Owning every constituent at its index weight is called full replication, and where it is feasible it is the right answer. Tracking error is zero by construction, the portfolio needs no model, and a cap-weighted index requires no trading when prices move. It stops being feasible when the index has thousands of names, when some of them are too thin to buy in size, when a foreign-ownership limit blocks a holding, or when the fund is small enough that a 0.03% position is a handful of shares.
How do stratified sampling and optimisation work?
Stratified sampling divides the universe into cells, usually sector by size or country by sector, matches each cell's aggregate weight exactly, and holds a few representative names within it. The sector bet is controlled by construction, which is the method's whole appeal, and everything else is left to chance. Whether that is a good trade depends on what drives the index's risk. Where sector membership explains most of the covariance, stratification works well. Where it does not, forcing a small cell's weight onto two or three names concentrates idiosyncratic risk badly, and Worked example 6 shows a stratified sample tracking worse than a naive top-30 holding on the same universe, at 190.74 basis points against 139.29.
Optimisation minimises forecast tracking error directly, subject to a name budget, position bounds and whatever else the mandate requires. It uses a risk model, and that is both its strength and its exposure. The strength is that it will happily hold a name for no reason other than that the name hedges a factor exposure the sample would otherwise carry. The exposure is that it is only as good as the covariance matrix behind it, and a risk model that understates a factor will produce a portfolio that looks precise and is not. On the same universe and the same 30-name budget, optimisation reaches 120.37 basis points of tracking error, against 139.29 for the naive sample and 190.74 for the stratified one.
Advertisement
How do the methods compare, and what does a sample leave out?
The comparison is not settled by tracking error alone, which is why the table in Worked example 6 carries four columns. Optimisation wins on tracking error and loses on turnover: at the stated reconstitution it trades 4.19% of net assets one way against 1.26% for the naive sample, because the optimiser's weights respond to every change in the risk model, not just to changes in membership. Full replication has the highest build cost, at 5.16 basis points against 4.00, because it must buy the thin tail the samples skip. There is no dominant method. There is a mandate, a fund size and a universe, and those three pick the method.
One number is worth carrying away from the sampled case because it is an identity rather than a result. When a tracker holds the top k names of a cap-weighted index at pro-rata weights, every held name is overweight and every excluded name is underweight by exactly its benchmark weight, so the active share equals the total weight of the excluded tail. Holding the top 30 of the 50-name index leaves out 11.619194% of the index, and the active share is 11.619194%. The tail's weight is the sample's risk, stated in one figure, before any covariance matrix is opened.