Taylor Series Foundation of Sensitivity Based Risk Measures
Overview
The
Taylor Series Approximation
is used to construct a number of risk measures in finance. In fixed income, the primary
such measures are
Function Approximation
The classical framework to fixed income risk approaches the problem
through a
function approximation lense.
That is, the price of fixed income instrument (referred to as a bond here),
is viewed as a function of a set of rate variables. The simplest approach
views the bond price as a function of bond yield, which may be a function
of time
{% Price = P(y(t)) %}
here {% y(t) %} is the bonds yield at time t.
Price as a function of yield is a smooth differentiable function, and
can therefore be approximated by a
Taylor Series
. Specifically,
{% P(y_1) - P(y_0) = (y_1 - y_0) \times \frac{dP}{dy} + \frac{1}{2} (y_1 - y_0)^2 \frac{d^2P}{dy^2} + ... %}
This shows the change in the price of the bond, given that its yield goes from
{% y_0 %} to {% y_1 %}. Notice that we retain only the first two
terms and suppress dependence of y on t.
Duration and Convexity
Duration and convexity are just names given to terms in the
Taylor series approximation above. The first term in the approximation
is
{% \Delta y \times \frac{dP}{dy} %}
where {% \Delta y %} refers to the change in the value of
y, or {% y_1 - y_0 %}. Duration {% D %} is usually defined as
{% \Delta P / P \approx -D \times \Delta y %}
then
{% D = - \frac{dP}{dy} / P %}
This shows that duration is the negative of the first term of
the Taylor series, divided by the bond price. The negative
is included because the derivative of the price with respect to
yield is negative, so this converts duration to be and positive number
and it is further divided by the bond price in order to convert duration
to be percentage based.
Convexity is treated similarly.
{% C = \frac{d^2 P}{\partial y ^2}/P %}
