You are given the following information about a fully immunized portfolio:

1. The liability is a single payment of 600,000 due in two years.
2. The asset portfolio consists of a one-year zero-coupon bond maturing for x and a four-year zero-coupon bond maturing for y.
3. The annual effective interest rate is 4.6%.

Calculate x.

• 218,800
• 325,400
• 365,600
• 382,400
• 402,800

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

Solution: D

The [math]P V[/math] of the liability is [math]\frac{600,000}{1.046^2}=548,387.92[/math] and its Macaulay duration is 2 . Then, equating present values:

[[math]] \frac{x}{1.046}+\frac{y}{1.046^4}=548,387.92 [[/math]]

And equating durations:

[[math]] \frac{(x / 1.046)}{548,387.92}(1)+\frac{\left(y / 1.046^4\right)}{548,387.92}(4)=2 [[/math]]

Solving the system of equations results in [math]x=382,409[/math]

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