Question

Find the duration of a 7.2% coupon bond makingsemiannually coupon payments if it has three years until maturity and has a yield to maturity of 6.0%. What is the duration if the yield to maturity is 10.0%? Note: The face value of the bond is $100.

Answer 1

Duration if YTM is 6%:

Coupon rate per Six Months = 7.2% / 2

= 3.6%

YTM per six Months

= 6% / 2

= 3%

Particulars	Values
YTM per period	3.000%
Coupon Rate per period	3.600%
Time in periods	6
Period/ Year	Period

= [ ( 1 + Y ) / Y ] - [ [ ( 1 + Y ) + T ( C - Y) ] / [ C [ [ ( 1 + Y )^ t ] - 1 ] + Y ] ]
= [ ( 1 + 0.03 ) / 0.03 ] - [ [ ( 1 + 0.03 ) + 6 ( 0.036 - 0.03 ) ] / [ 0.036 [ [ ( 1 + 0.03 ) ^ 6 ] - 1 ] +0.03 ] ]
= [ ( 1.03 ) / 0.03 ] - [ [ ( 1.03 ) + 6 ( 0.006 ) ] / [ 0.036 [ [ ( 1.03 ) ^ 6 ] - 1 ] +0.03 ] ]
= [ 34.3333 ] - [ [ ( 1.03 ) + ( 0.036 ) ] / [ 0.036 [ [ ( 1.1941 ] - 1 ] +0.03 ] ]
= [ 34.3333 ] - [ [ ( 1.066 ) ] / [ 0.036 [ [ 0.1941 ] +0.03 ] ]
= [ 34.3333 ] - [ [ ( 1.066 ) ] / [ 0.007 ] +0.03 ] ]
= [ 34.3333 ] - [ [ ( 1.066 ) ] / [ 0.037 ] ]
= [ 34.3333 ] - [ 28.8108 ]
= 5.5233 Periods

Duration in Years:
= Duration in Periods / 2
Duration in Years:
= Duration in Periods / 2
= 2.76 Years

If YTM is 10%:

Coupon rate per Six Months = 7.2% / 2

= 3.6%

YTM per six Months

= 10% / 2

= 5%

Particulars	Values
YTM per period	5.000%
Coupon Rate per period	3.600%
Time in periods	6
Period/ Year	Period

= [ ( 1 + Y ) / Y ] - [ [ ( 1 + Y ) + T ( C - Y) ] / [ C [ [ ( 1 + Y )^ t ] - 1 ] + Y ] ]
= [ ( 1 + 0.05 ) / 0.05 ] - [ [ ( 1 + 0.05 ) + 6 ( 0.036 - 0.05 ) ] / [ 0.036 [ [ ( 1 + 0.05 ) ^ 6 ] - 1 ] +0.05 ] ]
= [ ( 1.05 ) / 0.05 ] - [ [ ( 1.05 ) + 6 ( -0.014 ) ] / [ 0.036 [ [ ( 1.05 ) ^ 6 ] - 1 ] +0.05 ] ]
= [ 21 ] - [ [ ( 1.05 ) + ( -0.084 ) ] / [ 0.036 [ [ ( 1.3401 ] - 1 ] +0.05 ] ]
= [ 21 ] - [ [ ( 0.966 ) ] / [ 0.036 [ [ 0.3401 ] +0.05 ] ]
= [ 21 ] - [ [ ( 0.966 ) ] / [ 0.0122 ] +0.05 ] ]
= [ 21 ] - [ [ ( 0.966 ) ] / [ 0.0622 ] ]
= [ 21 ] - [ 15.5305 ]
= 5.47 Periods
Duration in Years:
= Duration in Periods / 2
Duration in Years:
= Duration in Periods / 2
= 2.74 Years