4.1.6 Advanced Extensions

Basic Prerequisites: 04 · Expected Shortfall and Multivariable Calculus (Euler’s homogeneous-function theorem).

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1. Intuition & Practical Objective

ES is not an isolated fix - it is the most conservative member of a whole family: the spectral risk measures. This page (a) places ES inside that family, (b) gives the Euler allocation that turns a portfolio ES into additive per-desk risk contributions (the basis of modern risk budgeting), and (c) traces the Basel FRTB switch from $99\%$ VaR to $97.5\%$ ES. The objective is to move from “which single number” to “which weighting of the tail,” and to see the regulatory consequences of that choice.

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2. Mathematical Ground Truth & Derivations

2.1 Spectral risk measures (Acerbi 2002)

A spectral risk measure is a weighted average of the VaR quantiles with a fixed, non-decreasing weight function $\phi :[0,1]\to {\mathbb{R}}_{\ge 0}$:

$$\boxed{M_{\phi }(L)={\int }_0^1\phi (u){\mathrm{VaR}}_u(L)du,{\int }_0^1\phi (u)du=1,\phi \text{non-decreasing}.}$$

Coherence ⇔ non-decreasing $\phi$. The monotonicity of $\phi$ encodes aversion to tail events: the worse the quantile (higher $u$), the more weight it receives. A flat $\phi \equiv 1$ gives the mean; a decreasing $\phi$ would reward tail risk and breaks subadditivity.

ES is the extreme spectral measure. Take

$${\phi }_{\mathrm{ES}}(u)=\frac{1}{1-\alpha }{1}_{\{u>\alpha \}}.$$

This is non-decreasing (a step), integrates to $1$, and gives exactly

$$M_{{\phi }_{\mathrm{ES}}}(L)=\frac{1}{1-\alpha }{\int }_{\alpha }^1{\mathrm{VaR}}_u(L)du={\mathrm{ES}}_{\alpha }(L).$$

So ES is the spectral measure that assigns zero weight to everything below the $\alpha$-quantile and uniform weight above it - the most tail-concentrated member of the family.

Kusuoka representation. Every law-invariant coherent risk measure is a mixture (supremum/convolution) of expected shortfalls at different levels:

$$\rho (L)=\underset{\mu \in \mathcal{M}}{\sup }{\int }_{[0,1]}{\mathrm{ES}}_{\alpha }(L)d\mu (\alpha ),$$

so ES is the atomic building block of all law-invariant coherent risk measures - the theoretical reason it is the canonical replacement for VaR.

2.2 Euler allocation (risk budgeting)

A positively homogeneous (Axiom PH) differentiable measure satisfies Euler’s theorem:

$$\rho (L)=\sum _{i=1}^n{w}_i\frac{\partial \rho }{\partial w_i},$$

so the terms ${\mathrm{EC}}_i=w_i\partial \rho /\partial w_i$ are additive risk contributions (“Euler allocation”, “component CVaR”, “expected shortfall contribution”). ES - being coherent and PH - admits this decomposition; VaR does not generally (its derivative is ill-defined on atoms). This is what lets a firm allocate one firm-wide ES budget down to desks so the parts sum to the whole.

2.3 Basel FRTB: from $99\%$ VaR to $97.5\%$ ES

• Old regime (1996 Amendment / Basel II.5): market-risk capital on $10$-day $99\%$ VaR, $\text{capital}=k\cdot \mathrm{VaR}$ with multiplier $k\ge 3$; plus a stressed-VaR add-on.
• FRTB (BCBS, Minimum Capital Requirements for Market Risk, Jan 2019, d457): capital on Expected Shortfall at $97.5\%$, computed with a stressed calibration and liquidity-horizon scaling (positions are assigned liquidity buckets and ES is aggregated across horizons), aggregated across risk classes and multiplied by a backtesting-derived multiplier. The $97.5\%$ level was chosen because, in the normal benchmark, ${\mathrm{ES}}_{97.5\%}\approx {\mathrm{VaR}}_{99\%}$ - a like-for-like calibration that swaps the measure while roughly preserving the headline level (and removes VaR’s tail blindness).

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3. Computational Implementation - spectral ES, Euler allocation, and the FRTB calibration

Two verifications: (i) the Euler contributions of a two-asset normal portfolio sum exactly to the portfolio ES (Euler’s theorem, Acerbi–Tasche); (ii) the FRTB $97.5\%$ ES / $99\%$ VaR normal ratio. Stdlib only.

The Euler contributions sum to the portfolio ES to machine precision (diff $1.4\times 10^{-17}$) - the additivity that makes ES usable for capital allocation. The FRTB calibration ratio is $1.0049$: ${\mathrm{ES}}_{97.5\%}$ sits within half a percent of the old $99\%$ VaR in the normal case, while capturing the tail in the non-normal case.

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4. Failure Modes & First-Principles Breakdowns

1. Spectral choice is a policy, not a fact. Acerbi’s family lets you pick any non-decreasing $\phi$; ES is the most conservative ($\phi =0$ below $\alpha$). A flatter $\phi$ dilutes tail sensitivity and approaches the mean - coherent but weak. The regulator’s $\phi$ and the firm’s may differ, and the gap is a governance risk.
2. Euler allocation needs homogeneity; liquidity breaks it. Axiom PH fails for large illiquid positions, so the additive Euler split under-states the true marginal risk of scaling up (03 · §2.1 (axiom PH)). Capital allocation must be liquidity-adjusted.
3. Non-elicability persists. Even the spectral/ES family shares the non-elicitable property; FRTB backtests ES jointly with VaR and via a stressed-calibration multiplier rather than a clean ES backtest (05 · §4).
4. Liquidity horizons complicate the “one number.” FRTB computes ES per liquidity bucket and aggregates with prescribed correlations; the choice of aggregation and the stressed window materially change capital - model risk in the regulatory layer, not just the pricing layer.
5. Estimation of the spectral measure. Estimating an integral $\int \phi {\mathrm{VaR}}_{u}du$ from finite data inherits both quantile noise and the weight’s tail emphasis; ES’s larger sampling error in small samples (05 · §3) applies to every spectral measure with tail-loaded $\phi$.

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5. References

• Acerbi, C., Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion, J. Banking & Finance 26(7):1505–1518 (2002)
• Acerbi, C. & Tasche, D., On the Coherence of Expected Shortfall / Expected Shortfall: A Natural Coherent Alternative to Value at Risk, JBF 26(7) (2002)
• Kusuoka, S., On Law Invariant Coherent Risk Measures, Advances in Mathematical Economics 3:83–95 (2001)
• Rockafellar, R.T. & Uryasev, S., Optimization of Conditional Value-at-Risk, J. Risk 2(3) (2000)
• BCBS, Minimum Capital Requirements for Market Risk (Jan 2019, BIS d457; FRTB)
• BCBS, Amendment to the Capital Accord to Incorporate Market Risks (1996) and Supervisory Framework for Backtesting (1996)
• McNeil, Frey & Embrechts, Quantitative Risk Management (2015)

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6. Connected Graph Bridges

• Back: 05 · Failure Modes · Index Hub
• Base: Multivariable Calculus (Euler’s theorem) · Probability & Measure Theory
• Sibling: Extreme Value Theory & Fat Tails · VaR Estimation Methods · Stress Testing
• Forward: Portfolio Risk Constraints