A company is required to pay 190,000 in 20.5 years. The company creates an investment portfolio using three bonds with annual coupons, so that its position is Redington immunized based on an annual effective interest rate of 7%. The table below shows the Macaulay duration for each of the bonds.

	Macaulay Duration
Bond A	10 years
Bond B	15 years
Bond C	30 years

The company invests twice as much money in Bond C as in Bond B.

Calculate the amount the company invests in Bond A.

• 6,640
• 8,630
• 11,075
• 13,308
• 14,240

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

Solution: E

Let x be the amount invested in Bond A and y the amount invested in Bond B. Then 2y is invested in Bond C. To match the present value of the assets and liabilities:

[[math]] \begin{array}{l}{{x+y+2y=190,000(1.07)^{-20.5}}}\\ {{x+3y=47,466.39.}}\end{array} [[/math]]

To match the Macauley durations, . Then,

[[math]] \begin{array}{c}{{20.5(47,466.39)=10(47,466.39-3y)+75y}}\\ {{\phantom{M}}}\\ {{y=\frac{20.5(47,366.39)-10(47,466.39)}{75-30}=11,075.49}}\end{array} [[/math]]

and

[[math]]X = 47,466.39 – 3(11,075.49) = 14,239.92[[/math]]

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