Value at Risk — three ways
How much could this book lose on a bad day? VaR draws a line in the loss tail at confidence 95%. The parametric line assumes a normal curve, VaR = (z·σ·√h − μ·h)·V with z = 1.645 (one-tailed). The historical and Monte-Carlo methods read VaR off a simulated loss distribution instead. Under normality the three roughly agree; fat tails pull the historical figure deeper.
1-day 95% parametric VaRUS$3.29M
Parametric (normal) · largestUS$3.29M
Historical (fat tails)US$3.05M
Monte-Carlo (5,000 draws)US$3.27M
z (95%, one-tailed) 1.645Horizon σ√h 2.00%Largest VaR Parametric
Portfolio value VUS$100M
Daily volatility σ2.0%
Daily expected return μ+0.00%
Horizon h1 day
Confidence level c
The three estimates land near US$3.29M, but the historical figure of US$3.05M is about the same, since this sample happens not to bite past the Gaussian tail here. The largest of the three is Parametric.
Discuss: which method would you report to a regulator, and which to your own risk committee?