SCENARIO ANALYSIS OF NON-LIFE TECHNICAL PROVISIONS: THE CZECH INSURANCE MARKET CASE STUDY

Transcription

1 The 10 h Inernaional Days of Saisics and Economics Prague Sepember SCENARIO ANALYSIS OF NON-LIFE TECHNICAL PROVISIONS: THE CZECH INSURANCE MARKET CASE STUDY Radek Hendrych Tomáš Cipra Absrac This conribuion inroduces and discusses a complex economeric model of non-life echnical provisions based on he Czech non-life insurance marke daa. Seleced economic-acuarial relaions among given insurance variables are described by means of he dynamic linear sysem of simulaneous equaions. In paricular he provision for ousanding claims he provision for unearned premium he oher (marginal) echnical provisions he acquisiion and adminisraive expenses he benefi expenses and heir muual ineracions are sudied in greaer deail. The suggesed economeric model is esimaed saisically verified and inerpreed wih special regard o he acuarial poin of view. The proposed modelling scheme can be furher employed for prognosing he considered non-life echnical provisions. Paricularly hese forecass can be aken ino accoun by non-life insurance companies in heir inernal calculaions (e.g. for financial planning purposes for verifying he sufficiency of non-life echnical provisions or for liabiliy adequacy ess LAT) or by an insurance regulaor (e.g. for performing sress ess). Alernaively his approach migh moivae developmen of an inernal model applicable in he Solvency II framework. Boh deerminisic and randomly generaed scenarios are analysed; hey are capable of delivering relevan oupus for formulaing various recommendaions and conclusions. Key words: economeric sysem of simulaneous equaions non-life insurance scenario analysis Solvency II echnical provisions. JEL Code: C30 C32 C39. Inroducion Technical provisions are undoubedly key insurance variables. They represen he amoun of money mainained by an insurance company needed o mee all is fuure liabiliies owards he cliens (under a cerain measuremen of presen obligaions). The echnical provisions mus be sufficien o cover all hese anicipaed commimens a all imes. I should be ensured 554

2 The 10 h Inernaional Days of Saisics and Economics Prague Sepember by various regulaory principles inroduced e.g. by he Solvency I or Solvency II framework. The sufficiency of echnical provisions is coninuously moniored by naional regulaors and oher supervisory auhoriies (e.g. in he Czech case by he Czech Naional Bank). Generally one disinguishes beween he life and he non-life echnical provisions (according o he underlying insurance conracs). All he provisions are regularly recalculaed verified and repored in he annual (quarerly monhly) balance shees on he liabiliy side. There exis several exacly specified caegories of he echnical provisions given by he legal framework of each counry. The presen paper inroduces and discusses a complex economeric model of he mos significan non-life echnical provisions based on he Czech non-life insurance marke daa. Paricularly he dynamic linear sysem of simulaneous equaions is employed in order o describe differen ineracions among he seleced economic-acuarial insurance variables. Namely he provision for ousanding claims he provision for unearned premium and he oher (marginal) echnical provisions are sudied in greaer deail. Afer saisical verificaion he suggesed modelling scheme can be furher applied for prognosing he considered non-life echnical provisions. Paricularly hese forecass can be aken ino accoun by non-life insurance companies in heir inernal calculaions (e.g. for financial planning purposes for verifying he sufficiency of non-life echnical provisions or for liabiliy adequacy ess) and by insurance regulaors (e.g. for performing sress ess or for verifying a prudency level). Alernaively hey migh be useful for formulaing an inernal model applicable in he Solvency II framework. Differen aspecs of economeric models which invesigaed cash flows or echnical provisions in he life insurance were discussed in various academically or pracically oriened works (Feilmeier & Junker 1982; Cipra 1998; Baranoff 2007; Hendrych 2011; Hendrych & Cipra 2015). However o he bes of our knowledge here has no been published any complex economeric model examining non-life echnical provisions (or even based on he Czech non-life insurance marke daa). On he conrary several paricular non-life echnical provisions and relaed issues have been analysed in he lieraure from he saisical or acuarial poins of view (Dahms 2012; Hürlimann 2009; and many ohers). 1 Model of non-life echnical provisions for he Czech insurance marke As was menioned above we shall concenrae on he following hree key caegories of he non-life echnical provisions: (i) he provision for ousanding claims (ii) he provision for 555

3 The 10 h Inernaional Days of Saisics and Economics Prague Sepember unearned premium and (iii) he oher (marginal) provisions (i.e. he sum of all oher marginal non-life echnical provisions represening only a minoriy of he oal volume of all non-life provisions). The provision for ousanding claims is he esimaed value of (fuure) compensaions for policyholders and policy beneficiaries. More specifically i involves he provision for IBNR (Incurred Bu No Repored) claims and he provision for RBNS (Repored Bu No Seled) claims. The provision for unearned premium corresponds o such a par of he wrien premium which relaes o fuure accouning periods. The oher (marginal) provisions involve e.g. he provision for bonuses and sales. For simpliciy le us consider only relaionships arising from he quarerly published summary balance shees (he liabiliy side) of all he Czech non-life insurance companies. Ineracions among hese accouning daa migh be furher invesigaed hrough economeric modelling conceps based on he acuarial heory (Cipra 2010). Noneheless one could possibly exend he inroduced daase by including oher insurance or economic variables. In paricular we assume he following non-life insurance variables: CS - he claims expenses in ime (in housands of CZK) EAC - he acquisiion expenses in ime (in housands of CZK) EAD - he adminisraive expenses in ime (in housands of CZK) EB - he exising business in ime (i.e. he number of exising non-life insurance conracs in pieces) NRC - he number of repored claims in ime (in pieces) TPC - he echnical provision for ousanding claims in ime (in housands of CZK) TPO - he oher non-life echnical provisions in ime (in housands of CZK) TPP - he echnical provision for unearned premium in ime (in housands of CZK) TPT - he oal repored non-life echnical provisions in ime (in housands of CZK) defined as TPT = TPC + TPO + TPP WP - he wrien premium in ime (in housands of CZK) = 1 28 ( = 1 refers o he Q and T = 28 o he Q3 2015). The quarerly based daase was obained from he quarerly repored summary balance shees (he liabiliy side) published by he Czech Naional Bank (ČNB) on he regular basis. 1 We can proceed o he formulaion of he dynamic linear economeric sysem of simulaneous equaions which describe relaionships among he paricular non-life insurance marke variables lised above. The considered modelling scheme simulaneously explains he casual relaions among more han one dependen variable. Therefore i enables o model he analysed phenomenon in greaer complexiy. To be more precise i reflecs muual 1 rerieved March

4 The 10 h Inernaional Days of Saisics and Economics Prague Sepember ineracions among he sudied insurance variables hrough he following complex modelling srucure (by assuming non-rivially correlaed residuals). We have considered he following simulaneous equaions model (for = 2 T): TPP TPC [ Q2] 1 22 [ Q2] 1 31 [ Q3] 1 32 [ Q3] 1 41 [ Q4] 1 42 [ Q4] TPP TPO 13 33TPO 1 53 EAD 43EAC 1 73NRC EAC EAC 14 14TPP 1 94EB 1 64WP 1 74NRC EAD EAD WP 55EAD 1 85EB 35TPO 1 CS CS 16 66CS 1 76NRC 56WP 86EB TPT TPC TPO TPP TPC WP 1 61 CS 62 TPP 1 82 TPO EB TPC (1) where 1 [ ] denoes he binary indicaor of he even Δ sands for he firs difference operaor and he superscrip indicaes ha he variable has been seasonally adjused (by using a simple rouine muliplicaive seasonal facor mehod). Moreover βs γs and ϕs wih various indices represen he parameers of he model and εs denoe he sochasic error erms. The considered dynamic economeric sysem (1) includes six sochasic equaions (i.e. he equaions wih he sochasic residual erms) and one deerminisic equaion (i.e. he ideniy for he oal provisions). In he suggesed model he inercep he seasonal dummies and he variables EB NRC WP (and hus also he lagged EB and WP ) are assumed o be sricly exogenous (i.e. uncorrelaed wih residual componens a all imes); hese variables ener ino he sysem from ouside. Such a paricular choice of exogenous variables seems o be pragmaic wih regard o he apparen exernal characer of hese variables. Furhermore he lagged endogenous variables CS EAC EAD TPC TPO and TPP are supposed o be predeermined (i.e. uncorrelaed wih curren and fuure residual disurbances); hey are fully deermined by he sysem (1) in ime -1. Noe ha each equaion in (1) saisfies he necessary condiion of idenificaion (Greene 2003; Cipra 2013). To be precise he model (1) (afer ignoring he las deerminisic equaion) follows he srucural form of he dynamic sysem of linear simulaneous equaions (Greene 2003): T T T T T y Γ y 1Φ1 x Β ε 0 (2) where: y x ε T ( TPP TPC TPO EAC EAD CS ) ( WP WP NRC EB EB ( [ Q2] [ Q3] TPP TPC [ Q4] TPO EAC 1 EAD CS T ) 1 ) T (3) 557

5 The 10 h Inernaional Days of Saisics and Economics Prague Sepember where y denoes he (6 1) vecor of endogenous variables x is he (9 1) vecor of sricly exogenous variables and ε sands for he (6 1) sochasic vecor of he srucural error erms (everyhing for all given imes ). Poin ou ha he indices of he parameers β γ and ϕ in (1) refer o he corresponding elemens of he parameer marices B (for sricly exogenous variables) Γ (for endogenous variables) and Φ 1 (for predeermined variables wih he ime lag 1) respecively. Moreover some a priori consrains mus be assumed namely β 61 = -β 51 γ 53 = -ϕ 53 β 65 = -β 55. Oher elemens of he marices of parameers are equal o zero. Several oher assumpions are usually inroduced: (A1) (ε ) = 0 for all var(ε ) = (ε ε T ) = Σ is a symmeric posiive definie marix for all and cov(ε s ε ) = (ε s ε T ) = 0 for all s ; (A2) ((y T -1 x T ) T (y T -1 x T )) = Q is a finie symmeric posiive definie marix for all ; (A3) he marix Γ conaining he parameers concerning (exclusively) endogenous variables is inverible wih elemens -1 on is diagonal (Greene 2003; Lükepohl 2005). To esimae he unknown parameers of he proposed sysem of economeric equaions (1) he hree-sage leas squares mehod (3SLS) migh be considered. This full informaion esimaion echnique is a special case of he generalized mehod of momens GMM exploiing all informaion available in he considered sysem. I guaranees suiable properies (under general assumpions). Namely he 3SLS esimaes are consisen asympoically normally disribued and asympoically efficien (Greene 2003). Table 1 presens he 3SLS esimaes of he model (1) joinly wih he esimaed sandard errors. Tab. 1: The 3SLS esimaes of he parameers of he model (1) Eq. for TPP Eq. for TPC Eq. for TPO Eq. for EAC Eq. for EAD Eq. for CS Par. Es. (Sd. Err.) β (451313) β (201651) β (194997) β (199541) β ( ) ϕ ( ) Par. Es. (Sd. Err.) β ( ) β (404924) β (405270) β (418687) β ( ) ϕ ( ) ϕ ( ) Source: Auhors (by EViews 8.0). Par. Es. (Sd. Err.) β ( ) β ( ) γ ( ) ϕ ( ) ϕ ( ) Par. Es. (Sd. Err.) β (565896) β ( ) β ( ) β ( ) Par. Es. (Sd. Err.) β (517488) β ( ) β ( ) ϕ ( ) ϕ ( ) ϕ ( ) Par. Es. (Sd. Err.) β ( ) β ( ) β ( ) β ( ) ϕ ( ) 558

6 Thousands of CZK Thousands of CZK Thousands of CZK Thousands of CZK Thousands of CZK Thousands of CZK The 10 h Inernaional Days of Saisics and Economics Prague Sepember One can see ha he esimaed model fis he daa suiably (see Figure 1). Moreover i demonsraes is saisical adequacy (refer o he assumpions lised above). For insance he sample correlaion marix of he esimaed 3SLS residuals delivers several relaively high correlaions (see he oupus displayed in Table 2). The Pormaneau ess and he empirical auocorrelaion funcions of he 3SLS residuals do no indicae he presence of residual auocorrelaions (Lükepohl 2005). Furhermore he join Jarque-Bera es canno rejec he mulivariae normaliy of he 3SLS residuals. Finally neiher he Hausman specificaion es comparing he wo and he hree-sage leas squares esimaes nor he Sargan es for overidenifying resricions rejec he proper model specificaion (Greene 2003; Cipra 2013). Noe ha he sandard 5% significance level has been applied. Fig. 1: The observed endogenous variables wih heir fied counerpars EAC EAD CS TPP TPO TPC Daa Fied Source: Auhors (by EViews 8.0). Tab. 2: The esimaed 3SLS residual correlaion marix TPP TPC TPO EAC EAD CS TPP TPC TPO EAC EAD CS Source: Auhors (by EViews 8.0). 559

7 The 10 h Inernaional Days of Saisics and Economics Prague Sepember Scenario analysis From he pracical poin of view prognosing he discussed non-life echnical provisions migh be ruly useful. For insance one could employ randomly generaed scenarios for sress esing he sufficiency of he paricular non-life echnical provisions by applying he modelling scheme (1). Consequenly his approach migh moivae developmen of an inernal model for non-life insurance companies which migh be inroduced in he Solvency II framework in order o prescribe he solvency capial requiremen SCR. Moreover i could be used by any supervisory auhoriy o se or revise he prudency level effeciveness. To illusrae he key idea of he sress esing discussed above wo differen sress scenarios for he exogenous variables EB NRC and WP are considered. Boh scenarios are raher pessimisic since one usually ess he sufficiency (or he prudency level effeciveness) of he provisions under (exremely) unfavourable condiions. The firs scenario is formulaed as follows: he number of exising non-life insurance conracs EB decreases by 3% each quarer he number of repored claims (seasonally adjused) NRC increases by 3% each quarer and he wrien premium (seasonally adjused) WP decreases by 5% each quarer = i.e. from Q o Q Noe ha he seasonally adjused wrien premium decreases faser han he exising business. The second underlying scenario follows analogous expecaions as before only wih he minor changes: he number of exising non-life insurance conracs EB decreases by 5% each quarer he number of repored claims (seasonally adjused) NRC increases by 3% each quarer and he wrien premium (seasonally adjused) WP decreases by 3% each quarer = i.e. from Q o Q Here he seasonally adjused wrien premium decreases slower han he exising business porfolio. Acceping hese wo underlying sress scenarios we have furher employed he suggesed modelling scheme (1). Paricularly we have calculaed realizaions (forecass) of all he endogenous variables for each given sress scenario and he whole predicion horizon by using he prescribed sricly exogenous variables EB NRC WP and randomly generaed vecor error erms ε ; refer o (1) and (2)-(3). We have applied he sandard residual boosrap mehod (Hendrych & Cipra 2015). In paricular i means ha he mulivariae disribuion of he disurbances ε has been deermined by he (cenred) empirical residuals compued during he realized 3SLS esimaion. See Table 1. All compuaions were performed in EViews version 8.0 by auhors calculaion procedures. 560

8 Thousands of CZK Thousands of CZK Thousands of CZK Thousands of CZK The 10 h Inernaional Days of Saisics and Economics Prague Sepember The resuls of simulaion sudy are summarized in Figure 2 and Figure 3. A firs sigh one can see ha boh scenarios have an impac on all he non-life echnical provisions; all he presened provisions have significanly decreased. Neverheless i corresponds o raional anicipaions for he considered scenarios. Furhermore one can idenify ha he differences beween boh assumed sress scenarios are ruly significan. Especially le us compare he resuls of prognosing he provision for ousanding claims (TPC) in Figure 2. The mean forecased provision is subsanially lower for he second underlying sress scenario. However i perfecly reflecs he cumulaive impac of he sricly exogenous variables on he paricular non-life echnical provisions (consul he model (1)). From he analysis of he cumulaive impac which goes beyond he scope of his paper one migh presume ha he number of exising non-life insurance conracs (EB) and he number of repored claims (NRC ) considerably influence he oal non-life echnical provisions (see Figures 2 and 3). From he acuarial viewpoin he generaed projecions provide several useful inerpreaions (see Figure 3). For insance we can observe ha he empirical probabiliy ha he oal non-life echnical provision TPT in Q will be less or equal o he wo hirds of he Q level is only 0.05% for he firs sress scenario bu 97.46% for he second sress scenario respecively. The subsanial difference is apparen (refer o he commens above). Such oupus could be furher employed e.g. by he regulaor for esing he sufficiency of he oal provisions or for calibraing he prudency level. Fig. 2: The resuls of prognosing he paricular non-life echnical provisions TPP TPC TPO TPT Scenario 1 (mean) Scenario 1 (mean +/- 2 s.d.) Scenario 2 (mean) Scenario 2 (mean +/- 2 s.d.) Source: Auhors (by EViews 8.0). 561

9 The 10 h Inernaional Days of Saisics and Economics Prague Sepember Fig. 3: The resuls of prognosing he oal non-life echnical provisions (Q4 2017) Source: Auhors (by EViews 8.0). Conclusion The conribuion presened he complex economeric model of he key non-life echnical provisions (and oher imporan acuarial variables) based on he Czech non-life insurance marke daa. In paricular he economeric sysem of dynamic linear simulaneous equaions was employed in order o describe economic-acuarial relaionships wihin he quarerly published summary balance shees of he Czech non-life insurers. The esimaed modelling scheme was furher employed in he sress esing analysis. I migh be used e.g. for financial planning purposes or for esing he sufficiency of he non-life echnical provisions. To illusrae he main idea of he discussed sress esing wo underlying (exremely) unfavourable scenarios of fuure developmen based on he prescribed sricly exogenous variables were invesigaed in deail. The simulaion resuls evaluaed by using he residual boosrap corresponded o pragmaic anicipaions (i.e. he decreasing endency of all he paricular non-life echnical provisions). Consequenly one could idenify ha he number of exising non-life insurance conracs and he number of repored claims influenced he oal non-life echnical provisions subsanially. Acknowledgmen This work was suppored by he gran GA P402/12/G097. References Baranoff E. G. Papadopoulos S. & Sager T. W. (2007). Capial and risk revisied: A srucural equaion model approach for life insurers. Journal of Risk and Insurance 74(3)

10 The 10 h Inernaional Days of Saisics and Economics Prague Sepember Cipra T. (1998). Economeric analysis of cash-flows in a life insurance company. Pojisné rozpravy 1998(3) Cipra T. (2010). Financial and insurance formulas. New York: Physica-Verlag/Springer. Cipra T. (2013). Financial economerics. Prague: Ekopress (in Czech). ČNB ARAD. Rerieved March from Dahms R. (2012). Linear sochasic reserving mehods. ASTIN Bullein Feilmeier M. & Junker M. (1982). Besandsorieniere Bilanzprojekion eines Lebensversicherungsunernehmens. Bläer Deusche Gesellschaf für Versicherungsmahemaik 15(3) Greene W. H. (2003). Economeric analysis. New Jersey: Prenice Hall. Hendrych R. (2011). Economeric sysem of simulaneous equaions in life insurance. In Dlouhý M. Skočdopolová V. (Eds.) Proceedings of he 29h Inernaional Conference on Mahemaical Mehods in Economics 2011 (pp ). Prague: Professional Publishing. Hendrych R. & Cipra T. (2015). Economeric model of he Czech life insurance marke. Prague Economic Papers 24(2) Hürlimann W. (2009). Credible loss raio claims reserves: he Benkander Neuhaus and Mack mehods revisied. ASTIN Bullein Lükepohl H. (2005). New inroducion o muliple ime series analysis. New York: Springer. Conac Radek Hendrych Charles Universiy in Prague Faculy of Mahemaics and Physics Deparmen of Probabiliy and Mahemaical Saisics Sokolovská Prague 8 Czech Republic hendrych@karlin.mff.cuni.cz Tomáš Cipra Charles Universiy in Prague Faculy of Mahemaics and Physics Deparmen of Probabiliy and Mahemaical Saisics Sokolovská Prague 8 Czech Republic cipra@karlin.mff.cuni.cz 563