Index replication: full replication, optimized sampling and swaps

Advanced12 min read · updated 12 October 2026 · 22 sources

An index fund can replicate its index in three ways: buy all the securities, hold a well-chosen selection, or obtain the index return from a bank via a swap. The type of index replication determines costs, tracking error, transparency and risks. Here you will see why a fund tracking an index with thousands of securities does not need all of them, how much tracking error the last securities actually save, and what lies behind a swap ETF.

Three methods at a glance

With full replication, the fund holds every index constituent at its index weight. With optimized sampling, it holds only some of the securities, chosen and weighted so that countries, sectors, size and other characteristics match the index. With synthetic replication, it enters into a total return swap: a bank pays it the index return, and it pays the return of its own basket of securities or hands over the cash. The first two together are called physical replication; the article ETF basics gives an overview for choosing an ETF.

Full replication

Full replication is worthwhile when the index is manageable and liquid, for example a few hundred large stocks. The fund then tracks the index exactly, apart from costs, cash and taxes. It gets expensive at the margins: at every index rebalancing, the fund has to trade all the changes, including tiny positions in thin markets, and incoming dividends sit briefly as cash. For indices with thousands of small securities or tens of thousands of bonds, full replication is hardly practicable: many bonds trade rarely, and minimum denominations prevent exact weights.

Optimized sampling: stratify and optimize

Sampling usually begins with stratification: the index is divided into cells by country, sector and size, or for bonds by currency, duration, rating and issuer group. In each cell, the fund holds a few representative securities and weights them so that the cell receives its index weight. A rulebook might, for example, require that no sector and no country deviates from the index by more than ±10 basis points. That is an example, not an industry standard.

The second stage is optimization with a risk model. It looks for the weights with the smallest expected tracking error, subject to constraints such as a maximum number of securities, a minimum position size and turnover. A factor model splits the deviation into a part arising from different factor exposures and a specific part (Grinold and Kahn 2000):

If the fund matches the index's factor exposures (x ≈ 0), only the specific part remains. It answers the central question: how many securities do you need?

How many securities do you need? Diminishing marginal benefit

Because the deviations enter squared, a security's weight counts disproportionately. A large security that you leave out costs a lot of tracking error; hundreds of small securities together cost little. A comparison with an assumed specific risk of 25% a year:

For an entire index, a simplified model shows the diminishing marginal benefit. Assumptions: 1,250 securities whose weights fall with rank as 1/(rank + 1.5)^0.98. The largest security then weighs 5.85% and the ten largest 26.4%. This is similar to the MSCI World with 1,249 securities, 5.81% for the largest and 27.85% for the ten largest (MSCI World Factsheet (CAD) 2026). The fund holds the 250 largest securities (73.9% of the index) in full. It divides the rest into eight size classes of 125 securities each, holds the largest k in each class and scales them up to the weight of the class. Specific risk: 25% for all securities.

From 330 to 650 securities, the tracking error falls to about a third; the next 400 securities up to 1,050 reduce it by only 16.1 basis points. The model is optimistic: it assumes exactly matched factor exposures and independent specific returns. In reality, small securities in the same industry move together, the risk model is estimated, and costs, cash and taxes come on top. For an index like the MSCI World, the model gives no reason to expect a tracking error of a few basis points with only 200 to 300 securities: holding only the 250 largest would leave out 26.1% of the index weight and would not even match the size structure.

What funds actually hold

Sources: providers' factsheets (SPDR ACWI IMI Factsheet (Dist) 2026; SPDR ACWI IMI Factsheet (Acc) 2026; Vanguard VT Factsheet 2026; Vanguard BND Factsheet 2026; Vanguard BNDX Factsheet 2026). The tracking error comes from the factsheet of the accumulating share class (as of summer 2026); returns: fund NAV versus index, average per year. All four funds state sampling as their method.

Two patterns stand out. According to its factsheet, Vanguard's large global equity fund uses sampling and yet holds almost all the securities. Here, sampling probably serves mainly to leave out tiny or hard-to-trade positions. With bonds, the difference is larger: the international bond fund holds less than half of the bonds in its index and lagged the index by 0.12 percentage points a year over ten years, with ongoing charges of 0.07% (Vanguard BNDX Factsheet 2026).

Synthetic replication via swaps

A swap ETF does not buy the index constituents but enters into a total return swap with one or more banks. Ramaswamy (2011) distinguishes two structures (funded and unfunded). With an unfunded swap, the fund buys its own basket of securities, the substitute basket, and exchanges its return for the index return. With a funded swap, it hands the cash to the bank and receives collateral in a segregated account; it is not the owner of this collateral, which can delay its realization in the event of a default. The basket and the collateral can look quite different from the index.

The risk lies with the counterparty (counterparty risk): if the bank defaults, the fund loses the value of the swap to the extent that the basket or collateral does not cover it. For European funds, the EU's UCITS Directive limits counterparty risk from derivatives to 10% of fund assets if the counterparty is a credit institution, and to 5% otherwise (UCITS Directive 2009). Since the amendment by Directive (EU) 2024/2994, which had to be transposed by June 25, 2026, this limit applies explicitly to derivatives that are not cleared through an authorized central counterparty (Directive (EU) 2024/2994 2024). Invesco explains that for its swap ETFs, the value of the swap is reset to zero as soon as it exceeds a threshold or shares are issued or redeemed, and that up to six banks act as counterparties (Mellor 2022). How such a risk can be valued is explained in the article CVA and issuer risk.

The swap costs a fee that fluctuates over time and can even be negative in some markets (Mellor 2022). For funds domiciled outside the US, one tax difference concerns US equities: physical Irish ETFs pay 15% withholding tax on US dividends (Crowley 2023). Under the US rules on dividend equivalents (Section 871(m)), by contrast, a broad, qualified index is treated as a single security that is not an underlying security (US Section 871(m) Rules 2015); swaps on such indices therefore do not trigger withholding tax, and the swap can deliver the gross return (Mellor 2022). Whether the advantage persists depends on legislation and swap costs. When looking inside the fund, bear in mind that the holdings list of a swap ETF shows the basket, not the economic exposure (see look-through).

Pros and cons compared

Advanced: optimization in practice

• Checking the formula: for a cell of n equally weighted securities, TE = σ W √(1/k − 1/n) follows directly from the sum: k securities deviate by W/k − W/n, n − k securities by −W/n, giving Σ Δw² = W² (1/k − 1/n) in total.
• Integer constraints: an upper limit on the number of securities turns the optimization into a combinatorial problem. Methods range from heuristics (Beasley et al. 2003) to mixed-integer optimization (Canakgoz and Beasley 2009).
• Estimation error: the risk model is estimated, and optimizers exploit precisely its errors. Out-of-sample tests are a must; Meyer-Bullerdiek (2022) found for portfolios of ten stocks from Germany's DAX index that simple equal weighting performed best out of sample.
• Tracking error is not efficiency: Roll (1992) showed that portfolios with minimum tracking error are not efficient in the Markowitz sense. For an index fund this is intended, because its goal is the index, not the efficient frontier.
• Bonds: sampling is done by risk contributions rather than by name: duration per maturity bucket, spread duration per sector and rating, share per issuer. Because bonds from the same issuer are closely related, one bond is often replaced by another from the same issuer with a similar maturity.

What the research shows

Frino and Gallagher (2001) document that even S&P 500 index funds cannot track their index at no cost because of market frictions, but that on average they performed better than active funds after costs. For US equity index funds from 2010 to 2020, Dyer and Guest (2022) compare the two physical methods: funds using sampling traded three to four times as much, had 30 to 50% higher costs and fees and returned 50 to 70 basis points less a year, and not only because of costs but also because of poor security selection. The authors' conclusion:

The study is a working paper, and most of the indices examined have fewer than 1,000 securities; for very broad indices, the authors see the point at which sampling can pay off somewhere between 1,000 and 3,000 securities. Jansen and van Dijk (2002) and Montfort et al. (2008) show that optimization can track an index closely with relatively few securities; in a test with the MSCI Europe, the optimized portfolio behaved similarly to the index. A comparison with Statman (1987) shows that tracking error requires more securities than diversifying total risk does: according to that study, 30 to 40 stocks are enough for a well-diversified portfolio of randomly chosen stocks. For a gap of a few basis points to an index with thousands of securities, that is nowhere near enough. On swap ETFs, Ramaswamy (2011) warns that long chains of derivatives obscure risks and that baskets with little relation to the index can lead to sales in periods of high counterparty risk.

Limitations and common mistakes

• Overrating the replication method: for most investors, the index and costs matter more than the question of physical or synthetic; the tracking difference over the years shows what actually reaches you.
• Number of holdings as a quality mark: more securities does not automatically mean less deviation; what matters are the weights and factor exposures of the missing securities.
• Confusing model and measurement: the tracking error from the risk model is an estimate; the factsheet shows measured values, which also include costs, taxes and valuation times (see index providers and valuation).
• Leaving collateral unchecked: with swap ETFs, it pays to look at the reports: which banks, how much collateral, how often is the swap reset?
• Assuming tax rules are fixed: advantages such as those on US dividends depend on laws that can change. This is not tax advice.

How Portfolio Lab calculates

Portfolio Lab does not replicate indices and does not optimize sampling portfolios. Under Securities, you can see the replication method of ETFs and filter by it. In the Look-through, the holdings data table shows for each fund how many positions the provider's list contains, where it comes from and what share it covers. This lets you see whether a fund uses sampling; for swap ETFs, the basket appears. How closely a fund follows its index can be measured in the Backtest with a benchmark, or in Assets with tracking error, beta and R² against a comparison fund. The prices are exchange prices, not NAVs; the tracking error from monthly returns is therefore more meaningful than that from daily returns.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.