Value at Risk (VaR) and Expected Shortfall (ES)

One-line definition

VaR: the loss not expected to be exceeded over a horizon at a given confidence level. ES: the average loss given that VaR is breached.

Formula (parametric / Gaussian)

VaR = PortfolioValue * sigma * z

ES = PortfolioValue * sigma * phi(z) / (1 - confidence)

where z = norm.ppf(confidence), phi is the standard normal density, and sigma is portfolio volatility over the same horizon as the desired VaR.

Units

Currency, over the same horizon as sigma (e.g. daily volatility gives daily VaR).

Why PMs care

VaR/ES turn a distribution of possible outcomes into a single risk-limit number a mandate or regulator can cap.

Worked example

PortfolioValue = $1,000,000, daily sigma = 2%, confidence = 95%:

z ≈ 1.6449 → VaR ≈ $32,897

ES ≈ $41,254 (always larger than VaR at the same confidence — it's an average over the worse tail, not a single percentile).

Common mistakes

• treating VaR as the worst-case loss — by construction, 5% of outcomes (at 95% confidence) are worse than VaR
• mixing volatility and confidence horizons/periods that don't match
• ignoring fat tails: real return distributions breach parametric VaR more often than the Gaussian assumption predicts

Limitations

This is the parametric (delta-normal) method only. It assumes returns are normally distributed with no autocorrelation — historical simulation and Monte Carlo VaR relax that assumption but aren't implemented here.

Related

• Portfolio volatility
• Stress testing