VaR using GARCH in MATLAB Calculator

An expert tool for estimating 1-day Value at Risk (VaR) using parameters from a GARCH(1,1) model, commonly implemented in MATLAB’s Econometrics Toolbox.

GARCH(1,1) VaR Calculator

Calculation Results

Results Analysis

Amortization Table: Projected Volatility and VaR over 10 days

Day (t)	Forecasted Variance (σ²_t)	Forecasted Volatility (σ_t)	Projected VaR (95%)

What is Calculating VaR using GARCH in MATLAB?

Calculating VaR using GARCH in MATLAB is a sophisticated financial risk management technique used to estimate the maximum potential loss a portfolio might face over a specific time horizon, within a given confidence level. Unlike simpler methods that assume constant volatility, this approach utilizes the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model to capture the reality of volatility clustering—the tendency for periods of high volatility to be followed by more high volatility, and calm periods by more calm. MATLAB, with its powerful Econometrics Toolbox, provides a robust environment for estimating GARCH model parameters and forecasting future volatility, which is the critical input for a more accurate Value at Risk (VaR) calculation.

This method is crucial for anyone involved in financial risk modeling because it acknowledges that risk is not static. The GARCH model provides a forward-looking estimate of volatility, which leads to a more dynamic and realistic VaR measure compared to historical simulation or simple variance-covariance methods. For a deep dive, consider our guide on GARCH model basics.

The GARCH(1,1) and VaR Formula Explained

The core of this technique lies in a two-step process: first, forecasting the next period’s variance using a GARCH model, and second, using that variance to calculate the VaR.

1. GARCH(1,1) Variance Forecast

The most common GARCH model, the GARCH(1,1), forecasts the conditional variance for the next period (t) as follows:

σ²_t = ω + α * ε²_{t-1} + β * σ²_{t-1}

This equation is fundamental to understanding how to go about calculating VaR using GARCH in MATLAB, as it forms the basis of the volatility forecast.

GARCH(1,1) Model Variables

Variable	Meaning	Unit	Typical Range
σ²_t	The forecasted variance for the next period (t).	Unitless (squared returns)	Positive value
ω (omega)	The long-run average variance component (constant term).	Unitless	Small positive value
α (alpha)	The ARCH term coefficient, reacting to last period’s market shock.	Unitless	0 to 1
ε²_{t-1}	The lagged squared residual (yesterday’s “surprise” squared).	Unitless	Positive value
β (beta)	The GARCH term coefficient, representing persistence in volatility.	Unitless	0 to 1
σ²_{t-1}	The lagged conditional variance (yesterday’s forecasted variance).	Unitless	Positive value

The sum of α and β determines the persistence of volatility shocks. A sum close to 1 implies that shocks are very persistent.

2. Value at Risk (VaR) Calculation

Once the next-period variance (σ²_t) is forecasted, the 1-day VaR is calculated using this formula:

VaR = Portfolio Value * Z-Score * σ_t

Where σ_t = sqrt(σ²_t) is the forecasted volatility (standard deviation), and the Z-Score corresponds to the desired confidence level (e.g., approximately 1.645 for 95% confidence assuming a normal distribution).

Practical Examples

Example 1: A Stable Market Day

Let’s assume a low-volatility scenario for a $1,000,000 portfolio.

• Inputs:
  • Portfolio Value: $1,000,000
  • GARCH Parameters (ω, α, β): 0.000002, 0.05, 0.94
  • Last Period’s Squared Residual (ε²_{t-1}): 0.00001 (low surprise)
  • Last Period’s Variance (σ²_{t-1}): 0.00008
  • Confidence Level: 99%
• Calculation Steps:
  1. Forecast Variance (σ²_t) = 0.000002 + 0.05 * 0.00001 + 0.94 * 0.00008 = 0.0000777
  2. Forecast Volatility (σ_t) = sqrt(0.0000777) ≈ 0.008815 or 0.88%
  3. Z-Score (99%): 2.326
  4. VaR = $1,000,000 * 2.326 * 0.008815 ≈ $20,505
• Result: The firm can be 99% confident that it will not lose more than approximately $20,505 in the next day.

Example 2: A Volatile Market Day

Now, consider a scenario after a significant market shock.

• Inputs:
  • Portfolio Value: $1,000,000
  • GARCH Parameters (ω, α, β): 0.000002, 0.12, 0.87 (higher alpha)
  • Last Period’s Squared Residual (ε²_{t-1}): 0.0009 (a 3% shock)
  • Last Period’s Variance (σ²_{t-1}): 0.0002
  • Confidence Level: 99%
• Calculation Steps:
  1. Forecast Variance (σ²_t) = 0.000002 + 0.12 * 0.0009 + 0.87 * 0.0002 = 0.000284
  2. Forecast Volatility (σ_t) = sqrt(0.000284) ≈ 0.01685 or 1.69%
  3. Z-Score (99%): 2.326
  4. VaR = $1,000,000 * 2.326 * 0.01685 ≈ $39,200
• Result: Following the shock, the 99% VaR estimate jumps to $39,200, reflecting the increased risk captured by the GARCH model.

How to Use This GARCH VaR Calculator

Using this calculator is straightforward if you have the output from a GARCH model estimation, which is a common task in quantitative finance.

1. Enter Portfolio Value: Input the total current value of your investment portfolio.
2. Input GARCH(1,1) Parameters: Enter the `omega` (Constant), `alpha` (ARCH), and `beta` (GARCH) coefficients. These are typically obtained by fitting a GARCH model to historical return data, for example, using the `garch` and `estimate` functions in MATLAB’s Econometrics Toolbox.
3. Enter Lagged Values: Input the most recent period’s squared residual (ε²_{t-1}) and conditional variance (σ²_{t-1}). These are also outputs from the GARCH estimation and forecasting process.
4. Select Confidence Level: Choose the desired confidence level for your VaR calculation (e.g., 95% or 99%).
5. Calculate and Interpret: Click “Calculate VaR”. The primary result is your 1-day Value at Risk. The intermediate values show the forecasted variance and volatility, which are key components of the calculating VaR using GARCH in MATLAB workflow.

Key Factors That Affect GARCH VaR

• Alpha (α) Parameter: A higher alpha means that volatility reacts more intensely to recent market news (shocks). This will cause VaR to spike more aggressively after a large price swing.
• Beta (β) Parameter: A higher beta indicates greater persistence in volatility. If beta is high, a shock will keep volatility (and thus VaR) elevated for a longer period.
• Sum of Alpha and Beta: The closer (α + β) is to 1, the longer the memory of past volatility shocks. This is a crucial metric for understanding risk persistence. See our analysis on time series forecasting for more.
• Current Volatility State: The calculation is path-dependent. If the previous day was highly volatile (high ε²_{t-1} and σ²_{t-1}), today’s VaR will be higher, all else being equal.
• Confidence Level: A higher confidence level (e.g., 99% vs. 95%) will always result in a higher VaR, as it covers a larger range of potential negative outcomes.
• Choice of Distribution: While this calculator assumes a Normal distribution for the innovations (Z-score), MATLAB allows fitting GARCH models with other distributions like the Student’s t-distribution. A t-distribution has fatter tails and would result in a higher VaR for the same parameters.

Frequently Asked Questions (FAQ)

1. Why use GARCH for VaR instead of historical simulation?

GARCH is forward-looking and conditional. It adapts to the current volatility environment. Historical simulation is unconditional and assumes the future will be a random sample of the past, which can be inaccurate if market conditions have changed.

2. How do I get the GARCH parameters in MATLAB?

You would use the Econometrics Toolbox. First, define a model `Mdl = garch(‘GARCHLags’,1,’ARCHLags’,1);`. Then, use `EstMdl = estimate(Mdl, returns);` where `returns` is your time series of asset returns. The estimated parameters will be in the `EstMdl` object.

3. What does it mean if α + β is greater than 1?

This implies a non-stationary, or “integrated,” GARCH (IGARCH) process. In this case, the volatility shocks are permanent, and the unconditional variance is not defined. The model suggests volatility will not revert to a long-run mean.

4. Are the inputs percentages or decimals?

All inputs for GARCH parameters and variances should be in decimal form. For example, a 1% return shock (residual) squared would be (0.01)^2 = 0.0001.

5. Can I use this for multi-day VaR?

Directly, no. This calculator provides a 1-day VaR. To calculate a T-day VaR, one would typically scale the 1-day VaR by the square root of time (e.g., VaR_10day ≈ VaR_1day * sqrt(10)). However, a more accurate method involves multi-step-ahead volatility forecasting, which is more complex.

6. Why is the Z-score used?

The Z-score (or quantile) translates the confidence level into a number of standard deviations. It assumes the standardized residuals (innovations) follow a standard normal distribution. For a 95% confidence level, we look for the point that leaves 5% in the tail.

7. What is the difference between conditional and unconditional variance?

Conditional variance (σ²_t) is the forecast for the next period, given today’s information. Unconditional variance is the long-term average variance that the model will revert to, calculated as ω / (1 – α – β).

8. Does this calculator account for mean-reversion in returns?

No, this is a pure GARCH VaR calculator focusing on volatility. A full model in MATLAB might include an ARMA component for the mean return (`arima` model with a `garch` variance component), but the VaR calculation is primarily driven by the volatility component.

Related Tools and Internal Resources

Expand your knowledge of econometric modeling and risk analysis with these related resources:

• Econometrics for Finance: An overview of key models and applications.
• Volatility Modeling Techniques: A comparison of ARCH, GARCH, EGARCH, and other models.
• Monte Carlo Simulation Guide: Learn another powerful method for risk analysis and VaR.
• Portfolio Optimization Strategies: Understand how risk metrics like VaR are used in constructing efficient portfolios.
• Introduction to Value at Risk: A foundational article on the different methods to calculate VaR.
• MATLAB Econometrics Toolbox Tutorial: A practical guide to getting started.