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arxiv: 2604.16438 · v1 · submitted 2026-04-06 · 💱 q-fin.RM · math.PR· q-fin.MF

Recognition: no theorem link

Ranking Metrics: Extending Acceptability and Performance Indexes

Asmerilda Hitaj , Elisa Mastrogiacomo , Ilaria Peri , Marcelo Righi
Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:36 UTC · model grok-4.3

classification 💱 q-fin.RM math.PRq-fin.MF
keywords ranking metricsacceptability indicesrisk measurescash-quasiconcavityperformance measuresacceptance setsportfolio rankinginsurance risk
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The pith

Ranking metrics can be represented through families of acceptance sets and risk measures using monotonicity and cash-quasiconcavity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces ranking metrics as functionals that assign a performance level to each financial or insurance position instead of producing a normalized reward-to-risk ratio. It assumes monotonicity together with a new property called cash-quasiconcavity and derives representation theorems that connect these metrics to families of acceptance sets and risk measures. The approach extends the existing theory of acceptability indices, recovers classical ratios such as the Sharpe ratio as special cases, and supplies new examples built from expected-loss and quantile-based constructions. Empirical illustrations on portfolio ordering and climate-risk insurance show how the framework supports concrete ranking tasks.

Core claim

We develop an axiomatic framework for ranking metrics, functionals that evaluate and order positions by assigning them a performance level. Relying on monotonicity and cash-quasiconcavity, we obtain representation results that link ranking metrics to families of acceptance sets and risk measures, thereby extending the theory of acceptability indices. Classical performance ratios arise as special cases, while new constructions based on expected loss, Lambda-quantiles, and bibliometric indices demonstrate the framework's flexibility.

What carries the argument

Cash-quasiconcavity, a property introduced for ranking metrics that, combined with monotonicity, yields representation theorems connecting the metrics to families of acceptance sets and risk measures.

If this is right

  • Classical risk-adjusted performance measures such as the Sharpe ratio, RAROC, and Omega arise directly as special cases of the ranking metrics.
  • New ranking metrics can be constructed from expected-loss, Lambda-quantile, and bibliometric indices while preserving the axiomatic structure.
  • The same representation applies uniformly to both portfolio ranking and climate-risk insurance evaluation.
  • Any ranking metric satisfying the axioms admits a description in terms of a family of acceptance sets and associated risk measures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could allow practitioners to replace separate performance ratios with a single family of metrics whose ordering behavior is governed by explicit acceptance sets.
  • If cash-quasiconcavity proves empirically stable, optimization routines that maximize ranking metrics might inherit convexity or quasiconcavity properties from the underlying acceptance sets.
  • The representation might extend naturally to other domains that require ordering of uncertain outcomes, such as sustainability scoring or operational-risk ranking, by reinterpreting the acceptance sets accordingly.

Load-bearing premise

Cash-quasiconcavity is a natural and useful property for ranking metrics in financial and insurance settings, and the derived representations hold without further restrictions that would limit their practical application.

What would settle it

A monotonic ranking metric that violates cash-quasiconcavity yet still produces consistent orderings of real portfolios or insurance contracts, or a classical ratio such as the Sharpe ratio that fails to fit the representation under the stated axioms.

Figures

Figures reproduced from arXiv: 2604.16438 by Asmerilda Hitaj, Elisa Mastrogiacomo, Ilaria Peri, Marcelo Righi.

Figure 1
Figure 1. Figure 1: Portfolios’ performance rankings using RAROC, Om [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Portfolios’ performance rankings using bibliome [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Construction of bibliometric indices for portfol [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Ranking of European zones climate-related-losse [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Ranking of European zones climate-related-losse [PITH_FULL_IMAGE:figures/full_fig_p034_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ranking of zones smaller-than-mean losses based o [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Portfolio allocation using different ranking metr [PITH_FULL_IMAGE:figures/full_fig_p036_7.png] view at source ↗
read the original abstract

This paper develops an axiomatic framework for ranking metrics, a general class of functionals for evaluating and ordering financial or insurance positions. Unlike traditional risk-adjusted performance measures-such as the Sharpe ratio, RAROC, or Omega-that express reward per unit of risk, ranking metrics assign each position a performance level rather than a normalized return. Relying on monotonicity and a new property called cash-quasiconcavity, we derive representation results linking ranking metrics to families of acceptance sets and risk measures, extending the theory of acceptability indices. Classical ratios arise as special cases, while new examples-based on expected-loss, Lambda-quantile, and bibliometric indices-illustrate the framework's flexibility. Empirical applications to portfolio ranking and climate-risk insurance demonstrate its practical relevance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops an axiomatic framework for ranking metrics, a class of functionals that assign performance levels to financial or insurance positions rather than normalized returns. Relying on monotonicity and the newly introduced cash-quasiconcavity property, it derives representation results that connect ranking metrics to families of acceptance sets and risk measures, thereby extending the theory of acceptability indices. Classical ratios such as the Sharpe ratio, RAROC, and Omega ratio arise as special cases, while new examples based on expected-loss, Lambda-quantile, and bibliometric indices are constructed. The paper concludes with empirical applications to portfolio ranking and climate-risk insurance.

Significance. If the representation theorems hold, the work provides a unified axiomatic foundation that systematically generates and justifies ranking metrics from acceptance-set and risk-measure primitives. This extends acceptability-index theory without introducing free parameters, as classical ratios are recovered exactly by satisfying the stated axioms and new examples are shown to do likewise. The empirical illustrations in portfolio and insurance contexts add practical value by demonstrating how the framework can be applied to real data for ordering positions.

minor comments (3)
  1. The definition and motivation for cash-quasiconcavity in the axiomatic section would benefit from an explicit comparison to standard quasiconcavity and an intuitive example showing why the cash-specific variant is required for the representation to hold.
  2. In the empirical applications, the portfolio-ranking exercise reports ordering results but does not include robustness checks (e.g., sensitivity to estimation windows or out-of-sample performance), which would strengthen the claim of practical relevance.
  3. Notation for the families of acceptance sets and the associated risk measures should be introduced with a summary table to avoid repeated re-definition across the representation theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, positive assessment of significance, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Axiomatic derivation from stated properties shows no circularity

full rationale

The paper introduces ranking metrics via an axiomatic setup relying on monotonicity and the new cash-quasiconcavity property, then derives representation theorems linking them to acceptance sets and risk measures. These steps are presented as direct consequences of the axioms rather than reductions to fitted inputs, self-definitions, or prior self-citations. Classical ratios (Sharpe, RAROC, Omega) are recovered as special cases satisfying the axioms, and new examples (expected-loss, Lambda-quantile, bibliometric) are constructed to illustrate flexibility without forcing the general results by construction. No load-bearing self-citation chains, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are indicated in the provided structure. The framework is self-contained against the stated axioms and extends acceptability-index theory without internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework is built on standard monotonicity from decision theory plus the newly introduced cash-quasiconcavity; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Monotonicity
    Higher outcomes must receive strictly higher ranking scores; standard in financial decision theory.
  • ad hoc to paper Cash-quasiconcavity
    New property introduced in the paper to enable the representation results.

pith-pipeline@v0.9.0 · 5434 in / 1235 out tokens · 41136 ms · 2026-05-10T18:36:59.516336+00:00 · methodology

discussion (0)

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Reference graph

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