The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 124 Friday, March 12, 2004 12:39 PM

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124 The Mathematics of Financial Modeling and Investment Management

The first term in the square brackets on the right-hand side of the equa-

tion is the first approximation and is the approximation based on the

duration of the bond. The second term in the square brackets on the

right-hand side is the second derivative, the convexity measure, multi-

plied by one half. The third term is the residual. Its size is responsible

for the quality of the approximation.

The residual is proportional to the third power of the interest rate

shift x. The term in the square bracket of the residual is a rather com-

plex function of C,M,N, and i. A rough approximation of this term is

N(N + 1)(N + 2). In fact, in the case of zero-coupon bonds, i.e., C = 0,

the residual can be written as

1

32 ⋅

• x^3 32 C
  ( 1 ++i ξ)
  4
  
  
  • ... NN ( +^1 )( N +^2 )( CM+ )
    ( 1 ++i ξ)
    N + 3
  
  
  
  


• 
  
  
  
  • 
    
    
    
    • 
      
      
      
      • 
        
      
      
      
      
    
    
    
    
  
  
  
  

1

C

( 1 +i)

1

C

( 1 +i)

2

... CM+

( 1 +i)

N

+ + + -

 (

(^1) NN +^1 )(N +^2 )M
 1
R 3 = –------------x^3

---------------------------------------------------------------------------
32 × ( 1 ++i ξ)N +^3 
M
( 1 +i)
N
( 1 +i)
N
=NN ( + 1 )(N + 2 )------------------------------------
( 1 ++i ξ)
N + 3
which is a third order polynomial in N.
Therefore, the error of the second order approximation is of the
order 1/(3 ×2)^3. For instance, if x = 0.01 and N = 20 years, the
approximation error is of the order 0.001. The following numerical
example will clarify these derivations.
In Chapter 2 we discussed the features of bonds. In our illustration
to demonstrate how to use the Taylor series, we will use an option-free
bond with a coupon rate of 9% that pays interest semiannually and has
20 years to maturity. Suppose that the initial yield is 6%. In terms of