A liability consists of a series of 15 annual payments of 35,000 with the first payment to be made one year from now. The assets available to immunize this liability are five-year and ten-year zero-coupon bonds. The annual effective interest rate used to value the assets and the liability is 6.2%. The liability has the same present value and duration as the asset portfolio.

Calculate the amount invested in the five-year zero-coupon bonds.

• 127,000
• 167,800
• 208,600
• 247,900
• 292,800

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

Solution: C

First, the present value of the liability is

[[math]] \mathrm{PV} = 35,000 a_{\overline{15}|6.2\%} = 335,530.30. [[/math]]

The duration of the liability is:

[[math]] \overline{d} = \frac{\sum tv^tR_t}{\sum v^tR_{t}} =\frac{35,000v+2(35,000)v^{2}+\cdots+15(35,000)v^{15}}{335,530.30}={\frac{2,312,521.95}{335.530.30}}=6.89214 [[/math]]

Let X denote the amount invested in the 5 year bond. Then

[[math]] \frac{X}{335,530.30}(5)+\left(1-\frac{X}{335,530.30}\right)(10)=6.89214 \implies X=208,556. [[/math]]

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