A construction firm is facing three liabilities of 1000, due at the end of each of the periods 1, 2,and 3. There are three bonds available to match these liabilities, as follows:

Bond I: A bond due at the end of period 1 with a coupon rate of 1% per period, valued at a periodic effective yield rate of 14%

Bond II: A bond due at the end of period 2 with a coupon rate of 2% per period, valued at a periodic effective yield rate of 15%

Bond III: A zero-coupon bond due at the end of period 3 valued at a periodic effective yield rate of 18%

Calculate the total purchase price of the three bonds required to exactly match the liabilities.

• 2182
• 2202
• 2222
• 2242
• 2283

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

Solution: D

Let [math]F_1, F_2, F_3[/math] be the redemption amounts of each bond for purchase. To exactly match the liabilities with cash income:

[[math]] \begin{aligned} & 1000=F_3 \\ & F_3=1000 \\ & 1000=F_2(1.02) \\ & F_2=980.39 \\ & 1000-980.39(0.02)=F_1(1.01) \\ & F_1=970.69 \end{aligned} [[/math]]

The total purchase price is

[[math]] \frac{970.69(1.01)}{1.14}+\frac{980.39(0.02)}{1.15}+\frac{980.39(1.02)}{1.15^2}+\frac{1000}{1.18^3}=2241.82 [[/math]]

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