Question

Consider a 3-year bond with 14 percent semi-annual coupon payments and currently priced to yield 12 per cent per annum.

1. Calculate duration of the bond.
2. What is the percentage of price change if interest rate increases to 12.15% per annum?

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.

Particulars	Values
YTM per period	6.000%
Coupon Rate per period	7.000%
Time in periods	6
Period/ Year	Period

= [ ( 1 + Y ) / Y ] - [ [ ( 1 + Y ) + T ( C - Y) ] / [ C [ [ ( 1 + Y )^ t ] - 1 ] + Y ] ]
= [ ( 1 + 0.06 ) / 0.06 ] - [ [ ( 1 + 0.06 ) + 6 ( 0.07 - 0.06 ) ] / [ 0.07 [ [ ( 1 + 0.06 ) ^ 6 ] - 1 ] +0.06 ] ]
= [ ( 1.06 ) / 0.06 ] - [ [ ( 1.06 ) + 6 ( 0.01 ) ] / [ 0.07 [ [ ( 1.06 ) ^ 6 ] - 1 ] +0.06 ] ]
= [ 17.6667 ] - [ [ ( 1.06 ) + ( 0.06 ) ] / [ 0.07 [ [ ( 1.4185 ] - 1 ] +0.06 ] ]
= [ 17.6667 ] - [ [ ( 1.12 ) ] / [ 0.07 [ [ 0.4185 ] +0.06 ] ]
= [ 17.6667 ] - [ [ ( 1.12 ) ] / [ 0.0293 ] +0.06 ] ]
= [ 17.6667 ] - [ [ ( 1.12 ) ] / [ 0.0893 ] ]
= [ 17.6667 ] - [ 12.542 ]
= 5.13 Periods

Duration in Years:
= Duration in Periods / 2
Duration in Years:
= Duration in Periods / 2
= 2.56 Years

Modified duaration :

Modified duration is a measurable change in the value of a security in response to a change in interest rates.

Modified duration = Duration / [ 1 + YTM ]
It specifies% change in Price in opposite direction due to 1% change in YTM.

Particulars	Values
Duration	2.56
YTM	12.0000%

Modified Duration = Duration / [ 1 + YTM ]
= 2.56 / [ 1 + 0.12 ]
= 2.56 / [ 1.12 ]
= 2.2857 %

I.e 1% change in disc rate leads to 2.2857 % change in Bond Price

Change in Disc Rate is 0.15 %
% Change in Bond price for change in disc rate by 0.15 % is [ 2.2857 % * 0.15 ]

I.e0.15 %change in Disc Rate leads to 0.3429 %