Discrimination-free insurance pricing

Discrimination-free insurance pricing

Andreas TsanakasCass Business School, City, University of London

joint work withM. Lindholm, R. Richman, M. V. Wüthrich

Full paper on SSRN

OICA, 28/04/2020

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Discrimination-free insurance pricing

THANKS TO THEORGANISERS!

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Discrimination-free insurance pricing

Discrimination in insurance pricing

When we price insurance, should we use as rating factorsprotected characteristics, such as sex or ethnicity?

Key considerations:• Legal• Ethical• Systemic

Trade-off: predictive accuracy v non-discrimination• Beyond insurance: credit rating; health care rationing

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Discrimination-free insurance pricing

Discrimination in insurance pricing

When we price insurance, should we use as rating factorsprotected characteristics, such as sex or ethnicity?

Key considerations:• Legal• Ethical• Systemic

Trade-off: predictive accuracy v non-discrimination• Beyond insurance: credit rating; health care rationing

3

Discrimination-free insurance pricing

Discrimination in insurance pricing

When we price insurance, should we use as rating factorsprotected characteristics, such as sex or ethnicity?

Key considerations:• Legal• Ethical• Systemic

Trade-off: predictive accuracy v non-discrimination

• Beyond insurance: credit rating; health care rationing

3

Discrimination-free insurance pricing

Discrimination in insurance pricing

When we price insurance, should we use as rating factorsprotected characteristics, such as sex or ethnicity?

Key considerations:• Legal• Ethical• Systemic

Trade-off: predictive accuracy v non-discrimination• Beyond insurance: credit rating; health care rationing

3

Discrimination-free insurance pricing

Article 2, Council Directive 2004/113/EC

For the purposes of this Directive, the following definitionsshall apply:

a) direct discrimination: where one person is treated lessfavourably, on grounds of sex, than another is, has beenor would be treated in a comparable situation;

b) indirect discrimination: where an apparently neutralprovision, criterion or practice would put persons of onesex at a particular disadvantage compared with persons ofthe other sex [. . . ]

Indirect discrimination relates to proxying characteristics likegender from other policyholder features

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Discrimination-free insurance pricing

Article 2, Council Directive 2004/113/EC

For the purposes of this Directive, the following definitionsshall apply:

a) direct discrimination: where one person is treated lessfavourably, on grounds of sex, than another is, has beenor would be treated in a comparable situation;

b) indirect discrimination: where an apparently neutralprovision, criterion or practice would put persons of onesex at a particular disadvantage compared with persons ofthe other sex [. . . ]

Indirect discrimination relates to proxying characteristics likegender from other policyholder features

4

Discrimination-free insurance pricing

Article 2, Council Directive 2004/113/EC

For the purposes of this Directive, the following definitionsshall apply:

a) direct discrimination: where one person is treated lessfavourably, on grounds of sex, than another is, has beenor would be treated in a comparable situation;

b) indirect discrimination: where an apparently neutralprovision, criterion or practice would put persons of onesex at a particular disadvantage compared with persons ofthe other sex [. . . ]

Indirect discrimination relates to proxying characteristics likegender from other policyholder features

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Discrimination-free insurance pricing

Research question

Our aims are to:1 Formulate an actuarial definition of indirect

discrimination (we are not lawyers!)2 Propose an adjustment to insurance prices that

avoids indirect discrimination

Literature• Actuarial: [De Jong and Ferris, 2006, Guillén, 2012,

Chen and Vigna, 2017, Chen et al., 2018]• Causal inference (indicative):

[Pearl et al., 2016, Kusner et al., 2017]

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Discrimination-free insurance pricing

Research question

Our aims are to:1 Formulate an actuarial definition of indirect

discrimination (we are not lawyers!)2 Propose an adjustment to insurance prices that

avoids indirect discrimination

Literature• Actuarial: [De Jong and Ferris, 2006, Guillén, 2012,

Chen and Vigna, 2017, Chen et al., 2018]• Causal inference (indicative):

[Pearl et al., 2016, Kusner et al., 2017]

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Discrimination-free insurance pricing

Some notation

Y – claims costs

X – non-discriminatory covariates

D – discriminatory covariates

µ(X,D) = E[Y |X,D] – best-estimate prices

• Best predictor of Y in L2 sense

µ(X) = E[Y |X] – unawareness prices

• What insurers end up doing

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Discrimination-free insurance pricing

Some notation

Y – claims costs

X – non-discriminatory covariates

D – discriminatory covariates

µ(X,D) = E[Y |X,D] – best-estimate prices

• Best predictor of Y in L2 sense

µ(X) = E[Y |X] – unawareness prices

• What insurers end up doing

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Discrimination-free insurance pricing

Some notation

Y – claims costs

X – non-discriminatory covariates

D – discriminatory covariates

µ(X,D) = E[Y |X,D] – best-estimate prices

• Best predictor of Y in L2 sense

µ(X) = E[Y |X] – unawareness prices

• What insurers end up doing

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Discrimination-free insurance pricing

Example portfolio

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Discrimination-free insurance pricing

Best-estimate prices

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Discrimination-free insurance pricing

Unawareness prices

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Discrimination-free insurance pricing

What should a ‘discrimination-free’ price look like?

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Discrimination-free insurance pricing

What should a ‘discrimination-free’ price look like?

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Discrimination-free insurance pricing

Definition of discrimination-free prices

Let D be discrete univariate. Then write unawareness prices as

µ(X) =∑d

E[Y |X, D = d] · P(D = d|X)

A construction of discrimination-free prices

µ∗(X) =∑d

E[Y |X, D = d] · P∗(D = d)

no proxying of D by X!

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Discrimination-free insurance pricing

Definition of discrimination-free prices

Let D be discrete univariate. Then write unawareness prices as

µ(X) =∑d

E[Y |X, D = d] · P(D = d|X)

A construction of discrimination-free prices

µ∗(X) =∑d

E[Y |X, D = d] · P∗(D = d)

no proxying of D by X!

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Discrimination-free insurance pricing

Formal definitions

DefinitionA price avoids direct discrimination, if it can be written as

µ∗(Z) = E∗[Y |Z],

where Z is σ(X)-measurable, and where the expectation istaken w.r.t. a probability measure P∗ on (Ω,F)

DefinitionA price µ∗(Z) that avoids direct discrimination is said to alsoavoid indirect discrimination if Z and D are independentunder P∗.

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Discrimination-free insurance pricing

Formal definitions

DefinitionA price avoids direct discrimination, if it can be written as

µ∗(Z) = E∗[Y |Z],

where Z is σ(X)-measurable, and where the expectation istaken w.r.t. a probability measure P∗ on (Ω,F)

DefinitionA price µ∗(Z) that avoids direct discrimination is said to alsoavoid indirect discrimination if Z and D are independentunder P∗.

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Discrimination-free insurance pricing

A more complicated example (health product)CovariatesX1 = age ∈ 15, . . . , 80X2 ∈ smoker, non-smokerD ∈ woman, man

3 types of claims: birthing related; cancer; other

PortfolioP(D = woman) = 0.45P(X2 = smoker) = 0.3P(D = woman | X2 = smoker) = 0.8

We will calculate discrimination-free prices, after fitting aneural network to (synthetic) data

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Discrimination-free insurance pricing

A more complicated example (health product)CovariatesX1 = age ∈ 15, . . . , 80X2 ∈ smoker, non-smokerD ∈ woman, man

3 types of claims: birthing related; cancer; other

PortfolioP(D = woman) = 0.45P(X2 = smoker) = 0.3P(D = woman | X2 = smoker) = 0.8

We will calculate discrimination-free prices, after fitting aneural network to (synthetic) data

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Discrimination-free insurance pricing

The ‘true model’

20 30 40 50 60 70 80

0.15

0.20

0.25

0.30

smokers

age

true

pric

es

best−estimate price (women)best−estimate price (men)discrimination−free priceunawareness price

20 30 40 50 60 70 80

0.15

0.20

0.25

0.30

non−smokers

age

true

pric

es

best−estimate price (women)best−estimate price (men)discrimination−free priceunawareness price

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Discrimination-free insurance pricing

The fitted model

20 30 40 50 60 70 80

0.15

0.20

0.25

0.30

smokers

age

estim

ated

pric

es

best−estimate price (women)best−estimate price (men)discrimination−free priceunawareness price

20 30 40 50 60 70 80

0.15

0.20

0.25

0.30

non−smokers

age

estim

ated

pric

es

best−estimate price (women)best−estimate price (men)discrimination−free priceunawareness price

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Discrimination-free insurance pricing

Concluding remarks

Direct and indirect discrimination were formally definedDiscrimination free prices can be calculated by a simpleadjustment to best-estimate prices• The method is not dependent on the complexity of the

pricing model• But information on discriminatory features is needed!

How to choose the measure P∗?• P∗ = P roots in causal inference• Choose P∗ to get unbiased prices, E[µ∗(Y )] = E[Y ]

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Discrimination-free insurance pricing

THANK YOU FOR YOURATTENTION!

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Discrimination-free insurance pricing

Chen, A., Guillén, M., and Vigna, E. (2018).Solvency requirement in a unisex mortality model.ASTIN Bulletin: The Journal of the IAA,48(3):1219–1243.

Chen, A. and Vigna, E. (2017).A unisex stochastic mortality model to comply with EUGender Directive.Insurance: Mathematics and Economics, 73:124–136.

De Jong, P. and Ferris, S. (2006).Adverse selection spirals.ASTIN Bulletin: The Journal of the IAA, 36(2):589–628.

Guillén, M. (2012).Sexless and beautiful data: from quantity to quality.Annals of Actuarial Science, 6(2):231–234.

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Discrimination-free insurance pricing

Kusner, M. J., Loftus, J., Russell, C., and Silva, R. (2017).

Counterfactual fairness.In Advances in Neural Information Processing Systems,pages 4066–4076.

Pearl, J., Glymour, M., and Jewell, N. P. (2016).Causal inference in statistics: A primer.John Wiley & Sons.

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