A note on the exact replication of a stock index with a multiplier rounding method

For full replication of a stock index one has to multiply the weights of the index with the desired index multiple in order to receive the corresponding share volumes. If rounding of the theoretical values is done the usual way, the desired replication result is missed to an extent greater than necessary. This paper presents an easy-to-implement algorithm based on a multiplicator rounding method which is suited to cope with the problem. As an application it is shown how this algorithm can be utilized to solve the rounding problem arising from full replication of the German Stock Index. Trading restrictions of the German Xetra system are also considered. The algorithm often cuts down the replication error and thus also the tracking error to a small fraction of the value achieved by ordinary rounding.