1. Definition

  • Value-at-Risk (VaR) is a quantile-based risk measure that estimates the worst expected loss over a given time horizon at a specified confidence level.

Formally:
For confidence level $\alpha$ (e.g., 95% or 99%), $VaR_\alpha = – \inf \{x : P(L \leq x) \geq \alpha \}$

where $L$ = loss distribution.

Intuition:

  • “At most, we expect to lose VaR amount with probability $1-\alpha$.”

2. Example

  • A bank holds a portfolio.
  • 1-day 95% VaR = \$10M.
  • Interpretation:
    • With 95% probability, daily losses will not exceed \$10M.
    • With 5% probability, losses may exceed \$10M.

3. Key Components

  • Confidence level: usually 95% or 99%.
  • Time horizon: e.g., 1 day, 10 days.
  • Loss distribution: estimated using historical data, variance–covariance, or simulation.

4. Methods to Calculate VaR

Parametric (Variance–Covariance) Method

  • Assume returns are normally distributed.
  • If portfolio return $N(\mu, \sigma^2)$, $VaR_\alpha = \mu – z_\alpha \sigma$ where $z_\alpha$​ is the standard normal quantile (e.g., 1.645 for 95%).

Historical Simulation

  • Use actual historical returns.
  • VaR = empirical quantile of losses at level $1-\alpha$.

Monte Carlo Simulation

  • Simulate many possible portfolio return paths from a model.
  • Compute loss distribution and take quantile.

5. Limitations

  • VaR gives only a threshold, not the size of losses beyond it.
  • Example: if VaR = \$10M, it does not tell you if the actual loss could be \$11M or \$100M.
  • Not subadditive → risk of portfolio may appear larger than sum of components.
  • Led to development of Expected Shortfall (CVaR), which looks at the average loss beyond VaR.

6. Applications

  • Banking regulation: Basel Accords use VaR for capital requirements.
  • Risk management: portfolio risk, trading desks.
  • Stress testing: understanding downside risk exposure.

Summary:

  • VaR (Value-at-Risk) = quantile of loss distribution at a chosen confidence level.
  • Answers: “How much could I lose, with X% confidence, over Y time horizon?”
  • Calculated via parametric, historical, or simulation methods.
  • Limitation: ignores losses beyond threshold → often complemented by Expected Shortfall (CVaR).