VaR and Expected Shortfall: three routes to one number

A risk report's VaR number looks like a single fact about a portfolio. It isn't — it's the output of a chosen method, and this module builds the same number three genuinely different ways on the same book: treat the P&L as linear in the risk factors, treat it as quadratic, or just resample it directly and read off the empirical tail. All three are "correct." They don't agree, and the gap between them is itself information about the book.

The whole story

Linear: the adjoint's sensitivity vector, squared

Delta-normal VaR treats the portfolio's P&L as linear in the risk-factor moves — s · x for sensitivities s and a Gaussian move x ~ N(0, Σ) — which makes the loss itself Gaussian, with a closed-form standard deviation:

double variance = 0.0;
for (int i = 0; i < n; i++) {
  for (int j = 0; j < n; j++) {
    variance += sensitivities[i] * covariance[i][j] * sensitivities[j];
  }
}
double sd = Math.sqrt(Math.max(variance, 0.0)) * Math.sqrt(horizonDays);
return Normal.inverseCdf(alpha) * sd;

s' Σ s is a quadratic form in the sensitivity vector, and multi-day figures scale the standard deviation by √horizonDays — not the variance — because variance itself is what scales linearly with time under i.i.d. daily moves.

Quadratic: the same eigensolver, a different job

A real book isn't linear — options carry gamma — so deltaGammaCornishFisher models the P&L as delta'x + 0.5x'Γx instead. Diagonalizing Σ^½ Γ Σ^½ turns that quadratic form into a sum of independent a_i y_i + 0.5 b_i y_i² terms, whose first four cumulants have closed forms:

Pca eig = Pca.of(m);              // m = Sigma^{1/2} Gamma Sigma^{1/2}
double[] b = eig.eigenvalues();
double[][] u = eig.loadings();

That's Pca — the same class 7.6's GARCH showcase used for a term- structure level/slope/curvature decomposition — repurposed here for a completely different job: diagonalizing a P&L quadratic form and, in matrixSqrt, computing Σ^½ itself. It's not the same eigensolver 8.1 used for Gauss-Hermite quadrature, though — that one was tqli, a QL algorithm with implicit shifts; Pca uses cyclic Jacobi rotations instead. Two independently-written symmetric eigensolvers in this codebase, each reused for more than the one thing it was originally built for.

The real run

VarEsShowcase builds a three-factor book — equity spot, a 5y rate, and EURUSD — with sensitivities {2.4M, −18M, 1.1M} and a gamma matrix that's short (negative) convexity in all three factors, at 99% one-day confidence:

route	1-day VaR
delta-normal	67,294
delta-gamma (Cornish-Fisher)	70,175
historical (4,000-day empirical)	67,945
expected shortfall (historical)	76,744

Delta-normal and historical land close together, as they should for a book that's only mildly non-linear. Delta-gamma sits meaningfully above both: short gamma in every factor produces negative skew (−0.108, computed alongside the VaR) — a fatter loss tail than a pure Gaussian approximation sees — and Cornish-Fisher's expansion pushes the quantile out to match. Expected shortfall, the mean loss beyond the 99% quantile, sits above every VaR figure, which it always must: it's an average over outcomes already worse than VaR, never a quantile itself. Ten-day delta-normal VaR comes out to 212,804 — exactly the one-day figure times √10, the scaling confirmed directly in source.

Try it yourself

Flip the sign of every entry in GAMMA (long gamma instead of short) and predict whether Cornish-Fisher VaR lands above or below delta-normal's 67,294 this time. (It flips to below: 64,427, with skewness flipping sign too, from −0.108 to +0.108 — a positively-convex book has a thinner loss tail than the Gaussian approximation sees, so the tail expansion pulls the quantile in instead of pushing it out.)

▶️ Run it

mvn -o -q -pl nablatensor-examples exec:java \
  -Dexec.mainClass=com.nablatensor.examples.VarEsShowcase -Dwindow=4000

Nothing here touches the tape or an engine string — every route in this module is host-side linear algebra and closed-form tail mathematics, once the sensitivity vector already exists.

⚠️ What this doesn't do

Delta-gamma Cornish-Fisher assumes the risk factors themselves move jointly Gaussian — real factor returns have their own fat tails the Cornish-Fisher expansion doesn't correct for, only the portfolio's own convexity does. The historical route here draws its 4,000-day sample from the same quadratic model being tested, not real historical data, so its agreement with delta-normal is a consistency check on the mathematics, not a claim about a real book's empirical tail. And this page stops at the number itself — whether 67,294 or 70,175 is the right one-day 99% VaR to report, and whether either one clears regulatory backtesting, is Module 9.2's question, not this one's.

What's next

→ Deeper: Value at Risk and Expected Shortfall has the full method table and the pinned test tolerances this page's numbers are checked against. → Next: Backtesting a VaR model — Module 9.2 asks whether any of these three numbers actually held up against what happened next.