An investor purchases two bonds having the same positive annual effective yield rate. 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 d years, with a < d < b. Determine which of the following is an expression for the Macaulay duration of the other bond, in years.

• bd / a
• ad / b
• ab / d
• b + d – a
• a + d – b

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

Solution: A

Let i represent the common yield rate of the two bonds. Since the modified duration is the Macaulay duration divided by (1 + i) and i > 0, the Macaulay duration of each bond is greater than its modified duration. Since a < d < b, the Macaulay duration of d years must be associated with the bond with modified duration a years.

Since the bonds have the same yield rate, the ratio of the two types of duration is the same for each bond. So if x represents the Macaulay duration of the other bond in years, we have d / a = x / b implies ax = bd implies x = bd / a. The Macaulay duration of the other bond is bd / a years.

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