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GARCH-Based VaR

Asset returns show volatility clustering: calm days follow calm days and violent days cluster together. A GARCH(1,1) model captures this by letting today’s conditional variance depend on yesterday’s squared shock and yesterday’s variance. Crucially, GARCH forecasts the conditional variance only. To turn that into VaR you still need two more ingredients: a chosen confidence quantile and an assumed distribution for the standardised innovations (normal or, better for fat tails, Student-tt). VaR is then the quantile multiplier times the GARCH conditional sigma. This makes GARCH-VaR a fully dynamic measure whose risk number rises and falls with the forecast volatility, addressing the fat-tails and clustering that break a static normal VaR.

Why it matters

Volatility is sticky, so the best guess for tomorrow’s turbulence is a blend of today’s surprise and today’s volatility level, which is exactly what GARCH encodes. But a variance forecast is not a loss quantile. GARCH hands you sigma; you still choose how far into the tail to go (the confidence level) and what shape the standardised shocks have (normal or Student-tt). Only after layering a quantile and a distribution onto the forecast sigma do you get a VaR number. Forgetting that last step is the classic error.

Formulas

GARCH(1,1) conditional variance
ht=ω+α εt−12+β ht−1h_t = \omega + \alpha\,\varepsilon_{t-1}^{2} + \beta\,h_{t-1}
ω>0\omega>0, α,β≥0\alpha,\beta\ge 0. The term αεt−12\alpha\varepsilon_{t-1}^2 is the reaction to yesterday’s shock; βht−1\beta h_{t-1} is persistence. This forecasts variance only, not a quantile.
VaR from the GARCH sigma
VaRα,t=− ⁣(μ+qα ht )\mathrm{VaR}_{\alpha,t} = -\!\left(\mu + q_{\alpha}\,\sqrt{h_t}\,\right)
where qαq_{\alpha} is the α\alpha-quantile of the assumed innovation distribution (e.g. −1.645-1.645 normal at 95%, or the heavier Student-tt quantile). The conditional sigma ht\sqrt{h_t} comes from GARCH; the quantile and distribution are added on top.
Volatility persistence
α+β<1\alpha + \beta < 1
Stationarity requires α+β<1\alpha+\beta<1. Values near 1 (often above 0.95 for daily equities) mean shocks decay slowly, so high volatility persists for many days.

Worked examples

Scenario

A daily GARCH(1,1) fit gives ω=0.000002\omega = 0.000002, α=0.08\alpha = 0.08, β=0.90\beta = 0.90. Yesterday’s return shock was εt−1=−0.03\varepsilon_{t-1} = -0.03 and yesterday’s variance was ht−1=0.0004h_{t-1} = 0.0004. Find today’s 95% VaR on a $10m position under a normal innovation assumption.

Solution

First the variance: ht=0.000002+0.08(0.032)+0.90(0.0004)=0.000434h_t = 0.000002 + 0.08(0.03^2) + 0.90(0.0004) = 0.000434, so ht=0.0208\sqrt{h_t} = 0.0208 (about 2.08% daily). Then apply the 95% normal quantile to the $10m position: VaR=1.645×0.0208×10 m\mathrm{VaR} = 1.645 \times 0.0208 \times 10\,\text{m}, which is about $0.343m. The GARCH step delivered sigma; the normal 95% quantile of 1.645 and the normal assumption converted it to VaR.

Common mistakes

  • ✗GARCH outputs a VaR directly. GARCH forecasts the conditional variance only; you still must apply a confidence quantile and an assumed innovation distribution to obtain VaR.
  • ✗GARCH-VaR removes the need to assume a distribution. The standardised shocks still need a distribution (normal or Student-tt); the choice changes the tail quantile and hence the VaR.
  • ✗A normal-innovation GARCH fully solves fat tails. GARCH captures volatility clustering, but standardised residuals are often still fat-tailed, so a Student-tt innovation is frequently needed for the extreme quantiles.
  • ✗The parameters α\alpha and β\beta can sum above one without consequence. Covariance stationarity requires α+β<1\alpha + \beta < 1; otherwise the variance forecast is non-stationary and explodes.

Revision bullets

  • •Returns cluster in volatility; GARCH(1,1) models conditional variance
  • •h_t = omega + alpha * shock^2 + beta * h_{t-1}
  • •GARCH forecasts variance ONLY, not a loss quantile
  • •VaR needs a confidence quantile plus an assumed innovation distribution on the GARCH sigma
  • •Stationarity requires alpha + beta < 1; near 1 means persistent volatility

Quick check

A GARCH(1,1) model directly provides which quantity for VaR?

In ht=ω+αεt−12+βht−1h_t = \omega + \alpha\varepsilon_{t-1}^2 + \beta h_{t-1}, what does a value of α+β\alpha + \beta close to 1 indicate?

Connected topics

VaR DefinitionVaR Time ScalingFat TailsDynamic VaR

Sources

  1. Bollerslev (1986)
    Bollerslev, T. "Generalized Autoregressive Conditional Heteroskedasticity." Journal of Econometrics, 31(3), 307-327, 1986.
    Introduces the GARCH model of time-varying conditional variance underlying GARCH-VaR.
  2. Van Nguyen et al. (2022)
    Van Nguyen, P., et al. "Stock liquidity and firm value: evidence from an emerging market." 2022.
    Dr. Nguyen’s emerging-market volatility and liquidity work motivating conditional-volatility risk measures on the VNIndex.
  3. Jorion (2007), Ch. 9
    Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007.
    Covers GARCH volatility forecasting and its use in time-varying VaR.
How to cite this page
Dr. Phil's Quant Lab. (2026). GARCH-Based VaR. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-garch-var
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