Robust Long-Term Growth Rate of Expected Utility for Leveraged ETFs

ArXiv ID: 2310.02084 “View on arXiv”

Authors: Unknown

Abstract

This paper analyzes the robust long-term growth rate of expected utility and expected return from holding a leveraged exchange-traded fund (LETF). When the Markovian model parameters in the reference asset are uncertain, the robust long-term growth rate is derived by analyzing the worst-case parameters among an uncertainty set. We compute the growth rate and describe the optimal leverage ratio maximizing the robust long-term growth rate. To achieve this, the worst-case parameters are analyzed by the comparison principle, and the growth rate of the worst-case is computed using the martingale extraction method. The robust long-term growth rates are obtained explicitly under a number of models for the reference asset, including the geometric Brownian motion (GBM), Cox–Ingersoll–Ross (CIR), 3/2, and Heston and 3/2 stochastic volatility models. Additionally, we demonstrate the impact of stochastic interest rates, such as the Vasicek and inverse GARCH short rate models. This paper is an extended work of \citet{“Leung2017”}.

Keywords: Leveraged Exchange-Traded Funds (LETF), Robust Optimization, Worst-Case Analysis, Stochastic Volatility, Kelly Criterion, Equities

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, featuring advanced stochastic calculus, PDE theory, and robust optimization techniques across multiple complex models (e.g., Heston, 3/2, CIR). However, it is purely theoretical with no backtesting, code, or empirical data; the focus is on deriving analytic expressions and optimal leverage ratios in a worst-case setting.

  flowchart TD
    RQ["Research Goal: Find robust long-term growth rate & optimal leverage for LETFs under model uncertainty"]
    M["Methodology: Worst-case analysis, Comparison Principle, Martingale Extraction"]
    I["Input Models: GBM, CIR, 3/2, Heston, 3/2 Stochastic Volatility, Vasicek, Inverse GARCH"]
    C["Computation: Solve for robust growth rate & optimal leverage ratio"]
    F["Outcomes: Explicit formulas for worst-case growth rates, Optimal leverage strategies, Impact of stochastic interest rates"]