Value at Risk is the loss you will not exceed with 95% confidence — and the reason it can lie. VaR at level α is the α-quantile of the loss distribution. It is easy to read, and it is not a coherent risk measure: it can punish diversification and it says nothing about how bad the tail beyond it gets. Expected Shortfall — the average loss past VaR — repairs both. Rendered, not quoted. (A risk model with known blind spots, not investment advice.)
Losses L have a distribution. Fix a confidence α=0.95.
VaRα(L) = inf{ l : P(L ≤ l) ≥ α } — the smallest loss the tail cannot beat with probability 1−α.
ESα(L) = VaRα + &frac1;⁄(1−α) · E[(L−VaRα)+] — the mean loss inside that tail (Rockafellar–Uryasev CVaR).
For a Normal loss L~N(μ,σ²) the quantile is closed form: VaR0.95 = μ + 1.645σ. assumes Normal
RiskMetrics (1994) gave the desk one number: how much could I lose, with 95% confidence? — a quantile of loss.
Five years later Artzner et al. (1999) named the four axioms a risk measure should obey (monotonicity, translation-invariance, positive homogeneity, subadditivity) and proved a quantile fails the last one. Expected Shortfall is coherent — the neighbouring sphere to this quantile.
Neighbour: the-coherent-measure · the four axioms as the wall this sphere leans on.
Live re-check of the coherence claim the panel prints. Green while the counterexample stands; flips red the instant the tamper (window 6) forces the false claim that VaR is subadditive.
A loss position. Normal for the picture: mean μ and volatility σ. Plus a fixed two-bond counterexample the closed form cannot see.
ES ≥ VaR always: the shaded tail can only sit further out than its own cutoff.
| two i.i.d. bonds | VaR95 | ES95 |
|---|---|---|
| A (loss 100 w.p. 0.045) | 0 | 90.00 |
| B (identical, independent) | 0 | 90.00 |
| A + B (pooled) | 100 | 104.05 |
| sum of parts | 0 | 180.00 |
proving…
VaR is a threshold; ES is the mean beyond it. The threshold is legible and incoherent; the mean beyond it is coherent and dearer to compute. Models, not recommendations.
VaR names a cutoff and then goes blind. Past that line it does not care whether you lose 10 or 10 million — same VaR. It also assumes the loss shape you fed it: Normal VaR presumes constant volatility, log-normal / thin tails, no jumps, frictionless markets. Real crashes are none of these. assumed
“VaR measures risk, so lower VaR is safer.”
→ VaR is not subadditive: pooling two positions can raise it (0 + 0 → 100 below). It can penalise diversification.
“Two books with the same 95% VaR carry the same risk.”
→ Same VaR, tail of 10 vs 1,000,000. VaR cannot tell them apart; ES multiplies them apart.
“VaR is a worst case.”
→ It is the best of the worst 5% — the tail’s near edge, never its floor.
Planted void, disclosed: force the panel to assert VaR is always subadditive — VaR(A+B) ≤ VaR(A)+VaR(B). The two-bond counterexample (100 > 0) contradicts it, and the witness in window 7 catches it live.