Stratified sampling for a bond index

Stratified sampling divides the index into cells along the dimensions that drive returns, then holds enough securities in each cell to reproduce the cell's weight and its risk characteristics. The standard cell dimensions for a bond index are duration or maturity bucket, sector, and credit quality, sometimes crossed with issuer or with an optionality flag for callable and mortgage exposure.

The construction has three steps. Reproduce each cell's weight, so the portfolio's sector and quality mix matches. Reproduce each cell's duration and spread duration, so the portfolio's sensitivity to rates and to spreads matches inside the cell as well as in aggregate. Then choose the individual bonds inside the cell for liquidity and for issuer diversification, since within a cell the securities are, by the model's own assumption, substitutes.

How is tracking error calculated and reduced?

Tracking error is the standard deviation of the difference between portfolio return and index return, and for a sampled bond portfolio it decomposes into three sources that can be estimated separately and added in quadrature:

TE^2  =  (duration mismatch x sigma_rates)^2
       + sum over cells of (weight mismatch x spread duration x sigma_spread)^2
       + sum over cells of (cell weight x idiosyncratic sigma / sqrt(names held))^2

Worked Example 4 runs that decomposition on a 26-bond sample of a 5,248-issue index and gets 22.819666 basis points a year. The interesting part is where it comes from: 96.05% of the variance is idiosyncratic, from holding eight bonds where the index holds 1,900 in the BBB cell, and only 3.89% comes from the 0.05-year duration mismatch. Almost nothing comes from cell-weight mismatches, because those are the easy thing to fix and the manager fixed them.

The decomposition rests on one assumption that should be stated whenever the number is quoted. Adding the terms in quadrature treats the rate factor, the cell spread factors and the idiosyncratic residuals as mutually independent, and in a stress they are not: spreads widen across every credit cell at once and the correlation with the rate factor stops being zero. A tracking error built this way is therefore a fair estimate of an ordinary year and an understatement of a bad one, and a manager who reports it without that sentence has reported half a number.

That decomposition tells you what to do next. Idiosyncratic risk falls with the square root of the number of names, so doubling the sample from 26 to 52 cuts tracking error from 22.819666 to 16.451574 basis points, a reduction of 6.368092. The trade to get there turns over half the portfolio and costs 5.500000 basis points once, or 1.100000 basis points a year amortised over five years, which is 0.172736 basis points of cost per basis point of tracking error removed. On those numbers the trade is worth doing. On a portfolio one tenth the size, with position sizes below round lots and wider spreads, it is not, and the same arithmetic tells you so.

What is enhanced indexing?

Enhanced indexing sits between full sampling and active management. The manager matches the index's primary risk factors, duration above all, and takes small deliberate positions in the secondary ones: a modest overweight to a sector, a preference for slightly cheaper issues within a cell, a bias toward bonds about to enter the index. The defining constraint is that active duration stays near zero, because duration is the largest single risk in a bond book and taking it is no longer indexing. A typical enhanced mandate runs 25 to 75 basis points of tracking error against 5 to 25 for a pure sampled tracker, and it is judged on information ratio rather than on tracking error alone.