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 zero-coupon 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
coupon-bearing bond
is signifi-
cantly shorter than
it is for the
zero-coupon bond.
The first cash flow from
the cou-
pon-bearing bond comes after
one year,
the second after two
years,
and so on. On average, the
bondholder receives
the cash
flow in five
and one-half years
(l-l-2-^•3 + . .
. + 10)/10).
In the
case
of the zero-coupon 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.
It makes sense to weight the time