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Journal of Economics and Finance

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01-07-2012

Principal component analysis of yield curve movements

Authors: Joel R. Barber, Mark L. Copper

Published in: Journal of Economics and Finance | Issue 3/2012

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Abstract

An important issue in interest rate modeling is the number and nature of the random factors driving the evolution of the yield curve. This paper uses principal component analysis to examine (1) the inherent dimension of historical yield curve changes indicated by the significance of eigenvalues of the covariance matrix, (2) the practical dimension determined by a variance threshold, (3) the shape of the yield curve change associated with the first principal component, and (4) the persistence of this shape over time. We find that although the first two components explain 93% of the sample variation within a 90% confidence interval, the remaining components make statistically significant contribution to the covariance matrix. Consequently, we can establish a practical limit on the dimension only if we are willing to designate a threshold error variance. Further, our results on the persistence of the shape of the yield curve shift associated with the first component depend upon this threshold. If all components are included, the hypothesis that the shape persists between two sample time periods is rejected. On the other hand, if all but the first six components are eliminated, the hypothesis is not rejected.

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See for example, Litterman and Scheinkman (1991); Barber and Copper (1996); Falkenstein and Hanweck (1997); Geyer and Pichler (1999); Golub and Tillman (2000); Dungey et al. (2000); Lekkos (2001); Soto (2004); Reisman and Zohar (2004); Abad and Benito (2007); and Novosyolov and Satchkov (2008).

By which we are referring to principal (or common) factor anaylsis and maximum likelihood factor analysis.

With principal factor analysis, one starts with an assumed trial diagonal error covariance matrix and then backs out the implied common factor covariance as the difference between the sample and error covariance matrix. The next step involves applying PCA to the implied common factor covariance matrix. Based upon this result, one can determine a second trial error covariance matrix. The process continues until some convergence criterion is satisfied.

One could argue that in mean-variance portfolio optimizaton the diagonal structure is useful because it simplifies the technique. The dimension of the problem is reduced to the number of common factors.

This is especially true when the determinant of the covariance matrix is small. This would be the case if a small number of factors explain most of the movement in the term structure. Indeed, we found that Matlab’s maximum likelihood factor analysis was extremely slow when applied to interest rate data.

In other words

$$ U_{i}^{\prime}U_{j}=\left\{ \begin{array}{ccc} 0 & \text{if} & i\neq j \\ 1 & \text{if} & i=j \end{array} \right. . $$

Details are given in Bliss’s (1997) paper; Mr. Bliss kindly provide updated data.

The sign of the principal component is arbitrary. Depending upon the sign of the coefficient at time t the shift could be up or down.

Note that 0 ≤ L _k ≤ 1 since the l _i are eigenvalues of a positive matrix and the geometric mean of these values is always less than the arithmetic mean.

Described in Muirhead (1982).

Also compare to Soto (2004, Note 16) for Spanish bonds.

See Muirhead (1982) Section 9.6.

Abad P, Benito S (2007) Value at risk in fixed income portfolios: a comparison between empirical models of the term structure. Bus Rev Cambridge 7(2):342

Barber JR, Copper ML (1996) Immunization using principal component analysis. J Portf Manage 23(1):99–105CrossRef

Bliss RR (1997) Testing term structure estimation methods. Adv Futures Options Res 9:197–231

Dungey M, Martin VL, Pagan AR (2000) A multi-variate latent factor decomposition of international bond yield spreads. J Appl Econ 15(6):697–715CrossRef

Falkenstein E, Hanweck JJ (1997) Minimizing basis risk from non-parallel shifts in the yield curve—part ii: principal components. J Fixed Income 7(1):85CrossRef

Geyer ALJ, Pichler S (1999) A state-space approach to estimate and test multifactor Cox–Ingersoll–Ross models of the term structure. J Financ Res 22(1):107–130

Golub BW, Tillman LM (2000) Risk management: approaches for fixed income markets. Wiley, New York

Lekkos I (2001) Factor models and the correlation structure of interest rates: some evidence for usd, gbp, dem and jpy. J Bank Financ 25(8):1427–1445CrossRef

Litterman R, Scheinkman J (1991) Common factors affecting bond returns. J Fixed Income 1(1):54–61CrossRef

McCulloch JH, Kwon H-C (1993) US term structure data, 1947–1991. Working Paper 93-6, Ohio State University

Muirhead RJ (1982) Aspects of multivariate statistical theory. Wiley, New YorkCrossRef

Novosyolov A, Satchkov D (2008) Global term structure modelling using principal component analysis. Journal of Asset Management 9(1):49CrossRef

Reisman H, Zohar G (2004) Short-term predictability of the term structure. J Fixed Income 14(3):7CrossRef

Reitano RR (1996) Non-parallel yield curve shifts and stochastic immunization. J Portf Manage 22(2):71CrossRef

Soto GM (2004) Duration models and IRR management: a question of dimensions? J Bank Financ 28(5):1089CrossRef

Strang G (1980) Linear algebra and its applications. Academic, New York

Title: Principal component analysis of yield curve movements
Authors: Joel R. Barber
Mark L. Copper
Publication date: 01-07-2012
Publisher: Springer US
Published in: Journal of Economics and Finance / Issue 3/2012
Print ISSN: 1055-0925
Electronic ISSN: 1938-9744
DOI: https://doi.org/10.1007/s12197-010-9142-y

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