Hedging Strategy and Electricity Contract Engineering - IFOR

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108 Hedging strategies
has to wait before receiving cash payments. A zero coupon bond that matures
in n years also has a duration of n years, since all payments are made then. A
coupon bearing bond however, maturing in n years has a duration of less than
n years, since some cash flows occur prior to maturity.
Let current time be 0 and suppose there is a bond providing the holder with n
yearly payments c i at time t i (i 1 ž?ž@ž š n). The price of the bond B, and the
continuously compounded yield y, are related by
š
B
n
i 1
c i e«
yt i
ž (5.9)
The duration D, of the bond is then given by
D
n
i 1
t i
c i e«
yt i
B ž (5.10)
From (5.9) we can derive the bond price’s sensitivity to changes in the yield y
B
y
n
i 1
c i t i e«
yt i š (5.11)
which together with (5.10) can be written as
B
y
B Dž (5.12)
The relative change in the bond price Å B¡ B will for small parallel shifts in the
yield curve Å y, therefore be given by
Å B¡ B
Å y
DÅ yž (5.13)
From (5.13) one can see that the relative bond price changes are proportional
to the duration, why duration is often used as a measure of risk in the fixed
income market. By hedging a position such that the duration equals zero, the