A scholarship foundation is committed to make a payment of 20,000 exactly two years from today. It plans to use two zero-coupon bonds to fully immunize the liability based on a flat curve and an annual effective yield rate of 5.5%. One of the bonds will have a one-year maturity and the other will have a three-year maturity.

Calculate the face value of the one-year zero-coupon bond

• 9,479
• 9,779
• 10,079
• 10,379
• 10,679

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

Solution: A

Let [math]h(i)=P V_A(i)-P V_L(i)[/math]. Full immunization of a single liability requires both equations:

[[math]] \begin{aligned} & h(i)=0, h^{\prime}(i)=0 \\ & A_1 v+A_3 v^3-20,000 v^2=0 \\ & A_1 v^2+3 A_3 v^4-40,000 v^3=0 \\ & v=\frac{1}{1.055} \end{aligned} [[/math]]

Solve these two equations in two unknowns to get A₁ = 9478.67

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