An investor purchases a portfolio consisting of three bonds. Bond A has annual coupons of 6% and matures for its face amount of 1000 in ten years. It is purchased for 1000. Bonds B and C are zero-coupon bonds, maturing for 1000 each in five and ten years, respectively. All three bonds have the same yield rate.

Calculate the Macaulay duration in years at the time of purchase of the portfolio with respect to the common yield rate.

• 7.23
• 7.43
• 7.60
• 8.33
• 8.38

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

Solution: B

Because Bond A sold for its fact amount, the yield rate is the coupon rate of [math]6 \%[/math] per year. The present values of the three bonds are:

Bond A: 1000 (given) Bond B: [math]1000(1.06)^{-5}=747.26[/math] Bond C: [math]1000(1.06)^{-10}=558.39[/math] The durations are: Bond A: 7.8017 Bond B: 5 Bond C: 10 The portfolio duration is the average of these three durations, weighted by the bond prices.

[[math]] \text { Duration }=\frac{1000(7.8017)+747.26(5)+558.39(10)}{1000+747.26+558.39}=7.426 [[/math]]

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