Value-at-Risk

Definition and Concept

Value-at-Risk (VaR) measures the maximum expected loss of a portfolio or asset over a defined time period at a given confidence level \(\alpha\). It quantifies the point beyond which losses are unlikely to occur. Mathematically, VaR is defined as the \(\alpha\)-quantile of the loss distribution:

Equivalently, using the quantile (generalized inverse) function:

Historical context

VaR emerged in the 1970s and became widely adopted in the 1990s through J.P. Morgan's RiskMetrics framework. It was later integrated into Basel II as a standardized measure for market risk and capital adequacy.

Limitations

• Often assumes normality or simple parametric forms, which can underestimate extreme losses (fat tails).
• Reports only the probability of exceeding a threshold, not the magnitude of losses beyond it.
• VaR lacks subadditivity in general (non-coherent): portfolio VaR can exceed the sum of individual VaRs.

Categories of VaR methods

Common families of VaR estimators include:

Assumes a known return distribution (e.g., Normal, Student-t, GARCH). Efficient and simple to compute. For normally distributed returns the formula is:

Combines theoretical models with empirical tail modeling (for example, Extreme Value Theory). Better tail capture but computationally more involved.

Relies solely on historical returns with no parametric assumptions. The empirical VaR is the appropriate order statistic:

Mathematical framework

Model uncertainty using a probability space \((\Omega,\mathcal{F},P)\). Let \(S_t\) denote the asset price at time \(t\) and define the log-return \(\displaystyle X_{t+1} = \ln\!\left(\frac{S_{t+1}}{S_t}\right)\).

The one-period loss may be written as

and, under a single-factor exponential pricing approximation, a simplified loss function is

Conditional vs. unconditional VaR

Unconditional VaR assumes stationarity of returns, whereas conditional VaR conditions on current information and recent volatility (e.g., GARCH). Empirical evidence (Mandelbrot, Fama, Engle) shows volatility clustering, so conditional models often provide more accurate, time-varying VaR estimates.

This section presents 11 methods for computing Value-at-Risk (VaR) at a 95% confidence level, divided into four main categories:

• Rolling Normal Methods (3): observation windows of 50, 100, and 252 days.
• Historical Simulation Methods (4): windows of 100, 150, 250, and 500 days.
• Normal GARCH Methods (3): windows of 70, 100, and 500 days.
• Optimized Student-t Method (1): window of 250 days.

Each method estimates potential losses under different statistical assumptions and time horizons.

1. Rolling Normal Method

Assumes log-returns \(X_t\) follow a normal distribution \(X_t \sim N(\mu_t,\sigma_t^2)\) with rolling mean and variance estimated over a window of length \(n\):

The VaR at confidence level \(\alpha\) (loss form) can be expressed as:

This model captures short-term volatility through rolling estimation windows.

2. Historical Simulation Method

A non-parametric approach relying on past data without assuming any distribution. Compute period losses via \(l(x)=1-e^x\), order them, and take the empirical quantile:

This method is simple, data-driven, and captures actual market behavior directly.

3. Normal GARCH (1,1) Method

Models volatility clustering with a time-varying variance process:

Parameters \(\alpha_0,\alpha_1,\beta_1\) are estimated by maximum likelihood. The GARCH model captures persistence in volatility and supports VaR forecasting across horizons (70–500 days).

4. Optimized Student-t Method

Assumes returns follow a Student-t distribution \(X_t \sim t(\mu_t,\sigma_t^2,\nu)\), where \(\nu\) denotes degrees of freedom (fat tails). Parameters are estimated by MLE and VaR is computed via the inverse Student-t CDF:

This model is robust to heavy-tailed returns and delivers more realistic risk estimates under turbulent market conditions.

In summary, the methods range from simple statistical models (Rolling Normal, Historical Simulation) to advanced volatility-based and heavy-tailed approaches (GARCH, Student-t), enabling a broad comparative evaluation of VaR performance under different market and statistical assumptions.