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Zero-coupon yields estimated by zero-degree splines

Simerský, Mojmír (2018): Zero-coupon yields estimated by zero-degree splines.

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Abstract

The paper addresses the problem of zero-coupon yield curve (YC) estimation from a portfolio of coupon-bearing instruments, primarily coupon bonds. A fast and stable iterative procedure is proposed and implemented. The optimization problem is formulated in a matrix form, the principal cashflow matrix having the dimension given by the number of instruments (bonds) and the number of knots on the time axis. The number of instruments is arbitrary, as well as the number of time knots. In our concept of “equivalent cashflows”, each future cashflow at a time t is replaced by two cashflows, one at the left, the other at the right knot respective to the time t. We solve then a simplified problem of estimating the YC from a portfolio of instruments whose future cashflows occur only at predefined times. The method allows for further additional constraints, e.g., an ultimate forward rate fixing or predefined discount factor at some time. We touch also on the asymptotic case of a very dense partitioning of the time axis. The optimization is carried out in the space of forward rate functions of the simplest form – zero-degree splines, i.e., piecewise constant functions. Our approach is thus a generalization of the bootstrap method with no requirements on the bond maturity ladder and, at the same time, with optional smoothing. This work relates to our previous article ([Sim1]), where the idea of equivalent cashflows was introduced. Czech bond yields estimated by the proposed method can be found in ([Sim2]).

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