Paper: arXiv 2604.27287

Authors: Stephen W. Bianchi, Lisa R. Goldberg

Abstract

Counterintuitively, the S&P 500 Index rose between January 1, 2022, and December 29, 2023, while exchange-traded funds (ETFs) seeking to deliver 2x and 3x daily returns of the index delivered substantially negative returns. Roughly two-thirds of the difference between the returns of the index and the levered ETFs can be attributed to compounding and volatility. The remaining difference is explained by the covariance between the ETFs’ deviations from constant leverage and the index’s return.

Complexity vs Empirical Score

  • Math Complexity: 5.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical derivation for the geometric return decay of levered ETFs using second-order Taylor approximations while justifying the findings with empirical analysis of specific ticker data (VOO, SSO, UPRO) over a defined two-year window.

Research Flowchart

  flowchart TD
    A[<b>Goal:</b> Explain S&P 500 Levered ETF Underperformance] --> B[<b>Data:</b> 2022-2023 Index & Levered ETF Returns]
    B --> C[<b>Analysis:</b> Decompose Return Differences]
    C --> D{<b>Key Drivers Identified</b>}
    D --> E[Compounding & Volatility (2/3)]
    D --> F[Covariance/Path Dependency (1/3)]
    E --> G[<b>Outcome:</b> Quantitative Explanation of 'Volatility Drag']
    F --> G

Cited References (16)

  1. Determinants of levered portfolio performance (2014)
  2. Mean-variance approximations to the geometric mean (2012)
  3. Exchange-traded funds 101 for economists (2018)
  4. Passive in name only: Delegated management and index investing (2019)
  5. Will my risk parity strategy outperform? (2012)
  6. The Implied Futures Financing Rate (2021) — link
  7. Diversification returns and asset contributions (1992)
  8. ProShares UltraPro S&P500 Prospectus (2025)
and 8 more
  1. ProShares Ultra S&P500 Prospectus (2025)
  2. The dynamics of leveraged and inverse exchange-traded funds (2009)
  3. Exchange-traded funds and the new dynamics of investing (2016)
  4. Levered Single-Stock ETF s (2025)
  5. Path-dependence of leveraged ETF returns (2010)
  6. Understanding the risk of leveraged ETFs (2010)
  7. The market impact of leveraged ETF s: A Survey of the Literature (2024)
  8. Multi-day return properties of leveraged index ETFs (2025)

0 of 16 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.