An investor purchases two bonds. The bonds have the same annual effective yield rate i, with i > 0. With respect to the annual effective yield rate, their modified durations are a years and b years, with 0 < a < b. One of these two bonds has a Macaulay duration of c years, with a < c < b.

Determine which of the following is an expression, in years, for the Macaulay duration of the other bond.

• bc/a
• ac/b
• ab/c
• b + c – a
• a + c – b

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

Solution: A

Because the interest rate is greater than zero, the Macaulay duration of each bond is greater than its modified duration. Therefore, the bond with a Macaulay duration of c must be the bond with a modified duration of a and a = c/(1 + i) which implies 1 + i = c/a. The Macaulay duration of the other bond is b(1 + i) =bc/a.

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