The one-year non-life insurance risk

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1 The one-year non-life insurance risk Ohlsson, Esbjörn & Lauzeningks, Jan Abstract With few exceptions, the literature on non-life insurance reserve risk has been evote to the ultimo risk, the risk in the full runoff of the liabilities. This is in contrast to the short time horizon in moels for the total risk of the insurance company, an in particular the one-year risk perspective taken in the Solvency II project, an in the computation of risk margins with the Cost-of- Capital metho. This paper aims at clarifying the methoology for the one-year risk; in particular we escribe a simulation approach to the one-year reserve risk. We also iscuss the one-year premium risk an the premium reserve. Finally, we initiate a iscussion on the role of risk margins an iscounting for the reserve an premium risk. 1 Keywors Reserve risk, premium risk, Solvency II, IFRS 4 phase II, risk margin, Dynamic Financial Analysis, stochastic reserving. Länsförsäkringar Alliance, SE Stockholm, Sween, Gothaer Allgemeine Versicherung AG, Germany 1 Any opinions expresse in this paper are the authors own, an o not constitute policy of our employers. 1

2 1 INTRODUCTION 2 1 Introuction In most risk moels, non-life insurance risk is ivie into reserve risk an premium risk. Reserve risk concerns the liabilities for insurance policies covering historical years, often simply referre to as the risk in the claims reserve, i.e., the provision for outstaning claims. Premium risk relates to future risks, some of which are alreay liabilities, covere by the premium reserve, i.e. the provision for unearne premium an unexpire risks; others relate to policies expecte to be written uring the risk perio, covere by the corresponing expecte premium income. (For technical reasons, catastrophe risk is often single out as thir part of non-life insurance risk, but that lies outsie the scope of this paper.) The mentione risks are also involve in the calculation of risk margins for technical reserves; we will consier the Cost-of-Capital (CoC) approach, which is manatory in the Solvency II Draft Framework Directive, EU Commission (2007). In the Discussion paper IASB (2007) on the forthcoming IFRS 4 phase II accounting stanar, CoC is one of the liste possible approaches to etermine risk margins. In the Solvency II framework, the time horizon is one year, escribe by the EU Commission (2007) as follows: all potential losses, incluing averse revaluation of assets an liabilities over the next 12 months are to be assesse. In the actuarial literature, on the other han, reserve risk is almost exclusively iscusse in terms of the risk that the estimate reserves will not be able to cover the claims payment uring the full run-off of toay s liabilities, which may be a perio of several ecaes; we call this the ultimo risk. If R 0 is the reserve estimate at the beginning of the year an C are the payments over the entire run-off perio, this risk is measure by stuying the probability istribution of R 0 C. This is the approach of the so calle stochastic claims reserving which has

3 1 INTRODUCTION 3 been extensively iscusse in the actuarial literature over the last two ecaes, by Mack (1993), Englan & Verall (2002) an many others. Towars this backgroun, it may not be surprising that it is note in a stuy from the mutual insurers organization AISAM-ACME (2007), that Only a few members were aware of the inconsistency between their assessment on the ultimate costs an the Solvency II framework which uses a one year horizon. It is also note that the use of innovative actuarial methoologies is require to replace the classical ones which are inappropriate. AISAM-ACME (2007, Press release). This is the starting point for the present paper. Our first aim is to help in clarifying the methoological issues for the one-year approach to reserve risk, an in particular to write own a general simulation approach to the problem. A special case of this approach is the bootstrap methos in the context of Dynamic Financial Analysis which are implemente in some commercial software; a short escription of that methoology was given by Björkwall, Hössjer & Ohlsson (2008), Section 3.6. In particular, it is note that the one year reserve risk is the risk in the run-off result, as will be escribe in the next section. Another paper on the one-year reserve risk is Wütrich, Merz & Lysenko (2007). In the special case of a pure Chain-laer estimate, they give analytic formulae for the mean square error of preiction of the run-off result, by them calle the claims evelopment result, uner an extension of the classic Mack (1993) moel. After escribing the simulation metho for the reserve risk in the next section, we will turn to the one-year perspective for the premium risk in Section 3, followe by a iscussion on the role of the risk margin for both types of risks in Section 4. To the best of our knowlege, these issues have not previously been iscusse in the literature. We en this introuction with a iscussion of the time horizon in the

4 1 INTRODUCTION 4 QIS3 calibration 2 within the Solvency II evelopment. In the Technical Specifications for QIS3, CEIOPS (2007), we rea: I Reserve risk stems from two sources: on the one han, the absolute level of the claims provisions may be mis-estimate. On the other han, because of the stochastic nature of future claims payments, the actual claims will fluctuate aroun their statistical mean value. It is not clear from this efinition what the time perspective is; it is easy to get the impression that the quotation is about the ultimo risk: the risk over the entire run-off perio of the liabilities. However, at other places in the QIS efinitions it is clearly state that the perspective is strictly one-year, see e.g. CEIOPS (2006) paragraph 46, where the volatility is escribe as the stanar eviation of the run-off result of the forthcoming year. On the other han, AISAM-ACME (2007) states that QIS3 seems to be consistent with a full run-off approach rather than a one-yearhorizon volatility. The AISAM-ACME stuy shows that for longtaile business, the full run-off (ultimo) approach gives risk estimates that are 2 to 3 times higher than the ones for the one-year run-off result. Of course, the fact the QIS3 figures are consistent with the ultimo risk level for the investigate insurance companies oes not necessarily mean that the calibration has been one with the ultimo perspective in min. 2 When this is written, only a raft version of the technical specifications for QIS4 is publishe: we will therefore refer to QIS3 only, even though it is likely that much of what is sai on QIS3 here will also be vali for QIS4.

5 2 THE ONE-YEAR RESERVE RISK 5 2 The one-year reserve risk The reserve risk over a one-year time horizon is the risk in the oneyear run-off result. If C 1 is the amount pai uring next year, R 0 the opening reserve at the beginning, an R 1 the closing reserve estimate at the en of the year, then the technical run-off result is T = R 0 C 1 R 1. (2.1) The one-year reserve risk is capture by the probability istribution of T, conitione on the observations by time 0. (Note that T is also the ifference between the estimate of the ultimate cost at times 0 an 1.) This is in contrast to the ultimo or full run-off risk, which was escribe above as the risk in R 0 C. The AISAM-ACME (2007) stuy istinguishes between the shock perio, which is the one projection year when averse events occur, an the effect perio, which is the full length of the run-off of the liabilities. The irect effect of a shock is capture by C 1, while the variation in R 1 relates to the effect perio. Hence the one-year risk is, to some extent, affecte also by the risk uring the entire life-span of the liabilities, but not as much as the ultimo risk is. Note. We tacitly assume that an expenses reserve is inclue in the claims reserve above. The question of whether this reserve shoul inflate the reserve risk or not is outsie the scope of this paper. In risk moels, the insurance portfolio is ivie into more or less homogenous segments, e.g. lines of business (LoB). To be able to calculate the reserve risk for a segment an combine it with other risks to the total risk of the insurer, we nee the probability istribution of the run-off result T for this segment. In some cases, this may be one analytically, see the alreay mentione result for the Chain laer metho in Wütrich et al. (2007). In most practical situations, simulation methos will be the only possibility; in the rest of this section we

6 2 THE ONE-YEAR RESERVE RISK 6 will escribe the steps of such a simulation, in the case when we neither use iscounting nor a a risk margin to the reserves. This simulation algorithm is by no means new, but, to the best of our knowlege, it has not been iscusse in the literature before, except for the short escription in a bootstrap context in Björkwall et al. (2008). Wütrich et al. (2007) conition on the observe part of the claims triangle. Along the same line, we will conition on all observations up to time 0, the start of the risk year, an enote the collection of these ranom variables by D 0. For simplicity, we will assume that the one-year risk is compute on January 1 for the next calenar year. 2.1 Step 1, the opening reserve We have available the actuary s best estimate of the outstaning claims at the beginning of the year, the opening reserve R 0, which is not consiere stochastic here since it is base on observe values. We assume that this reserve was compute by a ocumente algorithm A that coul be repeate in the simulations. For a long-taile business such an algorithm might, e.g., be to use a evelopment factor metho on pai claims, with the factors smoothe an extene beyon the observation years by some regression moel; then a Generalize Cape Co (GCC) metho may be use to stabilize the latest years, while the earliest years reserves might be ajuste somehow by the claims incurre. For a escription of the GCC, see Struzzieri & Hussian (1998). The actuary s subjective jugement on iniviual figures can usually not be inclue in such an algorithm. If such jugement was use, we woul have to fin an approximate algorithm A, capturing the main features of the estimate, since the algorithm will be use in the simulation, see Step 3.

7 2 THE ONE-YEAR RESERVE RISK Step 2, generating the new year The next step is to simulate the events of the risk year, conitional on D 0. As a minimum, this will inclue claims pai uring the year, for each origin year. Let C ij be claims pai for origin year i an evelopment year j an let n be the ultimo year, when all claims are finalize. We nee to simulate a new iagonal in the evelopment triangle; in Figure 2.1, the simulate ranom variables are marke with bol-face. For long-taile business, it might be that no origin years are finalize, Development years origin years n 1 n 1 C 11 C 12 C 13 C 1,n 1 C 1,n 2 C 21 C 22 C 23 C 2,n 1 C n,n 3 C 31 C 32 C 33 C n,n n 1 C n 1,1 C n 1,2 C n,3 n C n,1 C n,2 Table 2.1: Development triangle with simulate variables in bol-face; the other variables belong to D 0. in which case we may (or may not) want to a an observation to year 1 as well. There are several possibilities to get the simulate iagonal an this is really outsie the scope of this paper; one possibility is bootstrapping, see for e.g. Björkwall et al. (2008), Section 3.6; another possibility is to simulate from a normal or log-normal istribution with mean given by the Chain-laer an variance given by formula (3) of Mack (1993). The sum along the new iagonal yiels the claims pai uring the year for historical origin years, enote C 1 above. For some reserving methos A, we will also nee a new iagonal of claims incurre or other quantities; we will not go into etails here,

8 2 THE ONE-YEAR RESERVE RISK 8 but just assume that all variables neee by the actuary for reserving can be simulate for the new year. 2.3 Step 3, the closing reserve In this step we generate the closing reserve R 1, the reserve as estimate by the en of the risk year. The iea is, for each simulate outcome in Step 2, to calculate R 1 by the metho A from Step 1, which is (an approximation of) the reserving metho use in practice when calculating R 0. It is easy to let the reserving metho inclue tail estimation by regression or ajustments mae by an algorithmic such as the Cape Co metho. The Bornhuetter-Ferguson metho might be a problem though: are the prior ratios really fixe or are they in effect a function of observe claims an other ranom variables? In the latter case the functions shoul be inclue in the simulation set-up an thus the ratios woul contribute to the risk; if not, they are constants which o not a anything to the risk in our opinion, popular as it might be to view Bornhuetter-Ferguson prior ratios as fixe, this is strange from a risk perspective since, of course, the ratios o not give a perfect estimate of the liabilities. We now have all the components neee to calculate the run-off result T in equation (2.1), for each of our, say, B simulations; the empirical istribution of these run-off results is our estimate of the probability istribution of T. The B run-off results can also be use further on in a risk moel to interact with other risks in orer to get the total risk an other aggregate risks. From the probability istribution of T we can, of course, get the stanar eviation, any Value-at-Risk (VaR) figure, or whatever risk measure we choose. Note. In many cases one woul wish to inclue (claims) inflation

9 3 THE ONE-YEAR PERSPECTIVE ON PREMIUM RISK 9 in the above calculations by initially ajusting the pai claims triangle for historic inflation an at the en of the calculations recalculate the result in running prices, by using some assume future inflation. The outcome of the inflation for the risk year, as well as the assume future inflation at the en of that year, shoul preferably be stochastic. We will not go into further etail on inflation here. 2.4 Discussion A problem with the one-year is risk that the reserves for long-taile business might change so little over one year that /... / it shoul not be a surprise that some long tail business where averse movements in claims provisions emerge slowly over many years require less solvency capital than some short tail business /... /. (AISAM-ACME, 2007). In our experience, this really happens in practice. This is a general problem with the one-year horizon of the Solvency II frame-work, which relates to risks that coul appear in the financial statements over one year an oes not take the long-term nature of insurance into account, right or wrong. Of course, mixing an ultimo perspective for liabilities with a one-year perspective for assets is not an alternative, if we are intereste in the combine total risk of the company. These problems inicate that for internal risk moels, it might be a goo iea to consier a longer risk perio, say three or five years. Note that it is straight-forwar to exten the simulation metho escribe above to two or more years. 3 The one-year perspective on premium risk In the QIS3 technical specifications, CEIOPS (2007), premium risk is introuce as follows.

10 3 THE ONE-YEAR PERSPECTIVE ON PREMIUM RISK 10 I Premium risk relates to policies to be written (incluing renewals) uring the perio, an to unexpire risks on existing contracts. In the specification for input calculations in I.3.231, the historic loss ratios, from which the volatility is calculate, are given by the estimate cost at the en of the first evelopment year, ivie by earne premiums. This inicates that the premium risk here is the risk in the cost ( C 1 + R 1 ), where C 1 are first year payments for the current origin year an R 1 is the (closing) claims reserve for the same year. In a simulation moel, risks are formulate in terms of a profit/loss result; if P is the earne premium expecte for the year an E the operating expenses, the result is T = P E ( C 1 + R 1 ), (3.1) which gives a premium risk of the above type, if P an E are nonranom. While we agree with the fact that the technical results for one year are mainly affecte by the first years payment an the initial claims reserve, we woul also like to clarify the role of the premium reserve in this context. In toay s accounting, the premium reserve is compute pro rata temporis an then an aitional provision for unexpire risks is ae, if so require by the outcome of a liability aequacy test. From an economic perspective, however, this reserve is not very ifferent from the claims reserve, only that it relates to claims that have not yet occurre, but for which we have a contractual liability; another ifference is that it only covers a part of the expecte liabilities for next origin year, in our experience % epening on the LoB. By this view, the premium reserve shoul only cover expecte claim costs (incluing hanling expenses). In particular, the insurance company might recognize part of the profit at the inception of an insurance contract, while the rest of

11 3 THE ONE-YEAR PERSPECTIVE ON PREMIUM RISK 11 the profit is recognize as the liabilities are run off. 3 If this perspective is taken on the premium reserve, one-year risk moels shoul ieally inclue the risk in that reserve for existing contracts plus the risk in contracts we expect to write/renew uring next year, the latter with expecte premium income P. Let Ũ t ; t = 0, 1 be the opening an closing premium reserve (the reserve for unexpire risks on existing contracts). Then the technical result for the current year is T = Ũ 0 + P E ( C 1 + R 1 ) Ũ 1. (3.2) If premium cycle variation is moele, P is stochastic. We will consier it as fixe here, i.e. we moel the risk inherent in the premium volume the company expects to receive; we also assume that expenses will equal the bugete value so that E is fixe. There are several ways to simulate the claim cost ( C 1 + R 1 ), one possibility being to simulate the corresponing loss ratio, with volatility estimate as in QIS3, as iscusse above. Another possibility suggeste by Kaufmann, Gamer & Klett (2001) is to simulate the ultimo loss ratio (from historical loss ratios) an thus get the total claims cost, then split that cost into pai the first year an claims reserve, by using a beta istribution for the proportion pai the first year, see their Equation (2.27). Yet another possibility is to simulate frequency an severity separately. Since we conition on D 0, Ũ 0 is non-ranom. Similar to the case with the claims reserve, we suggest that Ũ 1 is compute by the same rule as Ũ 0, but on the ata incluing the simulate year. (That rule might be similar to a liability aequacy test.) Until the new accounting rules are clear, one might choose to stick to the simplifie premium risk in (3.1). 3 In fact, the name premium reserve is not really aequate uner this interpretation; inee, we might just have one reserve for all liabilities, where the former premium reserve correspons to an extra origin year in the claims reserve, with yet no observation. For convenience, we stick to the term premium reserve in this paper, though.

12 4 RISK MARGINS AND DISCOUNTING 12 4 Risk margins an iscounting In the Solvency II frame-work as well as the IASB iscussion paper on the forthcoming accounting stanars IFRS 4, phase II, reserves are iscounte by a yiel curve of risk-free interest rates. Furthermore, a risk margin is ae to the reserves. In this section, we shall investigate the relation between the risk margin an the one-year insurance risk. The reaer shoul note that this section is tentative an not in all parts base on our own risk moelling practice. For reserve risk, let R t ; t = 0, 1 be the iscounte best estimate an M Rt ; t = 0, 1 the iscounte risk margin of the opening an closing claims reserve, respectively. Then the iscounte run-off result with risk margin is T R = (R 0 + M R0 ) C 1 (R 1 + M R1 ), (4.1) where C 1 is the payments uring year 1. For premium risk, let R 1 be the iscounte claims reserve for the current origin year an M R1 the corresponing iscounte risk margin. Furthermore, let U t ; t = 0, 1 be the iscounte premium reserve an M Ut ; t = 0, 1 the corresponing iscounte risk margin. Then the iscounte technical result for the current year, incluing risk margins, is T P = (U 0 + M U0 ) + P E ( C 1 + R 1 + M R1 ) (U 1 + M U1 ). (4.2) By aing (4.1) an (4.2) we get the result for the segment, from which we fin the risk of the segment, conitional on D 0. Note that, ue to iscounting, the expecte result from any segment woul ten to be negative; a remey is to a an investment income transferre from financial operations, calle I here. In our opinion, this quantity shoul be etermine to meet the iscounting, so that if all other things were equal (in particular the cash-flows equal their expecte values), the run-off result woul be zero.

13 4 RISK MARGINS AND DISCOUNTING 13 Conceptually the risk margin is the extra amount, besies the expecte value, a thir party woul eman for a transfer of the liabilities. (Note that the risk margin is sometimes calle the market value margin.) If reserves where trae on a liqui market, the probability istribution of any risk margin coul be estimate from observations of that market. In reality such markets o not exist an some proxy must be use. In the Solvency II preliminaries, the preferre metho is Cost-of-Capital (CoC), which is also one of the options set up by IASB (2007) for IFRS 4, phase II. In QIS3, the risk margin is compute per segment an iversification effects between segments are not taken into account when aggregating; we will use the same approach. Within segments, we will assume that only one risk margin is compute, i.e. iversification effects between the two reserves are taken into account. If separate margins are still require, these can be erive by some special metho for allocating the iversification effect to the two reserves, such as the Shapley metho, see e.g. Lan, M., Vogel, C. & Gefeller, O. (2001). Technically this means that we use a combine risk margin M t M Rt + M P t for the premium reserve an claims reserve.. = Now, by aing (4.1) an (4.2) we get T = (U 0 + R 0 + M 0 ) + P + I E C 1 (U 1 + R 1 + R 1 + M 1 ), (4.3) where we have chosen to let C 1 now enote all payments mae uring year 1. In the CoC metho, M t is the cost of the solvency capital require for running off the liabilities completely, i.e. of the sum of the consecutive capital amounts require to run the business for each run-off year until the ultimate year. Since that capital for each year is the risk in the one-year result, it may look as though there is a circular reference here: the risk epens on the risk margin while the risk margin epens on the risk. However, this is not the case, as will be shown in the next subsection, where we go into etails for the CoC

14 4 RISK MARGINS AND DISCOUNTING 14 approach. 4.1 The CoC metho an the one-year risk perspective In theory, the CoC metho requires a risk calculation for all years until complete run-off, so the time scale will now be t = 0, 1, 2,..., n, where n is the ultimate year of the run-off of the company s liabilities at time t = 0. Denote the collection of the ranom variables that are observe up to an incluing calenar year t by D t. This generalizes our previous notation D 0 enoting the history up to the point in time at which we o our calculations. To compute the risk margin, we consier the company to be in run-off at the beginning of year 1; in particular, no premiums are written. Let ˆR t be the iscounte best estimate of the entire liabilities for the segment at the closing of year t; M t is the corresponing risk marginal. The run-off result T t T t = for year t is ˆR t 1 + M t 1 + I C t ˆR t M t. (4.4) Note that the payments C t uring the year now relate to the contractual liabilities only, since there are no new policies written in the run-off situation. Let VaR(L) enote the Value-at-Risk for the loss L at the chosen level in Solvency II the level is 99.5%, an so VaR is the 99.5% quantile of the loss istribution: if L is continuous then P r{l VaR(L)} = 99.5%. Let SCR t 1 enote the solvency capital set up at the closing of year t 1, require to run the business uring year t. Let us try to unwin this risk backwars. Since liabilities are com-

15 4 RISK MARGINS AND DISCOUNTING 15 pletely run off by year n, ˆRn = M n = 0 an T n = ˆR n 1 + M n 1 + I C n ; SCR n 1 D n 1 = VaR( T n D n 1 ) = VaR(C n D n 1 ) ˆR n 1 ˆR n 1 M n 1 I, where we have use the fact that an M n 1 are non-ranom when we conition on D n 1. Note that, since the best estimate is unbiase, an I is the capital income that meets the iscounting, we have E(T n ) = M n 1, which is exactly the cost of proviing the capital SCR n 1, so that after this capital has been withrawn, the expectation is zero. From now on, we raw this annual CoC from the one-year result. In the current case of year n, the result is SCR n 1 D n 1 = VaR( T n D n 1 ) = VaR(C n D n 1 ) ˆR n 1 I, (4.5) where I is not the same amount as before, but still efine to meet the iscounting for the terms left in the equation. For t = 1, 2,..., n 1, first note that the risk margin in the accounts at t 1 is the one we expect to nee at the closing of year t, plus the CoC for year t, i.e. M t 1 = E(M t D t 1 ) + α SCR t 1, (4.6) where α is the CoC rate above risk-free interest rate, often chosen to be 6%. The term α SCR t 1 is again rawn from the result, as iscusse above. Then SCR t 1 D t 1 = VaR( T t D t 1 ) = (4.7) = VaR(C t + ˆR t + M t D t 1 ) ˆR t 1 E(M t D t 1 ) I. Together with (4.6), we get an equation system that is, in principle, solvable by backwars recursion, where the starting value is the SCR n 1 that was foun in (4.5). This shows that there really is no circular reference in letting the risk margin enter into the SCR calculation, as claime earlier. However,

16 4 RISK MARGINS AND DISCOUNTING 16 solving the above equations is impracticable, even with simulation: let us look at the next-to-last year an try to fin SCR n 2 : SCR n 2 D n 2 = VaR[C n 1 n 1 + ˆR + α VaR( T n D n 1 ) D n 2 ] n 2 ˆR α E[VaR( T n D n 1 )] I. If the computation of VaR( T n D n 1 ) requires simulation, then for each outcome of the ranom variables in year n 1, we must o a complete simulation of year n. So if we use B simulations in general, we here nee B B iterations. Going one more year backwars requires B B B simulations, etc. This calls for simplification Simplifie CoC metho, using uration A common simplification of the CoC calculation is to assume that the SCR in each year is run off at the same expecte rate as the reserve. For some c, the risk margin before iscounting is then M 0 = α c SCR 0. Here the CoC rate α is fixe (say 6%) an the calculation of SCR 0 will be iscusse below. The factor c is the grossing up factor for all future SCR. It is assume that there is a known payment pattern p 1, p 2,..., p n, with t p t = 1; in practice this woul be estimate from the claims triangle, but it is not subject to ranom fluctuation in this moel. Since the capital require in any year t is the sum of all future SCR, it is reaily seen that n n n s n c = p s = p s = sp s, (4.8) t=1 s=t s=1 t=1 s=1 which is simply the (average) uration of the reserve 4. Hence another way of looking at this well-known approximation of the SCR neee for CoC is to say that we are holing the entire initial SCR for as long as the (average) uration of the liabilities. The simplifie approach implies that M t is not stochastic an so the two terms containing it in (4.7) cancel an we coul leave the risk 4 There may be a ifference of 0.5, if we assume payments to be mae at July 1.

17 4 RISK MARGINS AND DISCOUNTING 17 margin out of the SCR-calculation, as is one in QIS3. The initial SCR 0 that we nee for the CoC calculation woul then result from SCR 0 = VaR(C 1 + ˆR ) 1 ˆR 0 I, (4.9) where we have eliberately use a notation without the efault conitioning on the claims history D 0. The simulation approach to this kin of risk calculation was iscusse in the first two sections of the paper. This finishes the specification of the simplifie CoC approach. Note. While the risk margin can be left out of the SCR-calculation, it must be taken into account when calculating the own funs in the balance sheet, since it is a funamental part of the approximation of a market value of the reserves. Acknowlegements This work has benefitte a lot from iscussions with Peter Englan, EMB, who introuce the basic ieas of the one-year reserve risk. The authors are also grateful to Jörgen Olsén, Guy Carpenter, an Susanna Björkwall, Länsförsäkringar Alliance, for valuable comments on a preliminary version of this paper.

18 5 REFERENCES 18 5 References AISAM-ACME (2007). AISAM-ACME stuy on non-life long tail liabilities. 17 october Available online at Björkwall, S., Hössjer, O. & Ohlsson, E. (2008). Non-parametric an parametric bootstrap techniques for arbitrary age-to-age evelopment factor methos in stochastic claims reserving.. Mathematical Statistics, Stockholm University, Research Report 2008:2. Available online at: CEIOPS (2006). Quantitative Impact Stuy 2: Questions & Answers. CEIOPS-FS-12/06. CEIOPS (2007). QIS3 technical specifications. CEIOPS-FS-11/07. Available online at: IASB (2007). Preliminary Views on Insurance Contracts. Discussion paper. Available online at EU Commission (2007). Draft Solvency II Framework Directive. European Commission, July Available as COM(2007) 361 online at: Englan, P. D. & Verrall, R. J. (2002). Stochastic claims reserving in general insurance. Presente to the Institute of actuaries, 28 January Available online at: Kaufmann, R., Gamer, A. & Klett, R. (2001). Introuction to Dynamic Financial Analysis. ASTIN Bulletin, 31, pp

19 5 REFERENCES 19 Lan, M., Vogel, C. & Gefeller, O. (2001). Partitioning methos for multifactorial risk attribution. Statistical Methos in Meical Research 2001, 10, pp Mack, T. (1993). Distribution-free calculation of the stanar error of chain-laer reserve estimates. ASTIN Bulletin, 23, pp Struzzieri, P. J. & Hussian, P. R. (1998). Using Best Practices to Determine a Best Reserve Estimate. The CAS Forum, Fall Available online at: Wütrich, M. V., Merz, M. & Lysenko, N. (2007). Uncertainty in the claims evelopment result in the chain laer metho. Preprint, ETH, Zürich. Available online at: wueth/papers2.html

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