What Practitioners Need
To
Know
.
. . About Duration and Convexity
Mark KHtzman,
Windham
Capital
Management
In 1938, Frederick Macaulay pub
lished his classic book.
Some
The
oretical Problems Suggested
by
the Movements
of
Interest
Rates,
Bond
Yields
and
Stock
Prices
in
the
United States
Since
1865
^
Al
though Macaulay focused prima
rily on the theory of interest rates,
as
an
aside
he
introduced
the
concept
of
duration
as a
more
precise alternative
to
maturity
for
measuring
the
life
of a
bond.
As
with many
of
the important inno
vations
in
finance,
the
investment
community
was
slow
to
appreci
ate Macaulay's discovery
of
dura
tion.
It was not
until
the
1970s
that professionai investors began
to substitute duration
for
maturity
in order
to
measure
a
fixed
in
come portfolio's exposure
to in
terest rate risk.^ Today, duration
and convexity—the extent
to
which duration changes
as
inter
est rates change—are indispens
able tools
for
fixed income inves
tors.
In
this column,
I
review
these important concepts
and
show
how
they
are
applied
to
manage interest rate risk.
Macaulay's Duration
A bond's maturity measures
the
time
to
receipt
of
the final princi
pal repayment and, therefore,
the
length
of
time
the
bondholder
is
exposed
to the
risk that interest
rates will increase
and
devalue
the remaining cash flows.
Al
though
it
is typically the case that,
the longer
a
bond's maturity,
the
more sensitive
its
price
is to
changes
in
interest rates, this
re
lationship does
not
always hold.
Maturity
is an
inadequate
mea
sure
of
the sensitivity
of
a bond's
price
to
changes
in
interest rates,
because
it
ignores
the
effects
of
coupon payments
and
prepay
ment
of
principal.
Consider
two
bonds, both
of
which mature
in 10
years.
Sup
pose
the
first bond
is a
zero
coupon bond that pays $2000
at
maturity, while
the
second bond
pays
a
coupon
of $100
annually
and $1000
at
maturity. Although
both bonds yield
the
same total
cash flow,
the
bondholder must
wait
10
years
to
receive
the
cash
flow from
the
zerocoupon bond,
while
he
receives almost half
the
cash flow from
the
coupon
bearing bond prior
to its
matu
rity. Therefore,
the
average time
to receipt
of the
cash flow
of the
couponbearing bond
is
signifi
cantly shorter than
it is for the
zerocoupon bond.
The first cash flow from
the cou
ponbearing bond comes after
one year,
the
second after
two
years,
and so
on.
On
average,
the
bondholder receives
the
cash
flow in five
and
onehalf years
(ll2^•3 +
.
.. +
10)/10).
In the
case
of the
zerocoupon bond,
the bondholder receives
a
single
cash flow after 10 years.
This computation
of the
average
time
to
receipt
of
cash flows
is an
inadequate measure
of the
effec
tive life
of
a
bond, because
it
fails
to account
for the
relative magni
tudes
of
the cash
flows.
The
prin
cipal repayment
of the
coupon
bearing bond
is 10
times
the
size
of each
of
the coupon payments.
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 Three '10
 YIPPIE
 Bond duration, duration

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