Question

Do equity indices (SPY, QQQ, IWM), emerging markets (EEM), bonds (AGG, TLT), commodities (GLD, USO) exhibit a common dominant eigenvalue and spectral density structure that would enable cross-asset replication via categorical morphism, or do asset-class boundaries create distinct spectral regimes that preclude such equivalence?

Method

We computed the eigenvalue spectrum of the return correlation matrix for eight liquid ETFs spanning four traditional asset classes over 2010-01-01 through 2024-12-31 (3,772 daily observations). The data source is yfinance daily adjusted-close returns. The universe comprises:

  • Equities: SPY (S&P 500), QQQ (Nasdaq-100), IWM (Russell 2000)
  • Emerging markets: EEM
  • Fixed income: AGG (aggregate bonds), TLT (long Treasuries)
  • Commodities: GLD (gold), USO (oil)

We applied principal component analysis to the 8×8 correlation matrix and compared the resulting eigenvalues against the Marchenko–Pastur (MP) null distribution. Under the MP random-matrix model, eigenvalues arising purely from sampling noise in an uncorrelated system lie within the interval [λ₋, λ₊], where the bounds depend on the ratio q = n_assets / n_obs = 8 / 3772 ≈ 0.002. Eigenvalues exceeding the upper bound λ₊ = 1.0942 are statistically distinguishable from noise and represent genuine common factors. The lower bound is λ₋ = 0.91.

To assess temporal stability, we recomputed the factor count annually on the same data within each calendar year (in-sample per-year splits), yielding a time series of significant factor counts from 2010 through 2024.

Result

Full-period spectrum

The eight eigenvalues in descending order are:

  1. 3.7566
  2. 1.9169
  3. 0.9640
  4. 0.6933
  5. 0.2865
  6. 0.2081
  7. 0.1291
  8. 0.0455

The Marchenko–Pastur upper bound is 1.0942. Exactly two eigenvalues (3.7566 and 1.9169) exceed this threshold, establishing n_significant_factors = 2. The remaining six eigenvalues fall within or below the MP noise band and are statistically indistinguishable from random fluctuation.

Variance decomposition

  • The top eigenvalue (factor 1) accounts for 46.96% of total variance.
  • The two significant factors together explain 70.92% of variance.
  • The residual 29.08% is distributed across six noise-level modes.

Factor structure and loadings

Factor 1 (eigenvalue 3.7566) loads most heavily on:

  • SPY: 0.494
  • IWM: 0.467
  • QQQ: 0.465

This factor captures a common equity-market mode, with near-equal weight on large-cap (SPY), mid/small-cap (IWM), and tech-heavy large-cap (QQQ) U.S. indices. EEM's loading on factor 1 is not reported in the top-three list, indicating it is materially lower than 0.465.

Factor 2 (eigenvalue 1.9169) loads most heavily on:

  • AGG: 0.671
  • TLT: 0.596
  • GLD: 0.406

This factor represents a fixed-income and safe-haven mode, dominated by bonds (AGG, TLT) with a secondary contribution from gold. The absence of equity tickers in factor 2's top loadings confirms a structural separation between the equity regime (factor 1) and the bond/defensive regime (factor 2).

Temporal dynamics

The per-year factor count series is:

Year Significant factors
2010 2
2011 1
2012 1
2013 2
2014 2
2015 2
2016 2
2017 2
2018 2
2019 2
2020 2
2021 2
2022 2
2023 2
2024 2

The two-factor regime is stable across 13 of 15 years. In 2011 and 2012, only one factor exceeded the MP bound, suggesting a temporary collapse of the equity–bond distinction (plausibly during the European sovereign debt crisis and subsequent central-bank intervention, when correlations across asset classes rose). From 2013 onward, the two-factor structure re-emerges and persists without interruption, including through the 2020 pandemic shock and the 2022 inflation/rate regime shift.

Interpretation

What the result supports

  1. Asset-class boundaries are real and persistent. The spectral decomposition reveals two statistically significant, economically interpretable factors that align cleanly with traditional asset-class divisions: an equity factor (SPY, QQQ, IWM) and a bond/defensive factor (AGG, TLT, GLD). The eigenvalue gap between factor 2 (1.9169) and factor 3 (0.9640) is substantial—factor 3 lies firmly within the MP noise band—indicating no third regime of comparable strength.

  2. No universal dominant eigenvalue. The hypothesis of a single common factor enabling categorical equivalence across all eight assets is rejected. Factor 1 accounts for 47% of variance, but factor 2 contributes an additional 24 percentage points that are orthogonal to factor 1. A categorical morphism that treats all assets as equivalent under a single spectral density would discard this second dimension and fail to replicate bond or gold behavior from equity returns alone.

  3. Cross-asset replication is structurally limited. To replicate the full covariance structure, one must account for both factors. A portfolio constructed to hedge equity exposure (factor 1) using only equity instruments cannot synthesize the bond/defensive exposure (factor 2) without introducing instruments that load on factor 2. Conversely, a bond-heavy portfolio cannot replicate equity beta from its own constituents. The two factors are not interchangeable via a simple morphism; they represent distinct risk premia.

  4. Temporal stability of the two-regime structure. The rolling factor count shows that the two-factor model is not an artifact of a single market regime. It holds across the post-crisis recovery (2013–2019), the pandemic (2020–2021), and the inflation/tightening cycle (2022–2024). The brief one-factor episodes in 2011–2012 are the exception, not the rule, and likely reflect a crisis-driven correlation spike rather than a fundamental collapse of asset-class boundaries.

What the result does NOT support

  1. It does not support categorical equivalence in the strong sense. If assets were equivalent under a categorical morphism preserving spectral density, we would observe either (a) a single dominant eigenvalue with all others in the noise band, or (b) eigenvalues of comparable magnitude across all modes, indicating no preferred factorization. We observe neither: two factors dominate, and they partition the universe along asset-class lines.

  2. It does not validate cross-asset hedging via spectral matching alone. A hedge constructed by matching the spectral density of a subset of assets (e.g., replicating SPY's spectrum using QQQ and IWM) will succeed within the equity factor but will not capture the bond/defensive dimension. Any claim that one can hedge a multi-asset portfolio by trading only within one factor's span is contradicted by the two-factor structure.

  3. It does not imply that emerging markets or commodities are redundant. EEM and USO do not appear in the top-three loadings for either factor, suggesting their variance is distributed across both factors and the noise modes. This is consistent with emerging-market equity having both equity beta (factor 1) and idiosyncratic country/currency risk, and with oil having both commodity-specific supply/demand dynamics and a weak defensive component. Neither is fully spanned by the two dominant factors, though their incremental explanatory power is small (the two factors already capture 71% of variance).

Economic interpretation of the factors

Factor 1 is the equity risk premium. The near-equal loadings on SPY, QQQ, and IWM (0.494, 0.465, 0.467) indicate that this factor captures broad U.S. equity market exposure, not a style tilt. The absence of bonds and the low loading on gold confirm that this is a growth/risk-on mode.

Factor 2 is the duration/safe-haven premium. The high loadings on AGG (0.671) and TLT (0.596) reflect interest-rate sensitivity and flight-to-quality dynamics. Gold's moderate loading (0.406) is consistent with its role as a partial inflation hedge and crisis asset, though it is less purely a duration play than long Treasuries. The orthogonality to factor 1 (by construction of PCA) means that bond returns are not mechanically driven by equity returns; they respond to a distinct set of shocks (monetary policy, inflation expectations, risk aversion).

Implications for hedging and replication

A portfolio manager seeking to hedge a multi-asset position faces a two-dimensional problem. Hedging equity exposure requires instruments with high factor-1 loadings (equity index futures, for example). Hedging duration or safe-haven exposure requires instruments with high factor-2 loadings (Treasury futures, gold). A single hedge instrument cannot address both dimensions unless it has non-negligible loadings on both factors—and no such instrument appears in this universe with sufficient weight.

Replication strategies must respect the same constraint. One cannot replicate a 60/40 equity/bond portfolio using only equity ETFs, even with leverage or dynamic rebalancing, because the bond component's factor-2 exposure is orthogonal to the equity factor. Conversely, one cannot replicate equity exposure using only bonds. The two factors are not fungible.

The stability of the two-factor structure over 13 of 15 years suggests that this is not a transient phenomenon but a persistent feature of the cross-asset covariance matrix. Any hedging or replication strategy that assumes a single common factor or categorical equivalence will systematically underperform during periods when the two factors diverge—precisely the scenarios (equity drawdowns with bond rallies, or simultaneous equity and bond losses) where hedging is most valuable.

Limitations

  1. Universe size and composition. The analysis covers eight ETFs, a small subset of the investable universe. The two-factor structure may be an artifact of the specific asset-class representatives chosen. A broader universe including international equities, corporate credit, real estate, and additional commodities might reveal additional factors or alter the loadings. The absence of high-yield bonds, for example, leaves open the question of whether credit risk constitutes a third factor or blends into factors 1 and 2.

  2. In-sample factor count. The rolling factor count is computed in-sample within each calendar year, not out-of-sample. This overstates the stability of the factor structure if the MP threshold is sensitive to the sample period. An out-of-sample test—computing eigenvalues on year t and validating on year t+1—would provide stronger evidence of persistence.

  3. Daily frequency and linear correlation. The analysis uses daily returns and Pearson correlation. Intraday data might reveal higher-frequency factors (e.g., liquidity shocks) that are averaged out at the daily level. Nonlinear dependence (tail correlation, copula structure) is not captured by the correlation matrix; a crisis-driven spike in tail dependence would not necessarily elevate an eigenvalue if it occurs in a small fraction of days.

  4. No economic shock attribution. The factors are statistical constructs. We interpret factor 1 as equity risk and factor 2 as duration/safe-haven risk based on loadings, but we do not formally attribute them to observable economic shocks (GDP growth, inflation surprises, monetary policy). A structural model linking eigenvalues to macro factors would strengthen the economic interpretation.

  5. Assumption of stationarity within years. The per-year factor count treats each calendar year as a stationary block. If the factor structure shifts mid-year (e.g., a regime change in March 2020), the annual estimate averages over distinct regimes and may undercount the true number of factors active at any given time.

  6. No transaction costs or liquidity constraints. The result establishes that cross-asset replication is structurally limited by the two-factor regime, but it does not quantify the cost of implementing a two-factor hedge. In practice, trading costs, bid-ask spreads, and liquidity constraints (especially for TLT and USO during stress) may make even a theoretically feasible hedge prohibitively expensive.

Conclusion

The eigenvalue spectrum of an eight-asset cross-asset universe over 2010–2024 exhibits exactly two statistically significant factors, corresponding to equity risk (SPY, QQQ, IWM) and duration/safe-haven risk (AGG, TLT, GLD). The two-factor structure is stable across 13 of 15 years, including through major market dislocations. This result rejects the hypothesis of categorical equivalence: asset-class boundaries are real, persistent, and orthogonal in the spectral sense. Cross-asset replication or hedging via a single common factor is structurally infeasible; any such strategy will fail to capture the bond/defensive dimension and will underperform precisely when diversification is most valuable. A rigorous hedging framework must account for both factors, requiring instruments from both the equity and fixed-income/defensive regimes.

Research evidence, not investment advice.