Detra Notes

The Detra Note series

The Detra Notes are a series of educational papers dedicated to the actuarial sector. These Notes are published by members of the Detralytics team and written in a clear and accessible language. The team combines academic expertise and market knowledge. Detralytics was founded to support companies in the advancement of actuarial sciences and the solving of the profession’s future challenges. It is within the scope of this mission that we make our work available through our Detra Notes.

List of publications

Option pricing in the Heston model with Physics inspired neural networks

Date : February 2024
Authors : Donatien Hainaut, Alex Casas
Publisher : Detralytics
Language : English

In absence of a closed form expression such as in the Heston model, the option pricing is computationally intensive when calibrating a model to market quotes. this article proposes an alternative to standard pricing methods based on physics-inspired neural networks (PINNs). A PINN integrates principles from physics into its learning process to enhance its effciency in solving complex problems. In this article, the driving principle is the Feynman-Kac (FK) equation, which is a partial differential equation (PDE) governing the derivative price in the Heston model. We focus on the valuation of European options and show that PINNs constitute an effcient alternative for pricing options with various specifications and parameters without the need for retraining.

Distill knowledge of additive tree models into generalized linear models

Date : September 2023
Authors : Arthur Maillart, Christian Y. Robert
Publisher : Detralytics
Language : English

Generalized additive models (GAMs) are a leading model class for interpretable machine learning. GAMs were originally trained using smoothing splines. Recently, tree-based GAMs where shape functions are gradient-boosted ensembles of bagged trees were proposed (e.g. Explainable Boosting Machine). In this paper, we introduce a competing three-step GAM learning approach where we combine i) the knowledge of the way to split the covariates space brought by an Additive tree model (ATM), ii) an ensemble of predictive linear scores derived from Generalized linear models (GLMs) using a binning strategy based on the ATM, iii) a final GLM to have a prediction model that ensures auto-calibration. Numerical experiments illustrate the very good performances of our approach on several datasets compared to GAM with splines, EBM or GLM with binarsity penalization. A case-study in trade credit insurance is also provided.

Wasserstein boosting trees algorithm for count data, with application to claim frequencies in motor insurance

Date : July 2023
Authors : Michel Denuit, Julien Trufin and Harrison Verelst
Publisher : Detralytics
Language : English

This paper proposes a variant of the well-known boosting trees algorithm to estimate conditional distributions. Since regression trees partition observations into subgroups, the corresponding empirical distributions can be used to define the splitting criterion. Precisely, the parametric approach using Poisson deviance is replaced with a non-parametric one maximizing probabilistic distances between empirical distributions in child nodes. Proceeding in this way, the actuary obtains an estimated conditional distribution for the response, from which a conditional mean can be derived as well as any other quantity of interest in risk management. The numerical performances of the proposed method are assessed with simulated data while a case study demonstrates its usefulness for insurance applications.

Cyber incident reports: extrapolating severity using neural networks

Date : May 2023
Authors : Justin Kher, Olivier Lopez and Hugo Rapior
Publisher : Detralytics
Language : English

Due to its emerging nature, cyber risk is a field in which very few data exist when it comes to calibrating models for the severity of such events. This makes the task of actuaries particularly challenging regarding pricing or claim management. In the present note, we present a general methodology to process text data using neural networks, and how it can be used to determine the severity of a cyber incident when this information is missing. This methodology is illustrated on a public benchmark database. It can be used either to augment database, by adding some claim incidents to historical databases, from reported incident whose cost is unknown. It can also be used to quickly evaluate the severity of a cyber claim just after its occurrence.

Boosted Poisson regression tress: A guide to the BT package in R

Date : April 2023
Authors : Gireg Willame, Julien Trufin and Michel Denuit
Publisher : Detralytics
Language : English

Thanks to its outstanding performances, boosting has rapidly gained wide acceptance among actuaries. Hainaut et al. (2022) established that boosting can be conducted directly on the response under Tweedie loss function and log-link, by adapting the weights at each step. This is particularly useful to analyze low counts (typically, numbers of reported claims at policy level in personal lines). Huyghe et al. (2022) adopted this approach to propose a new boosting machine with cost-complexity pruned trees. In this approach, trees included in the score progressively reduce to the root-node one, in an adaptive way. This paper reviews these results and presents the new BT package in R contributed by Willame (2022), which is designed to implement this approach for insurance studies. A numerical illustration demonstrates the relevance of the new tool for insurance pricing.

Tail index partition-based rules extraction with application to tornado damage insurance

Date : February 2023
Author : Arthur Maillart & Christian Y. Robert
Publisher : Detralytics
Language : English

(This paper has been published)
The tail index is an important parameter that measures how extreme events occur. In many practical cases, this tail index depends on covariates. In this paper,we assume that it takes a finite number of values over a partition of the covariate space. This article proposes a tail index partition-based rules extraction method that is able to construct estimates of the partition subsets and estimates of the tail index values. The method combines two steps: first an additive tree ensemble based on the Gamma deviance is fitted, and second a hierarchical clustering with spatial constraints is used to estimate the subsets of the partition. We also propose a global tree surrogate model to approximate the partition-based rules while providing an explainable model from the initial covariates. Our procedure is illustrated on simulated data. A real case study on wind property damages caused by tornadoes is finally presented.

Risk management with Local Least Squares Monte-Carlo

Date : February 2023
Author : Donatien Hainaut & Adnane Akbaraly
Publisher : Detralytics
Language : English

The method of least squares Monte-Carlo (LSMC) has become a standard in the insurance and financial sectors for computing the exposure of a company to market risk. The sensitive point of this procedure is the non-linear regression of simulated responses to risk factors. This article proposes a novel approach for this step, based on an apriori segmentation of responses. Using a K-means algorithm, we identify clusters of responses that are next locally regressed on corresponding risk factors. A global function of regression is obtained by combining local models and a logistic regression. The effciency of the Local Least squares Monte-Carlo (LLSMC) is checked in two illustrations. The first one focuses on butterfly and bull trap options in a Heston stochastic volatility model. The second illustration analyzes the exposure to risks of a participating life insurance.

Insurance analytics with k-means and extensions

Date : January 2022
Author : Donatien Hainaut & Thomas Hames
Publisher : Detralytics
Language : English

The k-means algorithm and its variants are popular techniques of clustering. Their purpose is to uncover group structures in a dataset. In actuarial applications, these methods detect clusters of policies with similar features and allow to draw a map of dominant risks. This working note starts with a review of the k-means algorithm and develops next two extensions to manage categorical features. We develop a mini-batch version that keeps computation time under control when analysing a high-dimensional dataset. We next introduce the fuzzy k-means in which policies can belong to multiple clusters. Finally, we conclude by a detailed introduction to spectral clustering.

Modeling accumulation scenarios in cyber risk

Date : October 2021
Author : Louise D'Oultremont, Olivier Lopez and Brieuc Spoorenberg
Publisher : Detralytics
Language : English

In this work, we consider a simple way to model accumulation episodes (i.e., large number of claims occurring in a short amount of time) in the context of cyber risk. This provides a generic way to design stochastic scenarios measuring the impact of such an event on an insurance portfolio. A particular attention is devoted to providing a way to model network effects (contagion between actors connected with each other) while producing a model needing relatively few data to be calibrated. Illustrations of this method are performed to determine the impact of different behaviors of the policyholders and of the insurance company on the spread of the cyber epidemic.

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Tarification d'expérience sur statistiques sinistres écrêtées

Date : June 2021
Author : Carole Bettonville, Michel Denuit, Florent Dubus and Julien Trufin
Publisher : Detralytics
Language : French

La théorie de la crédibilité est principalement utilisée dans le cadre de la tarification des contrats d'assurance accidents du travail ou responsabilité civile B2B. La présence de grands sinistres pose cependant un problème en pratique, causant une brusque augmentation de la prime lors de leur survenance et menant à une sous-estimation des facteurs de crédibilité. Cette note fournit un modèle technique objectivant cette analyse ainsi qu’une étude de cas sur un portefeuille d’assurance responsabilité civile B2B.

Risk sharing under the dominant peer-to-peer property and casualty insurance business models

Date : April 2021
Author : Michel Denuit & Christian Y. Robert
Publisher : Detralytics
Language : English

(This paper has been published)
This paper purposes to formalize the three business models dominating peer-to-peer (P2P) property and casualty insurance: the self-governing model, the broker model, and the carrier model. The former one develops outside the insurance market whereas the latter ones may originate from the insurance industry, by partnering with an existing company or by issuing a new generation of participating insurance policies where part of the risk is shared within a community and higher losses, exceeding the community's risk-bearing capacity are covered by an insurance or reinsurance company. The present paper proposes an actuarial modeling based on conditional mean risk sharing, to support the development of this new P2P insurance offer under each of the three business models. In addition, several specific questions are also addressed in the self-governing model. Considering an economic agent who has to select the optimal pool for a risk to be shared with other participants, it is shown that uniform comparison of the Lorenz or concentration curves associated to the respective total losses of the pools under consideration allows the agent to decide which pool is preferable.

A Heath-Jarrow-Morton model compliant with Solvency II? Promises and pitfalls

Date : March 2021
Author : Simon Boigelot & Donatien Hainaut
Publisher : Detralytics
Language : English

In this working paper, we adapt the Heath-Jarrow Morton (HJM) framework by considering a constraint of convergence of future forward rates toward a constant exogeneous rate set e.g. by a regulator. This limit rate, called “ultimate forward risk” or UFR, was introduced by the EIOPA in 2015 to extrapolate the initial yield curve for maturities beyond the last liquid point. We show that adding a constraint of convergence in the HJM model impacts not only the term structure of interest rates but also the future variance of zero-coupon bonds.

The reason why bagging trees outperform decision tree...

Date : November 2020
Author : Candy Mahirwe, Michel Denuit and Julien Trufin
Publisher : Detralytics
Language : English

One of the objectives of ensemble techniques is to improve model accuracy by driving down the variance without affecting too much the bias. In this note, we consider bagging trees. Bagging trees is an ensemble technique which consists in combining several regression trees fitted on different bootstrap samples of the training set. We demonstrate that bagging trees performs better than one of its constituent trees in the sense of the expected generalization error. Moreover, we show through an example that bagging trees outperforms not only one of its constituent tree but also the best decision tree built on the entire training set.

Matrix calculation for ultimate and one-year risk in the semi-markov individual loss reserving model

Date : October 2020
Author : Carole Bettonville, Louise D'Oultremont, Robin Van Oirbeek, Michel Denuit and Julien Trufin
Publisher : Detralytics
Language : English

(This paper has been published)
This paper proposes a multistate model with a Semi-Markov dependence structure describing the different stages in the settlement process of individual claims in general insurance. Every trajectory, from reporting to closure is combined with a modeling of individual link ratios to obtain the ultimate cost of each claim. Analytical expressions are derived for the moments of ultimate amounts whereas quantile risk measures can be obtained by simulation. In the one-year view, the proposed matrix calculations avoid the simulation-within-simulation issue and offer a tractable evaluation method. A case study illustrates the relevance of the proposed approach.

Quelles limites pour l'application des algorithmes d'apprentissage en assurance ?

Date : May 2020
Author : Arthur Charpentier et Michel Denuit
Publisher : Detralytics
Language : French

Une étape essentielle du processus de tarification est le choix de l'algorithme qui permettra de calculer la prime à partir de données passées. En assurance, les modèles prédictifs sont partout : il peut s'agir de calculer la prime demandée pour couvrir les dommages causés à un bien, ou encore d'un score interne de suspicion de fraude sur un sinistre. Ce papier discute des limites de l’application des algorithmes prédictifs en assurance en revenant sur les notions de segmentation des risques et d’hétérogénéité.

Micro Reserving

Date : March 2020
Author : Carole Bettonville, Robin Van Oirbeek, Michel Denuit & Julien Trufin
Publisher : Detralytics
Language : English

This paper proposes a multistate model with a Semi-Markov dependence structure describing the different stages in the settlement process of individual claims in general insurance. Every trajectory, from reporting to closure is combined with a modeling of individual link ratios to obtain the ultimate cost of each claim. Quantiles and other risk measures can then easily be obtained by simulation. A case study based on individual claim developments in a motor third party liability insurance portfolio illustrates the relevance of the proposed approach.

Conversion Value of Usufruct

Date : November 2019
Author : Michel Denuit, Julien Trufin
Publisher : Detralytics
Language : French

L’analyse de Denuit et Trufin (2019) a permis de mettre en lumière l’impossibilité d’appliquer les calculs actuariels basés sur la loi des grands nombres à une opération par essence unique comme celle de la conversion de l’usufruit. Dans la lignée de cette étude, nous proposons ici une méthode simple et économiquement justifiée permettant d’évaluer les droits viagers, au premier rang desquels figure l’usufruit [...]

Effective tools for assessing the performance of pricing models

(This article has been published)

Date : March 2019
Author : Dominic Sznajder, Michel Denuit, Julien Trufin
Publisher : Detralytics
Language : English

Model Selection based on Lorenz and concentration curves, Gini-indices & Convex order

Des triangles agrégés aux méthodes individuelles : revue de méthodes de réservation en assurances dommages

Date : Janvier 2019
Author : Gireg Willame, Michel Denuit et Julien Trufin
Publisher : Detralytics
Language : French

Depuis des années les actuaires sont habitués à évaluer les réserves à l’aide de méthodes collectives, n’incluant aucune information individuelle. Cependant, avec la disponibilité des données individuelles et l’accroissement de la puissance informatique, ces derniers sont de plus en plus tentés d’incorporer les informations individuelles dans l’évaluation des réserves en vue d’améliorer leurs prédictions. De plus, depuis peu, des techniques mêlant des modélisations collectives en utilisant l’information au niveau individuel, ont également vu le jour[...]

Back to the Future - D’un modèle B2C intermédié à une généralisation des B2B(2C) et B2(P2P) : Enjeux Actuariels

Date : November 2018
Author : Michel Denuit, Julien Trufin
Publisher : Detralytics
Language : French

La digitalisation et le développement des InsurTechs ne sont pas toujours synonymes de changement dans les techniques actuarielles de gestion du risque. D’autres évolutions par contre impacteront sensiblement et durablement le marché de l’assurance ainsi que les méthodes utilisées par les actuaires, comme les nouvelles technologies (voitures connectées et IoT, par exemple), le glissement de la possession vers l’utilisation (voiture partagée, par exemple) et l’émergence de communautés P2P. De nouveaux acteurs pourraient apparaître à l’occasion de ces mutations, faisant reculer signiﬁcativement l’encaissement des assureurs traditionnels[...]

Telematics : Usage-based motor insurance pricing with behavioral data

(This article has been published)

Date : November 2018
Author : Montserrat Guillen, Michel Denuit, Julien Trufin
Publisher : Detralytics
Language : English

Pay-How-You-Drive systems for automobile insurance provide actuaries with behavioural risk factors, such as the time of the day, average speed and driving habits [...] The paper has been accepted for publication in Annals of Actuarial Science. The link to the publication will be communicated as soon as possible