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Portfolio Optimization under Heavy Tails and Asymmetric Volatility: Evidence from Taiwan-Exposed ETFs

This paper analyzes thirty U.S.-listed Taiwan-exposed ETFs from 2015 to 2025 to demonstrate that while semiconductor concentration drives heavy-tailed risks and asymmetric volatility, tail-risk optimization (CVaR) yields more concentrated portfolios and different performance rankings compared to traditional mean-variance frameworks, highlighting the insufficiency of variance-based models for technology-concentrated investments.

Original authors: Ting-Jung Lee, Abootaleb Shirvani, Farzana Afroz, Svetlozar T. Rachev, Frank J. Fabozzi

Published 2026-07-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ting-Jung Lee, Abootaleb Shirvani, Farzana Afroz, Svetlozar T. Rachev, Frank J. Fabozzi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a treasure hunter navigating a stormy sea. For centuries, sailors have used a simple map that assumes the ocean is mostly calm, with waves that rise and fall in a predictable, symmetrical pattern. This map works well for gentle breezes, but it fails miserably when a hurricane hits. In the world of finance, this "calm ocean" map is called variance, a tool that measures how much prices jump around. It treats a sudden price drop (a disaster) and a sudden price spike (a jackpot) as equally risky, just because they are both big moves. But in reality, investors care much more about losing their ship than about finding extra gold.

When you look at markets driven by high-tech industries, like the one in Taiwan, the ocean isn't just stormy; it's full of hidden, massive waves that can capsize a ship without warning. These are called heavy tails (extreme events happen more often than the calm map predicts) and asymmetric volatility (bad news makes the waves get bigger faster than good news makes them shrink). To navigate this, experts use a special radar called CVaR (Conditional Value-at-Risk), which doesn't just ask, "How big is the biggest wave?" but rather, "If the worst wave hits, how much water will actually flood the boat?" This paper dives into the specific waters of Taiwan-exposed funds to see if the old maps are lying to us and if the new radar can help us build a better boat.

The Taiwan Semiconductor Storm

The authors of this study decided to test these ideas on thirty different investment funds (ETFs) that have a heavy connection to Taiwan. Why Taiwan? Because it is the heart of the global semiconductor industry, the place where the chips for your phone, your AI, and your computers are made. This makes these funds incredibly sensitive to the tech world's ups and downs, as well as geopolitical tensions. The researchers looked at data from February 2015 to February 2025, a period that included the calm before the pandemic, the chaotic crash of 2020, and the massive boom in AI that followed.

They asked a simple but crucial question: If we build a portfolio using the old "variance" map, will it protect us from the real dangers of these tech-heavy funds, or will it leave us exposed?

The Old Map vs. The New Radar

First, the team checked the "shape" of the risk. They used a tool called the Hill estimator to measure how "heavy" the tails of the risk distribution were. Think of this as measuring how likely it is to find a tsunami in the ocean. Surprisingly, they found that the shape of the danger was actually quite similar across all the funds, whether they were focused on just one semiconductor company or spread out across the whole world. The "tsunami probability" was roughly the same.

However, the size of the waves was very different. The funds focused on semiconductors (like SMH and SOXX) had much bigger waves overall because their prices moved more wildly. This is a key finding: the extra danger in these tech funds wasn't because the type of risk was weird, but simply because the scale of the movement was huge. The old variance-based maps missed this nuance, making the semiconductor funds look only slightly riskier than the safe ones, when in reality, their potential losses were massive.

The Mystery of the "Long Memory"

Next, the researchers investigated a spooky phenomenon called long memory. In finance, this is the idea that if a wave crashes today, it might keep crashing for a very long time, like a ghost that refuses to leave. Some models suggest this is a permanent feature of the market. But the authors tested this by filtering out the "noise" of daily volatility using a model called GJR-GARCH (which accounts for the fact that bad news scares people more than good news excites them).

After filtering out the noise, the "ghost" vanished. The long memory disappeared. This suggests that the market isn't haunted by a permanent curse; rather, the waves just look scary because they cluster together when things go wrong. The authors conclude that we don't need complex "long-memory" models to predict these funds; a simpler model that accounts for how bad news spikes volatility is enough.

Building the Boat: Two Different Blueprints

The most exciting part of the study was building the portfolios. The researchers tried two different blueprints:

The Mean-Variance Blueprint: The classic approach that tries to balance return against total movement (up and down).
The CVaR Blueprint: The new approach that tries to minimize the specific damage from the worst-case scenarios.

When they built the Mean-Variance boat, it was a balanced, diversified vessel. It spread its weight across many different funds to smooth out the ride. It didn't bet heavily on anything.

But when they built the CVaR boat, the result was shocking. The new blueprint decided that the best way to survive the worst storms was to load up heavily on the SMH fund (a semiconductor-focused ETF). Why? Because during the post-pandemic AI boom, SMH was so good at generating returns that, even with its huge waves, it offered the best "survival-to-reward" ratio. The CVaR boat became a concentrated, high-speed racer, betting big on the tech sector, whereas the old boat was a slow, steady cruise ship.

Who Wins the Race?

Finally, they raced these boats over the last ten years to see which one performed best. The results depended entirely on how you judged the winner:

If you judge by the old rules (Sharpe Ratio): The Equally Weighted Portfolio (a simple boat where you buy a little bit of everything) won. It was robust, didn't make big mistakes, and rode the AI wave well enough to beat the complex, optimized boats. The fancy math of the optimized boats actually hurt them because the math made small errors that got magnified.
If you judge by the new rules (Rachev Ratio): The CVaR-based boats won. When you look specifically at the balance between extreme gains and extreme losses, the portfolios that focused on tail risk performed better. They managed the worst days more effectively.

The Takeaway

The paper suggests that in a world dominated by technology and heavy tails, the old "variance" map is incomplete. It hides the true scale of the danger in semiconductor funds. While the simple "buy everything" strategy was the most robust winner in this specific race, the study proves that if you care deeply about avoiding catastrophic losses, you need a different kind of map.

The authors don't claim to have found a magic bullet that solves all investing problems. Instead, they show that how you measure risk changes the boat you build. If you only look at average waves, you build a cruise ship. If you look at the tsunamis, you might end up building a high-speed racer. In the volatile world of Taiwan's tech sector, knowing which map you are using is the difference between a smooth sail and a shipwreck.

Technical Summary: Portfolio Optimization under Heavy Tails and Asymmetric Volatility: Evidence from Taiwan-Exposed ETFs

Problem Statement
Taiwan's central role in global semiconductor manufacturing creates a unique investment environment for Taiwan-exposed Exchange-Traded Funds (ETFs). These assets are subject to technology concentration, geopolitical uncertainty, and supply-chain disruptions, leading to return distributions characterized by heavy tails, volatility clustering, and asymmetric responses to negative shocks. Traditional portfolio frameworks, primarily based on the mean–variance optimization of Markowitz (1952), treat positive and negative deviations symmetrically and rely on variance as the sole measure of risk. This approach is insufficient for markets where returns exhibit skewness, excess kurtosis, and large downside movements. The paper addresses whether variance-based measures systematically understate downside risk in this context, whether apparent long-memory in volatility reflects genuine fractional integration or conditional heteroskedasticity, and whether tail-sensitive optimization (Conditional Value-at-Risk, or CVaR) yields materially different portfolio allocations and performance rankings compared to mean–variance optimization.

Methodology
The study analyzes thirty U.S.-listed ETFs with exposure to Taiwan, emerging markets, Asia–Pacific, and the technology/semiconductor sectors over the period from February 20, 2015, to February 20, 2025. The empirical framework integrates four distinct methodological components:

Tail-Risk Diagnostics: The authors employ the Hill estimator to estimate the tail index ( $\alpha$ ) of the return distributions, characterizing the asymptotic decay of extreme losses. This is complemented by historical Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) calculations at 95% and 99% confidence levels.
Volatility Modeling: To investigate volatility persistence and asymmetry, the paper utilizes GJR-GARCH(1,1) models with Student-t innovations to capture asymmetric volatility responses (leverage effects). Additionally, rolling-window FIGARCH(1,d,1) models are estimated to distinguish between genuine long-memory (fractional integration) and persistence driven by conditional heteroskedasticity. Long-memory diagnostics include the Geweke–Porter–Hudak (GPH) estimator, the Local Whittle estimator, and the Hurst exponent, applied both to raw squared returns and to standardized residuals filtered by the GJR-GARCH model.
Portfolio Optimization: Two optimization frameworks are compared using a four-year (1,008 trading day) rolling window:
- Mean–Variance: Minimizing portfolio variance subject to expected return constraints (Markowitz, 1952).
- CVaR Optimization: Minimizing CVaR and maximizing the STARR ratio (excess return over CVaR) using the scenario-based linear programming formulation of Rockafellar and Uryasev (2002).
  Both frameworks are evaluated under long-only and long–short constraints.
Performance Evaluation: Portfolio rankings are assessed using three distinct reward-to-risk measures: the Sharpe ratio (variance-based), the Rachev ratio (comparing extreme gains to extreme losses), and the STARR ratio (excess return relative to CVaR).

Key Contributions and Results

Tail Risk and Scale vs. Thickness: The analysis confirms that Taiwan-exposed ETFs exhibit heavy-tailed behavior, with Hill tail-index estimates generally falling between 2.5 and 3.5. While the asymptotic tail-decay behavior (tail index) is broadly similar across the ETF universe, semiconductor-focused ETFs (e.g., SMH, SOXX) exhibit substantially larger VaR and CVaR estimates than diversified benchmarks. The paper concludes that these cross-sectional differences in extreme downside risk are driven primarily by differences in return scale (volatility) rather than differences in tail thickness. Variance-based measures like the Sharpe ratio fail to capture these scale-driven differences in tail risk.
Nature of Volatility Persistence: Initial diagnostics on raw squared returns suggest the presence of long-memory behavior. However, after filtering the return series through a GJR-GARCH(1,1) model, the long-memory diagnostics (GPH, Local Whittle, Hurst) collapse toward zero and lose statistical significance. This indicates that the apparent long memory is largely attributable to asymmetric conditional heteroskedasticity and volatility clustering rather than genuine fractional integration. Consequently, standard asymmetric GARCH models are deemed sufficient for modeling volatility in this context without the need for complex fractionally integrated specifications.
Divergence in Portfolio Allocations: CVaR-based optimization produces materially more concentrated portfolios compared to mean–variance optimization. While the mean–variance tangent portfolio remains diversified across multiple ETFs, the CVaR tangent portfolio allocates a dominant weight to the semiconductor ETF SMH, particularly during the post-COVID AI-driven expansion. This suggests that tail-risk optimization favors assets with favorable expected-return-to-tail-risk characteristics, even if they are highly concentrated.
Dependence of Rankings on Performance Measures: The study finds that portfolio performance rankings are highly sensitive to the chosen evaluation metric.
- The Sharpe ratio and STARR measure favor the equally weighted portfolio (EWP), which benefits from robustness against estimation error and broad exposure to the semiconductor-driven market appreciation.
- The Rachev ratio favors CVaR-based portfolios, as these strategies explicitly manage the balance between extreme gains and extreme losses.
- Under long-short constraints, optimized portfolios generally underperform the EWP in terms of cumulative returns and Sharpe ratios, likely due to estimation error amplification, though they may offer different tail-risk profiles.

Significance and Claims
The paper claims that in technology-concentrated investment environments, variance-based frameworks provide an incomplete characterization of risk. The results demonstrate that tail-sensitive risk measures (VaR, CVaR) reveal economically significant differences in downside risk that variance-based metrics obscure. Furthermore, the study highlights that the choice of performance measure fundamentally alters the assessment of portfolio quality; a portfolio that is optimal under a tail-sensitive criterion (Rachev ratio) may be suboptimal under a variance-based criterion (Sharpe ratio).

The authors assert that for Taiwan-exposed ETFs, the apparent long memory in volatility is better explained by asymmetric conditional heteroskedasticity than by fractional integration, suggesting that simpler GJR-GARCH models are preferable for risk management. Finally, the paper posits that while CVaR optimization leads to more concentrated allocations (specifically in semiconductor assets), the equally weighted portfolio remains a robust benchmark that often outperforms optimized strategies in out-of-sample performance, particularly when estimation error is a concern. The findings underscore the necessity of using multiple performance measures to evaluate portfolios in markets characterized by heavy tails and asymmetric volatility.