  Sequential updating (SU) decompositions are a well-known technique for
creating profit and loss (P&L) attributions, e.g., for a bond portfolio, by
dividing the time horizon into subintervals and sequentially editing each risk
factor in each subinterval, e.g., FX, IR or CS. We show that SU decompositions
converge for increasing number of subintervals if the P&L attribution can be
expressed by a smooth function of the risk factors. We further consider average
SU decompositions, which are invariant with respect to the order or labeling of
the risk factors. The averaging is numerically expensive, and we discuss
several ways in which the computational complexity of average SU decompositions
can be significantly reduced.
