The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 126 Friday, March 12, 2004 12:39 PM

126 The Mathematics of Financial Modeling and Investment Management

first term (duration) is +21.32%. Since the percentage price change is

underestimated, the new value of the bond is underestimated. The change

due to the second term of the Taylor series is the same in magnitude and

sign since when –0.02 is squared, it gives a positive value. Thus, the

approximate change is 21.32% + 3.28% = 24.6%. Using the terms of the

Taylor series does a good job of estimating the change in the bond’s value.

We used a relatively large change in interest rates to see how well the

two terms of the Taylor series approximate the percentage change in a

bond’s value. For a small change in interest rates, duration does an effec-

tive job. For example, suppose that the change in interest rates is 10 basis

points. That is, di is 0.001. For an increase in interest rates from 6% to

6.1% the actual change in the bond’s value would be –1.06% ($134.6722

to $133.2472). Using just the first term of the Taylor series, the approxi-

mate change in the bond’s value gives the precise change:

–10.66 ×0.001 = –1.066%

For a decrease in interest rates by 10 basis points, the result would be

1.066%.

What this illustration shows is that for a small change in a variable,

a linear approximation does a good job of estimating the change in the

value of the price function of a bond. A different interpretation, how-

ever, is possible. Note that in general convexity is computed as a num-

ber, which is a function of the term structure of interest rates as follows:

Dollar convexity = [ 2 C( 1 + i 1 )–^3 + 23 ⋅⋅C( 1 + i 2 )–^4 + ...

+ NN ⋅ ( + 1 ) ⋅ (CM+ )( 1 + iN)

• N – 2
  ]

This expression is a nonlinear function of all the yields. It is sensitive to

changes of the curvature of the term structure. In this sense it is a mea-

sure of the convexity of the term structure.

Let’s suppose now that the term structure experiences a change that

can be represented as a parallel shift plus a change in slope and curva-

ture. In general both duration and convexity will change. The previous

Maclaurin expansion, which is valid for parallel shifts of the term struc-

ture, will not hold. However, we can still attempt to represent the

change in a bond’s value as a function of duration and convexity. In par-

ticular, we could represent the changes in a bond’s value as a linear

function of duration and convexity. This idea is exploited in more gen-

eral terms by assuming that the term structure changes are a linear com-

bination of factors.