4.1.2 Value at Risk

Basic Prerequisites: 01 · From Zero and Probability & Measure Theory.

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1. Intuition & Practical Objective

Value at Risk is the most-used and most-misused risk number in finance. This page gives its exact definition (including the quantile subtleties Artzner is careful about), then isolates the precise defect that motivated two decades of reform: VaR is not subadditive, and its acceptance set is not convex. The practical objective is to know exactly when a VaR limit is safe to use and when it will be gamed by the portfolio it is supposed to police.

The fault is not estimation sloppiness - it is structural. VaR is a quantile, and quantiles of sums do not respect quantiles of parts. Everything downstream (capital allocation, desk limits, decentralised risk governance) inherits the flaw.

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2. Mathematical Ground Truth & Derivations

2.1 Definitions (Artzner §3.3; Hull §22.1)

For a loss $L=-\Delta V$ over a horizon and confidence $\alpha \in (0,1)$:

$${\mathrm{VaR}}_{\alpha }(L)=\inf \{l\in \mathbb{R}:\mathbb{P}(L>l)\le 1-\alpha \}=F_L^{-1}(\alpha ).$$

Artzner’s net-worth form makes the capital interpretation explicit: for a future net worth $X$ and reference return $r$ (usually $r=1$ over a short horizon),

$${\mathrm{VaR}}_{\alpha }(X)=-\inf \{x\in \mathbb{R}:\mathbb{P}(X\le xr)>\alpha \}.$$

The quantile trap (Artzner Def. 3.2). Define the left/right quantiles $q_{\alpha }^{-}=\inf \{x:\mathbb{P}(X\le x)\ge \alpha \}$ and $q_{\alpha }^{+}=\inf \{x:\mathbb{P}(X\le x)>\alpha \}$. These differ on at most countably many $\alpha$ (where the CDF has a flat spot or a jump). VaR is pinned to $q_{\alpha }^{+}$. With discrete loss distributions (defaults, options, integer positions) the flat spots are common, and the choice of $q^{-}$ vs $q^{+}$ changes the number - a real reconciliation headache between desks.

2.2 The subadditivity counterexample (Artzner §3.3)

Artzner’s own example: two digital options on a stock with the same expiry. $A$ pays $1000$ if $S_T>U$; $B$ pays $1000$ if $S_T<L$ (with $L<U$). Choose $L,U$ so $\mathbb{P}(S_T<L)=\mathbb{P}(S_T>U)=0.008$. A trader writes two $A$‘s and two $B$‘s.

• Writing $2A$ alone: the $1\%$ VaR of the net worth is $-2u$ (essentially the premium - the $0.8\%$ chance of a $2000$ payout sits inside the $1\%$ quantile when it is the adverse tail; the point is the isolated positions look cheap).
• Writing $A+B$ together: now the union of the two tail events has probability $0.016$, and the $1\%$ VaR jumps to the positive number $1000-l-u$.

The consequence Artzner stresses: the set of acceptable net worths is not convex - “an even worse feature than the non-subadditivity of the measurement.” A non-convex acceptance set means the risk measure can have multiple local minima as a function of portfolio weights, so a VaR-constrained optimiser can be trapped away from the true safe portfolio (Rockafellar–Uryasev make the same point: VaR “can exhibit multiple local extrema”).

2.3 When does VaR look subadditive? (Artzner §3.3, Remark 1)

If all prices are jointly normal and exceedance probabilities are below $0.5$, then

$${\mathrm{VaR}}_{\alpha }(X)=-({\mathbb{E}}_P[X]+{\Phi }^{-1}(\alpha ){\sigma }_P(X)),$$

and ${\sigma }_{X+Y}\le {\sigma }_X+{\sigma }_Y$ (standard deviation is subadditive) so VaR is subadditive. This is a trap: it makes VaR look fine in Gaussian textbooks, while the failure appears precisely in the fat-tailed, discrete, and option-heavy portfolios that dominate real books. (Rockafellar–Uryasev: “VaR is coherent only when it is based on the standard deviation of normal distributions.”)

2.4 Two more first-principles failures (Artzner §3.3)

• Concentration blindness (credit). With zero base rate, $2\%$ bond spreads, $1\%$ independent default probability, the $5\%$ VaR of a $1m single-name position is -$20,000 (apparently riskless). Spreading across $100$ names makes $\mathbb{P}(\ge 2\text{ defaults})>0.18$, so the same money has a $>5\%$ chance of negative net worth: diversification increased the VaR. Meanwhile the pile-up in one name went undetected.
• Bad risk allocation. VaR can prefer a Pareto-dominated allocation of risks across two agents (Artzner’s $3$-state example): a capital level “found sufficient” for $X$ is “more than sufficient” after a risk exchange that all risk-averse agents dislike - VaR does not encourage sensible risk sharing.

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3. Computational Implementation - the counterexample, exactly and by Monte Carlo

Exact pmfs (no simulation noise), plus a Monte Carlo confirmation. Stdlib only.

The exact and Monte Carlo numbers agree; the Monte Carlo path also shows the estimator noise that any production VaR carries (see 05 · Failure Modes).

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4. Failure Modes & First-Principles Breakdowns

1. Subadditivity violation → aggregation failure. Risk cannot be decentralised: a supervisor cannot split one cash limit $m=m_1+m_2$ across two desks and trust that $\rho (X_1)+\rho (X_2)$ covers $\rho (X_1+X_2)$ (Artzner’s Axiom S motivation). With VaR the split can under-reserve the firm.
2. Non-convex acceptance set → non-convex optimisation. VaR-constrained portfolio optimisation has multiple local minima; solvers return arbitrary local optima. Rockafellar–Uryasev’s fix: optimise the convex $F_{\alpha }$ and read VaR off as a by-product ($\S 3.3$ of this hub, and 04 · ES).
3. Gaussian complacency. Because VaR is subadditive under normality, back-tests on normal data pass while the real (fat-tailed) book fails silently. Always stress VaR on discrete/jump/option payoffs.
4. Quantile-edge ambiguity. $q_{\alpha }^{-}\ne q_{\alpha }^{+}$ on discrete books; two desks reporting “the same” VaR can differ by a full position. Fix the convention ($q_{\alpha }^{+}$ per Artzner) in the risk policy.
5. Tail blindness. The single quantile is insensitive to the size of the worst outcomes - demonstrated numerically in 05 · Failure Modes.

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5. References

• Artzner, Delbaen, Eber & Heath, Coherent Measures of Risk (1999)
• Rockafellar & Uryasev, Optimization of Conditional Value-at-Risk, J. Risk 2(3):21–41 (2000)
• Hull, Options, Futures, and Other Derivatives
• Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk, 3rd ed. (2006)
• Kupiec, P., Techniques for Verifying the Accuracy of Risk Measurement Models, J. Derivatives 3(2) (1995)

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Coherent Risk Measures · 04 · Expected Shortfall
• Sibling: Parametric, Historical & Monte Carlo VaR · Credit Risk & the Merton Model