The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 113 Friday, March 12, 2004 12:39 PM

Principles of Calculus 113

V = C C CM+

-------+ -------+ ...+ ----------------

1 i 2 i Ni

e e e

Application of the First Derivative

The sensitivity of the bond price V to a change in interest rates is given

by the first derivative of V with respect to the interest rate i. The first

derivative of V with respect to the interest rate i is called dollar dura-

tion. We can compute dollar duration in each case using the derivation

formulas defined thus far. In the discrete-time case we can write

dV i() d  C C CM+ 

-------------- = -----------------------+ ------------------+ ...+ --------------------

di di( 1 + i)^1 ( 1 + i)^2 ( 1 + i)N

d C d CM+

= ----------------------- + ...+ -------------------------

di ( 1 + i)^1 di ( 1 + i)N

d 1  d 1

= C ----- ------------------ + ...+ (CM+ )----- --------------------

di ( 1 + i)^1 di ( 1 + i)N

We can use the quotient rule

d 1 1

------- ---------- = – -------------f ' ()x

dx fx() f^2 () x

to compute the derivatives of the generic summand as follows:

d 1 1    1

----- ----------------- = – --------------------i( 1 + i)i –^1 = –i-------------------------

di ( 1 + i)i ( 1 + i)^2 i ( 1 + i)i +^1

Therefore, the derivative of the bond value V with respect to the interest

rates is

dV – 1  

(

• N
  -------- = –( 1 + i)


• 1
  [C( 1 + i) + 2 C( 1 + i)


• 2

• ...+ NC + M)( 1 + i) ]
  di

Using a similar reasoning, we can slightly generalize this formula,

allowing the interest rates to be different for each period. Call it the

interest rate for period t. The sequence of values is called the yield

curve. We will have more to say about the yield curve in Chapter 20.