Papers

Mortgage credit risk may exhibit dependence across economically connected areas, even when they are geographically distant. However, current models typically represent cross-area dependence through fixed geographical neighbourhoods, potentially overlooking economic connections that extend beyond physical proximity and evolve over time. These models may fail to capture shared risks among geographically distant but economically similar areas, and cannot describe a changing dependence of risk; it therefore risks retaining economically obsolete connections while omitting newly emerging ones. This paper uses similarities of industrial composition as a measure of economic proximity and develops a discrete-time survival model with dynamic industrial graphs to capture evolving cross-area dependence in mortgage default risk. The resulting graphs are reconstructed annually, allowing both their topology and edge weights to change as regional industrial compositions evolve. Jointly combining a proper conditional autoregressive process with a first-order autoregressive process governs the temporal cross-area dependence and the temporal persistence in latent risk simultaneously. The model is estimated using Integrated Nested Laplace Approximation and evaluated on U.S. mortgage loan-level data against the static industrial graph specifications. The results show that the proposed dynamic model achieves better in-sample fit and out-of-sample discrimination, while its estimated parameters indicate substantial graph-induced cross-area dependence and temporal persistence. These findings demonstrate the value of allowing economically defined neighbourhoods to evolve when modelling mortgage default risk.

The most widely used investment risk measures - variance, standard deviation, and tracking error - treat upside deviations as equally undesirable as downside losses. In stable markets with assumed symmetric return distributions, this simplification is potentially harmless. In the climate transition age, it is not. Climate risk is two-dimensionally asymmetric, across assets and over time. Both dimensions are negatively skewed against investors. Hence, climate transition investment strategies deliberately deviate from benchmarks that reflect an insufficiently decarbonising global economy. When fiduciaries enforce tracking error budgets that penalise outperformance just as heavily as underperformance, they inadvertently constrain exactly the portfolio return opportunities that climate-aware investing seeks. We argue that replacing symmetric deviation-based risk measures with their downside-only counterparts - semi-variance, standard risk and trailing error - resolves this conflict without sacrificing any fiduciary discipline. The distinction is not merely semantic: as the gap between climate transition trajectories and market benchmarks widens over the coming decades, symmetric risk budgets will become binding constraints on transition strategies well before actual investment risk warrants it. We propose specific changes to mandate design that would allow institutional investors to pursue climate transition opportunities within genuinely risk-aware (rather than deviation-averse) frameworks.

Expected Shortfall (ES) measures the average size of tail losses. Its sequential backtests typically also require a Value-at-Risk (VaR) forecast. We develop two e-backtesting methods using only observed losses and ES forecasts: one for iid losses only and the other for time series under an AR-GARCH conditional VaR structure. Under the stated assumptions, both control false rejections at any stopping time. For the time-series method, validity requires a fixed parameter–quantile pair representing the conditional VaR path and a positive report margin. The time-series statistic takes an infimum over possible parameter and VaR values. We choose the betting process $(\lambda_t)$ by maximizing an empirical growth rate averaged across candidate parameters. We compute the statistic by continuous VaR profiling on a dense parameter grid. For grid approximations, we prove convergence to the continuous infimum, together with convergence of the probability of false rejection, as the grid is refined over a fixed monitoring period. Lower and upper bounds for the VaR infimum on the parameter grid control the accuracy of the numerical profiling. Simulations, including ES reports that change from over-reporting to under-reporting, and an application to daily NASDAQ losses provide numerical comparisons with VaR–ES e-processes and fixed-sample ES backtests, assessing detection frequencies and the timing of rejections.

Estimation error in short, correlated pricing panels is dominated by trajectory dispersion rather than sampling noise, creating an identification shortfall that only independently assigned variation can resolve. We propose a guided iteration methodology to decide how much variation to generate, where, and for how long: a cyclic price-perturbation scheme under a revenue-at-risk budget, framed as variance minimization over design-specific error covariances, analogous to portfolio selection. We establish four key results: (1) a closed-form minimum-variance combination rule; (2) proof that a convex cost of intensity yields an interior, spread-out allocation, whereas under a concave tolerable-volume-loss cost, concentrating intensity into fewer perturbations pays off only with heterogeneous candidates, priced-in fixed costs, or perturbations beyond the cost curve's inflection point; (3) a discounted recursion yielding a data-dependent exit timeline from the experimentation zone; and (4) an application to Dominick's retail data, in which about five designed independent units match the precision of approximately 700 passive equivalents in the anchor category, while a fast proxy across ten categories achieves a median 97% variance reduction relative to passive panels. Sixteen comparisons (eleven pre-specified) support the theory: greedy selection achieves 90-93% of the global optimum across more than fifty exhaustively verifiable instances, and the recursion reaches its predicted 90%-of-steady-state benchmark two cycles early, though a parametric bootstrap indicates that its delta-method interval understates the right-tail risk. Boundary conditions show that regularized covariance estimators extend plug-in noise tolerance from 29% to beyond 50%, submodularity holds up to a clear curvature threshold, and a shared shock imposes a fundamental precision floor.

Multiplicative background risk models (MBRMs) provide a tractable framework for describing dependent risks. We construct a class of MBRMs with Pareto margins by combining Marshall–Olkin exponential and additive gamma risk factor structures, each allowed to incorporate idiosyncratic and systemic background risk factors. Our main result establishes, under an admissibility condition on the risk factor systems, a finite sub-mixture representation of the joint survival function in terms of Pareto survival functions, whose weights are Dirichlet probabilities of orderings induced by the background risk factors and sum to at most one. This yields a new interpretation of such MBRMs, in which dependence is expressed through Pareto survival functions and their Dirichlet weights. The resulting family encompasses many known multivariate Pareto constructions, is closed under marginalization, and admits characterizations of independence and, under a non-degeneracy condition, of its singular and absolutely continuous parts. It is associated, its covariance decomposition separates the contributions of the two risk factor structures and their interaction, and its joint tails are regularly varying along rays. The presence of asymptotic tail dependence and the decay rates of joint exceedance probabilities are governed by the gamma risk factors alone; the exponential risk factors affect only the tail constants.

AI agents built on different language models can still share the software that presents their choices. We study one shared component, the order in which a borrowing agent's actions are listed, in a simulated collateralized lending market populated by three open-weight model families. In fixed decision states, moving the borrowing action from last to first raises valid borrowing requests by 27 percentage points. In 12 matched pairs of 15-agent markets, all agents in one market see the same order; the other spreads the six possible orders across agents. A common order moves whole markets: the four markets that listed borrowing first raised net debt by 1.8 to 2.7 times their initial unused borrowing capacity, while those that listed debt reduction first cut debt. These groups account for 96 percent of the sample variation among common markets, whose standard deviation is three times the balanced one. The borrowing cap stops 40 percent of valid borrowing requests made when borrowing is listed first, but it values collateral at the pool price, so correlated purchases raise borrowing capacity and leave positions above 70 percent of reference value. Under a 30 percent shock, an 8 percent liquidation bonus produces write-offs on loans that are fully collateralized before liquidation, and liquidation triggers that read the pool price add exposure. Common ordering does not put more borrowers into the write-off zone; it concentrates them in the markets whose shared order favored borrowing. Collateral valuation and liquidation rules determine how much of it creditors fail to recover.

A defined-benefit pension is only as safe as its sponsor, yet sponsor distress is under-instrumented relative to plan-side modelling. We present a transparent, reproducible early-warning pipeline built entirely on public data. A Sponsor Vulnerability Index (SVI), a gradient-boosted classifier over point-in-time SEC EDGAR balance-sheet fundamentals and two macro series, predicts a filings-based distress state one to two quarters ahead. On 36 U.S. sponsors in five sector groups (1,418 firm-quarters, 2015Q1-2025Q3) it reaches an out-of-fold AUC-ROC of 0.887 and PR-AUC of 0.711 (base rate 11.9%) under leave-one-firm-out cross-validation. A Monte Carlo VaR with a stylized transition-stress overlay shows that climate-type stress is negligible for a diversified plan but adds 20-43% to one-year 95% VaR when 22.5% of assets are held in sponsor securities. A dynamic self-investment cap maps the SVI to a limit that falls from 8% to below 1% in acute distress and recovers with the sponsor, which we illustrate on PG&E's 2019 bankruptcy and Boeing's persistent fragility. We report the method's failure modes alongside its results (book-equity labels that also fire on non-credit events such as spin-offs, positives concentrated in one firm, and stylized stress parameters) and release code, data and results under a concept DOI.

Correlation matrices combine nonlinear positive-definiteness constraints with a simultaneous row-column permutation symmetry. Their generation therefore involves two distinct problems. First, every output must remain on the correlation elliptope, irrespective of how its variables are ordered. Second, the generator must learn the domain-specific distribution occupied by empirical data inside that valid space. Existing financial image generators mix these problems: they learn ordered matrix entries and repair invalid draws afterward. We introduce CorrFlow, which separates them. A generalized Fisher transform maps full-rank correlation matrices bijectively to unconstrained coordinates, guaranteeing valid outputs without projection. A graph flow-matching network learns the empirical distribution in those coordinates and is permutation equivariant: reordering the variables only reorders the output. We compare a conditional CorrGAN reimplementation, image diffusion, and CorrFlow on $80\times80$ Taiwan equity matrices using five seeds per neural method. CorrFlow produces 100% valid draws without repair and obtains feature sliced-Wasserstein distance $.712\pm.141$, versus $4.339\pm.252$ for CorrGAN and $4.610\pm.191$ for image diffusion after their invalid outputs are projected back to the correlation space. It also improves this distance over historical lookup, while lookup retains higher local coverage. These results support the combined construction under the evaluated training budget; they do not establish superiority over fully tuned image models or isolate the contribution of each component.

A realized tail-risk conditional autoregressive expectile (RTCARE) model is proposed for joint forecasting of value-at-risk (VaR) and expected shortfall (ES). Realized tail measures such as the realized quantile and the realized ES, constructed from intraday high-frequency returns, are incorporated into the dynamics and measurement equations of the conditional expectile, allowing information about lower-tail risk to enter the forecasts. The parameters are estimated by Markov chain Monte Carlo, and we evaluate the resulting VaR-ES forecasts in numerical examples and in an empirical application using U.S. stock-return data. Across the considered tail levels, the RTCARE models deliver competitive and often superior forecasting performance relative to existing alternatives, with stable results across Monte Carlo replications and rolling forecast evaluations.

Background: Credit risk modelling is fundamental in financial institutions, typically using credit history, repayment behaviour, income, and outstanding debt to estimate a borrower's probability of default. Although transactional data are sometimes incorporated through aggregated measures such as recency, frequency, and monetary indicators, behavioural information embedded in raw transaction narratives remains largely unexplored due to its unstructured nature. This limits opportunities to extract richer behavioural insights. Aim: This research investigates payment narratives from transactional data as an alternative data source for credit risk modelling. Setting: The study uses transactional data from a large South African financial institution focused on small and medium enterprises. Method: Techniques such as natural language processing and regular expressions are used to transform unstructured transaction descriptions into meaningful, explainable behavioural features by categorising transactions into interpretable spending categories. A novel categorisation algorithm was developed and evaluated for classification accuracy. The extracted features are incorporated into a credit risk scorecard to assess their predictive value. Results: The categorisation algorithm demonstrates high classification accuracy. Using these features in the scorecard resulted in a significant increase in Gini compared to a scorecard without them. Conclusion: Text-based transactional information can be systematically extracted, categorised, and operationalised to enhance credit risk model performance. Contribution: The study develops a novel, integrated, explainable methodology for transforming transactional text into behavioural features, introduces a structured RegEx-based categorisation approach, and demonstrates their predictive value within traditional scorecard frameworks.

I study how intermediary liquidity affects mortgage resolution using 1.9 million Ginnie Mae buyout options from 2019 to 2022. When funding markets froze in 2020, nonbank issuers repurchased far fewer delinquent loans than depositories servicing loans in the same pools and, within the same issuer and month, shifted repurchases away from large balances, the loans that require the most cash. About $19 billion of repurchases were delayed, and most were completed later. A model of rationing by value per dollar of cash explains this pattern and shows that borrower information is most valuable when funding is intermediate.

Large language models are stochastic by nature. Repeated executions under the same experimental conditions may produce different responses. We propose a framework that can be applied across the model lifecycle, from task design to residual behavior and continuous monitoring. The method consists of building an empirical response distribution estimated through Monte Carlo simulation. Semantic interventions are introduced to study how these distributions change. We illustrate the framework with two different tasks, borrower classification and music album recommendation, and quantify the resulting distributional changes. The framework is linked to model risk management and governance by measuring robustness and control effectiveness, and characterizing residual behavior. Since it relies only on observed responses, it can also be applied to black-box models accessed through commercial APIs.

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.

Regulated credit scoring requires scores monotone non-decreasing in every exposure input. Deployed pipelines – hand-crafted monotone aggregates feeding sign-constrained gradient boosting – already meet this by composition; the open question is what learned temporal aggregation is worth inside one. We answer on five production-scale credit datasets at matched admissibility (one priced baseline convention excepted), with a monotone recurrent architecture whose per-input guarantee we extend, with proofs, to vector-valued inputs and to exogenously macro-conditioned decay gates, severities, thresholds, and peak memory. Two findings result. First, a strictness ladder: the value of learned monotone features rises with governance-frame strictness – zero on unconstrained engineered panels, maximal in summaries-only frames – replicated across two datasets and an official temporal-stability metric, though unconditioned features degrade on externally adjudicated later weeks. Second, a conditioning-delivery asymmetry under regime shift. On a train-on-boom, test-on-crisis mortgage design, two public macroeconomic series hurt as input columns, yet conditioning the recurrence on them delivers the paper's only learned-block crisis-cohort uplifts. The confirmed effect: +0.006 to +0.013 AUC on an internally pre-registered Freddie Mac replication, at all five held-out seeds. The discovery estimate: +0.015 to +0.021 on Fannie Mae (three of five seeds post hoc), worth 10-27 basis points of defaulted balance at an 80% approval cutoff, and grows with early-prepaid loans excluded. A state-level test identifies the mechanism: between-cohort calibration transfer. A pandemic-band episode bounds scope: under forbearance-distorted labels the gain generalizes at a quarter to a third of crisis size on Fannie Mae, on Freddie Mac only against the capacity control.

This paper proposes a framework for modeling probability of default via functional data analysis. By representing a series of credit variables as functions, we investigate whether intra-monthly information improves default predictability in linear models. We further show how a range of widely used variables, among them available funds, utilization rate, and overdraft, can all be derived from three quantities observed over time. Namely, 1) the type of each account, 2) the balance of the account, and 3) the size of the credit limit on the account. Retaining these quantities as continuous-time processes, rather than reducing them to monthly aggregated values, yields a continuous faithful representation of the borrower. When analyzing these three processes, we discovered that recurring events associated with the ordinal position among banking days created strong cyclical patterns. We therefore develop a relative time framework that aligns the recurring events across borrowers, ensuring that borrowers possess the same cyclical pattern, regardless of real time. We assess the framework using functional logistic regression. This approach accommodates the continuous representation while remaining closely related to a logistic regression model commonly used in credit risk practice, due to strict regulatory constraints. We show that, when equipped with an effective functional representation of transactional trajectories, the proposed model attains predictive performance on par with XGBoost while consistently outperforming logistic regression. Importantly, the model balances predictive performance of a machine learning model with the interpretability of linear default models, potentially enabling financial institutions to use the model, even under strict regulation.

In actuarial science and quantitative finance, convex order provides a natural way to compare risks with the same mean. In dimension one, convex order is well understood through several characterisations. In higher dimensions, a natural approach is to compare all one-dimensional projections, but, although necessary, the resulting condition is in general not sufficient for multivariate convex order. A simple counterexample due to Pinelis exploits Popoviciu's inequality. We revisit this example using gauge functions, and show that gauge functions themselves do not suffice to characterise multivariate convex order for symmetric distributions, even in dimension two. We then show that the radial case makes the source of the failure transparent and construct radial counterexamples in every dimension at least two. Nevertheless, projection tests are sufficient for some important families of distributions. In particular, this is the case for linear transformations of a common radially distributed random vector. This framework includes elliptical distributions generated from a common radial distribution and, in particular, centred Gaussian distributions. Beyond comparisons between Gaussian distributions, Jourdain and Pagès recently studied the comparison of a Gaussian distribution with a Gaussian mixture, obtaining necessary conditions and sufficient conditions whose equivalence was left open in general. In this paper, we settle the comparison of a centred Gaussian distribution with a mixture of two centred Gaussian distributions: in every finite dimension, the Gaussian is dominated by the mixture in multivariate convex order if and only if all one-dimensional projections are ordered in convex order.

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.

Credibility theory combines individual experience with portfolio information for insurance pricing, but classical formulations focus primarily on conditional means and expected premiums. We propose a Dirichlet mixed-membership model (DMMM) for multivariate distributional credibility. Policyholder risk is represented by a stable composition over latent risk classes: baseline characteristics determine its a priori assessment, while repeated multivariate experience progressively updates its a posteriori assessment. Support-specific expert distributions accommodate heterogeneous outcomes, with dependence induced through the shared latent structure. The resulting posterior predictive distribution can be written exactly as a convex combination of portfolio and experience components, extending the familiar credibility-factor structure beyond the conditional mean. We study the framework through a simulation experiment and a vehicle telematics application. The simulation shows that relatively simple experts can capture complex multivariate insurance distributions and learn policyholder-specific risk as experience accumulates. In the telematics application, recent claims and driving behaviour provide complementary information for future claim-frequency prediction, allowing similar policyholders to receive different experience-rated assessments. The DMMM therefore provides a flexible and interpretable framework for combining heterogeneous insurance experience while retaining the structure of classical credibility.

Risk regulation imposes directional constraints on scores; we adopt their strict per-input form – the score monotone non-decreasing in every exposure input – as a normative commitment. Deployed pipelines – monotone hand-crafted aggregates feeding sign-constrained gradient boosting – already satisfy it by composition, so constrained-versus-unconstrained comparisons price a guarantee the incumbent has for free. We instead hold admissibility fixed on both sides and measure what learning the aggregation is worth. Our instrument is a recurrent network whose state is classical risk statistics (an exponentially weighted moving average and a high-water mark with learned transforms), monotone by construction in every input and per MC-dropout sample. The central finding, by functional regression, is a subsumption boundary: a learned monotone channel reproduces the geometrically weighted separable family of hand-crafted statistics, one channel per member, to Spearman $\rho \ge 0.996$, approximates window statistics with measurable ceilings, and fails at consecutivity ($\rho = 0.924$) and time localization (0.628), both structural, and at the exposure floor (0.829), a learnability boundary. One explicit admissible basis repairs each failure (rank correlation 1.000). In or near the separable family, learned and engineered aggregation are substitutes, and the learned channel is never statistically behind at full sample size and specified capacity. Its advantages are incumbent-specific: a committed grid pays up to 0.019 AUC in decay regions it leaves uncovered (the learned channel stays within 0.004 of the strongest engineered consumer at every swept point); the highest-dimensional comparator degrades fastest with scarce data; and beyond the training support, grid-fed tree-ensemble scores go flat while a strictly increasing head keeps ranking. No single incumbent is dominated on all three axes.

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.

Feature selection for imbalanced classification tasks such as credit card fraud and consumer default detection requires balancing predictive relevance, inter-feature redundancy, and computational feasibility. We benchmark three computing paradigms, classical branch-and-bound optimization (Gurobi), photonic entropy computing (QCI Dirac-3), and simulated photonic boson sampling (Piquasso), across thirteen feature-selection methods on two datasets: ULB Credit Card Fraud (30 features) and AmEx consumer default (159 features). Each method is routed to the solver matched to its mathematical structure. On ULB, Dirac-3 MI-Spearman matches the all-features model using 13 of 30 features (mean F1 0.873 +/- 0.023 over five runs, best run 0.896), and Piquasso is the best method at k=5. On AmEx, performance rises steadily with the feature budget and every paradigm approaches F1 = 0.80 only near the full feature set. Most differences between Gurobi and Dirac-3 on identical methods fall within run-to-run variation; the large gaps occur where the certified optimum generalizes poorly, most sharply for distance correlation on AmEx at k=25 (Gurobi F1 = 0.422 vs. a Dirac-3 mean of 0.746). At matched budgets, F1 varies about ten times more across methods on ULB than on AmEx, which we trace to how concentrated the predictive signal is in each feature space.

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.

Bank treasury portfolios must balance yield, liquidity, and interest-rate risk across bonds of different maturities. Static allocation rules are ill-suited to this task: portfolios concentrated in long-duration securities with no dynamic adjust- ment mechanism can accumulate large mark-to-market losses and liquidity stress under rising interest rates, as illustrated by the failure of Silicon Valley Bank in 2023. We develop a tractable simulation-based framework for multi-period bond port- folio optimization under interest-rate risk and proportional transaction costs. Yield-curve dynamics are modeled using the Dynamic Nelson-Siegel parameter- ization with Vector Autoregressive factor dynamics, from which we construct a time-inhomogeneous discrete-state Markov chain approximating the joint yield process across bond maturities. This chain forms the state space of a finite- horizon Markov Decision Process in which the investor maximizes expected terminal wealth subject to proportional rebalancing costs. The optimal portfolio policy is obtained by backward induction. We also quantify the approximation error introduced by truncating the transition kernel, and show that it leaves mean terminal wealth almost unchanged while substantially distorting drawdown and tail statistics.

We compare two natural proportional notions of a risk component's contribution to the aggregate tail risk of a collection of risks: the fraction of aggregate tail risk capital allocated to the component under Conditional Tail Expectation (CTE), and the component's expected realized share of aggregate risk under Geometric Tail Expectation (GTE). The resulting proportional allocations generally differ. For arbitrary random vectors-allowing atoms, signed risks, and any tail domain–we establish when the allocations agree, determine their ordering when they do not, and characterize their asymptotic separation. Both proportional allocations are weighted averages of the conditional risk share of a component given the aggregate: the proportional CTE allocation weights tail scenarios by severity, whereas the proportional GTE allocation weights them uniformly. Their difference is therefore a normalized tail covariance. The proportional CTE and GTE allocations agree throughout a tail exactly when the conditional risk share is constant there; under a mild unimodality condition, dominance is characterized by its monotonicity. Among independent exponential dispersion models with heterogeneous natural parameters, exact agreement is possible only for the scaled Poisson family; under comonotonicity, it is equivalent to proportional quantile functions. In the extreme tail, the limiting relationship between the proportional CTE and GTE allocations is governed by the ratio of Expected Shortfall to Value-at-Risk: boundedness ensures common limits, convergence to one forces the allocations to merge, and divergence can cause their limits to separate.

We introduce a deep kernel hedging framework that combines the flexibility of deep learning with the structural inductive bias of kernel methods. The hedging functional is restricted to a reproducing kernel Hilbert space whose kernel is parameterized through a neural network embedding of the input features. The framework minimizes a regularized empirical risk under convex loss functions and can accommodate path-dependent information through truncated time-augmented signature features. We derive a generalized representer theorem for the joint hedging problem, reducing the empirical optimization to a finite-dimensional problem. To further reduce the computational cost associated with large kernel matrices, we develop a scalable random Fourier feature approximation and establish convergence guarantees. The random Fourier parameters are sampled once and remain fixed throughout training, while the deep kernel adapts to market data through the learned neural representation. We evaluate the performance of the proposed deep kernel approach on both synthetic and real data and compare it with standard kernel methods and classical deep hedging architectures. Numerical results indicate competitive and robust hedging performance, particularly in low-data regimes, which highlights the benefits of combining expressive neural representations with the inductive bias of kernel methods.

The internal-models approach of the Fundamental Review of the Trading Book (FRTB-IMA) determines market-risk capital at the aggregate level of a bank's approved trading desks, but it does not say how that charge should be attributed to individual trades. Risk managers, capital planners and validators often need one, and it is hard to obtain: capital passes through expected-shortfall, stress-scaling, non-modellable, default-risk, history and standardised-approach layers that are nonsmooth and depend on past positions. We present a formula-complete attribution of FRTB-IMA capital to (observation date, trade) positions that reconciles to computed capital. The calculation is written as a computational graph; after one forward evaluation, a single reverse pass propagates allocation vectors through every node by local Euler rules (expected-shortfall dual weights, active-branch weights at floors and maxima, the default-quantile scenario), avoiding one capital re-evaluation per trade. Under stated homogeneity assumptions the ledger provably sums to capital and to any desk or product grouping, and equals the Euler (gradient) allocation at smooth points; it is exact relative to the implemented calculation and this rule, not a regulatorily prescribed attribution. Its past-date entries capture capital allocated to earlier observation dates, which today's-book sensitivities miss by an exact identity at smooth points. At regulatory ties, where the marginal is not unique, we report the realisable marginal vertices, identified by exact linear-programming tests, instead of one selection. On 14 smooth 96-trade synthetic benchmark books, the median share of capital allocated to past observation dates is 61% (range 21%-99%), and ledgers reconcile to within 5.6e-16 relative error.

We consider an insurance market with hidden information, where the agent's type is private information and is drawn from an arbitrary type space. We study implementability of a collection of retention functions, namely, how to select premium schedules so that the resulting menu of contracts is incentive compatible, or truthful. Specifically, for general type spaces, implementability is equivalent to cyclical monotonicity of the collection of retention functions. For compact interval type spaces, we show that submodularity is a sufficient condition for implementability, under suitable type ordering assumptions. Moreover, for any implementable collection of retention functions, we characterize all corresponding premium schedules, up to a common additive constant. Finally, we apply our results to several standard classes of insurance contracts, for which the general implementability conditions admit simpler characterizations, and we provide several numerical illustrations.

Researchers increasingly use generative large language models (LLMs) to convert corporate text into empirical variables. We examine the extent to which LLM-based textual measures are invariant to model choice using thirteen measures, including sentiment, management clarity, uncertainty, answer specificity, and climate and political risk. Seven LLMs from different providers score earnings call transcripts of S&P 500 companies on these constructs. Cross-model rank correlations average only 0.52, and transcript-level differences common across providers account for only 34% of total score variation. Cross-model disagreement does not predict subsequent analyst or market disagreement, consistent with a substantial model-specific component rather than common ambiguity in the underlying disclosure. Model choice significantly affects downstream inference, with coefficient magnitudes, signs, and statistical significance varying substantially across models. Averaging across providers makes transcript rankings more stable for most constructs, but score levels remain sensitive to the models included in the ensemble. LLM-generated variables should therefore be treated as model-contingent measurements and validated across providers.

Brazil's Pix settles about 5.9 billion instant, irreversible transfers a month. A fraudulent transfer can be recovered only while the funds remain in a traceable account, and in 2025 the Central Bank's recovery mechanism (MED) returned 9% of accepted contested value. Interdiction therefore has to happen before settlement, by routing each transaction to pass, human review or block, under a finite analyst team and a regulatory hold window. To our knowledge no public simulator jointly models irreversible settlement, a regulated recovery mechanism, downstream fund dispersal and capacity-constrained review. We present PixSim, an open-source simulator of the Pix rail with these elements, calibrated to Banco Central do Brasil open data, with every parameter sourced, calibrated to one published observable, or registered as an assumption. With the model frozen, full-scale runs reproduce the 2025 recovery rate within 0.006 and its decomposition within 0.02; the February-April 2026 window is reported as a misfit and the May 2026 tracing regime as a projection. On a benchmark with a payer-side scorer, four reference policies and ten scenarios, within the simulated mule model: recovery after settlement is constrained by dispersal speed; staffing by the arrival profile cuts a fixed rule's alert expiry from 52% to 2% at constant hours; halving the team removes a fixed threshold-and-block rule's advantage over a queue-aware rule, on loss and on loss plus false-block harm (+0.106 of victim value, positive on all twenty paired seeds), while a reversal at two thirds of the team was not confirmed on independent seeds; and a synthetic scorer of held-out AUC 0.82 cuts lost value by about a quarter. Code and data: https://doi.org/10.5281/zenodo.22948895

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.