Using Tail Conditional Expectation for capital requirement calculation of a general insurance undertaking

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1 Usng Tal Condtonal Expectaton for captal requrement calculaton of a general nsurance undertakng João Duque 1, Alfredo D. Egído dos Res 2, and Rcardo Garca 3 Abstract: In ths paper we develop a solvency model to estmate the necessary economc captal of a real nsurance undertakng operatng solely n the Automoble branch, applyng the Tal Condtonal Expectaton rsk measure. The model assumes a one year tme horzon statc approach wth an unchanged asset and lablty structure for the company. After dscussng the man factors affectng the whole of the nsurance actvty and ther nfluence on the assets and labltes on that real nsurance undertakng used n the study, we calculate ts necessary economc captal, by usng the Monte Carlo smulaton technque to generate the probablty dstrbuton of the possble future proft and losses wth mpact on the company s far value. Ths paper ntroduces an applcaton of a set of technques that are usually appled to manage asset and lablty rsks to captal requrements. Wth a smulated exercse appled to a real nsurance undertakng we show ts feasblty, ts advantages and how useful t may be for nvestors, regulators and remanng stakeholders when the technque s explored n depth. Key words: Tal Condtonal Expectaton, Value-at-Rsk, captal requrement, resamplng, Monte Carlo smulaton, rsk management. 1 ADVANCE Research Centre and ISEG, Techncal Unversty of Lsbon. Partal fnancal support from FCT-Fundação para a Cênca e Tecnologa s gratefully acknowledged. 2 CEMAPRE and ISEG, Techncal Unversty of Lsbon. Partal fnancal support from FCT-Fundação para a Cênca e Tecnologa (Programme FEDER/POCI 2010) s gratefully acknowledged. 3 ISP- Insttuto de Seguros de Portugal (Portuguese Insurance and Penson Fund Supervson Authorty) 1

2 1. Introducton The determnaton of the economc captal requrement to ensure, wth hgh probablty, the development of operatons, even n adverse envronments, s a prmer queston, due to the role of nsurance undertakngs n the economy. In ths artcle we develop a solvency model to estmate the necessary economc captal for a real portfolo of a partcular nsurer, usng a set of specfc rsk analyss tools that have been wdely used for dfferent purposes and ams. It allows us to calculate the economc captal requrement for an nsurance undertakng, n order to face adverse stuatons wth a chosen hgh probablty, gven hs current asset and labltes structure and consderng a one year tme horzon. We lmt our study to the automoble branch, dentfyng the man assets, labltes and operatons that cause uncertanty on the economc value of the nsurer under study. We measure the causes of uncertanty by the mpact on the economc results and enhance the nterest and practcal applcatons of the model to the nsurance ndustry. Here we only consder stocks and bonds as manageable assets, whle for labltes we wll account for premum and clams reserves. The stocks and bonds consdered are those that the nsurance undertakng actually shows n the balance sheet and the smulatons are based on the assumpton that they are kept constant wthn the tme nterval under study. As far as the reserves are concerned, they relate to the underwrtng of nsurance contracts and respectve clams settlng. There are several types of rsks n the nsurance actvty that affect assets, labltes or both. In ths paper we model the equty rsk, nterest rate rsk, credt rsk, reserve rsk and 2

3 the premum rsk, determnng the exposure of the assets and labltes to the dfferent types of rsks. We suggest the captal requrement to be determned based upon the Tal Condtonal Expectaton (smply, TCE) rsk measure, assumng a (future) smulated proft and loss dstrbuton for the company, whch n turn, s estmated by means of Monte Carlo smulaton. We assume that n the case of run before the end of the perod, the nsurance undertakng has the ablty to provde addtonal captal to ensure the contnuty of ts actvtes. We consder a statc approach, stressng the need for a perodcal re-evaluaton, snce t s not expectable that, wthn a reasonable tme horzon, nether the asset and labltes structure nor the future proft and loss dstrbuton remans unchanged. Furthermore, for the sake of smplcty, we assume that the far value of the nsurer s labltes equals the best estmate,.e., the market value rsk margn s null. Thus, we present VaR and TCE rsk measures, defne the rsk factors affectng the whole of the nsurance ndustry and the partcular nsurance undertakng studed, model ther ndvdual and aggregate behavour and detal the smulaton procedures. Fnally, these procedures are appled to a non lfe nsurer operatng n the motor branch, and are used to calculate hs economc captal requrement. The mportance of these sort of rsk measures to compute captal requrement s enhanced by the newly proposed regulatons for the nsurance ndustry n the European market, under the programme Solvency II. For more detals, please see Lnder and Ronkanen (2004). 3

4 The body of ths artcle s dvded nto fve sectons. In the next secton we formulate the solvency model, ntroducng the ndvdual rsk factors nvolved, ther modellng and smulaton procedures. In Secton 3 we develop our applcaton under the assumptons consdered. In Secton 4 we show the results of our applcaton and calculate the captal requrement for the perod, rsk by rsk and for the aggregate. Fnally, n the last secton we present the man conclusons. 2. The model 2.1. VaR and TCE The Value-at-Rsk (smply, VaR α ( X ) or VaR α ) s defned as the quantle of order α of the probablty dstrbuton of the random varable X that represents the future results, proft and losses, of the nsurer. That s, VaRα = nf x R FX ( x) α, where ( x) F X s the dstrbuton functon of X. The Tal Condtonal Expectaton, denoted as TCE α ( X ) or smply TCE α, s defned as TCE ( X ) E[ X X VaR ( X )] =. That s, whle wth the α < VaR we are nterested n knowng how much can a frm lose wthn a certan tme horzon, under certan set of consderatons, the TCE allow us to estmate the expected loss whenever the occurred loss s greater than the VaR (assumng that the VaR s negatve). Therefore, t seems that TCE s a more conservatve but safer rsk measure to adequately protect the nsurance undertakng ndustry, ther shareholders and remanng stakeholders. α 4

5 Consderng the advantages of the TCE over the VaR as presented by Artzner (1999) and Lynn Wrch and Hardy (1999), ths wll be the chosen rsk measure to determne the necessary economc captal. Nevertheless, snce VaR s nowadays the most commonly used rsk measure and that we need the VaR for the computaton of the TCE, we wll present both rsk measures for comparatve purposes Modellng the ndvdual rsks As far as equty rsk s concerned, t s defned as the rsk assocated wth stock prce returns fluctuaton, assumng a well dversfed nsurer s portfolo. If ths s the case, we may then consder the Sharpe (1964) and Lntner s (1965) Captal Asset Prcng Model (CAPM). Assumng a captal market n equlbrum, and that a set of assumptons s fulflled (see for nstance n Elton et al, 2007), the expected rate of return of the portfolo wll then be gven by where E ( R ) R + ( E( R ) R ) p = β, (Eq. 1) f p R f stands for the rsk free rate of return, m represents the N assets portfolo beta that equals β = = f R m s the market rate of return, and N p 1 w β β p, wth each ndvdual beta component determned by β = σ, denotng σ m the covarance between the 2 m / σ m, 2 rate of return of stock and the market rate of return; σ m stands for the equty market varance and w s the weght of stock n the portfolo. Consderng a portfolo wth a large number of assets we can assume, wthout loss of accuracy that σ β σ (see for nstance n Elton et al, 2007). In order to estmate the p parameters β we use the Market Model. p m 5

6 For smulaton purposes we assumed that the smulated nstantaneous market rate of return follows a Geometrc Brownan moton, whose dynamcs are gven by the equaton 2 d ln S = ( µ σ / 2) dt + σdw where S t s the stock market prce level at tme t, W t a t t standard Brownan moton, µ s the drft constant, σ s the volatlty, and dw Z( dt) 1/ 2 t = s the ncrement of W t, W Normal(0; dt) ; Z s a standard normal random varable. t For practcal purposes the above stochastc process s dscretzed n short tme ntervals, say = k t / p, where p s the number of ncrements and k the chosen tme horzon. Thus, the smulated stock prce level at t + t wll be gven by 2 S = S exp{( µ σ / 2) t + σ z t }. (Eq. 2) t+ t t In order to model the prcng behavour of the debt nstruments that are sgnfcant for the assets sde of the balance sheet of the nsurance undertakng, we need to consder both the credt rsk and the nterest rate rsk. Startng wth the credt rsk, we use the J.P. Morgan s (1997) CredtMetrcs to model the credt rsk of the debt nstruments under consderaton. The credt spreads are extracted from the companes ratng scores. The better the ssuer s ratng, the lesser the credt spread requred and, consequently, the larger the dscounted value of the debt. The model assumes that f the market value of the ssuer s debt follows beyond a gven threshold, the entty wll enter n default. Ths reasonng s extended n a way that wll allow us to determne a relatonshp between the assets value and the ssuer s ratng. We assume that the rate of return of the ssuer s asset follows a Normal dstrbuton n the case of a sngle credt, or a Multvarate Normal dstrbuton for a portfolo of credts. For the smulaton procedure of the assets rate of returns we generate 6

7 a set of correlated (pseudo-) Normal random values and a new ratng and ts respectve credt spread s assgned, f necessary. The analyss s even refned consderng scenaros for default. The credt rates of recovery are hghly volatle, whch means that, for each scenaro of default, we smulate a value for the credt recovery rate assumng a Beta dstrbuton wth parameters n accordance wth the credt s level of the subordnaton. If the debt s smulated to default, then the credt value wll equal the smulated rate of recovery tmes the nomnal value of the credt. Otherwse, the ssuer s credt rsk consderng the smulated ratng s added up to the respectve rsk free dscount rate n order to estmate the value of the debt nstrument. In order to model the debt nterest rate rsk we use smulated zero coupon bonds wth maturty equal to the duraton of each bond portfolo, as suggested by J.P. Morgan and Reuters (1996). We assume that the daly short term nterest rate behavour follows a modfed one perod one factor, short term nterest rate model of Cox, Ingersol and Ross (1985) (CIR model) suggested by Fsher, May and Walther (2002), and whose parameter estmaton method s easer to mplement. The notaton, the parameter estmaton and the smulaton procedure s accordng to those proposed by these authors. The short term nterest rate behavour s assumed to follow the stochastc dynamcs gven by ( ) dr() t = b a r() t dt + σ r() t dw() t, (Eq. 3) 7

8 where b, a and σ are postve constants and W(t) s a standard Brownan moton process. As far as bond prces for the several maturtes are concerned, these are assumed to value { p ( t r( t), T )}, defned by p B( T ) r( t ) ( t r t T ) A( t T ) e t,, ( ),, =, (Eq. 4) ( a+ h)( T t ) 2 2he where A( t, T ) = 2 ( ) ( T t ) ; ( t, T ) h h + a + h e 1 ( ) 2b σ 2 ( T t ) h 2( e 1) B = ; T t h 2h + a + h e 1 ( ) ( ) ( ) 2 2 h = a + 2σ, t s the tme moment when the value s determned and T s the maturty. The mplct yeld curve of the CIR model s estmated from r ( t T ) ( p( t, r() t, T )) log, =. (Eq. 5) T t If equaton (3) holds, t means that the process s not drectly observed, snce t s developed under the rsk neutral probablty measure. However, as we need to smulate the stochastc nterest rate process under the real world probablty measure, an acceptable market estmate for a can be ã (see Fsher, May and Walther, 2002). The parameters b, ã and σ can be emprcally determned usng market data. The estmaton method for the parameters b, ã and σ s based upon the Martgales Estmaton Functons as presented n Fsher, May and Walther (2002). The actual short term nterest rate, r(0), and the estmates of bˆ and σˆ are then used to estmate a, assumng that market prces ( M ) p equal the theoretcal prces ( ) T at tme zero (t = 0). The estmate for a s then obtaned by p for, = 1,..., n zero coupon bonds, wth maturty Mn a n M ( p p ) =

9 Short term nterest rates have to be smulated usng the estmate ã, nstead of a, snce we are nterested n generatng real world scenaros for r(t), as n Fsher, May and Walther (2002). Gven a tme dscretzaton nto equally tme spaced nstants we splt the tme nterval [,T ] 0 nto N equal tme ntervals: = T N. Then, we smulate the future values of the short term nterest rate, r n, usng the recurson method, for n = 1,..., N wth the startng pont r(0) accordng to r n ( bˆ ~ ar ˆ~ n ) ˆ 1 + rn 1 Wn = rn 1 + σ (Eq. 6) Wth the yeld curve, consderng the credt rsk, and usng the nverted verson of equaton (5) t s then possble to determne a smulated future value of each bond, gven both credt and nterest rate rsks. The dfference between the smulated future value of each bond and ts present (dscounted) value corresponds to the smulated result (gan or loss) of holdng each bond for the one year tme perod. The reserve rsk s related to the rsk of adverse development of the clams reserve. It corresponds to an estmate of the total cost that the nsurer wll have to bear n order to settle all clams occurred untl the end of the year, whether they have been reported or not. Ths s a net value, after the deducton of all payments already done concernng those clams. Defne I j as the ncremental payments made n the development year j regardng clams occurred n the year and R = I, s the total reserve. Here, represents the (, j) j set of ndexes assocated to the total future ncremental payments dsplayed n the usual development matrx,.e., = {( j, ) : 0 NN ; + 1 j N+ 1}, N s the observed development perod, we assume the clams development stops at N+1. For more detals please see Taylor (2000). We wll use a Generalsed Lnear Model (smply, GLM) 9

10 where the varables I j are consdered to be ndependent and dentcally dstrbuted (d), whose dstrbuton belong to the Exponental famly, descrbed n McCullagh and Nelder (1989). A GLM s characterzed by a random component and a systematc component. Regardng the frst component, consder a set of ndependent r.v. s Y, = 1, 2,...,n wth densty f (, φ), where θ j s the canoncal shape of a locaton parameter and φ s a scale y θ parameter. As for the random component, consder a matrz X (n p), whose elements, x j, are the n observatons of p explanatory varables X j, j=1, p. The -th observaton of these varables generates a lnear predctor (lnear combnatons of the explanatory varables) η, gven by η p x j 1 jβ = j =, = 1,..., n, where the β j, j=1,,p, are unknown parameters, to be estmated from the data. The two components relate each other through µ = h ( η ) = h( z β ) and η = g ( µ ), where h s a monotonous and dfferentable functon; g = h 1 s the lnk functon; z s a vector of dmenson p, functon of the vector of explanatory varables, say, x ; µ = EY ( ) and Var( Y ) = φ V ( µ ) / w, where w s a constant and V µ ) the varance functon. Then, consder a trangle of development of ncremental payments I j, wth 0 N, 0 j N+1. Suppose w j = 1, j, ζ the followng shape ( ) µ gven by, thus we have ( I ) φ V ( ) V µ j, j η = ln µ = µ + α + β V, j µ, j T ( =. The varance functon has, =, ζ 0. The lnk functon and the lnear predctor are, j j j, (Eq. 7) 10

11 whereα denotes the effect caused by the occurrence perod, development perod j and µ the global average. The estmates for the future ncremental payments (, j n 1) { ˆ µ + ˆ α β } ˆ µ ˆ, + j = exp j, where µˆ, αˆ, β j the effect caused by the I, j + are gven by βˆ j are estmates of the maxmum quas-lkelhood. In order to avod the over parametersaton the constrants α = β = are ntroduced. Estmate Rˆ s obtaned throughr ˆ = µ ( j, ) ˆj. The standard error (SE) of Rˆ wll be, SE ( Rˆ ) = E ( R Rˆ ) 2 { } = AQE ˆ ( ˆ ) + ˆ, j µ ˆ, jµ x, ycov( ˆ η ˆ, j, η x, y ) (, j ) { } µ,, j ) ( x, y ) (, j ) ( x, y) ( ) ˆ ζ 2 ˆ µ = φ ˆ µ + { V ( ˆ µ ) + V ( ˆ α ) + V ( ˆ β ) + 2{ Cov( ˆ, µ ˆ α ) + Cov( ˆ, µ ˆ β ) + Cov( ˆ α, ˆ β ) } A QE ˆ µ. ˆ, j, j j j j, j Renshaw and Verrall (1998) proposed a stochastc verson of the Chan Ladder method, whch assumes that the ncremental payments follow an over-dspersed Posson dstrbuton, and a lnear predctor wth shape as n Equaton (7) and a logarthmc lnk functon. (In ths GLM the parameter ζ of the varance functon assumes the value 1, but the scale parameter φ s estmated nstead of beng pre-determned. The model has the E I, µ, followng assumpton: ( j ) j =, V ( I ) V ( ) = ϕ µ = ϕµ, ζ = 1, φ > 0, = 0,...,n ; j j j N j Ij 0 and 0 j N. = 0 In the model the scale parameter s estmated usng the approxmaton of the Generalsed Pearson s statstc proposed by McCullagh and Nelder (1989). The model ft s done wth two tests: (1) Wald s nullty test of the lnear predctor parameter and (2) the global 11

12 sgnfcance test usng the scale quas-devance statstc. In addton, we wll analyse the graphcal representaton of Person resduals usng a Normal probablty plot and the graphcal representaton of the resduals aganst the adjusted values of explanatory varables of the lnear predctor. µˆ and aganst each We wll smulate the possble values of R, usng a bootstrap method n assocaton wth the over-dspersed Posson GLM. The Bootstrap method requres the exstence of a set of observatons of d random varables. However, the I, j s do not satsfy ths assumpton snce they depend on the parameters, therefore we wll use the Pearson s resduals of the model, r,j, 0 N, 0 j N, snce they can be consdered as observatons of the random varables. The resduals r 0, N, N, 0 r and r, N 1 wll be dropped snce by defnton they are equal to zero, as exposed by Pnhero, Andrade e Slva and Centeno (2003). The new trangle of resduals wll be converted n a pseudo-data trangle I bs, j usng 0 + bs bs I j r ˆ,, j µ, j + ˆ µ, j = wth r,, satsfyng bs j {(, j) :0 N,0 j N } (, j) ( 0, N 1) { } = + and ˆµ, j as the estmated values. We wll apply the over-dspersed Posson GLM to the pseudo-data trangle n order to bs obtan the reserve estmate, called pseudo-reserve. We use the notaton Rˆ ( b ), 1 b B, for the pseudo-reserves and Rˆ for the orgnal estmate. Ths process s repeated a large number of B tmes. As far as the computaton of the SE s concerned, we ll have to add to the standard devaton of the Bootstrap results, say ˆ σ ( Rˆ ) bs, a volatlty measure of the stochastc process nherent to the over-dspersed Posson GLM. Accordng to England 12

13 and Verral (1999) ths s ˆ φ Rˆ, where φˆ s an estmate of the scale parameter. Fnally, n we obtan the SE of the bootstrap estmates for Rˆ, SEˆ ( Rˆ ) ˆ Rˆ 2 φ + ˆ σ ( Rˆ ) bs = bs. n + p Gven that the total reserve s a sum of the random future payments, ts estmate should equal the dscounted value of the ncremental future payments, dscounted wth an approprate rate, therefore the clams reserve s also subject to nterest rate rsk. In ths paper we wll use the rsk-free nterest rate as an approxmaton to the approprate dscount rate for labltes. We wll smulate the rsk-free nterest rate term structure n one year s tme usng the CIR model as explaned n the prevous secton. The dfference between the expected value of the dscounted reserve today and the dscounted value of the smulated reserve wthn one year wll be the result assocated of the development of the clams reserve (ncludng the correspondng nterest rate rsk). Premum Rsk s assocated wth the premum reserves. In motor nsurance contracts are usually done on a annual bass and premums are receved upfront. Insurers are requred to buld premum reserves to cover future clams of the set of polces n force. Premum rsk s the rsk that those reserves are not suffcent to face these future payments. To calculate the rsk t s necessary to model the future annual clam payments. To fnd the dstrbuton of the aggregate clams cost n the tme nterval, say (0, t], we use the well known Collectve Rsk Model, for detals please see, for nstance, Bowers et al. (1997). Under ths model the aggregate clams cost s wrtten as a random sum of ndvdual clams, denoted S(t): S () t = () t N X = 0, where X 0 0, (Eq. 8) 13

14 wherex s the -th ndvdual random clam and N(t) s the number of clams n (0, t]. { Nt (), t 0} s a stochastc countng process, { } X, = 1,2,... s a sequence of d random varables, wth common dstrbuton functon G(x), and ndependent of N(t), therefore { St (), t 0} s a compound process. In the classcal model N () t follows a Posson dstrbuton. In the applcaton we test both 2 a Posson and a negatve bnomal dstrbutons usng the classcal χ test. For the clam 2 amount dstrbuton we use both the χ and the Kolmogorov tests, for Gama, Pareto and Lognormal dstrbutons. For detals see Klugman, Panjer and Wlmot (1998), Once the dstrbutons chosen, we smulate the process gven by (8). For each smulated path we frst generate a number N ( t) = n, then generate the n values X = x, = 1,..., n. The premum rsk for the set of polces n force, consderng ts remanng tme, s calculated from the dfference between the aggregate clams cost and the expected premum reserve. For the sake of smplcty, and snce the duraton of the premum reserves n the automoble nsurance s usually less than sx months, we wll neglect the nterest rate rsk of premum reserve. For the Aggregaton of Rsks we assume that the jont dstrbuton of rsks follows a multvarate dstrbuton belongng to the Ellptc Dstrbuton Famly, as n Embrechts, McNel and Straumann (1999) and Embrechts, Lndskog and McNel (2003). Thus, the dependence between rsk factors s measured by ther lnear correlaton coeffcents and t wll be so n the smulaton procedures, whenever applcable. 14

15 3. Applcaton 3.1 General consderatons Based on the statstcal and fnancal nformaton of an nsurance undertakng at December 31, 2002, we modelled the fve rsk factors earler explaned. All the necessary techncal nformaton, the prospectus related to the nsurer bond portfolo, the hstorcal bond and stock prces, stock ndex fgures and nterest rates were collected from Bloomberg delvery nformaton system. Gven the lack of nformaton to study the jont behavour of the major rsk factors we assumed that all rsks were ndependent, wth the excepton of the equty and nterest rate rsks for whch we studed ther correlaton. For confdental ssues all monetary values related to the real nsurance undertakng used n ths study are masked. 3.2 Equty rsk The stock portfolo held by the nsurer under study s composed by 11 lsted companes from a sngle Euro Zone country. Therefore, we chose the man representatve stock ndex for that country as a proxy of ts relevant market portfolo wth mpact on the stocks systematc rsk. When estmatng the stocks beta coeffcents usng the Market Model, we used daly closng prces from January 2, 1998 to December 31, 2002 adjusted for dvdends, stock splts and other prce factors. We tested all regresson equatons for ther global sgnfcance (F test) and for all the ndvdual parameters (t test) rejectng the null hypothess at 5% sgnfcance level for all the cases. The results are presented n Table 1. 15

16 Table 1 - Estmates of the ndvdual and portfolo's betas Stock #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 Port. Beta Beta The average and standard devaton of the stock ndex nstantaneous rate of return for the same tme nterval were, respectvely, 3.59% and 18.46% per annum, and the lnear correlaton coeffcent between the nstantaneous short-term nterest rate and the nstantaneous stock ndex rate of return was postve but small, and not statcally sgnfcant. Then, usng the process gven by equaton 2 we ran 5,000 smulatons for the one year stock ndex daly fgures, assumng a tme step = 1/260 per year. From the stock ndex smulated paths we could then estmate the smulated one year portfolo returns for the real nsurance undertakng stock portfolo, accordng to the estmated CAPM parameters. The random component of the smulaton process was based on the generaton of standard Normal ndependent r.v. s. As a proxy for the rsk free nterest rate we used a one year maturty German Treasury Bll yeld, observed at December 31, Interest rate and credt rsks We started by dvdng the nsurer s bond portfolo (entrely composed by Euro Zone bonds) nto three sub-portfolos regardng the ssuer s type: government bonds; bonds ssued by banks and other fnancal nsttutons; and a sngle bond ssued by one telecommuncatons company. These three groups were those actually observed wthn the real portfolo of the nsurance undertakng that we are studyng. Then, we estmated the weghted Fsher-Wel duraton for each sub-portfolo. The rsk free yeld curve was 16

17 extracted from a seres of German zero coupon bonds (coupon strps) wth dfferent maturtes, rangng from 1 to 27 years, and whose prces were observed at December 31, Any ntermedate maturty yeld was estmated by lnear nterpolaton. For the 3 and 6-months maturtes we used the German Treasury Bll yelds observed at December 31, 2002 for these maturtes. Bond cash-flows were dscounted by usng a dscount rate that adds the correspondng rsk free maturty to the relevant credt spread, estmated n accordance to the ndustry sector and the ssuers ratng of the bond. Credt spreads are regularly suppled by J.P. Morgan and could be found n The credt spread for German Treasury Bll and Bonds was assumed to be neglgble and, therefore, null. In order to smplfy the smulaton of the bond portfolo, we assumed that the nterest rate and the credt rsk of holdng any of the mentoned sub-portfolos was smlar to the rsk of holdng an equvalent zero coupon bond wth analogous duraton. Then we smulated the three zero coupon bond prces usng the nterest rate model explaned n Secton 2. As a proxy for the rsk free short term nterest rate we used the German Treasury Bll yeld wth 3-months maturty. In the estmaton process of the parameter a we used the market prces of German Treasury Blls maturng n 3 and 6 months tme and the market prces of Coupon Strps of German Treasury Bonds maturng n 1, 2, 3, 4, 5, 6, 7, 8 e 9 years tme. Hstorcal parameters were estmated usng hstorcal data from January 1, 1998 to at December 31, 2002 and the results were: â ~ =3.0411, bˆ =0.1068, â = and σˆ =

18 We started by usng equaton (6) to smulate the daly behavour of the rsk free short term nterest rate. The random component of the stochastc process was based on the generaton of standard Normal ndependent varables. After, we smulated the prces of the three zero coupon bonds usng equaton (4) consderng no credt rsk. Then, from equaton (5), we calculated the correspondng rsk free yeld and the whole process was repeated 5,000 tmes, havng generated 5,000 values for the one year rsk free yeld. Afterwards, we appled the Credt Metrcs model consderng each actual sub-portfolo average credt ratng (Table 2) n order to ncorporate the credt rsk spread nto the smulatons. Credt ratngs were collected from Standard and Poor s and based upon the JP Morgan ( we bult a ratng transton probablty matrx for the one year tme frame. Table 2 - Ratng and duraton of the bond's portfolo Portfolo Ratng Duraton Government Bonds AAA 2.71 Bonds ssued by banks and other fnancal nsttutons AA 5.08 Bonds ssued by telecommuncaton frms A 2.05 In order to estmate the correlaton coeffcent matrx for the 3 sub-portfolos we used several proxes: the Dow Jones Euro Stoxx Bank Index rate of return as proxy for the banks and fnancal nsttutons bond portfolo; the bond tself for the telecom company bond; and the German coupon strp wth 2.5 years maturty for the government bonds portfolo, whose duraton was The correlaton matrx among the three sub-portfolos of bonds s shown n Table 3 and all the fgures are sgnfcant at a 5% level. 18

19 As a result, we manage to generate, a set of correlated Standard Normal r.v. s from a set of ndependent Standard Normal r.v. s by applyng the Cholesky decomposton, as n Horn and Johnson (1985). Table 3 - Varance-covarance matrx of asset's rate of return proxes Governmental Bank & fn. Inst. Telecom. Company Governmental Bank and fnancal nst Telecom. company Addng up the smulated credt spread to the smulated rsk free yeld for all zero coupon bonds we found the approprate yeld consderng the credt rsk. From ths latter rsky yeld and nvertng (5) we got the future value of each zero coupon yeld, takng nto account both nterest rate and credt rsks. Whenever the smulated ratng was consdered a default, we assumed the bond value to equal the credt recovery rate (smulated by a Beta dstrbuton) tmes the face value of the bond. The process ends up by comparng each smulated value to ts ntal prce n order to calculate the annual rate of return for each zero coupon bond and then by multplyng ths rate of return by ts respectve market value at December 31, Reserve rsk Our nsurer portfolo was recent and not yet stable. Thus, n order to apply any stochastc methods to the clams payments we had to exclude the occurrences for 1997 and 1998, because we know that the payment pattern of those years was sgnfcantly dfferent. We appled the Renshaw and Verral s (1998) model to the clams payments matrx, occurred between 1999 and We dd the quas-devance scale test and concluded 19

20 that the model was globally sgnfcant. We also dd the ndvdual parameter test, and concluded they were sgnfcant wth one excepton, the parameter assocated wth the effect of the occurrence year of One possble economc reason mght be the fact that n ths year clams were almost fully developed. Nevertheless, gven that the matrx s not fully stable and that we only consdered four occurrence years, we decded to keep ths parameter n the model. The above results are presented n Tables 4 and 5. Table 4 - Estmates of the parameters of the over-dspersed Posson's model Test of the nullty of the parameters Standard Parameter Estmate 2 Error W χ() 1 at 5% Concluson U , Statstcally sgnfcant β Statstcally sgnfcant α Not Statstcally sgnfcant β Statstcally sgnfcant. α Statstcally sgnfcant β Statstcally sgnfcant α Statstcally sgnfcant β Statstcally sgnfcant Table 5 - Scale devance test H 0 : The model s adequate Scale devance (D*) 3.02 n 16 p 8 20

21 2 at 2.5% χ( n p ) D* < χ( n p ) at 2.5%: Accept H0 The SE of the total reserve was 14%, whch s a value that we fnd acceptable snce the matrx s not fully stable. The graphcal representatons of the resduals aganst each of the explanatory varables do not seem to show any systematc standards. As a concluson, the over-dspersed Posson model has an acceptable ft to the data. We then appled a Bootstrap procedure assocated wth the valdated model to the pad clams matrx. We smulated 5,000 pad clams matrces and calculated 5,000 values for undscounted value of the clams reserve. We smulated 5,000 tmes the rsk-free nterest rate term structure n a one year perod and determned the dscounted value of the smulated clams reserve n December 31, We got the reserve rsk results subtractng the smulated values at December 31, 2003 from the expected value of the dscounted clams reserve at December 31, 2002 (for clams occurred 1999 and 2002). Regardng the bootstrap results, the SE of the estmated reserve s 15%, n lne the results of the analytc model. The graphcal representatons of the resduals aganst each of the explanatory varables dd not evdence any systematc pattern. 3.5 Premum rsk Frst we ft the dstrbuton of the number of clams per year of the whole portfolo, based on data consstng the number of clams occurred per polcy n the last year. We appled the Ch-square test (wth a 5% sgnfcance level) to the mentoned Posson and negatve bnomal dstrbutons. The parameters of the dstrbutons were estmated by maxmum lkelhood estmaton (MLE). 21

22 From Table 6 we can see that the Posson dstrbuton was clearly rejected. The negatve bnomal was accepted, wth a p-value of 24,3%. Assumng that the number of clams follows a negatve bnomal we determned the parameters of the dstrbuton of the number of clams for the set of polces n force at December 31, 2002, α = 58,211 and p = 0.925, correspondng to the sum m dd negatve bnomal, where m s the number of polces n force at that date. Table 6 - Dstrbutons for the number of clams per polcy No. of clams per polcy Dstrbuton Parameters Estmates p-value Degrees of freedom Posson ( λ ) λ Negatve bnomal ( α, p ) α, p 0.809, Next, we studed, usng a Ch-square test, at 5% level, the ft for the ndvdual clam amount dstrbuton, based on a lst of total cost, clam by clam, of all the clams occurred and reported n We tested a Lognormal, Pareto and a Gamma dstrbuton. We use the MLE for the Lognormal and both the moment and ML estmates for the Pareto and Gamma. From Table 7 we see that the Gamma dstrbuton was clearly rejected as well as the Pareto wth ML estmaton. The dstrbuton that better fts the data s the Pareto, wth parameters estmated by the moments method (MME), however wth just one degree of freedom. The Lognormal was rejected, but f we exclude ts tal (clam amounts above 30,000) one observes that ths dstrbuton has a better ft than the Pareto. Hence, we also performed the Ch-square test to a Lognormal dstrbuton truncated at 30,000 wth a 2 Pareto tal. The latter dstrbuton gves a p-value of 20% and the χ has two degrees of freedom. 22

23 Lognormal, MLE Pareto, MME Pareto, MLE Gamma, MME Gamma, MLE Table 7 - Dstrbutons for modelng the ndvdual clam amount Dstrbuton Parameters Estmates p-value µ σ α k 2, α k α β 89, α β Lognormal, MLE, µ truncated at wth σ Pareto tal, MME α β 2, Degrees of freedom In addton, we performed Kolmogorov tests, at 5% level, for the dstrbutons Lognormal, Pareto and truncated Lognormal wth Pareto tal. All dstrbutons were accepted. Based on the results of both tests we chose the Lognormal dstrbuton wth Pareto tal to model the ndvdual clam amounts. Results are shown n Table 8 Assumng that the number of clams followed a Negatve Bnomal dstrbuton and that ndvduals clams amounts followed a Lognormal dstrbuton wth the Pareto tal, we smulated, usng equaton (8), 5,000 values for the aggregate total cost of clams of the polces n force at December 31, 2002, consderng the remanng tme that they wll be n force. We deducted from the expected value of premum provson, the smulated 23

24 values and acheved the results of the premum rsk (suffency/nsuffency of premum provson). Table 8 - Results of the Kolmogorov test Dstrbuton Test statstc Crtcal value (5%) Concluson Lognormal, MLE 20.75% 33.84% Accepted Pareto, MME 16.33% 33.84% accepted Lognormal, MME, truncated at 30,000, wth Pareto tal, MME 20.75% 33.84% Accepted 3.6 Aggregaton of rsks For the aggregaton of Rsks and calculaton of the VaR and TCE, the global results for the nsurer wll be the result of the aggregaton of all ndvdual rsk factors. We consdered, rsk by rsk, each of the smulated values, obtanng 5,000 possble global results for the next year. Results are shown n the next secton. 4. Results In ths secton, we present the results from our applcaton, consderng the VaR and the TCE measures for one year tme horzon and dfferent alpha levels. We start by presentng the ndvdual analyss for each rsk factor and we conclude wth the aggregated results for the entre portfolo of assets and labltes of the company. 4.1 Equty rsk Table 9 - Equty rsk results (euros) Level 95.0% 97.5% 99.0% 99.5% 24

25 VaR -21,437-29,686-39,189-43,119 TCE -31,885-38,571-45,432-50,258 Startng wth the equty rsk we see from Table 9 that the results are n lne wth the expectatons, gven the model n use and the parameters estmates. Takng nto account the reduced exposure to the stock market and the smulated results, the equty rsk does not seem to be a menace to ths partcular nsurer s solvency. The worst VaR loss scenaro n a one year tme perod wth a 99.5% level s 43,119 and the correspondng TCE s an expected loss of 50,258. Ths s qute small n relatve terms as wll we see later n ths secton after comparng these fgures wth the Reserve and Premum rsks. 4.2 Interest rate and credt rsks Table 10 - Interest rate and credt rsk results (euros) Statstcs Bonds ssued by Banks Bonds ssued by the Governmental and other Fnancal Telecommuncatons Bonds Insttutons company VaR 95.0% 429,139 84,444 29, % 428,076 83,523 29, % 426,668 82,561 29, % 425,892 25,619 18,346 TCE 95.0% 427,684 67,355 20, % 426,694 50,734 10, % 425,490 2,164-17, % 424,638-75,536-54,387 Table 10 shows that both nterest rate and credt rsks are also small for the modelled bond portfolo. In the worst VaR loss scenaro n a one year tme perod wth a 99.5% level the fgures are all postve expressng no defaults, nor sgnfcant losses n the bond portfolo. Addtonally, as the governmental bond portfolo (a hgh ratng and low 25

26 duraton portfolo) was of much hgher sgnfcance the total bond portfolo doesn t seem strongly affected by nterest rate or credt rsk. However, when the TCE s computed, we experence a potental captal loss n terms of both corporate bonds portfolo. As we see, the TCE measure for the smulated fgures n a one year tme perod wth a 99.5% level s negatve ether for the portfolo of bonds ssued by the banks and other fnancal nsttutons (- 75,536) ether for the bond portfolo ssue by the telecommuncatons company (- 54,387). Even though, the total bond portfolo s stll postve n ths scenaro, as a result of the strong weght of the governmental bond portfolo. The negatve results shown n the TCE measure are the result of the smulated bond ratngs downgradng wth a consequent rase n the requred credt spreads. 4.3 Reserve rsk Table 11 - Reserve rsk results (euros) Level 95.0% 97.5% 99.0% 99.5% VaR -1,944,351-2,409,572-2,771,884-2,931,341 TCE -2,488,839-2,743,515-3,005,122-3,158,818 Analysng Table 11, we can observe that there s some reserve rsk arsng from the most adverse development scenaros. The reserve rsk follows approxmately a Normal dstrbuton and even tough t s the second more severe sngle rsk factor; t does not have a heavy tal. 4.4 Premum rsk Table 12 - Premum rsk Results (euros) Level 95.0% 97.5% 99.0% 99.5% VaR -1,510,020-1,916,298-2,658,329-3,146,273 26

27 TCE -2,503,282-3,319,229-4,950,018-7,020,996 From Table 12 we see that the premum rsk s the sngle rsk factor that presents future results more severe to the nsurer. Ths s due to the heavy tal of the estmated dstrbuton. 4.5 Determnaton of the aggregate VaR and TCE Table 13 - Insurance undertakng's aggregate results (euros) Level 95.0% 97.5% 99.0% 99.5% VaR -1,988,506-2,488,230-3,493,925-3,981,377 TCE -3,061,498-3,920,409-5,471,164-7,201,006 From Table 13 we observe that the TCE s expectedly more conservatve as a rsk measure than VaR, by presentng captal requrements (clearly) hgher. Takng TCE as rsk measure at December 31, 2002, the nsurance undertakng would need, at the confdence level of 99.5%, an economc captal of 7,201,006, to be solvent. The dfference between VaR and TCE becomes more sgnfcant as the confdence level ncreases, ths s due to the heavy tal of the global proft and loss dstrbuton, and very nfluenced by the premum rsk heavy tal. TCE s much more senstve to heavy taled dstrbutons. The study of such dstrbutons requres very specal technques and care, whch dscusson s beyond the scope of ths work. 5. Conclusons The model presented had the objectve of showng that t s possble to buld up a solvency model that determnes the economc captal requrement usng the rsk measure 27

28 TCE, based upon the man rsk factors that affect the nsurance actvty and the balance sheet structure of an nsurance undertakng at a gven tme, for a selected tme horzon. In order to buld up the model t s necessary to dentfy the assets, labltes and operatons that generate value, the rsks that affect them, as well as the dependences among them. Ths procedure wll lead to a more sound knowledge of the whole actvty and structure of an nsurance undertakng. The constructon and applcaton of a solvency model for the automoble branch allowed observng that the results obtaned depend heavly on the estmates of the nvolvng parameters and on the data used n the estmaton. The constructon of a solvency model lke ths one wll force nsurance companes to nvest consderably n human resources tranng, nformaton technology, and on the access to databases wth relevant and accurate nformaton. Also mportant and senstve are the correlatons between rsks for the calculaton of captal requrements, snce the benefts of an ncreased dversfcaton mght result nto a consderable lower captal need. As far as the model practcal results are concerned, we conclude that the nsurer had a conservatve nvestment portfolo wth lmted nterest rate and credt rsks. Gven the estmates of the parameters and the reduced exposure to the stock markets, the equty rsk dd not seem to nfluence the nsurer s solvency. Nevertheless, by consderng the volatlty observed n stock markets n recent years, t s possble that the actual future results became less favourable than the smulated ones. From the jont applcaton of the over-dspersed Posson model wth the bootstrap procedure we concluded that the reserve rsk s materal to solvency of the nsurance undertakng. Indvdually, the more potentally demandng rsk s the premum rsk, snce 28

29 the total aggregate clams cost dstrbuton has a heavy tal, as we have already underlned at the end of the precedng secton. As we notced, heavy tal dstrbutons need specal care and an accurate estmaton s not an easy task. An mproper ft can lead to unfar captal requrement calculaton, ether excessve or defectve. Fnally, we remark, that ths s a statc approach that assumes the mantenance of the current asset and labltes structure, not takng nto account, namely, new busness underwrtngs. Thus, ths analyss should be conducted perodcally. In addton, we should pont out that the TCE rsk measure can be shown to be a lot more conservatve than the VaR rsk measure. References Artzner, P. Applcaton of Coherent Rsk Measures to Captal Requrements n Insurance, North Amercan Actuaral Journal, Vol.2 No.2 (1999), Bowers, N.L., Gerber, H.U., Hckman, J.C., Jones, D.A., and Nesbtt, C.J. (1997); Actuaral Mathematcs, 2 n Ed., Shaumburg, Il: The Socety of Actuares Cox, J., J. Ingersoll and S. Ross, A Theory of the Term Structure of Interest Rates, Econometrca, Vol.53, No.2 (1985), Elton, E., Gruber, M., Brown, S. and Goetzmann, W. (2007), Modern Portfolo Theory and Investment Analyss, 7 th Ed., New York: John Wley & Sons. Embrechts, P., A. McNel and D. Straumann, Correlaton: Ptfalls and Alternatves, RISK, Vol.12, No. 5 (1999), Embrechts, P., Lndskog, F. and McNel, A. (2003), Modellng Dependence wth Copulas and Applcatons to Rsk Management. In: S. Rachev (Ed.) Handbook of Heavy Taled Dstrbutons n Fnance. Elsever. England, P. and R. Verral, Analytc and Bootstrap Estmates of Predcton Errors n Clams Reservng, Insurance: Mathematcs and Economcs, Vol.25, No.3 (1999), Fscher, T., A. May and B. Walther. Smulaton of the Yeld Curve: Checkng a Cox- Ingersoll-Ross Model, Pre-prnt No. 2226, Stochastk und Operatons Research, Technsche Unverstät Darmstadt,

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