An insurance company has a liability of 2662 that is due at the end of three years. The present value of this liability is 2000. There are two investments available: a one-year zero-coupon bond and a four-year zero-coupon bond. The company wants to find an investment plan that satisfies Redington immunization.

Calculate the amount the company invests in the one-year zero-coupon bond.

• 400
• 667
• 858
• 1,000
• There is no investment plan that satisfies Redington immunization

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

Solution: B

Let [math]x[/math] be the amount invested in the one-year bond and [math]y[/math] the amount invested in the four-year bond. First match the present value of assets and liabilities:

[[math]] \begin{aligned} & P V_A=P V_L \\ & x+y=2000 \end{aligned} [[/math]]

Second, the durations of assets and liabilities should also match:

[[math]] \begin{aligned} & D_A=\frac{1 x+4 y}{x+y} \\ & D_A=\frac{1 x+4(2000-x)}{2000}=D_L=3 \\ & x=666.67 \end{aligned} [[/math]]

Convexity of the assets is:

[[math]] \frac{666.67\left(1^2\right)+1333.33\left(4^2\right)}{2000}=11 [[/math]]

Convexity of the liability is: [math]3^2=9[/math]. Convexity of assets is greater than convexity of liabilities so Reddington immunization is met.

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