Quantitative Methods & Risk·Value at Risk & Stress Testing
What is Value at Risk?
The single number every desk reports
Value at Risk — VaR — answers a question every risk committee asks: 'How much could we lose, on a really bad day, with X% confidence?' It is the single number that summarizes a portfolio's downside exposure over a horizon, at a confidence level. The 95% one-day VaR of a portfolio is the loss that is exceeded only 5% of the time.
What VaR is — and isn't
VaR tells you the loss threshold at a confidence level. It does NOT tell you how bad the worst case is when you cross that threshold. For that, you need Expected Shortfall (CVaR) or scenario stress testing.
Parametric VaR (single asset)
VaR(α) = μ + σ · Φ⁻¹(α) · √Δt · V
Three flavors, three trade-offs
Closed-form
Assumes returns are normally distributed. Cheap to compute, transparent, but underestimates tail risk badly when returns are fat-tailed (which they almost always are in markets).
VaR's blind spot
VaR is silent on what happens past the threshold. A 95% VaR of $1M doesn't distinguish between 'lose $1M to $2M' and 'lose $1M to $50M.' Always pair VaR with a tail metric (CVaR / Expected Shortfall) and discrete scenario stress tests.
Knowledge check
A portfolio has a 1-day 95% VaR of $500k. Which statement is correct?