4.1 Value at Risk & Expected Shortfall

Basic Prerequisites: Probability & Measure Theory (CDFs, quantiles, expectation) and 01 · From Zero (this folder). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

This folder answers one question with two competing instruments: how much can we lose, and how bad is the part we did not plan for? Value at Risk (VaR) is a quantile - the loss level that is exceeded only $(1-\alpha )$ of the time. Expected Shortfall (ES, a.k.a. Conditional VaR / CVaR / Tail-VaR) is the average of the losses beyond that quantile - the answer to “and then what?”

This folder is a hub: it (a) gives the fast formula lookup and the coherence scorecard below (job #1 of this pillar), and (b) routes to six sub-pages that walk from raw intuition through the definition and its flaws, the coherence axioms, ES, the failure modes, and the spectral/regulatory extensions.

The one-sentence essence. “VaR tells you the door to the tail but not what is behind it; ES prices what is behind the door, and unlike VaR it is coherent - a merger of independent bets never looks riskier than the sum of its parts.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). Notation: $L=-\Delta V$ is the portfolio loss over horizon $\Delta t$; $\alpha \in (0,1)$ the confidence level; $F_L$ the loss CDF; $\Phi ,\phi$ the standard-normal CDF/density; $z_{\alpha }={\Phi }^{-1}(\alpha )$.

The coherence scorecard (Artzner et al. 1999, Def. 2.4). A risk measure $\rho$ is coherent iff it satisfies all four axioms. VaR satisfies three and fails subadditivity; ES satisfies all four.

Subadditivity failure in one line (after Artzner §3.3, made discrete; §3 of this hub). Two independent bonds each default with probability $4\%$ losing $100: ${\mathrm{VaR}}_{95\%}=0$ for each alone, but $1-0.96^2=7.84\%>5\%$, so the merged ${\mathrm{VaR}}_{95\%}=100$. A merger created measured risk from nothing.

Regulatory switch (Basel FRTB). The 1996 Amendment / Basel II.5 set market-risk capital on 10-day $99\%$ VaR, $\text{capital}=k\cdot \mathrm{VaR}$ with multiplier $k\ge 3$ (Hull BS 22.1). The Fundamental Review of the Trading Book (BCBS 2019, d457) replaces VaR with $97.5\%$ Expected Shortfall - chosen so that, in the normal benchmark, ${\mathrm{ES}}_{97.5\%}\approx {\mathrm{VaR}}_{99\%}$ while capturing the tail VaR ignores.

3. Computational Implementation - the formula engine

Standard library only ($\Phi$ from math.erf; a rational ${\Phi }^{-1}$). Reproduces every number above.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Tail blindness. VaR is a single quantile: it is silent about the magnitude of losses beyond it (change the worst loss from $-3$ to $-15$ and a $99\%$ VaR does not move - §5/§3). Optimizing against a VaR constraint therefore rewards selling deep-OTM tails.
2. Non-coherence / non-convexity. VaR violates subadditivity and its acceptance set is non-convex; it can have multiple local extrema in portfolio weights (Rockafellar–Uryasev), making it hazardous to optimize and unsafe to decentralise.
3. Estimation error. Both measures must be estimated; $99\%$ VaR from a 250-day window has a sampling standard deviation of order $0.23\sigma$, and empirical ES is noisier still because it averages a handful of tail points.

5. References

• Artzner, Delbaen, Eber & Heath - Coherent Measures of Risk, Mathematical Finance 9(3):203–228 (1999). Defines the four coherence axioms (Def. 2.4), proves VaR fails subadditivity (§3.3 with the digital-option and normal-distribution remarks), and proposes tail conditional expectation (§5.1)
• Rockafellar & Uryasev - Optimization of Conditional Value-at-Risk, Journal of Risk 2(3):21–41 (2000). Introduces the convex function $F_{\alpha }(x,\zeta )=\zeta +\frac{1}{1-\alpha }\mathbb{E}[(L-\zeta {)}^{+}]$ whose minimum is CVaR and whose minimiser is VaR (Thms 1–2), reducing CVaR optimisation to linear programming. Read in full from the corpus PDF.
• Acerbi & Tasche - On the Coherence of Expected Shortfall, J. Banking & Finance 26(7):1487–1503 (2002). ES is coherent under general (non-normal) distributions; the formal justification for Basel’s shift to ES.
• McNeil & Frey - Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: An Extreme Value Approach, J. Empirical Finance 7(3–4):271–300 (2000). Conditional/dynamic VaR-ES via GARCH + EVT
• Hull - Options, Futures, and Other Derivatives
• McNeil, Frey & Embrechts - Quantitative Risk Management (2015)

6. Connected Graph Bridges

• Foundational base: Probability & Measure Theory · 01 · From Zero
• Sub-pages (in-folder): 01 From Zero · 02 VaR Definition & Flaws · 03 Coherent Risk Measures · 04 Expected Shortfall · 05 Failure Modes · 06 Advanced Extensions
• Sibling topics: Parametric, Historical & Monte Carlo VaR · Extreme Value Theory & Fat Tails · Stress Testing & Scenario Analysis
• Forward: Portfolio Risk Constraints & Mean–Variance

Beginner: start at 01 · Practitioner: start at 05