The Mathematics Behind Leveraged ETF Performance: Why Expected Multiples Don't Materialise

A Follow-Up to Our Leveraged ETF Evolution Analysis
In our previous article, we explored how leveraged and inverse ETFs are evolving from retail speculation tools to sophisticated institutional hedging instruments. We discussed why these products require deep understanding of their mechanics and aren't suitable for buy-and-hold strategies.

But what exactly are those mechanics? And why don't these products deliver their stated multiples over time?

Using actual performance data from SPY-based leveraged and inverse products, we can demonstrate the mathematical reality that makes these instruments fundamentally different from traditional investments.

The Expected vs Actual Performance Gap

Over a recent one-year period, SPY delivered a 16.1% total return. If leveraged ETFs performed as their names suggest, the mathematics would be straightforward:

Expected Returns Based on Stated Multiples:
•    3x leveraged products: 48.3% return
•    2x leveraged products: 32.2% return
•    -2x inverse products: -32.2% return
•    -3x inverse products: -48.3% return

Actual Returns:
•    3x leveraged products (SPXL, UPRO, HIBL): 28.8% to 29.5%
•    2x leveraged product (SPUU): 25.1%
•    -2x inverse product (SDS): -30.0%
•    -3x inverse products (SPXU, HIBS): -44.2% to -65.6%

The divergence is substantial. Even in a strongly positive market environment where SPY gained 16%, leveraged long products delivered only 58-61% of their expected multiples. Inverse products showed similar or worse divergence from expectations.

Why the Mathematics Don't Work as Expected

Three mathematical realities create this performance gap:

1. Daily Rebalancing Effects

Leveraged ETFs reset their leverage ratio daily. This means they deliver the stated multiple of the daily return, not the cumulative return over longer periods.

Consider a simplified two-day scenario:
•    Day 1: Index rises 10%, 3x ETF rises 30%
•    Day 2: Index falls 9.09%, returning to starting point

The 3x ETF would fall 27.27% on Day 2, leaving it down 2.1% overall despite the underlying index being flat. The daily rebalancing creates path dependency – the sequence and magnitude of returns matter, not just the end result.


2. Volatility Drag (Geometric vs Arithmetic Returns)

This is the mathematical phenomenon that causes the most significant performance degradation.
When returns compound, the geometric mean (actual return) is always lower than the arithmetic mean (average return) – and the gap widens with volatility. For leveraged products, this effect is amplified by the leverage ratio.

A simple example:
•    An index alternates +10% and -10% days
•    Average arithmetic return: 0%
•    Actual geometric return: -1% (due to compounding)
•    For a 3x leveraged version: approximately -9%

The higher the volatility and the greater the leverage, the larger this drag becomes. This is pure mathematics, not a flaw in the products' design.


3. Compounding Effects Over Time

Daily rebalancing means returns compound differently than intuition suggests. A 3x leveraged ETF doesn't deliver 3x the buy-and-hold return – it delivers 3x the daily return, which compounds very differently over time.

This effect works in both directions. During sustained uptrends with low volatility, leveraged products can slightly outperform their stated multiples on a daily basis. During volatile or choppy markets, they significantly underperform.

Visualising the Complexity: When Tracking Error Varies

To understand just how unpredictable these mathematical effects can be, we analysed tracking error across different holding periods and market conditions for SPY-based leveraged products.

The results demonstrate why these instruments require constant professional monitoring rather than simple buy-and-hold approaches.

Tracking error heatmap – 20-day holding period: 

Tracking error heatmap – 60-day holding period: 
 

What This Analysis Reveals

• SPY cumulative trend (horizontal axis): whether the market moved up, down, or stayed flat
• SPY daily volatility (vertical axis): how choppy the daily movements were
  
  

• Dark regions (lower tracking error – closer to 0 difference between stated multiple of SPY and actual multiple attained) cluster around strong directional moves with moderate volatility. Even here, "lower" doesn't mean zero – these products still deviate from stated multiples.
• Bright regions (higher tracking error – further from 0 difference between stated multiple of SPY and actual multiple attained) appear when markets are flat or choppy, especially with elevated volatility. This is where the mathematical effects we described earlier compound most severely.
• The variability itself is the key insight. Tracking error isn't consistent or predictable – it depends on the specific path the market takes, which cannot be known in advance.

Why This Matters for Risk Management

• Unpredictability Requires Active Management: The tracking error varies dramatically based on market conditions that cannot be forecast. This is why sophisticated institutional users monitor these positions continuously and adjust exposures dynamically, rather than setting and forgetting.
• Pattern Recognition Isn't Strategy: While the heatmaps show patterns in historical data, using them as a timing guide would require accurately predicting both future trend and volatility – a notoriously difficult combination. Professional users understand these patterns exist, but don't rely on them for decision-making.
• Complexity Validates Professional Use: The fact that we need multi-dimensional analysis across trend, volatility, and time just to understand historical tracking error demonstrates why these products are positioned for institutional use. Retail investors seeing "3x leverage" cannot be expected to understand these dynamics.
• Even "Good" Conditions Show Deviation: Notice that even in the darkest regions of the heatmap, tracking error exists. There are no conditions where these products reliably deliver their stated multiples over multi-week periods.

The Real-World Impact

Looking at our one-year data across nine SPY-based products, every single one showed substantial deviation from its stated multiple:

• Delivered only 58-61% of expected returns
• Despite a favourable trending market environment
• The 19-20 percentage point shortfall represents significant performance drag

• Showed even more dramatic divergence
• Some inverse products performed "better" than expected (lost less than the stated multiple)
• Others performed worse (lost more than expected)
• The variability itself demonstrates the unpredictability of tracking error

Why This Matters for Investors

These mathematical realities explain several critical facts about leveraged ETFs:

They're Not Long-Term Investment Vehicles: The compounding effects and volatility drag make extended holding periods increasingly unreliable for delivering expected multiples. This is mathematical certainty, not market opinion.

Professional Use Requires Sophisticated Understanding: Institutional investors who use these products understand they're not buying "3x SPY" – they're buying a tool that delivers 3x daily exposure with specific mathematical properties that must be actively managed.

The Name Can Be Misleading: A "3x S&P 500 ETF" sounds like it should deliver three times the S&P 500's return. The reality is far more complex, and the name doesn't capture the daily rebalancing and compounding effects.

Conclusion: Understanding the Mechanics

These products do exactly what they're designed to do: provide leveraged exposure to daily index returns. The performance "gap" isn't a flaw – it's the mathematical consequence of daily rebalancing combined with compounding and volatility.

This analysis reinforces why our previous article positioned these as institutional tools requiring sophisticated understanding. The mathematics are unforgiving for investors who don't fully comprehend how daily rebalancing, compounding, and volatility interact over time.

Understanding these mechanics explains why these products have evolved toward professional use rather than retail investment. Sophisticated users can employ them effectively within proper risk frameworks that account for these mathematical realities. Retail investors expecting simple "leveraged returns" will consistently be disappointed by the mathematical reality.

Data Note: Analysis based on SPY and related leveraged/inverse products over a recent one-year period. All figures represent actual historical performance and are provided for educational purposes to illustrate mathematical principles.
Disclaimer: This analysis is educational and examines mathematical principles behind leveraged ETF performance. It does not constitute investment advice or recommendations for any strategy. Leveraged and inverse ETFs involve substantial risk including complete loss of invested capital.