Component GARCH vs GARCH(2,2) and Risk Measures

Introduction to Component GARCH and GARCH(2,2) Equivalence

The Component GARCH (CGARCH) model is a popular extension of the standard GARCH framework, often used in quantitative finance to capture long-run and short-run volatility components. In the FN3142 assignment, you are asked to show that CGARCH is equivalent to a GARCH(2,2) model. This equivalence is crucial for understanding how volatility persistence can be decomposed. Let's break it down step by step.

The CGARCH model is defined by:

Rt = μ + εt, εt = σt zt, zt ~ N(0,1)
σt² = qt + α (εt-1² - qt-1) + β (σt-1² - qt-1)
qt = ω + ρ qt-1 + φ (εt-1² - σt-1²)

Here, qt represents the long-run component, while σt² - qt is the short-run component. To show equivalence to GARCH(2,2), substitute qt into the σt² equation and rearrange. You'll end up with an ARMA(2,2) representation for εt², which corresponds to a GARCH(2,2) model with certain parameter restrictions. This exercise demonstrates how CGARCH nests within the GARCH family, highlighting its flexibility in modeling volatility dynamics.

In practice, CGARCH is useful for assets with persistent volatility, such as cryptocurrencies or meme stocks like GameStop (2021). Just as a long-term trend in a video game's player base (e.g., Fortnite) can coexist with short-term spikes due to events, CGARCH separates these components.

Value-at-Risk (VaR) and Expected Shortfall Derivation

Question 2 focuses on risk measures under the assumption of normally distributed returns with constant mean and variance. This is a classic setup in risk management, especially for portfolios with daily rebalancing.

Deriving 1-Day VaR at Critical Level α

Value-at-Risk at the α critical level is the quantile of the loss distribution. For a normally distributed return R ~ N(μ, σ²), the 1-day VaR at level α (e.g., 95%) is:

VaR_α = -μ - σ Φ⁻¹(α)

where Φ⁻¹ is the inverse standard normal CDF. Since α is typically small (e.g., 0.05 for 95% VaR), Φ⁻¹(α) is negative, making VaR positive. This formula is fundamental for setting capital requirements under Basel III.

Deriving 1-Day Expected Shortfall (ES)

Expected Shortfall, also known as Conditional VaR, measures the average loss beyond VaR. For normal returns, the 1-day ES at level α is:

ES_α = -μ + σ * φ(Φ⁻¹(α)) / α

where φ is the standard normal PDF. This formula is derived by integrating the tail of the distribution. ES is a coherent risk measure and is preferred over VaR because it accounts for tail severity.

Convergence of VaR and ES Proportion as α → 0

You are asked to prove that (VaR_α - ES_α) / VaR_α → 0 as α → 0. Intuitively, for extremely small α, the tail becomes very thin, so the average loss in the tail approaches the quantile. Mathematically, using the asymptotic properties of the normal distribution, one can show that the ratio tends to zero. This implies that for very high confidence levels, VaR and ES give similar relative risk assessments.

This concept is analogous to extreme weather forecasting: as the event becomes rarer (e.g., a 1-in-1000 year flood), the difference between the worst expected loss (VaR) and the average of worst losses (ES) diminishes in relative terms.

Optimality and Forecasting Power of Economic Forecasts

Question 3 tackles the relationship between statistical optimality (e.g., unbiasedness, efficiency) and practical forecasting performance (R²). This is a nuanced topic in econometrics.

Can an Optimal Forecast Have Low R²?

Yes. An optimal forecast minimizes expected loss (e.g., mean squared error), but if the variable is inherently noisy (high variance), the R² from regressing actuals on the forecast will be low. For example, forecasting daily stock returns with a model that uses only past returns: even if the model is the best possible (optimal), the R² might be less than 5% because returns are nearly unpredictable. This illustrates that optimality does not guarantee high explanatory power.

Can a Non-Optimal Forecast Have High R²?

Yes. A forecast that is biased or inefficient can still have high R² if it captures a large portion of the variance, even if it is not the best possible. For instance, a naive forecast that always predicts the sample mean will have zero R², but a forecast that overfits to historical data might achieve high in-sample R² but poor out-of-sample performance. More practically, a model that uses future information (in-sample) can have high R² but is not optimal for real-time forecasting.

Consider a sports analogy: a betting model that predicts game outcomes based on player social media activity might have high R² in training data but fail in live matches due to overfitting. Conversely, a simple model based on team rankings may be optimal (minimizing error) yet have low R² because games are unpredictable.

These examples show that statistical optimality (e.g., unbiasedness, minimum variance) is not synonymous with forecast quality measured by R². Practitioners must consider both aspects.

Maximum Likelihood Estimation for GARCH(1,1)

Question 4 covers MLE for a simple normal model and then for GARCH(1,1).

MLE for σ² with μ=0

Given iid normal data with mean zero, the log-likelihood is:

log L = -T/2 log(2π) - T/2 log(σ²) - (1/(2σ²)) Σ xt²

Maximizing with respect to σ² gives the estimator:

σ²_hat = (1/T) Σ xt²

This is the sample variance (without Bessel's correction).

Log-Likelihood for GARCH(1,1)

For GARCH(1,1) with conditional normal distribution: xt | Ft-1 ~ N(0, σt²), where σt² = ω + α xt-1² + β σt-1². The log-likelihood is:

log L = -T/2 log(2π) - 1/2 Σ [log(σt²) + xt²/σt²]

This is computed recursively: start with an initial σ1² (e.g., sample variance), then iterate for t=2,...,T.

Obtaining Estimates and Issues

To estimate (ω, α, β), maximize the log-likelihood numerically (e.g., using BFGS). Common issues include:

Stationarity constraints: α + β < 1, ω > 0, α ≥ 0, β ≥ 0.
Local maxima: The likelihood surface may have multiple peaks; try different starting values.
Non-normality: If returns have fat tails, quasi-MLE (using normal likelihood) still yields consistent estimates but standard errors need adjustment.
Initial values: Poor initial σ1² can affect early observations; use a large pre-sample or estimate as a parameter.

In practice, GARCH(1,1) is widely used for volatility forecasting in risk management and options pricing. For example, during the 2020 COVID crash, GARCH models captured volatility clustering, aiding in portfolio hedging.

Conclusion

This tutorial covered key concepts from the FN3142 assignment: CGARCH equivalence, VaR and ES derivation, forecast optimality, and GARCH MLE. Understanding these topics is essential for quantitative finance professionals. As of June 2026, with AI-driven trading and decentralized finance expanding, these foundational models remain relevant for risk assessment and volatility modeling.