SOME FORMS OF RISK REGULATION IN SOLVENCY II

Transcription

1 SOME FORMS OF RISK REGULATION IN SOLVENCY II Tomáš Cpra, 1 Rade Hendrych * Abstract The contrbuton deals wth the rs regulaton n the framewor of Solvency II, whch s the new regulatory system n nsurance vald n majorty of the EU countres snce It concentrates on the underwrtng rs (n partcular, on the reserve rs) and on the counterparty default rs (.e. manly on the rensurers default rs), snce such rss are crucal for nsurance actvtes. Varous actuaral approaches to the underwrtng rs appled by subjects respected by nsurance regulators and supervsors are surveyed. Moreover, one of them suggests by means of a real data example a smplfed approach to the reserve rs, whch may be apprecated n practce just for ts smplcty. As to the counterparty default rs, the paper presents a method that can be sutable when the rensurers form a small group of heterogeneous subjects mperlled by a common shoc as a fnancal crss or a natural catastrophe; ths methodologcal approach s also demonstrated by a numercal example. Keywords: rs regulaton, Solvency II, underwrtng rs, reserve rs, techncal provsons, counterparty default rs, actuaral methods JEL Classfcaton: G22, G28, C02 1. Introducton The preparatory wor for Solvency II goes on snce the begnnng of the mllennum. Solvency II has been the new regulatory system n nsurance vald n majorty of the EU countres snce It s an nnovatve approach n the nsurance regulaton, whch allows to quantfy the correspondng captal requrements really accordng to underlyng rss n partcular nsurance companes. It regulates manly by means of solvency captal requrements all types of rss common n the modern nsurance ndustry (underwrtng rs, maret rs, credt rs, lqudty rs, operatonal rs and others). Moreover, t copes the successful organzaton to three pllars from ban regulatory systems Basel II and III (see BIS, 2010). The followng ponts characterze the regulatory system Solvency II n a nutshell (see e.g. Cpra, 2015; Sandström, 2011): *1 Tomáš Cpra, Department of Probablty and Mathematcal Statstcs, Faculty of Mathematcs and Physcs, Charles Unversty n Prague, Prague, Czech Republc (cpra@arln.mff.cun.cz); Rade Hendrych, Department of Probablty and Mathematcal Statstcs, Faculty of Mathematcs and Physcs, Charles Unversty n Prague, Prague, Czech Republc (hendrych@arln.mff.cun.cz). Ths wor was supported by the Czech Scence Foundaton (the Grant No. GA P402/12/G097: DYME Dynamc Models n Economcs ). 722 Prague Economc Papers, 2017, 26(6), ,

2 Tmng: The Solvency II regme has been vald n majorty of EU countres snce It ncludes transtonal arrangements n a number of areas tmed up untl Jursdcton: Solvency II s an oblgatory regme for most nsurers and rensurers wth ther head offce n the EU, ncludng mutuals and companes n run-off unless ther annual premum ncome s less than a prescrbed sum dependng on the natonal legslatve. However, Solvency II should be appled n a way whch s proportonate to the nature, scale and complexty of the nsurer. Rs-based captal: Solvency II s a rs-based captal regme, smlar n concept to Basel II, based on three pllars. Pllar 1 s a maret consstent calculaton of nsurance labltes and rs-based calculaton of captal. Pllar 2 s a supervsory revew process. Pllar 3 ncludes reportng and transparency requrements. Captal calculaton: The calculaton of nsurance labltes under Solvency II denoted as techncal provsons ncludes a best estmate of labltes and a rs margn (where the lablty s not approprately hedged). Captal s nown as own funds and s dvded nto three Ters (1 3) reflectng the ablty to absorb losses. The rs based captal requrement, the solvency captal requrement SCR, s calculated usng ether () a standard formula; or () an nternal model approved by the nsurer s supervsor, or a mxture of both. The standard formula wll cover the underwrtng rs, maret rs, credt rs and operatonal rs n a formulac way. The calculaton wll be calbrated to ensure a 99.5 per cent confdence level over a one year perod (the BBB ratng accordng to Standard and Poor s should correspond to ths rs level). Breach of the SCR results n supervsory nterventon desgned to restore the SCR level of captal. There s also the mnmum captal requrement MCR set at a lower threshold (e.g. about 85 per cent confdence level). The MCR should not be less than 25 per cent of the SCR. Breach of the MCR (unless corrected qucly) leads to a loss of the nsurer s authorsaton. Assets: The rules requrng nvestment of assets n a lst of admssble assets and the counterparty and asset lmts contaned n the prevous regulatory regme Solvency I regme for EU nsurers are replaced by the prudent person prncple. Ths wll place greater responsblty for nvestment decsons on the nsurer ncludng sgnfcantly ncreased reportng requrements n relaton to assets. Groups: In the case of groups one can calculate the group SCR requrement based on a consoldated bass. Ths can reflect dversfcaton but there may be problems, e.g. non-eu subsdares should calculate the solvency on a Solvency II bass wth exceptons of some countres outsde of the EU (e.g. Swtzerland). Rs mtgaton: Solvency II also ncludes rules relatng to rs mtgaton technques and rensurance. For non-eu based rensurers they should have credt of equvalence accordng to ther countres (e.g. EIOPA has assessed Bermuda, Swtzerland and Japan as equvalent for the purposes of rensurance). Prague Economc Papers, 2017, 26(6), , 723

3 Fgure 1 Rss Covered by Solvency II Structured accordng to the Standard Formula (SLT denotes Smlar to Lfe Technques,.e. rs components of health nsurance smlar from the techncal pont of vew to lfe nsurance, and Non-SLT n the opposte case; other symbols and terms are descrbed n the text above) SCR Adjustment for techncal provsons Basc SCR Operatonal rs Rs module: Lfe underwrtng Rs module: Non-lfe underwrtng Rs module: Health underwrtng Rs module: Counterparty default Rs module: Maret rs Rs module: Intangble assets Mortalty Premum and reserve SLT Non-SLT Catastrophe Interest rate Longevty Lapse Mortalty Premum and reserve Equty Dsablty Morbdty Catastrophe Longevty Lapse Property Lapse Dsablty Morbdty Spread rs Expenses Lapse Currency Revson Expenses Maret rs concentraton Catastrophe Revson Countercyclcal premum Source: EIOPA Source: EIOPA 724 Prague Economc Papers, 2017, 26(6), ,

4 Governance: Solvency II mposes formal governance requrements establshng roles such as a rs management functon, an audt functon, a complance functon and an actuaral functon. In partcular, the rs management should be set out n an Own Rs and Solvency Assessment (ORSA). The ORSA should nclude a rs-based assessment of the nsurer s solvency needs based on ts busness and ts own rs appette (the supervsor wll revew t n the framewor of the Pllar 2). Governance deals also wth the outsourcng n the nsurance company. Reportng: Pllar 3 requres an ntensve reportng nsurer s actvty ncludng a prvate regular supervsory report, a publc annual report called the solvency and fnancal condton report (SFCR) and quanttatve reportng templates (QRT). Supervson: The role of the nsurance supervsor s very mportant and the success of Solvency II depends on ts mplementaton by nsurance supervsors n a consstent manner. It s ntended that n group stuatons the relevant supervsors led by the group supervsor wll cooperate and act n a coordnated manner. To present the whole spectrum of rss taen nto account by Solvency II one can use the survey n Fgure 1 structurng the partcular rss accordng to the standard formula. Ths paper concentrates only on the underwrtng rs (n partcular, on the reserve rs) and on the counterparty default rs (.e. manly on the rensurers default rs), snce such rss are crucal for nsurance actvtes. Secton 2 surveys from the actuaral pont of vew varous approaches to the underwrtng rs suggested by subjects respected by nsurance regulators and supervsors; such a survey, not summarzed n lterature so far, can be useful for the European actuaral practce. Sectons 3 and 4 concentrate on the reserve rs ncludng the role of techncal provsons n Solvency II. Partcularly, n Secton 4 one also suggests by means of a real data example a smplfed approach to the reserve rs, whch may be apprecated n practce just for ts smplcty. Fnally, Secton 5 deals wth the counterparty default rs and suggests a method that can be sutable when the counterpartes (.e. manly rensurers) form a small group of heterogeneous subjects mpacted by a common shoc as a fnancal crss or a natural catastrophe (so far only asymptotc methods for large portfolos of homogenous rss are usual n lterature); ths method s also demonstrated by a numercal example. 2. Varous Approaches to Underwrtng Rs The underwrtng rs plays the ey role n nsurance ndustry. As the underwrtng rs s concerned we usually dstngush premum rs and reserve rs. The premum rs conssts of a possblty that the premum wll not be suffcent to cover the nsurer s oblgatons followng from nsurance contracts (manly the clam expendtures). The reserve rs (or techncal provson rs or run-off rs) s the rs of losses due to nadequate techncal provsons and ther spendng. The approaches to underwrtng rs descrbed n ths secton usually result n calculaton of the rs captal (or regulatory captal) RC, whch s capable to mtgate losses caused by ths rs (one can loo upon t as a charge for the rs). Moreover, t s sutable to dstngush here the lfe and non-lfe nsurance as t could be seen n the followng examples. Prague Economc Papers, 2017, 26(6), , 725

5 2.1 GDV model n non-lfe nsurance The fnal verson of ths model was publshed by GDV (Gesamtverband der Deutschen Verscherungswrtschaft) n 2005 as the German contrbuton to the dscusson of the future form of the standard formula n Solvency II. However, as soon as the CEIOPS (.e. the predecessor of EIOPA) publshed other suggestons, the development of the GDV model was stopped. In the context of underwrtng rs of LoB (lne of busness), the GDV model uses the combned rato NCE OE CR, (1) NP where NCE are the net clams expendtures (.e. the run-off results for clams related to the nsurance perod plus the clams expendtures for the accountng year), OE are the operatng expenses and NP are the net earned premums for the gven LOB. For the combned rato of the whole portfolo t could be wrtten CR ( NCE OE) NP w CR wth w NP, (2) NP where CR denotes the combned rato of the whole nsurance portfolo and CR denotes the combned rato of the -th LOB (we use symbols from GDV (2005)). If the partcular combned ratos are looed upon as random varables wth expectatons μ = E(CR ) and varances σ = 2 var(cr ), then t holds for μ = E(CR) and σ2 = var(cr), (3) w, w w 2 j j j j where ρ j s the correlaton coeffcent between CR j and CR. Let SR = RC/NP be the correspondng solvency rato. The prncple of GDV model conssts n the requrement that the loss of each nsurer NCE + OE NP should not exceeds the rs captal RC wth a hgh probablty α (e.g %),.e. P(NCE + OE NP RC) α,.e. P(CR 1 SR) α. It can be rewrtten by means of the α-quantle q of CR as SR 1. (4) In practce one can approxmate ths quantle as q α μ + a. σ, where the coeffcent a would be prescrbed by the regulator applyng global maret data. Therefore the requred rs captal must be at least RC NP ( ˆ a ˆ 1), (5) where ˆ and ˆ are estmates of μ and σ delvered ndvdually by partcular nsurers over ther LOBs usng the estmated portfolo relatons (3). q 726 Prague Economc Papers, 2017, 26(6), ,

6 2.2 CEA model n non- lfe nsurance Smlarly as n the prevous case, the CEA model (Comte Europeen des Assurances) was suggested durng the dscusson of the standard formula n Solvency II (see CEA, 2006). All calculatons are agan made for separate LOBs and aggregated usng correlatons: The reserve rs n LOB requres the rs captal RC RR, calculated as RC f CP, (6) RR, RR, where CP are net clams provsons and f RR, s a reserve rs factor prescrbed as a standard for all nsurers (smlarly as n the GDV model f RR, s taen as a multple of the standard devaton of the dstrbuton of the techncal provsons such that the maret value of techncal provsons wth the correspondng rs captal are 99.5 % suffcent to cover the ncurred clams n the run-off sense). The premum rs n LOB requres the rs captal RC PR, calculated as RCPR, fpr, ( WP UP ), (7) where WP s the net wrtten premum, UP s the net unearned premum and f s a premum PR, rs factor prescrbed as a standard for all nsurers (agan t s a multple of the standard devaton of the dstrbuton of the loss rato such that the net premum wrtten n the comng year plus the net unearned premum reserve altogether wth the correspondng rs captal are 99.5 suffcent to cover all nsurer s expendtures (.e. the clams, the clam reserves % and operatng expenses). 2.3 IAA model n lfe nsurance Though the Internatonal Actuaral Assocaton deals wth all types rs (see e.g. IAA, 2004) we shall present here only ts contrbuton to mortalty rs as an mportant bometrcal component of the underwrtng rs n the framewor of lfe nsurance. In any case, the mortalty rs plays a ey role for underwrtng actvtes of lfe companes and ts rs components are volatlty, catastrophes, level uncertanty and trend uncertanty. The IAA model supposes the common stuaton that the portfolo of a gven lfe nsurer s a set of ndvdual contracts wth death probabltes q and clams X n the case of death ( = 1,, n). The aggregate clam s modelled as a random varable S wth the compound Posson dstrbuton and the frst three moments n 3 n n qx 2 1 qx, qx,. (8) Due to the equvalence prncple, the premum P should fulfll P = μ. Then there can be calculated the rs captal for partcular components of the mortalty rs n the followng way: () Volatlty: The term volatlty means n the context of IAA model n lfe nsurance the rs that the aggregate loss n the gven portfolo fluctuates around the expected Prague Economc Papers, 2017, 26(6), , 727

7 value μ and the premum may not cover the aggregate clam S. The IAA model suggests the applcaton of VaR prncple so that the rs captal RC fulflls wth a hgh vol confdence α PS ( RC P). (9) vol From (9) one can express the rs captal RC vol by means of the approxmaton denoted as NP2 (Normal Power 2, sometmes called also Cornsh-Fsher extenson) as RC vol ( ( )) 1 ( ), (10) 6 where Φ 1 (α) s the α-quantle of N(0, 1). Moreover, the IAA (2004) recommended the confdence level 99.5 % so that RCvol ( ). (11) Applyng the smplfyng approxmaton wth an average mortalty q = 2.5 o /oo one obtans even RC vol ( ) P n n, (12) where n s the number of contracts n the portfolo (see Table 1 for three varous portfolos wth dfferent n and sewness of clams). A less sophstcated approxmaton whch maes use of assumptons q ~ q and X ~ X leads to the formula RCvol ( ) ( ( )) 1 P qn 6qn. (13) Table 1 The Rs Captal Coverng Volatlty Component of Mortalty Rs wth Confdence Level 99.5 % for Three Varous Portfolos Portfolo n Sewness of clams RC vol (% of P) 1 125, , , Source: IAA (2004) () Catastrophe: Snce t s hard to model the catastrophe component of the mortalty rs mathematcally the IAA suggested that the correspondng rs captal could be based on the fxed rato of the expected number of death n the gven portfolo. () Level uncertanty: Ths uncertanty of the mortalty rss conssts n an nadequate estmate of the average mortalty. Accordng to the study of IAA the Table 2 presents the ncrease of the sngle premum caused by the shoc ncrease of mortalty by 728 Prague Economc Papers, 2017, 26(6), ,

8 10%. Besdes ths scenaro approach the rs captal coverng the level uncertanty n the framewor of the mortalty rs can be calculated by means of the formula (12) wth the confdence 99.5%: f e.g. the average mortalty was estmated by means of a sample of yearly observatons then the correspondng mortalty shoc should be 77.4/ / = 0.14 = 14% of the appled reference mortalty. Table 2 The Increase of the Sngle Premum Caused by the Shoc Increase of Mortalty by 10% Term of contract The ncrease of the sngle premum caused by the shoc ncrease of mortalty by 10 % Endowment nsurance Pure endowment nsurance Term nsurance % 0.73% 9.70% % 1.14% 9.49% % 1.54% 9.33% Source: IAA (2004) (v) Trend uncertanty: Ths uncertanty of the mortalty rss conssts n an nadequate estmate of the mortalty trend. The IAA (2004) suggested a smplfed factor approach where the correspondng rs captal RC trend can be obtaned f multplyng the sngle premum by the factor mn(α, n. β) (see Table 3 for the coeffcents α and β). In the case of lfe annutes one suggested to tae 4 % of the captalzed annuty value nstead of ths factor. Table 3 The Coeffcents α and β for the Calculaton of the Rs Captal Coverng Trend Uncertanty Component of Mortalty Rs α (%) β (%) Endowment nsurance Pure endowment nsurance Term nsurance Source: IAA (2004) 3. Techncal Provsons n Solvency II Solvency II promotes a dfferent approach to the techncal provsons n comparson wth the prevous practce. Also the regulatons conform to ths new practce (e.g. one dscusses no upper lmt of the techncal nterest rate n the lfe nsurance, no nvestment lmts for assets coverng the techncal provsons and other changes). It s motvated by the effort to construct the techncal provsons n a maret consstent way composng them from ther best estmate and the rs margn (only n specfc stuatons the techncal provsons can be constructed as a whole by the mar-to-maret method, e.g. when they can be relably replcated or they are hedged by a sutable fnancal portfolo). Prague Economc Papers, 2017, 26(6), , 729

9 The techncal provsons are constructed to cover the future oblgatons of nsurers. In comparson wth the prevous practce, Solvency II dstngushes only three types of techncal provsons, namely for oblgatons from lfe nsurance (the so called lfe nsurance oblgatons); for premum n non-lfe nsurance (the so called premum provson); for clams n non-lfe nsurance (the so called provson for clams outstandng). If an oblgaton cannot be replcated by means of sutable fnancal nstruments, then the techncal provsons TP must by constructed as the best estmate BE of ths oblgaton dscounted applyng the rs-free nterest rate and augmented by the rs margn RM: TP dscounted BE RM. (14) 3.1 Best estmate The best estmate BE s the probablty-weghted average (.e. the expected or mean value) of future cash flows generated by the gven oblgaton and covered by the techncal provsons. When calculatng BE, such actuaral or statstcal methods should be used that reflect all rss affectng the correspondng cash flows. The calculaton methods for BE can be classfed n the followng way: Smulaton methods consst n generatng future scenaros, whch are representatve n an approprate way for all future scenaro alternatves. The recommended smulaton procedures are e.g.: Smulaton Monte Carlo: one smulates TP applyng realzatons of sutable random varables, and the mean of obtaned values s regarded as the expected value of TP; n some cases, e.g. when smulatng the lapses, the correlatons wth the development of relevant macroeconomc varables can be used. Bootstrap method: ths method s popular n the actuaral context and s based on the resamplng of estmated resduals n the correspondng actuaral model, whch enables n the context of the techncal provsons to obtan nformaton on ther dstrbuton (see e.g. Hendrych and Cpra (2015) for lfe nsurance). Bayes approach: when smulatng TP one combnes explct assumptons on ther pror dstrbuton wth the observed nsurance clams. Analytcal methods are based on explct formulas for probablty dstrbuton of future cash flows. Examples can be gven both for the lfe nsurance (e.g. stochastc models of mortalty) and for the non-lfe nsurance (e.g. the dstrbuton free Chan-Ladder models, see Secton 4). Determnstc methods produce the projectons of cash flows necessary for the calculaton of BE by means of determnstc procedures, the stochastc aspects beng contaned n the approprate assumptons (e.g. Chan-Ladder, Cape Cod, Bornhuetter-Ferguson, separaton method, see Cpra, 2015). 730 Prague Economc Papers, 2017, 26(6), ,

10 3.2 Dscountng The dscountng produces the present value of BE usng sutable rs-free nterest rates. Accordng to the Drectve 2014/51/EU (the so called Omnbus II), EIOPA n a poston of the central nsurance subject n Europe wll publsh regularly for ths purpose the so called basc rs-free nterest rate, whch s supposed to be modfed n each domestc nsurance maret by means of the so called matchng and volatlty adjustment respectng the maret credtblty and other factors. 3.3 Rs margn Rs margn RM must guarantee that the techncal provsons truly cover the correspondng oblgatons. RM s usually calculated as the present value of captal costs necessary for such a guarantee (also here one apples the rs-free nterest rate to dscount these captal costs): SCR RM CoC, (15) t t 1 t0 (1 rt 1) where CoC s the cost captal rate (Cost-of-Captal) wth a suggested startng value 6 %; SCR t s the solvency captal requrement after t years to cover the oblgatons n techncal provsons and r t + 1 s the rs-free nterest rate for the maturty t + 1 (see above). Snce the procedure accordng to (15) s too general, and there may be problems wth ts practcal applcaton, one admts some smplfcatons for the calculaton of RM. The hghest possble smplfcaton conssts n the naïve formula RM BE, (16) LOB net 0 where BE net s the best estmate of the net TP calculated n tme t = 0,.e. excludng 0 rensurance, and α LOB s a fxed rato for the gven LOB. The more sophstcated formulas for the rs margn are created ndvdually for partcular rs modules and submodules (e.g. separately for underwrtng, maret and credt rs). Moreover, the constructon of techncal provsons accordng to Solvency II respects some general prncples: () Segmentaton: When calculatng the techncal provsons the nsurers segment ther nsurance oblgatons nto homogeneous rs groups. Moreover, the assgnment of an oblgaton to a LOB should reflect the nature (substance) of the rss relatng to the gven oblgaton, whch may not reflect the legal form: e.g. oblgatons for health nsurance should be allocated to lfe or non-lfe accordng to whether the busness s carred out on a smlar techncal bass to that of lfe nsurance (SLT health) or to that of non-lfe nsurance (non-slt health, see also Fgure 1). The process of segmentaton of nsurance contracts nto partcular rss s called unbundlng (e.g. contracts coverng lfe nsurance can be unbundled to SLT health nsurance, lfe nsurance wth proft partcpaton, ndex-lned lfe nsurance, unt-lned lfe nsurance and other lfe nsurance). Prague Economc Papers, 2017, 26(6), , 731

11 () Contract boundary: There are varous oblgatons relatng to a gven nsurance contract, and the prncple of contract boundary determnes whch of them really belong nto the contract framewor ( boundares ). () Recoverables from rensurance contracts: The best estmate of techncal provsons s calculated n the gross form,.e. wthout subtractng sums recoverable from rensurance contracts. However, the results concernng the recoverables must be adjusted to tae account of expected losses due to the default of counterparty (.e. due to the default of rensurer). That adjustment s based on an assessment of the probablty of default PD and on the average loss resultng there from (the so called loss-gven-default LGD). For nstance, let the recoverables n the frst, second and thrd year amount to R 1, R 2 and R 3, respectvely, wth the same LGD of 70 %,.e. the recovery amounts to 30 % of the loss. Then the recoverables should be reduced by the amount 0.7 [ PD ( R R R ) PD ( R R ) PD R ] (17) (for smplcty we gnore the tme value of money). (v) Proporconalty of TP method: Insurers apply for the calculaton of techncal provsons such methods that are proportonate to nature, scale and complexty of rss related to the gven nsurance oblgatons. Judgng the proportonalty of a method for TP calculaton the nsurer should assess the admnstratve and calculaton burden of the method (both n a quanttatve and qualtatve way) respectng the complexty of all rss affectng the techncal provsons. In partcular, accordng to ths prncple the nsurer s n some cases allowed to use sutable smplfcatons (see Secton 4). When calculatng TP one must also regard a lot of further aspects, e.g. all overhead expenses and costs related to the management of nsurance and rensurance oblgatons, the (cost) nflaton, future proft shares (even f they are not guaranteed) and others. 4. Calculaton of Rs Captal for Coverng the Reserve Rs In ths secton we present three approaches to the calculaton of rs captal RC to cover the reserve rs. These approaches have been chosen snce all of them are applcable n the framewor of Solvency II (e.g. as the nternal model), but they were unusual for the techncal provsons n the perods before Solvency II (we gnore a lot of other methods, e.g. Hendrych and Cpra, 2015; Merz and Wüthrch, 2008; SST, 2006). 4.1 One-year tme horzon approach The classcal clam reservng methods for the non-lfe nsurance have an ultmo tme horzon,.e. they loo at the total run-off. In spte of ths fact, the one-year tme horzon reservng methods have been suggested supported by the practcal opnon that they are suffcent for the regulatory purposes of solvency (see e.g. Hürlmann, 2008; and Ohlsson and Lauzenngs, 2008): 732 Prague Economc Papers, 2017, 26(6), ,

12 The reserve rs n such a one-year tme horzon model s taen nto account by means of the rato LR (reserve loss rato) that expresses the consdered loss n relaton to the clam reserve: L R C R R C LRR 1 X , (18) R R R where R 0 s the clam reserve n the begnnng of the current year (.e. n the begnnng of the consdered one-year perod), R 01 s the best estmate of ths reserve at the end of the current year (the upper ndex ndcates that R 01 ncludes the changes occurred n the reserve durng ths one-year perod) and C01 s the estmate of the amount pad out durng the current year. Obvously the rato LR relates the techncal loss L (the run-off result) to the reserve R 0 n the begnnng of the current year. Due to the actuaral prncple of equvalence t must hold E(LR) = 0, or equvalently E(X) = 1. Emprcal data show (e.g. Hürlmann, 2008) that the random varable X can be modelled by means of the logarthmc normal dstrbuton X LN. (19) 2 ~ ( X, X) Snce then E(X) = exp(μ + σ 2 /2) = 1 and var(x) = exp(2μ + σ 2 2 )(expσ X X X X X 1) = σ 2, t must 2 hold μ X = σ X2 /2 and σ X = ln(1 + σ 2 ). By means of these results we can fnd an analytcal formula for the value-at-rs VaR α (L) of the loss L n the numerator of the rato LR,.e. the analytcal formula for the rs captal requred wth a gven confdence to cover the reserve rs n the model wth one-year tme horzon (see also Cpra, 2015): RC VaR ( L) VaR ( L / R ) R VaR ( X 1) R VaR ( X ) 1 R exp{ ( ) ln(1 )} 0 exp{ X X ( )} 1R 1 R. 2 1 Qute analogously we can derve the reserve rs captal RC when one apples the expected shortfall ES α (L) as the rs measure (Cpra, 2015): (20) 1 2 ( ) ln(1 ) 0 RC ES ( L) R. (21) 1 Table 4 presents the rs captal for the confdence levels 99 % and 99.5 % n the one-year tme horzon dependng on the standard devaton σ of the rato X and on the clam reserve R0 n the begnnng of the current year. E.g. the reserve rs captal for the estmated standard devaton 0.12 of the rato X calculated as the expected shortfall ES 0.99 s RC ES R (the standard devaton must be estmated usng the hstorcal observatons of X). 0 Prague Economc Papers, 2017, 26(6), , 733

13 Table 4 Reserve Rs Captal for the Confdence Levels 99 % and 99.5 % n the One-Year Tme Horzon Dependng on the Standard Devaton σ of the Rato X and on the Clam Reserve R 0 n the Begnnng of the Current Year VaR α ES α σ α = 0.99 α = α = 0.99 α = σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R σ R0 Source: authors 4.2 Analytcal approach to techncal provsons Analytcal methods are based on explct formulas for probablty dstrbuton of future cash flows. For nstance, Mac (1993) has derved the formula for the mean squared error MSE of the clam reserve constructed by the method Chan-Ladder (e.g. Cpra, 2015), whch s mportant for descrpton of the stochastc behavour of ths reserve both of the type RBNS (.e. Reported But Not Settled) and the type IBNR (.e. Incurred But Not Reported). In any case, t s a typcal example of the analytcal approaches to techncal provsons n practce: MSE( Rˆ ) I 1 2 ˆ 2 ˆ 1 1 CI ˆ2 ˆ2 I I1f C j1 C j, (22) where C s a random varable denotng the cumulatve clam amount pad out tll the development year snce the accdent year (, = 1,, I); one gnores the dependences among varous accdent years so that the random vectors C,......, C and C,, C are 1 I j1 ji supposed to be ndependent for j; R s the clam reserve for the accdent year,.e. R = C C ( = 2,, I); I, I +1 f s the development coeffcent n E(C, +1 C 1,, C ) = C f ( = 1,, I, = 1,,, I 1) whch can be estmated n the unbased way by the method of Chan-Ladder (e.g. Cpra, 2010) as I I j, 1 j, j1 j1 (23) fˆ C C, 1,..., I Prague Economc Papers, 2017, 26(6), ,

14 (moreover, these estmates of development coeffcents are mutually uncorrelated); then n the Chan-Ladder method one estmates future cumulatve clam amounts as Cˆ C fˆ... fˆ, I 1 and Cˆ C ; (24) I, 1 I1 I, 1 I, 1 σ s the unnown parameter n var(c, +1 C 1,, C ) = C σ ( = 1,, I, = 1,, I 1) 2 whch can be estmated n the unbased way as 2 I 2 1 C, 1 ˆ ˆ C f, 1,..., I 2 ; (25) I 1 1 C MSE( R ˆ ) s the estmated mean squared error of the clam reserve for the clam amounts wth the accdent year defned as 2 E( Rˆ R C 1,..., CI 1 ). Snce the nsurer needs the amount of total reserve R = R + + R, one must extend 2 I the formula (22) to the form MSE( Rˆ) MSE( R ) C C 2 ˆ /fˆ I I I 1 2 ˆ ˆ ˆ I ji I 2. (26) j1 I1 C n 1 n Table 5 Cumulatve Run-Off Trangle Accdent year Development year , , ,599 1,027,692 1,360,489 1,647,310 1,819,179 1,906,852 1,950, , , ,288 2,142,656 2,961,978 3,683,940 4,048,898 4,115, , ,682 1,522,637 3,203,427 4,445,927 5,158,781 5,342, , ,828 2,900,301 4,999,019 6,460,112 6,853, , ,394 2,920,745 4,989,572 5,648, , ,994 4,210,640 5,866, , ,480 1,954, , , ,121 Source: Mac (1993) For nstance, let us consder the run-off trangle n Table 5 (for the buldng nsurance n the framewor of mortgage portfolo) appled for the IBNR reserve creaton. The estmated reserves for partcular accdent years ncludng the total reserve estmated over all development years by the Chan-Ladder method are gven n Table 6. Moreover, Prague Economc Papers, 2017, 26(6), , 735

15 the correspondng standard devatons (.e. root mean squared errors MSE( Rˆ ) Rand MSE( R ˆ) presented as the percentage of the estmated reserves) are calculated n Table 6 accordng to the formulas (22) and (26). One can see that the errors of the Chan-Ladder reserves can be unexpectedly hgh so that the correspondng rs captal RC must tae ths fact nto account n the proper way (see also Wüthrch and Merz, 2008). Table 6 The IBNR Reserves Estmated by the Mac s Method Includng Ther Standard Devatons (.e. the root mean squared errors presented as the percentage of the estmated reserves) Rˆ Standard devaton (%) , , , , , Total 14, Source: Mac (1993) 4.3. Some smplfcatons Reasonable smplfcatons for the TP calculatons are acceptable n some cases n the framewor of Solvency II. For nstance, one can calculate the IBNR reserve smply as TPIBNR flob TPRBNS, (27) where f LOB s a factor specfc for the gven LOB of the gven nsurer. Ths factor expresses the approxmate rato of the IBNR reserve from the RBNS reserve where the RBNS reserve s composed by partcular case estmates provded by experts. Here we suggest a more sophstcated approach usng observatons over several years (n partcular over three years t, t 1 and t 2). The correspondng formula has the form TP N A, (28) IBNR, t t t where N / p N / p N Nt Rt t Rt R R R t1 1 t2 2 t3 t1 t2 t3, (29) 736 Prague Economc Papers, 2017, 26(6), ,

16 A t s the mean clam amount of the type IBNR n year t; N t s the number of clams of the type IBNR n year t ndependently on ther accdent year (ths value s observed for = 1, 2, 3 and N must be calculated addtonally t accordng to (29)); p 1 s the percentage of clams of the type IBNR n the end of year t 3 reported durng year t 2; p 2 s the percentage of clams of the type IBNR n the end of year t 3 reported durng years t 2 and t 1; R t s the number of clams reported n year t ndependently on ther accdent year (ths value s observed for = 0, 1, 2, 3). When calculatng the rate coeffcent t n (29) one must regard the fact that e.g. the observed number of N t 1 n year t 1 s ncomplete and one should augment t artfcally to the actual (but unobserved) number of clams of ths type by means of the observed percentage p 1. Moreover, the proper calculaton s more sophstcated, snce t should nclude the estmaton bas, the nflaton, the tme value of money and other relevant facts (see the followng numercal example). Let us demonstrate the second approach to smplfed IBNR reserves by means of the data tll 12 December 2015 and addtonal nformaton from Tables 7 and 8. Usng ths nformaton one constructs the best estmate of IBNR reserve to ths date. Table 7 Smplfed Calculaton of the IBNR Reserve on 31 December 2015: Data and Projectons t N t (pcs) R t (pcs) Estmated nflaton (%) Estmated dscount rate (%) , , , , Source: authors The clams reported to 31 December 2015 amount to 11,000,000 EUR. For one half of ths sum (.e. for 5,500,000 EUR) one has created the RBNS reserve requrng the rs margn due to a possble bas of the estmate to 90 % of actual value so that the orgnal amount of 11,000,000 EUR s corrected to 5,500,000 / ,500,000 = 11,611,111 EUR (the so called bas correcton). Prague Economc Papers, 2017, 26(6), , 737

17 Table 8 Smplfed Calculaton of the IBNR Reserve on 31 December 2015: Addtonal Informaton t Delay n clams (%) Delay n clams (EUR) Inflaton (EUR) Dscountng (EUR) ,805,556 6,095,833 5,861, ,063,889 4,480,438 4,142, ,741,667 2,016,197 1,792,392 Total 11,611,111 11,796,186 Source: authors The clams reported each year are characterzed approxmately by the followng delay n ther payments: 50 % n the actual year, 35 % n the next year and 15 % n the year after next. Table 8 contans also further adjustments of the amount necessary to cover the nsurance clams reported to 31 December 2015 due to the nflaton (e.g = ) and due to the tme value of money adjusted by dscountng (e.g /1.042 = ). Then one calculates easly the mean clam of the type IBNR n the year 2015 A / R / EUR Snce t was found emprcally n the consdered stuaton that startng wth N 2012 = 100 clams of the type IBNR n the year 2012 one has reported 85 clams n the year 2013 and 10 clams n the year 2014, one can put p 0.85, p Therefore after substtutng to (29) one obtans N 2015 = 129. Fnally, accordng to (28) one obtans the best estmate of the IBNR reserve n the year 2015 as TPIBNR, EUR. 5. Counterparty Default Rs As to the counterparty default rs, we suggest a method that can be sutable when the rensurers form a small group of heterogeneous subjects mpacted by a common shoc as a fnancal crss or a natural catastrophe (so far only asymptotc methods for large portfolos of homogenous rss are usual n lterature, see e.g. CEA, 2006; McNel et al., 2005; Sandström; 2011). 738 Prague Economc Papers, 2017, 26(6), ,

18 5.1 Descrpton of method based on common shoc The suggested method can be descrbed n several steps: () Let us denote the common shoc affectng all rensurers of a gven nsurance company (e.g. the arrval of the fnancal and economc crss or recesson, a legslatve change or reform, a catastrophc event) as a random varable R rangng n the nterval between zero and one. For the values of R near to zero or one, the common nfluence of the gven phenomenon s low or hgh, respectvely. The behavour of R can be descrbed by the probablty densty functon of the form 1 f() r r, 0 r 1, (30) where β s a parameter (0 < β < 1). Such a probablty densty functon can be looed upon as a specal case of the beta dstrbuton (e.g. Cpra, 2015), and t s acceptable from the practcal pont of vew snce accordng to t (1) the small shocs are the most probable ones, whle (2) the probablty of more ntensve shocs declnes to small postve values (however, the zero value s never acheved). Another nterpretaton s also possble: the maxmal shoc among n mutually ndependent annual shocs (.e. durng n years) has the probablty densty functon of the form (30) but wth the parameter nβ nstead of β. Therefore e.g. for β = 0.05 and n = 20, the probablty densty functon (30) corresponds to the unform dstrbuton so that the maxmal shoc durng the perod of 20 years attans each of ts values wth the same probablty (one can mae use of ths property when calbratng the parameter β). () The probabltes of default wll depend on the common shoc R. A sutable functonal relaton seems to be /p PD () r p (1 p ) r, 0 r 1, (31) where p s a basc level of the probablty of default of the -th counterparty ( = 1,, ) n the credt portfolo (t s a benchmar f excludng the nfluence of the common shoc R). Moreover, n (31) one adds to ths basc level a component that depends on R by means of a postve parameter γ. The power exponent n (31) s a decreasng functon of the basc level of the probablty of default p snce counterpartes wth a low p are not very senstve to the random shocs whle the hgher probablty of default ncreases the default senstvty even f the attaned values of R are not hgh. In any case, the functon (31) ncreases from the basc level p to the value 1. Moreover, ths functon s concave for γ < p (and the probablty of default PD s consdered more lely as large), t s convex for γ > p (and the probablty PD s consdered more lely as small); for γ = p one obtans an ncreasng lne. () By ntegratng the functon (31) over r usng the probablty densty functon (30) one obtans the formula for the probablty of default of the -th counterparty as 1 ( ) p PD E( PD ( R)) PD ( r) f ( r) dr. (32) p 0 Prague Economc Papers, 2017, 26(6), , 739

19 Conversely, f one taes the numercal values of default probabltes PD of partcular counterpartes as an external ratng provded by specalzed agences, e.g. Standard & Poor s (S&P) or other subjects whch rate regularly rensurance companes, then one can fnd the basc level of the probablty of default p of the -th counterparty evdently as PD p. (33) (1 PD ) One can summarze that the behavour of partcular counterpartes n the credt portfolo may be modelled n a sutable parametrc way usng two parameters β and γ. (v) Fnally, one should extend the results for partcular counterpartes to the whole credt portfolo. Let I s a random default ndcator of the -th counterparty,.e. I = 1 or I = 0 dependng whether the default has occurred or not, respectvely. The loss L generated by the whole credt portfolo can be expressed n the form, (34) 1 L LGD I where LGD s the partcular loss followng from the default of the -th counterparty. It holds where E( L) LGD PD, var( L) LGD LGD, (35) j j 1 1 j1 (1 p ) (1 p ) PD (1 PD ), ( PD p ) ( PD p ), j. (36) j j 1 1 j j p pj For nstance, to derve σ j for j one can wrte usng (32) cov( I,I) EI (, I) EI ( ) EI ( ) EI (, I) PDPD j j j j j j ( ) p ( ) p 1 /p /p j 1 j [ p (1 p) r ][ pj (1 pj) r ] r ds p 0 pj (1 p) (1 pj) ( PD p) ( PD j p j). p p 1 1 j In the specal case of a sngle counterparty (.e. = 1) the relatons (35) and (36) are smplfed to 2 E( L) LGD PD, var( L) LGD PD (1 PD). (37) Hence the captal requrements calculated usng the quantle rs measures can be obtaned (e.g. value-at-rs VaR and others, see Cpra, 2015). 740 Prague Economc Papers, 2017, 26(6), ,

20 5.2 Numercal demonstraton Table 9 contans numercal values that are used for the calculaton of the credt captal requrements as a numercal example of the methodology descrbed n Secton 5.1. In ths example an nsurer s rensured by three rensurers ( = 1, 2, 3) wth the default probabltes correspondng to ther external ratng accordng to S&P (the appled values of parameters β and γ are taen from Cpra (2015), but n practce they should be prescrbed by the regulator). Table 9 Numercal Values Used for the Calculaton n Secton 5.2 Symbol Numercal value Parameters Probablty of default (S&P) β 0.05 γ 0.1 PD 1 (~AA) PD 2 (~A) PD 3 (~BBB) LGD m CZK Loss gven default LGD m CZK LGD 3 80 m CZK Source: authors In Table 10 we have calculated the basc levels of the probabltes of default of partcular counterpartes (accordng to (33)) and the varances σ and covarances σ j (accordng to (36)) necessary for the calculaton of varance of loss n (35). Table 10 Auxlary Values Calculated n Secton 5.2 p σ j j: Source: authors It enables to calculate accordng to (35) the expected value and the standard devaton of the loss L followng from the credt default of counterpartes Prague Economc Papers, 2017, 26(6), , 741

21 E( L) 9.3m CZK, var( L) 36.4 m CZK. Assumng the normal dstrbuton one obtans fnally e.g. VaR = m CZK (.e. the value-at-rs for the confdence level 99.5%). 6. Conclusons The paper concentrated on mportant aspects of the underwrtng rs and counterparty credt rs n the framewor of Solvency II. In partcular, the problem of techncal provsons was nvestgated carefully snce the results of the quanttatve mpact studes QIS (e.g. Cpra, 2015) have shown that the techncal provsons constructed accordng to ths new methodology (e.g. (14)) may be substantally lower (sometmes even by 15 %) than the ones constructed accordng to the practce before Solvency II. The smplfed calculaton of techncal provsons shown n the paper can be useful n the actuaral practce. A specfc approach to the counterparty credt rs based on the concept of common shoc was also suggested. Ths approach should be extended n a future research n order to enable not only the calculaton of VaR wthout the assumpton of normalty but also the calculaton of other quantle rs measures (e.g. the condtonal value-at-rs CVaR, expected shortfall ES, margnal expected shortfall MES) snce the VaR recommended n the so called standard formula of Solvency II s overcome n the new rs regulatory scheme Basel III for bans (see BIS, 2010). References BIS (2010). A Global Regulatory Framewor for More Reslent Bans and Banng Systems. Basel: Ban for Internatonal Settlements. ISBN CEA (2006). CEA Worng Document on the Standard Approach for Calculatng the Solvency Captal Requrement. Brussels: Comte Europeen des Assurances. Cpra, T. (2010). Fnancal and Insurance Formulas. Hedelberg, Dordrecht, London, New Yor: Physca-Verlag / Sprnger. ISBN , org/ / Cpra, T. (2015). Rs n Fnance and Insurance: Basel III and Solvency II. Prague: Eopress (n Czech). GDV (2015), Dscusson Paper for a Solvency II Compatble Standard Approach (Pllar I) Model Descrpton. Berln: Gesamtverband der Deutschen Verscherungswrtschaft. Hendrych, R., Cpra, T. (2015). Econometrc Model of the Czech Lfe Insurance Maret. Prague Economc Papers, 24(2), , Hürlmann, W. (2008). On the Non-lfe Solvency II Model. Manchester: 38th ASTIN Colloquum. IAA (2004). A Global Framewor for Insurance Solvency Assessment. Ontaro: Internatonal Actuaral Assocaton. Mac, T. (1993). Dstrbuton-Free Calculaton of the Standard Error of Chan Ladder Reserves Estmates. ASTIN Bulletn, 23(2), , McNel, A. J., Frey, R., Embrechts, P. (2005). Quanttatve Rs Management. Prnceton and Oxford: Prnceton Unversty Press. 742 Prague Economc Papers, 2017, 26(6), ,

22 Merz, M., Wüthrch, M. V. (2008). Modelng the Clams Development Result for Solvency Purposes. Manchester: 38th ASTIN Colloquum. Ohlsson, E., Lauzenngs, J. (2008). The One-Year Non-lfe Insurance Rs. Manchester: 38th ASTIN Colloquum. Sandström, A. (2011). Handboo of Solvency for Actuares and Rs Managers. Theory and Practce. New Yor: Chapman and Hall / CRC Press. SST (2006). Techncal Document on the Swss Solvency Test. Swtzerland: Federal Offce of Prvate Insurance. Wüthrch, M. V., Merz, M. (2008). Stochastc Clams Reservng Methods n Insurance. Chchester: Wley. Prague Economc Papers, 2017, 26(6), , 743