Permanent and Transitory Factors Affecting the Dynamics of the Term Structure
of Interest Rates
PÉRIGNON - Anderson School, UCLA
Christophe VILLA - ENSAI, CREST-LSM and CREREG-Axe
This paper has now been published under a new title and is no longer
available as a part of our Research Paper Series. The published text can be found with the following
Christophe Pérignon, Christophe Villa, "Sources
of Time Variation in the Covariance Matrix of Interest Rates", in Journal of Business, 2005, vol.
paper proposes a novel methodology, based on the Common Principal Component analysis, allowing one to
estimate the factors driving the term structure of interest rates, in the presence of time-varying covariance
structure. The advantages of this method are first, that, unlike classical principal component analysis,
common factors can be estimated without assuming that the volatility of the factors is constant; and
second, that the factor structure can be decomposed into permanent and transitory common factors. We
conclude that only permanent factors are relevant for modeling the dynamics of interest rates, and that
the common principal component approach appears to be more accurate than the classical principal component
one to estimate the risk factor structure.
This paper proposes a novel methodology, based on the Common
Component analysis, allowing one to estimate the factors driving the term structure
of interest rates, in the presence of time-varying covariance structure. The main advantages of the
CPC frame work can be presented as follows.
First, unlike in the classical
principal component analysis, the covariance matrix is not supposed to be constant over the considered
period, an unrealistic assumption in the case of bond yields. On the other hand, the CPC approach allows
the covariance matrix to change from subperiod to subperiod, while still estimating a single common
factor structure over the whole sample period.
Second, both permanent and transitory,
or subperiod-specific, factors can be estimated. In this paper, a factor is said to be permanent if
it has the same financial meaning, captured by the factor loadings (eigenvectors), over the whole time
Since our methodology is more flexible than the principal component
analysis in the case of time-varying covariance matrix structure, it has the potential to advantageously
replace it in many financial applications. First, it could be relevant in certain aspects of risk management.
For instance, immunization strategies, durations and Value-at-Risk computations, and the reduction in
dimension for scenario simulation can be achieved by decomposing the covariance matrix into principal
components. Second, the interest-rate derivative literature may also benefit from the framework proposed
in this paper. Recently, Driessen, Klaassen and Melenberg (2002) have priced and hedged caps and swaptions,
employing successively the Heath, Jarrow and Morton (1992) and the Libor market models, using principal
component analysis to estimate the volatility functions. In a closely related paper, Fan, Gupta and
Ritchken (2001) investigate the performance of Gaussian, proportional and square-root multi-factor models
also basing their estimation procedure on principal component analysis. Longstaff, Santa-Clara and Schwartz
(2001a) study the relative pricing of European-style caps and swap options, and Longstaff, Santa-Clara
and Schwartz (2001b) quantify the cost of using a mispecified model of the term structure to price American
swap options. In the latter two papers, the authors use a principal component analysis of the historical
covariance matrix to estimate the pricing factors, and make the identification assumption that these
pricing factors generate also the covariance matrix implied by interest rate derivative prices. Our
methodology could be very useful to calibrate all these interest rate derivative pricing models.
main contribution of the present paper is to propose a new methodology allowing one to estimate the
permanent and transitory factors driving the term structure of interest rates. This methodology is based
on the CPC model, which is an extension of the classical principal component analysis in the case of
several groups. In this paper, we associate for the first time the groups to successive time periods.
By initially running a separate principal component analysis on each subperiod, we observe that the
factor loadings remain fairly constant across subperiods whereas the volatility of the factors fluctuate
extensively through time. These results stay valid regardless of the number and the nature (non-overlapping
vs. overlapping and equal size vs. unequal size) of subperiods considered. We also notice that the variance
accounted for by the first factors changes substantially from subperiod to subperiod. We then propose
different analyses allowing one to estimate either only permanent factors, or both permanent and transitory
factors, using successively two, three, four and eight non-overlapping subperiods. We conclude that
the factor structure has not changed appreciably and that permanent factors should be estimated using
the common principal component approach.