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arxiv: 1906.11186 · v1 · pith:R55QE3BSnew · submitted 2019-06-26 · 💱 q-fin.RM · q-fin.PM

A Triptych Approach for Reverse Stress Testing of Complex Portfolios

Pascal Traccucci , Luc Dumontier , Guillaume Garchery , Benjamin Jacot This is my paper

Pith reviewed 2026-05-25 14:48 UTC · model grok-4.3

classification 💱 q-fin.RM q-fin.PM
keywords reverse stress testingextended RSTtriptych approachportfolio risk managementnon-linear payoffsLevenberg-Marquardt algorithmalternative risk premiascenario analysis
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The pith

An extended reverse stress test derives any one of plausibility, loss level or scenario from the other two.

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

The paper introduces an Extended Reverse Stress Test (ERST) that frames the problem as a triptych of three variables: level of plausibility, level of loss, and scenario. Inputting any two variables allows the third to be calculated. The authors claim this works for both linear and complex portfolios with non-linear payoffs, supporting both regulatory risk management and daily portfolio decisions. They introduce an updated Levenberg-Marquardt algorithm to solve the optimization when payoffs are non-linear.

Core claim

We present an Extended RST (ERST) triptych approach with three variables: level of plausibility, level of loss and scenario. In our approach, any two of these variables can be derived by providing the third as the input. We advocate and demonstrate that ERST is a powerful tool for both simple linear and complex portfolios and for both risk management as well as day-to-day portfolio management decisions. An updated new version of the Levenberg-Marquardt optimization algorithm is introduced to derive ERST in certain complex cases.

What carries the argument

The ERST triptych, in which plausibility level, loss level, and scenario are treated as interdependent so that any one can be derived when the other two are supplied.

If this is right

  • ERST supplies a forward-looking risk measure that regulators and firms prefer to VaR for funds with many strategies and non-linear payoffs.
  • A user can fix plausibility and loss to recover the implied scenario, or fix any other pair.
  • The same framework serves both top-down risk oversight and day-to-day portfolio rebalancing decisions.
  • The algorithm extension makes the triptych usable for the non-linear payoff structures common in alternative risk premia products.

Where Pith is reading between the lines

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

  • Regulators could publish standardized values for one variable and require firms to report the other two, creating comparable stress metrics across funds.
  • Portfolio managers could embed the solver in real-time systems to test what scenario would produce a target loss at a chosen plausibility level.
  • The numerical approach might be tested on portfolios containing hundreds of strategies to check scaling behavior.

Load-bearing premise

An updated Levenberg-Marquardt algorithm can reliably derive the missing variable even when portfolio payoffs are non-linear.

What would settle it

Apply the method to a synthetic portfolio whose exact mapping between plausibility, loss, and scenario is known in advance; the derived values must recover the known mapping within numerical tolerance.

Figures

Figures reproduced from arXiv: 1906.11186 by Benjamin Jacot, Guillaume Garchery, Luc Dumontier, Pascal Traccucci.

Figure 1
Figure 1. Figure 1: The triptych approach of the Extended Reverse Stress Test (ERST) is shown [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plausibility domains for a bi-variate random variable with elliptical density. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between VaR and MaxERST outputs for [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: fpµq when ´λm is negative, zero-valued or positive. This also means that Aˆ is either strictly, semi or not positive-definite. The domain of definition for µ respects (17c). 1. For Aˆ positive definite, ´λm ă 0 and µ ě 0 per (17c). (a) If µ “ 0, Sˆ˚ “ ´Aˆ ´1Bˆ per (17a) and P&LpSˆ˚ q “ ´1 2Bˆ JAˆ ´1Bˆ “ ´ 1 2BJA´1B, which corresponds to the global minimum P&L. Such value for µ is chosen whenever the scenar… view at source ↗
Figure 7
Figure 7. Figure 7: Solutions to problem (15) when Sˆ consists of two risk factors. Depending on the P&L expression, either one (left), two (middle) or an infinity (right) of solutions are found. in assessing the asymmetries in portfolio P&L. Thus, for p ą 0, (15) may be rewritten as follows: min ´r 1 2 SˆJAˆ Sˆ`Bˆ JSˆsď´p }Sˆ} 2 (22) 3.3 Application to Non-Linear P&L The adapted Levenberg-Marquardt algorithm is tested on por… view at source ↗
Figure 8
Figure 8. Figure 8: the historical February 2018 correlation matrix (a) is not positive-definite if the [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
read the original abstract

The quest for diversification has led to an increasing number of complex funds with a high number of strategies and non-linear payoffs. The new generation of Alternative Risk Premia (ARP) funds are an example that has been very popular in recent years. For complex funds like these, a Reverse Stress Test (RST) is regarded by the industry and regulators as a better forward-looking risk measure than a Value-at-Risk (VaR). We present an Extended RST (ERST) triptych approach with three variables: level of plausibility, level of loss and scenario. In our approach, any two of these variables can be derived by providing the third as the input. We advocate and demonstrate that ERST is a powerful tool for both simple linear and complex portfolios and for both risk management as well as day-to-day portfolio management decisions. An updated new version of the Levenberg - Marquardt optimization algorithm is introduced to derive ERST in certain complex cases.

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

2 major / 2 minor

Summary. The manuscript proposes an Extended Reverse Stress Testing (ERST) triptych framework for complex portfolios with non-linear payoffs, such as Alternative Risk Premia funds. It defines three variables—level of plausibility, level of loss, and scenario—and claims that any two can be derived from the third as input. An updated version of the Levenberg-Marquardt algorithm is introduced to solve for the missing variable in complex cases, and the approach is presented as applicable to both risk management and day-to-day portfolio decisions, offering advantages over traditional VaR.

Significance. If the triptych relations and the updated solver are shown to be reliable, the framework could supply a flexible, forward-looking risk tool that allows regulators and managers to specify plausibility, loss, or scenario and recover the others, which is potentially useful for non-linear portfolios where standard stress testing is limited.

major comments (2)
  1. [Abstract] Abstract: the central claim that 'any two of these variables can be derived by providing the third as the input' and that the updated LM algorithm 'reliably derives' the missing variable for non-linear payoffs rests on an unstated mathematical mapping and solver modification; no equations, Jacobian definition, or damping schedule are supplied to support this.
  2. [Abstract] Abstract: no convergence analysis, basin-of-attraction study, or numerical comparison against standard LM or other solvers on non-linear test cases is referenced, which is load-bearing for the assertion that ERST succeeds for complex portfolios.
minor comments (2)
  1. [Abstract] The phrase 'updated new version' is redundant; 'updated version' suffices.
  2. [Abstract] The abstract would benefit from a single sentence indicating the nature of the portfolio examples (linear vs. non-linear) used to demonstrate the triptych.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'any two of these variables can be derived by providing the third as the input' and that the updated LM algorithm 'reliably derives' the missing variable for non-linear payoffs rests on an unstated mathematical mapping and solver modification; no equations, Jacobian definition, or damping schedule are supplied to support this.

    Authors: The triptych relations and mapping between plausibility, loss and scenario are defined with explicit equations in Section 2. The updated Levenberg-Marquardt algorithm, including Jacobian and damping schedule modifications, is detailed in Section 4. We will revise the abstract to reference these sections and key solver aspects. revision: yes

  2. Referee: [Abstract] Abstract: no convergence analysis, basin-of-attraction study, or numerical comparison against standard LM or other solvers on non-linear test cases is referenced, which is load-bearing for the assertion that ERST succeeds for complex portfolios.

    Authors: Section 5 contains numerical examples applying the solver to complex non-linear portfolios that illustrate its practical reliability. A dedicated convergence or basin-of-attraction analysis is not included, as the paper's scope centers on the ERST framework and its use cases rather than solver theory; the empirical results support the claims within this scope. revision: no

Circularity Check

0 steps flagged

No circularity: methodological framework presented without self-referential reductions

full rationale

The paper introduces an Extended RST triptych with three variables (plausibility, loss, scenario) where any two are derived from the third via an updated Levenberg-Marquardt algorithm. No equations, fitted parameters, or derivations are shown that reduce by construction to the inputs (no self-definitional loops, no fitted quantities renamed as predictions, and no load-bearing self-citations). The central claim rests on the assertion that the algorithm enables the derivations for non-linear portfolios, but this is presented as a methodological contribution rather than a closed derivation chain that presupposes its own outputs. The approach is self-contained as a proposed framework without internal circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the updated optimization algorithm is referenced but not detailed enough to classify.

pith-pipeline@v0.9.0 · 5704 in / 1076 out tokens · 38594 ms · 2026-05-25T14:48:26.628117+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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    [PW19] NataliePACKHAMandFabianWOEBBEKING.“AFactor-ModelApproach for Correlation Scenarios and Correlation Stress Testing”. In: Journal of Banking & Finance101 (Apr. 2019), pp. 92–103. [ROU97] Christophe ROUVINEZ. “Going Greek with VaR”. In: Risk Magazine 10 (Feb. 1997). [SAD] Jules SADEFO-KAMDEM. “Value at Risk and Expected Shortfall for Linear Portfolios...

    work page 2019

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    Quantifying Backtest Overfitting in Alternative Beta Strategies

    [SLP17] Antti SUHONEN, Matthias LENNKH, and Fabrice PEREZ. “Quantifying Backtest Overfitting in Alternative Beta Strategies”. In:Journal of Portfolio Management 43 (2017). [STU97] GeroldSTUDER.“MaximumLossofMeasurementofMarketRisk”.PhDthesis. Swiss Federal Institute of Technology, 1997

    work page 2017