are we in a bear or bull market with beta dynamics?

Beyond Contagion: Understanding Beta Dynamics Across Market Regimes

Abstract

Traditional asset pricing models assume stable relationships between assets and the market across all market conditions to tell are win a bear or bull market with beta dynamics. However, empirical evidence increasingly suggests that these relationships, particularly market beta, can vary significantly across different market regimes. While increased comovement during market downturns has been extensively studied under the framework of "contagion," the parallel phenomenon during extreme positive markets remains less defined. This article examines the dynamics of time-varying and quantile-dependent betas, with particular focus on characterizing and interpreting increasing comovements during extreme positive market periods. Using quantile regression analysis, we identify distinct patterns of beta behavior and propose a conceptual framework for understanding right-tail comovement, including "Speculative Comovement" and "Euphoric Coupling." The implications for portfolio construction, risk management, and asset pricing theory are discussed.

1. Introduction

In financial markets, the relationship between individual assets and the broader market is fundamental to modern portfolio theory, asset pricing, and risk management. The Capital Asset Pricing Model (CAPM) introduced by Sharpe (1964) and Lintner (1965) operationalizes this relationship through beta, a measure of an asset's sensitivity to market movements. Traditionally, beta has been treated as a stable parameter—a single value that describes an asset's market sensitivity across all market conditions.

However, a growing body of evidence suggests that the assumption of beta stability may be oversimplistic. Market relationships often change dramatically during periods of stress or exuberance, with many assets exhibiting increased sensitivity to market movements during extreme conditions. This phenomenon challenges conventional asset pricing models and has profound implications for investment strategies and risk management.

The increased correlation between assets during market downturns has been extensively studied under the framework of "contagion," as formalized by Forbes and Rigobon (2002). Contagion describes how market shocks can propagate across assets and markets, leading to synchronized declines that exceed what would be expected based on fundamental relationships. However, a parallel phenomenon exists during extreme positive market conditions that has received comparatively less attention and remains inadequately characterized in the financial literature.

This article addresses this gap by examining the dynamics of beta across different market regimes, with particular focus on understanding and interpreting increasing comovements during extreme positive market periods. We employ quantile regression techniques to estimate conditional betas across the distribution of market returns, enabling us to identify distinct patterns of beta behavior across market regimes.

The article is structured as follows: Section 2 provides the theoretical framework for understanding conditional betas and reviews relevant literature. Section 3 outlines the methodological approach based on quantile regression. Section 4 examines empirical patterns of beta variation. Section 5 offers economic interpretations of right-tail beta increases, proposing terminology and conceptual frameworks for understanding this phenomenon. Section 6 discusses the financial implications for portfolio management and risk assessment. Section 7 concludes with a summary of findings and directions for future research.

2. Theoretical Framework

2.1 Beta in Asset Pricing Theory

The concept of beta is central to the Capital Asset Pricing Model (CAPM), which posits that an asset's expected return is determined by its systematic risk, measured by its covariance with the market portfolio. Mathematically, beta is defined as:

βi=Cov(ri,rM)Var(rM)\beta_i = \frac{\text{Cov}(r_i, r_M)}{\text{Var}(r_M)}βi​=Var(rM​)Cov(ri​,rM​)​

Where $r_i$ represents the return on asset $i$ and $r_M$ represents the market return. In the traditional CAPM framework, beta is estimated using Ordinary Least Squares (OLS) regression:

ri=αi+βirM+εir_i = \alpha_i + \beta_i r_M + \varepsilon_iri​=αi​+βi​rM​+εi​

This approach implicitly assumes that beta remains constant across all market conditions—whether the market is experiencing extreme gains, extreme losses, or normal fluctuations. The resulting beta represents an average sensitivity across the entire distribution of market returns.

2.2 Conditional Betas and Market Regimes

The assumption of beta stability has been challenged by numerous empirical studies showing that systematic risk can vary with market conditions. This has led to the development of conditional asset pricing models that allow for time-varying betas. Ferson and Schadt (1996) pioneered this approach by modeling betas as functions of observable state variables.

Building on this foundation, researchers have increasingly recognized that beta may not only vary over time but also across different market regimes. These regimes can be characterized by market volatility (high vs. low), market direction (bull vs. bear), or the magnitude of market movements (normal vs. extreme).

Ang and Chen (2002) demonstrated that correlations between stocks and the aggregate market are asymmetric, with stronger correlations during market downturns. Similarly, Pettengill et al. (1995) found evidence of a conditional relationship between beta and returns that depends on whether the market excess return is positive or negative.

2.3 Quantile Regression Approach

Quantile regression, introduced by Koenker and Bassett (1978), offers a powerful framework for examining how beta varies across different market regimes. Unlike OLS regression, which focuses on the conditional mean, quantile regression estimates the relationship between variables at different points in the conditional distribution.

In the context of asset pricing, quantile regression allows us to estimate how an asset's sensitivity to market movements changes across different quantiles of the market return distribution. The quantile regression model can be specified as:

Qri∣rM(τ)=αi(τ)+βi(τ)rM+εi(τ)Q_{r_i|r_M}(\tau) = \alpha_i(\tau) + \beta_i(\tau)r_M + \varepsilon_i(\tau)Qri​∣rM​​(τ)=αi​(τ)+βi​(τ)rM​+εi​(τ)

Where $Q_{r_i|r_M}(\tau)$ represents the $\tau$-th conditional quantile of asset returns given market returns, and $\beta_i(\tau)$ is the quantile-specific beta that may vary across different values of $\tau$.

This approach allows us to distinguish between beta in normal market conditions (e.g., $\tau \in [0.1, 0.9]$), extreme negative conditions (e.g., $\tau < 0.1$), and extreme positive conditions (e.g., $\tau > 0.9$). By comparing these conditional betas, we can identify patterns of varying systematic risk across market regimes.

3. Methodology

3.1 Model Specification

To investigate beta dynamics across market regimes, we employ a quantile regression framework that allows beta to vary across the distribution of market returns. The model specification follows:

Qri∣rM(τ)=αi(τ)+βi(τ)rM+εi(τ)Q_{r_i|r_M}(\tau) = \alpha_i(\tau) + \beta_i(\tau)r_M + \varepsilon_i(\tau)Qri​∣rM​​(τ)=αi​(τ)+βi​(τ)rM​+εi​(τ)

Where:

• $Q_{r_i|r_M}(\tau)$ is the $\tau$-th conditional quantile of asset $i$'s return given the market return

• $\alpha_i(\tau)$ is the quantile-specific intercept

• $\beta_i(\tau)$ is the quantile-specific beta coefficient

• $r_M$ is the market return

• $\varepsilon_i(\tau)$ is the error term for the $\tau$-th quantile

By estimating this model across a range of quantiles $\tau \in (0,1)$, we can trace out the entire conditional distribution of asset returns and examine how beta varies across different market regimes.

3.2 Defining Market Regimes

For analytical purposes, we partition the market return distribution into three regimes:

1. Extreme Negative Returns (Left Tail): Corresponding to $\tau < 0.1$, representing the bottom 10% of market returns

2. Normal Market Conditions: Corresponding to $\tau \in [0.1, 0.9]$, representing the middle 80% of market returns

3. Extreme Positive Returns (Right Tail): Corresponding to $\tau > 0.9$, representing the top 10% of market returns

This partitioning allows us to compare beta behavior across regimes and identify patterns of varying systematic risk. The choice of 10% thresholds strikes a balance between capturing truly extreme market conditions and ensuring sufficient data points for reliable estimation.

3.3 Beta Pattern Classification

Based on how beta varies across these regimes, we classify assets into several distinct patterns:

1. Stable Beta: Beta remains relatively constant across all market regimes

2. Increasing Beta in Both Tails: Beta increases during both extreme positive and extreme negative market conditions

3. Increasing Beta in Left Tail Only: Beta increases during extreme negative market conditions but remains stable or decreases during extreme positive conditions

4. Increasing Beta in Right Tail Only: Beta increases during extreme positive market conditions but remains stable or decreases during extreme negative conditions

These patterns provide a framework for understanding how assets respond to different market environments and form the basis for our economic interpretations.

4. Empirical Patterns of Beta Variation

4.1 The Stable Beta Case

The stable beta case serves as our benchmark scenario, reflecting the traditional CAPM assumption of constant systematic risk. In this scenario, an asset maintains consistent market sensitivity regardless of whether the market is experiencing extreme losses, normal fluctuations, or extreme gains.

Mathematically, a stable beta implies:

βi(τ1)≈βi(τ2) for all τ1,τ2∈(0,1)\beta_i(\tau_1) \approx \beta_i(\tau_2) \text{ for all } \tau_1, \tau_2 \in (0,1)βi​(τ1​)≈βi​(τ2​) for all τ1​,τ2​∈(0,1)

While perfectly stable betas are rare in practice, many defensive stocks and diversified portfolios exhibit relatively stable betas across market regimes. These assets tend to have consistent market exposure and are less susceptible to regime-specific factors.

From a risk management perspective, assets with stable betas provide predictable market exposure and are valuable for constructing portfolios with well-defined risk characteristics. However, they may not offer opportunities for dynamic risk management or regime-specific strategies.

4.2 Increasing Beta in Both Tails

Some assets exhibit increased market sensitivity during both extreme negative and extreme positive market conditions. This pattern is characterized by a U-shaped beta profile across quantiles:

βi(τ)>βi(0.5) for τ near 0 or 1\beta_i(\tau) > \beta_i(0.5) \text{ for } \tau \text{ near 0 or 1}βi​(τ)>βi​(0.5) for τ near 0 or 1

This pattern reflects a general amplification of market sensitivity during extreme conditions, regardless of market direction. It can arise from leverage effects, liquidity constraints, or market microstructure factors that become more pronounced during periods of market stress or exuberance.

Assets exhibiting this pattern include highly leveraged companies, small-cap stocks, and certain financial institutions. From a risk perspective, these assets may appear moderately risky during normal market conditions but can become significantly more volatile during extreme periods—both positive and negative.

4.3 Asymmetric Tail Behavior

Many assets exhibit asymmetric beta profiles, with different behavior in the left and right tails of the market return distribution. Two common asymmetric patterns are:

4.3.1 Increasing Beta in Left Tail Only (Contagion)

βi(τ)>βi(0.5) for τ near 0\beta_i(\tau) > \beta_i(0.5) \text{ for } \tau \text{ near 0}βi​(τ)>βi​(0.5) for τ near 0βi(τ)≈βi(0.5) for τ near 1\beta_i(\tau) \approx \beta_i(0.5) \text{ for } \tau \text{ near 1}βi​(τ)≈βi​(0.5) for τ near 1

This pattern, commonly associated with contagion effects, reflects increased market sensitivity during downturns but not during rallies. It can arise from risk aversion, flight-to-quality effects, and forced liquidations during market stress.

4.3.2 Increasing Beta in Right Tail Only

βi(τ)≈βi(0.5) for τ near 0\beta_i(\tau) \approx \beta_i(0.5) \text{ for } \tau \text{ near 0}βi​(τ)≈βi​(0.5) for τ near 0βi(τ)>βi(0.5) for τ near 1\beta_i(\tau) > \beta_i(0.5) \text{ for } \tau \text{ near 1}βi​(τ)>βi​(0.5) for τ near 1

Conversely, some assets show increased market sensitivity specifically during strong positive markets. This pattern, which we explore in detail in the next section, reflects a unique set of economic mechanisms distinct from contagion.

4.4 Quantifying Beta Variations

To quantify the magnitude of beta variations across regimes, we can compute several metrics:

1.     Absolute Beta Differential: The difference between extreme and normal betasΔβabsolute=βi(τextreme)−βi(τnormal)\Delta\beta_{\text{absolute}} = \beta_i(\tau_{\text{extreme}}) - \beta_i(\tau_{\text{normal}})Δβabsolute​=βi​(τextreme​)−βi​(τnormal​)

2.     Relative Beta Change: The percentage change in beta between regimesΔβrelative=βi(τextreme)βi(τnormal)−1\Delta\beta_{\text{relative}} = \frac{\beta_i(\tau_{\text{extreme}})}{\beta_i(\tau_{\text{normal}})} - 1Δβrelative​=βi​(τnormal​)βi​(τextreme​)​−1

3.     Beta Range: The difference between maximum and minimum beta across all quantilesRange(βi)=max⁡τβi(τ)−min⁡τβi(τ)\text{Range}(\beta_i) = \max_{\tau}\beta_i(\tau) - \min_{\tau}\beta_i(\tau)Range(βi​)=maxτ​βi​(τ)−minτ​βi​(τ)

These metrics help quantify the economic significance of beta variations and identify assets with substantial regime-dependent risk profiles.

4.     Economic Interpretation of Right-Tail Beta Increases

5.         

While increasing beta during market downturns has been extensively studied under the framework of contagion, the parallel phenomenon during market rallies requires its own conceptual framework. We propose several economic interpretations for increasing beta in the right tail:

5.1 Speculative Comovement

Speculative Comovement refers to the amplified sensitivity of certain assets to market movements during rallies, driven by speculative trading, momentum strategies, and performance chasing. This phenomenon reflects how investor behavior changes during bull markets, with increased focus on short-term gains and greater willingness to take on risk.

During market rallies, investors often allocate capital to assets that have shown recent outperformance or are perceived as high-beta plays that will outperform in a rising market. This behavior leads to capital concentration in certain assets or sectors, amplifying their response to positive market movements.

Key drivers of Speculative Comovement include:

• Momentum trading strategies that invest in recent winners

• Increased use of leverage during bullish periods

• Performance chasing by both retail and institutional investors

• Short-covering rallies that accelerate price movements

Empirically, Speculative Comovement is often observed in growth stocks, emerging markets, and specific sectors that become "market darlings" during bull markets.

5.2 Euphoric Coupling

Euphoric Coupling describes how investor optimism during bull markets leads to decreased discrimination between assets, resulting in tighter coupling between individual securities and the market. Unlike Speculative Comovement, which focuses on trading behavior, Euphoric Coupling emphasizes changes in risk perception and valuation standards.

During periods of market euphoria, investors tend to:

• Reduce their focus on fundamental analysis

• Apply higher valuation multiples across the board

• Downplay risk factors and potential weaknesses

• Exhibit herding behavior and confirmation bias

This decreased discrimination leads to a convergence in pricing mechanisms, where market sentiment becomes the dominant factor driving returns across assets. The result is increased synchronization specifically during positive market periods.

Euphoric Coupling has been observed during multiple historical bull markets, including the dot-com boom, the pre-2008 housing bubble, and more recently in certain technology and

cryptocurrency markets.

5.3 Upside Contagion

Upside Contagion represents the positive counterpart to downside contagion, describing how positive sentiment spreads across assets during market booms, creating synchronized amplified upward movements. While traditional contagion focuses on the propagation of negative shocks, Upside Contagion examines how positive market developments can cascade through the financial system.

This phenomenon involves:

• Cross-market spillovers of positive sentiment

• Reinforcing feedback loops between market segments

• Liquidity cascades that amplify positive movements

• Information contagion where positive news in one sector affects perceptions in others

Upside Contagion has important implications for diversification strategies, as assets that appear uncorrelated during normal or negative markets may become highly synchronized during positive extremes.

5.4 Perception-Driven Amplification

Perception-Driven Amplification describes how investors' perception of certain assets as "high growth" or "momentum plays" can create self-reinforcing patterns where these assets show higher beta specifically in bullish markets. This interpretation focuses on the role of asset classification and labeling in driving market dynamics.

When assets are perceived through specific mental models or categorizations:

• Investors assign them roles within portfolio construction (e.g., "growth accelerator")

• Trading algorithms target them based on factor exposures

• They attract specific investor segments with particular behavior patterns

• They become proxies for broader market sentiment or economic expectations

These perception effects can amplify market sensitivity selectively during positive periods, as investors look for vehicles to express bullish views or capture upside potential.

5.5 Empirical Evidence

Empirical evidence for right-tail beta increases has been documented across various markets and time periods:

• Technology stocks during bull markets often exhibit increasing right-tail betas

• Emerging markets frequently show stronger coupling with global markets during positive periods

• Small-cap stocks typically demonstrate more pronounced right-tail sensitivity

• Certain sectors (e.g., consumer discretionary, financial services) historically display stronger upside participation

These patterns challenge traditional asset pricing models that assume stable relationships and highlight the importance of considering regime-dependent risk when constructing portfolios and evaluating investment strategies.

6.     Financial Implications

The existence of varying betas across market regimes has profound implications for financial theory and practice:

6.1 Portfolio Construction Considerations

Traditional mean-variance optimization assumes stable correlations and betas, potentially leading to suboptimal portfolio construction when these parameters vary across regimes. To address this limitation, investors should:

1. Incorporate Regime-Dependent Risk: Use conditional betas that reflect different market environments

2. Balance Regime Exposures: Combine assets with complementary beta profiles across regimes

3. Stress Test Portfolios: Evaluate portfolio performance under various market scenarios

4. Consider Higher Moments: Look beyond variance to incorporate skewness and kurtosis

For example, a portfolio exclusively containing assets with increasing right-tail betas might outperform during bull markets but fail to provide diversification when needed most during downturns.

6.2 Risk Management Strategies

Regime-dependent betas necessitate more sophisticated risk management approaches:

1. Dynamic Hedging: Adjust hedge ratios based on prevailing market regime

2. Tail Risk Protection: Implement asymmetric hedging strategies focusing on specific tails

3. Scenario Analysis: Develop comprehensive stress tests incorporating regime-specific correlations

4. Liquidity Management: Plan for changing liquidity conditions across market regimes

Particularly important is the recognition that assets exhibiting right-tail beta increases may provide superior returns during bull markets but fail to offer diversification benefits during market stress.

6.3 Asset Pricing Implications

The existence of regime-dependent betas challenges traditional asset pricing models and suggests the need for conditional approaches:

1. Conditional CAPM: Incorporate regime-switching or state-dependent betas

2. Multi-Factor Models: Include factors that capture regime-specific risk exposures

3. Quantile-Based Pricing: Develop asset pricing frameworks based on quantile relationships

4. Asymmetric Risk Premiums: Recognize that investors may price left-tail and right-tail risks differently

These considerations may help explain empirical puzzles in asset pricing, such as the low beta anomaly and the flat security market line.

6.4 Investment Strategies

Understanding beta dynamics across regimes opens opportunities for investment strategies:

1. Regime-Switching Allocation: Dynamically adjust portfolio allocations based on identified market regimes

2. Tail Arbitrage: Exploit mispricing in assets with misunderstood tail behavior

3. Conditional Momentum: Incorporate regime considerations into momentum strategies

4. Smart Beta Construction: Design factor portfolios with targeted regime exposures

For example, investors might overweight assets with right-tail beta increases during early bull markets while shifting to more stable or left-tail-resistant assets as the cycle matures.

7. Conclusion

This article has examined the dynamics of beta across different market regimes, with particular focus on understanding and interpreting increasing comovements during extreme positive market periods. Using quantile regression analysis, we identified distinct patterns of beta behavior and proposed a conceptual framework for understanding right-tail comovement.

While increasing beta in negative market regimes (left tail) is commonly referred to as "Contagion," we have characterized the parallel phenomenon in positive market regimes (right tail) through several complementary concepts: Speculative Comovement, Euphoric Coupling, Upside Contagion, and Perception-Driven Amplification. These concepts provide a vocabulary and theoretical framework for understanding how and why assets exhibit increased market sensitivity during bullish periods.

The existence of regime-dependent betas has significant implications for portfolio construction, risk management, and asset pricing theory. Traditional approaches that assume stable

relationships across all market conditions may lead to suboptimal investment decisions and inadequate risk assessments. By incorporating regime-specific considerations into financial models and investment strategies, practitioners can develop more robust approaches that account for the complex dynamics of market relationships.

Future research directions include developing formal asset pricing models that incorporate regime-dependent betas, examining the determinants of right-tail beta increases at the firm and sector levels, and exploring the interaction between left-tail and right-tail behavior across different asset classes and market environments.

Understanding the asymmetric nature of market relationships across different regimes represents an important frontier in financial research, with the potential to enhance our understanding of market dynamics and improve investment outcomes across the full spectrum of market conditions.

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