You are given the following information regarding Company J.

1. It has a single liability of 1,750,000 to be paid 12 years from now.
2. Its asset portfolio consists of a zero-coupon bond maturing in 5 years for 242,180 and a zero-coupon bond maturing in n years.
3. At an annual effective interest rate of 7%, Company J’s position is fully immunized.

Calculate n.

• 13
• 14
• 15
• 16
• 17

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

Solution: B

Use the full immunization equations and let [math]N[/math] be the maturity value of the asset maturing in [math]n[/math] years.

[[math]] \begin{aligned} & 242,180(1.07)^7+N(1.07)^{-(n-12)}-1,750,000=0 \\ & 242,180(7)(1.07)^7-N(n-12)(1.07)^{-(n-12)}=0 \end{aligned} [[/math]]

From the first equation:

[[math]] N(1.07)^{-(n-12)}=1,750,000-242,180(1.07)^7=1,361,112 \text {. } [[/math]]

Substituting this in the second equation: [math]n-12=242,180(7)(1.07)^7 / 1,361,112=2[/math] and so [math]n=14[/math]

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