Sampling

Copying a 50-stock index is easy. Copying a 500-stock index is not.

Two things get in the way. Some of the smaller names barely trade. And buying 500 lines of stock, then reinvesting every dividend, costs a lot in brokerage.

Sampling is the way around it. The manager holds most of the index, but swaps the awkward names for stocks that behave like them. The stand-in comes from the same sector or industry, or has a similar size or a similar risk profile.

The result is a portfolio that is not the index. It is a close cousin of the index. Its risk and return look almost the same, which is what the client is paying for.

The other technique, full replication, holds every single constituent.

When full replication breaks down (section 18.1.4.2). Take an index of the Nifty500 type. Two problems appear: some constituent stocks have low liquidity, and transaction costs are higher — both because of the sheer number of stocks and because every dividend has to be reinvested. Exchange traded funds have reduced both problems for portfolio managers, but when inflows or outflows are very large an ETF itself runs into the same two difficulties while replicating its benchmark.

What the manager actually does. Some index stocks are replaced by similar ones — from the same sector or industry, of similar market capitalisation, or of a similar risk profile. The portfolio then does not hold the same, or all, constituents, but it is very close to the index in mean-variance terms, that is in its return-and-risk profile.

The four problems the workbook says sampling solves:

1. Indices with a large number of stocks, of the Nifty500 type, can be mirrored.
2. An index with many segments, including multi-country and multi-capitalisation ones, can be mirrored.
3. Liquidity and transaction cost problems in illiquid constituents are avoided by picking similar liquid stocks instead.
4. An index can be mirrored even when AUM is below the threshold size that full replication needs.

No number is attached to the technique. Section 18.1.4.2 gives no minimum number of stocks to hold, no maximum permitted substitution and no tracking-error limit. The only quantities in the section are the index names themselves — full replication is advised where the constituent count is limited, as with Nifty 50 or the S&P BSE Sensex, and sampling where it is not, as with Nifty500.

Illustrative figures. Kalpana manages a Rs 5 crore portfolio indexed to a 500-stock benchmark. Full replication would mean 500 trades, and the smallest 150 names between them carry under 4% of the index.

She samples instead, holding 130 stocks.

One constituent, a small cement maker with a 0.12% index weight, trades about Rs 8 lakh a day. Her required holding is 0.12% of Rs 5 crore = Rs 60,000 — small, but getting in and out of it on every rebalance would cost her more in impact than the position is worth.

So she drops it and adds Rs 60,000 to a larger, liquid cement company already in the index, taking that holding above its index weight.

Her portfolio now differs from the index in two lines. In sector terms it does not: cement exposure is unchanged at Rs 60,000 plus the original weight. That is the whole idea — the substitution preserves the index's risk profile while removing the illiquid name.

Across the 150 smallest constituents the saving is real. At Rs 25 per trade on both legs of a quarterly rebalance, dropping 370 lines saves roughly Rs 74,000 a year in brokerage on a Rs 5 crore portfolio — about 0.15% of assets, which on a passive mandate is a meaningful share of the total fee.

Chapter 18, section 18.1.4, gives the two indexing portfolio construction techniques, with 18.1.4.1 on full replication and 18.1.4.2 on sampling.

The examinable points are the pairing itself — full replication and sampling are the two major ways named — and the conditions under which each applies. Full replication is advisable when the number of constituents is limited, each constituent is liquid, and the portfolio has the threshold minimum size. Sampling is what you use when one of those three fails. Expect a question asking which technique suits a broad 500-stock index, or which problems sampling solves.

• Sampling is still passive. The substitute stock is chosen for similarity, not because the manager likes it better. There is no attempt to beat the index.
• Sampling raises tracking error; full replication minimises it. That is the price paid for lower cost and better liquidity.
• Three conditions, not one, decide between the techniques. Number of constituents, liquidity of each constituent, and whether the portfolio has the threshold minimum size. Candidates remember only the first.
• ETFs reduce the problem but do not remove it. The workbook is explicit that an ETF facing very large inflows or outflows meets the same liquidity and cost difficulties.
• This is not the sampling unit of statistics. Same word, unrelated idea — one is index construction, the other is survey design.