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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.
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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) (^) 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
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 -
\
\
\
\
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.
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 ) "
( l + y )
As shown in Appendix A. the first derivative (negative of modified duration) can be written as equation (4).
- ' 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
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)
_{n+_
Par Bond
Annuity
A v i E y I D
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
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
Appendix A
Using the Geometric Series Theorem, the price of a bond can be expressed as
( l + y )
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 ) "
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
Appendix B
From equation (8) in the text we have
Substituting the derivatives with d = ( I + y) results in
P/d V
2Cd"'*'^-2Cd^
Cd"""
-2nCyd+ n
(n+1) (F-C/y)y^
By the rule of ratios (and cancelling 2) we have