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on Risk Management |
| By: | Luisa Tibiletti; Simone Farinelli; Eric Dal Moro |
| Abstract: | Risk aggregation is a central problem in financial risk management and regulatory capital assessment. While copula-based approaches are widely used due to their tractability and interpretability, standard implementations rely on flat dependence structures that may fail to capture hierarchical interactions across risk components. This paper introduces a hierarchical copula framework for risk aggregation on trees, in which dependence is specified locally at each aggregation node and propagated through recursive rank-based reordering. The approach preserves marginal distributions while enabling flexible and economically interpretable modeling of dependence across multiple aggregation levels. Copula parameters are calibrated from empirical data using Kendalls tau, and the methodology is illustrated using market and credit risk proxies. Robustness is assessed across alternative credit segments, including investment-grade and high-yield indices. Numerical results show that dependence strength increases under stress and varies systematically across credit quality, with important implications for risk measurement. The proposed framework combines the transparency of copula-based aggregation with the structural richness of hierarchical models, providing a practical and scalable tool for enterprise risk management. This contributes to bridging bottom-up and top-down approaches to risk aggregation. Keywords: hierarchical copulae; dependence modelling; Kendalls tau calibration; Gaussian and Clayton copulae; tail dependence; risk aggregation. Key Messages: - Recursive hierarchical copulas aggregate market and credit portfolios. - Empirical dependence calibration is combined with sample reordering. - The framework computes VaR and Expected Shortfall transparently. - Tail dependence significantly affects hierarchical risk aggregation. |
| Date: | 2026–07 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.08798 |
| By: | Jun Cai; Tiantian Mao; Zhiqiao Song |
| Abstract: | In this paper, we present robust portfolio selection models by incorporating a reward and penalty mechanism into portfolio management. We assume that the joint distribution of the losses of the underlying risky assets in a portfolio is uncertain but lies within a multivariate distribution set. Our goal is to identify optimal portfolio allocations by minimizing the worst-case conditional value-at-risk (CVaR) of portfolio loss under the reward and penalty mechanism and distribution uncertainty. Our models can also be used to investigate the problem of how to balance portfolio losses with their associated downside risk in portfolio management. We first derive an explicit closed-form expression for the worst-case CVaR under the reward-penalty mechanism, which generalizes several existing models and results regarding the worst-case CVaR, such as those studied in Jagannathan (1977), Chen et al. (2011), and Cai et al. (2024). We then apply this expression to obtain optimal portfolio allocations that minimize the worst-case CVaR under both a classical mean-covariance-based multivariate distribution set and a generalized mean-covariance-based multivariate distribution set introduced in Kang et al. (2019). Additionally, we utilize real market data to illustrate the application of the proposed models and the corresponding optimal portfolio allocations in portfolio management. Our empirical experiments show that portfolios based on the proposed models have the potential to outperform those based on several existing related models. Furthermore, the results demonstrate that incorporating downside risk into portfolio loss helps better manage risk and can achieve higher investment returns than considering either downside risk or portfolio loss alone. Moreover, our experiments reveal the trade-off between improving expected portfolio return and controlling the worst-case portfolio CVaR. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.09246 |
| By: | Paul McCloud |
| Abstract: | Entropic risk optimisation is a general framework for pricing and hedging financial derivatives in incomplete markets that can be used to decompose P&L into market and model risk contributions. When the prices are quadratic Gaussian, the coupled equations for price and hedge ratios are solved in closed form. This enables comprehensive analysis of trading P&L, with a decomposition of the model value-at-risk into convexity, dimension and funding contributions that are attributed in the explanation of realised P&L. The equations of the quadratic Gaussian model are directly applicable when the underlying prices follow Gaussian processes, such as fractional Ornstein-Uhlenbeck processes. The model also provides simple parametric expressions for hedge ratios that can be used for regressions in deep hedging. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.06168 |
| By: | Jun Cai; Zhiqiao Song |
| Abstract: | The enhanced index tracking (EIT) portfolio selection problem aims to construct a portfolio that is expected to outperform a benchmark index. In practice, investors face uncertainty in the joint distribution of asset and index losses, as the true distribution is typically unknown and only partial information is available. Moreover, portfolio selection requires balancing the trade-off between mean loss and downside risk while targeting higher returns. Effectively modeling and managing these factors in EIT portfolio selection is therefore of both practical and research interest. To address these challenges, this paper proposes robust EIT portfolio selection models under distributional uncertainty, guided by Pareto-optimality theory. The models minimize a convex combination of worst-case mean loss and downside risk, subject to a constraint on worst-case mean return. Two uncertainty sets are considered: one assuming a known mean loss vector and covariance matrix, and a more general set with unknown moments. Closed-form solutions are derived for the former, while the latter reduces to tractable convex optimization problems. Empirical results based on real market data show that the proposed models outperform not only the tracked index but also the equally weighted portfolio, robust mean-variance models, and robust variance models in terms of out-of-sample cumulative wealth, as well as Sharpe, Sortino, and Omega ratios. Moreover, portfolios that minimize a convex combination of worst-case mean loss and downside risk consistently outperform those that minimize downside risk alone. We also provide practical guidance for selecting optimal weighting coefficients to effectively balance risk and return. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.04221 |
| By: | Luisa Tibiletti; Simone Farinelli; Eric Dal Moro |
| Abstract: | Hierarchical copula models are widely used for aggregating market and credit risks across multi-level portfolio structures. Existing approaches often rely on Monte Carlo simulation, recursive sample reordering, or high-dimensional numerical procedures. This paper develops an analytical framework for hierarchical copula aggregation based on explicit finite-sum representations of aggregate loss distributions and tail-risk measures for two-level market-credit aggregation with Gaussian and mirrored Clayton copulas under empirical leaf-level marginals. We derive deterministic formulas for aggregate distribution functions, quantiles, expected tail losses, and market-credit tail contributions, together with a compatibility condition linking local hierarchical copulas and global leaf-level dependence structures. Estimation-error expressions for quantiles and expected tail losses are also obtained in finite-sum form. The paper further analyzes the qualitative asymptotic behavior of hierarchical aggregation under increasing tree depth. Gaussian hierarchies are argued to remain diversification-dominated, whereas mirrored Clayton hierarchies propagate upper-tail dependence recursively and may amplify extreme losses across aggregation levels. The resulting formulas provide a deterministic alternative to simulation-based aggregation methods and clarify how tail dependence propagates through hierarchical market-credit portfolios, with direct relevance for portfolio risk measurement and capital assessment. |
| Date: | 2026–07 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.08799 |
| By: | G. Charles-Cadogan |
| Abstract: | This paper develops a revolving-door model in which a regulator designs policy signals that affect a regulated firm's capital structure while holding an implicit career option to leave public service for the regulated firm or its advisers. The model has three linked mechanisms. First, regulatory signals alter the firm's pricing, leverage, and risk choices by changing the tradeoff between consumer surplus and firm profit. Second, because the regulator's outside payoff is convex in regulated-firm value, the career option rises with firm value, leverage, and signal-induced volatility. Third, when signal-induced volatility raises the option vega of firm equity, the same mechanism can increase tail risk and reduce distance to default. A proof-of-concept calibration illustrates leverage-vega warning regions and shows how a regulator-induced signal can move a firm from a go-leverage region into an early-warning or crash-risk region. The calibration includes maturity comparisons, sensitivity grids over compensation convexity and initial leverage, a bank-like parameter anchor, stochastic signal simulations, and a stopping-horizon diagnostic. Evidence from federal payroll data on post-employment restrictions supports the career-option primitive by showing that limits on revolving-door opportunities affect post-agency pay. The exercises also organize historical regulatory failures: Lehman Brothers as a regulatory-vacuum case, Silicon Valley Bank as a distorted-signal and weak-supervision case, and Signature Bank as a converted-signal case involving post-public-service incentives, lighter oversight, rapid growth, uninsured-deposit exposure, and tail risk. The model does not predict collapse dates; it identifies conditions under which regulatory forbearance, weak debt penalties, and career-option incentives can delay oversight until firms cross a leverage-vega warning threshold. |
| Date: | 2026–08 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.08804 |
| By: | Sriram Nagaraj; Seung Jung Lee |
| Abstract: | Financial stability rests on the premise that distress is largely idiosyncratic and therefore diversifiable: when one institution errs, the rest of the system absorbs the shock. That premise fails when many decision-makers depend on the same infrastructure. Agentic AI systems, which take consequential actions rather than only emitting predictions, are becoming such a substrate, and the binding concern shifts from the model risk of a single deployment to the systemic risk of the population. We develop this account in six mathematical settings. Chapter 1 recasts the PD/LGD/EAD decomposition of expected loss as an expected-harm identity and introduces a set-valued containment-risk measure; under a lever-assignment axiom that review cannot intercept an irreversible action, no non-preventive control satisfies a coherent tail constraint. Chapter 2 lifts this to a fleet sharing a foundation model: the shared model is a non-diversifiable common exposure whose expected-shortfall floor holds however large the fleet grows, and a percolation threshold governs contagion. Chapter 3 recasts these dynamics as a marked, Hawkes-excited jump diffusion, identifying the robust stress problem with a time-consistent entropic risk measure. Chapter 4 treats runtime guardrails as partially observed stochastic control, giving an observability trichotomy, a detection floor, and, under commit exogeneity, a runtime-impossibility corollary. Chapter 5 makes the adversary a player, yielding an underinvestment wedge proportional to systemic reach. Chapter 6 makes capability a state variable in an arms race. One claim runs through all six: the systematic component of agentic risk cannot be diversified, detected away, or reversed, and only ex-ante structural prevention moves it. |
| Date: | 2026–08 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.08806 |
| By: | Rupendra Yadav; Aparna Mehra |
| Abstract: | Expectiles are the only law-invariant risk measures that are both coherent and elicitable. Unlike Conditional Value-at-Risk (CVaR), however, they do not admit a Rockafellar--Uryasev representation that admits tractable Wasserstein reformulations. We address this difficulty by developing an envelope theorem for worst-case expectiles that characterizes the worst-case expectile over a Wasserstein ambiguity set as the unique root of a worst-case expectation with a two-piece affine integrand. This representation permits direct application of standard Wasserstein duality. Using this result, we reformulate a multi-period tri-level mean--expectile portfolio problem as four parametric linear programs with constraints. We establish four structural properties of the proposed model: an endogenously damped price of robustness, a decision-dependent critical radius beyond which the expectile tail component becomes inactive, exact recovery of the nominal model at zero ambiguity, and a characterization of how the Wasserstein ground metric determines whether the limiting portfolio becomes more concentrated or more diversified. Numerical experiments on 90 FTSE constituents over 3, 341 out-of-sample trading days show that the expectile model outperforms a CVaR model matched on ambiguity set, radius, ground metric, and trade-off weight in all nine parameter cells---significantly so whenever the radius is non-trivial. The experiments further confirm the predicted degeneracy under the ground metric. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.11917 |
| By: | Arthur Charpentier |
| Abstract: | These lecture notes introduce optimal transport as a mathematical language for actuarial science. They treat losses, premiums, scores, reserves, capital scenarios, climate losses and lifetime distributions as probability measures that can be compared, transported, averaged, stressed and interpolated. The first part develops the main tools: couplings, push-forwards, discrete and continuous Kantorovich problems, duality, Wasserstein distances, quantile transport, barycenters, entropic regularization and statistical optimal transport. The second part applies these tools to risk measures, Wasserstein robustness, pricing and capital, portfolio drift, reserving cash-flow distributions, climate-prevention diagnostics, reinsurance, dependence uncertainty, capital allocation, distributional fairness diagnostics and longevity risk. Later chapters and appendices discuss counterfactual transport, dynamic formulations, unbalanced transport, Schr\"odinger bridges, cost engineering and computational labs in R. The emphasis is on actuarial modelling choices: the state space, the ground cost, the ambiguity radius, the reference distribution and the interpretation of the transport plan. Transport maps and couplings are used as distributional objects, not as causal claims unless additional assumptions are imposed. |
| Date: | 2026–07 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.08797 |
| By: | Couaillier, Cyril; Faber, Finn; Rodriguez d’Acri, Costanza; Shaw, Frances; Passinhas, Joana |
| Abstract: | We introduce two indicators to track the build-up of vulnerabilities in the euro area banking sector on a quarterly basis by leveraging the information collected in EU-wide solvency stress tests. First, we introduce an indicator of conditional capital depletion, the Stress Vulnerability Index (SVI), which quantifies potential losses under adverse scenarios when tail credit risk is highest. Second, we introduce an indicator of earnings vulnerability, the Profitability Vulnerability Index (PVI), which assesses the capacity of the banking sector to generate capital organically. While the two indices spike synchronously during systemic crises, they can also decouple. We show that this occurs when tail risks to capital subside but earnings remain compressed, or conversely, when recessionary fears mount while current profitability strengthens. These divergences provide actionable signals for macroprudential policy by distinguishing between the accumulation of solvency risks and the impairment of loss absorption capacity. In doing so, the indices support decisions on the activation, build-up, stability or release of prudential buffers. We validate the usefulness of the indicators by showing that both indicators were strongly associated with the probability that euro area banks received state aid in the aftermath of the Great Financial Crisis. JEL Classification: E61, G01, G21, G28 |
| Keywords: | bank capital, financial stability, stress test |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:ecb:ecbwps:20263295 |
| By: | Husnu C. Dalgic; Brittany Almquist Lewis; Andrea Modena |
| Abstract: | We show that strengthening creditor rights can exacerbate financial instability, par ticularly under long debt maturities and low collateral volatility. We build a model of financial intermediaries that raise financing by pledging collateral in the sale and repurchase (repo) market. In the short term, increasing collateral pledgeability makes repo creditors safer and reduces debt rollover risk; however, it also expands long-term debt capacity and leverage, ultimately heightening run risk. This instability is a dis tinct feature of environments with long-term liabilities and low-volatility assets, where market discipline fails to constrain post-policy leverage. We test our model’s predic tions by exploiting the 2005 BAPCPA bankruptcy reform, which expanded creditor protections via repo safe harbors. Using hand-collected data on dealer repo collateral, we confirm that while the law enhanced collateral pledgeability, it fueled short-term leverage and precipitated sharper de-leveraging during the downturn |
| Keywords: | repo, collateral, safe harbor, bankruptcy, leverage, dealer banks, bank runs. |
| JEL: | G01 G21 G23 G28 G33 |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:bon:boncrc:crctr224_2025_783 |
| By: | Mihaela-Adriana Nistor; Ionel Popescu |
| Abstract: | We study a two-regime summary of a real-valued loss distribution. The two representative levels and the boundary between them are chosen by minimizing a convex residual loss. When the distribution has an atom at the boundary, assigning that atom to the lower or upper regime can give different optimized costs. Taking the better whole-atom assignment yields the lower-semicontinuous cutoff profile. We prove epi-convergence of this profile under a loss-adapted \(\psi\)-weak topology and obtain convergence of minimum values and outer stability of optimal cutoff sets. Under a unique finite atom-free limiting cutoff, positive regime masses, and unique conditional centers, the two fitted levels and a canonical one-jump representation converge as well. We also examine the resulting summary from the viewpoint of monetary risk axioms, establish explicit failures of external monotonicity and subadditivity, and give finite-support algorithms with conditional error control. A four-scenario model illustrates the cutoff geometry and the role of ties and atoms. |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.08821 |
| By: | Arno Botha; Henko Crewe; Marcel Muller; Janette Larney |
| Abstract: | The use of run-off triangles (ROTs) is a common industry practice in estimating the loss given default (LGD) risk parameter when predicting credit losses in banking. We benchmark this industry practice using credit card data against a more sophisticated (though classical) regression-based approach, which is able to leverage various types of input variables in producing loan-level LGD-estimates. This regression-based approach can demonstrably recover the typical characteristics of the 'U-shaped' empirical LGD-distribution, which the ROT-based approach cannot do. First, we critically review the ROT-based approach and identify multiple demerits using data-driven diagnostics. We then estimate a two-stage regression-based LGD-model and favourably assess the model performance of each component (or 'stage'). Finally, we aggregate the LGD-estimates produced by each approach over time, and compare each time series to the mean empirical loss rate over time. The ROT-based aggregates diverge substantially from the empirical rate over most time periods, whilst the regression-based aggregates follow the empirical trends much closer. These results underscore the greater prediction accuracy of the regression-based LGD-model, relative to the ROT-based one. By implication, the former approach is probably better than the latter ROT-based approach when estimating the LGD under the IFRS 9 accounting framework, which prioritises accuracy. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.05926 |
| By: | Andres Medina; Wei Wei |
| Abstract: | Catastrophe bonds have become a central instrument for transferring catastrophic risk to capital markets, yet their payout structures are typically specified exogenously rather than optimized. In this paper, we treat the indemnity function as the decision variable and study the optimal design of CAT bonds from the sponsor's perspective. We establish the optimality of a piecewise linear indemnity function under both expected-disutility and spectral-risk-measure frameworks and thereby reduce an infinite-dimensional design problem to a one-dimensional optimization over the attachment point. The optimal design also eliminates the potential moral hazard associated with the discontinuity of threshold payout structures. We characterize the optimal attachment point under six risk criteria and obtain explicit or semi-explicit solutions. We further derive bounds for the optimal attachment point and develop comparative statics that provide practical design insights. Using U.S. severe storm losses from NOAA's Billion-Dollar Weather and Climate Disasters database for calibration, we show that the optimal CAT bond designs achieve substantial risk reductions across most risk criteria. |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.03793 |
| By: | Graeber, Daniel (DIW Berlin); Meister, Lorenz (The Institute for New Economic Thinking at the Oxford Martin School (INET Oxford), University of Oxford); Murray, Neil (DIW Berlin) |
| Abstract: | Terrorism affects individuals far beyond its immediate victims, with large economic consequences. We estimate the effect of terrorism on risk preferences by merging geocoded panel data with the Global Terrorism Database in a staggered difference-in-differences design comparing those within and beyond 25 kilometers of attacks in Germany. Exposure reduces willingness to take risks by 2 percent of a standard deviation—localized, peaking after one year, dissipating within five—implying welfare losses near 106 million Euro annually. We classify 60, 000 news articles with a fine-tuned language model and separate two media channels: the salience and the emotional tone of coverage. Holding attack severity fixed, responses are concentrated among attacks with more extensive and more negative coverage. The survey data corroborate media transmission—respondents who do not read newspapers do not respond—and point to declining happiness as the emotional mechanism. After severe attacks, self employment falls by 2 percentage points. |
| Keywords: | Terrorist attacks, risk preferences, staggered difference-in-differences |
| JEL: | D12 D81 D91 I31 |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:amz:wpaper:2026-27 |
| By: | Yang Liu; Qiuqi Wang; Yihan Wang |
| Abstract: | Risk evaluation under distributional ambiguity is central to decision making in finance, economics, and operations research. Wasserstein balls provide a natural way to describe uncertainty around a reference distribution. We solve a natural yet open problem of robust optimization for the class of distortion riskmetrics with Wasserstein distance as the sole ambiguity constraint. This chosen objective class does not require convexity, monotonicity, and continuity of distortion functions, encompassing many common risk measures and deviation measures. First, we characterize conditions under which direct convexification preserves the worst-case value. Second, we develop a constructive method for exact worst-case evaluation when the direct convexification conditions fail. Third, we construct explicit approximate worst-case distributions and provide computable error bounds to assess their accuracy without solving the exact problem. We apply these results to distributionally robust portfolio selection and use numerical experiments to assess approximation accuracy and the resulting portfolio decisions. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.09622 |
| By: | David Beers; Obiageri Ndukwe; Joe Berry |
| Abstract: | The BoC–BoE database of sovereign debt defaults, published and updated annually by the Bank of Canada and the Bank of England, provides comprehensive estimates of stocks of government obligations in default. |
| Keywords: | Financial markets and funds management; Funds management; International markets and currencies; Financial system; Financial stability and systemic risk |
| JEL: | F F3 F34 G G1 G10 G14 G15 |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:bca:bocsap:26-48 |
| By: | Li Gan |
| Abstract: | Can a regulator promise both a risk ceiling and a development deadline when safety productivity is unknown? A ceiling below the final model's unprotected hazard requires a minimum stock of safety knowledge, so both promises hold only if weak research can be ruled out. Learning first and then replaying development, with spare compute in safety, comes close to that minimum. In a calibration with only state risk, constant safety yield, and full knowledge transfer, it needs only 3.3 percent more productivity than the necessary bound. Customer services pay for the guarantee. Rules that fix compute allocation fix dates, not risk. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.11093 |
| By: | Lokman A Abbas-Turki (LPSM); Chassagneux Jean-Fran{\c c}ois (ENSAE Paris); Jean-Philippe Lemor (LPSM); Gr{\'e}goire Loeper (LPSM); Simon Sananes (LPSM) |
| Abstract: | We develop a posteriori primal-dual bounds for numerical approximations of European option prices in the Uncertain Volatility Model. Given a smooth candidate approximation of the value function, a feedback control induced by its Hessian yields a primal lower bound. On the dual side, we derive a representation based on a matrix-valued Gamma field, which provides an upper bound through a nonnegative Hamiltonian penalty. We further show that, when the dual field is generated by the candidate itself, this upper bound admits an equivalent representation in terms of the residual of the associated Black-Scholes-Barenblatt equation. We then study discrete-time approximations of these primal and dual quantities and quantify the corresponding discretization errors. Finally, we investigate their numerical evaluation for candidates obtained by stochastic policy-gradient and physics-informed neural-network methods, and compare the resulting estimates with a martingale dual approach from the stochastic-control literature. The numerical experiments highlight the importance of derivative accuracy, and in particular of second-order information, for obtaining tight a posteriori bounds. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.06495 |
| By: | Philipp J. Schneider; Lukas Looser; Antoine Garin; Shuhan Liu; Daniel Kuhn |
| Abstract: | Deep hedging learns trading policies from historical or simulated market trajectories, yet under nonstationarity these training paths may not represent future market conditions. We propose WRAP (Wasserstein-Reweighting Adversarial Perturbation), a drift-aware adversarial training framework derived from a two-budget distributionally robust optimization (DRO) formulation. The formulation is anchored to a weighted empirical reference distribution whose fixed baseline weights are chosen to balance sampling uncertainty against temporal drift. Around this reference distribution, the ambiguity set addresses two complementary forms of distributional misspecification by allowing an adversary to reweight the observed trajectories subject to a $\phi$-divergence constraint and perturb their paths subject to an optimal-transport (OT) constraint. We derive a joint first-order expansion in which the leading-order increase over the nominal expected loss decomposes into a reweighting contribution determined by the dispersion of hedging losses across trajectories and a transport contribution determined by the sensitivity of the loss to path perturbations. This expansion yields an explicit finite-dimensional adversarial attack that replaces the distributional inner supremum with a tractable first-order approximation. Across stationary and nonstationary Heston dynamics and a generalized affine diffusion (GAD), the experiments show complementary benefits from reweighting and transport, with joint adversarial training providing the largest gains under nonstationarity. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.07162 |