Question
Does the correlation between a mega-cap technology stock (AAPL, ~$3T market cap) and a mega-cap energy stock (XOM, $400B market cap) exhibit statistically significant distance decay consistent with a gravity model where correlation strength inversely relates to market-cap distance? Specifically, we test whether two stocks separated by substantial market capitalization "distance" ($2.6T) and operating in fundamentally different sectors (technology vs. energy) display the weak positive correlation predicted by a gravity analogy, rather than near-zero or negative correlation.
Method
We computed the Pearson correlation coefficient between daily adjusted-close returns for Apple Inc. (AAPL) and Exxon Mobil Corporation (XOM) over the window 2010-01-01 through 2024-12-31, using yfinance daily data (n = 3,772 trading days). 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-iteration bootstrap resampling of the paired return series. To assess temporal stability, we recomputed the Pearson correlation separately within each calendar year (2010–2024) using the same method, producing a rolling per-year series that reveals time variation in the relationship. This approach is reproducible: the data source (yfinance daily adjusted close), the ticker pair (AAPL, XOM), the window (2010-01-01 to 2024-12-31), and the inference procedure (2,000-permutation p-value, 2,000-bootstrap CI) are fully specified.
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: the null hypothesis of zero correlation is rejected at conventional levels. The positive point estimate indicates that AAPL and XOM returns move together more often than they move apart, despite their sectoral and market-cap differences.
The per-calendar-year Pearson correlations reveal substantial time variation:
- 2010: 0.527
- 2011: 0.542
- 2012: 0.283
- 2013: 0.055
- 2014: 0.183
- 2015: 0.420
- 2016: 0.272
- 2017: 0.056
- 2018: 0.436
- 2019: 0.413
- 2020: 0.455
- 2021: 0.051
- 2022: 0.278
- 2023: 0.126
- 2024: –0.033
The correlation is positive in 14 of 15 years, ranging from a high of 0.542 (2011) to a low of –0.033 (2024). The series exhibits pronounced swings: correlations above 0.4 in 2010–2011, 2015, 2018–2020, and correlations near zero or slightly negative in 2013, 2017, 2021, 2023–2024. The mean per-year correlation is approximately 0.27, consistent with the full-sample estimate, but the standard deviation across years is large (≈0.19), indicating that the relationship is not constant. The 2024 value is the only negative observation, suggesting a recent decoupling.
Interpretation
What the numbers support
The full-sample correlation of 0.31 with a narrow confidence interval [0.26, 0.36] provides strong evidence that AAPL and XOM returns are positively associated over the 15-year window. This positive association is consistent with a gravity-model prediction: two large-cap equities, despite sectoral differences and a $2.6 trillion market-cap gap, exhibit non-negligible co-movement, as if their "mass" (market cap) generates a weak attractive force that prevents complete independence. The correlation is moderate rather than strong, which aligns with the hypothesis that market-cap distance attenuates the relationship—AAPL and XOM are not close neighbors in cap-weighted space, so the correlation is positive but far below the 0.7–0.9 range typical of same-sector mega-caps.
The per-year series reveals that this positive association is not mechanically stable. The correlation was strongest in 2010–2011 (0.53–0.54), a period of post-financial-crisis recovery when broad market factors (monetary policy, risk appetite) likely drove synchronous movements across sectors. It weakened sharply in 2013 and 2017 (0.05–0.06), years of divergent sector performance (technology outperformance vs. energy underperformance), and rebounded in 2018–2020 (0.41–0.46), a span including the COVID-19 shock, which induced high cross-sectoral correlation. The near-zero or negative values in 2021, 2023, and 2024 (0.05, 0.13, –0.03) coincide with periods of sector rotation and energy-price volatility, when technology and energy stocks moved on distinct fundamental drivers.
What the numbers do NOT support
The result does not demonstrate that market-cap distance is the dominant or sole determinant of correlation. The time variation is large: the correlation ranges from –0.03 to 0.54 across years, a 0.57-unit swing. This variability indicates that macroeconomic regimes, sector-specific shocks, and risk-on/risk-off dynamics modulate the relationship far more than a static gravity analogy would predict. A pure gravity model would imply a stable, distance-determined correlation; the data show a time-varying correlation that responds to external factors.
The result does not establish causality or directionality. The correlation is symmetric and does not identify whether AAPL returns lead XOM, vice versa, or both respond to a common factor (e.g., market-wide risk premia, oil-price shocks affecting both tech supply chains and energy revenues). The positive association could reflect shared exposure to aggregate demand, interest-rate changes, or dollar strength, rather than a direct "gravitational" link between the two firms.
The result does not generalize beyond this pair. AAPL and XOM are both mega-caps with global operations and high liquidity; the correlation structure may differ for mid-cap or small-cap pairs, or for pairs with larger or smaller market-cap gaps. The gravity hypothesis requires testing across a range of cap distances and sectors to establish a systematic distance-decay pattern.
The out-of-sample predictive power is not assessed. The per-year correlations are in-sample recomputations within each calendar year; we do not have an out-of-sample R² or a forecast of future correlation. The 2024 negative correlation suggests the relationship may be weakening or reversing, but this is a single year and could be noise.
Dynamics and structure
The rolling per-year series is the core empirical contribution. It shows that the AAPL-XOM correlation is regime-dependent: high during crises and recoveries (2010–2011, 2018–2020), low during sector-divergence periods (2013, 2017, 2021, 2023–2024). This pattern is economically interpretable. During market stress, correlations across assets rise as investors sell broadly (flight to safety, deleveraging); during calm periods with sector-specific growth, correlations fall as fundamentals dominate. The gravity analogy, if valid, operates conditionally: the "gravitational force" (positive correlation) is strong when common factors dominate, weak when idiosyncratic factors dominate.
The 2024 negative correlation (–0.03) is notable. It coincides with a year of divergent sector narratives: technology stocks (AAPL) faced concerns over AI capex and China demand, while energy stocks (XOM) benefited from geopolitical oil-price support. This decoupling suggests that the gravity model's predictive power is limited in periods of strong sector rotation. The negative value is within sampling error of zero (the 95% CI for the full sample is [0.26, 0.36], but that is a 15-year aggregate; a single-year estimate has much wider uncertainty), but it signals that the positive association is not a law—it is a central tendency that can reverse.
Relation to the literature
The gravity model originates in spatial economics and international trade, where bilateral flows (trade, FDI, migration) are modeled as proportional to the product of two "masses" (GDP, population) and inversely proportional to distance (geographic, cultural, institutional). [P1] applies a gravity model to higher-education demand in the Netherlands, finding that distance decay governs university choice: students prefer nearby institutions, controlling for quality. [P2] extends the gravity framework to foreign direct investment, showing that bilateral FDI positions are well-predicted by GDP products and geographic distance, with cultural and institutional distance as additional frictions. [P8] uses the Zollverein customs union as a natural experiment, demonstrating that removing border frictions (reducing "distance") increased urban growth in towns near the liberalized border, consistent with gravity predictions.
Our result—positive correlation between two large-cap stocks separated by market-cap and sector distance—echoes the gravity model's core insight: entities with large "mass" (here, market cap) exhibit non-negligible interaction even when separated by distance (here, sectoral and cap differences). The moderate correlation (0.31) is consistent with distance attenuation: AAPL and XOM are not close in cap-weighted or sector space, so the correlation is positive but weak. The time variation, however, introduces a dynamic element absent from static gravity models. [P5] analyzes global financial markets as a network with time-varying community structure, finding that correlations increase during crises and decrease during calm periods, with different behavior in equity vs. currency markets. Our per-year series mirrors this: the AAPL-XOM correlation is high in crisis years (2010–2011, 2018–2020) and low in divergence years (2013, 2017, 2021, 2023–2024), suggesting that the "gravitational force" is modulated by macroeconomic regime.
[P9] studies cross-correlations between individual stock risk and market risk, finding asymmetric lead-lag relationships and regime-dependent correlation during crises. Our result complements this: we examine cross-stock correlation rather than stock-market correlation, but the regime-dependence is similar. The gravity analogy, if interpreted as a baseline model, must be augmented with regime indicators (crisis vs. calm, sector rotation vs. broad rally) to capture the observed dynamics.
[P3] applies a modified gravity model to carbon-emission intensity in China's construction industry, using spatial correlation networks to identify core-periphery structure. The "core-edge" finding—economically developed provinces act as hubs—suggests that in a market-cap gravity model, mega-caps like AAPL might act as hubs, with correlations to other stocks (including XOM) reflecting their centrality. Our result is consistent with this: AAPL, as the largest-cap stock, may exert a "gravitational pull" on XOM's returns via shared market-factor exposure, even though the two are sectorally distant.
[P4] examines dynamic conditional correlations between green bonds, renewable energy, and crypto markets, finding no short-run volatility linkage but time-varying return correlations. [P6] studies institutional and cultural determinants of financial market development in emerging markets, emphasizing that correlation structures depend on institutional quality and cultural distance. These papers underscore that correlation is not a fixed parameter but a function of underlying structural and regime variables. Our per-year series is direct evidence of this: the AAPL-XOM correlation is not a constant 0.31 but a time-varying quantity that responds to macroeconomic and sector-specific shocks.
The literature does not provide a direct precedent for testing gravity models in equity return space using market cap as mass and correlation as force. The gravity framework is typically applied to flows (trade, FDI, migration) or network formation (citations, collaborations), not to return correlations. Our result is thus an exploratory extension: we find that the gravity analogy has some descriptive power (positive correlation between large-cap stocks), but the time variation and regime-dependence indicate that a static gravity model is insufficient. A dynamic gravity model, with time-varying distance or mass parameters, would be needed to capture the observed swings.
Limitations
Sample and universe
The result is based on a single pair of stocks (AAPL, XOM) over a single 15-year window (2010–2024). Generalization requires testing across multiple pairs with varying market-cap distances (e.g., $3T vs. $400B, $3T vs. $50B, $400B vs. $50B) and sector combinations (tech-energy, tech-finance, energy-utilities). The gravity hypothesis predicts a systematic distance-decay pattern: correlation should decrease as market-cap distance increases, controlling for sector. Our result is consistent with this (moderate positive correlation for a large-cap pair with substantial distance), but it is a single data point. A cross-sectional regression of pairwise correlations on market-cap distance, using all S&P 500 pairs, would provide a stronger test.
The choice of AAPL and XOM is not random—both are mega-caps with global operations and high liquidity—but it is not a representative sample. Mid-cap or small-cap pairs may exhibit different correlation structures due to lower liquidity, higher idiosyncratic volatility, or different factor exposures. The gravity model may apply only to large-cap stocks, where market-wide factors dominate idiosyncratic noise.
Window and frequency
The 2010–2024 window spans multiple macroeconomic regimes (post-crisis recovery, taper tantrum, COVID-19, inflation surge, rate hikes), which is a strength for assessing time variation but a limitation for isolating the gravity effect. The per-year correlations show that the relationship is regime-dependent; a longer window (e.g., 1990–2024) would include additional regimes (dot-com bubble, financial crisis) and test whether the positive association is a long-run feature or a post-2010 artifact.
Daily frequency is standard for return correlations, but it may miss intraday dynamics or longer-horizon co-movement. High-frequency (minute-level) data could reveal lead-lag relationships or liquidity-driven correlation; monthly or quarterly data could reveal business-cycle co-movement. The gravity model does not specify a frequency, so the choice is pragmatic, but the result may be frequency-dependent.
Assumptions and inference
The permutation test and bootstrap CI are distribution-free and robust to non-normality, but they assume that the paired returns are exchangeable under the null (permutation) and that the bootstrap sample approximates the population distribution. These are standard assumptions, but they may fail if the return distribution has heavy tails, time-varying volatility, or structural breaks. The per-year correlations suggest some non-stationarity (the 2024 value is negative), which could bias the full-sample estimate if the relationship is trending.
The correlation is a linear measure and may miss nonlinear dependence (e.g., tail dependence, asymmetric response to shocks). Copula-based measures or quantile correlations could reveal richer structure. The gravity model, as typically formulated, is linear in logs (log flow ~ log mass – log distance), so Pearson correlation is a natural analog, but the analogy is imperfect.
Causality and mechanism
The result establishes association, not causation. The positive correlation could arise from shared exposure to a common factor (e.g., market risk premium, oil prices affecting both tech supply chains and energy revenues, dollar strength affecting both multinational revenues) rather than a direct "gravitational" link. A factor model (e.g., Fama-French five-factor) could decompose the correlation into common-factor and residual components; if the residual correlation is near zero, the gravity analogy is spurious (the correlation is entirely due to shared factor loadings). If the residual correlation is positive, it suggests a direct link, but the mechanism remains unclear.
The market-cap "mass" is not a physical quantity, and the analogy to gravitational force is metaphorical. Market cap reflects investor expectations of future cash flows, not a fundamental property like mass. The correlation could reflect information spillovers (AAPL earnings news affecting market-wide risk appetite, which affects XOM), liquidity linkages (both stocks are in the S&P 500 and subject to index flows), or behavioral factors (momentum, herding). The gravity model provides a descriptive framework but not a causal mechanism.
Strengthening the result
A stronger test would:
Cross-sectional validation: Compute pairwise correlations for all S&P 500 stocks, regress correlation on market-cap distance (log |cap_i – cap_j|) and sector distance (dummy for same sector), and test whether the coefficient on cap distance is negative and significant. This would establish whether distance decay is a systematic pattern.
Out-of-sample prediction: Use the first half of the sample (2010–2017) to estimate the correlation, predict the second half (2018–2024), and compute out-of-sample R². This would test whether the gravity model has predictive power.
Factor decomposition: Estimate a factor model (e.g., market, size, value, momentum, sector factors), compute residual returns, and test whether the residual correlation is positive. This would isolate the "gravitational" component from common-factor exposure.
Alternative distance metrics: Test whether sector distance (number of GICS levels separating the two stocks), geographic distance (headquarters location), or supply-chain distance (input-output linkage) better predicts correlation than market-cap distance. This would clarify what "distance" means in equity space.
Longer window: Extend the sample to 1990–2024 or earlier, if data are available, to test whether the positive association is a long-run feature or a post-2010 artifact.
Nonlinear and tail dependence: Compute copula-based measures (e.g., tail dependence coefficients) to test whether the gravity model applies to extreme co-movements (crashes, rallies) or only to average co-movements.
The current result is a proof-of-concept: it demonstrates that two large-cap stocks separated by substantial market-cap and sector distance exhibit a statistically significant positive correlation, consistent with a gravity analogy. The time variation and regime-dependence indicate that the analogy is incomplete—correlation is not a fixed function of distance but a dynamic quantity that responds to macroeconomic and sector-specific shocks. A full validation of the gravity model in equity markets requires cross-sectional and out-of-sample tests, factor decomposition, and alternative distance metrics. The present result is a necessary first step, establishing that the phenomenon exists and is quantifiable, but it is not sufficient to validate the gravity model as a general framework for equity correlations.
Research evidence, not investment advice: This is a research finding quantifying the historical correlation between two equity return series. It is not a trading signal, a recommendation to buy or sell AAPL or XOM, or a prediction of future returns or correlations. The result describes past co-movement and does not imply that the relationship will persist or that it can be exploited for profit. Investors should consult financial advisors and conduct independent due diligence before making investment decisions.