An insurance company has a single liability due in three years. The company fully immunizes its position by purchasing one-year and four-year zero-coupon bonds. The face value of the one- year bond is 20,000 and the face value of the four-year bond is 50,000. Assume that the yield curve is flat.

• 40,000
• 55,699
• 69,624
• 73,333
• 97,500

Calculate the amount of the liability.

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

Solution: C

Macaulay duration of the liability is 3 . Asset duration must equal 3. Let [math]P_1[/math] and [math]P_4[/math] be the present values of the two assets.

[[math]] \begin{aligned} & \frac{P_1 \cdot 1+P_4 \cdot 4}{P_1+P_4}=3 \text {, then } P_4=2 P_1 \\ & P_1=\frac{20,000}{1+i}, P_4=\frac{50,000}{(1+i)^4}, \frac{P_1}{P_4}=\frac{P_1}{2 P_1}=\frac{1}{2}=\frac{\frac{20,000}{\frac{1+i}{50,000}}}{(1+i)^4}: 1=0.8(1+i)^3: \\ & (1+i)^3=1.25 ;(1+i)=1.077217 \end{aligned} [[/math]]

PV of assets must equal PV of liabilities. So, PV of assets equals:

[[math]] P_1+2 P_1=3 P_1=3 \frac{20,000}{1.077217}=55,699.07 \text {. } [[/math]]

Amount of liability equals: 55,699.07(1.077217) [math]=69,623.83[/math].

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