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19 pages, 1696 KB

Open AccessArticle

The Kerper–Bowron Method: Additional Notes on Manufacturer’s Warranties and Collateralization Applications

by Lee Bowron, John Kerper, Alice Lightfoot and Wheeler Bowron

Risks 2026, 14(9), 213; https://doi.org/10.3390/risks14090213 - 14 Sep 2026

Viewed by 143

The Kerper–Bowron (KB) Method (patent pending) projects expected claims at the individual contract level. This paper extends that cash-flow engine to manufacturer-warranty accruals and to three proposed uses: collateralized risk transfer, lending against service-contract equity, and risk-adjusted customer lifetime value (CLV). Manufacturer warranties [...] Read more.

The Kerper–Bowron (KB) Method (patent pending) projects expected claims at the individual contract level. This paper extends that cash-flow engine to manufacturer-warranty accruals and to three proposed uses: collateralized risk transfer, lending against service-contract equity, and risk-adjusted customer lifetime value (CLV). Manufacturer warranties and separately priced service contracts are treated as related but distinct products under ASC 460, ASC 450, and IAS 37. Expected cost per unit of exposure is formed with a generalized linear model; a Tweedie mean–variance function is used as a working choice, not as a tested warranty distribution. Accident-month estimates are allocated to payment months, incurred-but-not-reported cost on pre-valuation months is isolated, and remaining paid cash flow is split into pre-valuation runoff and post-valuation occurrence. One present-value risk margin is taken from the predictive distribution as the present value of the gap between a stated percentile and the mean; a constant loading on the discounted mean is an illustrative substitute when simulation is not run. The contribution is contract-level granularity and a single paid path that can be refreshed as experience and assumptions change. The same paid path can be used as a financial-monitoring tool: expected claims, equity, and risk-adjusted values can be refreshed as time passes, actual results emerge, and model or economic assumptions change. Accuracy, balance sheet derecognition, investor diversification, and lendable capacity would be the subject of further research. Full article

(This article belongs to the Special Issue Advances in Risk Models and Actuarial Science)

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42 pages, 1362 KB

Open AccessFeature PaperArticle

Dynamic Non-Life Insurance Pricing Under Delayed Claim Reporting: A Partially Observed Risk-Sensitive Control Framework

by Desmond Marozva, Selah Tanaka Marozva and Ştefan Cristian Gherghina

Risks 2026, 14(9), 208; https://doi.org/10.3390/risks14090208 - 10 Sep 2026

Viewed by 173

Abstract

Non-life insurance pricing is forward-looking, yet its principal cost signal, reported claims, is delayed by the occurrence-to-reporting process. The rating cell is treated as a stylised, homogeneous unit without renewal, lapse, expiry, or cohort dynamics; a fully annual-contract formulation would require these dynamics [...] Read more.

Non-life insurance pricing is forward-looking, yet its principal cost signal, reported claims, is delayed by the occurrence-to-reporting process. The rating cell is treated as a stylised, homogeneous unit without renewal, lapse, expiry, or cohort dynamics; a fully annual-contract formulation would require these dynamics to be modelled jointly with the risk and claims processes. This paper develops a partially observed risk-sensitive control framework in which premium-sensitive exposure generates claims in a latent risk regime, while unreported claims form an atomic population governed by an age-structured transport equation. The numerical instance solved and validated in this research restricts the general model to a memoryless (one-phase) reporting process for tractability. It is best matched to lines with predominantly short-to-medium reporting tails, rather than to the most extreme long-tailed liability or cyber exposures the general model is designed to eventually accommodate. A finite-state reporting reservoir and nominal Bayesian filter provide the decision state, and compound Poisson–Gamma loss enters an entropic Bellman recursion through a closed-form exponential-tilting identity, checked against an independently coded Bellman-residual test. Every dynamic policy is benchmarked against alternatives matched at the same risk-sensitivity parameter and evaluated using common random numbers. This is a general feature of the results, not a single statistic:

C E_{0.8}

is a standardised yardstick applied uniformly across strategies, while

C E_{γ}

at the policy’s own

γ

is what that policy actually optimises, and the two need not agree. In 3000 out-of-model paths at

γ

= 1.2, the delay-aware dynamic policy increases mean profit by EUR 0.148 million and a standardised

γ_{eval}

= 0.8 certainty equivalent by EUR 0.034 million relative to a matched static price. Its fifth percentile and TVaR 5%, by contrast, are lower by EUR 0.056 and 0.078 million. At the policy’s own optimisation level, however, the

γ_{eval}

= 1.2 certainty-equivalent difference is EUR −0.006 million, with a 95% interval reaching zero. The dynamic policy, therefore, does not clearly outperform the matched static price on the exact objective it was optimised to maximise. Matched comparisons attribute EUR 0.039–0.043 million of mean profit to reporting-delay modelling and EUR 0.017 million to dynamic continuation. Separating state observation from transition-law knowledge attributes EUR 0.161–0.182 million to observing the regime exactly, and a small, sign-changing EUR −0.010 to +0.003 million to knowing the true transition law itself. Across an eight-scenario misspecification stress suite, the dynamic policy’s mean-profit advantage over the matched static price is directionally robust in seven of eight scenarios, but it reverses sign under a +30% true-severity shock, indicating that this advantage is sensitive to substantial severity misspecification specifically. Grid, Bellman-residual, and out-of-model filter diagnostics indicate that state reconstruction is economically primary, while entropy risk sensitivity is a secondary overlay whose apparent benefit depends materially on which certainty-equivalent level and evaluation model are used to judge it. Full article

(This article belongs to the Special Issue Advances in Risk Models and Actuarial Science)

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19 pages, 748 KB

Open AccessArticle

Modeling Data with Nonlinear LogNormal–Pareto Regression via the Approximate Bayesian Computation

by Mostafa S. Aminzadeh

Risks 2026, 14(7), 165; https://doi.org/10.3390/risks14070165 - 16 Jul 2026

Viewed by 518

Abstract

The development of regression models for composite distributions has received insufficient attention in the literature. The purpose of this research is to provide maximum likelihood (ML) and approximate Bayesian computation (ABC) estimators for the parameters of a regression model with a response variable [...] Read more.

The development of regression models for composite distributions has received insufficient attention in the literature. The purpose of this research is to provide maximum likelihood (ML) and approximate Bayesian computation (ABC) estimators for the parameters of a regression model with a response variable following the LogNormal–Pareto composite distribution. Composite models such as Exponential–Pareto, LogNormal–Pareto, and Inverse-Gamma–Pareto, which separate small-to-moderate and significant losses using a threshold parameter, have been developed using classical and Bayesian methods and applied to insurance data. We derive closed-form formulas for MLEs in regression models with two and three covariates. For models with more than three covariates, we provide MLEs using Mathematica code. Simulation studies show that the ABC method is more accurate than the ML method. Mathematica code written specifically for the computations of the proposed methods is provided. Full article

(This article belongs to the Special Issue Advances in Risk Models and Actuarial Science)

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33 pages, 820 KB

Open AccessArticle

The Kerper–Bowron Method: A Foundational Change for Service Contract Claim Estimation and Accounting

by John Kerper and Lee Bowron

Risks 2026, 14(3), 44; https://doi.org/10.3390/risks14030044 - 24 Feb 2026

Viewed by 2496

Abstract

The Kerper–Bowron Method (KB Method) is a patent-pending approach that revolutionizes service contract loss estimation and accounting by introducing a precise, contract-level approach to forecasting expected losses and cancellations. Building on a prior 2007 paper, this update presents the Earned Contract formula, aligning [...] Read more.

The Kerper–Bowron Method (KB Method) is a patent-pending approach that revolutionizes service contract loss estimation and accounting by introducing a precise, contract-level approach to forecasting expected losses and cancellations. Building on a prior 2007 paper, this update presents the Earned Contract formula, aligning with Solvency II and modern accounting standards. By leveraging a probabilistic exposure base and Generalized Linear Models, the KB Method enhances accuracy in claims and cancel liabilities as well as other liability and asset estimates across global service contract markets. This methodology offers superior precision, automation, and compliance, redefining actuarial and financial practices for vehicle and other service contracts. Full article

(This article belongs to the Special Issue Advances in Risk Models and Actuarial Science)

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45 pages, 9784 KB

Open AccessArticle

Building a Life Table for Lebanon: Towards a Deeper Understanding of Our Future

by Natalia Bou Sakr, Stéphane Loisel, Gihane Mansour and Yahia Salhi

Risks 2026, 14(2), 34; https://doi.org/10.3390/risks14020034 - 5 Feb 2026

Viewed by 1631

Abstract

Lebanon does not have a national mortality table that reflects its demographic and health conditions. Despite ongoing changes in mortality patterns driven by economic crises, political instability, and social changes, outdated foreign tables such as AM80 remain in use in the insurance and [...] Read more.

Lebanon does not have a national mortality table that reflects its demographic and health conditions. Despite ongoing changes in mortality patterns driven by economic crises, political instability, and social changes, outdated foreign tables such as AM80 remain in use in the insurance and public sectors. This dependency introduces significant risks in actuarial calculations, policy design, and long-term planning. This study addresses this gap by building a mortality table specifically adapted to the Lebanese insurance context, together with a first estimation of population-level mortality. In the absence of any official mortality database, we collaborated directly with local insurance companies to access and organize internal records of insured lives. These data, which represent one of the few available structured sources of mortality information in the country, form the core of our analysis. We apply actuarial methods to estimate age-specific death rates and life expectancy and benchmark the results against national and international references to assess consistency and range. By offering a locally grounded, data-driven alternative to imported mortality assumptions, this work fills a critical statistical need. The resulting table supports more accurate forecasting, pricing, and demographic modeling, with applications across insurance, pensions, and public health planning in Lebanon. Full article

(This article belongs to the Special Issue Advances in Risk Models and Actuarial Science)

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Graphical abstract

17 pages, 1157 KB

Open AccessFeature PaperArticle

Numerical Calculation of Finite-Time Ruin Probabilities in the Dual Risk Model

by Rui M. R. Cardoso and Andressa C. O. Melo

Risks 2025, 13(9), 174; https://doi.org/10.3390/risks13090174 - 11 Sep 2025

Viewed by 1708

Abstract

In the dual risk model, while the ultimate ruin probability has an exact and straightforward formula, the mathematics becomes significantly more complex when considering a finite time horizon, and the literature on this topic is scarce. As a result, there is a need [...] Read more.

In the dual risk model, while the ultimate ruin probability has an exact and straightforward formula, the mathematics becomes significantly more complex when considering a finite time horizon, and the literature on this topic is scarce. As a result, there is a need for numerical approximations. To address this, we develop two numerical algorithms that can accommodate a wide range of distributions for the amount of individual earnings with minimal adjustments. These algorithms are grounded in the methodologies proposed by Cardoso and Egídio dos Reis (2002) and De Vylder and Goovaerts (1988), which involve approximating the continuous risk process with a discrete-time Markov chain framework. We work out some examples, providing approximate values for the density of the time to ruin, and we compare, in the long run, our approximations with the exact values for the ultimate ruin probability to evaluate their accuracy. We also benchmark our results against the few existing figures available in the literature. Our findings suggest that the proposed approaches offer an efficient and flexible methodology for computing finite-time ruin probabilities in the dual risk model. Full article

(This article belongs to the Special Issue Advances in Risk Models and Actuarial Science)

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22 pages, 1833 KB

Open AccessArticle

Survival Analysis for Credit Risk: A Dynamic Approach for Basel IRB Compliance

by Fernando L. Dala, Manuel L. Esquível and Raquel M. Gaspar

Risks 2025, 13(8), 155; https://doi.org/10.3390/risks13080155 - 15 Aug 2025

Viewed by 4701

Abstract

This paper uses survival analysis as a tool to assess credit risk in loan portfolios within the framework of the Basel Internal Ratings-Based (IRB) approach. By modeling the time to default using survival functions, the methodology allows for the estimation of default probabilities [...] Read more.

This paper uses survival analysis as a tool to assess credit risk in loan portfolios within the framework of the Basel Internal Ratings-Based (IRB) approach. By modeling the time to default using survival functions, the methodology allows for the estimation of default probabilities and the dynamic evaluation of portfolio performance. The model explicitly accounts for right censoring and demonstrates strong predictive accuracy. Furthermore, by incorporating additional information about the portfolio’s loss process, we show how to empirically estimate key risk measures—such as Value at Risk (VaR) and Expected Shortfall (ES)—that are sensitive to the age of the loans. Through simulations, we illustrate how loss distributions and the corresponding risk measures evolve over the loans’ life cycles. Our approach emphasizes the significant dependence of risk metrics on loan age, illustrating that risk profiles are inherently dynamic rather than static. Using a real-world dataset of 10,479 loans issued by Angolan commercial banks, combined with assumptions regarding loss processes, we demonstrate the practical applicability of the proposed methodology. This approach is particularly relevant for emerging markets with limited access to advanced credit risk modeling infrastructure. Full article

(This article belongs to the Special Issue Advances in Risk Models and Actuarial Science)

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12 pages, 1125 KB

Open AccessArticle

Algorithmic Trading System with Adaptive State Model of a Binary-Temporal Representation

by Michal Dominik Stasiak

Risks 2025, 13(8), 148; https://doi.org/10.3390/risks13080148 - 4 Aug 2025

Viewed by 4981

Abstract

In this paper a new state model is introduced, an adaptative state model in a binary temporal representation (ASMBRT) as well as its application in constructing an algorithmic trading system. The presented model uses the binary temporal representation, which allows for a precise [...] Read more.

In this paper a new state model is introduced, an adaptative state model in a binary temporal representation (ASMBRT) as well as its application in constructing an algorithmic trading system. The presented model uses the binary temporal representation, which allows for a precise analysis of exchange rates without losing any informative value of the data. The basis of the model is the trajectory analysis for the ensuing changes in price quotations and dependencies between the duration of each change. The main advantage of the model is to eliminate the threshold analysis, used in existing state models. This solution allows for a more accurate identification of investor behavior patterns, which translates into a reduction of investment risk. In order to verify obtained results in practice, the paper presents a concept of creating an algorithmic trading system and an analysis of its financial effectiveness for the exchange rate most popular among investors, namely EUR/USD. Full article

(This article belongs to the Special Issue Advances in Risk Models and Actuarial Science)

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22 pages, 426 KB

Open AccessArticle

Uncovering Systemic Risk in ASEAN Corporations: A Framework Based on Graph Theory and Hidden Models

by Marc Cortés Rufé, Jordi Martí Pidelaserra and Cecilia Kindelán Amorrich

Risks 2025, 13(5), 95; https://doi.org/10.3390/risks13050095 - 13 May 2025

Cited by 3 | Viewed by 2434

Abstract

In the context of an ever-evolving global economy, ASEAN companies face dynamic systemic risk that reshapes their financial interrelationships. This study examines the transmission of these risks using advanced graph theory techniques, particularly the measurement of eigenvector centrality based on Euclidean distances, combined [...] Read more.

In the context of an ever-evolving global economy, ASEAN companies face dynamic systemic risk that reshapes their financial interrelationships. This study examines the transmission of these risks using advanced graph theory techniques, particularly the measurement of eigenvector centrality based on Euclidean distances, combined with a hidden model that incorporates macroeconomic variables, such as GDP. The research focuses on identifying critical nodes within the corporate network, evaluating their contagion potential—both in terms of reinforcing resilience and amplifying vulnerabilities—and analyzing the influence of external factors on the network’s structure and behavior. The findings offer an innovative framework for managing systemic risk and provide strategic guidelines for the formulation of economic policies in emerging ASEAN markets. Full article

(This article belongs to the Special Issue Advances in Risk Models and Actuarial Science)

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