Tag Archives: insurance

Fairness and discrimination, PhD Course, #2 Insurance and risk classes

For the second course, we will get back a little bit on insurance pricing in a context of heterogeneous portfolio, and risk classification (slides are still online on the github repository). The starting point will be the pure premium.

See our online textbook, with Michel Denuit, Non Life Insurance Mathematics, for additional motivation. If we have some risk related variables \boldsymbol{x}=(x_1,\cdots,x_k), the pure premium will be the conditional expectation,

Here also, we have some law of numbers, for the conditional expected value,

This relationship, which defines the conditional expected value using the limiting value of a conditional frequency cannot be used to define properly \mathbb{P}[Y|\boldsymbol{X}=\boldsymbol{x}] and \mathbb{E}[Y|\boldsymbol{X}=\boldsymbol{x}]. One can consider a limit,\mathbb{P}\big(Y\in \mathcal{A}\big\vert X = x\big)=\lim_{\epsilon\to0}\frac{\mathbb{P}(\{Y\in \mathcal{A}\}\cap\{|X -x|\leq \epsilon\})}{\mathbb{P}(\{|X -x|\leq \epsilon\})}or\mathbb{P}\big(Y\in \mathcal{A}\big\vert X = x\big)=\lim_{\epsilon\to0}\mathbb{P}\big(Y\in \mathcal{A}\big\vert |X -x|\leq \epsilon\big)as in the law of the unconscious statistician or as Proschan and Presnell (1998) wrote it

statisticians make liberal use of conditioning arguments to shorten what would otherwise be long proofs

We can now compute conditional frequency, given some risk characteristics, for some quantity of interest y, such as the age of death, in life insurance contracts.

Demographic risk and heterogeneity

First, we will see some gender-based life tables, starting with the one obtained by Nicolaas Struyck (see e.g. Alberts et al. (2014))

More recently, in France, some wealth based life tables were obtained, with various quantiles

And finally, we will see some life tables obtained 50 years ago in the US, with racial distinction

Mean and variance decomposition

About pure premiums, an important property is the law of total expectations, and a desirable property, that we will name “balance property”

We will also mention variance and variance decomposition, depending if we take into heterogeneity, or not. With homogenous pricing, we have

If we use the “true” underlying risk factor, \Theta, we have the standard variance decomposition, also called law of total variance

i.e.

And finally, if we do not observe \Theta, but we have a collection of covariates, \boldsymbol{X}=(X_1,\cdots,X_k),

Some historical perspectives

In the textbook, Insurance: Biases, Discrimination and Fairness, I have several paragraph about an historical perspective, starting with insurance as clubs, without segmentation. Then segmentation started, with risk classes and groups. For example, according to Issues And Needed Improvements In State Regulation Of The Insurance Business, by Harry Havens, in 1979,

The price which a person pays for automobile insurance depends on age, sex, marital status, place of residence and other factors. This risk classification system produces widely differing prices for the same coverage for different people. Questions have been raised about the fairness of this system, and especially about its reliability as a predictor of risk for a particular individual. While we have not tried to judge the propriety of these groupings, and the resulting price differences, we believe that the questions about them warrant careful consideration by the State insurance departments. In most States the authority to examine classification plans is based on the requirement that insurance rates are neither inadequate, excessive, nor unfairly discriminatory. The only criterion for approving classifications in most States is that the classifications be statistically justified — that is, that they reasonably reflect loss experience. Relative rates with respect to age, sex, and marital status are based on the analysis of national data. A youthful male driver, for example, is charged twice as much as an older driver all over the country} (…) t has also been claimed that insurance companies engage in redlining – the arbitrary denial of insurance to everyone living in a particular neighborhood. Community groups and others have complained that State regulators have not been diligent in preventing redlining and other forms of improper discrimination that make insurance unavailable in certain areas. In addition to outright refusals to insure, geographic discrimination can include such practices as: selective placement of agents to reduce business in some areas, terminating agents and not renewing their book of business, pricing insurance at un-affordable levels, and instructing agents to avoid certain areas. We reviewed what the State insurance departments were doing in response to these problem. To determine if redlining exists, it is necessary to collect data on a geographic oasis. Such data should include current insurance policies, new policies being written, cancellations, and non-renewals. It is also important to examine data on losses by neighborhoods within existing rating territories because marked discrepancies within territories would cast doubt on the validity of territorial boundaries. Yet, not even a fifth of the States collect anything other than loss data, and that data is gathered on a territory-wide basis.

According to The Role of Risk Classification in Property and Casualty Insurance: A Study of the Risk Assessment Process : Final Report, by Barbara Casey, Jacques Pezier and Carl Spetzler, in 1976,

On the other hand, the opinion that distinctions based on sex, or any other group variable, necessarily violate individual rights reflects ignorance of the basic rules of logical inference in that it would arbitrarily forbid the use of relevant information. It would be equally fallacious to reject a classification system based on socially acceptable variables because the results appear discriminatory. For example, a classification system may be built on use of car, mileage, merit rating, and other variables, excluding sex. However, when verifying the average rates according to sex one may discover significant differences between males and females. Refusing to allow such differences would be attempting to distort reality by choosing to be selectively blind. The use of rating territories is a case in point. Geographical divisions, however designed, are often correlated with socio-demographic factors such as income level and race because of natural aggregation or forced segregation according to these factors. Again we conclude that insurance companies should be free to delineate territories and assess territorial differences as well as they can. At the same time, insurance companies should recognize that it is in their best interest to be objective and use clearly relevant factors to define territories lest they be accused of invidious discrimination by the public. (…) One possible standard does exist for exception to the counsel that particular rating variables should not be proscribed. What we have called `equal treatment’ standard of fairness may precipitate a societal decision that the process of differentiating among individuals on the basis of certain variables is discriminatory and intolerable. This type of decision should be made on a specific, statutory basis. Once taken, it must be adhered to in private and public transactions alike and enforced by the insurance regulator. This is, in effect, a standard for conduct that by design transcends and preempts economic considerations. Because it is not applied without economic cost, however, insurance regulators and the industry should participate in and inform legislative deliberations that would ban the, use of particular rating variables as discriminatory.

And then, more recently, we started to talk about personalization, as in Barry and Charpentier (2020). And next week, we will start talking about predictive modeling, and machine learning.

Melting contestation: insurance fairness and machine learning

Nice review of our paper , with Laurence Barry, on montrealethics.ai,

Machine learning tends to replace the actuary in the selection of features and the building of pricing models. However, avoiding subjective judgments thanks to automation does not necessarily mean that biases are removed. Nor does the absence of bias warrant fairness. This paper critically analyzes discrimination and insurance fairness with machine learning.

Melting contestation: insurance fairness and machine learning

Présentation sur l’équité et la discrimination en assurance, pour Intact

Demain matin, avec Olivier et Marie-Pier Côté, on sera chez l’assureur Intact pour parler équité et discrimination. Olivier présentera ses travaux récents sur l’utilisation de modèles causaux pour proposer des modèles “équitables” en assurance. Le papier (a fair price to pay: exploiting directed acyclic graphs for fairness in insurance) sera bientôt disponible !

Insurance, biases, discrimination and fairness, v2

In the Summer 2022, my report Insurance, biaises, discrimination and fairness (v1) was officially uploaded on the website of the Institut Louis Bachelier. I have spent another year to add illustrations and examples, and I sent the manuscript to the publisher at the beginning of the Summer 2023. Because of delays, the book is not out yet, but the publisher allowed me to upload Insurance, biaises, discrimination and fairness v2 of the document. Note that it will be the lecture notes of the doctoral course I will give this Winter at ENSAE, in Paris, France.

The R functions (and package) will be uploaded on https://github.com/freakonometrics/InsurFair soon.

Mitigating Discrimination in Insurance with Wasserstein Barycenters

Our new paper, with François Hu and Philipp Ratz, Mitigating Discrimination in Insurance with Wasserstein Barycenters is now available on ArXiv.

The insurance industry is heavily reliant on predictions of risks based on characteristics of potential customers. Although the use of said models is common, researchers have long pointed out that such practices perpetuate discrimination based on sensitive features such as gender or race. Given that such discrimination can often be attributed to historical data biases, an elimination or at least mitigation is desirable. With the shift from more traditional models to machine-learning based predictions, calls for greater mitigation have grown anew, as simply excluding sensitive variables in the pricing process can be shown to be ineffective. In this article, we first investigate why predictions are a necessity within the industry and why correcting biases is not as straightforward as simply identifying a sensitive variable. We then propose to ease the biases through the use of Wasserstein barycenters instead of simple scaling. To demonstrate the effects and effectiveness of the approach we employ it on real data and discuss its implications.

(fictitious maps used in the article)

Workshop “Machine Learning and Data Mining in Insurance and Finance ” at the SSC 2023

At the end of May, on Sunday May 28th, I will participate to the workshop Machine Learning and Data Mining in Insurance and Finance, at the Statistical Society of Canada Annual Meeting

Machine learning and data mining are hot research topics in insurance and finance and are heavily used in the industry of insurance and finance as well. In this workshop, the four invited speakers will introduce methods of machine learning and data mining and discuss their applications in insurance and finance. Each of the four speakers has 60 minutes for the presentation. There in a ten-minute break after a presentation. The total duration of the workshop is four and half hours.

Collaborative Insurance Sustainability and Network Structure

A second version of Collaborative Insurance Sustainability and Network Structure is now available on ArXiv,

The peer-to-peer (P2P) economy has been growing with the advent of the Internet, with well known brands such as Uber or Airbnb being examples thereof. In the insurance sector the approach is still in its infancy, but some companies have started to explore P2P-based collaborative insurance products (eg. Lemonade in the U.S. or Inspeer in France). The actuarial literature only recently started to consider those risk sharing mechanisms, as in Denuit and Robert (2021) or Feng et al. (2021). In this paper, describe and analyse such a P2P product, with some reciprocal risk sharing contracts. Here, we consider the case where policyholders still have an insurance contract, but the first self-insurance layer, below the deductible, can be shared with friends. We study the impact of the shape of the network (through the distribution of degrees) on the risk reduction. We consider also some optimal setting of the reciprocal commitments, and discuss the introduction of contracts with friends of friends to mitigate some possible drawbacks of having people without enough connections to exchange risks.

What is the future of predictive probabilities in insurance?

This post was written with Laurence Barry and Ewen Gallic, in French, in November 2019 (see hal-02350006)

Insurance policies are classic examples of random contracts. This forces insurers to regularly quantify this uncertainty, to calculate probabilities in order to propose “fair” premiums for the commitments they are going to make. Isn’t it time to question this practice, at a time when artificial intelligence is exploding, offering predictive algorithms of a precision never seen before? At a time when big data / big brother could mean the disappearance of uncertainty itself?
Continue reading What is the future of predictive probabilities in insurance?

Insurance and discrimination, what role for actuaries?

This post was initially published in French in September 2021.

The essential role of an actuary in charge of pricing is the segmentation of the portfolio (or “insurance classification” in English), corresponding to a discrimination activity (mathematically speaking) in the sense that the actuary will look for the most “discriminating” variables, to explain another one (in relation with the loss experience). But in the legal sense, discrimination is forbidden by law, which places the actuary in an often delicate and complex position.
Continue reading Insurance and discrimination, what role for actuaries?