## parallel session 5

## Wednesday april 29th

## 9:00 – 10:15

## lyon room - Actuarial Finance

Variable Annuity products, which make up a substantial part of retirement products sold by insurers, have become increasingly complex over the past decades. In this paper, we investigate the drivers for these product innovations. We distinguish “virtuous” innovations that complete the market and “obfuscating” innovations that increase complexity without benefitting consumers. We demonstrate that both forms are potentially relevant in the Variable Annuities market, both theoretically and empirically. In particular, we document a recurring pattern where innovative products that expand the scope of attainable consumption paths in retirement are followed by many more complex variants in the same product class.

This paper studies intertemporal asset pricing in network economies when distress shocks can propagate through the network, similarly to epidemic outbreaks. Two classes of equilibria exist. In the first, idiosyncratic shocks are diversifiable and do not affect investor asset valuations. In the second, they generate non-diversifiable cascades of shocks that introduce a new risk premium component which is not explained by traditional systematic factors. We present a methodology that allows us to derive closed-solutions for asset prices as a function of the properties of the network and to discuss their properties.

Modern legislation has increased the amount of quantities that insurance companies should report in order to prove solvent as well as prudent. More of these quantities require not just simple bookkeeping but a mere projection of the future. For a given simulated development of the financial market, insurance companies have to compute their benefits and balances. In this presentation, I provide a solid base for this crystal ball exercise by presenting differential equations for the retrospective reserves of a pension company, in a setting where the surplus and the dividends are modelled. The differential equations rely on dynamics of the stochastic reserve that are affine functions of the stochastic reserve themselves. The retrospective reserves are defined as conditional expected values, given limited information, leading to computational tractable differential equations for the reserves. I discuss practical uses in terms of considering validation of guarantees and discretionary benefits at future time points.

## douala room - Data Science

We construct the Copula Recursive Tree (Cort) estimator: a flexible, consistent, piecewise linear estimator of a copula, leveraging the patchwork copula formalization and various piecewise constant density estimators. While the patchwork structure imposes the grid, the Cort estimator is data-driven and construct the (possibly irregular) grid recursively from the data, minimizing a chosen distance on the copula space. Furthermore, while the addition of the copula constraints makes solutions available for density estimation unusable, the Cort estimator is only concerned with dependence and guaranties the uniformity of margins. Refinements such as localized dimension reduction and bagging are developed, analyzed, and tested through applications on simulated data.

Prediction of the evolution of a claim is a challenging problem in insurance, especially for guarantees associated with high volatility of the cost such as third-party insurance.

Identifying, soon after occurrence, the claims that require more attention, is particularly interesting for the company since it allows to better adapt its response to the specificity of a claim. With the increase of available data on a claim in order to analyze its severity, artificial intelligence techniques are a promising direction to deal with this problem.

In this paper, we propose an ensemble method using Neural Networks as an early warning system for predicting the cost which is not directly observed due to censoring. The model is fed by informations of various types (such as texts reports about the circumstances of claims and nature of the damage) obtained at the opening of the claim. A particular attention is devoted to deal with the unbalanced characteristic of our data, with minority classes representing 2% of our observations. We combine bagging with a rebalancing method to improve our results and reduce the variance of the estimator.

We applied our methodology to a classification of the gravity of an accident

For calculating non-life insurance premiums actuaries traditionally rely on seperate severity and frequency models using covariates to explain the claims loss exposure. The aim of this paper is to model the pure premium as a combination of the severity and frequency models. The frequency and severity models are calibrated with the help of two model families, the gradient boosting models (GBM) and generalized additive models (GAM). GBM are optimized with the help of the exponential family deviance as a loss function, i.e. the Gamma deviance in the case of the severity and with the help of the traditional root mean squared error. Their performance is measured with the help of goodness-of-fit and goodness-of-lift statistics. The goodness-of-lift statistics, such as the area between Lorenz and concentration curves, allows to measure the accuracy of the predictions in relation to the customer past expenses. In our application, we rely on a dataset covering the loss exposure of a Swiss collision insurance portfolio covering the period from 2011 to 2015.

Keywords: gradient boosting models, machine learning, performance analysis, goodness-of-lift, goodness-of-statistics

## bogota room - Reinsurance and analytics

The crisis caused by COVID-19 revealed the global unpreparedness to handle the impact of a pandemic. In this paper, we present a statistical analysis of the data related to the COVID-19 outbreak in China, specifically the infection speed, death and fatality rates in Hubei province. By fitting distributions of these quantities we design a parametric reinsurance contract whose trigger and cap are based on the probability distributions of the infection speed, death and fatality rates. In particular, fitting the distribution for the infection speed and death rates we provide a measure of the effectiveness of a state’s action during an epidemic, and propose a reinsurance contract as a supplement to a state’s social insurance to alleviate financial costs.

We consider a stochastic optimisation problem for a diffusion approximation to an insurance risk model. In this context the drawdown of the process is defined as the absolute distance from the running maximum. The insurer is allowed to buy proportional reinsurance to minimise the expected discounted time the drawdown process exceeds some critical value d. Both insurer and resinsurer charge premiums which are calculated via the expected value principle. We obtain explicit results for the value function, the optimal strategy and their dependence on the safety loading of the reinsurance premium.

The optimal strategy resulting from the minimisation of drawdowns is solely influenced by negative deviation from the running maximum. As an extension to the model we introduce an incentive to grow. In particular, we assume that the insurer pays out dividends following a barrier strategy. We consider the post-dividend process under proportional reinsurance and maximise the value of the expected discounted dividends minus a penilisation for time spent in drawdown.

Unstructured data such as text remain quite untapped nowadays in the (re)insurance industry. The basic reason of this probably comes from the unawareness of the way to handle well these texts. Natural language processing (NLP) field propose various techniques to address text analysis tasks such as extraction or classification. However, insurance industry suffers from inconvenient domain specificities that often limit the setting up of such methodologies. The presentation aims at presenting through different (re)insurance case studies, some of these issues while presenting and applying deep learning techniques to illustrate how to overcome them.

Data collection will firstly be discussed to highlight insurance lack of data as regards of a common NLP projects and data augmentation techniques such as synonym replacement, random insertion, etc. will be presented and applied to balance this lack. Then data preparation will be introduced to illustrate insurance vocabulary specificities. Pretrained word embeddings (through Word2Vec, GloVe or ConceptNet) will be compared to custom and contextualised embeddings approaches (Elmo, BERT). Data quality aspects and impact on data extraction tasks will be described. Named Entity Recognition (NER) will be explained and custom neural network approaches (Bi-LSTM CRF) will be applied to demonstrate the effectiveness of deep learning systems. Annotation part and uncertainty measure will be also discussed. Active learning models (CNN least confidence, DO-BALD and BB-BALD) will be experimented.

Results will be finally presented through different insurance modelling scenarios including sequence representation techniques such as RNN, CNN, etc. to illustrate domain customization complexity.

## montreal room - Finance

The comparison of systemic risk in banking and insurance is a complex question, since the range of possible systemic risk indicators is wide. This presentation suggests that results of classical portfolio theory could contribute to develop new systemic risk indicators that could be compared between the banking and insurance sectors. Asset return comovement related indicators may theoretically be associated with some aspects of systemic risk, and based on classical portfolio theory it is possible to quantify the extent of diversification effects resulting from the correlation characterictics of assets. This presentation aims at exploring whether efficient portfolio calculations in classical portfolio theory could contribute to calculate asset return comovement related systemic risk measures that could be applied to compare systemic risk in banking and insurance.

It is general said that out-of-the-money call options are expensive and one can ask the question from which moneyness level this is the case. Expensive is actually meaning that the price one pays for the option is more than the (discounted) average payoff one gets for it. In finance, the P-world is the physical (actual) world in which payoffs are realized, whereas the Q-world is an artificial setting under which one determines the price. For a contingent claim, the arbitrage-free price is the discounted expectation of the payoff in the Q-world, whereas the expected realized payoff is the expectation of the payoff in the P-world. The objective of the presentation is to investigate the break-even moneyness level of a European call option, i.e. the strike where expectations in both worlds are equal. For a fixed maturity, this break-even strike determines the option with an expected return equal to zero. In order to calculate the expected return and corresponding break-even level, we need information on the pricing and physical probability measure of the asset underlying the call option at the given maturity. To fully exploit the insights of the option market we deploy the so-called Tilted Bilateral Gamma option pricing model which is able of simultaneously estimating both measures based on option data of the underlying asset. As such, the physical measure is not estimated based on backward looking, historical data, as it was traditionally done, but is jointly estimated with the pricing measure from option prices. We illustrate the proposed pricing strategy on a variety of option surfaces of certain stock indices, assessing the stability over time of the break-even strike or moneyness level of a European call option.

In order to prevent financial risks and promote the diversification of bank card applications，China has proposed an bank card “chip migration” plan of which electronic cash policy (e-cash) is a main component. This paper seeks to validate a comprehensive model of consumer acceptance in the context of e-cash policy using the extended unified theory of acceptance and use of technology (UTAUT) model and Structural Equation model (SEM). Based on 4,482 surveys from the e-cash pilot cities in China, the empirical results show that perceived security (PS) and cost of use (Cost) are beneficial extensions to the traditional UTAUT model. In addition, after controlling the effects of demographic factors, the classical technology acceptance factors, such as perceived usefulness, ease of use and so on, are key antecedents to users’ intention to adopt e-cash.

## sydney room - Non-Life insurance

Regression modelling involving heavy-tailed response distributions, which have heavier tails than the exponential distribution, has become increasingly popular in many insurance settings including non-life insurance. Mixed Exponential models can be considered as a natural choice for the distribution of heavy-tailed claim sizes since their tails are not exponentially bounded. This paper is concerned with introducing a general family of mixed Exponential regression models with varying dispersion which can efficiently capture the tail behaviour of losses. Our main achievement is that we present an Expectation-Maximization (EM) type algorithm which can facilitate maximum likelihood (ML) estimation for our class of mixed Exponential models which allows for regression specifications for both the mean and dispersion parameters. Finally, a real data application based on motor

insurance data is given to illustrate the versatility of the proposed EM type algorithm.

Typical risk classification procedure in insurance is consists of a priori risk classification determined by observable risk characteristics, and a posteriori risk classification where the premium is adjusted to reflect the policyholder’s claim history. While using the full claim history data is optimal in a posteriori risk classification procedure, i.e. giving premium estimators with the minimal variances, some insurance sectors, however, only use partial information of the claim history for determining the appropriate premium to charge. Classical examples include that auto insurances premium are determined by the claim frequency data and workers’ compensation insurances are based on the aggregate severity. The motivation for such practice is to have a simplified and efficient posteriori risk classification procedure which is customized to the involved insurance policy. This paper compares the relative efficiency of the two simplified posteriori risk classifications, i.e. based on frequency versus severity, and provides the mathematical framework to assist practitioners in choosing the most appropriate practice.

Bonus-Malus systems (BMS) are widely used in actuarial sciences. These systems are applied by insurance companies to distinguish the policyholders by their risks. The most known application of BMS is in automobile third-party liability insurance. In BMS there are several classes, and the premium of a policyholder depends on the class he/she is assigned to.

The classification of policyholders over the periods of the insurance depends on the transition rules. In general, optimization of these systems involves the calculation of an appropriate premium scale considering the number of classes and transition rules as external parameters.

We present a mixed integer LP formulation for determining the premium scale and the transition rules. Furthermore, numerical examples will also be given to demonstrate that the mixed integer LP technique is suitable for handling existing Bonus-Malus systems.