This paper studies dynamical and cross-sectional structures of bonds, typically used as riskfreeassets in mathematical finance. After reviewing a mathematical theory on commonfactors, also known as principal components, we compute empirical common factors for 10US government bonds (3month, 6month, 1year, 2year, 3year, 5year, 7year, 10year, 20year,and 30year) from the daily data for the period 19932006 (data for earlier period is notcomplete) obtained from the official web site www.treas.gov. We find that the principalcommon 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 findthat the simple AR(1) models produce sample paths that look almost indistinguishable (incharacteristic) from the empirical ones, although the AR(1) models do not seem to pass thenormality based Portmanteau statistical test. Slightly more complicated ARMA(1,1) modelspass the test. To see the independence of the first two common factors, we calculate theempirical copula (the joint distribution of transformed random variables by their marginaldistribution functions) of the first two common-factors. Among many commonly used copulas(Gaussian, Frank, Clayton, FGM, Gumbel), the copula that corresponds to independentrandom variables is found to fit the best to our empirical copula. Loading coefficients (that ofthe 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 governmentbonds from 1993 to 2006 can be simply modelled by two independent common factors which,in turn, can be modelled by ARMA(1,1) processes.
