On Term Structure of Yield Rates. 7. Another Temporal Variable for Maturities

In paper it is offered to consider a time variable that describes term to maturity of zero-coupon bonds as result of nonlinear transformation of the temporary terms that are independent on parameters of interest rate dynamics model, allowing to map the time numerical axis into an interval of unit length. This way has advantages before application as a measure of time of a duration of a short-term interest rate because at the duration application the time variable depends on parameters of considered models that complicates a comparison of yields for the same real terms to maturity. It is shown that resulting yield functions possess practically the same properties as a yield to maturity curve and a forward curve, except for (in certain cases) properties connected with the second derivative. At the same time they it is more convenient because allow to analyze visually the yields on all time axis. Use of such approach is illustrated in the analysis of properties of the yield curve and the forward curve for one-factor model of Duffie – Kan, Fong – Vasicek two-factor model and three-factor models of interest rates: Fong – Vasicek expanded model, Chen model and the BDFS model. In paper the mathematical models of dynamics of the state variables for all these cases (six various models) are formulated, the equations for functions of term structure are deduced and (when it is possible) their analytical solutions are found. As the main part of the equations can be solved only by calculations, numerical calculations for all six models are carried out and comparisons of yield functions characterizing their term structure are carried out. Calculations were carried out for a set of the parameters based on estimates, published by D. Ahn and B. Gao, fitting one-factor Duffie – Kan model for the description of dynamics of process of an annualized one-month U.S. Treasury bill rate for the supervision period from January, 1960 to February, 1991. Calculations showed that the increase of model dimension implies the decrease of yield rate