4-PrincipCalculus Page 122 Friday, March 12, 2004 12:39 PM
122 The Mathematics of Financial Modeling and Investment ManagementSuch series, called power series, generally converge in some interval,
called interval of convergence, and diverge elsewhere.
The Taylor series expansion is a powerful analytical tool. To appre-
ciate its importance, consider that a function that can be expanded in a
power series is represented by a denumerable set of numbers even if it is
a continuous function. Consider also that the action of any linear oper-
ator on the function f can be represented in terms of its action on pow-
ers of x.
The Maclaurin expansion of the exponential and of trigonometric
functions are given by:2 n
e x = 1 ++ x x------+ ...+ x ------+ R
n
2! n!x^3 x^5 (– 1 )nx^2 n +^1
sin x = x – ------+ ------+ ...+ ------------------------------+ Rn
3! 5! ( 2 n + 1 )!2 4
x x (– 1 )
n
x
2 n
cos x = 1 – ------+ ------+ ...+ -----------------------+ Rn
2! 4! ( 2 n)!Application to Bond Analysis
Let’s illustrate Taylor and Maclaurin power series by computing a sec-
ond-order approximation of the changes in the present value of a bond
due to a parallel shift of the yield curve. This information is important
to portfolio managers and risk managers to control the interest rate risk
exposure of a position in bonds. In bond portfolio management, the first
two terms of the Taylor expansion series are used to approximate the
change in an option-free bond’s value when interest rates change. An
approximation based on the first two terms of the Taylor series is called
a second order approximation, because it considers only first and sec-
ond powers of the variable.
We begin with the bond valuation equation, again assuming a single
discount rate. We first compute dollar duration and convexity, i.e., the
first and second derivatives with respect to x evaluated at x = 0, and we
expand in Maclaurin power series. We obtain()= V 0
1
Vx ()– (Dollar duration)x + ---(Dollar convexity)x
2
+ R 3
2We can write this expression explicitly as: