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Provisions for claims outstanding, incurred but not reported, with generalized linear models: prediction error formulated according to calendar year Cálculo de provisiones para prestaciones pendientes de declaración con modelos lineales generalizados: formulación del error de predicción por año de calendario Eva Boj del Val1 Teresa Costa Cor1 Universidad de Barcelona (España) Recibido el 27 de febrero de 2015, aceptado el 19 de octubre de 2015 Publicado online el 25 de enero de 2017

Nº de clasificación JEL: C13, C15 DOI: 10.5295/cdg.150526eb Abstract:

In the current context of Solvency II, insurance companies are required to implement demanding business risk management systems. An important aspect of this risk management is the problem of technical provisions in non-life insurance and, as such, it is in the interest of insurers to calculate the prediction error that has occurred when using methodology to estimate a company’s future payments. Furthermore, the predictive distribution of the fitted values, which is descriptive of the risk, allows us to estimate, for example, its Value at Risk at a given confidence level. In this paper we focus on the application of generalized linear models to the amounts of claim losses of a run-off triangle. In order to achieve error distribution, a parameter dependent parametric family is assumed, along with the logarithmic link function. The parametric family has as particular cases the Poisson, the Gamma and the Inverse Gaussian distributions. The particular case which assumes an (over-dispersed) Poisson distribution with the logarithmic link is widely known because it offers the same provision estimation as the deterministic Chain-Ladder method. In this study we develop formulas of the prediction error of future payments by calendar years for the general parametric family. This allows us to perform calculations that consider a financial environment, whether employing analytical formulation or bootstrap estimation. In practice, the presented formulations allow a determination to be made of the present value of the incurred but not reported claim of future payments including a risk margin with statistical significance. Keywords: Technical provisions, generalized linear model, calendar year, Solvency II.   Departamento de Matemática Económica, Financiera y Actuarial, Facultad de Economía y Empresa, Avenida Diagonal 690, 08034 Barcelona (España). [email protected]; [email protected] 1

ISSN: 1131 - 6837

Cuadernos de Gestión Vol. 17 - Nº 2 (2017), pp. 157-174

157

Provisions for claims outstanding, incurred but not reported, with generalized linear models Resumen: El actual contexto de Solvencia II requiere una exigente gestión empresarial del riesgo de las Entidades Aseguradoras. En el problema de cálculo de provisiones técnicas en seguros de no-vida es de interés calcular el error de predicción cometido con la metodología utilizada para la estimación de los pagos futuros de la Entidad. Además, la distribución predictiva de las estimaciones, que es descriptiva respecto del riesgo, permite obtener, por ejemplo, su valor en riesgo a un nivel de confianza fijado. En este trabajo nos centramos en la aplicación de los modelos lineales generalizados a las cuantías de siniestros de un triángulo de desarrollo. Asumimos para la distribución del error una familia paramétrica dependiente de un parámetro, junto con la función de enlace logarítmica. La familia paramétrica tiene como casos particulares las distribuciones de Poisson, Gamma e Inversa Gaussiana. Es conocido el caso particular en que se asume una distribución de Poisson (sobredispersa) junto con el link logarítmico, que ofrece la misma estimación de provisiones que el método determinista Chain-Ladder. En este estudio desarrollamos las fórmulas del error de predicción de los pagos futuros por años de calendario para la familia paramétrica general, que nos permiten realizar cálculos teniendo en cuenta un ambiente financiero, tanto para el caso de utilizar formulación analítica como para el caso de realizar estimación bootstrap. En la práctica, las formulaciones presentadas nos ponen en disposición de poder calcular el valor actual de los pagos futuros para siniestros pendientes incluyendo márgenes de riesgo con significado estadístico. Palabras clave: Provisiones técnicas, modelo lineal generalizado; año de calendario, Solvencia II.

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Cuadernos de Gestión Vol. 17 - Nº 2 (2017), pp. 157-174

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presentadas nos ponen en disposición de incluyendo poder calcular el valor actual de los pagos futuros para siniestros pendientes márgenes de riesgo con significado estadístico. Palabrasincluyendo Clave: Provisiones Modelo Lineal Generalizado; año de calendario, para siniestros pendientes márgenestécnicas, de riesgo con significado estadístico. Solvencia II. Palabras Clave: Provisiones técnicas, Modelo Lineal Generalizado; año de calendario, Eva Boj Del Val / Teresa Costa Cor

Solvencia II.

1. INTRODUCTION1. INTRODUCTION This paper takes a similar starting point to that of Boj et al. (2014) in that it considers a

1. INTRODUCTION This paper takes a portfolio similarofstarting point to that of Boj et al. (2014) in that it considers a risks and assumes that each claim is settled either in the accident year or in the portfolio of risks and assumes that each claim is settled either in the accident year or in the This paper takes a following similar starting point years. to that of Boj et al.a family (2014)ofinrandom that itvariables considers It then k development {c }ia, j∈{0,1,...k}, following k development years. It then aconsiders family random variables portfolio of risks and assumes that each considers claim is settled either of in the accident year or in ijthe wherefollowing cij is thek amount of claim of accident yearofi random which variables is paid with delay development years. losses It then considers a family , of j {cijwith }i, ja∈{0,1,... where cij is the amount of claim losses of accident year i which is paid a delay of j k} years and hence in development year j and in calendar year i+j . cij refers to the incremental where cij is the amount of claim losses of accident year i which is paid with a delay of j loss of accident year iyears andand development year j year and j itand is assumed hence in development in calendar that year ithese refers to the + j . c incremental and≤ink calendar to athe hence in development cij refersij in + j . collected lossesyears cij areand observable for calendaryear yearsj i+j and thatyear theyi are run-off incremental loss of accident year and development year j andyear it isj assumed that these loss ofi accident year i and development and it is assumed that these triangle as in Figure 1.incremental incremental losses cij are observable for calendar years i + j ≤ k and that they are collected incremental losses c are observable for calendar years i + j ≤ k and that they are collected in in a run-off triangle as in Figure 1. ij Figure 1 Figure 1 a run-off triangle as in Figure 1. Run-off incrementallosses losses Run-offtriangle triangle with with incremental Accident year

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Source: Own elaboration.

In the run-off triangle, the numbers for year of origin i are grouped into rows while those for development year j are grouped into columns. The numbers on the antidiagonals with i+j = t denote the payments that were made in the calendar year t. The incremental losses are unobservable for calendar years i+j ≥ k + 1 and are thus very difficult to predict. There are various kinds of incurred but not reported (IBNR) claim provisions which are of interest. The provisions for the different accident years i = 1,...,k are obtained by adding the future incremental losses predicted in the corresponding row of the square:

Pi =

k

∑

j = k −i +1

(1)

Pde= Gestión Cuadernos ∑ ∑Vol.cˆij17. - Nº(2)2 (2017), pp. 157-174 159 k

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cˆij . (1)

k

i =1 j = k −i +1 k

Provisions for claims outstanding, incurred but not reported, with generalized linear models k

Pi = incremental ∑ cˆij . (1) The total provision is calculated by adding all the future losses predicted in

the bottom-right part of the run-off triangle:

j = k −i +1 kk

k

= P =Pi∑

∑

i =1 jj==kk −−ii++11

ccˆˆij .. (1) (2) ij

(2)

k k k And the future payments for the different calendar years t = k + 1,...,2k are obtained by (2) P = (3) i.e., the values FP = cˆt − cjˆ, ijj .. years, adding the incremental losses that were made in the future calendar t i =1j =jt = k i − + 1 −k of the same against-diagonal t :

∑∑∑ k

θ = (3) ˆt −µj ,ijj ,. (4) FPtV=( µ∑ ij ) c

(3)

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µij = E ⎡⎣cij ⎤⎦ and Var ⎡⎣cij ⎤⎦ = φ wij V µij = φ wij µθijθ , (5) The principal focus of this paper is to calculate the present (4)future payments V µij value = µij ,of the for the different calendar years, taking into account the Solvency II Directive (see Parla(6) 2010, Moreno log µij =and ηij .Alonso mento Europeo y Consejo de la Unión Europea 2009, Albarrán θ and , (5) ⎡ ⎤ ⎡ ⎤ Var c φ w V µ φ w µ = = µ E c = 2013). InijSolvency thatij the best technical provisions ij estimateij of the ij ⎣ ijstated ⎦ ⎣ ij ⎦ II, it is clearly must be calculated by including the time value ofηmoney, which is, to (7) say, that they must = c + α + β ij 0 i j include the expected present value of future payments. log Calculating by calendar (6) µij = ηij .payments year allow us to work in a financial environment because each amount is situated in the corˆij = exp acˆ0conservative (8) + αˆ i + βˆ j . scenario responding future calendar year. In order to cconsider Solvency II ηij = c0 + α i + β j , (7) recommends adding a risk margin to the best estimate to obtain a correct Solvency Capital Requirement (SCR). If possible, the SCR should also derive directly from the probability 2 2 k ⎤⎦=−exp ⎤⎦ βˆ ⎤the . (9) − cˆij ⎥⎤ = by − E ⎡⎣ccˆijapplied −cˆ E+⎡⎣αcˆ ijusing E ⎢⎡ cij generated E ⎢⎡thecijstatistic cˆijmodel, distribution forecast . (8) + ij (1) at Risk (VaR) P0i = i cˆj ij⎥⎦ . Value ⎣ ⎦ ⎣ measure. j = k −i +1 Section 3 describes two means of determining an adequate Solvency Capital Require2 2 2 2 ⎡cˆ c⎤ of ⎤ .prediction ⎡ c⎤ˆ a−percentage ⎤ kof ⎡first ⎤− ⎤ of these consists adding the error to the k ⎡ c .(10) ˆ ˆ ⎤⎦ ⎤ −While ⎡ ⎤ ⎡ ⎡ ⎤ − + E ⎡ cij − E ⎡⎣cij ment. E c c E E E c 2E ⎡ cthe ˆ ˆ ⎡ ⎤ (9) − = − − E c c E c c E ij ⎦ij ij⎣ ij ⎦ ⎣ ⎥ij ⎦ ⎣⎣⎢ ij ⎦ij ⎦ ⎣ ⎣⎢ij ⎦ij ⎣ ij ⎣⎢byij⎣calendar ⎦⎥ij years, ⎣⎢ ⎦⎥ payments ⎦ ⎦⎥ij . (2) P =of directly cˆusing future the second consists the estimated VaR i =1 j = k − of the predictive distribution of the future payments by calendar k i +1 years for a given signifi2 2 2 2 2 (1) P = ⎡ ⎤⎡+ E ⎡ cˆ − Ei ⎡cˆ⎤ ∑ ⎤cˆij. . (11) ⎤ ⎡ ⎤ ⎤ c .(10) ˆ ˆ ⎤⎦ ⎤ −level. ⎡⎣cˆE ⎤ ⎡ ⎤ + − E ⎡ cij − E ⎡⎣cij cance cˆij⎢−cEij − E c E c 2EE⎡⎢⎡ ccijij −−Ecˆij⎡⎣cij⎥⎤⎤⎦≈ E ij ij ij ⎦ ij ⎦⎣ ij ⎢ ij ⎦ ⎣ kj = k⎦−i +1⎥ ⎣ ⎥ ⎣ ⎦ ⎣ ⎣ (GLM) ⎦is asssumed ⎦⎣⎢ ⎣ (see, e.g.,⎦⎥ McCullagh ⎦ linear model and Nelder 1989; ⎣⎢ ⎦⎥A generalized FPt = cˆt − j , j . (3) Boj et al. 2004; Boj and Costa 2014) to model the incremental losses of the run-off triangle. kj =t − k k 2 for the2 ⎤ 2 2 of error distributions The parametric family variance function is assumed ⎡ . (2) ˆ P = c the variance is the sum Eof⎡ cprocess .ij (11) ⎡c⎡ˆc∑ ⎤ ⎤ ⎤and − cˆ ⎤ variance ≈ E ⎡ cij −EE⎢ ⎡⎣cijij ⎤⎦− E⎤⎡⎣+cijE⎤⎦ ⎡ c⎥ˆij=−Var E∑ ij i =⎣1 ⎣jij= k⎦−⎦ i +1⎥ ⎢⎣ ⎥⎦ ⎢⎣ ⎦ ⎣ ⎣⎢ ij ij ⎦⎥ θ ⎦ V µij = µij , (4) (4)

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⎤ = Var ⎡cˆ ⎤ . The MSE of predictions ⎡ c is−given ⎤ FPVar . the (3) = ∑ cˆ − E ⎡⎣cˆ is⎤⎦ ) the sum⎣ of⎦“power process variance ⎡ ⎤⎦by: c c⎤⎦ˆ and (variance function”, on ⎥⎦ named and Var (5)This parametric ⎡⎣cE ⎣⎢⎤⎦(= φ Ew⎣c)depends V )( µ⎦⎥ =) = w ) µ , Ɵ. µ = E ⎡⎣c ⎤⎦variance (which (φ ⎡⎣parameter

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φ αwi ij+) µ µij = E ⎡⎣cij ⎤⎦ and Var ⎡⎣cij ⎤⎦ = (φ wij )V η( µij ij=) 2c=0(+ (7) βijθj , (5) (5) 2 ⎤ ≈ Var ⎡∂cµij⎤ + Var ⎡cˆ ⎤ . (12) MSE (cij ) = E ⎡⎢(cij − cˆVar ) ij ⎥⎦⎡⎣cˆij ⎤⎦ ≅ ⎣ ij ⎦ Var ⎡⎣⎣ηijíj ⎤⎦⎦ . (13) ⎣ where Ø is the dispersion parameter and of the data, assumed w are ∂ηa prioriˆµweights (8) cˆij ij= exp ijcˆ0 log + α i ij+ =βˆηj ij .. (6)

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equal to one, wij = 1 , for the incremental claim losses of2 a run-off triangle. The estimation

∂µij = c + α + β , (7) ∂µ ij 0 i ⎡⎣cˆij ⎤⎦ of ⎡⎣η ⎤⎦2 .j (13) ≅ η VarMSE Var ijcould logarithmic function ⎡ = µ , and then, the predictions be 2 ⎡ 160 link ⎤ ∂ η ˆ ˆ ⎡ ⎤ ⎡ ⎤ −17 E∂⎢η(cVol. c -)Nº 2=(2017), E ⎢ cpp. E ⎣cij ⎦ − cij ij− E ⎣cij ⎦ ⎤⎥ . (9) Cuadernos de Gestión 157-174 ij − ⎣ ij ij ⎦⎥ ⎣ ⎦ cˆij = exp cˆ0 + αˆ i + βˆ j . (8) ∂µ 2 function 2 could be logarithmic link = µ , and then, the MSE of predictions E ⎡ c − E ⎡c ⎤ ⎤ − 2E ⎡ c − E ⎡c ⎤ cˆ − E ⎡cˆ ⎤ ⎤ + E ⎡ cˆ − E ⎡cˆ ⎤ ⎤ .(10)2

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)

ISSN: 1131 - 6837

Pi = Eva Boj Del Val / Teresa Costa Cor

k

∑

j = k −i +1

k

cˆij . (1)

kk

(2) =Pi∑ . exponential (1) = ∑ in cˆijijthe process of this kind of GLM, which does not have aPdistribution family,

i =1 jj==kk −−ii++11 uses extended quasi-likelihood equations. A detailed description of the fitting algorithm may be found in McCullagh and Nelder (1989). k k k Several actuarial methods which are frequentlyFP a run-off triangle (3) cˆt − cjˆ, ijj . (2) Pused =t = to complete can be described by a GLM. Among these are the Chain-Ladder method, the arithmetic j t k = − i =1 j = k −i +1 and geometric separation methods, and de Vylder’s least squares method. In particular, the classical Chain-Ladder deterministic method can V be derived a GLM assuming , (4) µk = µijθfrom . e.g., (3) Haberman and FPtfunction = ij kcˆt −(see, (over-dispersed) Poisson errors and logarithmic link j, j Pi =j =t −k cˆij . (1) Renshaw 1996; England and Verrall 1999, 2002 and 2006; England 2002; Kaas et al. 2008; j = k −i +1 µij2014). = E ⎡⎣cij ⎤⎦ and Var ⎡⎣cij ⎤⎦ = φ wij V µij = φ wij µijθ , (5) Boj et al. Assume for the GLM the logarithmic link function V k µij k = µijθ , (4) P = log µ = ηcˆij .. (2) (6) (6) ij ij i =1 j = k −i +1 θ µij = E ⎡⎣cij ⎤⎦ and Var ⎡⎣cij ⎤⎦ = φ wij V µij = φ wij µij , (5) c0 +kαthe β j , (7) cij are modelled Next, we can define log (μij)= co+αi+βj, a GLMηin which ij = i +responses (3) ˆt − j , j .. link FPta log =logarithmic c as random variables with variance function (4), with µ = ηij (6)function (6) and j =t − kij with linear predictor cˆij = exp cˆ0 + αˆ i + βˆ j . (8) (7) ηij =Vc0 µ + α i =+ µβθj ,, (7) (4)

∑∑∑

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(

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ij

2 ⎡(c − cˆ ) ⎤ = E ⎡ c − E ⎡c ⎤ − cˆ − E ⎡c ⎤ ⎤ . (9) E ij ij ij ij ij ij where αi is the to ⎣the i ⎦= ˆ1,...,k and β is the factor ⎢ ⎥θ . (8) ⎢ factor corresponding ⎥ cˆij⎦accident = exp cˆ=0year +⎣αˆw i + βµj ⎦ , and ⎦Var ⎡⎣⎣cijyear (5) j ⎡⎣⎣cijthe ⎤⎦ development ⎤⎦ = jφ=w1,...,k. µij = E to ij V µ ij ij isijthe term corresponding corresponding The c φvalue

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to the accident year 0 and development year 0. In the Poisson case, where Ø =1 is as-

2 2 2 ⎤ + E ⎡ by .(10) ˆ ˆ ⎤⎦ ⎤ − 2over-dispersion ⎡⎣cˆij ⎤⎦account ⎡ ⎤ ⎤ (6) scale parameter ≈ E ⎡ cij − E ⎡⎣cij sumed, − E ⎡ cijE−⎡Ec⎡⎣c−ij c⎤⎦ˆ is 2cˆtaken E c E c into estimating unknown log µ⎥⎤the ⎤ ij − ij ij ij⎤ = η ij .. ⎣ ⎦ ˆ ⎡ ⎤ ⎡ = − − − E c E c c E c ⎣ ⎦ ⎢⎣ ⎥⎦ ⎢ ( ) ij ij ij ⎣ ⎣ij ⎦ ⎣ ⎦ij ⎦ ⎥⎦ (9) ⎢⎣ ij ⎢⎣ fitting ⎥⎦ Ø as a part of the procedure. The predicted values cˆij of the IBNR 2provisions (1), (2) and future (7)payments (3) are 2 ⎡ c − E ⎡c ⎤ ⎤ + E ⎡ηcˆij =− cE0 ⎡+2cˆα⎤i +2 ⎤β.j , (11) ⎡ ⎤ 2 estimated from ˆ − ≈ E c c E ( ) ⎦ ⎦⎥ ≈ E ⎡ cij − E ⎡⎣cij ⎤⎦ ⎤ − 2E ⎡⎢⎣ cijij − Eij⎡⎣cij⎥⎦⎤⎦ cˆij⎣⎢− Eij ⎡⎣cˆij ⎤⎦⎣ ⎤ij +⎦ E⎦⎥⎡ cˆij −⎣⎢ Eij⎡⎣cˆij ⎤⎦ ⎣ ⎤ij.(10) ⎣ ⎦ ⎢⎣ ⎥⎦ ⎣⎢ ⎦⎥ cˆij = exp cˆ0 + αˆ i + βˆ j . (8) (8) 2 ⎡ ⎤ 2 2 variance is the sum of⎡ process2 ⎤ variance and the ⎡ ⎤ ⎡ ⎤ c ⎦⎤ ⎡ c −EE⎢ ⎡ccij ⎤− E⎤⎣+cijE⎦ ⎡ c⎥ˆ =−Var . (11) E ⎢(cij − cˆijerror E ⎡⎣the c⎣ˆij ij⎤⎦accident ) ⎥⎦ ≈ofE2 (8) ⎣ ⎣ ij ⎦(4)⎥is studied ⎦ij for The prediction assuming year provisions (1) ij ⎢ ⎢ ⎣ ⎡ ⎣ ⎦ ⎣ ⎦⎥2 ⎤ ⎡ ⎤ ˆij − E(1999, ⎡⎣cand ⎤⎦ −Verrall ⎡⎣cij ⎦⎤ 2002, . (9) and the total provision in, e.g., England 2006) and England = − E ⎢(cij − cˆij(2) E c E c ) ij ij ⎢⎣ ⎥⎦ ⎣ has been⎥⎦ extended This study to the IBNR future payments by calendar years (3) ⎡ cˆ − E ⎡cˆ ⎤ 2 ⎤(2002). 2 is given by:⎤ = Var ⎡⎣cˆij ⎤⎦ . The MSE of predictions ⎡ c(2014) ij ⎣ ijis⎦ the sumet al. of (2014) process ⎡cij ⎤⎦ only=for ⎡⎣cijparticular ⎤⎦ and the Varthe andvariance Espejo etEal. case in which the ⎣⎢variance ⎦⎥in 2Boj ij − E ⎣but ⎢⎣ ⎥⎦ 2 assumed gives the same estimation of ⎤⎦ ⎤ − 2E ⎡ cij − Poisson ⎡⎣cˆij ⎤⎦that⎤ .(10) ≈ E ⎡ cij − E ⎡⎣cij (over-dispersed) E ⎡⎣cij ⎤⎦ distribution cˆij − E ⎡⎣cˆij ⎤⎦ is⎤ + E ⎡ cˆij −(aEcase ⎣as the deterministic Chain-Ladder ⎦ 2 ⎢⎣ method). The ⎥⎦novelty of the current study is ⎣⎢ ⎦⎥ provisions ⎡ ⎤ MSE (cij ) = E ⎢(cij − cˆij ) ⎥ ≈ Var ⎡⎣cij ⎤⎦ + Var ⎡⎣cˆíj ⎤⎦ . (12) ⎡ cˆ − E ⎡cˆ ⎤ 2 ⎤that the ⎡formulas the prediction forgiven of predictions by: payments by calendar years (10) and ⎣ error is ⎦the future = Var cˆij ⎤⎦ . TheofMSE ij ij ⎦ ⎣ ⎣ 2 2 ⎣⎢ ⎦⎥(14) of Boj⎡et al. (2014) 2 are extended to the general (4). ⎡ ⎤ ⎤ . (11) ⎤ ⎡ ˆ ⎤ family E ⎢(cij − cˆij ) ⎥ ≈ E cij − E ⎡⎣cij ⎤⎦ + E ⎡ cˆparametric ij −2E ⎣cij ⎦ ⎥ 2 develops ⎢ ⎥ for the prediction error The paper formulas ⎣ is organized ⎦ as⎣⎢ follows. Section ⎦ ⎣ ⎦ ∂µij 2 ⎡ by calendar of a GLM for theMSE future family of (13) . parametric (12) ˆij ) ⎤⎡⎣c≈ˆijVar ⎤⎦years ⎤ +assuming ≅ ⎡⎣c(3) Var cpayments Var ⎡⎣⎡⎣ηcˆijíj ⎤⎦the ( ij ) = E ⎢( cij − cVar ⎣ these formulas ⎦⎥ ηij ij⎦ data and distributions (4). Section 3 applies to2∂⎤real a practical demonstration ⎡ variance is the sum of process variance E cij − E ⎡⎣cij ⎤⎦ ⎡⎣cij payments ⎤⎦ and theby calendar year = Var of Solvency Directive II is given by⎢⎣ working with ⎥⎦the future 2 studied in Section ∂µ 2 (using an interest rate and including ∂µij a risk margin to the provisions). , and then, the MSE predictions be 5 and 6 contain he logarithmic 2 link function = µconclusions . (13) ⎡⎣cˆij are ⎤⎦ of ⎡⎣ηijcould ⎤⎦Sections ≅ presented Var Varwhile In Section 4 the main of the paper ⎡ cˆ − E ⎡cˆ ⎤ ⎤ = Var ⎡cˆ ⎤ . The ∂η MSE of predictions is given by:∂ηij and references. ⎣ ij ⎦ ⎦⎥acknowledgments ⎣ ij ⎦ ⎣⎢ ij

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∂µ 2 of ⎡predictions be = µ ,c and 161 ⎡(c −the ⎤≈ . - (12) ˆij )MSE ⎤⎦ +Gestión ⎤⎦ 17 MSE c Var cijde Var ⎡⎣Vol. cˆíjcould ISSN: 1131 - 6837 ∂ η Cuadernos Nº 2 (2017), pp. 157-174 ( ij ) = Ethen, ij 2 ⎣⎢ MSE c⎦⎥ ≅ φµ⎣θ + ( ij ) ij µijVar ⎡⎣ηij ⎤⎦ . (14)

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µij = E ⎡⎣cij ⎤⎦ and Var ⎡⎣cij ⎤⎦ = (φ wij )V ( µij ) = (φ wij ) µij , (5) P = ∑ cˆ . (1)P = θ θ, (4) ∑ cˆ . =)µ ) ijw ij ,, (5) µ =c(0(φµ µij = E ⎡⎣cij ⎤⎦ and Var ⎡⎣cij ⎤⎦ = (φ wij )V (ηµijij )=V (7) + α + β ij ij log µiji = ηijj . (6) V ( µij ) = µijθ , (4) log µij = ηij . (6) P = ∑ ∑ cˆ . (2) θ P =cˆ∑. ∑ cˆ . P= ∑ , (5) V ( µijVar φ w µ = µij = E ⎡⎣cij ⎤⎦ and Var ⎡⎣cµij ⎤⎦ ==E(φ⎡c w⎤ij )and θ (1) . (6) ) ( ) log µ = η ij ij ˆ , (5) ij ij ⎡ ⎤ c φ w V µ φ w µ = = ( ) ( ) ( ) ˆ ˆ ˆ . (8) , (7) exp η = c + α + β = c + α + β ij ij ij ij ij ij ij ⎦c ⎣ ⎦ ⎣ ij incurred Provisions for claims outstanding, linear models , FPgeneralized (7) ηijij =but0c00not+ reported, αi ii + βj jjwith . (3) = ∑ cˆ . FP = (2) cˆ =∑ ∑ µ = η2 ijj ., (6) (7) Plog . (6) ηij = clog µij = ηcˆ ij.∑ 0 + αiji +ˆβ 2 2. PREDICTION ERROR FOR FUTUREcˆPAYMENTS YEARS . (8) = exp cˆ + αˆBY + βCALENDAR V ( µ ) = µ , (4) E ⎢⎡(cij − cˆij ) ⎥⎤ = E ⎢⎡ cij − E ⎡⎣ccˆijijij⎤⎦=−exp cˆij −cˆ00E+⎡⎣αcˆ iiij +⎤⎦ βˆ jj ⎤⎥ . (9) (8) V (µ ) = µ , ⎣ ⎦ ⎣ ⎦ ηij = FP c0 += α∑i +cˆ β j. , (3)(7) , (7) η = c + α + β ˆ ij 0 i j ˆ ˆ ˆ . (8) cEij⎡⎣c=⎤⎦exp cVar and value ⎡⎣cα⎤⎦ i=cˆ+ Consider a random variable (φ.βw2The ) = (φ wsquared ) µ , (5) error 0 + j )V ( µmean cij and µa =predicted 2 E ⎣⎡c ⎦⎤ ijand 2 ⎤Var ⎣⎡c ⎦⎤ = (φ w )V ( µ ) = (φ w ) µ , 2 ⎡2c ⎤ . (9) E ⎡(c −is:cˆ )2 ⎤ = E ⎡ c − E ⎡c ⎤ − cˆµ −= E (MSE) of prediction k

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) ( (( )) )) ( ) ( ) ( ) ⎦⎤ ) ⎤⎥cˆ . = exp (9) cˆ + α(ˆ +) βˆ ) . (8) = E ⎡⎣⎢⎣⎢⎡c((ˆ(c⎤⎦ )⎤−+EE⎣⎡⎣c⎡(c⎦⎤⎦ˆ ) − (Ecˆ ⎡⎣c− ˆE ⎤ ⎣⎡c⎤ .(10) E ⎡(c − E ⎡⎣c ⎤⎦ ) ⎤ − 2E ⎡(c E−⎢⎣⎢⎡(Ec⎡⎣c− c⎤⎦ˆ )()cˆ⎥⎦⎥⎤ − (9) ⎣ ⎦ ⎢⎣cˆ = exp ( cˆ ⎦+)⎣αˆ⎥⎦ +⎦ )β)ˆ ⎦⎥⎦⎤) . (8) ( ⎣⎡ ⎦⎤ ⎣⎢ ⎦⎥ ⎣⎡ ) ( ) ( ) E ⎢(c − cˆ ) ⎥ = E ⎢((c − E ⎡⎣c ⎤⎦ ) − (cˆ − E ⎡⎣c ⎤⎦ )) ⎥ . ( (9) ⎣ ⎦ ⎣ ⎦ − E ⎤⎣⎡c ⎦⎤ ) − (cˆ − E ⎣⎡c ⎦⎤ )) ⎤⎥ . (9) E ⎡⎡(c − E ⎡⎣c ⎤⎦ ) ⎤⎤ − 2E ⎡⎡⎡(c − E ⎡⎣c ⎤⎤⎦ )(cˆ ⎡− E ⎡⎣cˆ ⎤⎦ )⎤⎤ E+ ⎢⎡E(c⎤⎡⎡(−cˆcˆ −⎡) E⎥⎤ =⎡⎣cˆE ⎤⎦⎡⎢()(c⎤⎤ .(10) ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ˆ ˆ ˆ ⎡ ⎤ ⎡ ⎤ ( gives: ) .(10) ˆ ˆ ˆ ˆ − ≈ − + − E c c E c E c E c E c ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎦ − 2E ⎢((cinE−⎡cˆ(Ecinstead +⎦E)E⎡c⎣expectation E ⎣(c − E ⎣c ⎦ ) ⎦Plugging c c⎤ − c − E c and ( E ⎢c((the ⎦⎤⎦)))⎦⎥ ⎤⎥. . (11) (9) c⎦⎣)⎦−final ⎦⎣ )⎣⎡⎣c⎦⎥⎦expanding ⎥⎢⎣( ⎤⎦ ) −⎣⎢((cˆ ⎣ − E ⎣⎣ ⎣⎢ ⎣)−⎦⎥c⎦ˆ )() of ( α β) ⎣⎢=cE ⎣in ⎦ ⎣ ⎣⎢⎡ ⎦⎥⎤ ⎥ ⎦ ⎡ ⎤ E (c − E ⎡⎣c ⎤⎦ ) − 2E ⎡(c − E ⎡⎣c ⎤⎦ )(cˆ − E ⎡⎣⎣cˆ ⎤⎦ )⎤ + E (cˆ − E ⎡⎣cˆ ⎤⎦ ) .(10)⎦ ⎣ ⎦ ⎥ ⎢ c ⎤( ⎡ cˆ )− E ⎡cˆ ((⎤⎦⎥ ⎤ + E⎤ ⎡) cˆ( − E ⎡cˆ ⎤)) ⎤ .(10) (10) ⎣⎢ E ⎢⎡(⎦c − cˆE)⎡⎥⎤ c≈ E−⎡cˆ(c −⎤E≈⎡⎣cE ⎤⎦⎡) c⎤ −−2E E ⎡⎡(cc ⎤− E⎤⎣⎡⎣+ E⎦ )( cˆ − E⎣ ⎡⎣(c⎦ˆ)⎦⎤⎦ ) )⎤ ⎢⎣. ( (11) ⎣ ( ⎣ ⎦( ) )⎥⎦ ( ) )) ⎢ ⎥ (the ( ) ⎣ ⎦ ( ) ⎡ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ variance is the sum of⎢ process ⎥ variance EˆE⎤(⎡⎣c⎤ ⎤⎦−)E⎦⎥⎡⎡⎣+cˆE⎤⎦ ⎣⎢⎡)((c⎤ˆ ⎡=−ˆVar . (11) ⎤⎦ )⎤⎦ ⎦⎥and E E⎤ ⎡⎣c⎡⎣ˆ⎤c.(10) ⎣⎢−(cE − ˆ ⎡ + − E ⎡(c − E ⎡⎣c ⎤⎦ ) ⎤ − 2EE⎡⎣⎢⎣((cc −−Ecˆ(⎡⎣)c ⎦⎥⎦⎤⎦≈ c c E c E c ⎢ ⎥ ⎦⎥( ( ⎣⎢⎡ ⎦)(⎣ ⎦ ) ⎦⎥) ⎦⎥⎤( ))( ⎣⎢⎡ ( ⎣ ⎣⎦ )) ⎦ ) ⎤≈ ⎣⎢ ⎦⎥The assumption (11) ⎡⎣)cˆ of⎤⎦ )past − )E ⎡⎣⎡c ⎤⎦( ) ˆ⎤⎣⎢are + ⎤Eindependent E⎣⎢⎡(c − cˆ )that E (( cobservations cˆ⎡ − ( (E ( ))⎤ ) future ( ( ⎤. +observations ⎡ c)ˆ) −( E(⎡cˆ ⎤gives: ) )( ⎥ ⎡ ⎤ − ≈ − E c c E c E c E ⎢ ⎥ ⎢ ⎥ ( ⎦ ) ⎦⎥ ⎣ ⎤⎣⎢( ⎣ ⎦ ( ⎣ ⎣ ⎦ ) ⎦⎦⎥ ⎣ ⎦ ) ⎦⎥ . (11) ⎢ ⎡ ⎢ ⎣ ⎣ variance is the sum of process variance E ⎡(c − E (⎡⎣c ⎤⎦ )) ⎤ = Var ⎡c ⎤ and( the ) ⎤ = Var ⎢ is given MSEvariance of predictions sum⎡⎣cˆof⎡⎤⎦ . The process the cˆ − E ⎡⎣cˆ is⎤⎦ ) the ⎡ ⎤by:⎥ ( ⎡)(⎡⎣⎣ˆc⎤ ⎤⎦⎦) ⎤and (variance ( ) ) ( (11)) E ⎢(c − cˆ ) ⎥⎤ ≈ (E ⎡(c) −EE⎣⎢⎣((⎡⎣cc ⎤⎦−)E⎤)⎣+c E⎦ ⎡)(( c⎦⎥⎦ˆ =−Var E ⎣c )⎦ ) .() (11) ⎥⎦ ( ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎣ ⎦ ⎡ ⎣ ⎦ ⎣ ⎦ varianceThus, is the sum of process variance and the ⎡ ⎤ ⎡ ⎤ E c − E c = Var c the prediction variance is the sum of E ⎣((c ⎦− E ⎡⎣c) ⎤⎦ ) ⎤ = Var ⎡⎣c ⎤⎦ and the ( process⎣ variance ⎦ ) ⎦⎥ ⎥ ⎣⎢ ( ⎣⎢ ⎤ ( ) ) (⎦ ( The MSE variance of predictions given by: variance ⎡⎣cˆthe ⎤⎦ . prediction Thus, is theˆ is sum of process and ) ) ((ccˆˆ −− EE ⎡⎣⎡ccˆˆ ⎤⎦⎤ )) ⎥⎦⎤ == Var ⎡ ⎤ ⎡⎣⎤c ⎤⎦ + Var ⎡⎣cˆ ⎤⎦ . (12) MSE cof) =predictions E ⎢(c ⎡− c is) given ≈ Var . The MSE by: ˆ ( ⎡ ⎤ Var c ⎣ of⎦ process variance⎣ is⎦ the sum variance ⎤ ) MSE ⎡⎣c ⎤⎦ and the = Var estimation of predictions is given by: ⎣ E) (c − ⎦⎥E ⎡⎣.cThe ⎥⎦the variance (⎤ = Var ⎡ variance The MSE⎦by: of ⎦⎥predictions is given by: ) ⎤⎦ ) predictions ⎡⎣cˆ⎣⎢ ⎤⎦ . is cˆ − E ⎡⎣cˆ of ( given ⎡⎣cˆ ⎤⎦E. The cˆ ⎤⎦ ) ⎤ = Var ⎢⎣( MSE ⎥⎦ (cˆ − E ⎡⎣estimation ( ) ⎥⎦ (12) ( ⎡c)ˆ ⎤ . (12) MSE (c ) = E ⎡⎡⎢(c − cˆ ) ⎤⎤⎥ ≈ Var ⎡⎣∂(cµ)⎤⎦ + Var ⎡⎣c ⎤⎦ ⎡+ Var ⎣⎡⎣cˆ ⎤⎦⎤⎦ .. (12) MSE (c ) = E ⎣⎢(c( − cˆ ) )⎦⎥⎡ ≈ˆ Var ⎤ ((13) Var is given ⎡⎣cˆ ⎤⎦ . The MSE of predictions ⎡⎣c )⎤⎦ + Var ⎡⎣cˆ ⎤⎦ . (12) MSE c − cˆ⎡⎣η() )⎤⎦ ≈ Var ⎣ ⎦⎣c ⎤⎦(c≅by: ) =ηE ⎢⎣(Var (cˆ − E ⎡⎣cˆ ⎤⎦ ) ⎥⎦ = Var ⎤ + Var ⎡cˆ ⎥⎦⎤ . (12) c derived cˆ ) ⎤the ≈ Var The followingMSE can (be delta⎡∂cmethod: ) = E ⎡(cby−using i =1 j = k −i +1

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(4)

ij

θ (5) ij ij ij V µij = µijθ , (4) ij ij ij ij ij ij log µij0 = ηij . i (6) j ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij 2 log µij = ηij . (6) 2 ij 0 i j θ , (5) φ w=ij c V+ αµij + =β φ, w(7) µij = E ⎡⎣cij ⎤⎦ and Var ⎡⎣cij ⎤⎦ = η ij µij ij ij ij ij ij ij ij 0 i j 2 2 2 ηij = 2c0 + α i + β j , (7) 2 2 2 2 2 logijµ = η . (6) 2 ij ij ij ij ij ij ij ij ij ij ij ij 2 ij cˆij = exp cˆ0 + αˆ i + βˆ j . ij(8) ij ij ij ij ij ij ij ij ij ij ijij ij ij ij ij cˆij = exp cˆ0 + ˆ i + ˆ j . (8) ij ij ij ij ij ij 2 2 ij ηij = c0 + α i + β j , (7) 2 2 ij ij ij ij 2 ij ij E ⎡ c − E ⎡c ⎤ − cˆ − E ⎡c ⎤ 2⎤ . (9) 2 ij E ⎡⎢ij cij − cˆij ⎤⎥ = 2 2 2 ij ij ij ⎣2 ⎦ ⎣ ij ⎦ ⎦⎥ 2 ⎢ ⎣ ⎦ ⎡ ⎣ ij ij ij ij ij ij 2 ij 2 ⎤⎦ ⎤ . (9) E ij⎡⎢ cijij − cˆij2 ⎤⎥ = Eij⎢ cˆijij =− exp Eij⎡⎣cijcˆ⎤⎦0 +−αˆicˆ+ij β−ˆ jE .⎡⎣cij(8) ij ij 2 ij ij ij ⎥⎦ 2 ⎣2 ⎦ ⎣ ij 2 ij 2 ij ij ij ij ij ij 2 2 ⎡ cˆij − E ⎡cˆij ⎤ ⎤ .(10) ij ij ij E ⎡ cij 2−ijcˆij ⎤ ≈ijE ⎡ cij − E ⎡ijcij ⎤ ⎤ − 2E2 ⎡ cij −ijE ⎡cij ⎤ cˆij − ij E ⎡cˆij 2⎤ ⎤ + E 2 ⎣2 ⎦ ⎦⎥ ⎣2 ⎦ ⎡ ⎣ ⎦⎤ ⎦ ⎡2⎣⎢ ⎣ ⎦ ⎥ ⎣ ⎣⎢ ⎦⎥ ⎣⎢ ˆij E ij⎡c= ⎤E cˆ c− − E ⎡c ⎤⎤ −⎦ cˆ ⎡ − E ⎡c ⎤ 2 ⎤ .2 ⎤ (9) E ⎢⎡ cijij− cˆij ⎥⎤ ≈ Eij⎡ cij − E ⎡⎣2cij ⎤⎦ ⎤ − E2ijE⎣⎢ ⎡cijc−ij c− ij ij ⎦⎥⎣ ij ⎦ ⎣⎢ ij ij E ⎣⎡cˆ⎣ij ⎦⎤ij ⎦⎦ + Eij⎣⎢ cˆij −⎣ Eij ⎦⎣⎡cˆij ⎦⎤⎥ ⎦⎥ .(10) ⎣ij ⎣ ⎦ ⎣⎢ij ⎦⎥2 ij ij ij ij 2 ⎡ ij cˆ 22⎤ ≈ E ⎡ c − E ⎡cij ⎤ 2 ⎤ + E ⎡ cˆ − E ⎡cˆ ⎤ 2 ⎤ . (11) E ij ij ij ij ⎦ 2 ij ij ⎦ 2 ⎣⎢ 2cij − 2 ⎣ ⎣ 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎦ 2 2 ⎡ ⎤ ⎡ 2 ⎤ E ⎢⎡ cij − cˆij ⎥⎤ ≈ E cij −ij E ⎡⎣cij ⎤⎦ −ij2E ⎡ cij −EE⎣⎡⎡⎣ccij ⎤⎦− cˆcˆij − E ⎡⎦cˆ ⎡⎤ ⎤⎣+ E cˆ − E⎤⎡cˆ⎦ ⎤⎦ ⎡ ⎤cˆ.(10) ij ij ij ⎣2 ij ⎢ ij ijij ij⎤⎥ ≈⎣ Eij ⎢⎦ c⎦ij − E⎣⎢2⎡⎣cijij ⎤⎦ ⎥⎣ +ij E ⎥ − E ⎡⎣cˆij ⎤⎦ ⎥ . (11) ⎣ ⎣⎢ ⎦⎥ ij ij ij⎦ ij ⎣ ⎦ ⎣ 2 ⎦ ⎣⎢ ⎦ ij ⎦ ⎤ ij Thus, the prediction variance is the ijsum of process variance2 E ⎡ cij ij − E ⎡⎣cij ⎤⎦ ij 2= Var ⎡⎣cij ⎤⎦ and ijthe2 ij 2 ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ ⎡ c − cˆ ⎤ ≈ ⎣E c − E ⎡c ⎤ ⎦ ⎡+ E cˆ − E ⎡cˆ2 ⎤⎤ . (11) ij 2 ⎣ ij ⎦ E⎦⎥ cij ⎣⎢− Eij ⎡cij ⎤⎣ ij ⎦= Var 2 Thus, the prediction variance is theE ⎣⎢sum ofij process ⎦⎥ ⎣⎢ ijvariance ⎣ ⎦ ⎦⎥ ⎦⎥ ⎡⎣cij ⎤⎦ and the ij ij ij 2 ⎣⎢ 2 ij ij ij ij íj ij ij ij estimation variance E ⎡2 cˆ − E ⎡cˆ ⎤ ⎤ = Var ⎡cˆ ⎤ . The MSE of predictions is given by: 2 2 ij ⎣ ij ⎦ ⎥⎦ ij ⎣ ij ⎦ ij ⎢⎣ ij 2 is the sum of process variance E ⎡ cij − E ⎡⎣cij ⎤⎦ ⎤ = Var ⎡⎣cij ⎤⎦ and the ij Thus, the ij prediction variance is given by: estimation variance Eij ⎡ cˆij − E ⎡⎣cˆij ⎤⎦ ⎤ = Var ⎡⎣cˆij ⎤⎦ . The MSE of predictions ⎣⎢ ⎦⎥ ij ij ij ⎢⎣ ⎥ 2 ⎦ 2 ⎡ c − cˆ 2 ⎤ ≈ Var ⎡c ⎤ + Var ⎡cˆ ⎤ . (12) MSE c = E 2 ijij ij ij íj 2 ⎣ ij ⎦ ⎣ íj ⎦ ij ij ij ⎣⎢ of ⎦⎥ 2 MSE predictions is given by: estimation variance E ⎡ cˆij − E ⎡⎣cˆij ⎤⎦ ⎤ = Var ⎡⎣cˆij ⎤⎦ . The 2 2 ij ij ij ij ⎣⎢ ij ⎦⎥ MSE cíjij = E ⎡⎢ cij − cˆij ⎤⎥ ≈ Var ⎡⎣cij ⎤⎦ + Var ⎡⎣cˆíj ⎤⎦ . (12) 2 ij ij ij ij ij ij ij ij íj ⎣ ⎦ 2 ij ij ⎣ ij ij⎦MSE ⎣E ⎡íj(⎡⎣cc⎦ˆijij−⎤⎦ c≅ˆij )∂2µ⎤ ij≈ Var Var⎡⎡⎣cηij⎤ ⎤⎦+.Var(13) 2 (c ) =Var ˆ ⎡ ⎣⎢ ij ij ⎦⎥ c ij ij ∂η⎦⎥ ij ⎣ ⎦ ⎣ íj ⎤⎦2 . (12) ⎣⎢ ∂µ ∂ ij 2 ij Var 2⎣⎡cˆij ⎦⎤ ≅ 2 ⎡c . (13) ∂ ˆ ⎤ ⎡ ⎤ ⎣⎡ηij ⎦⎤ . (13) ≅ Var Var ∂µij 2 ∂ηij Var ij ij ⎦ ⎤⎡⎣c≈ˆijVar ⎦ ⎣ ∂ MSE cij = E ⎡ cij − cˆVar (12) ˆ ⎡ ⎤ ⎡ ∂ µ c + Var c . (13) ⎤ ⎡ ⎤ 2 ≅ Var (13) ˆ ⎡ ⎤ ⎡ ≅ be ∂µVar ηbeij ⎤⎦ .(13) ijíjcould ⎣ MSE In the case=where the logarithmic link , ijand ij could ij⎦ then, the , and then, theij function MSE logarithmic link function ⎣Var ⎦⎣cijofVar ⎢⎣ ⎥⎦⎣ ij ∂⎦ηof= µ⎣∂∂predictions ⎦ predictions ⎣ ⎡⎣cˆij∂⎤⎦η≅ Var ⎡⎣ηij ⎤⎦ . (13) ij ij ∂ η ∂ ∂µ ⎡ ij (13) ⎡⎣cˆij ⎤⎦ link ⎤⎦ . then, ≅ functionVar the MSE of predictions could be In the case where the Var logarithmic =⎣µ ,ijand ∂η ∂ approximated as: ij 2 ∂µ In the case ∂ where logarithmic function andthen, then, the MSE of predic∂MSE µlink the MSE of predictions could be In=thethe wherethen, the logarithmic function = µ ,, and ∂predictions logarithmic ,caseand thelink of could be , and predictions could be In the link case function where the ∂logarithmic link as: function ∂ijηthen, MSE µij2Var ⎣⎡ηij ⎦⎤ . (14) + be (13) , and then, the logarithmic link ⎡c=ˆ µ⎤ of ⎡ (cMSE ⎤) ≅. φµijθof ≅ predictions VarMSE Varthe ijcould ∂ = approximated tions function could be approximated as: ij

ij

ij

ij

µ µ η µ η ( ) η µ( ) µ η η η η µ µ µ µ η µ ∂η⎣ ij ⎦ ⎣ηij ⎦ ∂ηij ∂η µ approximated as: could MSE be (c ) ≅ φµ + µ Var logarithmic link function = µ , and then, the MSE of θpredictions ⎣⎡η ⎦⎤ . (14) MSE (cij ) ≅ φµij + µij2Var ⎡⎣ηij ⎤⎦ . (14) ∂η approximated as: MSE (c ) ≅ φµ + µ Var ⎡⎣η ⎤⎦ . (14) ∂µ logarithmic link function could be = µ , and then, the MSE of θpredictions θ ⎡⎣ηij ⎤⎦ . (14) Varcalculated + µij2be The prediction prediction with+i,µj2Var =MSE 1,...,k ijthud ∂η error (PE) for each .) ≅ φµ (14) ⎡⎣η(ijc⎤⎦ijcould MSE cij ) ≅ φµ ( ijθ ij2 . (14) ⎡ ⎤ MSE c φµ µ Var η ≅ + as the root of the MSE (14). Next, the formulas of the MSE are given for the accident year ( ij ) ij ij ⎣ ij ⎦ θpayments 2 provisions, for the total provision and for the future by calendar years. The PE of MSE (cij ) ≅ φµij + µijVar ⎡ηij ⎤ . (14) those amounts could be calculated as the corresponding root. ⎣ ⎦ ij

θ

ij

2 ij

θ

ij

ij

2 ij

ij

ij

The PE obtained when assuming the parameterθ dependent family of distributions (4) ⎡⎣ηij ⎤⎦ . (14) MSE cij ≅provision, φµij + µij2Var for the accident year provisions and the total has been studied in e.g. England and Verrall (1999). The PE for the future payments by calendar years in the (over-dispersed) Poisson case has been studied in Boj et al. (2014) and Espejo et al. (2014) where it is programmed for practitioners with RExcel. Next, the parametric family of distributions (4) is assumed, thereby extending the formulations for the future payments by calendar year (10) and (14) of Boj et al. (2014). The squared PE for the accident year provisions are:

( )

162

Cuadernos de Gestión Vol. 17 - Nº 2 (2017), pp. 157-174

ISSN: 1131 - 6837

(14)

(2014). Eva Boj Del Val / Teresa Costa Cor The squared PE for the accident year provisions are: 2 MSE ( Pi ) = E ⎡ Pi − Pˆi2 ⎤ ≈ ∑ φµθ ijθ +µTiTVar [ηi ] µi = MSE ( Pi ) = E ⎡ ⎣⎢Pi − Pˆi ⎤ ⎦⎥≈ ∑ j =1,..,φµ k ij +µi Var [ηi ] µi = ⎢⎣ ⎥ j =i1,.., + j >k 2⎦ k ⎡ ⎤ ˆ + > i j k MSE ( Pi ) = θE Pi − Pi 2 ≈ ∑ φµijθ +µiTVar [ηi ] µi =⎡ = ∑ φµij +⎣⎢ ∑ µij Var ⎦⎥ ⎣⎡ηj =ij1,.., ⎦⎤ +k 2 ∑ µij µij Cov ⎣ηij ,ηij ⎦⎤ . (15) =1,.., k µ µ Cov ⎡η ,η ⎤ . φµk ijθ + ∑j =1,..,µk ij 2Var ⎣⎡ηiji +⎦⎤j >+k2 j , j∑ = ∑j =1,.., ij1 ij2 ⎣ ij1 ij2 ⎦ (15)

( ( )) ( )

i + j >k j =1,.., k θ i + j >k ij j =1,.., k i + j >k

1

1

i + j >k j =1,.., k i + j >k

∑ k µijk2Var ⎡⎣ηij ⎤⎦ + 2 ij ==1,..,1,..., j >k i =i +1,..., k The squared PE for the total provision is: i = 1,..., k =

∑ φµ

+

2

1

(15)

2

2

j > j1 j1i,+2j 2j1=>1,.., k ,i +k j2 > k j2 > j 1 i + j1 > k ,i + j2 > k ij1 j1 , j 2 =1,.., k j2 > j 1 i + j1 > k ,i + j2 > k

µ µij Cov ⎡⎣ηij ,ηij ⎤⎦ . (15)

∑

2

1

2

2 The squared the MSE ( PPE E ⎡( P µ TVar − Pˆtotal ) = for [η ] µ = ) ⎤2≈provision ∑ k φµφµijθ θ+is: T MSE ( P ) = ⎡⎣⎢E ⎡( P −ˆ P2ˆ ⎤⎦⎥) ⎤ ≈ Var [η ] µ = i , j =1,..., ∑ ij + µ θ T ⎢ ⎥ i + j >k

(

)

MSE ( P ) = E P⎣ − P ≈⎦ i , j =1,...,φµ k ij + µ Var [η ] µ = ⎢⎣ ⎥⎦ i , j∑ . (16) i +1,..., j > kk = µi1 j1 µi2 j2 Cov ⎡⎣ηi1 j1 ,ηi2 j2 ⎤⎦ . (16) = ∑ φµijθ +θ ∑ µij 2Var i +⎡η j > ij k ⎤+2 ∑ 2 ⎣ ⎦ ⎡ηi j ,ηi j ⎦⎤ . (16) φµij i ,+j =1,...,∑ µij Var ⎡ηij ⎤ + i21 , j 1,i2 , ∑ i, = j =1,...,∑ k k j2 =1,..., k µi j µi j Cov ⎣ j >ik, j =1,..., j >i ,kj =1,..., > k kµ µ k,i,i,2 j+ =j21,..., ⎡ ⎤ φµk ijθ + i + ∑ µkij 2Var ⎡⎣η⎣ij ⎤⎦ +⎦ 2ii1 +ji ≠j,1ij>∑ = i+ ∑ i1 j1 i2 j2 Cov ⎣ηi1 j1 ,ηi2 j2 ⎦ j 1 1

i+ j >k i , j =1,..., k i+ j >k

2 2

1 1

(16)

2 2

1 1 2 2 1 1i + 2j > 2 k ,i + j > k i1 , i1j 1j ,i≠21i, jj2 =21,...,2 k i1 +1j11> k2,i22 + j2 > k i1 j1 ≠ i2 j2

i+ j >k i , j =1,..., k i+ j >k

And the squared PE for the future payments by2 calendar years are:

⎡ Pfuture And the squared calendar MSEPE µiTVar = Ethe ≈ ∑ φµijθ +by ( Pi )for [ηi ] µi =years are: ( i − Pˆi ) ⎤ payments

⎥⎦ j =1,.., k ⎣⎢ c − cˆ 2 ⎡ θ θ T T !2 ⎤ ≈⎤ ≈i + j > k φµφµ r⎤ijµP=== ij ij . (18) ⎡i η MSE Pi t−−PˆFP Var η µ MSE Var ( Pti))=θ=EE⎢⎡⎢ FP [ ] ⎥ ∑ ∑ ( FP i t ij +ijµ+ iµ i ⎣ t ⎦ t cij −cˆθcˆij t ⎣ µ 2Var⎦⎥⎡η⎦ j ⎤=1,.., k2 k ⎡⎣ηij1 ,ηijij2 ⎤⎦ . (15) , j =1,.., µij1 µij2 Cov = ∑ φµij + ⎣∑ (18) rijP = ∑ ij ⎣ iji +⎦iij +>+ kj =t j =1,.., k j =1,.., k j1 , j 2 =1,.., k cˆijθ + > i + j >k i j k > j j (17) 2 2 ⎡ ⎤ 1 ⎡η⎡η⎤ +⎤2+ 2i +2∑ 2 ⎡ ⎤ . (17) = =∑∑φµ µ Var µ µ Cov η , η . , φµijθ ijθ+ +∑ µ Var µ µ Cov η η k ,i + j2 > ki j ij i j ij ∑ ⎣ ⎣ij ⎦ij ⎦ 2 j1 >∑ ij ij ⎣ ⎣ci1 j1ij−1 cˆi 2ijj2 ⎦⎦ (15) 1 1 12 2 2 1 1 ij ij 2 = = = j k j k j j k 1,.., 1,.., , 1,.., i , j =1,..,k ˆ P i , j =1,..,k , 11 22 =1,..,k θ T ⎡tj > kP k− Pˆ ∑ ⎤ ≈ rijiiP11 ,ji12≠φµ (19) =ii +=Eij+=1,..., 2 . iMSE + j i=+t j > k( φ ij22= j>2 j 1 +µ Var [η ∑ Pi ) = i >kk − 1 i ] µi =c − > k ,i=+t j2 ˆθ ˆ i1+ j21i=+t ,ji1ij n −⎢⎣ 21ki − 1ii , j =⎥⎦1,...,k j =∑ 21 i , j =1,..., k 2 +nj 2− ij cijcij P Pk 1,.., ˆ j ≤ k i +rj > k i + j ≤k φ = = 1,..., k i + ∑ = ∑ cˆθ . (19) ij t = ki +1,...,2 n −2 2k k− 1 i , j =1,...,k θ n − 2k − 1 i , j =1,...,k ij T MSE ( P=) =∑ E ⎡φµ Pθ−+Pˆ ∑⎤ ≈µ 2Var φµ µ Var η µ + = i j k i j ≤k + ≤ + [ ] ⎡⎣η]ij1 ,ηisij2 ⎤⎦estimated ∑ . (15) 2 which µij1 µVar ∑ ij ij2 Cov[ĉ ⎢⎣ given ⎥⎦ iji ,the ⎣⎡ηijij⎦⎤ +in Next, formulas j =are for case the by means of 1,..., = j k n ij P' P j =1,.., 1,.., k , j 2 =1,.., k 2 k i + j >k by means ofrijstandard Next, formulas are giveni +for ⎡⎣µcˆjjTíj12Var , (20) θ r = > k ⎡the >⎤ jE k which >⎤ j 1is estimated ij ⎦ MSE(SE) P case Pˆi + jin φµVar η µ −predictive ≈ ∑the + = standard error the distribution of c . The predictive distribution of cij is ( P ) =θof [ j2 >]k ij ij i + j1 > k ,i + n − 2nk − 1 P . (16) ⎢ ⎥ P' =1,...,⎤k + 2 ⎡ ⎤ , φµij +⎣ ∑ µij 2⎦Varii ,+⎣⎡jη µ µ Cov η η = ∑ , (20) r = r ∑ ij i j i j i j i j ⎦ estimated by bootstrap methodology (see Efron and Tibshirani 1998). The treatment ij ij 1 1 2 2 ⎣ 11 2 2⎦ i = 1,...,of k cj > k. Thei1predictive ˆ cof − 2ckíj −is1estimated =1,..., k i , j =1,..., k , j 1 ,i2 , j2 =1,..., k distributionnof error (SE) of theii ,+jpredictive distribution ij − cij . (16) P íj '* + j >k j > k have > k. From (18)involve ipaper jP2frequently θ B iresamples 2 of in 1 + j1 > k ,i2 + . (18) r = and (20) we isolate In the bootstrap process estimation we the residuals, r the GLM and associated claims this the application of bootµi1 j1 µi2 j2 Cov ⎡⎣ηi1 j1 ,ηi2 j2 ⎤⎦ = ∑ φµij + ∑ µij Var ⎡⎣ηij ⎤⎦ + 2i1 j1 ≠i2∑ ij j2 ij θ ˆ c 2 P(see, '*k i , j =1,..., k i , j =1,..., k Pearson i1 , j 1 ,i2 , j2 =1,..., ij * P '* θ strapping residuals based on residuals e.g., Boj and Costa 2014). When (4) is ⎡we .The (20) we isolate In estimation have the residuals, rijj2 > kcorresponding +P i + we j> k( P iresamples j > kEfron by bootstrap methodology Tibshirani of the GLM andB >µ + k ,Ti2the and estimate amounts. The cij the = rbootstrap cˆij + cˆprocess MSE φµijθii1 +jand Var = EB(see − Pˆ ⎤and ≈ofthe +j11998). = (18) and )then [ηFrom ] µtreatment ij ij for each sample, ∑GLM 1 1 ≠ i2 j2 ⎣⎢ Pearson ⎦⎥ i , residuals j =1,..., k assumed and w = 1, the take the expression: * P '* θ boot ij i + j > kaGLM for eachgive sample, andthis then we estimate the the(for corresponding The B cij = rijof the cˆij +estimations cˆij associated each value),amounts. of the accident values the predictive distribution valueand cˆapplication ij claims in frequently involve of ˆ Pof . (16) could the be estimated by:bootstrapping residuals The scaleθpaper parameter 2 φ boot boot boot ⎡ ⎤ ⎡ ⎤ ˆ 2 , φµ µ Var η µ µ Cov η η = + + c − c ˆ ˆ ˆ (for each value), of the accident values of the estimations give the predictive distribution of a value c ∑ ∑ ij ij ⎦ i j i2 j2 ⎦ ij year provisions Pi (for each accident year), of theij total⎣provision , andi1 j1 ofi2 jthe future by P∑ 2 ij (18) .w (18) rij⎣P =1 1 ijpayments =1,..., k i , j =1,..., k j 1 ,i2 , j2 =1,..., k based on Pearsonii ,+j jresiduals (see, e.g., Boj and Costaii1 ,+2014). When (4) is assumed and θ ij = 1, boot boot >k i+ j >k k ,boot i2 + j2 > k 1 j1 >ˆ ˆ ˆ 2 c ∂ year provisions (for each accident year), of the total provision , and of the future payments by P P ˆ c − c ij ≠ i j i j calendar years FPi t (for each calendar year). 1 1 2 2 cij − cˆij rijP = ijP 2 ij . (18) 1 1 P boot ˆ 2 θ . (19) ∂ = (21) Pearson residuals take the expression: θ (for each calendar year). calendar years FPthe ij cˆij k t , i, j =r1,..., PE boot cij ≈ φφˆ P cˆ= SE cˆijboot 2 . ij + ˆijθ 2 1 2 1 n − k − n − k − c i , j =1,..., kc − c i , j =1,..., k ˆ 2 The scale parameterˆ Pôp could be estimated by: 1 1 ij ij Pˆ P2 θ +, j ≤ k i + j ≤k (21) ˆij + SE cˆijboot i∑ i, j = 1,..., k . (19) PE boot c∑ φ = ij ≈ rijφ c= 2 ˆijθc− 2 1 n − 2k − 1 i , j =1,...,k n − k − c i , j =1,..., k c c− 2 ˆ ˆ c boot 1 P i +≈j ≤k 1 Pˆ boot ij ij ij ij i + j ≤k , P i = ˆ2P= (22) , k . (18) cˆijθ + SE ( i ) ∑in ∑ (19) φˆ P the = PE rijP φbootstrap i In this paper, residuals the process for the degrees(19) of freedom as ∑2 rij =are1,cˆθKadjusted 2k − 1 i , j =1,...,kj =1,..,k ˆ P θ n − 2k −ˆ1boot n −boot cˆθ i , j =1,..., k PE ( P )i +≈j ≤k i + j >k φ cˆ + SE P i + j ≤k , i = 1,ijK ij, k . (22)

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)

)

i n rijP ' = rij2P , (20) n − k − 2 1 boot θ 2 (cijnP−ˆ bootcˆij )r2for 1in thePE 1 φˆPP'are P. In this paper, the residuals bootstrap (23) degrees of freeSE ∑ , the (20) rij cˆ=ij∑+adjusted (19) φˆ P = r(ijPP)) ≈= process ( ij2 . ∑ θ , j =2 1,..., k− 1P θ ˆ n − k − 2 1 Pn'*i− 2 1 n − k − k c boot boot n i j k i j k = = , 1,..., , 1,..., ˆ dom as follows: ˆ ij we. isolate > k φ (18) jFrom In the bootstrap process estimation we have B resamples of the residuals, (23) r P ' = ˆ and (20) ( P ) ≈rij i +. ∑ + ijj ≤ k+ SE P i +PE j ≤k ic rijP , (20) ij i , j =1,..., k n − k − 2 1 * P '* θ P '* + > i j k sample,we andhave thenBwe estimateofthe and the The B cij the = rbootstrap cˆij + cˆprocess . From2 (18) andamounts. (20) we isolate In estimation resamples theGLM residuals, rij corresponding ij ij for each boot ∂ boot ˆ P cˆθ +boot P ', t = k +n P. (24) (20) 1,..., 2 k PE FP ≈ φ SE FP ( ) t * P '* θ , (20) ∑ r = r t ij values the predictive distribution of aGLM valueand (for each of the ijaccident cˆcompatibility, ij2 value), amounts. sample, and then we estimate the B cij = rijof the cˆij +estimations cˆij for eachgive ij the corresponding as similarly done , for the sake for parameter φˆ P . i , j =1,.., k P of n −the 2k −scale 1 The θ boot ∂ boot ˆ + = i j t ˆ , . (24) boot boot t = k + 1,..., 2 k PE FP ≈ φ c + SE FP ( ) t ˆ , andeach ∑1,..,value ij P each the accident year),distribution of thet totalofprovision of the future by year Pˆi (for give value), of payments the accident valuesprovisions of the estimations predictive cˆijboot (for i , j =a k P '* i + j =t boot P '* In the bootstrap process estimation we have B resamples of the residuals, . From (18) and (20) we isolate r ∂Pˆ boot ij , and In thedone bootstrap process estimation we have resamples of the by residuals, (for calendar year). calendar years FP t year provisions (for accident year), of the the total of future Pˆ boot as each similarly , for sakeprovision of compatibility, forBthe the scalepayments parameter ôp . rij . From (18) and i * P '* θ 2 boot each sample, and then we estimate boot and B cij = rij cˆij + ∂ cˆij for ˆ Phave In each the calendar bootstrap process estimation we B resamples the residuals, (21) ˆθthe corresponding ˆboot ) , i, jamounts. =of 1,..., k . The PE the(cGLM rijP '* . From (for year). calendar years FP ij ) ≈ φ cij + SE ( cij t boot * P '* θ 2 (18) and we each sample, then estimate (for each value), of and the accident values of the estimations give (20) the predictive distribution of a value ˆ φˆˆcijˆPijcˆfor each wewe estimate the the GLM and the (20) we isolate isolate cij PE = rboot θfor ij ( c c)ij≈+ c ˆbootsample, 1,...,then k . (21) ) ,2 i, j =and ij ij + SE ( cij boot boot ˆ ˆ year provisions Pi (for each accident year), of the total the future by , i = 1,payments K , k . (22) PE boot φPˆ P cˆθ ,+and SE of Pˆ boot ( Pprovision )≈

follows:

i

j =1,.., k i + j >k

ij

( (

( (

∑

i

) )

) )

ij

( (

i

163

) )

=1,..,values k Cuadernos ISSN: 6837 year). amounts. The jB Gestión Vol. 17 - Nºthe 2 (2017), pp. 157-174 corresponding of thedeestimations give predictive distribution of a ∂ boot 2 (for1131 each-calendar calendar years FP j >k ˆ P θ t PE boot ( Pi ) ≈ i +∑ φ cˆij + SE Pˆi boot , i = 1,K , k . (22) 2 kˆ P θ ˆij + SE cˆijboot , i, j = 1,..., k . ˆ(21) PE boot cij i≈j+=1,.., boot kφ c 2 P ofj >the accident year provisions (for each accident year), of value cˆijboot (for each value), PE boot ( P ) ≈ ∑ φˆ P cˆijθ + SE Pˆ boot . i(23)

( )

(

i , j =1,..., k

)

(

)

cale parameter φˆ P . of the residuals, rijP '* . From (18) and then we estimate the GLM and the

Provisions for claims outstanding, incurred but not reported, with generalized linear models

give the predictive distribution of a

GLM and the corresponding amounts. The B values of the estimations give the predictive a value cˆijboot (for each value), of the accident year provisions Pˆi boot (for each accident year), of the total provision Pˆ boot , and of the future payments by calendar years boot ! by calendar years FP t (for (for each each calendar year). From the B values it is possible to estimate the variance of the distribution as the square of the standard error (SE) of the distribution. The SE estimated in this way replaces of the distribution asthe the part squareofofthe the estimation variance, Var [ĉ ], in formulas (14), (15), (16) and (17). ij Next, a description is provided of the formulations of the estimations of the PE, i.e., the d in this way replaces the part of the root of the MSE is given, for a value, for the accident year provisions, for the total provi2 sion and for theboot future payments ) and (17). P θby calendar ˆboot . (21) estimation of ˆij + SE cˆijbootyears, ini,the j =case 1,...,ofk bootstrap PE c ≈ φ c ij Var [ĉij]. We denote this by PE . e estimations of the PE,The i.e., the root bootstrap estimations of the PE for each value are:

ons Pˆi boot (for each distribution accident year), of of

(

( )

)

for total provision and provisions, estimations ofPE thefor PEthe for accident the accident year provisionsare: are:2 mations ofthethe year provisions PE boot (cij ) ≈ φˆ P cˆijθ + SE (cˆijboot ) , i, j = 1,..., k . (21)

(21)

22

θ boot ap estimations of the PE forbootthePaccident provisions ˆ are: boot P ˆ Pθ c ˆ+ijPE PE + for SE P 1,…, kk .. (22) ˆPboot ∑φˆyear (estimations ˆprovisions , , iiyear (22) i accident ==1,…, PEbootstrap Pi accident ≈i ) ≈ ∑year SE ofφcthe the provisions are: stimations of theThe PE for the ij i are:

( ) ( ( )) PE ( P ) ≈ ∑ φˆ cˆ + SE ( Pˆ ) , i = 1,…, k . PE ∑ φˆ cˆis:+ SE ( Pˆ ) , i = 1,…, k . ( Ptotal ) ≈ provision p estimation of the PE for the j =1,..,k

j =1,.., i +k j >k i + j >k

boot

boot

i

i

P

θ P θ ij j =1,..,k ij i + j >k j =1,..,k i + j >k

i

mation of the PE for the total provision is:

boot 2 i boot

2

The bootstrap estimation of the PE for the total provision is:

boot ap estimation of the PE for the total PE provision ≈ ( P )is:

stimation of the PE for the totalPE provision ( P ) ≈is: boot

2

∑ ˆφPˆ ˆθcˆθ + SE ( Pˆˆboot ) 2 .. P

∑

ij

(

boot

φ cij + SE P

i , j =1,..., k i + j >k

)

(22)

(22) (23)

(23)

(22)

(23)

2

PE boot ( P ) φˆ P cˆijθ + SE Pˆ boot 2 . (23) ∑ boot i , by j =1,...,calendar kˆ P θ tstrap estimationsAnd of the for theestimations future(payments years are: . by (23) PE P )of≈ the PE cˆij future + SE Pˆ boot the PE bootstrap payments calendar years are: + j >for i∑ k φ the i , j =1,..., k ≈i + j >k

(

(

)

)

j =1,..., k estimations of the PE for the future paymentsii ,+by j > k calendar years are:

2 otstrap estimations boot of the PE for the future payments by⎞calendar years are: ⎛ ! boot (24) PE ( FPt ) ≈ ∑ φˆ P cˆijθ +SE ⎜ FP t ⎟ , t = k +1,...,2k . (24) 2 ap estimations of the PE for the future payments by ⎞calendar years are: boot ⎝ ⎠ ⎛ i , j =1,.., k ! tˆ P cˆθ +SE FP t boot ,2 t = k + 1,..., 2k . i + j =φ PE boot ( FP )≈ (24) ⎜ ⎛! ⎟ ⎞ ijˆ P θ boott ˆ , . (24) PE ( FPt )i ≈ φ c + SE FP t = k +1,...,2 k t ⎝ ⎠ ⎜ ⎟ ∑ , j =1,..,k ij 2 boot (14) ⎝ (13), ⎠ and (15) of Boj et al. (2014), here SE of i + j =t i the ⎛ ⎞ Note that, unlike formulas (12), , j =1,.., kP θ ! boot nlike the formulas Bojt et al., (2014), the i + j =tand PE (12), ( FP )(13), ≈ (14) φˆ cˆ(15) +SEof FP t = k +here 1,..., 2SE k . of (24)

∑

∑

⎜ the bootstrap for⎟ degrees of freedom, because it is assumed t distributions is ijnot adjusted ⎝completed ⎠ with (20), the adjusted Pearson residuals. i , j =1,..,k that work in the bootstrap process is i + j and =t he formulas (12), (13), (14) (15) of Boj et al. here SE the stributions is not adjusted for degrees of freedom, because it(2014), is assumed that work unlike the formulas (13),made (14) of and of Boj et al.(2014), SEal.of of the If direct(12), use were the(15) Pearson residuals (18), as inhere Boj et (2014), it would be necessary to correct the SE multiplying it by n/n(n-2k−1), where 2k+1 is the number of rap completed with (20), adjusted Pearson residuals. If direct use e theprocess (12), (13), and (15) of Boj et al.it(2014), here SE the ons is formulas not adjusted for of the freedom, because that ofdegrees model (7). stributions isisparameters not adjusted for(14) degrees of freedom, because itisisassumed assumed thatofwork work

of theprocess Pearson residuals as(20), inofBoj et al. (2014), it residuals. be necessary to butions is completed notisadjusted for(18), degrees freedom, because itwould is assumed that ocess is with (20), the adjusted Pearson IfIf direct use strap completed with the adjusted Pearson residuals. directwork use 3. APPLICATION

SE multiplying itresiduals by (18), ,inthe where is theit number ofto + 1(2014), n (with n −(18), 2k(20), −as 1)Boj process is completed Pearson residuals. If direct use of the Pearson Boj et2kal. itwould wouldof necessary Pearson residuals as in etadjusted al. (2014), bebeparameters necessary to

The proposed methodology is illustrated using the triangle of Taylor and Ashe (1983)

of Figure with incremental losses. This dataset is used inbe many texts thatof he multiplying Pearson residuals as et al. (2014), it number would necessary todeal with IBNR SE it by n2(18), where is the parameters +is1 the k ,−in 1) ,Boj tiplying it byproblems, where number ofofand parameters of and England 2k +2k1(1989, n 2(kn − −as12)those (n −such of Renshaw 1994), England Verrall (1999)

multiplying it by n ( n − 2k − 1) , where 2k + 1 is the number of parameters of 164 Cuadernos de Gestión Vol. 17 - Nº 2 (2017), pp. 157-174 ISSN: 1131 - 6837

ATION

ed methodology is illustrated using the triangle of Taylor and Ashe (1983) of CATION

Eva Boj Del Val / Teresa Costa Cor

(2002). In particular, England and Verrall (1999), demonstrate (in Tables 1 and 2) the estimation of provisions and the PE as a percentage of the provision estimate (i.e., the called “coefficient of variation” in many text books and computer programs) for the accident year and total provisions, (1) and (2). This is calculated, in cases when the over-dispersed Poisson and the Gamma distributions are assumed using analytic formulas to calculate PE and when the over-dispersed Poisson distribution is assumed using bootstrap methodology for the estimation of PE. Figure 2 Run-off triangle with 55 incremental losses Accident year 0 1 2 3 4 5 6 7 8 9

Development year 0 357848 352118 290507 310608 443160 396132 440832 359480 376686 344014

1 2 766940 610542 884021 933894 1001799 926219 1108250 776189 693190 991983 937085 847498 847361 1131398 1061648 1443370 986608

3 482940 1183289 1016654 1562400 769488 805037 1063269

4 527326 445745 750816 272482 504851 705960

5 574398 320996 146923 352053 470639

6 7 8 9 146342 139950 227229 67948 527804 266172 425046 495992 280405 206286

Source: Taylor and Ashe (1983)

This Section completes the analysis conducted by England and Verrall (1999) by including the study for the future payments by calendar years (3). Poisson (over-dispersed) and Gamma distributions are assumed and an estimation is made of the PE with analytic formula and with bootstrap (using size B = 1000 resamples). Both distributions are particular cases of the parametric family (4) described in this paper: when Ɵ =1 we have the Poisson distribution and when Ɵ =2 we have the Gamma distribution. In addition, computations are made with the R software (R Development Core Team 2015) and models are fitted with the function glm of the stats package for R. Results are given in Tables 1 to 8. The future payments by calendar years enable to work in a financial environment and to calculate the best estimate of provisions. The present value of the future payments by calendar years can be determined by taking into account the time value of money (and thus, following the Solvency II Directive). The present value constitutes the “today real provision” of an insurance company. This Section provides an example which assumes a 1.5% fixed annual interest rate for the future nine years. Additionally, a risk margin can be added to calculate the SCR. Three possibilities are presented to show how to calculate the best estimate in the context of Solvency II: 1) In the first of these, the present value of the future payments by calendar years is computed without any risk margin:

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Cuadernos de Gestión Vol. 17 - Nº 2 (2017), pp. 157-174

165

Provisions for claims outstanding, incurred but not reported, with generalized linear models

IBNRactual =

2k

t − (t − k ) 1

∑ FP (1 + I ) t

t = k +1

,

(25)

where I1t for t = k + 1,...,2k are the annual interest rates for each of the next calendar 2k − (t − k ) years takenIBNR into account in the run-off . (26) FPt + δ triangle. MSE ( FPt ) 1 + I1t actual = 2) In the second, the value of the future payments by calendar years is com1 t = k +present puted plus a fixed percentage, δ, of the PE. In this way a risk margin is included for each 2k k ) In the application of calendar year equal to the 2fixed percentage of the corresponding k t −(t −PE. boot t − (t − k ) , (25) IBNR = FP 1 + I 1 actual t this SectionIBNR 25% of the PE is added, i.e., δ =0.25. Use of the analytic . formula (27) (17) to esti= FP + δ PE FP 1 + I ( ) 1 actual t t =2kk+1 t − − t k t = k +1 ( ) mate the MSE(FTP) derives: t

∑(

)(

∑(

∑ )(( )) = ∑ FP (1 + I )

2k

IBNRactual

)

t

1

,

(25)

(26) . (26) ) ( ) . (28) ( ) ( ) . (26) IBNR = ∑ ( FP + δ MSE ( FP ) ) (1 + I ) And use of bootstrap estimation of the PE with formula (24) to estimate the PE t = k +1 boot t t

− (t − k ) t − (t − k ) 1t 1 t − (t − k )

IBNRactual = ∑ FP + δVaR MSEFP + II ∂( FP ) (11 + IBNR = t∑ α t =actual k +1 2k

actual

t = k 2+k1

2k

t = k +1

t

t

1

boot

t − (t − k ) 1

(FTP), derives: . (27) IBNRactual = ∑ ( FPt + δ PE ( FPt ))(1 + I ) he third possibility, the present value 1 the values at risk, VaR, is computed t =2kk +of − (t − k ) . (27) IBNRactual = ∑ ( FPt + δ PE boot ( FPt ))(1 + I1t ) 2 kyear predictive t = k +1 boot future payments ed confidence level α for each calendar ∂ t (1 + I t )−(t −k ) . (28) IBNR = VaR FP

∂ on FP

(25)

∑ ∑

( (

) )(

(27)

3) And in the third possibility, present value of the values at risk, VaR, is comt =2kk+the 1 − (t − k ) puted at a fixed confidence level α for each year ∂ boot : . future (28) payments disIBNRactual = VaRα calendar FP 1 + predictive I1t t boot tribution PE : t = k +1 actual

boot t

boot

IBNRactual =

2k

α

)

− t −k ) ⎛ ! boot ⎞ t ( . FP I 1+ t ⎜ ⎟ α 1 ⎝ ⎠

∑ VaR

t =k +1

1

(

)

(28)

(28)

A risk margin is added because expected mean of the predictive distributions argin is thus added because thethus expected mean the of the predictive distributions

(which must be similar to the future payments by calendar years) is replaced by the quantile α of its predictive distribution. The example given in this Section uses the 99.5% quantile must be similar to the future payments by calendar years) is replaced by the of predictive distribution of the future payments by calendar years. Note that this way of adding a risk margin is only available when estimates are made by bootstrapping the predistribution. The payments. example given in this Section uses the α of its predictive dictive distribution of future Table 1 calculates the future payments by calendar years, the prediction errors and the uantile of predictive payments byover calendar years. coefficientsdistribution of variation of (i.e.,the thefuture proportion of the PE the amount estimate, in percentage) for the over-dispersed Poisson distribution using analytic formula to compute the as the aroot the MSEis(17). In available Table 2 we when have the same computations t this way ofPE adding riskof margin only estimates are madefor the Gamma distribution. Note thatdistribution in Table 1, inofthe second column, the estimations obtained of the future paytrapping the predictive future payments. ments by calendar years are the same as those obtained with the Chain-Ladder deterministic method. The reason for this, as explained in the paper, is because when an over-distes the future payments by calendar years, the prediction errors and the persed Poisson distribution is assumed with the logarithmic link function, the estimation with the GLM coincides with that of the Chain-Ladder method. variation (i.e., Inthe proportion of and thethePEstandard over deviations the amount Table 3, the means of theestimate, predictivein distributions are calculated for the future payments by calendar years, the prediction errors (24), and the

the over-dispersed Poisson distribution using analytic formula to compute the 166 Cuadernos de Gestión Vol. 17 - Nº 2 (2017), pp. 157-174 ISSN: 1131 - 6837 f the MSE (17). In Table 2 we have the same computations for the Gamma

Eva Boj Del Val / Teresa Costa Cor

coefficients of variation for the over-dispersed Poisson distribution using bootstrap methodology with B = 1000 resamples. Table 4 shows the same computations for the Gamma distribution. In Table 5, calculations are shown for the present value of the future payments by calendar years as in (25), and the present value of the future payments by calendar years plus 25% of the prediction error using analytic formula as in (26), assuming in both cases the over-dispersed Poisson distribution. Table 6 provides the same computations for the Gamma distribution. In all cases it is assumed that there is a 1.5% fixed annual interest rate for the future nine years. In Table 7, calculations are shown for the present value of the future payments by calendar years as in (25); the present value of the future payments by calendar years using bootstrap methodology (with 1000 resamples) plus 25% of the prediction error as in (27); and the present value of the Value at Risk as in (28) with a confidence level of 99.5%, assuming the over-dispersed Poisson distribution. In Table 8, the same computations are shown for the Gamma distribution. In both cases a 1.5% fixed annual interest rate is assumed. Table 1 Prediction errors for over-dispersed Poisson using analytic formula2 Calendar year 10 11 12 13 14 15 16 17 18

Payment 5226535.8 4179394.4 3131667.5 2127271.9 1561878.9 1177743.7 744287.4 445521.3 86554.6

Source: Own elaboration.

Prediction error 747369.6 710144.6 644139.5 479125.6 404967.7 364294.9 294424.6 250986.8 108268.8

Coefficient of variation 14.30 % 16.99 % 20.57 % 22.52 % 25.93 % 30.93 % 39.56 % 56.34 % 125.09 %

Table 2

Prediction errors for Gamma using analytic formula3 Calendar year 10 11 12 13

Payment 5096855.3 4050001.5 3064407.7 2078010.5

Prediction error 847281.6 749549.8 628141.0 431885.8

Coefficient of variation 16.62 % 18.51 % 20.50 % 20.78 %

  Future payments by calendar years, prediction errors and coefficients of variation for the over-dispersed Poisson distribution using analytic formula. 2

  Future payments by calendar years, prediction errors and coefficients of variation for the Gamma distribution using analytic formula. 3

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167

Provisions for claims outstanding, incurred but not reported, with generalized linear models

14 15 16 17 18

1510392.7 1095402.7 692118.4 416539.9 82075.9

345880.7 292255.7 220057.8 181226.5 47918.1

22.90 % 26.68 % 31.79 % 43.51 % 58.38 %

Source: Own elaboration.

Table 3 Prediction errors for over-dispersed Poisson using bootstrap4 Calendar year

Mean payment

10 11 12 13 14 15 16 17 18

5262187.6 4206004.9 3153556.8 2139244.8 1562523.4 1178586.1 771451.6 455633.0 91579.0

Source: Own elaboration.

Standard deviation 748660.0 718618.2 653134.1 504395.7 408073.2 364102.8 302919.9 250906.3 104329.2

Prediction error

Coefficient of variation

756563.2 721067.3 649753.2 487995.9 411005.7 365547.7 292974.4 254458.2 107988.8

14.48 % 17.25 % 20.75 % 22.94 % 26.31 % 31.04 % 39.36 % 57.11 % 124.76 %

Table 4 Prediction errors for Gamma using bootstrap5

Calendar year

Mean payment

10 11 12 13 14 15 16 17 18

5096897.2 4050047.9 3064465.1 2078021.4 1510393.3 1095419.1 692130.4 416548.9 82080.9

Standard deviation 1017.5 1014.2 920.9 694.1 580.7 493.1 395.9 342.7 152.7

Prediction error

Coefficient of variation

652964.8 545647.9 434825.7 297581.3 233914.7 194252.2 142601.5 109441.6 26649.2

12.81 % 13.47 % 14.19 % 14.32 % 15.49 % 17.73 % 20.60 % 26.27 % 32.47 %

Source: Own elaboration.

  Future payments by calendar years, standard deviations, prediction errors and coefficients of variation for the over-dispersed Poisson distribution using bootstrap methodology with 1000 resamples. 4

  Future payments by calendar years, standard deviations, prediction errors and coefficients of variation for the Gamma distribution using bootstrap methodology with 1000 resamples. 5

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ISSN: 1131 - 6837

Eva Boj Del Val / Teresa Costa Cor

Table 5 Present values of the future payments for over-dispersed Poisson using analytic formula6 Calendar year 10 11 12 13 14 15 16 17 18 Present value

Deferral (in years) 1 2 3 4 5 6 7 8 9

Payment 5226535.8 4179394.4 3131667.5 2127271.9 1561878.9 1177743.7 744287.4 445521.3 86554.6 17873967

Payment + 0.25 Prediction error 5413378.2 4356930.6 3292702.4 2247053.3 1663120.8 1268817.4 817893.5 508268.0 113621.8 18820197

Source: Own elaboration.

Table 6 Present values of the future payments for Gamma using analytic formula7 Calendar year 10 11 12 13 14 15 16 17 18 Present value

Deferral (in years) 1 2 3 4 5 6 7 8 9

Payment 5096855.3 4050001.5 3064407.7 2078010.5 1510392.7 1095402.7 692118.4 416539.9 82075.9 17310125

Payment + 0.25 Prediction error 5308675.7 4237389.0 3221442.9 2185982.0 1596862.9 1168466.7 747132.9 461846.5 94055.4 18199962

Source: Own elaboration.

  Present values of the future payments by calendar years and of the future payments by calendar years plus 25% of the prediction error for the over-dispersed Poisson distribution using analytic formula and assuming a 1.5% fixed annual interest rate. 6

  Present values of the future payments by calendar years and of the future payments by calendar years plus 25% of the prediction error for the Gamma distribution using analytic formula and assuming a 1.5% fixed annual interest rate. 7

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169

Provisions for claims outstanding, incurred but not reported, with generalized linear models

Table 7 Present values of the future payments and of the Value at Risk for over-dispersed Poisson using bootstrap8 Calendar year 10 11 12 13 14 15 16 17 18 Present value

Deferral (in years) 1 2 3 4 5 6 7 8 9

Source: Own elaboration.

Payment 5226535.8 4179394.4 3131667.5 2127271.9 1561878.9 1177743.7 744287.4 445521.3 86554.6 17873967

Payment + 0.25 Prediction Error 5415676.6 4359661.3 3294105.8 2249270.9 1664630.3 1269130.6 817531.0 509135.8 113551.8 18830614

VaR99.5 7417055.0 6364764.7 5207534.8 3682095.3 2735533.8 2209257.2 1841310.7 1262432.7 473412.3 29688278

Table 8

Present values of the future payments and of the Value at Risk for Gamma using bootstrap9 Calendar year 10 11 12 13 14 15 16 17 18 Present Value

Deferral (in years) 1 2 3 4 5 6 7 8 9

Payment 5096855.3 4050001.5 3064407.7 2078010.5 1510392.7 1095402.7 692118.4 416539.9 82075.9 17310125

Payment + 0.25 Prediction Error 5260096.5 4186413.5 3173114.1 2152405.9 1568871.3 1143965.8 727768.8 443900.3 88738.2 17938348

VaR99.5 5099652.6 4052656.1 3067038.9 2079754.3 1511867.6 1096662.3 693145.23 417478.6 82457.2 17324230

Source: Own elaboration.

It may be observed in Tables 1 to 4 that when an analysis is made of the future payments by calendar years, in general and for this dataset, lower coefficients of variation are   Present values of the future payments by calendar years, of the future payments by calendar years plus 25% of the prediction error and of the Value at Risk with a confidence level of the 99.5% for the over-dispersed Poisson distribution using bootstrap methodology with 1000 resamples and assuming a 1.5% fixed annual interest rate. 8

  Present values of the future payments by calendar years, of the future payments by calendar years plus 25% of the prediction error and of the Value at Risk with a confidence level of the 99.5% for the Gamma distribution using bootstrap methodology with 1000 resamples and assuming a 1.5% fixed annual interest rate. 9

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Eva Boj Del Val / Teresa Costa Cor

obtained for Ɵ =2 , the case of the Gamma distribution, if the results are compared with those of the over-dispersed Poisson where Ɵ =1. And, specifically, it becomes more evident when we compare Tables 3 and 4, the case in which an estimation is made of the real distribution by bootstrap. The same pattern may be observed in the results for accident year provisions and total provisions in Tables 1 and 2 of England and Verrall (1999). As may be seen, it preferable to use the bootstrap estimations (21) to (24) of the PE instead of the analytic estimation formulas (14) to (17), because in the run-off triangle there is usually a small dataset and the data do not always follow the hypothesis assumed in the GLM. It is thus preferable to simulate the real distribution of our portfolio than apply the theoretical formulations of the distribution. This will help to obtain more accurate information on future losses incurred by insurance companies. It may be observed in Tables 5 to 8 that the amount of the “today real provision”, i.e. the present value, depends on the added risk margin. While the lower amount always represents the simple present value the simple present value (25), the amount will vary depending on the added percentage δ for formulas (26) and (27), and depending on the confidence level α for (28). As might be expected, higher δ (or α ) will obtain higher present values, as will be seen below. Some additional present values are calculated with formulas (27) and (28) for the Gamma model. First, some percentages δ are used to complete the results of Table 8, with δ={0.1,0.4, 0.5, 0.95}, and the corresponding present values of 17561414, 18315283, 18566572 and 19697375, respectively, in monetary units. Second, the present values are calculated with formula (28) and taking into account the confidence levels for the VaR: α={75%, 85%, 90%, 95%} The corresponding present values are 17313858, 17315730, 17317046 and 17319257, respectively, in monetary units. Finally, Figure 3 shows the predictive distributions of the future payments by calendar years for the calendar years 13 and 14 to illustrate graphically the empirical results of the bootstrap process. Calculations of the percentiles for the VaR or the SE of formula (24) are drawn from these distributions. Figure 3

Histograms of the predictive distribution of the future payments with Gamma10 Histogram of payments[, cal]

6e-04 Density

0e+00

0e+00

1e-04

2e-04

4e-04

4e-04 3e-04 2e-04

Density

5e-04

6e-04

8e-04

7e-04

Histogram of payments[, cal]

1509000

1510000

1511000

payments[, cal]

1512000

1094000

1094500

1095000

1095500

1096000

1096500

1097000

payments[, cal]

Source: Own elaboration.   Histograms of the predictive distribution of the future payments for the calendar years 14 and 15 for the data of Taylor and Ashe (1983) assuming a GLM with Gamma distribution. 10

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Provisions for claims outstanding, incurred but not reported, with generalized linear models

4. CONCLUDING REMARKS This paper has deduced the formulas (17) and (24) related to the PE for the future payments by calendar years (3) for the GLM in the general case of the parametric distribution family (4) assuming the logarithmic link. Studying the use of calendar years for the IBNR provisions problem has provided results that enable the actuary to take decisions regarding the best estimate of the technical provisions and the risk margin, and, therefore, regarding the financial inversions of the SCR of the insurance companies in the current context of Solvency II. As regards the best estimate, this paper proposes that the present value (25) be calculated directly by taking account of the time value of money. Additionally, it is proposed that a risk margin be included by adding a percentage of the prediction error of the future payments by calendar years, as seen in formulas (26) and (27) or by calculating the VaR of the predictive distribution of the future payments by calendar years, as seen formula (28). While the analysis in the paper is illustrated with the triangle of Taylor and Ashe (1983), the theoretical and practical results complement the study of the use of GLM in the problem of provisions in other actuarial studies where the analysis is done only with the accident year provisions and total provisions (1) and (2), such as those of Haberman and Renshaw (1996), England and Verrall (1999), England (2002), England and Verrall (2002, 2006) and Kaas et al. (2008). In addition, there has been further analysis of the future payments by calendar years (3) for the Poisson case, conducted in Boj et al. (2014) for the general parametric family (4). 5. ACKNOWLEDGEMENTS Authors have been supported by the Spanish Ministerio de Educación y Ciencia under grant MTM2010-17323 and by the Generalitat de Catalunya, AGAUR under grant 2014SGR152. 6. REFERENCES Albarrán, I. and Alonso, P., 2010. Métodos estocásticos de estimaciones de las provisiones técnicas en el marco de Solvencia II. Cuadernos de la Fundación MAPFRE, 158. Madrid: Fundación MAPFRE Estudios. Boj, E., Claramunt, M. M. and Fortiana, J., 2004. Análisis multivariante aplicado a la selección de factores de riesgo en la tarificación. Cuadernos de la Fundación MAPFRE, 88. Madrid: Fundación MAPFRE Estudios. Boj, E. and Costa, T., 2014. Modelo lineal generalizado y cálculo de la provisión técnica. Depósito digital de la Universidad de Barcelona. Colección de objetos y materiales docentes (OMADO). http://hdl.handle.net/2445/49068 Boj, E., Costa, T. and Espejo, J., 2014. Provisiones técnicas por años de calendario mediante modelo lineal generalizado. Una aplicación con RExcel. Anales del Instituto de Actuarios Españoles, Tercera Época, 20, 83–116.

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Efron, B. and Tibshirani, J., 1998. An Introduction to the bootstrap. New York: Chapman & Hall/CRC. England, P. D. and Verrall, R. J., 1999. Analytic and bootstrap estimates of prediction errors in claims reserving. Insurance: Mathematics and Economics, 25, 281–293. England, P. D., 2002. Addendum to “Analytic and bootstrap estimates of prediction errors in claim reserving. Insurance: Mathematics and Economics, 31, 461–466. England, P. D. and Verrall, R. J., 2002. Stochastic claims reserving in general insurance (with discussion). British Actuarial Journal, 8, 443–544. England, P. D. and Verrall, R. J., 2006. Predictive distributions of outstanding liabilities in general insurance. Annals of Actuarial Science, 1 (II), 221–270. Espejo, J., Boj, E. and Costa, T., 2014. Una aplicación de RExcel para el cálculo de provisiones técnicas con modelo lineal generalizado. Depósito digital de la Universidad de Barcelona. Colección de Investigación- Software. http://hdl.handle.net/2445/56230 Haberman, S. and Renshaw, A. E., 1996. Generalized linear models and actuarial science. Journal of the Royal Statistical Society. Series D (The Statistician), 45 (4), 407–436. Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M., 2008. Modern actuarial risk theory: using R. Second edition. Heidelberg: Springer-Verlag. Moreno, F. P., 2013. Jornada sobre las Directrices de EIOPA de preparación a Solvencia II. Dirección General de Seguros y Fondos de Pensiones. http://www.dgsfp.mineco. es/sector/documentos/Jornada%20Directrices%20EIOPA%20de%20preparacion%20 Solvencia%20II_10-12-2013/Fernando%20Moreno_Jornada%20Directrices%20EIOPA.%20DGSFP-UNESPA.pdf McCullagh, P. and Nelder, J., 1989. Generalized linear models. Second edition. Londres: Chapman and Hall. Parlamento Europeo y Consejo de la Unión Europea, 2009. Directiva 2009/138/CE del Parlamento Europeo y del Consejo, de 25 de noviembre de 2009. Diario Oficial de la Unión Europea, L 335, 1–155. http://eur-lex.europa.eu/LexUriServ/LexUriServ. do?uri=OJ:L:2009:335:0001:0155:es:PDF R Development Core Team, 2016. R: a language and environment for statistical computing. Vienna. Austria. http://www.R-project.org/ Renshaw, A.E., 1989. Chain ladder and interactive modelling (claims reserving and GLIM). Journal of the Institute of Actuaries, 116 (III), 559–587. Renshaw, A. E., 1994. On the second moment properties and the implementation of certain GLIM based stochastic claims reserving models. Actuarial Research Paper, No. 65. Department of Actuarial Science and Statistics, City University, London. Taylor, G. and Ashe, F.R., 1983. Second Moments of Estimates of Outstanding Claims. Journal of Econometrics, 23, 37–61.

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