Focussed selection of the claim severitydistributioninnon ... and conference...Problem • Further, the Zis are assumed to be independent, and also independent of N. • Finally, the

Focussed selection of the claimseverity distribution in non-life in-suranceYinzhi Wang and Ingrid Hobæk Haff

May 24, 2018

Problem• Among the risks a non-life insurance company faces, insurance

risk is the most important.• Finding a good model for the total loss X due to claims from a

portfolio of policies is therefore essential in non-life insurance.• The classic way of modelling X is via the collective risk model :

X =N∑

i=1

Zi ,

where• N is the total number of claims over a certain period of time• Zi are the corresponding claim severities.

Ingrid Hobæk Haff Focussed selection of the claim severity distribution 1 / 22

Problem• Further, the Zis are assumed to be independent, and also

independent of N .• Finally, the Zis are assumed to be identically distributed [Kaas

et al., 2001, Klugman et al., 2012].• This is a simplification, but is adequate when only the sum X is of

interest.• Model selection then consists in finding

• a claim frequency distribution (for N )• a claim severity distribution (for Zi )


Problem• One of the main applications of the total loss distribution is the

estimation of the reserve.• The reserve q� is given by

P(X >q�) = �

for a small �.• It is therefore very important to capture the right tail of the total

loss distribution.• The claim severity distribution has much more influence on the tail

risk than the claim frequency distribution [Klugman et al., 2012].


Problem• Claim severities are positive random variables, that follow a skew

distribution with a heavier right tail.• Typical choices are:

• 2-parameter distributions: Gamma, Weibull, log-normal, Pareto• 3-parameter distributions: Burr, Beta prime.

• Typical strategies for selecting the claim severity distributioninclude [Gray and Pitts, 2013, Bakar et al., 2015, Reynkens et al.,2017, Brazauskas and Kleefeld, 2016]:• Q-Q plots• goodness-of-fit tests• AIC, BIC.


Problem• A disadvantage of such strategies is that they primarily test the fit

in the middle of the distribution where data is abundant.• Our main interest is however to estimate the reserve.• It is by no means certain that the best-fitting distribution overall is

the one providing the best reserve estimate.• Our idea is to select the claim severity distibution with the FIC,

using the reserve as the focus parameter.


FIC for claim severities• Since the typical candidate models are not nested, we use the

FIC version of Jullum and Hjort [2017], rather than the original FICfrom Claeskens and Hjort [2003].

• The desired focus parameter in this setting is the reserve q�,which is a quantile in the the total loss distribution, which dependson both the claim frequency and the claim severity distributions.

• There is no explicit expression for the pdf of this distribution.• Hence, the FIC cannot be computed analytically when q� is the

focus parameter.


Focus parameter• One would have to resort to Monte Carlo methods, which would

be computationally very heavy.• Further, the non-parametric estimate of the reserve would be

computed based on observations of X , of which there is only oneper year, resulting in 20 observations in total at the very best.

• As q� is a quantile far out in the right tail of the distribution of X ,the uncertainty would be huge.

• For these reasons, we have chosen to use other focus parametersas proxies for the reserve.


Focus parameter• One possibility is to use μ = E(X ) + k · sd(X ) for some k .• Since the number N of claims is assumed to be independent of

the individual claim sizes Zi , we have

E(X ) = E(N )E(Zi), sd(X ) =√

E(N )Var(Zi) + Var(N ) (E(Zi))2.

• As the aim is to select the claim severity distribution, the claimfrequency distribution is considered known, such that E(N ) andVar(N ) are known constants.

• If N ∼ Poisson(λ), then

μ = λE(Zi) + k√λE(Z 2

i ).

• This can easily be modified to other claim frequency distributions,such as the negative binomial.


Focus parameter• Another option is to let the α-quantile qZ

α of the claim severitydistribution, for some α ∈ (0,1), be the focus parameter.

• The reserve q� is not just a function of one quantile of the claimseverity distribution.

• A better selection criterion may be a weighted sum of FICs basedon quantiles in the right tail, qZ

αifor some weights wi and

αi ∈ (0,1), i = 1, . . . , k .• This results in an AFIC.


Aim of the study• We wanted to find out how the FIC performs for selection of the

claim severity distribution compared to other model selectionmethods for• different tail properties of the claim severity distribution• different sample sizes• different level � for the reserve.

• The goal was to find the distribution that produced the bestreserve estimate.

• Q-Q plots are quite subjective and cannot be automatised.• Goodness of fit tests are not really model selection methods.• The model selection methods used for comparison were therefore

the AIC and BIC.


Candidate modelsThe candidate models for the claim severity distribution were:• 1-parameter: exponential• 2-parameter: Gamma, Weibull, log-normal, Pareto, log-Gamma• 3-parameter: Burr, Beta prime.• 4-parameter distribution:

Z = β(

GθGα

)η,

where α, θ, β, η >0 and Gα and Gθ are Gamma distributed withmean 1 and shape α and θ, respectively.


Assessment of the model selectioncriteria• Each candidate claim severity distribution was used in turn as the

true distribution.• In each simulation, all candidate models were fitted.• Each model selection criterion was then used to find the "best"

claim severity distribution.• The model selected by each of the criteria was used to estimate

the reserve.• The reserve estimates were then assessed in terms of the bias

and root mean squared error (RMSE).


Parameter setting• Since we were mainly interested in the claim severity distribution,

the claim frequency distribution was simply Poisson(λ), withλ = 50,500.

• The parameters of the different claim severity distributions wereset so that they would have rather different tail behaviour: all haveE(Zi) ≈ 10, but sd(Zi) ranges from 2.9 to 24.6.

• Focus parameters:• μ = E(X ) + k · sd(X ), with k = 1.5,2,3• μ = qZ

�

• AFIC with μi = qZαi

, αi = 0.90,0.91, . . . , 0.99, wi = 1/10.

• Sample sizes: n = 5000,500,50.• Reserve: � = 0.05,0.01.• Number of simulations in each experiment: 1000.


� = 0.01 n = 5000 λ = 50AIC BIC FICmkd FIC� AFIC

k =1.5 2 3

BIAS

Ga. -2.7 -2.7 -2.0 -2.0 -2.0 -1.7 2.0We. -9.0 1.4 1.3 1.2 1.3 1.3 1.4L.-G. 15 -4.3 -57 -122 -155 18 90Pa. -15 10 -4.4 -6.0 -5.3 7.0 9.7Burr 6.6 -4.4 0.87 -0.91 -0.91 0.83 0.87

RMSEGa. 22 20 3.8 3.7 3.8 3.7 3.9We. 7.4 5.7 5.6 5.7 5.6 5.4 5.7L.-G. 85 42 137 176 193 101 160Pa. 65 40 49 48 47 42 55Burr 64 61 3.1 3.1 3.1 3.0 3.1



k =1.5 2 3

BIAS

Ga. -0.54 -0.55 0.04 -0.06 0.14 0.28 0.52We. -37 0.13 0.17 0.13 0.27 0.12 0.27L.-G. 5.3 0.13 40 -22 -50 24 62Pa. -61 1.1 -2.7 -4.4 -3.9 0.09 -6.9Burr -292 -4.8 0.21 0.26 0.16 0.09 0.31

RMSEGa. 20 20 3.7 3.6 3.7 4.0 3.8We. 153 5.7 5.6 5.6 5.5 5.5 5.5L.-G. 33 23 120 91 75 27 55Pa. 216 23 33 27 28 29 33Burr 426 56 3.5 3.6 3.5 3.4 3.7



k =1.5 2 3

BIAS

Ga. -16 -2.2 30 28 22 0.79 -1.7We. -3.5 11 29 29 21 -2.1 269L.-G. 163 55 -335 -336 42 28 88Pa. 149 213 -103 -98 97 47 131Burr -14 -0.88 24 21 -15 -0.64 -2.3

RMSEGa. 103 32 108 104 104 32 33We. 61 78 114 115 110 50 324L.-G. 2873 1749 382 380 367 724 1018Pa. 621 535 240 241 239 389 614Burr 97 28 94 90 94 28 39



k =1.5 2 3

BIAS

Ga. -9.0 -8.7 24 16 6.3 -2.5 -3.0We. -4.3 0.36 0.82 0.75 0.87 1.0 1.1L.-G. 150 38 -108 -69 -59 79 175Pa. -157 3.1 137 153 -15 -5.8 6.9Burr -54 -44 0.96 0.87 0.81 0.32 0.93




k =1.5 2 3

BIAS

Ga. -119 -5.9 89 88 85 3.3 4.8We. -39 -24 51 51 51 61 90L.-G. 3052 1194 -1349 -1354 1350 90 613Pa. 670 702 -497 -495 -502 245 469Burr -122 -13 64 63 61 -9.7 -20



Norwegian motor insurance claimsAIC BIC FICmkd FIC0.05 FIC0.01 AFIC

k =1.5 2 3

Exp. 5.4 5.4 1.5 1.5 1.7 1.2 2.7 1.3Ga. 457 457 1.5 1.6 3.0 53 1901 340L.-n. 5.3 5.3 4.8 5.2 5.8 2.8 19 10We. 5.4 5.4 0.92 0.96 1.1 16 1428 241L.-G. 7.5 7.5 481 581 813 370 8383 1297Pa. 5.4 5.4 4.7 7.8 16 3.5 296 59E. P. 5.3 5.3 607 635 694 14 104 44Burr 5.3 5.3 35 53 102 16 99 22F-par. 5.3 5.3 - - - 14 427 56


Summing up• We have explored how the focussed information criterion, FIC,

works as a tool for selecting the claim severity distribution.• As the reserve cannot be used directly as a focus parameter, we

have tried different proxy focus parameters.• The best of the FICs is FIC�, based on one single quantile from

the claim severity distribution.• The performance of FIC� is either comparable to or considerably

better than that of the BIC, except for a few cases.


Summing up• In particular, the FIC works well when

• the data are heavy-tailed and the sample size is small• the parameter of interest is a quantile far out in the tail of the total

loss distribution.

• It is surprising that the AFIC, which takes several quantiles of theclaim severity distribution into account, does not perform better.

• It might be improved by putting more weigth on quantiles furtherout in the tail.

• We would have expected the FICmkd to be better for larger claimfrequencies.

• That is however not the case in our simulations.


Further work• We have fixed the claim frequency distribution at the Poisson

distribution.• The FIC scores can however easily be modified to account for

other claim frequency distributions.• One could imagine doing a simultaneous selection of the claim

frequency and severity distributions using the FIC.• As the reserve typically is more influenced by the latter, it would

be interesting to see whether the reserve provides informationthat enables to discriminate between different claim frequencydistributions.


R. Kaas, M. Goovaerts, J. Dhaene, and M. Denuit. Modern Actuarial Risk Theory – Using R.Springer-Verlag Berlin Heidelberg, 2001.

S.A. Klugman, H.H. Panjer, and G.E. Willmot. Loss Models: From Data to Decisions. WileySeries in Probability and Statistics. Wiley, 2012.

R.J. Gray and S.M. Pitts. Risk modelling in general insurance: From principles to practice.International Statistical Review, 81(3):478–479, 2013.

S.A. Abu Bakar, N.A. Hamzah, M. Maghsoudi, and S. Nadarajah. Modeling loss data usingcomposite models. Insurance: Mathematics and Economics, 61:146 – 154, 2015.

Tom Reynkens, Roel Verbelen, Jan Beirlant, and Katrien Antonio. Modelling censored lossesusing splicing: A global fit strategy with mixed erlang and extreme value distributions.Insurance: Mathematics and Economics, 77:65 – 77, 2017.

Vytaras Brazauskas and Andreas Kleefeld. Modeling severity and measuring tail risk ofnorwegian fire claims. North American Actuarial Journal, 20:1–16, 2016.

M. Jullum and N.L. Hjort. Parametric or nonparametric: the fic approach. Statistica sinica., 2017.ISSN 1017-0405.

Gerda Claeskens and Nils Lid Hjort. The focused information criterion. Journal of the AmericanStatistical Association, 98:900–916, 2003.


Yinzhi Wang and Ingrid HobækHaff

Focussed selection of theclaim severity distribution innon-life insurance

Documents

Focussed selection of the claim severitydistributioninnon ... and conference...Problem • Further, the Zis are assumed to be independent, and also independent of N. • Finally, the