Generalising Interest Rate Duration with Directional Derivatives: Direction X and Applications

Conventional (or Fisher-Weil) duration is an ordinary derivative that measures the response of portfolio value to marginal parallel shifts in the term structure, while other proposed measures are generally specific to particular interest rate processes. In this paper, we show that portfolio responses to arbitrary shifts in the term structure may be handled by the use of Frèchet or directional derivatives, presenting a simple algorithm for the directional derivative of a fixed interest portfolio, viewed as a set of cash flows. For a given portfolio, one can locate the most sensitive areas along the term structure by computing a function or profile (“Direction X”) that gives the term structure movement to which the portfolio is most exposed. Immunisation techniques can be based on choosing ancillary assets that ensure that the portfolio directional derivative is zero, or as close to zero as possible; this generalises approaches based on factor models of the term structure. The analysis is applied to both continuous and discrete time.