Question

What is the Macaulay duration (D) of a 2-year bond with a $73 annual coupon (paid annually), $1,000 par, and a yield of 5.4%? Round to four decimals. (show work)

Answer 1

Duration:

Duration is a measure of the sensitivity of the price of a bond or other debt instrument to a change in interest rates.

Duaration = Sum [ Weight * Year ]

Particulars	Amount
Coupon Amount	$           73.00
Maturity Value	$     1,000.00
Disc Rate	5.4000%

Duration:

Year	Cash Flow	PVF@ 5.4 %	Disc CF	Weight	Wt * Year
1	$                73.00	0.9488	$      69.26	0.0669	0.0669
2	$                73.00	0.9002	$      65.71	0.0635	0.1270
2	$           1,000.00	0.9002	$    900.16	0.8696	1.7392
Duration in Years					1.9331

Thus Duration of Bond is 1.93 Years.

Alternatively:

= [ ( 1 + Y ) / Y ] - [ [ ( 1 + Y ) + T ( C - Y) ] / [ C [ [ ( 1 + Y )^ t ] - 1 ] + Y ] ]   
= [ ( 1 + 0.054 ) / 0.054 ] - [ [ ( 1 + 0.054 ) + 2 ( 0.073 - 0.054 ) ] / [ 0.073 [ [ ( 1 + 0.054 ) ^ 2 ] - 1 ] +0.054 ] ]  
= [ ( 1.054 ) / 0.054 ] - [ [ ( 1.054 ) + 2 ( 0.019 ) ] / [ 0.073 [ [ ( 1.054 ) ^ 2 ] - 1 ] +0.054 ] ]  
= [ 19.5185 ] - [ [ ( 1.054 ) + ( 0.038 ) ] / [ 0.073 [ [ ( 1.1109 ] - 1 ] +0.054 ] ]  
= [ 19.5185 ] - [ [ ( 1.092 ) ] / [ 0.073 [ [ 0.1109 ] +0.054 ] ]  
= [ 19.5185 ] - [ [ ( 1.092 ) ] / [ 0.0081 ] +0.054 ] ]  
= [ 19.5185 ] - [ [ ( 1.092 ) ] / [ 0.0621 ] ]  
= [ 19.5185 ] - [ 17.5845 ]  
= 1.93 Years