Modeling and measuring incurred claims risk liabilities for a multi-line property and casualty insurer
Carlos Andrés Araiza Iturria, Frédéric Godin, Mélina Mailhot
MModeling and measuring incurred claims risk liabilitiesfor a multi-line property and casualty insurer ∗ Carlos Andr´es Araiza Iturria † , Fr´ed´eric Godin ‡ , M´elina Mailhot § July 15, 2020
Abstract
We propose a stochastic model allowing property and casualty insurers with mul-tiple business lines to measure their liabilities for incurred claims risk and calculateassociated capital requirements. Our model includes many desirable features which en-able reproducing empirical properties of loss ratio dynamics. For instance, our modelintegrates a double generalized linear model relying on accident semester and develop-ment lag effects to represent both the mean and dispersion of loss ratio distributions,an autocorrelation structure between loss ratios of the various development lags, and ahierarchical copula model driving the dependence across the various business lines. Themodel allows for a joint simulation of loss triangles and the quantification of the overallportfolio risk through risk measures. Consequently, a diversification benefit associatedto the economic capital requirements can be measured, in accordance with IFRS 17standards which allow for the recognition of such benefit. The allocation of capitalacross business lines based on the Euler allocation principle is then illustrated. Theimplementation of our model is performed by estimating its parameters based on a carinsurance data obtained from the General Insurance Statistical Agency (GISA), and byconducting numerical simulations whose results are then presented.
KEYWORDS: IFRS 17, Loss triangles, Double Generalized Linear Models, Hierarchical cop-ulas, Risk measures, Capital allocation. ∗ Financial support from NSERC (Godin: RGPIN-2017-06837, Mailhot: RGPIN-2015-05447) and MITACS(Araiza Iturria, Godin and Mailhot: IT12099) is gratefully acknowledged. † University of Waterloo, Department of Statistics and Actuarial Science, 200 University Ave W, Waterloo,Ontario, Canada, N2L 3G1, [email protected] ‡ Concordia University, Department of Mathematics and Statistics, 1455 Boulevard de Maisonneuve O,Montr´eal, Qu´ebec, Canada, H3G 1M8, [email protected] § Concordia University, Department of Mathematics and Statistics, 1455 Boulevard de Maisonneuve O,Montr´eal, Qu´ebec, Canada, H3G 1M8, [email protected] a r X i v : . [ q -f i n . R M ] J u l Introduction
An important task in the practice of property and casualty insurance is the prediction offuture claims arising from incurred liabilities. These claims are commonly known as theunpaid claim liabilities. Such predictions are required for multiple purposes such as reservescalculations, financial reporting and the determination of economic capital requirements.The objective of this article is to provide a model allowing a multi-line insurance companyto forecast its unpaid claim liabilities, taking into account possible dependencies betweenbusiness lines. The model combines numerous desirable features and reproduces empiricalcharacteristics of loss ratio dynamics.First, a Tweedie distributed Double Generalized Linear Model (DGLM) is used to rep-resent the marginal distribution of loss ratios for each business line. The proposed modelis flexible and allows for the fluctuation of both the mean and dispersion across accidentsemesters and development lags. Furthermore, the Tweedie distribution allows for a massat zero, representing the situation where no loss is observed, which is frequent in propertyand casualty insurance. The Tweedie family of distributions was introduced in Tweedie [30].The flexibility of the Tweedie family and its ability to model null losses have made it anattractive distribution for loss reserving as in Avanzi et al. [5] and Smol´arov´a [26].Generalized linear models (GLM) are commonly used in the insurance industry to forecastfuture claims due to their advantageous trade-off in terms of flexibility, parsimony and easeof interpretation. A generalization of the GLM model called double generalized linear model (DGLM) is considered in Smyth and Jørgensen [28] as an alternative when the number ofclaims is not available. In recent actuarial literature, applications in non-life insurance ofTweedie DGLM can be found in Boucher and Davidov [6] and Andersen and Bonat [2].The latter generalization enables modeling the variability of the dispersion parameter jointlywith the mean, instead of considering a fixed dispersion parameter as in traditional GLMs.The current work makes the assumption that loss ratios for a given accident semester andbusiness line are autocorrelated across development lags. This assumption has been exploredin Hudecov´a and Peˇsta [14] where different correlation structures are compared in a claimreserving setting. An application of GLMs with correlated observations in the context ofproperty and casualty insurance can be found in Smol´arov´a [26] which illustrates the use ofsuch models for insurance pricing. Due to the autocorrelation assumption of our model, theestimation procedure used in the current paper relies on Generalized Estimating Equationswhich are presented among others in Liang and Zeger [20] or Hardin and Hilbe [13], andmore specifically in an insurance setting in Smol´arov´a [26].Another notable characteristic of the model developed herein is its convenient specifica-tion of the dependence between losses of the various business lines. Indeed, a hierarchical2opula is embedded in the model, which allows for a flexible and easily interpretable represen-tation of the global dependence structure. Copulas have been recently gaining in popularitydue to recent developments in the copulas theory and to increases in computational powerprovided by modern computers. In Shi and Frees [25], a Gaussian copula is used to measuredependence between personal and commercial auto lines. A hierarchical copula model is usedin Burgi et al. [7] to represent the dependence in an insurance portfolio and to study thecalculation of the diversification benefit for the insurer. A rank-based hierarchical copulamethod dealing with multiple property and casualty insurance lines with different paramet-ric copula families is used and compared to a nested Archimedean copula in Cˆot´e et al. [9].To perform simulations out of the hierarchical copula model, Arbenz et al. [3] establishesrigorous mathematical foundations and adapts the Iman-Conover reordering algorithm usedin the current paper.The usefulness of the model presented here is highlighted in the context of the new IFRS17 accounting standards. Indeed, such standards require the calculation of a quantity referredto as the risk adjustment for non-financial risks , which will be detailed subsequently. Amongthe admissible methods for the calculation of the latter quantity, some require a specificationof the entire portfolio loss distribution. Our modeling framework allows for the constructionof such a distribution; it even provides the joint distribution of unpaid claim losses over allbusiness lines. Hence, the current work illustrates how our loss triangles prediction modelcan be leveraged within a stochastic simulation to estimate the latter joint distribution,and therefore obtain estimates for the risk adjustment for non-financial risk and capitalrequirements. The allocation of reserves and capital requirements across the business linesbased on the Euler allocation principle is also illustrated.The paper is organized as follows. Section 2 discusses the valuation of insurance liabilitiesunder the new IFRS 17 standards. Section 3 describes the Canadian automobile insurancedataset used in the current study. In Section 4, the prediction model for loss ratios of a multi-line property and casualty insurer is presented. In Section 5, the estimation and simulationof the model are discussed. In Section 6, a numerical application of the model is illustrated.A stochastic simulation involving the use of traditional risk measures is compared to a costof capital approach with respect to the calculation of capital requirements, including itsallocation across the various business lines. Section 7 concludes.
The International Accounting Standards Board (IASB), an independent international non-profit group of experts in accounting and financial reporting, issued in May 2017 the IFRS17
Insurance Contracts standards, a new set of accounting standards for insurance contracts3uperseding the current regulatory framework IFRS 4. IFRS 17
Insurance Contracts estab-lishes principles for the recognition, measurement, presentation and disclosure of insurancecontracts.The effective date of IFRS 17 has officially been set by the IASB to January 1 st , 2023, meaning March 31 st , 2023 would correspond to the first quarter of reporting under IFRS 17.The unpaid claim liabilities, which are a very important part of the liabilities found inthe balance sheet of a property and casualty insurer, are referred to under IFRS 17 as theLiabilities for Incurred Claims (LIC). The LIC represent insurance events that have alreadyoccurred, but for which the claims have not been reported or have not been fully settled. Aparamount duty for insurers in the upcoming years will consist in measuring the LIC in amanner that is consistent with IFRS 17 Insurance Contracts standards. LIC are measuredwith the General Model which establishes in paragraph 32 of IASB [16] that upon initialrecognition, a group of insurance contracts should be measured as the sum of: • The fulfillment cash flow (FCF), which includes: – Estimates of future cash flows, – An adjustment to reflect the time value of money and the financial risks relatedto the future cash flows, – A risk adjustment for non-financial risk. • The contractual service margin (CSM).The CSM represents the unearned profit that the insurer will recognize as it providesservices in the future, as stated in paragraph 38 of IASB [16]. CSM applies for unexpiredcoverage. It is excluded from the scope of this work.The risk adjustment for non-financial risks is the compensation the entity requires forbearing the uncertainty of the amount and timing of the cash flows arising from non-financialrisks associated to claim losses, see Paragraph B88 of IASB [16]. The choice of the method-ology to calculate the risk adjustment for non-financial risks is not prescribed by IFRS 17standards. A few possible approaches, either based on the Cost of Capital (CoC) or riskmeasures (e.g. VaR and TVaR) are considered in subsequent sections of the current study.
The dataset used in our analysis comes from the General Insurance Statistical Agency (GISA)and corresponds to data for the entire Canadian automobile industry. The dataset contains The IASB had originally proposed the implementation of IFRS 17 to be effective in 2021, but it has beendelayed following public consultation. . Incremental incurred claim amounts and earned premiums are provided inthe dataset.Loss and Expense (L&E) semestrial claim amounts are available for each insurance lineand region combination from the first semester of 1997 to the second semester of 2017. Datafrom before 2003 are discarded, since notes on historical claims are only available startingfrom the first semester of 2003. Therefore, fifteen years of information are taken into account,i.e. data from 2003 to 2017.In order to work with stationary data, incremental semestrial loss amounts are scaledby the premium for the associated accident semester, which provides loss ratios. Indeed,the use of loss ratios instead of loss amounts removes the need to quantify trends related toyear-to-year changes in exposure.Previously observed loss ratios can be presented graphically in an upper triangle array,commonly known as a run-off triangle. This presentation can be done in two different ways;using either cumulative claims or incremental claims. Incremental claims are used in thecurrent study. Indeed, for a business line k , an accident semester i , and a development lag j , the loss ratio Y ( k ) i,j is defined as Y ( k ) i,j = C ( k ) i,j − C ( k ) i,j − p ( k ) i , C i, = 0 , where p ( k ) i is the amount of premiums collected for accident semester i and business line k ,and C ( k ) i,j represents the cumulative claims associated with accident semester i obtained untildevelopment lag j for business line k . The loss ratios run-off triangles are then used in thesubsequent modeling steps.For illustrative purposes, the current study considers a fictitious insurer whose exposureis the content of the entire aforementioned GISA dataset. Consider an insurance portfolio composed of K possibly dependent business lines. Themain objective is to model the joint distribution of loss ratios Y ( k ) i,j for all possible values ofaccident semester i , development lag j and business line k . Then, one can predict loss ratiosfor future periods based on the observed ones. We obtain the run-off triangle of all semestrial Atlantic Canada is made up of four provinces: Prince Edward Island, New Brunswick, Nova Scotia andNewfoundland & Labrador. The development lag is the number of semesters between the occurrence of an accident and the date inwhich the final payment is made (closure of case). i = { , , . . . , I } with I = 30, and development lag j = { , , . . . , J } with J = 30 such that i + j ≤ I + 1. Theupper and lower triangles are defined as T U := { ( i, j ) : i ∈ { , , . . . , I } , j ∈ { , , . . . , J } , i + j ≤ I + 1 } , T L := { ( i, j ) : i ∈ { , , . . . , I } , j ∈ { , , . . . , J } , i + j > I + 1 } , which represent respectively loss ratios that are observable, and those which need to bepredicted to perform loss reserving. Once a model is fitted to loss ratios in T U , loss ratiosfrom T L can be forecasted. The latter procedure is known as completing the square.In Section 4.1, the marginal distribution of loss ratios Y ( k ) i,j is modeled for each k , i and j . Then, the dependence between loss ratios across development lags and business lines ismodeled in subsequent sections. For the marginal distribution of loss ratios, a Tweedie distributed DGLM is considered.A DGLM model is a generalization of a GLM where both the mean and the dispersionparameters are dependent on explanatory variables. Explaining the dispersion parameterwith a nested GLM adds flexilibity to the model. The Tweedie DGLM is equivalent to thecombination of a GLM for modeling the frequency parameter and of another GLM to quantifyseverity in the classic actuarial Compound Poisson-Gamma model under certain conditionsdescribed in Quijano Xacur [23]. As explained in Smyth and Jørgensen [28], an additionalbenefit of using a DGLM with a Tweedie distribution is that it allows handling the case wherethe claim count has not been observed or recorded, or is not reliable; Tweedie distributionsare typically mixed distributions with a positive mass at zero.Before introducing the formal marginal model, we introduce the Tweedie distribution. Avariable Y is said to have the Tweedie distribution T W p ( µ, φ ) if its density is given by f Y ( y ; µ, φ, p ) = a ( y ; φ, p ) exp (cid:34) φ (cid:32) y µ − p − p − µ − p − p (cid:33)(cid:35) , y > a ( y ; φ, p ) = ∞ (cid:88) r =1 (cid:34) φ p − y (cid:96) (2 − p )( p − (cid:96) (cid:35) r r !Γ( r(cid:96) ) y, with (cid:96) = − − p − p and Γ( z ) = (cid:82) ∞ x z − e − x dx being the Gamma function. The point of mass atzero has a probability provided by f Y (0; µ, φ, p ) = exp (cid:34) − µ − p φ (2 − p ) (cid:35) . a) Mean parameters. (b) Dispersion parameters. Figure 1: Loss triangle’s coefficients for the mean GLM model (left panel) and the dispersionGLM model (right panel). Each row of the loss triangle corresponds to an accident semester,whereas each column corresponds to a development lag.The expectation and variance of Y are respectively E [ Y ] = µ and Var[ Y ] = φµ p .In our model, the set of predictors contains exclusively deterministic dummy variablesindicating the current accident semester i and development lag j . The assumption for ourmodel is that Y ( k ) i,j ∼ T W p ( k ) ( µ ( k ) i,j , φ ( k ) j ) with g ( µ ( k ) i,j ) = ι ( k ) + α ( k ) i + δ ( k ) j , (1) g ( φ ( k ) j ) = ι ( k ) d + γ ( k ) j , (2)where constants ι ( k ) and ι ( k ) d are respectively the intercept for the mean and dispersion equa-tions, the constants α ( k ) i and δ ( k ) j represent respectively the accident semester and develop-ment lag effects for the mean equation, and the constants γ ( k ) j reflect the development lageffect for the dispersion parameter. For any given business line, the dispersion parameter istherefore assumed to depend only on the development lag, and not on the accident semester.This decision is taken to avoid an over-parametrization of the model. Furthermore, unre-ported verification performed by the authors indicated that including an accident semestereffect in the dispersion parameter has a limited impact. Figures 1a and 1b illustrate param-eters driving respectively the mean and dispersion of each entry of the loss triangle.The log-link function g ( x ) = log( x ) is used for both the mean and dispersion equations,which is a standard choice.In actuarial literature as in Shi and Frees [25], Cˆot´e et al. [9], Smol´arov´a [26] insurance datais usually available for 10 years, making up a total of 55 loss ratios in their data set. Moreover,the GLM they consider has 20 parameters, i.e. the ratio of the number of parameters over thenumber of data points is 0.36. Within the 15 years of loss ratio data from the current study, we7re dealing with 465 data points for each line of business. Moreover, the DGLM we considerhas 89 parameters per business line. Thus, the ratio of the number of parameters over thenumber of data points is 0.19. Our model can therefore be considered relatively parsimoniousin comparison to literature benchmarks; it should therefore not be more prone to overfittingthat the latter. Nevertheless, the number of parameters versus number of observations ratiois still considerably high in comparison to many other applications in statistics; care mustbe applied during the review of the calibration in practice to ensure the variability of lossratios is not under-estimated due to over-fitting, which would be inconvenient from a riskquantification standpoint.The marginal distribution model for each business line k involves the following parametersto be estimated: the Tweedie index p ( k ) , intercepts ι ( k ) and ι ( k ) d , accident semester effects α ( k ) i , i = 1 , . . . , I , and development lag effects δ ( k ) j and γ ( k ) j , j = 1 , . . . , J . The next step in the model construction consists in specifying the dependence structureof loss ratios over the various development lags within a given business line for a givenaccident semester. The objective is to remove the marginal effects, which allows analyzingthe dependence across business lines in further steps.The first assumption made within the model is that loss ratios from different accidentsemesters are independent. Thus, we assume that for any business lines k , k and develop-ment lags j , j , when two different accident semesters i (cid:54) = i are considered, the associatedloss ratios Y ( k ) i ,j and Y ( k ) i ,j are independent. This is a standard assumption in the literature,see for instance Avanzi et al. [5] or Cˆot´e et al. [9].Within any given accident semester i , for a fixed business line k , a dependence struc-ture across development lags is considered. The assumption is that the correlation betweenloss ratios of development lags j and j (cid:48) is given by cor( Y ( k ) i,j , Y ( k ) i,j (cid:48) ) = ρ | j − j (cid:48) | k for some con-stants ρ , . . . , ρ K ; we assume that the correlation of loss ratios decreases exponentially as thedistance between their respective development period increases. The correlation intensity dif-fers for each line of business, but remains constant over the different accident semesters. ThePearson correlation only measures linear dependence. As such, possibly multiple dependencestructures (e.g. copulas) could lead to such correlation structures over the development lags.We choose not to explicitly specify the dependence structure of the loss ratios across thedevelopment lag dimension further than through its correlation structure. Note that sincepredictors in the DGLM model from the current paper are deterministic dummy variables,the dependence structure of loss ratios Y ( k ) i,j carries over to scaled innovations defined in thenext section. 8 .3 Dependence between business lines The remaining part of the model specification consists in detailing the dependence structureof loss ratios between business lines. A convenient approach to represent this dependencestructure in an easily interpretable way consists in using hierarchical copula models (HCM).Such a dependence structure is assumed to hold on the decorrelated loss ratio innovations onwhich autocorrelation impacts were removed.For that purpose, define the scaled innovations ˜ Y ( k ) i,j for loss ratios through˜ Y ( k ) i,j = Y ( k ) i,j − E (cid:104) Y ( k ) i,j (cid:105)(cid:114) Var (cid:104) Y ( k ) i,j (cid:105) . Jørgensen [18] shows the following result allowing to approximate the distribution of scaledinnovations by a normal distribution when Y ( k ) i,j is Tweedie distributed:˜ Y ( k ) i,j d −→ N (0 ,
1) as φ → , (3)where d −→ denotes convergence in distribution. Define the column vector ˜ Y ( k ) i = (cid:104) ˜ Y ( k ) i, , . . . , ˜ Y ( k ) i,J (cid:105) (cid:62) which contains loss ratio scaled innovations for all development lags associated with businessline k and accident semester i . As explained in Section 4.2, the correlation matrix of ˜ Y ( k ) i isgiven by R k,J which is defined as R k,J = ρ k ρ k . . . ρ J − k ρ k ρ k . . . ρ J − k ρ k ρ k . . . ρ J − k ... ... ... . . . ... ρ J − k ρ J − k ρ J − k . . . . (4)Define L − k being the inverse of the lower triangle matrix in the Choleski decompositionof R k,J , and U ( k ) i = L − k ˜ Y ( k ) i being the decorrelated innovations vector for accident semester i and business line k . Elements of the vector U ( k ) i , denoted respectively U ( k ) i, , . . . , U ( k ) i,J , areuncorrelated.The main assumption about the dependence between business lines is that for any ac-cident semester i and development lag j , the copula representing the dependence betweendecorrelated innovations U (1) i,j , . . . , U ( K ) i,j does not depend on i nor j . The model selected tocharacterize such dependence is presented in the next section. Hierarchical copulas are models which involve sequentially specifying the dependence be-tween subgroups of the population and eventually obtain a dependence model between all9ubgroups. They are convenient dependence models because they are easy to estimate, val-idate and interpret. Moreover, such models are adapted to frameworks where there existsa natural order in which subgroups can be aggregated. Such an approach is appropriate inan insurance setting where the portfolio is already subdivided, for instance by geographicalregions, dependence on legislation Burgi et al. [7], similarity of insurable risk types Shi andFrees [25], or according to some dependence distance metric as in Cˆot´e et al. [9].In order to ease the interpretation of the dependence structure, a bivariate approach ischosen. The six lines of business from the GISA dataset are paired first through a geographicalcriterion; for each geographical region, the personal and commercial auto lines are linkedtogether through first level copulas . The first level copulas C , C and C represent thedependence between the personal and commercial auto decorrelated innovations respectivelyfor Ontario, Alberta and Atlantic Canada. Then, for the three regional groups obtained, thesum of decorrelated innovations associated with each cluster is considered: U (ON) i,j = U (1) i,j + U (2) i,j , U (AB) i,j = U (3) i,j + U (4) i,j , U (ATL) i,j = U (5) i,j + U (6) i,j , where for each cluster the first and second components represent respectively the personaland commerical auto lines.The subsequent level pairing criteria are determined as in Cˆot´e et al. [9] by clusteringthe most dependent regions based on each pair’s Kendall τ . More precisely, a second levelcopula C is incorporated to link summed decorrelated innovations from the Alberta andAtlantic clusters, i.e. the copula C represents the dependence model between U (AB) i,j and U (ATL) i,j . Finally, summing the decorrelated innovations within the Alberta-Atlantic clusterthrough U (AB+ATL) i,j = U (AB) i,j + U (ATL) i,j , one last bivariate copula C is integrated to represent the dependence between U (AB+ATL) i,j and U (ON) i,j which correspond to the Alberta-Atlantic cluster and the Ontario cluster. For avisual representation of the hierarchical copula used in the model, see Figure 2.This aggregation approach is consistent for instance with the work of Arbenz et al. [3].For a complete specification of the dependence model, their work includes a conditionalindependence assumption, meaning that given the aggregate scaled innovation at a givennode, children of this node are independent from any node that is not a child of that givennode. This same assumption holds in the current work which allows to fit any copula at eachnode regardless of the parametric family. 10 C U (1) i,j U (2) i,j C C U (3) i,j U (4) i,j C U (5) i,j U (6) i,j Figure 2: Hierarchical copula model used in the current modelFor more details about the copulas selected to compose the hierarchical copula model,refer to Section 5.1.2.Since losses from dependent business lines are not comonotonic, the total loss aggregatedover all business lines is considered less risky than the set of all business line losses consideredin silo (i.e. separately); this leads to the existence of a diversification benefit. The recognitionof such diversification benefit for the risk adjustment for non-financial risks is allowed by IFRS17 standards, when one is able to show that diversification holds in periods of stress.
The current section details the implementation of the proposed model. The estimation of themodel parameters is first discussed. Then, we present a stochastic simulation algorithm togenerate future loss ratios and obtain loss distributions in order to compute capital require-ments and the risk adjustment for non-financial risks.
The current section details the estimation algorithms used for the estimation of the modelparameters. The estimation is done in two steps. The first step consists in estimatingparameters of the DGLM models representing the distribution of loss ratios independentlyfor each business line. The second step entails specifying the structure of the hierarchicalcopula model and estimating its parameters.
First, the parameters impacting a single business line are estimated for each business line k individually. The estimation approach relies on a Generalized Estimation Equations (GEE)method for parameters of the mean component of the DGLM and a Restricted Maximum11ikelihood (REML) approach for the dispersion parameters. The GEE is a convenient ap-proach to estimate parameters of a DGLM model in the presence of correlation betweenobservations, see for instance Liang and Zeger [20], Hardin and Hilbe [13] and Smol´arov´a[26].Indeed, the usual assumption of independence between observations of a DGLM does nothold in the current model. The REML allows circumventing a joint estimation of both meanand dispersion parameters and enables reducing the downward bias associated the traditionalmaximum likelihood estimates of dispersion parameters, see for instance Lee and Nelder [19].An iterative algorithm is used to obtain the estimates of the parameters. Indeed, param-eters impacting the business line k ∈ { , , . . . , K } are split in two subsets: Θ ( µ ) k and Θ ( φ ) k which contain, respectively, the parameters driving the mean and dispersion:Θ ( µ ) k := { ι ( k ) , α ( k ) j , δ ( k ) j : j = 2 , . . . , J } , Θ ( φ ) k := { ι ( k ) d , γ ( k ) j : j = 2 , . . . , J } . Note that the constraint α ( k )1 = δ ( k )1 = γ ( k )1 = 0 is imposed to avoid identifiability issues. Thesimultaneous estimation of the mean and dispersion for a Tweedie distribution is possible dueto the statistical orthogonality of the parameters, see for instance Cox and Reid [10], Smyth[27]. For a fixed value of k , the iterative procedure goes as follows until convergence: Algorithm 1
Estimates of the DGLM parameters
Step (1)
Keeping the current estimates of mean and dispersion parameters Θ ( µ ) k and Θ ( φ ) k fixed, refine the estimate of the development lag correlation parameter ρ k . Step (2)
Keeping the current estimates of dispersion parameters Θ ( φ ) k and developmentlag correlation parameter ρ k fixed, refine estimates of the mean parameters Θ ( µ ) k . Step (3)
Keeping the current estimates of mean parameters Θ ( µ ) k and development lagcorrelation parameter ρ k fixed, refine estimates of the dispersion parameters Θ ( φ ) k .Details about the selection of a suitable value of the Tweedie index p k are provided inAppendix B.More details are now provided on each step of Algorithm 1. To ease the notation, lossratios for a fixed accident semester i ∈ { , , . . . , I } are regrouped in a random vector: Y ( k ) i = (cid:104) Y ( k ) i, , . . . , Y ( k ) i,n i (cid:105) (cid:62) , which corresponds to the vector of the n i := J + 1 − i observedloss ratios from accident semester i (i.e. for all development lags j = 1 , ..., n i ). Recall thatthe unconditional distribution of each of its component is given by Y ( k ) i,j ∼ TW p k ( µ ( k ) i,j , φ ( k ) j )for a development lag j . Step (1)
12t the first step, the correlation parameter is refined according to the following formulaanalogous to the sample correlation of scaled innovations:ˆ ρ k = I − (cid:88) i =1 n i (cid:88) j =2 ˜ Y ( k ) i,j ˜ Y ( k ) i,j − I − (cid:88) i =1 n i (cid:88) j =2 (cid:16) ˜ Y ( k ) i,j − (cid:17) . Step (2)
The second step of Algorithm 1, where mean parameters Θ ( µ ) k are refined, is now discussed.Denote the column mean vector of Y ( k ) i by µ ( k ) i = (cid:104) µ ( k ) i, , . . . , µ ( k ) i,n i (cid:105) (cid:62) and its dispersion vector φ ( k ) i = (cid:104) φ ( k )1 , . . . , φ ( k ) n i (cid:105) (cid:62) . Mean parameters estimates that are being refined are chosen as thesolution to the following GEE: n i (cid:88) i =1 D ( k ) (cid:62) i V ( k ) − i (cid:16) Y ( k ) i − µ ( k ) i (cid:17) = 0 , (5)where D ( k ) i = ∂ µ ( k ) i ∂ β ( k ) is a matrix of dimension ( n i × q ) containing partial derivatives, β ( k ) isthe mean parameter vector of dimension q = 1 + ( I −
1) + ( J −
1) = 2 J − β ( k ) = (cid:104) ι ( k ) α ( k )2 · · · α ( k ) I δ ( k )2 · · · δ ( k ) J (cid:105) (1 × q ) , V ( k ) i = A ( k ) / i R n i ( ρ k ) A ( k ) / i is a variance matrix of dimension ( n i × n i ), where R n i ( ρ k ) = R k,n i is the correlation matrix of the random vector Y ( k ) i defined in (4), and the diagonal matrix A ( k ) i is given by A ( k ) i = φ ( k )1 V k ( µ ( k ) i, ) 0 . . . φ ( k )2 V k ( µ ( k ) i, ) . . . . . . φ ( k ) n i V k ( µ ( k ) i,n i ) ( n i × n i ) , with V k representing the variance function of the Tweedie family through equation V k ( µ ) = µ p k . The variance matrix V ( k ) i is key to capture the correlation between observations sincethe correlation matrix R n i ( ρ k ) is functionally related to the scalar ρ k .If the correlation matrix was the identity matrix, i.e., R n i ( ρ ) = I n i , where I n i is the iden-tity matrix of dimension n i , the estimation procedure would be equivalent to the traditionalDGLM estimation where the independence is assumed between observations. However, theindependence assumption does not hold in the current study as outlined in Section 4.2.13he use of the Generalized Estimating Equation (5) is equivalent to using a weightedleast squares estimator. This entails that the estimator of β ( k ) is consistent. It is relevant tonote that the combination of a DGLM model and a correlation structure between innovationsis a novel addition to loss triangle modeling literature. Step (3)
The third step of the estimation procedure entails refining the estimate of variance disper-sion parameters while keeping mean and correlation related parameter estimates fixed. Forsuch purposes, an approach similar to Smyth [27] is followed, where the dispersion parametersestimation relies on the construction of an auxiliary GLM. In this auxiliary GLM, measuresof disparity between realized and expected loss ratios, called deviances, are constructed andserve as the dependent variable. However, a modification to the latter approach proposedby Lee and Nelder [19] and implemented in the context of insurance claims modeling bySmyth and Jørgensen [28] is considered. Such a modification entails applying a correctionto the deviance associated with each observation based on its leverage; the correction allowsreducing the downward bias of small sample dispersion parameter estimates, especially fordevelopment lags for which very few observations are available. Lee and Nelder [19] statethat the leverage-based correction also provides the benefit of accelerating convergence of theestimation procedure while having an overall limited impact on resulting estimates.The procedure is largely inspired by Smyth and Jørgensen [28], and the reader is referredto the latter paper for more extensive details. First, unit deviances are defined as d ( k ) i,j = 2 (cid:32) y ( k ) i,j y ( k ) − pk i,j − µ ( k ) − pk i,j − p k − y ( k ) − pk i − µ ( k ) − pk i,j − p k (cid:33) . The objective consists in choosing dispersion parameters such that the various φ ( k ) j definedin (2) match unit deviances d ( k ) i,j as closely as possible; indeed, in the DGLM model, theexpected value of the deviance d ( k ) i,j is φ ( k ) j as stated in Lee and Nelder [19]. The parametersand predictors of the auxiliary GLM constructed for dispersion parameter estimation are14espectively given by the vector γ ( k ) and the dummy matrix Z defined according to Z γ ( k ) = . . . . . . . . . . . . . . . . . . ( n × J ) ι ( k ) d γ ( k )2 ... γ ( k ) J ( J × = ι ( k ) d ι ( k ) d + γ ( k )2 ... ι ( k ) d + γ ( k ) J ι ( k ) d ... ι ( k ) d + γ ( k ) J − ... ι ( k ) d ( n × . where n = Card( T U ) = J ( J +1)2 is the number of observed elements in the loss triangle T U associated with a given business line. Indeed, as seen in Figure 1b, each row of the matrix Z corresponds to an entry of the loss triangle so that the element in the same row of Z γ ( k ) contains the sum of all parameters characterizing its dispersion.Moreover, the dummy matrix X of dimension n × q is defined through X β ( k ) = I J . . . . . .
01 0 . . . . . . . . . . . .
11 1 . . . . . .
01 1 . . . . . . . . . . . . ( n × q ) ι ( k ) α ( k )2 ... α ( k ) I δ ( k )2 ... δ ( k ) J ( q × = ι ( k ) ι ( k ) + δ ( k )2 ... ι ( k ) + δ ( k ) J ι ( k ) + α ( k )2 ι ( k ) + α ( k )2 + δ ( k )2 ... ι ( k ) + α ( k ) I ( n × , where, as seen in Figure 1a, each row of the matrix in the right-hand side of the equationabove corresponds to the sum of all coefficients characterizing the mean for a given entry ofthe loss triangle. Therefore, the matrix X which contains predictors of the mean parametersGLM composing the DGLM model. Moreover, another matrix W of dimension n × n isdefined through W = diag (cid:32) ∂g ( µ ( k ) i,j ) ∂µ (cid:33) − Y ( k ) i,j ) = diag µ ( k ) i,j − p k φ ( k ) j , ( i, j ) ∈ T U Note that such definition of the weight matrix W disregards the presence of correlation between obser-vations across development lags. Indeed, the framework of Smyth and Jørgensen [28] was developed underthe assumption of independent observations. We leave the consideration of the correlation in this step as afuture refinement to our model. Thisallows defining the diagonal projection matrix H of dimension n × n as H = W / X ( X T W X ) − X T W / . Elements on the diagonal of H , known as the leverages, are denoted by h ( k ) i,j , ( i, j ) ∈ T U .The leverage matrix allows defining modified deviances as d ∗ ( k ) i,j = d ( k ) i,j − h ( k ) i,j . Using suchmodified deviances in the estimation procedures, the approach of Smyth [27] ultimatelyamounts to setting γ ( k ) = ( Z T W d Z ) − Z T W d z d where W d is the n × n diagonal matrix defined as W d = diag (cid:32) − h ( k ) i,j (cid:33) , ( i, j ) ∈ T U and the column vector z d of length n is given by z d = (cid:34) d ∗ ( k ) i,j − φ ( k ) j φ ( k ) j + log φ ( k ) j (cid:35) ( i,j ) ∈T U . Estimated parameters resulting from this three-step procedure can be found in Tablefound in Appendix A.
Once the marginal distribution parameters are estimated for all business lines, the estimationof the hierarchical model is then performed. To obtain the set of copula parameter estimates,maximum pseudo-likelihood is applied independently at each node of the copula hierarchicaltree representation due to the conditional independence assumption of the hierarchical modelmentioned in Section 4.4.Under this method, model residuals are transformed as approximate Uniform[0 ,
1] vari-ables, called the pseudo-uniform residuals, through the application of the decorrelated resid-ual’s empirical cdf to the decorrelated residual itself. This is equivalent to setting pseudo-uniform residuals equal to the scaled ranks decorrelated residuals. More precisely, for a givenaccident year i and development lag year j , pseudo uniform residuals V and V are defined as V ( k ) i,j = F k (cid:16) U ( k ) i,j (cid:17) , k = 1 , . . . , , V ( ON ) i,j = F ( ON ) (cid:16) U ( ON ) i,j (cid:17) , V ( AB ) i,j = F ( AB ) (cid:16) U ( AB ) i,j (cid:17) V ( AT L ) i,j = F ( AT L ) (cid:16) U ( AT L ) i,j (cid:17) , V ( AB + AT L ) i,j = F ( AB + AT L ) (cid:16) U ( AB + AT L ) i,j (cid:17) The order of indices ( i, j ) put into the diagonal of the matrix are respectively (1 , , , . . . , (1 , J ) , (2 , , (2 , , . . . F ’s denote empirical cdfs (with a scaling correction to avoid values exactly equalto 1), i.e. F k ( x ) = (cid:80) ( i,j ) ∈T U { U ( k ) i,j ≤ x } card( T U ) + 1 . Once the pseudo-uniform residuals are obtained, the parameters θ of a bivariate copuladefining the dependence between to given groups of business lines κ and κ is estimatedthrough ˆ θ = arg max θ (cid:88) ( i,j ) ∈T U log (cid:16) c θ ( v ( κ ) i,j , v ( κ ) i,j ) (cid:17) , where c θ is the parametric copula density and v ( κ ) i,j denotes the pseudo-uniform residuals fromthe group of business lines κ .The maximum pseudo-likelihood method differs from the traditional maximum likelihoodestimate by not considering the parametric estimates of marginal distributions in the func-tion to be maximized. Instead, one uses an empirical estimate of the marginal cumulativedistribution functions.Figure 3 illustrates the hierarchy used during the aggregation process whereas Figure 4provides the bivariate copula family chosen at each step of the hierarchical aggregation. t ν represents the t-copula with ν degrees of freedom and Π represents the independence copula.The bivariate t-copula with ν degrees of freedom and shape parameter ρ is given by, C ν,ρ ( u, v ) = t ν,ρ ( t − ν ( u ) , t − ν ( v ))= (cid:90) t − ν ( u ) −∞ (cid:90) t − ν ( v ) −∞ π (1 − ρ ) / (cid:18) s − ρst + t ν (1 − ρ ) (cid:19) − ν +22 dsdt, where t ν and t ν are the multivariate and univariate distribution functions of a student-t,respectively.ON+AB+ATLONPA CA AB+ATLABPA CA ATLPA CAFigure 3: HCM structure by province Π t PA CA t t PA CA ΠPA CAFigure 4: HCM structure by copula familyTable 1 provides the parameter estimates for each bivariate copula in the hierarchicalmodel. The results are obtained using the copula and
TwoCop package in R . The column The degrees of freedom ν is rounded to the nearest integer for the copula package. -value from the table provides the p -values from Cram´er-von Mises tests applied to verifythe goodness-of-fit of the copulas, see Genest and R´emillard [12], R´emillard and Scaillet [24].The null hypothesis of the test is H : C ∈ C θ , i.e., the copula C is indeed part of theparametric family C θ .Province Copulafamily Dependenceparameters Standard error of ρ p -valueON t ν = 8 , ρ = 0 .
166 0.050 0.59AB t ν = 5 , ρ = 0 .
290 0.049 0.77ATL Independence - - 0.07AB+ATL t ν = 4 , ρ = 0 .
228 0.050 0.39ON+AB+ATL Independence - - 0.54Table 1: Parameters and goodness-of-fit of copula models by province or province group
The procedure to perform a stochastic simulation of unobserved loss ratios in the loss trianglesbased on the model outlined in Section 4 is provided in this section.The first step consists in simulating independent decorrelated innovation vectors U i,j = (cid:104) U (1) i,j , . . . , U ( K ) i,j (cid:105) (6)for all accident semester and development lag combinations ( i, j ) that are yet unobserved,representing future observations. The dependence structure in the HCM is achieved throughthe Iman-Conover reordering algorithm proposed in Iman and Conover [17] and adapted byArbenz et al. [3]. Subsequently, the covariance structure of residuals across developmentlags is included in simulated innovations by applying a linear transformation, which providesscaled innovations. The latter are finally transformed through an inversion procedure toobtain simulated values for all missing loss ratios.A total of N realizations of the loss triangles must be performed. Each iteration consistingin the simulation of a single realization of the K loss triangles involves the following steps:1. First, independently for each ( i, j ) ∈ T L , simulate the decorrelated innovation vec-tor U i,j defined in (6), where the marginal distribution of each component U ( k ) i,j , k =1 , . . . , K is standard normal and where the copula driving the dependence betweenelements of U i,j is the aforementioned hierarchical copula model. Details on how to18erform such a simulation are found in Appendix C. Note that imposing the standardnormal distribution to decorrelated innovations is an approximation justified by (3).2. The second step consists in inducing the correlation structure by applying a transfor-mation on the decorrelated innovations so as to obtain the scaled innovations. This isdone independently for each business line k and accident semester i . Indeed, for each k = 1 , . . . , K and i = 2 , . . . , I , define the vector˘ U ( k ) i = (cid:104) U ( k ) i,I +2 − i , . . . , U ( k ) i,J (cid:105) (cid:62) which contains the simulated decorrelated innovations associated to each unobservedloss ratio. Then, as explained in Appendix D, the conditional distribution of unobservedloss ratio scaled innovations ˘˜ Y ( k ) i = (cid:104) ˜ Y ( k ) i,I +2 − i , . . . , ˜ Y ( k ) i,J (cid:105) (cid:62) given observed loss ratios (cid:104) ˜ Y ( k ) i, , . . . , ˜ Y ( k ) i,I +1 − i (cid:105) (cid:62) for the given accident year is approx-imately multivariate normal with mean vector ˘ M ( k ) i and covariance matrix ˘ V ( k ) i asdefined by (8)-(9) in Appendix D. This allows simulating the unobserved scaled inno-vations vector through ˘˜ Y ( k ) i = ˘ M ( k ) i + ˘ L − k,i ˘ U ( k ) i where ˘ L − k,i is the inverse of the lower triangle matrix in the Choleski decomposition of˘ V ( k ) i .3. Finally, the simulated scaled innovations are transformed such that they are properlyscaled and that marginal distributions match the true Tweedie one from the model(instead of being Gaussian): Y ( k ) i,j = F − i,j,k (cid:16) Φ( ˜ Y ( k ) i,j ); µ ( k ) i,j , φ ( k ) j , p ( k ) (cid:17) ≈ T W p ( k ) ( µ ( k ) i,j , φ ( k ) j )where F − i,j,k is the functional inverse of the CDF of the marginal distribution of the lossratio Y ( k ) i,j , and Φ is the standard normal CDF. Indeed, in the model, the unconditionaldistribution of the scaled innovations is approximately the standard normal one.Note that the correlation of loss ratios Y ( k ) i,j and Y ( k ) i,j in this simulation is only approx-imately equal to ρ | j − j | k ; indeed, in the third step of the simulation algorithm above, whenchanging the marginal distribution of loss ratios from normal to Tweedie, the correlationstructure also changes as the Pearson correlation between two random variables is not in-variant to changes in their marginal distributions.19 Numerical Results
The current section illustrates how the stochastic model presented in Section 4 can be used tocalculate the insurer’s risk adjustment for non-financial risks and its economic capital. Suchcalculation is performed through a Monte-Carlo simulation where multiple realizations of theunobserved (i.e. future) elements from the loss triangles are generated. Such realizations areused to construct a loss distribution for the insurer on which risk measures can be applied toquantify the insurer’s exposure. This approach is referred to as the confidence level method.It is subsequently compared to the alternative CoC approach for the determination of therisk adjustment for non-financial risk. Capital requirement allocation approaches are alsoexplored in the numerical results.
The confidence level approach relies on the use of a risk measure to determine the riskadjustment amount. We recall the definition of two traditional risk measures commonly usedin practice for such purpose, namely Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR).For a loss random variable X , its VaR at confidence level α is defined byVaR α ( X ) = inf { x ∈ R | P [ X ≤ x ] ≥ α } . VaR α ( X ) represents the α − quantile of the loss distribution. A drawback of VaR is its blindspot for risk scenarios beyond the confidence level α ; this risk measure is not sufficient tounderstand the spectrum of worst possible losses for insurers.This points toward considering an alternative risk measure. TVaR at confidence level α is defined by TVaR α ( X ) = 11 − α (cid:90) α VaR u ( X )d u. TVaR α ( X ) can be interpreted as the expected loss over the worst (1 − α )% scenarios. Thismeasure is meant to correct for the blind spot of VaR by providing additional insight onthe behavior in the tail of the loss distribution. Indeed, TVaR takes into account potentialoutcomes beyond any chosen confidence level. Furthermore, TVaR is a coherent risk measureas it complies with properties established in Artzner et al. [4], see Acerbi and Tasche [1] forthe proof of coherence of TVaR.Another way of determining the risk adjustment for non-financial risks is through theCoC method described in IAA [15], where the risk adjustment is the present value of thefuture costs of capital associated with the unpaid claim liabilities. In the CoC approach, the20isk adjustment for non-financial risks is calculated byRisk adjustment = T (cid:88) t =1 r t · C t (1 + d t ) t , (7)where C t represents the assigned capital amount for the period ending at time t , r t is theselected cost of capital rate for period ending at time t , d t is the selected discount rateallowing to discount from time t back to time 0, and T is the number of periods considered.The main advantage of the cost of capital method is its simplicity and interpretability. Inthe CoC method, the cost of bearing the uncertainty in the liabilities is reflected through theCoC rate, whereas it is represented by the loss distribution when using the confidence levelapproach. Nevertheless, when using the CoC approach, the IFRS 17 regulatory frameworkrequires the risk adjustment to be converted to a VaR confidence level; the insurer is requiredto disclose such equivalent confidence level. Moreover, the CoC technique requires settingadditional assumptions about the cost of capital rate.A first simulation based on the loss triangle model and parameter estimates obtained inSections 4 and 5 is now performed to calculate the economic capital based on TVaR at level α = 99% of the aggregate loss distribution as recommended by the Canadian regulatoryrequirements, OSFI [22]. Note that OSFI [22] allows using either TVaR at level α = 99% orVaR at level α = 99 .
5% as the economic capital requirement. We focussed TVaR measures,as it is more representative of tail events, and it possesses more advantageous theoreticalproperties. The aggregate discounted loss distribution obtained by generating 100 ,
000 sce-narios of unpaid claim liabilities loss triangles is used to calculate capital requirements andits allocation to the six business lines. For illustrative purposes, in what follows, the discountand cost of capital rates are assumed to be constant for all periods; for all t , we have d t = 2%and r t = 5%, respectively.The results of this simulation are found in Tables 2 and 3. In the first row ( Aggregate )of Table 2, the first six columns contain the allocation of the economic capital split to allbusiness lines according to the Euler allocation principle, that isTVaR α ( X i | S ) = E ( X i | S >
VaR α ( S )) , where S = X + ... + X . We refer the reader to Tasche [29] and McNeil et al. [21] for detailedexplanations concerning the TVaR-based allocation and the Euler allocation principle. Also,in this first row, the last column ( Total ) provides the economic capital based on the TVaRat confidence level α = 99% applied to the empirical aggregate loss distribution, where theaggregate loss is obtained by summing discounted liabilities over all accident years, devel-opment lags and business lines. The second row ( Silo ) of Table 2 provides the TVaR at21onfidence level α = 99% for each business line on a standalone basis. The element in thelast column ( Total ) for that second row is simply the sum of TVaRs over each business line.A diversification benefit of $482 million is therefore obtained by subtracting the Aggregatecapital requirement from the Silo total capital requirement. Such an amount seems verymodest due to being less than 0.5% of the economic capital of the fictitious insurer. This ishowever explained in the current case by the fact that the exposure is highly concentrated inthe Ontario Personal (ON/PA) insurance business line; diversification has a marginal impactdue to the much lesser exposure to other lines of business. The diversification benefit wouldhave been much higher if the insurer’s exposure had been more balanced.ON ON AB AB ATL ATL TotalPA CA PA CA PA CATVaR
Aggregate 83,539 6,121 15,954 1,760 7,636 637 115,647Silo 83,583 6,195 16,169 1,811 7,712 659 116,129Table 2: Economic capital and allocation to business lines (millions CAD)We now turn to the calculation of the risk adjustment for non-financial risks, which issummarized in Table 3. Two different methods are compared: the first based on TVaR atlevel α = 87%, and the second being the CoC method.The first row ( Aggregate ) of the TVaR panel presents the excess over the mean using theTVaR of the total portfolio loss along with the allocation of that amount to the variousbusiness lines according to the Euler allocation principle. The confidence level chosen is α = 87% because it leads to roughly similar results as the CoC method in terms of totalportfolio risk adjustment for non-financial risks given the assumptions made. The secondrow ( Silo ) of the TVaR panel presents the excess over the mean using the TVaR of theloss distribution for each standalone business line, along with the sum of such values acrossall business lines in the column
Total . The CoC panel contains the risk adjustment for thetotal portfolio, along with the values for standalone business lines. For individual businesslines, the capital C t considered is the standalone business line capital calculated from thesimulations. The ‘Equivalent α ’ row contains the confidence level for which the univariate(Silo) VaR would give the same amount than the CoC approach. In the CoC method (7),the capital requirement C t considered is C t =VaR ( X t ) − E ( X t ) where X t is the aggregateloss of year t . This is consistent with requirements of IFRS 17, see for instance p.59 IAA[15].Comparing the total portfolio risk adjustment for the Aggregate versus the Silo approachin Table 3, one observes that amounts obtained through the TVaR based approach aresmaller. This is due to the diversification of risks. Moreover, one sees that for a comparable22otal capital amount obtained with the CoC and TVaR methods, the risk allocation acrossbusiness lines exhibits less concentration with the CoC approach.ON ON AB AB ATL ATL TotalPA CA PA CA PA CA E ( X ) 82,502 6,109 15,891 1,755 7,629 637TVaR ( X ) − E ( X ) Aggregate 617 8 41 3 4 < α for VaR α ( X ) − E ( X ) 87.79 91.92 84.43 86.80 93.19 89.35Table 3: Risk adjustments for non-financial risks and allocation to business lines (millionsCAD)Table 4 performs a sensitivity analysis on the risk adjustment for non-financial risks withrespect to the cost of capital rate r t . Outcomes stemming from the baseline value r t = 5%are compared to figures obtained with either r t = 4% or r t = 6%. Such sensitivity is seen inthe table to be quite material.Cost of capital rate ON ON AB AB ATL ATL Total r t PA CA PA CA PA CA4% 361 36 81 18 37 8 5415% 451 45 101 23 46 10 6766% 541 53 121 27 55 12 809Table 4: Sensitivity of the risk adjustment for non-financial risks (millions CAD) to the costof capital rate, ceteris paribus
This article provides a statistical model for the prediction of loss ratios associated to lia-bilities for incurred claims risk of a multi-line property and casualty insurer. The modelwas designed based on a automibile insurance dataset from the General Insurance StatisticalAgency for which a history of loss ratios was available for combinations of two business linetypes (i.e. personal versus commercial) and three geographical regions. The model possessesadvantageous theoretical features allowing for the reproduction of empirical characteristicsof loss ratios identified in the dataset. A Tweedie distributed Double Generalized LinearModel is used to represent the marginal distribution of loss ratios, where accident semesterand development lag effects are taken into account when modeling both the mean and the23ispersion of the distribution. An autocorrelation structure represents the loss ratio depen-dence across the various development lags for a given accident semester and business line,whereas the dependence across business lines is represented by a hierarchical copula model.A two-step estimation procedure is followed: parameters for each standalone business linesare estimated separately first through Generalized Estimating Equations, and then the hi-erarchical copula model is constructed based on previously obtained marginal business linesparameter estimates.The model developed herein serves many purposes and can be used for reserving (e.g.determination of the risk adjustment for non-financial risks), financial reporting and economiccapital requirements calculations. A key attribute of our model is its consistency with IFRS17 reporting standards; the dependence structure between loss ratios of the various businesslines embedded in the model can be used to quantify the joint loss distribution across suchbusiness lines, hence allowing for the computation of the diversification benefit recognizedunder the IFRS 17 standards. The methodology for the quantification of the diversificationbenefit relies on a stochastic simulation using the loss ratio prediction model to generatemultiple cash flow scenarios for the insurer. Risk measures can then be applied to the set ofgenerated cash flow scenarios to measure either capital requirements or the risk adjustmentfor non-financial risks of the whole company and their allocation to the various business lines.The estimation procedure was applied on the current study’s dataset to obtain parameterestimates. The latter served as inputs to a stochastic simulation experiment which illustratedthe calculation of capital requirements and the risk adjustment for non-financial risks basedon the TVaR risk measure. In this experiment, values obtained for the latter quantities werecompared to a cost of capital approach. It was seen that the TVaR based method provided acapital allocation that exhibits more concentration to significant business lines, in comparisonto the CoC method which spread the allocation more evenly across the lines.
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30] Tweedie M (1984) An index which distinguishes between some important exponential families.Statistics: Applications and New Directions Proceedings of the Indian Statistical InstituteGolden Jubilee International Conference (Eds J K Ghosh and J Roy) pp 579–604 Parameter estimates for marginal business lines
Parameters for the accident semester and development lag effects which are denoted AS andDL, respectively. Furthermore, the lines of business are Personal Auto (PA) and CommercialAuto (CA) for the three regions of Ontario (ON), Alberta (AB) and Atlantic Canada (ATL).
No. Parameter PA ON CA ON PA AB CA AB PA ATL CA ATL1 Intercept -1.55 -1.80 -1.05 -1.12 -1.40 -1.482 AS = 2003-2 -0.13 0.03 0.01 -0.14 -0.10 -0.073 AS = 2004-1 -0.29 -0.24 -0.21 -0.36 -0.27 -0.294 AS = 2004-2 -0.10 -0.27 -0.12 -0.10 -0.09 -0.075 AS = 2005-1 -0.23 -0.40 -0.21 -0.42 -0.10 -0.486 AS = 2005-2 -0.06 0.04 -0.12 -0.24 0.05 -0.077 AS = 2006-1 -0.11 -0.23 -0.26 -0.37 -0.17 -0.348 AS = 2006-2 0.09 0.00 -0.01 -0.07 0.01 -0.389 AS = 2007-1 0.02 -0.16 -0.31 -0.42 -0.19 -0.5410 AS = 2007-2 0.08 0.10 -0.11 -0.19 -0.02 -0.3311 AS = 2008-1 -0.02 -0.07 -0.25 -0.45 -0.23 -0.4712 AS = 2008-2 0.09 0.36 -0.13 -0.29 -0.25 -0.3913 AS = 2009-1 0.06 0.18 -0.38 -0.84 -0.21 -0.5914 AS = 2009-2 0.27 0.16 -0.20 -0.51 0.01 -0.5415 AS = 2010-1 0.16 0.03 -0.53 -0.61 -0.10 -0.5716 AS = 2010-2 0.15 0.09 -0.23 -0.66 0.01 -0.1217 AS = 2011-1 -0.08 -0.12 -0.44 -0.60 -0.18 -0.6418 AS = 2011-2 0.00 0.00 -0.18 -0.34 0.01 -0.0619 AS = 2012-1 -0.13 -0.06 -0.29 -0.63 -0.16 -0.7320 AS = 2012-2 -0.03 0.03 -0.09 -0.21 0.13 -0.4521 AS = 2013-1 -0.13 -0.14 -0.26 -0.24 -0.15 -0.3622 AS = 2013-2 0.06 0.02 -0.02 -0.30 0.17 0.0023 AS = 2014-1 -0.10 -0.06 -0.25 -0.63 -0.09 -0.0724 AS = 2014-2 0.07 0.18 0.06 -0.25 0.07 -0.0325 AS = 2015-1 -0.03 -0.09 -0.13 -0.48 0.05 -0.1726 AS = 2015-2 0.13 0.12 0.05 -0.43 0.31 -0.0527 AS = 2016-1 -0.02 -0.09 -0.18 -0.66 0.09 -0.3128 AS = 2016-2 0.09 0.02 0.02 -0.33 0.16 -0.1129 AS = 2017-1 -0.12 -0.15 -0.27 -0.47 0.01 -0.1230 AS = 2017-2 0.15 0.13 -0.06 -0.17 0.20 0.03
Table 5: Mean model - Accident semester effects27 o. Parameter PA ON CA ON PA AB CA AB PA ATL CA ATL31 DL = 2 -0.33 -0.24 -0.63 -0.46 -0.60 -0.4732 DL = 3 -0.55 -0.44 -1.21 -1.14 -0.90 -0.8133 DL = 4 -0.61 -0.43 -1.36 -1.30 -0.98 -0.8734 DL = 5 -0.61 -0.37 -1.43 -1.37 -1.05 -0.9435 DL = 6 -0.65 -0.38 -1.51 -1.49 -1.15 -0.9936 DL = 7 -0.71 -0.43 -1.56 -1.55 -1.24 -1.0737 DL = 8 -0.82 -0.50 -1.66 -1.66 -1.35 -1.2038 DL = 9 -0.97 -0.63 -1.77 -1.78 -1.49 -1.3039 DL = 10 -1.14 -0.78 -1.90 -1.94 -1.67 -1.4640 DL = 11 -1.34 -0.96 -2.08 -2.14 -1.84 -1.6241 DL = 12 -1.56 -1.15 -2.25 -2.37 -2.02 -1.7542 DL = 13 -1.78 -1.42 -2.46 -2.61 -2.22 -1.9143 DL = 14 -2.02 -1.68 -2.68 -2.79 -2.42 -2.1444 DL = 15 -2.25 -1.91 -2.89 -3.01 -2.62 -2.2145 DL = 16 -2.45 -2.11 -3.15 -3.24 -2.82 -2.4046 DL = 17 -2.64 -2.34 -3.39 -3.39 -3.05 -2.7047 DL = 18 -2.84 -2.60 -3.58 -3.83 -3.25 -3.1448 DL = 19 -2.99 -2.77 -3.76 -4.03 -3.49 -3.3949 DL = 20 -3.15 -2.79 -3.98 -4.23 -3.63 -3.4550 DL = 21 -3.33 -3.03 -4.22 -4.42 -3.74 -3.9151 DL = 22 -3.48 -3.25 -4.51 -4.74 -3.95 -4.1152 DL = 23 -3.63 -3.46 -4.70 -4.89 -4.08 -4.1553 DL = 24 -3.74 -3.72 -5.10 -5.05 -4.34 -4.6754 DL = 25 -3.88 -3.78 -5.37 -4.96 -4.69 -5.0755 DL = 26 -4.03 -3.88 -5.82 -5.22 -4.74 -4.9556 DL = 27 -4.15 -4.58 -5.83 -5.28 -5.08 -6.5657 DL = 28 -4.25 -4.46 -5.84 -9.87 -5.09 -6.8058 DL = 29 -4.23 -4.55 -6.09 -13.40 -5.62 -5.7359 DL = 30 -4.57 -4.67 -6.16 -13.48 -5.72 -12.12
Table 6: Mean model - Development Lag effects28 o. Parameter PA ON CA ON PA AB CA AB PA ATL CA ATL60 Intercept -4.80 -5.78 -4.29 -2.94 -4.53 -4.8061 DL = 2 0.56 -1.36 -0.89 -2.08 -2.05 -0.6862 DL = 3 0.58 -1.75 -1.53 -2.79 -3.11 -1.0563 DL = 4 -0.16 -1.72 -1.32 -2.46 -3.43 -1.0364 DL = 5 -0.87 -1.81 -1.55 -2.79 -4.14 -2.7665 DL = 6 -1.33 -2.02 -2.00 -3.01 -4.71 -2.3466 DL = 7 -1.65 -2.20 -3.15 -4.35 -4.91 -2.1967 DL = 8 -2.47 -2.46 -3.53 -3.92 -4.68 -1.8768 DL = 9 -3.23 -1.76 -3.36 -3.44 -4.05 -1.1969 DL = 10 -2.88 -1.72 -2.86 -2.53 -3.26 -1.0070 DL = 11 -2.21 -1.54 -2.72 -2.40 -2.94 -0.7371 DL = 12 -1.50 -0.55 -2.52 -2.07 -2.46 -0.3872 DL = 13 -0.77 -0.35 -2.03 -1.72 -2.56 -0.5073 DL = 14 -0.43 -0.48 -1.58 -1.40 -2.04 0.0074 DL = 15 -0.11 -0.66 -1.83 -1.29 -2.01 0.2875 DL = 16 0.36 -0.23 -1.89 -0.98 -1.93 -0.0776 DL = 17 0.04 0.13 -1.45 -0.89 -1.87 -0.2077 DL = 18 -0.08 0.64 -1.26 -1.53 -1.82 -0.4778 DL = 19 0.22 0.43 -1.46 -0.97 -2.27 -0.3379 DL = 20 -0.03 0.81 -1.50 -0.66 -2.42 0.0680 DL = 21 0.48 0.16 -1.05 -0.65 -2.51 0.2781 DL = 22 0.47 0.49 -0.70 -0.36 -2.41 0.8282 DL = 23 1.06 0.66 -0.50 -0.50 -2.30 0.9583 DL = 24 1.25 0.98 -0.24 -0.23 -1.45 0.6484 DL = 25 0.80 0.51 -0.46 -0.07 -1.43 0.2585 DL = 26 1.39 0.01 -1.05 0.08 -1.92 0.6686 DL = 27 1.33 -1.81 -1.27 0.60 -2.21 -0.9587 DL = 28 0.27 -3.19 -1.56 -0.97 -2.57 -0.6488 DL = 29 0.71 -3.10 -4.10 -2.46 -5.21 0.2289 DL = 30 1.80 -8.96 -8.67 0.02 -9.00 -2.17
Table 7: Dispersion submodel - Development lag effects
Correlation PA ON CA ON PA AB CA AB PA ATL CA ATL ρ k Table 8: Estimated correlation parameter ρ k for each business line k Index parameter PA ON CA ON PA AB CA AB PA ATL CA ATL p k Table 9: Tweedie distribution index parameters p k for each business line k Selection of the Tweedie index p k Estimating p k is not a trivial endeavour, and therefore a procedure inspired from Dunn andSmyth [11] is considered in the current work.A set of fixed values of p k , namely p k = { . , . , . , . . . , . } , is considered.For each of these values, DGLM parameters Θ ( µ ) k and Θ ( φ ) k are estimated through maximumlikelihood while assuming a null development lag correlation i.e. ρ k = 0, the latter assump-tion considerably simplifying the estimation. The value of p k for which the loglikelihood ismaximized is the value selected as the parameter estimate.Recall that values of p k must lie in the (1 ,
2) interval. However, for stability consider-ations, values below 1 . p k is close to 2, the infinite sum approximation embedded in the Tweedie distribution includesa large number of terms that are materially different from zero, which makes computationsmore cumbersome. C The Iman-Conover procedure
Figure 2 provides an illustration of the modeled dependence structure of the GISA datasetlines of business, which is based on a hierarchical copula. The Iman-Conover reorderingalgorithm is used to simulate from such copula in numerical experiments and it goes asfollows:1. Simulate k independent samples of size m >> N composed of independent standardnormal random variables: U ( k ) ∼ N (0 , , k = { , , , , , } .
2. Simulate independent copula samples of size m from each bivariate copula C , . . . , C .3. Reorder the samples of each bivariate vector by merging the observed marginal rankswith the joint ranks in the copula sample. A brief example follows for the first node ofthe HCM. In Cˆot´e [8] it is pointed out that the empirical distribution functions of the marginals and the copulaconverge asymptotically to the true distributions. Thus, a larger sample size m provides a better estimate ofthe HCM sample. (1) Rank U (2) Rank C Ranks1.27 2 3.71 3 (0 . , .
4) (3 , . , .
9) (1 , . , .
3) (2 , → Reordered Sample(2 . , . . , . . , -2 . (cid:0) U (1) , U (2) (cid:1) ∼ C .4. Repeat step 3 for the first level copulas C and C .5. Aggregate the reordered data following the dependence structure to obtain samplesfrom U (1) + U (2) and respectively for U (3) + U (4) and U (5) + U (6) .6. Repeat step 3 to obtain sample from (cid:0) U (3) + U (4) , U (5) + U (6) (cid:1) ∼ C .7. Aggregate the reordered sample from C to obtain a sample from (cid:88) k =3 U ( k ) , and repeatstep 3 for (cid:32) U (1) + U (2) , (cid:88) k =3 U ( k ) (cid:33) ∼ C .8. To obtain a joint sample of (cid:0) U (1) , U (2) , U (3) , U (4) , U (5) , U (6) (cid:1) , perform the permutationsapplied to U (1) + U (2) back to U (1) and U (2) , the permutations applied to U (3) + U (4) back to U (3) and U (4) , and finally, the permutations applied to U (5) + U (6) back to U (5) and U (6) .9. Get a subsample of size N from the reordered sample of size m . D The conditional distribution of simulated scaled in-novations
The assumption made in the current paper’s model based on (3) is that for a given acci-dent semester i and business line k , the scaled innovations vector ˜ Y ( k ) i are approximatelymultivariate normal with a null mean vector and covariance matrix R k,J as defined in (4).A classic result on multivariate normal distributions is first recalled. Consider a mul-tivariate normal random column vector X which is decomposed into two blocks (i.e. two31tacked random vectors): X = [( X (1) ) (cid:62) ( X (2) ) (cid:62) ] (cid:62) . Denote respectively the mean vector andcovariance matrix of the entire vector X and of each of the two blocks X (1) and X (2) by µµµ = (cid:20) µµµ (1) µµµ (2) (cid:21) , Σ = (cid:20) Σ (1 , Σ (1 , Σ (2 , Σ (2 , (cid:21) . Then, the conditional distribution of X (2) given X (1) is multivariate normal with mean µµµ (2) +Σ (2 , (cid:2) Σ (1 , (cid:3) − (cid:16) X (1) − µµµ (1) (cid:17) and variance Σ (2 , − Σ (2 , (cid:2) Σ (1 , (cid:3) − Σ (1 , .We can decomposed the scaled innovation vector ˜ Y ( k ) i into two blocks: the unobservedone X (1) = ˜ Y ( k ) i,J +2 − i : J ≡ [ Y ( k ) i,J +2 − i , . . . , Y ( k ) i,J ] (cid:62) and the observed one X (2) = ˜ Y ( k ) i, J +1 − i ≡ [ Y ( k ) i, , . . . , Y ( k ) i,J +1 − i ] (cid:62) . In other words,˜ Y ( k ) i = (cid:34) ˜ Y ( k ) i, J +1 − i ˜ Y ( k ) i,J +2 − i : J (cid:35) . Since, the covariance matrix of ˜ Y ( k ) i can be decomposed as R k,J = (cid:34) R k,J +1 − i R (1 , k,i ( R (1 , k,i ) (cid:62) R k,i − (cid:35) with R (1 , k,i ≡ ρ J +1 − ik ρ J +2 − ik ρ J +3 − ik . . . ρ J − k ... ... ... . . . ... ρ k ρ k ρ k . . . ρ ik ρ k ρ k ρ k . . . ρ i − k . Setting µµµ = 000 in the previous result along with Σ (1 , = R k,J +1 − i , Σ (2 , = R k,i − andΣ (1 , = R (1 , k,i leads to approximate the conditional distribution of unobserved scaled inno-vations ˜ Y ( k ) i,J +2 − i : J given observed ones ˜ Y ( k ) i, J +1 − i by a multivariate normal with mean vectorand covariance matrix being respectively:˘ M ( k ) i ≡ E (cid:104) ˜ Y ( k ) i,J +2 − i : J | ˜ Y ( k ) i, J +1 − i (cid:105) = ( R (1 , k,i ) (cid:62) [ R k,J +1 − i ] − ˜ Y ( k ) i, J +1 − i , (8)˘ V ( k ) i ≡ Cov (cid:104) ˜ Y ( k ) i,J +2 − i : J | ˜ Y ( k ) i, J +1 − i (cid:105) = R k,i − − ( R (1 , k,i ) (cid:62) [ R k,J +1 − i ] − R (1 , k,i ..