Revision as of 18:32, 20 November 2023 by Admin (Created page with "You are given the following information about a fully immunized portfolio: # The liability is a single payment of 600,000 due in two years. # The asset portfolio consists of a one-year zero-coupon bond maturing for x and a four-year zero-coupon bond maturing for y. # The annual effective interest rate is 4.6%. Calculate x. <ul class="mw-excansopts"><li>218,800</li><li>325,400</li><li>365,600</li><li>382,400</li><li>402,800</li></ul> {{soacopyright | 2023 }}")
Nov 20'23
Exercise
You are given the following information about a fully immunized portfolio:
- The liability is a single payment of 600,000 due in two years.
- The asset portfolio consists of a one-year zero-coupon bond maturing for x and a four-year zero-coupon bond maturing for y.
- 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.
Nov 20'23
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.