Value at Risk (VaR)

Value at Risk (VaR) is a statistical measure that quantifies the level of financial risk within a portfolio over a specific time frame. It estimates the maximum expected loss at a given confidence level.

How to Read VaR: "95% 1-day VaR of $10,000" means:

• 95% of the time, daily loss won't exceed $10,000
• 5% of the time (1 in 20 days), loss could exceed $10,000

VaR Parameters:

• Confidence Level: Usually 95% or 99%
• Time Horizon: 1 day, 1 week, 1 month
• Dollar Amount or Percentage

Calculation Methods:

1. Historical Method:

• Use actual past returns
• Find the loss at the relevant percentile
• Simple but assumes history repeats

2. Parametric (Variance-Covariance):

• Assumes normal distribution
• VaR = Portfolio Value × Z-score × σ × √t
• Faster but may underestimate tail risk

3. Monte Carlo Simulation:

• Generate thousands of scenarios
• Most flexible but computationally intensive

Example: $1,000,000 portfolio, 15% annual SD, 95% 1-day VaR: VaR = $1M × 1.65 × 0.15 × √(1/252) = $15,575

95% confident daily loss won't exceed ~$15,575.

Limitations:

• Says nothing about losses beyond VaR
• Assumes normal distribution (fat tails exist)
• Backward-looking
• Different methods give different answers
• Can give false sense of precision

Conditional VaR (CVaR): Also called Expected Shortfall—average loss WHEN VaR is exceeded.