Question

Consider the following.

a. What is the duration of a four-year Treasury bond with a 9.5 percent semiannual coupon selling at par? (Do not round intermediate calculations. Round your answer to 2 decimal places. (e.g., 32.16)) Duration of the bond years

b. What is the duration of a three-year Treasury bond with a 9.5 percent semiannual coupon selling at par? (Do not round intermediate calculations. Round your answer to 2 decimal places. (e.g., 32.16)) Duration of the bond years

c. What is the duration of a two-year Treasury bond with a 9.5 percent semiannual coupon selling at par? (Do not round intermediate calculations. Round your answer to 2 decimal places. (e.g., 32.16)) Duration of the bond years

Answer 1

If the Bond is selling at par mean, coupon Rate and YTM are same.

Part A:

Particulars	Values
YTM per period	4.750%
Coupon Rate per period	4.750%
Time in periods	8
Period/ Year	Period

= [ ( 1 + Y ) / Y ] - [ [ ( 1 + Y ) + T ( C - Y) ] / [ C [ [ ( 1 + Y )^ t ] - 1 ] + Y ] ]
= [ ( 1 + 0.0475 ) / 0.0475 ] - [ [ ( 1 + 0.0475 ) + 8 ( 0.0475 - 0.0475 ) ] / [ 0.0475 [ [ ( 1 + 0.0475 ) ^ 8 ] - 1 ] +0.0475 ] ]
= [ ( 1.0475 ) / 0.0475 ] - [ [ ( 1.0475 ) + 8 ( 0 ) ] / [ 0.0475 [ [ ( 1.0475 ) ^ 8 ] - 1 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) + ( 0 ) ] / [ 0.0475 [ [ ( 1.4495 ] - 1 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) ] / [ 0.0475 [ [ 0.4495 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) ] / [ 0.0214 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) ] / [ 0.0689 ] ]
= [ 22.0526 ] - [ 15.2032 ]
= 6.8526 Periods

Duration in Years:
= Duration in Periods / 2
Duration in Years:
= Duration in Periods / 2
= 3.43 Years

Part B:

Particulars	Values
YTM per period	4.750%
Coupon Rate per period	4.750%
Time in periods	6
Period/ Year	Period

= [ ( 1 + Y ) / Y ] - [ [ ( 1 + Y ) + T ( C - Y) ] / [ C [ [ ( 1 + Y )^ t ] - 1 ] + Y ] ]
= [ ( 1 + 0.0475 ) / 0.0475 ] - [ [ ( 1 + 0.0475 ) + 6 ( 0.0475 - 0.0475 ) ] / [ 0.0475 [ [ ( 1 + 0.0475 ) ^ 6 ] - 1 ] +0.0475 ] ]
= [ ( 1.0475 ) / 0.0475 ] - [ [ ( 1.0475 ) + 6 ( 0 ) ] / [ 0.0475 [ [ ( 1.0475 ) ^ 6 ] - 1 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) + ( 0 ) ] / [ 0.0475 [ [ ( 1.3211 ] - 1 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) ] / [ 0.0475 [ [ 0.3211 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) ] / [ 0.0153 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) ] / [ 0.0628 ] ]
= [ 22.0526 ] - [ 16.6799 ]
= 5.3726 Periods

Duration in Years:
= Duration in Periods / 2
Duration in Years:
= Duration in Periods / 2
= 2.69 Years

Part C:

Particulars	Values
YTM per period	4.750%
Coupon Rate per period	4.750%
Time in periods	4
Period/ Year	Period

= [ ( 1 + Y ) / Y ] - [ [ ( 1 + Y ) + T ( C - Y) ] / [ C [ [ ( 1 + Y )^ t ] - 1 ] + Y ] ]
= [ ( 1 + 0.0475 ) / 0.0475 ] - [ [ ( 1 + 0.0475 ) + 4 ( 0.0475 - 0.0475 ) ] / [ 0.0475 [ [ ( 1 + 0.0475 ) ^ 4 ] - 1 ] +0.0475 ] ]
= [ ( 1.0475 ) / 0.0475 ] - [ [ ( 1.0475 ) + 4 ( 0 ) ] / [ 0.0475 [ [ ( 1.0475 ) ^ 4 ] - 1 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) + ( 0 ) ] / [ 0.0475 [ [ ( 1.204 ] - 1 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) ] / [ 0.0475 [ [ 0.204 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) ] / [ 0.0097 ] +0.0475 ] ]
= [ 22.0526 ] - [ [ ( 1.0475 ) ] / [ 0.0572 ] ]
= [ 22.0526 ] - [ 18.3129 ]
= 3.7426 Periods

Duration in Years:
= Duration in Periods / 2
Duration in Years:
= Duration in Periods / 2
= 1.87 Years