Dynamic Term Structure with Positive Spot Rate and Preferred Habitats as Explanatory State Factors

Abstract

The traditional term structure theories have explained the observed yield curve with essentially two types of factors: (1) the spot rate and (2) the preferred habitats of lenders (buyers) and borrowers (sellers), both can be randomized by the square root process of Feller (1951) as explanatory factors for the dynamic term structure. CIR (1985) have characterized the dynamic term structure by assuming that the spot rate follows a square root process. Dai and Singleton (2000) has extended the CIR (1985) model to a multidimensional affine model with state factors following a correlated square root (CSR) process, which is the only affine model with strictly positive spot rate in the family of affine term structure models of Duffie and Kan (1996). However, this model cannot allow for negative correlations among yields of different maturities. This paper modifies the pure square root model by interpreting a subset of even number square root factors as the preferred habitats of lenders and borrowers, which is technically equivalent to imposing the corresponding coefficients in the affine spot rate function and the affine risk neutral drift function to zero. The arbitrage free term structure is an affine yield function of strictly positive components of the spot rate plus a non affine yield spread function of the preferred habitats. Negative correlations among yields of different maturities are allowed because the preferred habitats of lenders and borrowers affect the yield spread function in opposite ways. The yield spread function is capable of generating complicated yield spreads with multiple pairs of preferred habitats, which clearly supports the preferred habitat theory of Modigliani and Sutch (1966) that lenders and borrowers can move away from their preferred maturity habitats upon adequate yield premiums/discounts. The liquidity preference theory of Hicks (1939) is a special case with one pair of preferred maturity preferences or habitats of a lender and a borrower, which has clear economic and intuitive interpretations. The market segmentation theory of Culbertson (1957) is also a special case, but its effect is only visible near the short end of the yield curve. The rich curvature of the yield spread function implies that the randomized preferred habitats hold significant explanatory power on the dynamic term structure, which can be investigated empirically in future research.