Portfolio Duration - Principal Components
Overview
One of the key weaknesses of duration is that it shows the sensitivity of an instrument or portfolio to
parallel shifts in yield, however, the curve in general does not make only parallel shifts. The curve
can twist and bend. The
key rate duration
methodology is designed to help overcome this challenge, however, it has it's own challenges.
One way to understand the types of shifts that the yield curve can undergo is to view changes in the
yield curve within the framework of
Principal Components.
(see
Nawalkha chap 11)
Implementation
First, a set of key rates are enumerated.
{% rates = \begin{bmatrix}
y_1, &
y_2, & ... &
y_n &
\end{bmatrix}
%}
Next, the history of each rate change is arranged in a matrix, where
{% \Delta y_{jk} %}
is the change in the kth key rate at time j.
{%
\begin{bmatrix}
\Delta y_{11} & \Delta y_{12} & ... & \Delta y_{1n} \\
...\\
\Delta y_{m1} & \Delta y_{m2} & ... & \Delta y_{mn} \\
\end{bmatrix}
%}
This matrix shows n key rates over m periods.
Next the principal component algorithm is applied to this matrix. That is, a covariance matrix is computed, and the
eigenvalues/vectors
are computed. Each eigenvector represents a principal component.
{% component_i =
\begin{bmatrix}
\Delta y_1, &
\Delta y_2, & ... &
\Delta y_n &
\end{bmatrix}
%}
The duration of the ith principal component should show the approximate change in the value of the
instrument or portfolio if rates move in the direction of the given component. That is,
we defined the {% D_i %} to be
{% D_i \approx -1 \times [PV(rates + \epsilon \times component_i) - PV(rates) ]/ \epsilon %}
where we then take the {% limit_{ \epsilon \rightarrow \infty } %}
