Using risk factors in insurance analytics data driven strategies...

transcript

Using risk factors in insurance analyticsdata driven strategies

Chaire DAMI workshop, Louvain-la-Neuve

Katrien AntonioLRisk - KU Leuven and ASE - University of Amsterdam

September 2017

K. Antonio, KU Leuven & UvA 1 / 28

Motivation

Claim frequency and claim severity

as function of

nominal / numeric ∼ ordinal / spatial

features

K. Antonio, KU Leuven & UvA Motivation 2 / 28

Research questions

I Generalized Linear Models (GLMs) for frequency (∼ Poisson) andseverity (∼ gamma).

I How to:

(1) select risk factors or features?

(2) cluster (or bin or fuse) levels within a risk factor?

age groups / postal code clusters / clusters of car models

I Procedure should be data driven, scalable to large (big) data.

I End product is interpretable, within actuarial comfort zone.

K. Antonio, KU Leuven & UvA Motivation 3 / 28

A data driven strategyfor the construction of insurance tariff classes

Henckaerts, Antonio, et al., 2017 (online)

GAM as a starting point

I Generalized Additive Model with predictor:

ηi = g(µi ) = β0 +

p∑j=1

βjxdij +

q∑j=1

fj(xcij ) +

r∑j=1

fj(xsij , y

I Information criteria:

AIC = −2 · logL+ 2 · EDF

BIC = −2 · logL+ log(n) · EDF,

balancing goodness of fit and complexity.

K. Antonio, KU Leuven & UvA Henckaerts, Antonio, et al., 2017 5 / 28

I MTPL data set from Denuit & Lang (2004), 163 231 records.

I Lowest BIC among exhaustive search with 1 024 fitted models:

log(E(nclaims)) =log(exp) + β0 + β1coveragePO + β2coverageFO + β3fueldiesel+

f1(ageph) + f2(power) + f3(bm) + f4(ageph, power) + f5(long, lat).

which combines offset and

categorical ∼ nominal continuous ∼ ordinal

interactions spatial

risk factors.

−0.25

25 50 75

0 50 100 150 200 250

powerf̂ 2

0 5 10 15 20

25 50 75

−0.5

0.5f̂4

−0.4

−0.2

Bin smooth GAM effects - spatial

I Bin or cluster f̂5(longi , lati ) for i ∈ {1, ..., 1 146}.

I We use: (also see classint package in R)

• equal intervals;

• quantile binning;

• complete linkage (see Kaufman & Rousseeuw, 1990)

• Fisher’s natural breaks (see Fisher, 1958 and Slocum et al., 2005).

I Tune number of clusters towards lowest AIC for the GAM with abinned spatial effect

Bin smooth GAM effects - spatial

[−0.48,−0.32)

[−0.32,−0.15)

[−0.15,0.012)

[0.012,0.18)

[0.18,0.34]

Quantile

[−0.48,−0.18)

[−0.18,−0.092)

[−0.092,−0.019)

[−0.019,0.067)

[0.067,0.34]

Complete

[−0.48,−0.32)

[−0.32,−0.079)

[−0.079,0.047)

[0.047,0.23)

[0.23,0.34]

Fisher

[−0.48,−0.27)

[−0.27,−0.14)

[−0.14,−0.036)

[−0.036,0.11)

[0.11,0.34]

Bin smooth GAM effects - continuous

I Bin f̂1(ageph), f̂2(power), f̂3(bm), f̂4(ageph, power).

I Use evolutionary tree (evtree, Grubinger et al, 2014) e.g. on:

Covariate: ageph Response: f̂1(ageph) Weight: w

18 0.495 1619 0.459 11620 0.424 393

I Evaluate tree with MSE + α · complexity penalty,

∑max(ageph)i=min(ageph)

wagephi(f̂1(agephi )− f̂ b1 (agephi ))

2∑max(ageph)i=min(ageph)

wagephi

and α tuning parameter.

Bin smooth GAM effects - continuous

25 50 75

−0.5

0.5f̂4

25 50 75

GLMcoefficients

−0.07

−0.021

−0.4

−0.2

f̂5GLMcoefficients

−0.329

−0.204

−0.155

Sparse modeling of risk factors in insurance analytics

Devriendt, Antonio, et al., 2017 (in progress)

I Less is more: (Hastie, Tibshirani & Wainwright, 2015)

a sparse model is easier to estimate and interpret than a dense model.

I Regularize (with budget constraint t, or regularization parameter λ):

minβ0,β{− logL} subject to ‖β‖1 ≤ t,

or equivalenty

minβ0,β

− logL+ λ ·p∑

|βj |

Shrinks coefficients and even sets some to zero.

K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 14 / 28

Lasso and friends

I Adjust lasso regularization to the type of risk factor:

• Determine type (nominal / numeric ∼ ordinal / spatial);

• Allocate logical penalty.

I Thus, for J risk factors, each with regularization term Pj(.), we wantto optimize:

− logL (β1, . . . ,βJ) + λ ·J∑

Pj (βj).

Matching regularization to type of risk factor

I Ordinal risk factors: fused lasso∑i

wi |βi+1 − βi |.

I Nominal risk factors: generalized fused lasso∑i>k

wi ,k |βi − βk |.

I Spatial risk factor: graph guided fused lasso∑(i ,k)∈G

wi ,k |βi − βk |.

Unified GLM framework with multiple type of penalties

I Gertheiss & Tutz (2010) and Oelker & Gertheiss (2017):

• GLMs with various penalties.

• R package available: gvcm.cat (not maintained).

I Uses local quadratic approximations of penalties and PIRLS:

• non-exact selection or fusion;

• computationally intensive.

Unified GLM framework with multiple type of penalties

I Our contribution:

• implements an efficient algorithm (with proximal operators);

• scalable to big data (splits into smaller sub-problems);

• flexibility of regularization

penalty takes type of risk factor into account;

works for all popular penalties;

• unifies penalty-specific (machine learning) literature with statistical (or:actuarial) literature.

MTPL claim frequency with multiple type of penalties

I Fit Poisson GLM with different penalties.

I Settings:

• Incorporate adaptive and standardization weights for better consistencyand predictive performance.

• Tune λ with BIC (λ̂BIC = 784).

I Re-estimate the final sparse GLM with standard GLM routines (from387 to 30 parms.).

1e−01 1e+01 1e+03 1e+05

period

lambda

BICAICGCV BIC.reestAIC.reestGCV.reest

−0.25

25 50 75

−1.00

0 50 100 150 200 250

−4.00

−3.00

−2.00

−1.00

0 10 20 30 40 50

carage

0 5 10 15 20

GAM pGLM pGLM refit

−0.20

fleet2 fourbyfour1 fuel2 monovolume1 sex2 sport2 use2

Binary variables

−0.10

coverage2 coverage3 period2 period3 period4

Coverage & Period variables

−0.25

25−S

Model variable

GAM pGLM pGLM refit

Wrap-up

I From multi-step to less is more.

I To do:

• further improve algorithm efficiency;

• implement penalties for spatial information, interaction effects;

• R package and paper in progress.

More information

For more information, please visit:

LRisk website, www.lrisk.be;

my homepage, www.econ.kuleuven.be/katrien.antonio.

Thanks to

Ageas Continental Europe Argenta

K. Antonio, KU Leuven & UvA More information 25 / 28

References

Henckaerts, R., Antonio, K., Clijsters, M. and Verbelen, R. (2017)A data driven strategy for the construction of insurance tariff classes.Submitted.

Wood, S. (2006)Generalized additive models: an introduction with R.Chapman and Hall/CRC Press.

Gertheiss, J. and Tutz, G. (2010).Sparse modeling of categorial explanatory variables.The Annals of Applied Statistics, 4(4), 2150-2180.

Oelker, M. and Gertheiss, J. (2017).A uniform framework for the combination of penalties in generalizedstructured models.Advances in Data Analysis and Classification, 11(1),97-120.

K. Antonio, KU Leuven & UvA List of references 26 / 28

References

Grubinger, T., Zeileis, A., and Pfeiffer, K.-P. (2014).evtree: Evolutionary learning of globally optimal classification andregression trees in R.Journal of Statistical Software, 61(1), 1-29.

Bivand, R. (2015).classInt: Choose Univariate Class Intervals.R package version 0.1-23.

Parikh, N. and Boyd, S. (2013).Proximal algorithms.Foundations and Trends in Optimization, 1(3):123-231.

Hastie, T., Tibshirani, R. and Wainwright, M. (2015)Statistical learning with sparsity: the Lasso and generalizations.Chapman and Hall/CRC Press.

K. Antonio, KU Leuven & UvA List of references 27 / 28

Extra plots

−4 −2 0 2 4

log(λ)

10 10 10 10 10

−7 −6 −5 −4 −3 −2

log(λ)

10 9 9 9 8 2

0 5 10 15 20

ordinal penalty example

var 1var 2var 3var 4var 5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

nominal penalty example

K. Antonio, KU Leuven & UvA Extra 28 / 28

Using risk factors in insurance analytics data driven strategies...

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