On Prudence of Risk Measures

Denny H. Leung; Foivos Xanthos; Niushan Gao

arxiv: 2606.21871 · v1 · pith:HCEQO6WMnew · submitted 2026-06-20 · 💱 q-fin.RM · math.FA

On Prudence of Risk Measures

Niushan Gao , Denny H. Leung , Foivos Xanthos This is my paper

Pith reviewed 2026-06-26 11:17 UTC · model grok-4.3

classification 💱 q-fin.RM math.FA

keywords prudencerisk measureslaw-invariantcash-additiveconvexstar-shapedinf-convolution

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The pith

Prudence coincides with weak prudence for convex law-invariant risk functionals and is preserved by hulls and inf-convolutions.

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

The paper establishes that weak prudence and prudence coincide for convex, law-invariant functionals. It proves that prudence is preserved by cash-additive hulls of star-shaped functionals under a simple asymptotic condition, and by inf-convolutions of convex, cash-additive, law-invariant prudent functionals. These results provide general methods for constructing prudent risk measures from existing ones. Readers would care as these properties help in ensuring stability of risk assessments in quantitative finance.

Core claim

Weak prudence and prudence coincide for a broad class of convex, law-invariant functionals. Prudence is preserved by cash-additive hulls of star-shaped functionals under a simple asymptotic condition, and by inf-convolutions of convex, cash-additive, law-invariant prudent functionals. The results provide general methods for constructing prudent risk measures from existing prudent functionals.

What carries the argument

Cash-additive hulls of star-shaped functionals under an asymptotic condition and inf-convolutions of prudent functionals as mechanisms that preserve prudence.

If this is right

Prudent risk measures can be constructed via cash-additive hulls of star-shaped functionals satisfying the asymptotic condition.
Inf-convolutions preserve prudence for convex cash-additive law-invariant functionals.
The equivalence allows reducing prudence checks to weak prudence checks for the broad class.

Where Pith is reading between the lines

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

The preservation results could apply to constructing prudent measures in portfolio risk management.
One might investigate if the asymptotic condition can be relaxed further.
These methods might help in verifying prudence for more complex functionals by reducing to simpler ones.

Load-bearing premise

The functionals are convex, law-invariant and cash-additive with the star-shaped case satisfying the asymptotic condition.

What would settle it

Finding a convex law-invariant functional that is weakly prudent but not prudent would falsify the coincidence.

read the original abstract

Prudence is a stability property of risk functionals recently introduced by Wang and Zitikis and subsequently studied by Amarante and Liebrich. In this paper, we first establish general relationships between prudence and other stability properties, showing, in particular, that weak prudence and prudence coincide for a broad class of convex, law-invariant functionals. We then prove that prudence is preserved by cash-additive hulls of star-shaped functionals under a simple asymptotic condition, and by inf-convolutions of convex, cash-additive, law-invariant prudent functionals. Our results provide general methods for constructing prudent risk measures from existing prudent functionals.

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.

Desk Editor's Note private letter to a colleague

The paper shows weak prudence equals prudence for convex law-invariant functionals and gives two preservation results, but the abstract alone leaves the proofs uncheckable.

read the letter

The main thing to know is that this paper proves weak prudence and full prudence coincide on convex law-invariant functionals, and it gives explicit preservation theorems for cash-additive hulls of star-shaped functionals (under one asymptotic condition) and for inf-convolutions of convex cash-additive law-invariant prudent ones.

Those two preservation results and the equivalence statement are new relative to Wang-Zitikis and Amarante-Liebrich. The work stays inside the usual setting of convex law-invariant risk measures and supplies concrete ways to build new prudent functionals from existing ones.

The abstract is clear about the scope and the load-bearing assumptions. The hull result depends on that asymptotic condition, which looks like the part that needs the most scrutiny once the full text is available.

The obvious limitation is that only the abstract is in front of us, so there is no way to inspect the arguments or see how often the asymptotic condition holds in practice. If the proofs are direct and the condition is mild, the results are useful inside the subfield; if either is fragile, the preservation claims shrink.

This is written for specialists already working on stability properties of risk measures. A reader who needs general construction tools for prudent functionals will get something concrete if the details check out. It is narrow enough that it does not reorganize the broader area, but the claims are precise enough to be worth referee time.

Referee Report

0 major / 1 minor

Summary. The paper studies the prudence stability property of risk functionals. It establishes that weak prudence and prudence coincide for a broad class of convex law-invariant functionals. It further proves that prudence is preserved under cash-additive hulls of star-shaped functionals subject to a simple asymptotic condition, and under inf-convolutions of convex cash-additive law-invariant prudent functionals, thereby providing general construction methods for prudent risk measures.

Significance. If the stated relationships and preservation results hold, the work supplies direct, non-circular methods for verifying and constructing prudent risk measures from existing ones. The coincidence of weak and strong prudence for convex law-invariant functionals simplifies analysis in a practically relevant setting, while the hull and inf-convolution results offer constructive tools that could be applied in quantitative risk management.

minor comments (1)

The abstract refers to a 'simple asymptotic condition' without stating it; while the main text presumably defines it, a brief inline reminder in the statement of the relevant theorem would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough reading and positive evaluation of the manuscript. Their recommendation to accept is appreciated, and we are pleased that the significance of the equivalence result for convex law-invariant functionals and the preservation properties under hulls and inf-convolutions was recognized.

Circularity Check

0 steps flagged

No significant circularity; derivations are direct from definitions

full rationale

The paper's core results consist of direct proofs relating prudence to weak prudence for convex law-invariant functionals, plus preservation under cash-additive hulls (with an explicit asymptotic condition) and inf-convolutions. These steps are presented as mathematical derivations from the stated functional properties without any reduction to fitted parameters, self-definitional equivalences, or load-bearing self-citations. Prior introductions of prudence are attributed to external authors (Wang-Zitikis, Amarante-Liebrich), and the abstract indicates no renaming of known results or smuggling of ansatzes. The derivation chain remains self-contained against the given assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no free parameters, ad-hoc axioms, or invented entities; the work rests on standard background notions of convexity, law-invariance, cash-additivity and star-shapedness from convex analysis and risk-measure theory.

pith-pipeline@v0.9.1-grok · 5623 in / 1005 out tokens · 32957 ms · 2026-06-26T11:17:56.357586+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references

[1]

Acciaio, B.: Short note on inf-convolution preserving the Fatou property,Annals of Finance5(2), 281–287 (2009)

2009
[2]

Ahn, J.Y., Shyamalkumar, N.D.: Asymptotic theory for the empirical Haezendonck–Goovaerts risk measure,Insurance: Mathematics and Economics55, 78–90 (2014)

2014
[3]

Amarante, M., Liebrich, F.B.: Distortion risk measures: Prudence, coherence, and the expected shortfall,Mathematical Finance34(4), 1291–1327 (2024)

2024
[4]

Ayg¨ un, M., Bellini, F., Laeven, R.J.: Generalized Orlicz premia, arXiv:2507.09181 (2025)

arXiv 2025
[5]

Bellini, F., Fadina, T., Wang, R., Wei, Y.: Parametric measures of variability induced by risk measures,Insurance, Mathematics and Economics106, 270-284 (2022)

2022
[6]

Bellini, F., Koch-Medina, P., Munari, C., Svindland, G.: Law-Invariant Functionals on General Spaces of Random Variables,SIAM Journal on Financial Mathematics12(1), 318–341 (2021)

2021
[7]

Bellini, F., Rosazza Gianin, E.: Haezendonck–Goovaerts risk measures and Orlicz quantiles, Insurance: Mathematics and Economics51, 107-114 (2012)

2012
[8]

Castagnoli, E., Cattelan, G., Maccheroni, F., Tebaldi, C., Wang, R.: Star-shaped risk measures, Operations Research70(5), 2637–2654 (2022)

2022
[9]

Chen, S., Gao, N., Leung, D., Li, L.: Automatic Fatou property of law-invariant risk measures, Insurance: Mathematics and Economics105, 41–53 (2022)

2022
[10]

Chen, S., Gao, N., Xanthos, F.: The strong Fatou property of risk measures,Dependence Mod- eling6(1), 183-196 (2018)

2018
[11]

Delbaen, F.: Coherent risk measures on general probability spaces, In:Advances in finance and stochastics, Springer Berlin Heidelberg, 2002, 1–37

2002
[12]

Dentcheva, D., Penev, S., Ruszczy´ nski, A.: Kusuoka representation of higher order dual risk measures,Annals of Operations Research181(1), 325–335 (2010)

2010
[13]

Filipovi´ c, D., Kupper, M.: Monotone and cash-invariant convex functions and hulls,Insurance: Mathematics and Economics41(1), 1–6 (2007)

2007
[14]

Filipovi´ c, D., Svindland, G.: Optimal capital and risk allocations for law- and cash-invariant convex functions,Finance and Stochastics12, 423–439 (2008)

2008
[15]

Gao, N., Munari, C.: Surplus-invariant risk measures,Mathematics of Operations Research45(4), 1342–1370 (2020)

2020
[16]

Gao, N., Munari, C., Xanthos, F.: Stability properties of Haezendonck–Goovaerts premium principles,Insurance: Mathematics and Economics94, 94–99 (2020)

2020
[17]

Gao, N., Leung, D., Munari, C., Xanthos, F.: Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces,Finance and Stochastics22(2), 395–415 (2018)

2018
[18]

Gao, N., Leung, D., Xanthos, F.: Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures,Studia Mathematica249, 329–347 (2019)

2019
[19]

Gao, N., Xanthos, F.: On the C-property andw ∗-representations of risk measures,Mathematical Finance28(2), 748–754 (2018)

2018
[20]

Gao, N., Xanthos, F.: A note on continuity and asymptotic consistency of measures of risk and variability,ASTIN Bulletin: The Journal of the IAA55(1), 168–177 (2025)

2025
[21]

Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou Property, Advances in Mathematical Economics9, 49–71 (2006)

2006
[22]

Krohmal, P.: Higher moment coherent risk measures,Quantitative Finance7(4), 373–387 (2007)

2007
[23]

Laeven, R.J., Rosazza Gianin, E., Zullino, M.: Dynamic return and star-shaped risk measures via BSDEs, arXiv:2307.03447 (2023)

arXiv 2023
[24]

ON PRUDENCE OF RISK MEASURES 17

Laeven, R.J., Rosazza Gianin, E., Zullino, M.: Law-invariant return and star-shaped risk mea- sures,Insurance: Mathematics and Economics117, 140–153 (2024). ON PRUDENCE OF RISK MEASURES 17

2024
[25]

Liebrich, F.: Risk sharing under heterogeneous beliefs without convexity,Finance and Stochastics 28, 999–1033 (2024)

2024
[26]

Liebrich, F., Munari, C.: Revisiting the Automatic Fatou Property of Law-Invariant Functionals, SIAM Journal on Financial Mathematics16(1), 2025

2025
[27]

Liu, P., Wang, R., Wei, L.: Is the inf-convolution of law-invariant preferences law-invariant? Insurance: Mathematics and Economics91, 144–154 (2020)

2020
[28]

O’Brien, M., Troitsky, V.G., van der Walt, J.H.: Net convergence structures with applications to vector lattices,Quaestiones Mathematicae46(2), 243–280 (2023)

2023
[29]

Rahsepar, M., Xanthos, F.: On the extension property of dilatation monotone risk measures, Statistics & Risk Modeling37, 139–152 (2020)

2020
[30]

Righi, M.B.: Star-shaped acceptability indexes,Insurance: Mathematics and Economics117, 170–181 (2024)

2024
[31]

Math.114, 175–183 (2020)

Tantrawan, M., Leung, D.H.: On closedness of law-invariant convex sets in rearrangement in- variant spaces.Arch. Math.114, 175–183 (2020)

2020
[32]

Wang, R., Zitikis, R.: An axiomatic foundation for the expected shortfall,Management Science 67(3), 1413–1429 (2021)

2021
[33]

Wu, Q., Mao, T., Hu, T.: Generalized optimized certainty equivalent with applications in the rank-dependent utility model,SIAM Journal on Financial Mathematics15(1), 255–294 (2024). Department of Mathematics, Toronto Metropolitan University, 350 Victoria Street, Toronto, Canada M5B 2K3 Email address:niushan@torontomu.ca Department of Mathematics, National...

2024

[1] [1]

Acciaio, B.: Short note on inf-convolution preserving the Fatou property,Annals of Finance5(2), 281–287 (2009)

2009

[2] [2]

Ahn, J.Y., Shyamalkumar, N.D.: Asymptotic theory for the empirical Haezendonck–Goovaerts risk measure,Insurance: Mathematics and Economics55, 78–90 (2014)

2014

[3] [3]

Amarante, M., Liebrich, F.B.: Distortion risk measures: Prudence, coherence, and the expected shortfall,Mathematical Finance34(4), 1291–1327 (2024)

2024

[4] [4]

Ayg¨ un, M., Bellini, F., Laeven, R.J.: Generalized Orlicz premia, arXiv:2507.09181 (2025)

arXiv 2025

[5] [5]

Bellini, F., Fadina, T., Wang, R., Wei, Y.: Parametric measures of variability induced by risk measures,Insurance, Mathematics and Economics106, 270-284 (2022)

2022

[6] [6]

Bellini, F., Koch-Medina, P., Munari, C., Svindland, G.: Law-Invariant Functionals on General Spaces of Random Variables,SIAM Journal on Financial Mathematics12(1), 318–341 (2021)

2021

[7] [7]

Bellini, F., Rosazza Gianin, E.: Haezendonck–Goovaerts risk measures and Orlicz quantiles, Insurance: Mathematics and Economics51, 107-114 (2012)

2012

[8] [8]

Castagnoli, E., Cattelan, G., Maccheroni, F., Tebaldi, C., Wang, R.: Star-shaped risk measures, Operations Research70(5), 2637–2654 (2022)

2022

[9] [9]

Chen, S., Gao, N., Leung, D., Li, L.: Automatic Fatou property of law-invariant risk measures, Insurance: Mathematics and Economics105, 41–53 (2022)

2022

[10] [10]

Chen, S., Gao, N., Xanthos, F.: The strong Fatou property of risk measures,Dependence Mod- eling6(1), 183-196 (2018)

2018

[11] [11]

Delbaen, F.: Coherent risk measures on general probability spaces, In:Advances in finance and stochastics, Springer Berlin Heidelberg, 2002, 1–37

2002

[12] [12]

Dentcheva, D., Penev, S., Ruszczy´ nski, A.: Kusuoka representation of higher order dual risk measures,Annals of Operations Research181(1), 325–335 (2010)

2010

[13] [13]

Filipovi´ c, D., Kupper, M.: Monotone and cash-invariant convex functions and hulls,Insurance: Mathematics and Economics41(1), 1–6 (2007)

2007

[14] [14]

Filipovi´ c, D., Svindland, G.: Optimal capital and risk allocations for law- and cash-invariant convex functions,Finance and Stochastics12, 423–439 (2008)

2008

[15] [15]

Gao, N., Munari, C.: Surplus-invariant risk measures,Mathematics of Operations Research45(4), 1342–1370 (2020)

2020

[16] [16]

Gao, N., Munari, C., Xanthos, F.: Stability properties of Haezendonck–Goovaerts premium principles,Insurance: Mathematics and Economics94, 94–99 (2020)

2020

[17] [17]

Gao, N., Leung, D., Munari, C., Xanthos, F.: Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces,Finance and Stochastics22(2), 395–415 (2018)

2018

[18] [18]

Gao, N., Leung, D., Xanthos, F.: Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures,Studia Mathematica249, 329–347 (2019)

2019

[19] [19]

Gao, N., Xanthos, F.: On the C-property andw ∗-representations of risk measures,Mathematical Finance28(2), 748–754 (2018)

2018

[20] [20]

Gao, N., Xanthos, F.: A note on continuity and asymptotic consistency of measures of risk and variability,ASTIN Bulletin: The Journal of the IAA55(1), 168–177 (2025)

2025

[21] [21]

Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou Property, Advances in Mathematical Economics9, 49–71 (2006)

2006

[22] [22]

Krohmal, P.: Higher moment coherent risk measures,Quantitative Finance7(4), 373–387 (2007)

2007

[23] [23]

Laeven, R.J., Rosazza Gianin, E., Zullino, M.: Dynamic return and star-shaped risk measures via BSDEs, arXiv:2307.03447 (2023)

arXiv 2023

[24] [24]

ON PRUDENCE OF RISK MEASURES 17

Laeven, R.J., Rosazza Gianin, E., Zullino, M.: Law-invariant return and star-shaped risk mea- sures,Insurance: Mathematics and Economics117, 140–153 (2024). ON PRUDENCE OF RISK MEASURES 17

2024

[25] [25]

Liebrich, F.: Risk sharing under heterogeneous beliefs without convexity,Finance and Stochastics 28, 999–1033 (2024)

2024

[26] [26]

Liebrich, F., Munari, C.: Revisiting the Automatic Fatou Property of Law-Invariant Functionals, SIAM Journal on Financial Mathematics16(1), 2025

2025

[27] [27]

Liu, P., Wang, R., Wei, L.: Is the inf-convolution of law-invariant preferences law-invariant? Insurance: Mathematics and Economics91, 144–154 (2020)

2020

[28] [28]

O’Brien, M., Troitsky, V.G., van der Walt, J.H.: Net convergence structures with applications to vector lattices,Quaestiones Mathematicae46(2), 243–280 (2023)

2023

[29] [29]

Rahsepar, M., Xanthos, F.: On the extension property of dilatation monotone risk measures, Statistics & Risk Modeling37, 139–152 (2020)

2020

[30] [30]

Righi, M.B.: Star-shaped acceptability indexes,Insurance: Mathematics and Economics117, 170–181 (2024)

2024

[31] [31]

Math.114, 175–183 (2020)

Tantrawan, M., Leung, D.H.: On closedness of law-invariant convex sets in rearrangement in- variant spaces.Arch. Math.114, 175–183 (2020)

2020

[32] [32]

Wang, R., Zitikis, R.: An axiomatic foundation for the expected shortfall,Management Science 67(3), 1413–1429 (2021)

2021

[33] [33]

Wu, Q., Mao, T., Hu, T.: Generalized optimized certainty equivalent with applications in the rank-dependent utility model,SIAM Journal on Financial Mathematics15(1), 255–294 (2024). Department of Mathematics, Toronto Metropolitan University, 350 Victoria Street, Toronto, Canada M5B 2K3 Email address:niushan@torontomu.ca Department of Mathematics, National...

2024