Relative Impact of Duration and
Convexity on Bond Price Changes
Robert Brooks and Miles Livingston
Robert Brooks is an Associate Professor at the University of Alabama, Tuscaloosa, AL and
Miles Livingston is a Professor at the University ofFlorida^ Gainesville, FL.
• Duration is a first order approximation of the magnitude of a
percentage change in a
bond"
s
price when interest rates change, and
convexity can be employed to improve the approximation to second
order. Duration and convexity are employed in a wide variety of
applications from immunizing future liabilities to hedging a
mortgage pipeline at a financial institution. Duration and convexity
are important tools to assess and manage interest rate risk exposure.
The percentage change in a bond's price with respect to a change
in interest rates can be expressed via a Taylor series expansion (see
section I below). An extensive literature has examined the first
derivative term in the Taylor expansion, namely modified
Macaulay's duration, as a measure of bond price volatility. Some
researchers have begun to examine the impact of
the
second deriva-
tive term, namely convexity, upon price risk." The purpose of this
paper is to examine the relative importance of duration and con-
vexity in approximating bond price
changes.
Specifically, we iden-
tify when it is particularly important to examine convexity. We find
that the relative importance of convexity rises with a decline in
interest rates.
I. Taylor Series Expansion
For a flat term structure, the price of a bond P, with annual
coupon C. par value F, maturity of n, and yield to maturity y is
equation (I).
where rm(y) is the remainder using the first m terms of the ap-
proximation. Setting m=2, expanding the summation, dividing by
the price (P) to get percentage changes, and omitting the remainder,
the approximation is written as equation (2).
(2)
The negative of the first term (without Ay) Is often called
modified Macautay's duration. Exhibit 1 .shows the relationship
between price and yield to maturity. Duration is a measure related
to the tangent to this curve at a particular level of interest rates.
Duration can be used as a measure of the sensitivity of bond price
to changes in yield to maturity. Geometrically, this means moving
along the tangent. For small changes in yield, moving along the
tangent is fairly close to moving along the curve
itself.
Adding higher derivative terms increases the accuracy of the
approximation, as shown in Exhibit I. For decreases in interest
3.
LelP=P(y), Then the Taylor Series may be expressed as
(1)where P'"*(y)denoteMhen[hderivative. Let Ay =y ' -y.and
A
P = Ply") - P(y),
and rearranging we have
P(y')
=
(Ay)''andAP
=
The change in price for a change in yield may be expressed in ""'
terms of a Taylor Series expansion as follows:
The Taylor Series may also be
expressed
as
I, See BiePAag [1.2]. Bierwag and Kaufman (3]. Boquist el. al. [4], Fabozzi and
Fabozzi |7]. Fisher and Weil [8], Grove 110], Hawawini [11], Hicks
]
12],
Homer and
Liebowiiz[
13],
Hopewell and Kaufman [14], Livingston and Caks[
16],
Livingston
[17].
[18], and [19]. Macaulay [20], Malkiel [21], Redington 123], and Samuelson
124].
P'"'(y)
ify'>y
y"<ty<y
2,
SeeDunetzandMahoney [5], Grantier [9], and Nawalka and Lacey [22], For an ,_ . , , _,,,. j>-. ,• i,n:i i i k i, *
, , „ • •.• I L
11
ci (For more mfomiation, see Ellis and Gulick [6] or any calculus bcok.)
application of convexity lo equities, see Johnson
[ 15
. ^
<
^ J
93
