The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 115 Friday, March 12, 2004 12:39 PM

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Principles of Calculus 115

d x x

-------()e = e

dx

Application of the Chain Rule

The above formulas express dollar duration which is the derivative of

the price of a bond with respect with the interest rate and which

approximates price changes due to small parallel interest rate shifts.

Practitioners, however, are more interested in the percentage change of a

bond price with respect to small parallel changes in interest rates. The

percentage change is the price change divided by the bond value:

dV 1

di V

The percentage price change is approximated by duration, which is the

derivative of a bond’s value with respect to interest rates divided by the

value itself. Recall from the formulas for derivatives that the latter is the

logarithmic derivative of a bond’s price with respect to interest rates:

dV (^1) d(log V)
Duration = -------- ----= ---------------------
diV di
Based on the above formulas, we can write the following formulas
for duration:
Duration for constant interest rates in discrete time:
dV (^11) C 2 C NC ( + M)
-------- ----= – -------------------- ----------------+ ------------------+ ...+ --------------------------
diV V( 1 + i) ( 1 + i) ( 1 + i)^2 ( 1 + i)N
Duration for variable interest rates in discrete time:
dV 1 1 C 2 C NC ( + M)
-------- ----= – -------------------------+ ---------------------+ ...+ -------------------------------
dxV V ( 1 + i
1 )
(^2) ( 1 + i
2 )
(^3) ( 1 + i
N)
N + 1