FinMA-ES: Risk Measures & Bilingual LLM

• FinMA-ES is a dual-concept framework that defines scenario-based expected shortfall risk measures under Swiss regulation and a Spanish-English financial LLM.
• It utilizes multi-scenario stress testing, coherent risk measures, and advanced backtesting techniques like multinomial VaR and e-backtesting.
• The approach employs a Kusuoka-type representation to ensure coherence while the bilingual LLM enhances performance in Spanish-English financial NLP tasks.

FinMA-ES refers to scenario-based Expected Shortfall methodologies and regulatory frameworks underpinned by the Swiss Financial Market Supervisory Authority (FinMA), centering on stress-tested, multi-scenario risk measures for banking market risk. This concept, formalized by Wang & Ziegel and later embedded in Swiss and Basel III/IV supervisory practice, also interfaces with backtesting (including multinomial and e-backtesting designs) and, in a parallel but unrelated context, denotes a Spanish-English bilingual financial LLM. The primary focus is the rigorous mathematical and regulatory specification of scenario-based ES for risk measurement, capital calculations, and backtesting.

1. Formal Definition and Structure of Scenario-Based ES

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $$\mathcal{Q} = \{ Q_1, \ldots, Q_n \} \subseteq \mathcal{P}$$ a finite family of scenario probability measures. For a random variable $X$ and confidence level $p \in (0, 1)$,

• The Value-at-Risk under $Q$ is $VaR_q^Q(X) = F_{X,Q}^{-1}(q)$.
• Expected Shortfall is $$ES_p^{Q}(X) = \frac{1}{1-p} \int_p^1 VaR_q^Q(X) \, dq$$.

Max-ES (stress-adjusted ES) is defined as:

$$MES_p^{\mathcal{Q}}(X) = \sup_{Q \in \mathcal{Q}} ES_p^Q(X).$$

Average-ES is

$$AES_p^{\mathcal{Q}}(X) = \frac{1}{n} \sum_{i=1}^n ES_p^{Q_i}(X).$$

General $\mathcal{Q}$-mixture of ES takes the form:

$$\hat{\rho}(X) = \sum_{i=1}^n w_i \int_0^1 ES_p^{Q_i}(X) \, dh_i(p),$$

with $w_i \ge 0$, $\sum w_i = 1$, $h_i$ cumulative functions on $[0,1]$.

Integral Max-ES and Replicated Max-ES augment this family:

• $$iMES_p^{\mathcal Q}(X) = \frac{1}{1-p}\int_p^1 MVaR_q^{\mathcal Q}(X)dq$$
• $$rMES_p^{\mathcal Q}(X) = ES_p^P\left( \max_{i=1,\ldots,n} X_i \right)$$, $X_i\overset{iid}{\sim}F_{X,Q_i}$

2. Axiomatic Foundations and Coherence Properties

Key risk measure properties:

• Cash invariance: $\rho(X + c) = \rho(X) + c$
• Monotonicity: $X \le Y \implies \rho(X) \le \rho(Y)$
• Positive homogeneity: $\rho(\lambda X) = \lambda \rho(X)$, $\lambda > 0$
• Subadditivity: $\rho(X + Y) \le \rho(X) + \rho(Y)$
• Comonotonic additivity: For comonotonic $X,Y$, $\rho(X+Y) = \rho(X) + \rho(Y)$

A risk measure is coherent iff it is cash-invariant, monotone, homogeneous, and subadditive.

Within the multi-scenario framework, $\mathcal Q$-based means:

$$X \stackrel{d}{=}_{Q_i} Y \; \forall i \implies \rho(X) = \rho(Y).$$

Summary of major scenario-based risk measures:

Risk Measure	Coherence	Comonotonic Additivity
$MES_p^{\mathcal Q}$	Yes	No
$MVaR_p^{\mathcal Q}$	No	Yes
$AES_p^{\mathcal Q}$	Yes	Yes
$iMES_p^{\mathcal Q}$	No	Yes
$rMES_p^{\mathcal Q}$	Yes	Yes

For all $X$:

$$AES_p^{\mathcal Q}(X) \le MES_p^{\mathcal Q}(X) \le iMES_p^{\mathcal Q}(X) \le rMES_p^{\mathcal Q}(X).$$

(Wang et al., 2018)

3. Representation Theorems and Mathematical Characterization

Suppose $\mathcal{Q}$ is a finite collection of mutually singular, atomless measures. Then $\rho: \mathcal{X} \to \mathbb{R}$ is coherent and $\mathcal Q$-based if and only if it can be written as a supremum of $\mathcal Q$-mixtures of ES:

$$\rho(X) = \sup_{\alpha \in A} \sum_{i=1}^n w_i^\alpha \int_0^1 ES_p^{Q_i}(X) \, dh_i^\alpha(p).$$

This statement, a Kusuoka-type representation, guarantees all coherent scenario-based risk measures permissible under the FinMA-ES regime are supremums over mixtures of scenario-wise ES functionals (Wang et al., 2018).

4. Implementation in Market Risk Regulation (FRTB and FinMA)

The Swiss implementation under FRTB (Basel III/IV) uses the following operational steps:

1. Stress Adjustment:
   • Identify a reduced risk-factor set $R$ and compute a scaling factor $$\theta = \max\{1, ES_p^{\rm full}(X)/ES_p^{R}(X)\} < 4/3.$$
   • Calculate $$ES_{R,S}(X) = \max_{Q \in \mathcal Q_{\rm stress}} ES_p^Q(\sum_{i \in R} X_i).$$
   • Set $\widetilde{ES}(X) = \theta \times ES_{R,S}(X).$
2. Dependence Adjustment:
   • Group risk factors into classes $C_1, ..., C_k$.
   • For each class, $$\widetilde{ES}_{C_j}(X) = MES_p^{\mathcal Q}(X_{C_j})$$.
   • Aggregate: $$\widetilde{ES}_C(X) = \sum_{j=1}^k \widetilde{ES}_{C_j}(X)$$.
3. IMCC (Internal Model Capital Charge):

$$IMCC(X) = \lambda \widetilde{ES}(X) + (1-\lambda) \widetilde{ES}_C(X), \quad \lambda = 0.5.$$

Each operation (max, sum, convex combination) preserves coherence due to the supremum-of-mixtures representation (Wang et al., 2018).

5. Backtesting Methodologies for Scenario-Based ES

Two prominent families of ES backtesting are intertwined with FinMA-ES adoption:

a) Multinomial VaR Backtesting:

Using the approximation

$$ES_\alpha \approx \frac{1}{(1-\alpha)N} \sum_{j=1}^{N} VaR_{u_j},$$

multinomial exception testing across $N$ quantiles replaces traditional binomial exception tests. Pearson, Nass, and likelihood-ratio tests are evaluated. For $N=4$–$8$, the power to detect misspecification is markedly superior to $N=1$, as established on real-data backtests (e.g., SP500 crisis data). A traffic-light system (green/yellow/red) guides escalation and capital adjustment (Kratz et al., 2016).

b) E-backtesting:

A model-free, sequential, anytime-valid mechanism based on e-processes is employed:

• The unique backtest e-statistic:

$$e_p^{ES}(x,r,z) = \frac{(x-z)_+}{(1-p)(r-z)}, \quad z < r,$$

with supermartingale properties under $H_0$ (forecast is not understated); reject if the capital process $M_t$ exceeds $1/\alpha$.

• GREE, GREL, and GREM strategies adapt the betting fraction using either past e-values, losses under current forecasts, or mixtures.
• Extensive simulations confirm high detection power and low type I error in realistic GARCH scenarios with roll-forward windows (Wang et al., 2022).

6. Integration in Practice and Regulatory Significance

FinMA-ES, as implemented in Swiss market risk regulation, operationalizes the scenario-based ES concepts underpinning FRTB. Regulatory guidelines require:

• Systematic stress scenario selection and recomputation,
• Bucketed risk-factor dependency aggregation,
• Coherent risk aggregation across variant risk landscapes,
• Explicit deployment of scenario-based ES and corresponding backtest regimes.

The mathematical underpinnings, notably the Kusuoka-type representation, guarantee adherence to the coherence axiom and formal justification of stress-testing procedures. The methods have enabled regulators and institutions to transition from VaR-centric to ES-centric market risk capital frameworks, incorporating real-world scenario diversity and robustness (Wang et al., 2018).

7. Contextual Remarks and Bilingual Model Homonym

The term "FinMA-ES" also appears in recent literature as the name for a Spanish-English financial LLM, unrelated to the risk-measure context discussed above. This usage refers to a 7B-parameter LLaMA2 derivative instruction-tuned on balanced Spanish/English financial datasets and evaluated on the FLARE-ES benchmark. It achieves state-of-the-art performance on Spanish financial NLP tasks and reduces the multilingual performance gap in practical applications, but is distinct from the scenario-based ES methodology foundational to FinMA regulatory frameworks (Zhang et al., 12 Feb 2024).