Value at Risk

Definition

Value at Risk (VaR) at confidence level $1-\alpha$ is the minimum loss $L_{\min}$ such that the probability of a loss exceeding $L_{\min}$ over a given holding period is at most $\alpha$ :

$\mathrm{VaR}_\alpha = \inf\{l : P(L > l) \leq \alpha\} = F_L^{-1}(1-\alpha)$

In practice for a portfolio with return $r_t$ , $\mathrm{VaR}_\alpha$ is the negative of the $\alpha$ -quantile of the return distribution: large $\mathrm{VaR}$ means more risk.

Key Ideas

VaR reduces the market risk of a portfolio to a single interpretable number: "with probability $1-\alpha$ , the loss will not exceed $L_{\min}$ ."
Under constant volatility, VaR is simply a fixed quantile of a static distribution — inadequate because empirical return volatility clusters and varies dramatically over time.
Generalized autoregressive conditional heteroskedasticity (GARCH)-based VaR uses the time-varying conditional distribution to compute a time-varying VaR estimate.
The Basel II/III regulatory framework uses VaR to set minimum capital requirements for market risk in bank trading books.

How It Works

GARCH-Based VaR

Under GARCH with conditionally normal errors, the conditional return distribution at time $t$ is $r_t | \mathcal{F}_{t-1} \sim \mathcal{N}(\mu_t, h_t)$ . The one-day VaR at level $\alpha$ is:

$\mathrm{VaR}_{t,\alpha} = -\mu_t + z_\alpha \sqrt{h_t}$

where $z_\alpha = \Phi^{-1}(1-\alpha)$ (e.g., $z_{0.01} = 2.326$ for 99% VaR). The key input is $h_t$ , estimated from the GARCH model. Because $h_t$ varies over time, so does VaR.

S&P 500 example (Nobel Committee 2003). Fitting GARCH(1,1) to daily S&P 500 returns (1995–2003):

$h_t = 2 \times 10^{-6} + 0.091\,\varepsilon_{t-1}^2 + 0.899\,h_{t-1}$

For a $1 million portfolio, the 99% one-day VaR ranges from:

$12,400 on September 1, 1995 (minimum conditional variance period)
$61,500 on July 31, 2002 (maximum conditional variance period)

The five-fold difference illustrates why time-varying volatility matters for risk management.

Dow Jones Industrial Average (DJIA)+bond portfolio example (Engle 2001). Fitting GARCH(1,1) to a mixed equity/bond portfolio (1990–2000) yields $\hat\alpha=0.0461$ , $\hat\beta=0.9461$ ( $\hat\alpha+\hat\beta=0.9922$ ) and long-run daily standard deviation (SD) $\hat{V}_L^{1/2}\approx1.07\%$ . The 1% VaR on a $1M portfolio ranged from $\approx$ $8,400 (calm periods) to $\approx$ $52,900 (crisis periods). Persistence of 0.9922 implies the half-life of a variance shock is $\ln(0.5)/\ln(0.9922)\approx88$ trading days — three times longer than the 30-day stylized half-life assumed by RiskMetrics.

IGARCH / EWMA

The integrated GARCH (IGARCH) (Engle and Bollerslev 1986) imposes $\alpha_1 + \beta_1 = 1$ , so shocks to variance are permanent. The special case $h_t = (1-\beta_1)\varepsilon_{t-1}^2 + \beta_1 h_{t-1}$ is an exponentially weighted moving average (EWMA) of past squared returns. This is the formula underlying the RiskMetrics methodology widely used by banks prior to the GARCH era; its simplicity (no intercept to estimate) made it practical for large portfolios. Under IGARCH, the unconditional variance does not exist, but conditional VaR forecasts are well-defined at short horizons.

Center vs. tails — a caution on EWMA for VaR (Alexander-Leigh 1997). Comparing the equally-weighted average, EWMA, and GARCH on equity and FX data, standard statistical criteria (which assess the center of the returns distribution) tend to favour EWMA at all but very short horizons. But VaR needs accuracy in the tails, and operational evaluation — backtesting under the BIS traffic-light rules — reverses the verdict: equally-weighted and GARCH forecasts sit in the "green zone" for almost all major equity and USD-FX series, whereas EWMA produces an unacceptably high number of exceptions (especially for FX). The lesson: a volatility model that fits the middle of the distribution best is not necessarily the safest for VaR. The paper also shows how the large covariance matrices needed for firm-wide risk systems can be built by combining univariate volatility forecasts with orthogonalization (PCA) of the risk factors — the idea developed into orthogonal/principal-component GARCH.

Multi-Step VaR

For $h$ -day VaR, one simulates or approximates the distribution of cumulative returns $\sum_{s=1}^{h}r_{t+s}$ . Under GARCH, the multi-step conditional variance has a closed-form recursion at horizon $h$ :

$E[h_{t+h}|\mathcal{F}_t] = \bar\sigma^2 + (\alpha_1+\beta_1)^{h-1}(h_t - \bar\sigma^2)$

where $\bar\sigma^2 = \omega/(1-\alpha_1-\beta_1)$ is the unconditional variance. For $\alpha_1+\beta_1$ close to 1 (high persistence), the conditional forecast mean-reverts slowly, so the GARCH-VaR remains elevated for many days following a volatility spike.

The square-root-of-time rule and its failure (Danielsson-Zigrand 2006)

In practice banks and the Basel Accords often scale a one-day VaR quantile to an $h$ -day horizon by the square-root-of-time rule, $\mathrm{VaR}_h \approx \sqrt{h}\,\mathrm{VaR}_1$ . This is exact only under IID normal returns. Daníelsson and Zigrand study time-scaling of quantiles when returns follow a jump diffusion — a natural model for the systemic risk the Basel capital rules exist to address — and show the $\sqrt{h}$ rule systematically underestimates* risk, with the under-estimation worsening as the horizon, the jump intensity, or the confidence level rises. So a rule adopted for prudence is, precisely in the tail and long-horizon regime that matters for capital adequacy, systematically imprudent.

Historical Simulation VaR — Pathology and Limits (Andersen et al. 2004)

Historical simulation (HS-VaR) computes the $p$ -th percentile of the empirical distribution of the most recent $T$ portfolio returns (typically $T = 250$ trading days), without any parametric model:

$\widehat{\mathrm{VaR}}_{t,p} = -r_{(\lceil Tp \rceil)}$

where $r_{(1)} \leq r_{(2)} \leq \cdots \leq r_{(T)}$ are ordered pseudo returns. HS-VaR is simple and assumption-free, but has two fundamental problems:

No conditionality. The estimate is the same whether yesterday was calm or a crisis. Pritsker (2001) and Andersen et al. (2004) demonstrate via simulation that the nominal 1% HS-VaR can have a true conditional coverage probability as high as 10% on any day following a volatility spike. After the October 1987 crash, the 1% HS-VaR barely moved because the second- and third-worst historical returns (which govern the 1st-percentile estimate) were largely unchanged.
No multi-day term structure. Scaling 1-day HS-VaR by $\sqrt{h}$ is only valid if returns are independent and identically distributed (i.i.d.) — the very assumption violated by volatility clustering. A 10-day 1% VaR from 250 daily observations would require only the 2–3 most extreme observations and is highly imprecise.

Filtered Historical Simulation (FHS)

Filtered Historical Simulation (Barone-Adesi, Bourgoin, and Giannopoulos 1998; Diebold et al. 1998; Hull and White 1998) combines a parametric variance model with nonparametric bootstrap on the standardized innovations:

Filter: estimate a GARCH or other volatility model; compute standardized returns $\tilde{r}_t = r_t / \hat\sigma_t$
One-day VaR: use the order statistic of $\{\tilde{r}_t\}$ times the current volatility forecast: $\widehat{\mathrm{VaR}}_{T+1,p} = -\hat\sigma_{T+1} \cdot \tilde{r}_{(\lceil Tp \rceil)}$
Multi-day VaR: simulate paths forward by drawing $\tilde{r}^*_{T+s}$ with replacement from $\{\tilde{r}_t\}$ , applying the GARCH recursion $\hat\sigma^2_{T+s} = \hat\omega + \hat\alpha({\tilde{r}^*_{T+s-1}}\hat\sigma_{T+s-1})^2 + \hat\beta\hat\sigma^2_{T+s-1}$ , and computing portfolio value.

FHS retains the distributional flexibility of HS (no parametric shape assumed for $\tilde{r}$ ) while inheriting the conditionality of GARCH (today's volatility governs tomorrow's risk). In the multivariate case, standardized returns $\tilde{z}_t = P_t^{-1} r_t$ (where $P_t$ is the Cholesky factor of the estimated covariance matrix $\Sigma_t$ ) are resampled as vectors to preserve constant cross-sectional dependence patterns including non-normal tails.

Log-Normal / Normal Mixture Model

Andersen, Bollerslev, Diebold, and Labys (2001) establish that returns standardized by realized volatility $r_t / \sqrt{RV_t}$ are approximately $\mathcal{N}(0,1)$ , whereas GARCH-standardized returns still show fat left tails (see Figure 6 vs. Figure 7 in Andersen et al. 2004). This motivates a mixture model for the conditional return distribution:

$\log(\sigma_t)\,|\,\mathcal{F}_{t-1} \sim \mathcal{N}(\mu_t, \tau^2), \qquad r_t\,|\,\sigma_t \sim \mathcal{N}(0,\sigma_t^2)$

Integrating out $\sigma_t$ gives a scale mixture of normals with fat tails. ABDL apply this to foreign exchange (FX) returns and find it produces well-calibrated VaR coverage rates. In the multivariate case with realized covariance matrices, the multivariate analogue applies directly when realized covariances are available for all assets.

Extreme Tail Correlations (Longin-Solnik 2001)

The bivariate normal distribution implies that correlations between extreme observations converge to zero as the threshold defining "extreme" grows. Empirically, equity return correlations between large negative returns are much higher than normal, not lower. Longin and Solnik (2001) show for major international equity indices that extreme-positive-return correlations taper off toward zero (consistent with normality), while extreme-negative-return correlations remain elevated even for tail thresholds beyond two standard deviations.

This asymmetric tail dependence means that VaR models based on normal multivariate distributions systematically underestimate portfolio tail losses: in a market crash, asset correlations rise precisely when diversification is most needed, and the normal-model-based VaR ignores this. Copula-based models (e.g., Clayton copula for lower tail dependence) or multivariate extreme value theory can capture this pattern, though scalability to large portfolios remains a challenge.

CAViaR (Engle and Manganelli 1999)

An alternative approach avoids specifying the full return distribution by directly modeling the conditional $\alpha$ -quantile $q_{t,\alpha} = Q_\alpha(r_t|\mathcal{F}_{t-1})$ as an autoregressive process:

$q_{t,\alpha} = \beta_0 + \beta_1 q_{t-1,\alpha} + \beta_2 g(r_{t-1})$

where $g(\cdot)$ is a function of lagged returns (e.g., $|r_{t-1}|$ or $\max(r_{t-1}, 0)$ ). Estimation uses quantile regression (minimizing the asymmetric check loss). Conditional autoregressive value at risk (CAViaR) requires no distributional assumption — it is consistent for the true conditional quantile even if the return distribution is non-normal.

Why It Matters

VaR is the industry and regulatory standard for quantifying market risk; GARCH provides its most rigorous foundation.
Time-varying VaR reveals that capital requirements should be counter-cyclical: higher during turbulent periods, lower during calm periods.
The gap between IGARCH-based (RiskMetrics) and GARCH-based VaR narrows when volatility is very persistent; when persistence is lower, the GARCH model's mean-reversion prevents indefinite risk escalation.

Open Questions

VaR is not a coherent risk measure (it lacks sub-additivity): a diversified portfolio can have higher VaR than the sum of its components in some scenarios (Artzner-Delbaen-Eber-Heath 1999). Expected Shortfall (conditional value at risk (CVaR)) is coherent and has replaced VaR in some Basel III applications.
Extreme-event risk (tail events beyond the 99th percentile) is poorly characterized by GARCH with normal innovations; heavy-tailed distributions (Student-t, generalized error distribution (GED)) improve coverage.