|Title||Theoretical and empirical analysis of common factors in a term structure model|
This paper studies dynamical and cross-sectional structures of bonds, typically used as risk-free assets in mathematical finance. After reviewing a mathematical theory on common factors, also known as principal components, we compute empirical common factors for 10 US government bonds (3month, 6month, 1year, 2year, 3year, 5year, 7year, 10year, 20year, and 30year) from the daily data for the period 1993-2006 (data for earlier period is not complete) obtained from the official web site www treas.gov. We find that the principal common factor contains 91% of total variance and the first two common-factors contain 99.4% of total variance. Regarding the first three common factors as stochastic processes, we find that the simple AR(1) models produce sample paths that look almost indistinguishable (in characteristic) from the empirical ones, although the AR(1) models do not seem to pass the normality based Portmanteau statistical test. Slightly more complicated ARMA(1,1) models pass the test. To see the independence of the first two common factors, we calculate the empirical copula (the joint distribution of transformed random variables by their marginal distribution functions) of the first two common-factors. Among many commonly used copulas (Gaussian, Frank, Clayton, FGM, Gumbel), the copula that corresponds to independent random variables is found to fit the best to our empirical copula. Loading coefficients (that of the linear combinations of common factors for various individual bonds) are briefly discussed. We conclude from our empirical analysis that yield-to-maturity curves of US government bonds from 1993 to 2006 can be simply modelled by two independent common factors which, in turn, can be modelled by ARMA(1,1) processes.
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