Worked research note
The diversification illusion: how many issuers do your ETFs really own?
Holdings concentration and historical co-movement answer different questions. Two examples show the difference.
By ArthaPilot Research
Published 2026-10-11
Research question
Compare issuer exposure in a VTI-and-BND example with common return patterns in a separate four-fund example.
Scope: The two calculations use different portfolios and cannot be combined into a causal explanation or composite diversification score.
Two wrappers resolved to 62 effective issuers
The holdings case begins with a sample VTI-and-BND portfolio and the funds' filings dated March 31, 2026. Duplicate issuers are recombined before concentration is measured. The response resolved 99.7550% of absolute exposure, leaving little, but not zero, unclassified exposure.
On that basis, the top ten issuers accounted for 33.3083% and the Herfindahl-Hirschman index was 0.0161268. Its reciprocal, the effective issuer count, was 62.0085. HHI sums squared issuer shares; its reciprocal is the number of equal-weight issuer exposures with the same concentration. The calculation turns a long constituent list into an economically weighted concentration measure.
Exhibit 01
Content-pinned result
The recorded look-through ledger
Issuer concentration and coverage for the VTI-and-BND example. No comparison portfolio is included.
How concentrated was resolved issuer exposure in the recorded VTI-and-BND household?
Construction. Recombine duplicate issuer exposure across both fund filings. Rank resolved issuer weights, sum the top ten, and calculate HHI and its reciprocal effective count. Divide resolved dollars by resolved plus unresolved absolute exposure for coverage.
| Category | Exposure share |
|---|---|
| Top ten issuers | 33.308% |
| Other resolved issuers | 66.692% |
Note: Coverage was 99.7550%; the chart partitions resolved exposure only.
Source and construction: Recorded ArthaPilot Look-Through fixture for VTI and BND filings dated 2026-03-31; response hash e91a0f5a1d0b… .
How this figure was calculated
Inputs
- The committed VTI-and-BND look-through response using filings dated 2026-03-31.
- The request's 180-day maximum filing age, which decides the filings the engine accepts.
- The request asks for the top 25 resolved issuers; the archived response includes the full 6,285-row issuer ledger.
- Resolved issuer weights and the response's absolute-exposure coverage fields.
Calculation
- Recombine duplicate issuer exposure across both fund filings.
- Rank resolved issuer weights, sum the top ten, and calculate HHI and its reciprocal effective count.
- Divide resolved dollars by resolved plus unresolved absolute exposure for coverage.
What this step cannot establish
The result describes one holdings snapshot. Effective issuer count is not a literal security count or a comparison with another portfolio.
Exact values behind the figure
| Series | Resolved issuer exposure | Share (%) |
|---|---|---|
| Exposure share | Top ten issuers | 33.308% |
| Exposure share | Other resolved issuers | 66.692% |
The result is not that the account owns only 62 issuers. Effective issuer count is a concentration equivalent, not a literal security count. Many small positions can coexist with a much smaller number of economically meaningful exposures.
Nor does this one case prove that adding a fund would improve or worsen diversification. That claim would require the same pinned snapshot before and after the portfolio change.
Four funds produced fewer than three effective return dimensions
The return case asks a different question. It aligns 120 monthly total-return observations for GLD, QQQ, SPY, and TLT from January 2015 through December 2024 and applies correlation-basis principal component analysis.
The first component explained 50.5649% of standardized variance. Two components were needed to cross 80%, while the effective component count was 2.9034. This entropy-based count weights all four components by their shares of explained variance. Four ticker symbols therefore did not behave like four equally important uncorrelated components in this sample.
Exhibit 02
Content-pinned result
The recorded return-dimension ledger
Principal component analysis (PCA) describes common movement in this four-fund monthly return panel. It does not measure issuer overlap.
How many historical return dimensions appeared in the recorded four-fund panel?
Construction. Standardize the aligned monthly return matrix through the correlation basis. Decompose the matrix and order components by explained variance. Report each variance share, the 80% threshold count, and the effective component count.
| Category | Explained variance |
|---|---|
| PC1 | 50.565% |
| PC2 | 33.054% |
| PC3 | 14.479% |
| PC4 | 1.902% |
Source and construction: Recorded ArthaPilot PCA fixture, monthly total returns from 2015-01-31 through 2024-12-31; engine 1.0.0; response hash 752b35f848b3… .
How this figure was calculated
Inputs
- Monthly total returns for GLD, QQQ, SPY, and TLT from 2015-01-31 through 2024-12-31.
- A price series requested from 2015-01-02, which the engine resolved to a 2015-01-31 effective start.
- The committed correlation-basis PCA response with 120 aligned observations.
Calculation
- Standardize the aligned monthly return matrix through the correlation basis.
- Decompose the matrix and order components by explained variance.
- Report each variance share, the 80% threshold count, and the effective component count.
What this step cannot establish
PCA describes co-movement in this return panel. It does not identify economic causes, issuer overlap, or future persistence.
Exact values behind the figure
| Series | Principal component | Explained variance (%) |
|---|---|---|
| Explained variance | PC1 | 50.565% |
| Explained variance | PC2 | 33.054% |
| Explained variance | PC3 | 14.479% |
| Explained variance | PC4 | 1.902% |
The panel was neither one-dimensional nor equivalent to four equally weighted uncorrelated return components. Its historical variation sat between those extremes.
PCA is sample-dependent and sign-indifferent. It can reveal common movement, but it cannot name the economic cause or forecast whether the same structure will persist.
The useful conclusion is a sequence, not a score
The two fixtures do not analyze the same holdings. That boundary is important: the issuer result cannot explain the PCA result, and the PCA result cannot be used to infer the issuer makeup of VTI and BND.
What the worked cases do establish is the analytical sequence. Resolve underlying issuers and disclose coverage. Then align return histories and measure common movement. Add a factor model only when the return drivers themselves need an economic label.
Exhibit 03
Method note
What each diversification lens can and cannot say
A scope ledger separates the claims supported by the two committed responses from claims outside this worked note.
How much of the recorded portfolio value did the issuer look-through resolve?
Construction. Plot resolved_dollars and unresolved_dollars as the two observed coverage amounts. Use the coverage ledger to qualify the issuer result before comparing its scope with the separate PCA case.
| Category | Recorded dollars |
|---|---|
| Resolved | $1,109,888 |
| Unresolved | $2,726 |
Note: The separate PCA fixture remains a distinct worked case; no cross-fixture causal inference is plotted.
Source and construction: ArthaPilot interpretation of the two recorded fixtures; no cross-fixture causal inference is made.
How this figure was calculated
Inputs
- Resolved and unresolved dollar amounts from the look-through response's coverage object.
- The separate PCA response, used only to define the boundary between the two worked cases.
Calculation
- Plot resolved_dollars and unresolved_dollars as the two observed coverage amounts.
- Use the coverage ledger to qualify the issuer result before comparing its scope with the separate PCA case.
What this step cannot establish
Dollar resolution measures the issuer dataset's coverage. It does not join the issuer and PCA portfolios or supply the absent factor analysis.
Exact values behind the figure
| Series | Look-through coverage | Portfolio value |
|---|---|---|
| Recorded dollars | Resolved | $1,109,888 |
| Recorded dollars | Unresolved | $2,726 |
A portfolio can be broad by issuer count and still dominated by a small number of return forces. It can also be concentrated by issuer weight while containing differentiated asset-class behavior. The measures are complementary because they are not interchangeable.
This note supports a transparent two-case demonstration. It does not compare any additional portfolio structures.
Method and constraints
Method
The displayed results reconcile to two archived calculations. Their request and result hashes identify the exact evidence reviewed on October 10, 2026.
Verification: Preserve the fixture paths, source commits, engine version where recorded, and request/response SHA-256 hashes with the published note.
Verification: Reconcile every displayed value to the recorded fixture facts without recalculating from a mutable provider response.
Verification: Keep the holdings and PCA inputs labeled separately throughout the report.
Assumptions held fixed
- The holdings example is the committed tutorial response for VTI and BND using filings dated 2026-03-31.
- The PCA example is a separate committed tutorial response for GLD, QQQ, SPY, and TLT using 120 monthly total-return observations from 2015-01-31 through 2024-12-31.
- HHI and effective issuer count use resolved issuer weights from the look-through response. PCA uses the correlation basis.
- No portfolio-structure ranking is inferred from the two separate fixtures.
What this study establishes
Diversification cannot be read from ticker count alone. In the recorded VTI-and-BND case, recombining fund constituents produced a 33.3083% top-ten issuer share and an effective issuer count of 62.0085, with 99.7550% resolved coverage. That is a holdings statement about one sample portfolio on one filing snapshot, not a universal verdict on two-fund portfolios.
A separate recorded return panel for GLD, QQQ, SPY, and TLT needed two principal components to reach 80% of standardized variance and had an effective component count of 2.9034. The two cases cannot be merged into one score. Together they show why a serious diversification review keeps holdings concentration and historical co-movement distinct.
Limitations
- The two calculations use different portfolios and cannot be combined into a causal explanation or composite diversification score.
- Constituent filings are point-in-time snapshots and do not reconstruct historical holdings turnover.
- PCA results depend on the selected assets, common window, return frequency, and correlation basis.
- Neither case is a forecast, portfolio recommendation, or representative sample of investor portfolios.
- The archives contain the recorded requests and results, not the full constituent inputs or monthly return matrix. Verification reconciles their figures and summary formulas; it does not recreate the original provider capture.
Sources and prior research
- U.S. Securities and Exchange Commission, Form N-PORT data sets (2026)
Authoritative description and downloads for registered-fund portfolio holdings disclosures.
- Ian T. Jolliffe and Jorge Cadima, Principal component analysis: a review and recent developments (2016)
Defines variance-ordered, uncorrelated components and the distinction between covariance and correlation PCA. Components need not be statistically independent.
- William F. Sharpe, Asset allocation: management style and performance measurement (1992)
Returns-based style analysis provides context for studying fund exposures. It is a different method from the PCA calculation used here.
- D. E. Shaw Group, The concentration game (2026)
Context on the distinction between holdings weights and contributions to portfolio risk. It does not validate the results of these two examples.
Evidence and reproducibility
diversification-illusion-recorded-demo-fixture
A descriptive worked note containing one VTI-and-BND look-through case and one separate four-fund PCA case, with no cross-case causal inference.
Evidence date: 2026-10-10. Basis and units are stated in each recorded fact.
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Source: web/src/data/research/portfolio/fixtures/lookthrough-vti-bnd.record.json, web/src/data/research/portfolio/fixtures/pca.record.json. Versions: fixture1=69daa3e2f3fd5efe2faa78e6b08748eee1bc250c, fixture2=69daa3e2f3fd5efe2faa78e6b08748eee1bc250c@1.0.0.
- A ranking of portfolio structures or a universal relationship between fund count and diversification
- A claim that the look-through and PCA cases describe the same portfolio
- A factor-regression result
Inspect the same question with your own inputs
The linked tools expose the assumptions, dates, and result tables behind this report.
- Measure common movement Run correlation-basis PCA on a selected ticker set or saved portfolio.
- Name the return drivers Estimate factor exposures, uncertainty, fit, and residual behavior after measuring co-movement.