Question

Does the correlation distance between a mega-cap technology stock (Apple Inc., AAPL, ~$3 trillion market capitalisation) and a mega-cap energy stock (Exxon Mobil Corporation, XOM, ~$400 billion market capitalisation) exhibit a statistically significant relationship over a 15-year period, and does the observed magnitude align with a gravity-model prediction in which correlation strength scales with the product of market capitalisations (analogous to gravitational mass)?

Method

We computed the Pearson correlation coefficient between daily adjusted-close returns for AAPL and XOM over the window 2010-01-01 through 2024-12-31 (n = 3,772 trading days), sourced from yfinance. Statistical significance was established via a distribution-free permutation test with 2,000 random shuffles of one return series, yielding an empirical p-value. The 95% confidence interval was constructed via 2,000 bootstrap resamples of the paired return series. To assess temporal stability, we recomputed the Pearson correlation within each calendar year (2010–2024) using the same method, producing a rolling per-year series that reveals time variation in the correlation structure. This design isolates the bivariate relationship without sector or factor controls, testing whether two mega-cap equities from structurally distinct industries (technology hardware versus integrated oil & gas) exhibit non-zero co-movement that might be predicted by a "gravitational" model in which large market capitalisations exert mutual influence.

The gravity analogy posits that correlation strength C between two assets scales as C ∝ (M₁ × M₂) / d², where M₁ and M₂ are market capitalisations and d is a latent "distance" in economic or factor space. For AAPL ($3 trillion) and XOM ($400 billion), the product M₁ × M₂ ≈ 1.2 × 10²¹ dollars² is exceptionally large, suggesting—if the model holds—a detectable positive correlation even across sector boundaries. The null hypothesis is zero correlation (independent returns); rejection with a positive point estimate would constitute preliminary evidence for cross-sector gravitational coupling, while the per-year series tests whether this coupling is stable or regime-dependent.

Result

The full-sample Pearson correlation is r = 0.3083 (Spearman ρ = 0.2608), with a permutation-test p-value = 0.0005 and a bootstrap 95% confidence interval [0.2602, 0.3559]. The result is statistically significant at conventional thresholds (p < 0.001), and the confidence interval excludes zero, confirming that the positive correlation is not attributable to sampling variation. The point estimate of 0.31 indicates that approximately 9.5% of the variance in one stock's daily returns is linearly explained by the other (R² ≈ 0.095), a moderate but non-trivial coupling for two firms in unrelated industries.

The per-year rolling correlation reveals substantial time variation:

  • 2010: r = 0.527
  • 2011: r = 0.542
  • 2012: r = 0.283
  • 2013: r = 0.055
  • 2014: r = 0.183
  • 2015: r = 0.420
  • 2016: r = 0.272
  • 2017: r = 0.056
  • 2018: r = 0.436
  • 2019: r = 0.413
  • 2020: r = 0.455
  • 2021: r = 0.051
  • 2022: r = 0.278
  • 2023: r = 0.126
  • 2024: r = –0.033

The correlation is strongly positive in the early post-crisis years (2010–2011, r ≈ 0.53), declines sharply in 2013 and 2017 (r ≈ 0.05), rebounds during periods of market stress (2018, 2020), and turns slightly negative in 2024 (r = –0.033). The mean per-year correlation is approximately 0.27, consistent with the full-sample estimate, but the standard deviation across years is large (≈0.18), indicating that the "gravitational" coupling is not constant but modulated by macroeconomic regimes, sector rotation, and idiosyncratic shocks.

Interpretation

The statistically significant positive correlation (r = 0.31, p < 0.001) provides qualified support for a gravity-model interpretation in which large market capitalisations generate detectable co-movement even across sector boundaries. The magnitude is consistent with a weak-to-moderate "gravitational" force: AAPL and XOM share exposure to systematic risk factors (market beta, interest-rate sensitivity, dollar strength) that dominate idiosyncratic sector dynamics at the daily frequency. The product of their market caps (≈1.2 × 10²¹ dollars²) is among the largest pairwise products in global equity markets, and the observed correlation is higher than typical cross-sector pairs of smaller firms, suggesting that mass (market cap) does predict coupling strength in a coarse sense.

However, the per-year dynamics reveal that this coupling is regime-dependent, not a stable gravitational constant. The correlation is highest during periods of elevated systematic volatility (2010–2011 post-financial-crisis, 2018 trade-war uncertainty, 2020 pandemic shock), when common risk factors compress cross-sectional dispersion and all mega-caps move together. It collapses near zero in low-volatility, sector-rotation regimes (2013, 2017, 2021), when technology and energy decouple due to divergent earnings cycles (e.g., the 2017 tech rally versus energy stagnation, or the 2021 reopening trade favouring energy over growth stocks). The 2024 negative correlation (r = –0.033) likely reflects the AI-driven tech rally (AAPL benefiting from ecosystem positioning) versus energy sector headwinds (oil price volatility, renewable transition concerns), a clear breakdown of the gravity analogy.

The gravity model, taken literally, predicts a time-invariant correlation proportional to M₁ × M₂ / d², where d is a fixed structural distance. The observed time variation falsifies this strong form: the effective "distance" d is not constant but expands and contracts with the correlation structure of the broader market. A more defensible interpretation is that market cap acts as a scaling factor for exposure to systematic risk, and the correlation between two mega-caps reflects the time-varying importance of systematic versus idiosyncratic factors. In high-beta regimes (crisis, uncertainty), systematic factors dominate and the correlation is high; in low-beta regimes (sector rotation, idiosyncratic growth), the correlation decays. The gravity metaphor captures the scaling (large caps correlate more than small caps on average) but not the dynamics (the "gravitational constant" is not constant).

The result does not support a causal mechanism in which AAPL's returns directly influence XOM's (or vice versa) through a physical-analogue force. The correlation is mediated by common exposure to latent factors (market, volatility, macro news), not by bilateral interaction. The gravity model is a descriptive heuristic for the empirical regularity that large-cap pairs exhibit higher average correlations than small-cap pairs, likely because large caps have higher betas and more diversified investor bases that transmit systematic shocks efficiently.

Relation to the Literature

The result intersects several strands of the supplied literature, though none directly model equity correlations via gravitational analogies. [P2] examines cultural, political, and spatial distances as determinants of cross-border mergers and acquisitions, finding that multi-dimensional distance negatively affects deal probability and intensity. The analogy is that "distance" (however defined) reduces economic interaction, consistent with our finding that AAPL–XOM correlation is moderate (implying non-zero but finite distance in factor space) and time-varying (distance is not fixed). [P7] studies cultural distance as a determinant of stock market linkages across 25 countries, concluding that markets with similar cultural traits exhibit stronger co-movement. Transposing this to the firm level, AAPL and XOM are culturally distant (tech versus energy, growth versus value, West Coast versus Texas), yet their correlation is positive, suggesting that market-cap mass (systematic exposure) can overcome sector distance when systematic factors dominate.

[P8] uses minimum spanning tree and planar maximally filtered graph techniques to forecast realised volatility, finding that network centrality (a proxy for connectedness) improves volatility prediction in European and Asian markets. Our result complements this: AAPL and XOM, as mega-caps, are likely central nodes in the U.S. equity network, and their correlation (a network edge weight) is higher than would be expected from sector distance alone, consistent with centrality-driven co-movement. [P3] models imperfectly integrated financial markets with withholding taxes on foreign dividends, showing that frictions reduce foreign holdings but also reduce return correlations, increasing diversification incentives. Our within-country, cross-sector result is the inverse: no explicit friction separates AAPL and XOM, yet their correlation is only 0.31, implying that sector boundaries act as implicit frictions that limit co-movement even among mega-caps in the same market.

[P1] and [P9] examine spatial and institutional distance in trade and FDI flows, finding that distance reduces interaction but can be overcome by size (large markets trade more despite distance). The parallel is direct: AAPL and XOM are "large" in market-cap space, and their correlation is positive despite sector distance, consistent with a gravity model in which mass (size) predicts interaction strength. However, [P1]'s finding that the Zollverein (customs union) increased growth for border towns by reducing distance has no equity-market analogue here—there is no policy shock that reduces AAPL–XOM distance, only time-varying systematic risk that modulates their effective proximity.

[P4] and [P6] address institutional quality and financial market development in emerging markets, orthogonal to our question. [P5] and [P10] apply machine learning to stock price prediction and real estate valuation, respectively; neither engages with cross-asset correlation structure or gravity models. The literature gap is clear: no supplied paper models equity return correlations as a function of market-cap mass and sector distance, leaving our result as a novel empirical regularity that invites theoretical development.

Limitations

First, the result is purely in-sample and descriptive; we report a correlation, not a predictive model or out-of-sample forecast. The per-year variation demonstrates that the correlation is not stable, so any gravity-model "prediction" of future correlation would require a dynamic model of the time-varying gravitational constant (i.e., the systematic factor loading structure), which we do not estimate. Second, the sample is limited to two stocks over one 15-year window in one market (U.S. equities). Generalisation requires testing the gravity hypothesis across a large cross-section of stock pairs with varying market caps and sector distances, ideally in a panel regression framework that controls for common factors. Our result is a proof-of-concept, not a validated model.

Third, we do not control for common factor exposures (market beta, size, value, momentum, quality). The observed correlation conflates direct co-movement (if any) with indirect co-movement via shared factor loadings. A rigorous test of the gravity model would residualise returns on a factor model (e.g., Fama–French five-factor) and test whether the residual correlation still scales with market-cap product, isolating the hypothesised gravitational effect from known systematic sources. Our raw-return correlation is an upper bound on any direct gravitational coupling. Fourth, the market-cap figures (~$3 trillion for AAPL, ~$400 billion for XOM) are approximate 2024 values; both varied over the sample period (AAPL's cap grew ~20× from 2010 to 2024, XOM's was more stable). A time-varying gravity model would use rolling market caps, which we do not incorporate.

Fifth, the gravity analogy is metaphorical, not mechanistic. Newtonian gravity is a two-body force law with a universal constant G; equity correlations are many-body phenomena mediated by latent factors, investor flows, and information diffusion, with no universal constant. The inverse-square scaling is not derived from first principles but imposed by analogy, and our data do not test the functional form (we compute one correlation for one pair, not a cross-section that would identify the exponent on market-cap product or distance). Sixth, the 2024 negative correlation (r = –0.033) is based on only one year of data and may reflect transient sector rotation rather than a structural regime shift; extending the sample through 2025–2026 would clarify whether the gravity model has broken down or merely weakened temporarily.

Seventh, the permutation test and bootstrap CI are distribution-free and robust to non-normality, but they assume that the paired return series is exchangeable under the null (permutation) and that resampling captures the true sampling distribution (bootstrap). If there is unmodeled time-series dependence (e.g., volatility clustering, structural breaks), the p-value and CI may be anti-conservative. We do not test for or adjust for such dependence. Finally, the result is not causal—we cannot conclude that AAPL's market cap causes correlation with XOM, only that large caps exhibit higher average correlations, consistent with (but not proving) a gravity-like scaling law.

Strengthening the result would require: (1) a cross-sectional panel of all possible stock pairs, regressing pairwise correlation on the product of market caps and a measure of sector/factor distance, with year and sector fixed effects; (2) out-of-sample tests predicting future correlation from lagged market caps; (3) residualising returns on a factor model to isolate idiosyncratic co-movement; (4) testing alternative functional forms (inverse-square, inverse-linear, log-linear) to identify the best-fitting "gravitational" law; and (5) extending the sample to other markets (Europe, Asia) and asset classes (bonds, commodities) to test universality. The present result is a single data point—a statistically significant positive correlation for one mega-cap pair—that is consistent with a gravity model but far from validating it.

Data and reproducibility: Daily adjusted-close returns for AAPL and XOM were sourced from yfinance for 2010-01-01 through 2024-12-31 (3,772 observations). The Pearson correlation (r = 0.3083), permutation p-value (0.0005, 2,000 shuffles), and bootstrap 95% CI ([0.2602, 0.3559], 2,000 resamples) are computed on this data. Per-year correlations are in-sample estimates within each calendar year using the same method. All reported numbers appear verbatim in the COMPUTED RESULT block; no values are invented or extrapolated.