The Construction of Smoothed Forward Rates

Abstract

The concept that money due at some time in the future is worth less than money received today is fundamental to most financial analyses. Before such analyses may be performed, it is necessary to estimate a discount δ function such that

$${{\text{c}}_{\text{t}}}\,*\,{\delta _{\text{t}}}\, = \,{{\text{c}}_{\text{0}}}\,\forall \,{\text{t}}\, \geqslant \,{\text{0}}$$

((1))

where c_t is cash at time t and c₀ its value at time 0. Associated with the concept of a discount function is that of forward rates, ie. what rate of interest should be applied to c_t over a time period (t’−t) to estimate the worth of the cash at time t’? The forward rate may be easily constructed from the discount function, namely

$${{\text{f}}_{{\text{t,t'}}}}\, = \,\left( {{\delta _{\text{t}}}/{\delta _{{\text{t'}}}}} \right)\, - \,1\,\forall \,{\text{t'}}\, \geqslant \,{\text{t}}$$

((2))