An investor purchases a 1200 face amount zero-coupon bond for a price of 1000. With respect to the bond’s annual effective yield rate, the Macaulay duration is four years and the modified duration is d years.

Calculate d.

• 3.33
• 3.82
• 3.86
• 4.00
• 4.19

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

Solution: B

Since the bond has no coupons, the Macaulay duration is the same as the amount of time until maturity, namely 4 years.

Thus, the effective annual yield rate, y, is

[[math]] \left(\frac{1200}{1000} \right)^{1/4} -1 = 0.046635. [[/math]]

The modified duration equals the Macaulay duration divided by (1 + y). Thus the modified duration is 4/1.046635 = 3.82177 years.

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

Solution: B

Since the bond has no coupons, the Macaulay duration is the same as the amount of time until maturity, namely 4 years.

Thus, the effective annual yield rate, y, is

[[math]] \left(\frac{1200}{1000} \right)^{1/4} -1 = 0.046635. [[/math]]

The modified duration equals the Macaulay duration divided by (1 + y). Thus the modified duration is 4/1.046635 = 3.82177 years.

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