Duration and Convexity
The
f
rst and second derivatives of bond prices with respect to interest rates
are crucial in hedging interest rate risk. These can be numerically calculated on a
computer. But they can be guessed by using the concepts of average life or duration
and average life squared or convexity.
1D
u
r
a
t
i
o
n
o
r
A
v
e
r
a
g
e
L
i
f
e
How much does price change when the interest rates change? By what percentage
does price change when the interest rate changes
?
Is there some way to see the
answer simply, in one’s head, rather than having to compute a complicated formula?
A brilliantly simple answer was provided by
John Hicks
in his famous book
Value and Capital
in 1939. Also discovered by
Macauley
in 1938.
They de
f
ned the
average life
or
duration
of a bond, and the
average life
squared
or
convexity
of a bond.
1.1
Average Life or Duration de
f
ned
Duration (Hicks, Macauley)
Average life of a bond should be the average
age at which the bond’s value is paid. The
average life
or
duration
of a bond with
payments
(
m
1
,m
2
,...,m
T
)
at
f
xed interes rate
r
is de
f
ned to be
AvgLife
(
m, r
)=
D
(
m, r
)
≡
m
1
1+
r
1+
m
2
(1 +
r
)
2
2+
m
3
(1 +
r
)
3
3+
···
+
m
T
(1 +
r
)
T
T
PV
(
m, r
)
=
T
X
t
=1
m
t
(1 +
r
)
t
t
PV
(
m, r
)
For example, let
m
=(0
,...,
0
,m
t
,
0
,...,
0)
beazerocouponbondw
ithbu
l
letpayment
at t. Then
D
(
m, r
)=
m
t
(1 +
r
)
t
t
PV
(
m, r
)
=
t
The
f
rst thing to notice about average life is that it does not depend on the level
of the payments
m
t
,
but rather on their relative sizes. Doubling all the
m
t
would
double the numerator, but also double PV(r) and therefore the denominator, leaving
average life the same. This is the
f
rst reason average life an easy number to work
with.
The second reason is that average life does not change much with the interest