15a_duration - Duration and Convexity The rst and second...

Duration and Convexity
The fi 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.
1 Duration or Average Life
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 fi 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 fi 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 fi xed interes rate r is de fi 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) be a zero coupon bond with bullet payment
at t. Then
D ( m, r ) =
m
t
(1 + r ) t
t
PV ( m, r )
= t
The fi 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 fi rst reason average life an easy number to work
with.
The second reason is that average life does not change much with the interest
rate. A higher interest rate lowers both numerator and denominator, leaving average
life nearly the same.
1

Third, average life is a weighted average of numbers, and most people are good
at doing such calculations in their head, by intuition.
Let us consider three examples showing that indeed average life is a very easy
number to guess in one’s head. A one year zero has average life 1, no matter whether
it is a Washington zero (paying $1) or a Lincoln zero (paying $5), and no matter
what the interest rate is. Similarly a ten year zero has average life 10, no matter
whether it is a Washington or Lincoln or Hamilton zero (paying $10), and no matter
what the interest rate is.
Consider a ten year 9% coupon bond at 8% interest. You should be able to guess
that its average life is around 7. Clearly the average life is above 5, because payments
are spread over ten years (which suggests 5), but the biggest payment is at the end,
which suggests more than 5. On the other hand the biggest payment is discounted
the most, so the average must be much less than 10. In fact, it can be computed
exactly as
AvgLife (8%) = 7 . 097508
(See average life spread sheet). If the interest rate were 10%, the average life would
be a little di ff erent. Is your intuition good enough to guess what it would be? Clearly