Modelling and Forecasting Risk Dependence and Portfolio VaR for Cryptocurrencies

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Cryptocurrencies have rapidly evolved from niche digital assets to major players in global financial markets. As Bitcoin, Ethereum, Litecoin, and Ripple continue to attract institutional and retail investors alike, understanding their risk dynamics has become critical for portfolio management and risk forecasting. This article explores the application of advanced statistical models—specifically the Generalized Autoregressive Score (GAS) model and the Dynamic Conditional Correlation GARCH (DCC-GARCH) model—to assess co-dependence structures and forecast Value-at-Risk (VaR) in cryptocurrency portfolios.

By analyzing daily price data from January 2016 to December 2021—a period encompassing the 2018 crypto bubble burst and the 2020 market turmoil caused by the pandemic—we evaluate how well these models capture volatility clustering, dynamic correlations, and tail risks. Our focus is on out-of-sample forecasting accuracy, a key benchmark for real-world applicability in financial risk modeling.

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Understanding Cryptocurrency Market Dynamics

The cryptocurrency market is characterized by extreme volatility, structural breaks, and rapidly shifting investor sentiment. Unlike traditional financial assets, digital currencies operate in decentralized ecosystems with limited regulatory oversight, contributing to heightened price swings and complex interdependencies.

Bitcoin (BTC), launched in 2009, remains the largest cryptocurrency by market capitalization. Ethereum (ETH) introduced smart contract functionality, enabling decentralized applications. Litecoin (LTC) was designed as a faster alternative to Bitcoin, while Ripple (XRP) focuses on cross-border payments. Despite their different use cases, these assets often move in tandem during periods of market stress.

Empirical studies show that cryptocurrencies exhibit leptokurtic return distributions, meaning they have fatter tails than normal distributions—indicating a higher probability of extreme outcomes. Additionally, traditional assumptions such as market efficiency and symmetric volatility responses do not always hold in this space.

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The Role of Multivariate Models in Risk Forecasting

To model the joint behavior of multiple cryptocurrencies, multivariate time series approaches are essential. Two prominent frameworks used in financial econometrics are:

  1. DCC-GARCH Model: A widely adopted method that captures time-varying volatilities and correlations. However, it can be sensitive to extreme shocks due to its reliance on squared residuals.
  2. GAS Model (Generalized Autoregressive Score): A more flexible observation-driven framework that updates parameters based on the score of the conditional likelihood function. It adapts more smoothly to large market movements.

The GAS model nests several classical models—including GARCH and ACD—and allows for time-varying location, scale, correlation, and shape parameters under a unified structure. This makes it particularly suitable for modeling fat-tailed and asymmetric return distributions commonly observed in crypto markets.

Why the GAS Model Excels in Crypto Risk Modeling

Data and Preliminary Analysis

We analyze daily closing prices for BTC, ETH, LTC, and XRP from January 2016 to December 2021, sourced via API from a major crypto data provider. Returns are calculated as logarithmic differences:

r_t = 100 × (log(P_t) - log(P_{t-1}))

To mitigate the influence of extreme outliers, returns are winsorized at the 0.5% and 99.5% levels. Descriptive statistics confirm high kurtosis across all series—evidence of fat tails—and reject normality via Jarque-Bera tests. The Augmented Dickey-Fuller test confirms stationarity.

Notably:

Rolling correlation analysis reveals that pairwise dependencies strengthened significantly post-2018, suggesting growing market integration among major cryptocurrencies.

In-Sample Estimation Results

We estimate both the multivariate GAS(1,1) and DCC-GARCH(1,1) models using a Student-t distribution to account for heavy tails.

Likelihood Ratio Testing Supports Full Time-Varying GAS Specification

A series of nested likelihood ratio tests favor a fully time-varying GAS model—where volatility, correlation, location, and shape parameters all change over time—over simpler specifications. This indicates that static or partially dynamic models fail to capture the full complexity of crypto return dynamics.

Parameter Estimates Show Strong Persistence

In the GAS model:

In contrast, asymmetric GARCH extensions (e.g., GJR-GARCH, EGARCH) show no significant leverage effects—unlike in equity markets. This suggests that negative returns do not systematically trigger higher future volatility in crypto, possibly due to a predominance of speculative rather than information-based trading.

Out-of-Sample Forecast Evaluation

We conduct rolling-window forecasts from 2019 to 2021 (1,096 days), comparing model performance using realized measures derived from high-frequency data.

Volatility and Correlation Forecast Accuracy

Using two robust loss functions:

Results show:

Diebold-Mariano tests confirm statistically significant gains for the GAS model in most cases.

Density Forecasting: Superior Probabilistic Predictions

We evaluate full density forecasts using three proper scoring rules:

All metrics indicate that the GAS model produces more accurate probabilistic forecasts. The energy and variogram scores—particularly sensitive to dependence structure—show large improvements, highlighting the GAS model’s ability to capture multivariate risk dynamics.

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Portfolio Value-at-Risk Forecasting Performance

We construct five portfolios (four long-only/long-short combinations) and simulate 10,000 paths to estimate 1-day-ahead VaR at 1% and 5% levels.

Backtesting via:

Reveals:

For example, in a long-only equally weighted portfolio:

This demonstrates the practical advantage of GAS: it avoids overreacting to transient shocks while still capturing systemic risk buildup.

Frequently Asked Questions (FAQ)

What is the main advantage of the GAS model over DCC-GARCH?

The GAS model uses the score of the likelihood function to update parameters, making it more robust to extreme observations. In contrast, DCC-GARCH relies on lagged squared returns, which can overamplify volatility following large shocks—common in crypto markets.

Does the study find evidence of contagion among cryptocurrencies?

Yes. Rolling correlation analysis shows increasing co-movement since 2018, particularly during crises like the 2020 pandemic selloff. This suggests growing market integration and reduced diversification benefits over time.

Why are asymmetric GARCH models ineffective for cryptocurrencies?

Unlike stocks, where bad news increases volatility more than good news (leverage effect), crypto markets show symmetric volatility responses. This may reflect speculative behavior rather than fundamentals-driven trading.

Can these models be applied to other digital assets like stablecoins or NFTs?

While this study focuses on major cryptocurrencies, the methodology can extend to other digital assets. However, stablecoins require different modeling due to their pegged nature, while NFTs lack sufficient high-frequency pricing data.

How does regulatory news affect model performance?

Events like the SEC lawsuit against Ripple caused sudden shifts in XRP’s volatility and correlation structure. Models with adaptive updating mechanisms—like GAS—handle such regime changes more effectively than rigid ones.

Is real-time implementation feasible?

Yes. The GAS model can be implemented in real-time using recursive filtering techniques. Combined with automated data pipelines, it supports dynamic risk monitoring systems used by exchanges and hedge funds.

Conclusion

This study provides compelling evidence that the multivariate GAS model outperforms traditional DCC-GARCH in forecasting risk dependence and portfolio VaR for cryptocurrencies. Its superior handling of volatility clustering, dynamic correlations, and tail events makes it a powerful tool for investors navigating the turbulent crypto landscape.

Key takeaways:

Future research could explore extensions incorporating macro-financial variables, machine learning enhancements, or regime-switching GAS models to further improve predictive power.

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