An institute has provided an early retirement incentive package to a 60-year-old retiree that pays 12,000 per year at the end of each year up to and including age 65, plus a lump sum payment of 150,000 at age 65. All payments are guaranteed whether or not the retiree is alive at age 65. The institute will create a portfolio of two five-year bonds to exactly match the payments under this package. The first bond has a face amount of 100,000 and an annual coupon rate of 10%.

Determine which of the following second bonds will exactly match the liability.

• A bond with a price of 42,015, an annual coupon rate of 4% and annual effective yield of 8%.
• A bond with a price of 50,000, an annual coupon rate of 4% and annual effective yield of 8%.
• A bond with a price of 41,588, an annual coupon rate of 8% and annual effective yield of 10%.
• A bond with a price of 50,000, an annual coupon rate of 8% and annual effective yield of 10%.
• A bond with a price of 55,451, an annual coupon rate of 8% and annual effective yield of 10%.

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

Solution: A

Cash flows (in thousands) are 12, 12, 12, 12, and 162. The first bond provides payments of 10,10, 10, 10, and 110. Therefore, the second bond must provide 2, 2, 2, 2, and 52. This implies a coupon rate of 2/50 = 4% and a face amount of 50. Only Answers A and B provide these. At an 8% yield, the price of this bond is 42.015 (or 42,015).

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