aug-23 15-Aug-23 14-Feb-24 14-Feb-24 15-Aug-24 15-Aug-24 15-Feb-25 15-Feb-25 15-Aug-25 15-Aug-25 15-Feb-26 15-Feb-26 100.00 100.67 100.69 100.03 100.22 100.38 100.81 102.03 98.16 102.28 97.41 101.72 99.16 109.13 101.41 107.84 99.41 103.00