FM for Actuaries

248 CHAPTER 8
Example 8.1: Calculate the Macaulay duration of a 4-year annual coupon bond
with 6% coupon and a yield to maturity of 5.5%.
Solution: The present values of the cash flows can be calculated using (8.1) with
i =5.5%. The computation of the Macaulay duration is presented in Table 8.1.
Table 8.1: Computation for Example 8.1
t (years) C t PV(C t ) w t tw t
1 6 5.6872 0.0559 0.0559
2 6 5.3907 0.0530 0.1060
3 6 5.1097 0.0502 0.1506
4 106 85.5650 0.8409 3.3636
Total 101.7526 1.0000 3.6761
The price of the bond P is equal to the sum of the third column, namely 101.7526.
Note that the entries in the fourth column are all positive and sum up to 1. The
Macaulay duration is the sum of the last column, which is 3.6761 years. Thus, the
Macaulay duration of the bond is less than its time to maturity of 4 years.
Example 8.2: Calculate the Macaulay duration of a 2-year semiannual coupon
bond with 4% coupon per annum and a yield to maturity of 4.8% compounded
semiannually.
Solution: The cash flows of the bond occur at time 1, 2, 3 and 4 half-years.
The present values of the cash flows can be calculated using (8.1) with i =2.4%
per payment period (i.e., half-year). The computation of the Macaulay duration is
presented in Table 8.2.
Table 8.2: Computation for Example 8.2
t (half-years) C t PV(C t ) w t tw t
1 2 1.953 0.0198 0.0198
2 2 1.907 0.0194 0.0388
3 2 1.863 0.0189 0.0568
4 102 92.768 0.9419 3.7676
Total 98.491 1.0000 3.8830
The price of the bond is equal to the sum of the third column, namely 98.491. The
Macaulay duration is the sum of the last column, namely 3.8830 half-years, which