A company must pay liabilities of 1000 at the end of year 1 and X at the end of year 2. The only investments available are:

1. One-year zero-coupon bonds with an annual effective yield of 5%
2. Two-year bonds with a par value of 1000 and 10% annual coupons, with an annual effective yield of 6%

The company constructed a portfolio that creates an exact cash flow matching strategy for these liabilities. The total purchase price of this portfolio is 1783.76.

Calculate the amount invested in the one-year zero-coupon bonds.

• 784
• 831
• 871
• 915
• 935

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Let A be the redemption value of the zero-coupon bonds purchased and B the number of two- year bonds purchased. The total present value is:

[[math]] 1783.76=A/1.05+B(100/1.06+1\,100/1.06^{2})=0.952384+1073.3357B. [[/math]]

To exactly match the cash flow at time one, A + 100B = 1000. Substituting B = 10 – 0.01A in the first equation gives 1783.76 = 0.95238A + 10733.357 – 10.733357A for A = 8949.597/9.780977 = 915. The amount invested is then 915/1.05 = 871.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.