GARCH(1,1) with Student-t Innovations: VaR and Tail Risk

Worked Solution

How to Think About It: GARCH(1,1) captures volatility clustering -- big moves beget big moves. Using Student-$t$ innovations instead of Gaussian ones adds heavier tails, which is crucial for risk management because real return distributions have fatter tails than the normal. The key objects are: (1) the conditional variance $\sigma_t^2$ (which updates with each new return), (2) the likelihood (needed to estimate parameters), and (3) the VaR (a direct function of $\sigma_t$ and the $t$-quantile).

Quick Sanity Checks: The VaR should scale linearly with $\sigma_t$ (if volatility doubles, VaR doubles). As $\nu \to \infty$, the Student-$t$ converges to the normal, so the VaR should approach the Gaussian VaR. For small $\nu$, the tails get fatter and VaR increases.

Derivation:

Part 1: Conditional log-likelihood.

The standardized innovation is $z_t = r_t / \sigma_t$. The Student-$t$ density with $\nu$ degrees of freedom (scaled to unit variance) is:

$f(z; \nu) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right) \sqrt{\pi(\nu - 2)}} \left(1 + \frac{z^2}{\nu - 2}\right)^{-(\nu+1)/2}$

The conditional density of $r_t$ given the past is $f(r_t/\sigma_t; \nu) / \sigma_t$. The conditional log-likelihood for observation $t$ is:

$\ell_t = \ln \Gamma\!\left(\frac{\nu+1}{2}\right) - \ln \Gamma\!\left(\frac{\nu}{2}\right) - \frac{1}{2}\ln[\pi(\nu-2)] - \ln \sigma_t - \frac{\nu+1}{2} \ln\!\left(1 + \frac{r_t^2}{(\nu-2)\sigma_t^2}\right)$

The total log-likelihood is $\mathcal{L} = \sum_{t=1}^{T} \ell_t$, maximized over $(\omega, \alpha, \beta, \nu)$.

Part 2: One-step-ahead VaR.

The one-step-ahead forecast of $\sigma_{t+1}^2$ uses the GARCH recursion:

$\hat{\sigma}_{t+1}^2 = \omega + \alpha r_t^2 + \beta \sigma_t^2$

Since $r_{t+1} = \sigma_{t+1} z_{t+1}$ and $z_{t+1} \sim t_\nu$ (standardized), the $(1-\alpha_{\text{VaR}})$-VaR is the $\alpha_{\text{VaR}}$-quantile of the loss distribution:

$\text{VaR}_{1-\alpha_{\text{VaR}}} = -\hat{\sigma}_{t+1} \cdot t_{\nu, \alpha_{\text{VaR}}}^{-1}$

where $t_{\nu, \alpha_{\text{VaR}}}^{-1}$ is the $\alpha_{\text{VaR}}$-quantile of the standardized Student-$t$ distribution (this is negative for $\alpha_{\text{VaR}} < 0.5$, so the VaR is positive).

Equivalently:

$\text{VaR}_{1-\alpha_{\text{VaR}}} = \hat{\sigma}_{t+1} \cdot |t_{\nu, \alpha_{\text{VaR}}}^{-1}|$

For example, for a 99% VaR ($\alpha_{\text{VaR}} = 0.01$), with $\nu = 5$, the quantile $|t_{5, 0.01}^{-1}| \approx 3.365$ vs. the Gaussian $|z_{0.01}| \approx 2.326$ -- the Student-$t$ VaR is about 45% larger.

Part 3: Tail risk and the $\nu \leq 4$ problem.

The Student-$t$ distribution with $\nu$ degrees of freedom has the $k$-th moment finite only if $k < \nu$. Specifically: - $\nu > 2$: variance exists (we need this for the GARCH model to make sense). - $\nu > 3$: skewness exists. - $\nu > 4$: kurtosis (and thus the fourth moment) exists.

How $\nu$ influences tail risk: Smaller $\nu$ means heavier tails and higher VaR/ES. The tail decays as $|r|^{-(\nu+1)}$ (power law), compared to the exponential decay of the normal. In practice, equity returns often give $\hat{\nu}$ in the range 4-8, meaning tails are much heavier than Gaussian.

Why $\nu \leq 4$ breaks ES: Expected Shortfall (Conditional VaR) is defined as:

$\text{ES}_{1-\alpha} = E[-r_{t+1} \mid r_{t+1} \leq -\text{VaR}_{1-\alpha}]$

This requires the conditional expectation of the tail to be finite. For the standardized Student-$t$, the ES integral involves $E[|z| \mid z < q]$, which converges only if $E[|z|] < \infty$ -- i.e., $\nu > 1$. However, the ES formula involves $E[z^2 \mid z < q]$ indirectly through the variance scaling $\sigma_{t+1}$, and the standard closed-form ES expression for the Student-$t$ is:

$\text{ES}_{\alpha} = \sigma_{t+1} \cdot \frac{f_\nu(t_{\nu,\alpha}^{-1})}{\alpha} \cdot \frac{\nu + (t_{\nu,\alpha}^{-1})^2}{\nu - 1}$

This formula is well-defined for $\nu > 1$, but the deeper issue is that when $\nu \leq 4$, the fourth moment of returns does not exist, which means: - Standard errors of ES estimates diverge (the variance of the ES estimator involves the fourth moment). - Backtesting ES becomes unreliable because the sampling distribution of tail averages has infinite variance. - Convergence of MLE for $(\alpha, \beta)$ parameters can be poor since the GARCH information matrix involves fourth moments of $z_t$.

In practice, when $\hat{\nu} \leq 4$, the ES point estimate may still be computable, but it is statistically unreliable -- you cannot meaningfully put confidence intervals on it.

Practical Interpretation: A trader using GARCH-$t$ for risk management should: - Always check $\hat{\nu}$. If $\hat{\nu}$ is close to 4, VaR estimates are still usable but ES estimates should be treated with skepticism. - Consider using a distribution with more controlled tail behavior (e.g., GED or skewed-$t$) if $\hat{\nu}$ is very low. - Remember that VaR scales as $\sigma_t \times |\text{quantile}|$, so the dynamic part is all in the GARCH volatility forecast.

Answer: The conditional log-likelihood is $\ell_t = \ln\Gamma(\frac{\nu+1}{2}) - \ln\Gamma(\frac{\nu}{2}) - \frac{1}{2}\ln[\pi(\nu-2)] - \ln\sigma_t - \frac{\nu+1}{2}\ln(1 + r_t^2/((\nu-2)\sigma_t^2))$. The one-step VaR is $\text{VaR} = \hat{\sigma}_{t+1} \cdot |t_{\nu,\alpha}^{-1}|$. Small $\nu$ fattens tails and inflates VaR; when $\nu \leq 4$, the fourth moment of returns is infinite, making ES statistically unreliable because its standard errors and backtesting statistics depend on moments that do not exist.