Actuarial Pricing Research Articles

Loss modelling from first principles

Abstract A common statistical modelling paradigm used in actuarial pricing is (a) assuming that the possible loss model can be chosen from a dictionary of standard models; (b) selecting the model that provides the best trade-off between goodness of fit and complexity. Machine learning provides a rigorous framework for this selection/validation process. An alternative modelling paradigm, common in the sciences, is to prove the adequacy of a statistical model from first principles: for example, Planck’s distribution, which describes the spectral distribution of blackbody radiation empirically, was explained by Einstein by assuming that radiation is made of quantised harmonic oscillators (photons). In this working party we have been exploring the extent to which loss models, too, can be derived from first principles. Traditionally, the Poisson, negative binomial, and binomial distributions are used as loss count models because they are familiar and easy to work with. We show how reasoning from first principles naturally leads to non-stationary Poisson processes, Lévy processes, and multivariate Bernoulli processes depending on the context. For modelling severities, we build on previous research that shows how graph theory can be used to model property-like losses. We show how the methodology can be extended to deal with business interruption/supply chain risks by considering networks with higher-order dependencies. For liability business, we show the theoretical and practical limitations of traditional models such as the lognormal distribution. We explore the question of where the ubiquitous power-law behaviour comes from, finding a natural explanation in random growth models. We also address the derivation of severity curves in territories where compensation tables are used. This research is foundational in nature, but its results may prove useful to practitioners by guiding model selection and elucidating the relationship between the features of a risk and the model’s parameters.

Artificial Intelligence-Based Automated Actuarial Pricing and Underwriting Model for the General Insurance Sector

Artificial Intelligence-Based Automated Actuarial Pricing and Underwriting Model for the General Insurance Sector

Generalised Additive Modelling of Auto Insurance Data with Territory Design: A Rate Regulation Perspective

Pricing using a Generalised Linear Model is the gold standard in the auto insurance industry and rate regulation. Generalised Additive Model applications in insurance pricing are receiving increasing attention from academic researchers and actuarial pricing professionals. The actuarial practice has constantly shown evidence of significantly different premium rates among the different rating territories. In this work, we build predictive models for claim frequency and severity using the synthetic Usage Based Insurance (UBI) dataset variables. First, we conduct territorial clustering based on each location’s claim counts and amounts by grouping those locations into a smaller set, defined as a cluster for rating purposes. After clustering, we incorporate these clusters into our predictive model to determine the risk relativity for each factor level. Through predictive modelling, we have successfully identified key factors that may be helpful for the rate regulation of UBI. Our work aims to fill the gap between individual-level pricing and rate regulation using the UBI database and provides insights on consistency in using traditional rating variables for UBI pricing. Our main contribution is to outline how GAM can address a more complicated functionality of risk factors and the interactions among them. We also contribute to demonstrating the territory clustering problem in UBI to construct the rating territories for pricing and rate regulation. We find that relativity for high annual mileage driven is almost three times that associated with low annual mileage level, which implies its importance in premium calculation. Overall, we provide insights into how UBI can be regulated through traditional pricing factors, additional factors from UBI datasets and rating territories derived from basic rating units and the driver’s location.

Explainable Artificial Intelligence (XAI) in Insurance

Explainable Artificial Intelligence (XAI) models allow for a more transparent and understandable relationship between humans and machines. The insurance industry represents a fundamental opportunity to demonstrate the potential of XAI, with the industry’s vast stores of sensitive data on policyholders and centrality in societal progress and innovation. This paper analyses current Artificial Intelligence (AI) applications in insurance industry practices and insurance research to assess their degree of explainability. Using search terms representative of (X)AI applications in insurance, 419 original research articles were screened from IEEE Xplore, ACM Digital Library, Scopus, Web of Science and Business Source Complete and EconLit. The resulting 103 articles (between the years 2000–2021) representing the current state-of-the-art of XAI in insurance literature are analysed and classified, highlighting the prevalence of XAI methods at the various stages of the insurance value chain. The study finds that XAI methods are particularly prevalent in claims management, underwriting and actuarial pricing practices. Simplification methods, called knowledge distillation and rule extraction, are identified as the primary XAI technique used within the insurance value chain. This is important as the combination of large models to create a smaller, more manageable model with distinct association rules aids in building XAI models which are regularly understandable. XAI is an important evolution of AI to ensure trust, transparency and moral values are embedded within the system’s ecosystem. The assessment of these XAI foci in the context of the insurance industry proves a worthwhile exploration into the unique advantages of XAI, highlighting to industry professionals, regulators and XAI developers where particular focus should be directed in the further development of XAI. This is the first study to analyse XAI’s current applications within the insurance industry, while simultaneously contributing to the interdisciplinary understanding of applied XAI. Advancing the literature on adequate XAI definitions, the authors propose an adapted definition of XAI informed by the systematic review of XAI literature in insurance.

Analytical View of Pricing Weather and Freight Derivatives: 1950–2020

A weather derivative is financial instrument that companies or individuals use to hedge against the risk of weather-related losses. Freight derivatives value is derived from the future levels of freight rates, like a dry bulk-a category of cargo stowed in bulk, consisting of grain, cotton, coal, etc., carrying rates, and oil tanker rates. Numerous empirical studies exist on weather and freight derivatives and their pricing models such as Indifference Pricing Approach, Arbitrage Pricing Model, Financial Pricing model, Benchmark Pricing Approach, Fair Pricing Approach, Actuarial Pricing, Consumer Based Pricing Method, and Index Modeling. This paper aims to address issues relating to the functioning of weather and freight derivatives. This paper also focuses on the studies published on weather and freight derivatives’ pricing models during the last seven decades (i.e., 1950–2020). Objectives: (1) To understand the vital issues relating to models of weather and freight derivatives. (2) To compare pricing models of weather and freight derivatives during the last seven decades (i.e., 1950–2020).

Actuarial pricing with financial methods

The objective of this paper is twofold. On the one hand, the optimal combination of reinsurance and financial investment will be studied under a general framework. Indeed, there is no specific type of reinsurance contract, there is no specific dynamics of the involved financial instruments and the financial market does not have to be free of frictions. On the other hand, it will be pointed out how the optimal combination above may provide us with new premium principles making the insurer global risk vanish. The risk will be managed with a coherent risk measure, and the new premium principles will seem to reflect several properties, which are desirable from both the analytical and the economic perspectives. From the analytical viewpoint, the premium principles will be continuous, homogeneous and increasing. From the economic viewpoint, the premium principles will lead to cheaper prices with respect to both the insurance market and the financial one. In other words, the premium principles will make the insurer more competitive in prices under a null risk. General necessary and sufficient optimality conditions will be given, as well as closed forms for the solutions under appropriate assumptions. Several methods preventing unbounded optimization problems will warrant special attention, and one particular case will be more thoroughly studied, namely, the combination of the Black–Scholes–Merton pricing model with the conditional value at risk.

Variable Selection Algorithm for a Mixture of Poisson Regression for Handling Overdispersion in Claims Frequency Modeling Using Telematics Car Driving Data

In automobile insurance, it is common to adopt a Poisson regression model to predict the number of claims as part of the actuarial pricing process. The Poisson assumption can rarely be justified, often due to overdispersion, and alternative modeling is often considered, typically zero-inflated models, which are special cases of finite mixture distributions. Finite mixture regression modeling of telematics data is challenging to implement since the huge number of covariates computationally prohibits the essential variable selection needed to attain a model with desirable predictive power devoid of overfitting. This paper aims at devising an algorithm that can carry the task of variable selection in the presence of a large number of covariates. This is achieved by generating sub-samples of the data corresponding to each component of the Poisson mixture, and wherein variable selection is applied following the enhancement of the Poisson assumption by means of controlling the number of zero claims. The resulting algorithm is assessed by measuring the out-of-sample AUC (Area Under the Curve), a Machine Learning tool for quantifying predictive power. Finally, the application of the algorithm is demonstrated by using data of claim history and telematics data describing driving behavior. It transpires that unlike alternative algorithms related to Poisson regression, the proposed algorithm is both implementable and enjoys an improved AUC (0.71). The proposed algorithm allows more accurate pricing in an era where telematics data is used for automobile insurance.

Time-consistent evaluation of credit risk with contagion

A time-consistent evaluation is a dynamic pricing method according to which a risk that will be almost surely cheaper than another one at a future date should already be cheaper today. Common actuarial pricing approaches are usually not time-consistent. Pelsser and Ghalehjooghi (2016) derived time-consistent valuation principles from time-inconsistent ones. The aim of this paper is twofold. Firstly, we propose a model for credit insurance portfolios taking into account the contagion risk via self-exciting jump processes. Secondly, we extend the approach of Pelsser and Ghalehjooghi to credit insurance in this framework. Starting from classical time-inconsistent actuarial pricing methods, we derive partial integro-differential equations (PIDE) for their time-consistent counterparts. We discuss numerical methods for solving these PIDEs and their results. We draw two conclusions from these results. On the one hand, we show that time-consistent evaluations tend to give higher prices, compared to time-inconsistent evaluations. On the other hand, our results show that the time-consistency of evaluations allows to better take into account the risk of contagion in credit insurance, if such a risk exists. Finally, we propose a method to calibrate our model and use it in practice.

Usage-Based Insurance—Impact on Insurers and Potential Implications for InsurTech

Insurers are increasingly embracing the InsurTech ecosystem. The most important InsurTech-related trend in automobile insurance is usage-based insurance (UBI), which enables insurers to access and incorporate drivers’ behavioral risk factors in actuarial pricing. Using a difference-in-difference research design with firm fixed effects, we provide evidence that UBI improves private passenger auto liability (PPAL) insurers’ underwriting performance by reducing their loss ratio. However, the improvement in underwriting performance is only significant among early UBI adopters, highlighting the early-mover advantage in InsurTech. Also, UBI produces benefits only when it matures. Our findings are robust to analyses that address potential reverse causality and self-selection bias. Additional tests show that early UBI adopters experience a significant increase in their market share, implying UBI’s advantage to attract low-risk drivers from other insurers. The overall performance effect of UBI programs for early adopters is a 1% increase in ROA and a 3% increase in ROE. The policy implications of our findings from the perspective of insurers should be of interest to firms’ management, actuaries, investors, and rating agencies.

Time-consistent longevity hedging with long-range dependence

Longevity securitization enables insurers to manage longevity or mortality risk in the life market. Recent empirical studies identify long-range dependence (LRD) in mortality rates using life tables, which casts doubt on the adequacy of Markovian models for actuarial pricing and risk management. This paper derives the first time-consistent mean–variance longevity hedging strategy for insurers using a stochastic mortality model with LRD. We adopt the open-loop equilibrium control framework and derive an analytical solution to the hedging strategy. Our numerical examples show the relevance of LRD to longevity hedging and the cost of ignoring it.

The costs and benefits of reinsurance

Purchasing reinsurance reduces insurers’ insolvency risk by stabilising loss experience, increasing capacity, limiting liability on specific risks and/or protecting against catastrophes. Consequently, purchasing reinsurance should reduce capital costs. However, transferring risk to reinsurers is expensive. The cost of reinsurance for an insurer can be much larger than the actuarial price of the risk transferred. In this article, we analyse empirically the costs and the benefits of reinsurance for a sample of U.S. property–liability insurers. The results show that the purchase of reinsurance significantly increases insurers’ costs but significantly reduces the volatility of the loss ratio. With purchasing reinsurance, insurers accept to pay higher costs of insurance production to reduce their underwriting risk.

Boosting Poisson regression models with telematics car driving data

With the emergence of telematics car driving data, insurance companies have started to boost classical actuarial regression models for claim frequency prediction with telematics car driving information. In this paper, we propose two data-driven neural network approaches that process telematics car driving data to complement classical actuarial pricing with a driving behavior risk factor from telematics data. Our neural networks simultaneously accommodate feature engineering and regression modeling which allows us to integrate telematics car driving data in a one-step approach into the claim frequency regression models. We conclude from our numerical analysis that both classical actuarial risk factors and telematics car driving data are necessary to receive the best predictive models. This emphasizes that these two sources of information interact and complement each other.

The Boundary of the Market for Biosecurity Risk.

Imported goods create value in destination countries but also create biosecurity risk. Although widely used in other domains of the economy, risk markets have not been created to manage losses that occur when exotic pests and diseases are introduced with traded goods. In this article we show that not all biosecurity risks are insurable. Losses arising from effort needed to detect and respond to exotic pests and diseases that breach national borders appear to be insurable because entry of these threats and consequent response costs, can be regarded as random events. As pests and diseases establish and spread, however, loss of access to export markets and productivity losses display systematic risk and appear to be uninsurable. Other insurability criteria support this definition of the boundary of biosecurity risk markets.We use the Australian biosecurity system as an example, although the framework described in this study will be applicable to biosecurity systems worldwide. We argue that biosecurity risk insurance could be incorporated into the current biosecurity system but would require legislation mandating importers to purchase insurance. Advantages of actuarial pricing of biosecurity risk are: (i) an increase in economic efficiency to the extent that importers respond to the price of biosecurity risk; (ii) financial sustainability would improve because actuarial pricing creates a structural link between funds available for biosecurity activities and risk exposure; and (iii) equity issues evident in the current biosecurity system could be addressed because risk creators (importers) would fund response activities through the purchase of insurance.

Mutual Risk Sharing and Fintech: The Case of Xiang Hu Bao

Xiang Hu Bao (XHB), meaning 'mutual treasury' in Chinese, is a novel online mutual aid platform operated by Alibaba's Ant Financial to facilitate mutual risk sharing of critical illness exposures. XHB reached nearly 100 million members in less than one year since its launch and so far has offered its members critical illness protections at significantly lower cost than traditional critical illness insurance. There are three major distinctions between XHB and traditional insurance products. First, XHB leverages the tech giant's platform and digital technology to lower enrollment and claim processing costs. Second, different from insurance applying sophisticated actuarial pricing models, XHB collects no premiums ex ante from members, but instead equally allocates indemnities and administrative costs among participants after each claims period. Third, XHB limits coverage amount, often below critical illness insurance products, particularly for older participants. We show this restriction potentially leads to separating equilibrium, a la Rothschild-Stiglitz, where low-risk individuals enroll in XHB while high-risk individuals purchase critical illness insurance. Data shows that the incidence rate of the covered illness among XHB members is well below that of comparable critical illness insurance. Our findings further suggest the role of advantageous selection in explaining the cost advantages of the Fintech-based mutual aid programs.

A Numerical Analysis to Study Whether the Early Termination of Reverse Mortgages is Rational

A reverse mortgage supports the aging society more sustainably by providing stable cash flows to elderly retirees. Although the reverse mortgage market has successfully grown in many developed countries, we have observed significantly increased early terminations due to the recent housing market boom in Korea. In this study, we provide the numerical solutions for the monthly payment from the actuarial pricing model of the reverse mortgage, reflecting the house price growth and mortality improvement to examine whether the early-termination-and-repurchase strategy is profitable. Findings suggest that in order for the strategy to be profitable, the realized growth rate of the house price should be significantly greater than the expected growth rate in the actuarial pricing model. Furthermore, for the older borrower, the greater growth rate of the house price is required for the strategy to be profitable.

Design and rating of risk-contingent credit for balancing business and financial risks for Kenyan farmers

ABSTRACTWeather-related agricultural risks and limited access to credit are serious impediments to agricultural productivity and growth in developing countries. This paper describes a novel insurance linked credit model piloted in Kenya, where insurance markets are effectively absent, and farmers do not borrow because of the risk of losing their collateral. One of the challenges in deigning bundled credit products, in the absence of traded securities, is the actuarial pricing and risk rating of the insurance and the loan product. We develop a rainfall linked risk-contingent credit that transfers drought risk related perils from borrower to lender via insurance mechanism that provide a balance between business and credit risks for smallholder farmers. We describe the methodology used to design and rating of a risk-contingent structured operating agricultural credit instrument using CHIRPS rainfall data from 1981–2016 in Kenya. We illustrate the use of Monte Carlo methods to risk modelling that can be integrated within the general insurance and credit rating framework. The innovative design and methodology presented in this paper are as important as the product delivery mechanism and will be of interest to specialists in development economics and agricultural finance.

Multivariate credibility modelling for usage-based motor insurance pricing with behavioural data

AbstractPay-how-you-drive (PHYD) or usage-based (UB) systems for automobile insurance provide actuaries with behavioural risk factors, such as the time of the day, average speeds and other driving habits. These data are collected while the contract is in force with the help of telematic devices installed in the vehicle. They thus fall in the category of a posteriori information that becomes available after contract initiation. For this reason, they must be included in the actuarial pricing by means of credibility updating mechanisms instead of being incorporated in the score as ordinary a priori observable features. This paper proposes the use of multivariate mixed models to describe the joint dynamics of telematics data and claim frequencies. Future premiums, incorporating past experience can then be determined using the predictive distribution of claim characteristics given past history. This approach allows the actuary to deal with the variety of situations encountered in insurance practice, ranging from new drivers without telematics record to contracts with different seniority and drivers using their vehicle to different extent, generating varied volumes of telematics data.

Incidence, Dependence Structure of Disease, and Rate Making for Health Insurance

In order to analyze the two goals under the national strategy of “Healthy China”, this paper attempts to solve the problem of coverage rate and guarantee level of health insurance, as well as the rational allocation of full life cycle health insurance resources. This paper uses pair copula to model the dependence of different disease incidence and proposes an actuarial model for rate making in health insurance based on the dependence captured by pair copula. These are far more accurate than any other model and more proper for covering a basket of several different diseases. The data for the paper was drawn from the experience incidence table of major diseases (malignant tumor, acute myocardial infarction, and stroke sequelae) from the ages 0-65 years in the Chinese life insurance industry. Extending the hypothesis of independence in actuarial modeling, the authors comprehensively use a hierarchical copula theory to extract the dependence structure of risk variable in insurance. The classification rate making technology and survival analysis method in traditional actuarial pricing were also considered. This paper applied the generalized linear model, which is commonly used in nonlife insurance pricing for empirical study of health insurance rate making. The authors discovered that the incidence of major diseases and the single premium rate calculated by the generalized linear model under HAC dependence structure were both significantly different from that calculated by the Manchester United method without dependency. The authors also stated that the rate based on the generalized linear model under HAC dependence structure was a bit different from that without dependency but both were generally the same as that of Care Expert in PICC Health. The underestimation or overestimation of systematic risks and the distortion of the rate system can be eliminated if we combine risk dependence into modeling.

A Critical Review of the Urban Institute Model of Financing Long-Term Services and Supports

Since the demise of the Community Living Assistance Services and Supports (CLASS) insurance program in the Patient Protection and Affordable Care Act (ACA) covering long-term care and since the majority report of the 2013 Commission on Long-Term Care, which did not endorse a new social insurance program, some advocates and analysts have been looking for other means to encourage policymakers to create a mandatory, tax-based social insurance program to finance long-term care. These advocates and analysts will find some comfort in the Urban Institute scoring model, which shows that the cost of such a program is surprisingly low. But a closer examination of the assumptions and methodology of the Urban Institute model reveals many sources leading to a significant underestimate of the cost. In particular, compared with the mainstream professional literature, Congressional Budget Office and Social Security Trustees models, and actuarial pricing, the Urban Institute assumes that the elderly (above age 65) population is healthier, experiences lower incidence and duration of disability, and gets significantly more unpaid care, both currently and, especially, projected into the future. The model also suffers from a policy bias—it is set up to consider only major federal programs, not the creation of varied incentives to encourage retired households to get private insurance and asset holdings to cover the long-term-care risk.

Equilibrium in Insurance Markets With Adverse Selection When Insurers Pay Policy Dividends

AbstractWe show that an equilibrium always exists in the Rothschild–Stiglitz insurance market model with adverse selection and an arbitrary number of risk types, when insurance contracts include policy dividend rules. The Miyazaki–Wilson–Spence state‐contingent allocation is an equilibrium allocation (defined as a set of type‐dependent lotteries sustained at a symmetric equilibrium of a market game), and it is the only one when out‐of‐equilibrium beliefs satisfy a robustness criterion. It is shown that stock insurers and mutuals may coexist, with stock insurers offering insurance coverage at actuarial price and mutuals cross‐subsidizing risks.