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The Relative Importance of Duration and Convexity in Approximating Bond Price Changes, Study notes of Finance

This document, authored by Robert Brooks and Miles Livingston, explores the significance of duration and convexity in assessing and managing interest rate risk exposure. The authors examine the Taylor Series expansion of a bond's price with respect to changes in yield and discuss how the addition of the second derivative term, or convexity, improves the approximation. They also provide insights into the relative importance of convexity for various bond types and its impact on the ratio of convexity to duration.

What you will learn

  • What is the role of duration and convexity in assessing and managing interest rate risk exposure?
  • What is the relative importance of convexity compared to duration for short-term and long-term bonds?

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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
pf3
pf4
pf5
pf8

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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).

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

  1. 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)

i f y ' > y y"<ty<y 2, SeeDunetzandMahoney, , „ • [5], Grantier [9], and Nawalka and Lacey [22], For an•.• I L 11 ci ,_(For more mfomiation, see Ellis and Gulick [6] or any calculus bcok.). , , _,,,. j>-. ,• i,n:i i i k i, * application of convexity lo equities, see Johnson [ 15. ^ < ^ J

94 FINANCIAL PRACTICE AND EDUCATION -- SPRING/SUMMER, 1992

Exhibit L Price-Yield Relation Based on Duration, Convexity and Actual

2 0 0

1 9 0

1 8 0

170

160

ISO

1 4 0

1 3 0

120

1 1 0

1 0 0

90

80-

70-

56-

4 0 -

\

  • • • •
    *- • • • * *' • • • • • * - • • • • • A

\

\

\

  1. 0 0 0. 0 2 0. 0 4 0. 0 6 0. 0 8 0. 10 0. 12 0. 1 4 0. 1 6 0. IB 0. 2 0 YIELD NOTE; - A c t u a l. — D u r a t i o n , ++ Convexity

rates, the approximation gets closer to the actual price from below as terms are added. The reason is that odd numbered derivatives are negative. When interest rates decrease. A y to an odd power is negative, making the product positive for odd terms. All the even terms are positive. This implies that as terms are added, the total approximation increases. That is, the total approximation ap- proaches the true value from below. For increases in interest rates, (A y)' is positive for all powers of i. Since the odd numbered derivatives are positive and the even ones are negative, adding the odd terms reduces the sum and adding the even terms increases tbe sum. This means that the approximation oscillates around the true value as terms are added to the Taylor Series expansion.

II. Duration and Convexity

Convexity sharpens the approximation of bond price changes since duration is only a first approximation. This paper will ex- amine the relative value of adding the second term, or convexity, for a variety of bond types. The goal is to determine the extent of improvement in accuracy by adding convexity. Assume that yield to maturity changes by A y. Then, the percentage change in price can be written as equation (3).

4, ii should be noted that the addition of higher order terms will always increase the precision of the approximation. Thus, although ascillation occurs when A y > 0, the estimation is increasingly accurate.

AP ( I + y + A y ) "

  • (^1) (3)

( l + y )

As shown in Appendix A. the first derivative (negative of modified duration) can be written as equation (4).

dP -^ri-. "(F-C/y)

- ' f (^) (4)

As shown in Appendix A. the second derivative (convexity) can be written as equation (5).

v= (^) 2P

2C [ 1 -

2Cn (^) n ( n + 1 ) ( F - C/v ) ( l + n ) n + 2 2P

% EINANCIAL PRACTICE AND EDUCATION -- SPRING/SUMMER, 1992

Exhibit 3. A Measure ofthe Relative Importance of Convexity with Comparative Statics

Security Type (Ay<0}

Comparative Statics (dR/dy)

Zero Coupon Bond

Perpetual Bond

- (n-l- 1) A y 2(1+y)

- A > 0

_{n+_

Par Bond

Annuity

A v i E y I D

y l ^.

yB

A[l/y+|-l-A'|>fl

dy

Exhibit 4. EfTect of Coupon and Maturity on the Raiio of Convexity to Duration. 10n the Raii % Initial Yield, I % Decline in Yield

C 10

BROOKS AND LIVINGSTON - RELATIVE IMPACT OF DURATION 97

Exhibit 5. Effect of Coupon and Yield on the Ratio of Convexity to Duration. 1 % Decline in Yield, 10 Year Bonds

t.091(1]

Exhibit 6. Ef»ect of Delta and Maturity on the Ratio of Convexity to Duration. 10% Coupon, 10% Initial Yield

D.45S

D-

DELTA - 0. 0 4 0

BROOKS AND LIVINGSTON -- RELATIVE IMPACT OF DURATION

Appendix A

Using the Geometric Series Theorem, the price of a bond can be expressed as

( l + y )

(A-1)

Thus, the first derivative is

dP dy.^

nC nF y ( l + y )^ , ,^ ,^ , n+^1 (l+y)/1^ ,^ ^ n-t-1 (A-2)

n ( F - C / y ) ri ' ( l + y ) "

(A-3)

and the second derivative is

2C nC (l+y)" y'(l+y) n+

( l + y ) " ' " - n (F-' C/y) (n+1) (1-fy)" (l+y)

2C , , 1

2 ( n + l )

2nC

n(n+l)(F-C/y)

(1+ y)

n+ 2

(A-4)

Appendix B

From equation (8) in the text we have

V

Substituting the derivatives with d = ( I + y) results in

P/d V

2Cd"'*'^-2Cd^

Cd"""

-2nCyd+ n

  • Cd + n (F y^d"-''

(n+1) (F-C/y)y^

  • C/y) y^

By the rule of ratios (and cancelling 2) we have

  • Cd^- nCyd+ ( 1/2) n ( n+1) ( F - C / y ) y'
    • Cd+
(B-1)
(B-2)
(B-3)