An insurance company has liabilities due at the end of each of the next two years. The liability due at the end of the second year is twice that for the first year. The company uses a combination of the following two bonds to match the liabilities. The second bond has annual coupons.

Term to maturity (in years)	Annual coupon rate	Annual effective yield
1	0%	6%
2	5%	8%

Assume both bonds are redeemed at par and the company invests 9465 in the two-year bond. Calculate the amount the company should invest in the one-year bond to construct an exactly matched portfolio.

• 4245
• 4398
• 4481
• 4717
• 4953

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

Let [math]Y[/math] indicate the nominal value of the two-year bond, then: [math]9465=\frac{0.05 Y}{1.08}+\frac{1.05 Y}{1.08^2}[/math], so [math]Y=10,000[/math].

Thus, the amount of liability at the end of the second year is 10,500 . Hence, the liability at the end of the first year is:

[[math]] \frac{10,500}{2}=5250 [[/math]]

So, the amount invested in the one-year bond is:

[[math]] \frac{5250-10,000(0.05)}{1.06}=4481 \text {. } [[/math]]

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