Modelling and Forecasting Risk Dependence and Portfolio VaR for Cryptocurrencies

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Introduction

In recent years, cryptocurrencies have captured significant attention from individual investors, regulatory authorities, and policymakers alike. These decentralized digital currencies operate independently of central banks and traditional monetary policies, offering a new paradigm in financial assets. Since the inception of Bitcoin in 2009, the cryptocurrency market has expanded rapidly, experiencing dramatic price increases between 2016 and 2020, including notable pricing bubbles in 2018. The market has also shown extreme volatility, with sharp declines during events such as the COVID-19 outbreak in March 2020.

Existing research has explored various aspects of cryptocurrency markets, including their hedging capabilities, market efficiency, and volatility patterns. Many studies have employed multivariate approaches such as GARCH-DCC, GARCH-BEKK, and GARCH-copula models to understand the interdependencies between cryptocurrencies and other financial assets. These studies confirm that both conditional volatilities and correlations among cryptocurrencies change over time, particularly during periods of market stress.

This article examines risk dependence and portfolio Value-at-Risk (VaR) for major cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Litecoin (LTC), and Ripple (XRP)—using the multivariate Generalized Autoregressive Score (GAS) model. We compare its performance against the traditional DCC-GARCH model, with a focus on out-of-sample forecasting accuracy during turbulent market conditions.

Understanding Cryptocurrency Risk Dynamics

The Evolving Cryptocurrency Market

The cryptocurrency market represents a fundamentally different asset class compared to traditional financial instruments. Unlike stock markets dominated by institutional investors, cryptocurrency markets feature a significant proportion of retail investors, which may explain the different volatility patterns observed. This market structure difference likely contributes to the absence of significant leverage effects (asymmetric volatility responses to price changes) commonly found in equity markets.

Cryptocurrencies exhibit unique characteristics that make their risk modeling particularly challenging:

Statistical Properties of Cryptocurrency Returns

Analysis of daily returns for major cryptocurrencies reveals several important characteristics:

These properties necessitate specialized modeling approaches that can capture the unique risk characteristics of digital assets.

Methodological Framework

Multivariate GAS Model

The Generalized Autoregressive Score (GAS) model, introduced by Creal et al. (2013), provides a flexible framework for modeling time-varying parameters in multivariate distributions. The model updates parameters using the score of the likelihood function, allowing it to naturally adapt to changing market conditions.

For our application, we specify a GAS(1,1) model with a multivariate standardized Student-t distribution to capture the fat-tailed nature of cryptocurrency returns. The time-varying parameter vector includes location, scale, correlation, and shape parameters, all evolving according to the score-driven mechanism.

The GAS framework offers several advantages for cryptocurrency modeling:

DCC-GARCH Model

The Dynamic Conditional Correlation GARCH model, developed by Engle (2002), represents the traditional approach to multivariate volatility modeling. The model decomposes the conditional covariance matrix into individual volatilities (modeled by GARCH processes) and a time-varying correlation matrix.

In our implementation, we use Student-t innovations to account for fat tails and employ a rolling window approach for out-of-sample forecasting. The DCC specification allows correlations to evolve smoothly over time according to a GARCH-like process.

Empirical Analysis

Data and Preliminary Analysis

We analyze daily price data for BTC, ETH, LTC, and XRP from January 2016 to December 2021, sourced from CryptoCompare. Returns are calculated as logarithmic differences and winsorized at the 0.5% and 99.5% levels to mitigate the impact of extreme outliers.

The sample is divided into two periods:

Preliminary analysis reveals several important patterns:

In-sample Estimation Results

Both GAS and DCC models were estimated using the in-sample period. Likelihood ratio tests support the GAS specification with time-varying volatilities, correlations, locations, and shape parameters. Interestingly, asymmetric GARCH specifications did not show significant leverage effects for any cryptocurrencies, consistent with previous research.

Parameter estimates for both models were statistically significant, confirming the presence of time-varying dependence structures. The GAS model produced more stable volatility estimates compared to the more reactive DCC model, particularly during the 2018 market crash.

Correlation Dynamics

Rolling correlation analysis reveals evolving interdependence patterns:

These patterns highlight the importance of modeling time-varying dependencies rather than assuming constant correlations.

Forecasting Performance Evaluation

Volatility and Correlation Forecasts

We evaluate out-of-sample forecasting performance using realized volatility and correlation measures computed from 5-minute intraday data. Two loss functions are employed: Mean Squared Error (MSE) and Gaussian Quasi-Likelihood (QLIKE).

The results show:

The GAS model's advantage stems from its more robust updating mechanism, which prevents overreaction to extreme observations while still capturing changing market conditions.

Density Forecast Evaluation

We assess the overall distributional forecasts using three proper scoring rules:

Results consistently favor the GAS model across all scoring rules, indicating better calibration of the entire predictive distribution. The differences are statistically significant for energy and variogram scores.

Value-at-Risk Forecasting

VaR forecasts are evaluated for individual cryptocurrencies and several portfolios with different weight structures. Backtesting results show:

The DCC model tends to overestimate risk, particularly during volatile periods, leading to excessive capital requirements. The GAS model provides more accurate risk estimates, especially for extreme quantiles.

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Practical Implications for Investors and Risk Managers

Portfolio Construction Benefits

The improved forecasting accuracy of the GAS model offers several practical advantages:

  1. More accurate capital allocation: Better VaR estimates lead to appropriate rather than excessive capital reserves
  2. Improved portfolio optimization: More reliable correlation forecasts enhance mean-variance optimization
  3. Enhanced risk monitoring: Early detection of changing dependence structures allows proactive risk management
  4. Better hedging strategies: More accurate correlation forecasts improve hedge ratio calculations

Risk Management Applications

Financial institutions and investors can apply these findings in several ways:

Frequently Asked Questions

What makes cryptocurrency risk modeling different from traditional assets?

Cryptocurrencies exhibit higher volatility, different market microstructure, and unique dependence patterns compared to traditional assets. Their 24/7 trading, sensitivity to regulatory news, and dominance of retail investors create distinct challenges for risk modeling that require specialized approaches like the GAS model.

How often should cryptocurrency risk models be updated?

Given the rapid evolution of cryptocurrency markets, risk models should be updated frequently—at least weekly for most applications, and potentially daily for active trading operations. The GAS model's score-driven framework automatically incorporates new information, making it particularly suitable for these dynamic markets.

Can these models be applied to other digital assets?

Yes, the methodological framework can be extended to other cryptocurrencies, tokens, and digital assets. The GAS model's flexibility allows it to accommodate different distributional assumptions and dependency structures appropriate for various digital assets.

How do regulatory changes affect cryptocurrency risk models?

Regulatory developments significantly impact cryptocurrency markets, often causing structural breaks in volatility and correlation patterns. Risk models should incorporate regulatory news indicators or use approaches like the GAS model that can automatically adapt to changing market regimes.

What are the limitations of these risk modeling approaches?

All risk models have limitations. These include: assumption of stationary processes over estimation windows, difficulty predicting unprecedented events, and model risk. Combining multiple approaches and maintaining conservative margin requirements can help mitigate these limitations.

How can investors use these models in practice?

Investors can use these models to: calculate position sizes based on risk estimates, diversify portfolios using correlation forecasts, set stop-loss levels based on VaR estimates, and stress test portfolios under extreme market scenarios derived from the models.

Conclusion

Our analysis demonstrates the superiority of the multivariate GAS model for modeling risk dependence and forecasting portfolio VaR in cryptocurrency markets. The model's robust updating mechanism, which uses the score of the likelihood function, provides more accurate volatility and correlation forecasts compared to the traditional DCC-GARCH approach.

These findings have important implications for investors, risk managers, and financial institutions operating in cryptocurrency markets. The improved forecasting accuracy can lead to better capital allocation, more effective hedging strategies, and enhanced risk monitoring systems.

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Future research could extend this work in several directions, including incorporating regulatory news indicators, exploring safe-haven properties during market stress, and developing more specialized scoring rules for evaluating multivariate density forecasts in specific regions of interest. As cryptocurrency markets continue to evolve, advanced modeling techniques like the GAS framework will play an increasingly important role in understanding and managing their unique risks.