Risk Margin for a Non-Life Insurance Run-Off

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1 Risk Margin for a Non-Life Insurance Run-Off Mario V. Wüthrich, Pau Embrechts, Andreas Tsanakas February 2, 2011 Abstract For sovency purposes insurance companies need to cacuate so-caed best-estimate reserves and a risk margin for non-hedgeabe insurance-technica risks. In actuaria practice, often the cacuation of the risk margin is not based on a sound mode but various ad-hoc methods are used. In the present paper we propery define these notions and we introduce insurance-technica probabiity distortions. We describe how the atter can be used to cacuate a risk margin for a non-ife insurance run-off in a mathematica consistent way. Key words. Caims reserving, best-estimate reserves, run-off risks, risk margin, market vaue margin, one-year uncertainty, caims deveopment resut, market-consistent vauation. 1 Market-consistent vauation The main task of an actuary is to predict and vaue insurance iabiity cash fows. These predictions and vauations form the basis for premium cacuations as we as for sovency considerations of an insurance company. As a consequence and in order to be abe to successfuy run the insurance business, actuaries need to have a good understanding of such cash fows. In most situations, insurance cash fows are not traded on deep and iquid financia markets. Therefore vauation of insurance cash fows basicay impies the pricing in an incompete financia market setting. Artice 75 of the Sovency II Framework Directive Directive 2009/138/EC states iabiities sha be vaued at the amount for which they coud be transferred, or setted, between two knowedgeabe wiing parties in an arm s ength transaction. The genera understanding is that this amount shoud consist of two components, namey the so-caed best-estimates reserves ETH Zurich, RiskLab, Department of Mathematics, 8092 Zurich, Switzerand Cass Business Schoo, City University, 106 Bunhi Row, London EC1Y 8TZ, UK 1

2 for the cash fows and a risk margin for non-hedgeabe risks. We wi discuss these two eements in detai by giving an economicay based approach how they can be cacuated. The cacuation of the best-estimate reserves is fairy straightforward. Artice 77 of the Sovency II Framework Directive says the best estimate sha correspond to the probabiity-weighted average of future cash-fows, taking account of time vaue of money... the cacuation of the best estimate sha be based upon up-to-date and credibe information.... This simpy means that the best-estimate reserves are a time vaue adusted conditiona mean of the outstanding oss iabiity cash fows, conditioned on the information that we have coected up to today. The cacuation of the risk margin has ed to more discussions as there is no genera understanding on how it shoud be cacuated. The most commony used approach is the so-caed cost-of-capita approach. The cost-of-capita approach is based on the reasoning that a financia agent provides for every accounting year the risk bearing capita that protects against adverse deveopments in the run-off of the insurance iabiity cash fows. Since that agent provides this yeary protection, a reward in the form of a yeary price is expected. The tota of these yeary prices constitutes the so-caed cost-of-capita margin which is then set equa to the risk margin. The difficuty with this cost-of-capita approach is that in amost a situations it is not reay tractabe. It invoves path-dependent, muti-period risk measures; see Sazmann-Wüthrich 8. The muti-period risk measure oadings can in most interesting cases not be cacuated anayticay, nor can they be cacuated numericay in an efficient way because they usuay invove arge amounts of nested simuations. Therefore, various proxies are used in practice. Probaby the two most commony used proxies are the proportiona scaing proxy and the spit of tota uncertainty proxy; see Sazmann-Wüthrich 8 and Wüthrich 11. Reated papers are Artzner-Eisee 1 and Möhr 7. In this paper we present a competey different, more economicay based approach. We argue that the risk margin shoud be reated to the risk aversion of the financia agent that provides the protection against adverse deveopments. This risk aversion can be modeed using probabiity distortion techniques and wi ead to a mathematicay fuy consistent risk margin. Under the proposed method, risk-adusted vaues of insurance cash-fows are cacuated as expected vaues after modifying distorting the probabiities used. This kind of ideas has been used in actuaria practice for a very ong time, however typicay in the fied of ife insurance mathematics, corresponding to the construction of first order ife tabes out of second order ife tabes. Second order ife tabes are expected death/surviva probabiities whereas for first order ife tabes a safety oading is added to insure that the ife insurance premium is sufficienty high. 2

3 We appy these ideas to the context of non-ife insurance iabiities. We study the run-off of outstanding oss iabiities in a chain adder framework. Using probabiity distortions, from the cassica chain adder factors, we deveop so-caed risk-adusted chain adder factors. These factors have a surprisingy simpe form and aow for a natura incusion of the risk margin into our considerations. Reated iterature to these probabiity distortion considerations are, among others, Bühmann et a. 2, Denuit et a. 4, Fömer-Schied 5, Tsanakas-Christofides 9, Wang 10 and Wüthrich et a. 12. The paper is organized as foows. In the next section we define the Bayesian og-norma chain adder mode for caims reserving. Within this mode we then cacuate the best-estimate reserves as required by the sovency directive; see Section 3 beow. In Section 4 we introduce genera insurance-technica probabiity distortions. An expicit choice for the atter then provides the positive risk margin. Finay, in Section 5 we provide a rea data exampe that is based on private iabiity insurance data. We compare our numerica resuts to other concepts used in practice. A the proofs of the statements are provided in the Appendix. 2 Mode assumptions We assume that a fina time horizon n N is given and consider the insurance cash fow vauation probem in discrete time t 0,..., n}. For simpicity we assume that the time unit corresponds to years. We denote the underying probabiity space by Ω, G, P and assume that, on this probabiity space, we have two fows of information given by the fitrations F = F t t=0,...,n and T = T t t=0,...,n. We assume F 0 and T 0 are the trivia σ-fieds. The fitration F corresponds to the financia market fitration and T corresponds to the insurance-technica fitration. In order to keep the mode simpe, we assume that these two fitrations are stochasticay independent under the probabiity aw P; see aso Section 2.6 in Wüthrich et a. 12. Of course, this ast assumption can be rather restrictive in appications, however, we emphasize that it can be reaxed by expressing insurance iabiities in the right financia units; see the vauation portfoio construction in Wüthrich et a. 12. This independent decouping into financia variabes adapted to F and insurance-technica variabes adapted to T impies that we can repicate expected insurance iabiity cash fows in terms of defaut-free zero coupon bonds; see Assumption 5.1 and Remark 5.2 in Wüthrich et a. 12. This is in-ine with Artice 77 of the Sovency II Framework Directive, but needs to be questioned 3

4 if we have no independent decouping into financia and insurance-technica variabes. Insurance iabiity cash fows are denoted by X i,, where i 1,..., I} is the accident year of the insurance caims and 0,..., J} is the deveopment year of these caims. We assume that a caims are setted after deveopment year J and that I J + 1. With this terminoogy, cash fow X i, is paid in accounting year k = i +. This provides the accounting year cash fows over a accident years i 1,..., I} X k = i+=k X i, = I k i=1 k J X i,k i = J k 1 =0 k I X k,. We denote the tota cash fow by X = X 1,..., X n and the outstanding oss iabiities at time t < n are given by X t+1 = 0,..., 0, X t+1,..., X n. Thus, our aim is to mode, predict and vaue the outstanding oss iabiity cash fow X t+1 for every t < n. For the modeing of the cash fows X we use the foowing Bayesian chain adder mode. Mode 2.1 Bayesian og-norma chain adder mode We assume n = I + J and T t = σ X i, ; i + t, i = 1,..., I, = 0,..., J} for a t = 1,..., I + J; conditionay, given Φ = Φ 0,..., Φ J 1 and σ = σ 0,..., σ J 1, we have X i, are independent for different accident years i; cumuative payments C i, = =0 X i, satisfy def. Ci,+1 ξ i,+1 = og 1 C i, Ti+,Φ,σ N Φ, σ 2 for = 0,..., J 1 and i = 1,..., I; σ > 0 is deterministic and Φ, = 0,..., J 1, are independent with Φ N φ, s 2, with prior parameters φ and s > 0, and X 1,0,..., X I,0 and Φ are independent. 4

5 We assume that the insurance-technica fitration T is generated by the insurance iabiity cash fows X i,. This suggests that this is the ony insurance-technica information avaiabe to sove the caims reserving probem. Moreover, since we have assumed independence between F and T we know that the time vaue adustments of cash fows need to be done with defaut-free zero coupon bonds. This immediatey impies that the best-estimate reserves for the outstanding oss iabiities at time t < n have to be defined by R t Xt+1 = k t+1 E X k T t P t, k = E X i, T t P t, k, 2.1 k t+1 i+=k where P t, k is the price at time t of the defaut-free zero coupon bond that matures at time k. This definition of best-estimate reserves provides the necessary martingae framework for the oint fitration of F and T under the measure P which in these terms provides an arbitrage free pricing framework; see Chapter 2 in Wüthrich et a. 12. We have chosen a Bayesian Ansatz in the assumptions of Mode 2.1. The advantage of a Bayesian mode is that the parameter uncertainty is, in a natura way, incuded in the mode, and parameter estimation is canonica using posterior distributions. Moreover, we have chosen an exact credibiity mode see Bühmann-Giser 3, Chapter 2 which has the advantage that we obtain cosed form soutions for posterior distributions. However, our anaysis is by no means restricted to the Bayesian og-norma chain adder mode. Other modes can be soved competey anaogousy, but in some cases one has the rey on simuation methods such as Markov Chain Monte Caro MCMC simuation methodoogy. 3 Best-estimate reserves cacuation In 2.1 we have defined the best-estimate reserves. In this section we cacuate these bestestimate reserves expicity for Mode 2.1. We assume that t I, which impies that at time t a initia payments X i,0 have been observed for accident years i 1,..., I}. For i + > t we then obtain, using the tower property for conditiona expectations note that we aso condition on the mode parameters Φ, 2 E X i, T t, Φ = C i,t i =t i exp Φ + σ 2 /2} + 1 exp Φ 1 + σ 2 1/2 }. 3.1 For a proof, we refer to Lemma 5.2 in Wüthrich-Merz 13. Formua 3.1 impies that we woud ike to do Bayesian inference on Φ, given the observations T t. That is, we woud ike to determine 5

6 the posterior distribution of Φ at time t. This then provides the Bayesian predictors E X i, T t = C i,t i E 2 =t i exp Φ + σ 2 /2} + 1 exp Φ 1 + σ 2 1/2 } T t In Mode 2.1 we can expicity provide the posterior density of Φ, given the observations T t : h Φ T t J 1 =0 exp } I 1 2s 2 Φ φ 2 i=1 t i J =1 exp } 1 2σ 1 2 ξ i, Φ 1 2. The first term on the right-hand side is the prior information about the parameters Φ, the second. term is the ikeihood function of the observations, given the parameters Φ. density immediatey provides the foowing theorem. This posterior Theorem 3.1 In Mode 2.1, the posteriors of Φ, given T t with t I, are independent normay distributed random variabes with Φ Tt N φ t, st 2, and posterior parameters φ t = s t 2 φ s σ 2 t 1 I i=1 ξ i,+1 and s t 2 = 1 s t 1 I σ 2. Theorem 3.1 impies that φ t = E Φ T t = β t ξ t + 1 β t φ, 3.2 with sampe mean and credibiity weight given by ξ t = t 1 I 1 t 1 I i=1 ξ i,+1 and β t = t 1 I s2 σ 2 + t 1 I. s2 Hence, the posterior mean of Φ is a credibiity weighted average between the sampe mean ξ t and the prior mean φ with credibiity weight β t. For non-informative prior information we et s and find that β t 1 which means that we give fu credibiity to the observation based parameter estimate ξ t. For perfect prior information we et s 0 and concude that β t 0, i.e. we give fu credibiity to the prior estimate φ. Using the posterior independence and Gaussian properties of Φ we obtain the foowing coroary for the Bayesian predictors. 6

7 Coroary 3.2 In Mode 2.1 we obtain, for i + > t I, with posterior chain adder factors 2 E X i, T t = C i,t i =t i f t f t 1 1, f t = E exp Φ + σ 2 /2} + 1 } T t = exp φ t + s t 2 /2 + σ 2 / Moreover, f t t=0,...,n are P, T-martingaes for a = 0,..., J 1. This emma has the consequence that, in Mode 2.1, the best-estimate reserves at time t I are given by I R t Xt+1 = i=t+1 J C i,t i J =t i+1 2 =t i f t f t 1 1 P t, i This soves the question about the cacuation of best-estimate reserves for outstanding oss iabiities: it is a probabiity-weighted, time vaue adusted amount that considers the most recent avaiabe information. We now turn to the more chaenging cacuation of the risk margin which covers deviations from these best-estimate reserves. 4 Risk-adusted reserves and risk margin In this section we define the risk margin using the economic argument that a risk averse financia agent wi ask for a premium that is higher than the conditionay expected discounted caim. This wi be achieved by introducing a probabiity distortion on the payments X i, which wi ead to the so-caed risk-adusted reserves R + t Xt+1 at time t. The risk margin at time t can then by defined as the difference RM t Xt+1 = R + t Xt+1 Rt Xt+1, 4.1 which wi be stricty positive under an appropriate probabiity distortion. Before doing this expicity for the Bayesian chain adder mode, we describe the probabiity distortions that we are going to use in more generaity. The crucia idea is that we introduce a density process ϕ = ϕ 0,..., ϕ n that modifies the probabiities in an appropriate way. The probabiity distortion functions introduced by Wang 10 reate to our framework in sufficienty smooth cases. 7

8 4.1 Insurance-technica probabiity distortions An insurance-technica probabiity distortion ϕ = ϕ 0,..., ϕ n is a T-adapted and stricty positive stochastic process that is a P, T-martingae with normaization ϕ 0 = 1. This is exacty the definition given in of Wüthrich et a. 12 and means that ϕ is a density process w.r.t. P, T. For a cash fow X we can then define the risk-adusted units by Λ t,k = 1 ϕ t E ϕ k X k T t. In view of 2.1, the risk-adusted reserves are then defined by R + t Xt+1 = k t+1 Λ t,k P t, k = k t+1 i+=k 1 ϕ t E ϕ k X i, T t P t, k. 4.2 For the choice ϕ 1 the best-estimate reserves and the risk-adusted reserves coincide, but for an appropriate risk averse choice of ϕ we wi obtain a stricty positive risk margin RM t Xt+1. For the atter, it is required that ϕ k X k T t is positivey correated, where in this case using the martingae property of ϕ Λ t,k = 1 ϕ t E ϕ k X k T t 1 ϕ t E ϕ k T t E X k T t = E X k T t. This correation inequaity is often achieved by using the Fortuin-Kasteeyn-Ginibre FKG inequaity from 6, which sometimes is aso caed the supermoduar property. The positive correatedness impies that more probabiity weight is given to adverse scenarios. In order to have time-consistency w.r.t. to risk aversion, we require that Λ t,k t=0,...,n is a P, T supermartingae. This impies that E Λ t+1,k E X k T t+1 T t Λ t,k E X k T t, 4.3 which says that, in expectation, the risk margin is constanty reeased over time. 4.2 Risk-adusted reserves for the Bayesian chain adder mode In the previous section, using insurance-technica probabiity distortions, we have given the genera concept for the cacuation of a positive risk margin. In the present section we give an expicit exampe for the insurance-technica probabiity distortion ϕ that wi fit to our Bayesian chain adder mode. We make the foowing choice: J I ϕ n = exp α 1 ξ i, + α 2 Φ 1 Iα 1 + α 2 φ 1 Iα 1 + α 2 2 s2 1 2 =1 i=1 Iα 2 1 σ }, 4.4 8

9 where α 1, α 2 0 are fixed constants. As wi become apparent beow, the parameters α 1 and α 2 characterize risk aversion: α 1 reates to the process risk in ξ i, and α 2 to the parameter uncertainty in Φ. We then define the insurance-technica probabiity distortion ϕ by ϕ t = E ϕ n T t. Lemma 4.1 ϕ is a stricty positive and normaized P, T-martingae. The proof of the emma is provided in the appendix. theorem. We are now ready to state our main Theorem 4.2 In Mode 2.1 we have, for k > t I and i k J,..., I}, 1 ϕ t E ϕ k X i,k i T t = C i,t i k i 2 =t i f +t f +t k i 1 1, with risk-adusted chain adder factors } f +t = exp φ t + st 2 + σ2 2 2 exp α 2 + I t 1α 1 s t 2 + α 1 σ 2 } + 1. The theorem is proved in the appendix. In view of Coroary 3.2 and Theorem 4.2 we obtain, for t I, the inequaity f +t f t. The posterior chain adder factors f t the best-estimate reserves at time t, the risk-adusted chain adder factors f +t provide provide riskadusted reserves that consider both process risk in ξ i, and parameter uncertainty in Φ. The risk-adusted reserves are then given by R + t Xt+1 = I i=t+1 J C i,t i J =t i+1 2 =t i f +t f +t 1 1 P t, i +, 4.5 and we obtain a positive risk margin RM t Xt+1. Remarks. We observe that it is fairy easy to cacuate the risk-adusted reserves in the Bayesian og-norma chain adder Mode 2.1, a that we need to do is to modify the chain adder factors appropriatey: f +t = f t 1 exp α 2 + I t 1α 1 s t 2 + α 1 σ 2 } The foowing function for t I 0, } τ,t α 1, α 2 = exp α 2 + I t 1α 1 s t 2 + α 1 σ 2, 0 9

10 exacty refects this modification according to the risk aversion parameters α 1 0 and α 2 0. Note that τ,t α 1, α 2 is deterministic and, as stated before, represents the eve of prudence simiar to the construction of the first and second order ife tabes in ife insurance. The parameter α 2 refects risk aversion in the parameter uncertainty and the parameter α 1 refects risk aversion in the process risk. However, α 1 aso infuences parameter uncertainty because in the Bayesian anaysis we do inference on the parameters from the observed information T t. This concept of constructing risk-adusted chain adder factors is by no means excusive to the Bayesian og-norma chain adder mode. It can be appied to other chain adder modes, or even more broady, to every caims reserving and pricing mode. It hence yieds a very genera concept for constructing a risk margin. We have chosen the Bayesian og-norma chain adder mode because of its practica reevance and because it aows for cosed form soutions, heping interpretation. Note that 4.4 gives a specia type of probabiity distortion, other choices coud have been made. The remaining, more economic and reguatory, question then is: which are aternative constructions of insurance-technica probabiity distortions used in practice, and how shoud these be caibrated? 4.3 Expected run-off of the risk margin In this subsection we woud ike to study the expected run-off of the best-estimate and of the risk-adusted reserves. For this, we need to foowing emma. Lemma 4.3 For t I s I 0 we have f +t,s = E f +t T s = f s 1 τ,t α 1, α The proof of this emma immediatey foows from 4.6 and the martingae property of the chain adder factors f t t=0,...,n. Observe that τ,t α 1, α 2 is decreasing in t which gives the super-martingae property 4.3. Moreover, we have the foowing theorem. Theorem 4.4 For t > s I we have for the expected best-estimate reserves E R t Xt+1 T s, F s = I i=t+1 J J C i,s i =t i+1 2 =s i f s f s 1 1 E P t, i + F s, 10

11 and for the expected risk-adusted reserves E R + I t Xt+1 Ts, F s = i=t+1 J J =t i+1 C i,s i t i 1 2 =t i =s i f +t,s f s f +t,s 1 1 E P t, i + F s Note that, in order to proect the expected run-off of the best-estimate reserves and the risk margin for t s I, we aso need to mode the expected future zero coupon bond prices E P t, i + F s. In the next section we give a numerica exampe for this run-off.. 5 Rea data exampe We present a rea data exampe. The data set is a private iabiity insurance cash fow triange. In Tabe 3 we provide the cumuative payments C i, = =0 X i, for i We choose the fina accident year under consideration I = 17 and we assume that a caims are setted after deveopment year J = 16. We then consider the run-off situation at time I for t = I,..., n = 33. Using the parameter choices from Tabe 3 we are abe to cacuate the credibiity weights β t and the posterior means φ t at time t = 17. In Figure 1 we present the prior means φ, sampe prior mean sampe mean posterior mean Figure 1: Prior mean φ, sampe mean ξ t and posterior mean φ t for = 0,..., 15 and t = 17. means ξ t and posterior means φ t based on the data T t with t = 17. We see that the posterior mean smooths the sampe mean using the prior mean with credibiity weights 1 β t ; see aso the credibiity formua

12 Next, we need to provide the term structure for the zero coupon bond prices at time t = 17 in order to cacuate the best-estimate and the risk-adusted reserves. We choose the actua CHF bond yied curve avaiabe from the Swiss Nationa Bank. Finay, we choose the risk aversion parameters: α 1 = 0.02 and α 2 = 1. Now we are ready to cacuate the best-estimate and the risk-adusted reserves, they are given in Tabe 1. These reserves are cacuated under the actua R 17 X 18 R + 17 X 18 RM 17 X 18 reserves under actua ZCB prices nomina reserves, i.e. P 17, k discounting effect discounting effect in % 2.82% 2.90% 4.64% Tabe 1: Best-estimate reserves R 17 X 18, risk-adusted reserves R 17 X 18 and risk margin RM 17 X 18 for the data set given in Tabe 3. CHF bond yied curve and for nomina prices, i.e. P 17, k 1. We observe that the discounting effect is quite sma which comes from the fact that we are currenty in a ow interest rate period. On the other hand we obtain a risk margin of which is 4.54% in terms of the best-estimate reserves. Of course, the size of this risk margin heaviy depends on the choice of the risk aversion parameters. In our case we have chosen these such that we obtain a simiar risk margin as in the cost-of-capita approach under the parameter choices used for Sovency II. If we choose the spit of tota uncertainty approach from Sazmann-Wüthrich 8 with security oading φ = 2 and cost-of-capita rate c = 6.5% see formua 4.2 in 8 we then obtain for nomina reserves a risk margin of which is comparabe to the in Tabe 1. Finay, the baancing between α 1 and α 2 was done such that if we turn off one of these two parameters then the risk margin has simiar size; see Tabe 2. The question of the choice of the risk aversion parameters aso needs input from the reguator. The atter gives the ega framework within which a oss portfoio transfer needs to take pace. This question concerns whether or not the insurance portfoio is sent into run-off. The reguator needs to decide at which state of the economy this transfer shoud take pace between so-caed wiing financia agents. Finay, we cacuate the expected run-off of the best-estimate reserves and the risk margin. Therefore, we need a stochastic mode for the deveopment of the term structure which determines future zero coupon bond prices; see Theorem 4.4. For simpicity we ony consider nomina Swiss Nationa Bank s website: 12

13 R 17 X 18 R + 17 X 18 RM 17 X 18 α 1 = 0.02 and α 2 = α 1 = 0 and α 2 = α 1 = 0.02 and α 2 = Tabe 2: Best-estimate reserves R 17 X 18, risk-adusted reserves R 17 X 18 and risk margin RM 17 X 18 for different risk aversion parameter choices. cash fows for the run-off anaysis which avoids modeing future zero coupon bond prices, i.e. we set P t, k 1 for t, k 17. Figure 3 provides for this case the expected run-off of the bestestimate reserves and the risk margin. 30'000 25'000 20'000 15'000 10'000 5' best-estimate reserves risk margin Figure 2: Expected run-off of the best-estimate reserves E R k Xk+1 T 17, F 17 and the risk margin E RM k Xk+1 T17, F 17 for k = 17,..., n 1. Finay, we cacuate the expected reative run-off of the risk margins defined by w k = E RM k Xk+1 T17, F 17 RM 17 X18 for k 17. We observe that the spit of tota uncertainty approximation v k 1, as defined in Sazmann- Wüthrich 8, gives a simiar picture to the risk margin run-off pattern w k. On the other hand, the proportiona scaing proxy v k 2 ceary under-estimates run-off risks. This agrees with the findings in Wüthrich 11 and refects that the expected caims reserves as voume measure for the run-off risks is not appropriate. 13

14 100% 80% 60% 40% 20% 0% run-off w_k run-off v_k 1 run-off v_k 2 Figure 3: Expected reative run-off of the risk margins w k, k 17, compared to the spit of tota uncertainty approximation v k 1 of Sazmann-Wüthrich 8 and the proportiona scaing proxy v k 2 see aso Sazmann-Wüthrich 8. 6 Concusion We have introduced the concept of insurance-technica probabiity distortions for the cacuation of the risk margin in non-ife insurance. This concept is based on the assumption that financia agents are risk averse which is refected by a positive correation between the insurance-technica probabiity distortions and the insurance cash fows. This then provides, in a natura and mathematicay consistent way, a positive risk margin. For our specific choice within the Bayesian og-norma chain adder mode we have found that this concept resuts in choosing prudent chain adder factors. The prudence margin refects the risk aversion in process risk and parameter uncertainty. We have compared our choice of the risk margin to the ad-hoc methods used in practice and we have found that the quaitative resuts are simiar to the more advanced methods presented in Sazmann-Wüthrich 8. In the present paper we have chosen one specific insurance-technica probabiity distortion because this choice has ed to cosed form soutions. Future research shoud investigate aternative constructions of probabiity distortions according to market behavior of financia agents and it shoud aso investigate the question how these choices can be caibrated. In our exampe, we 14

15 have assumed that the insurance cash fow is independent from financia market deveopments. This has resuted in the choice of the defaut-free zero coupon bond as repicating financia instrument. Future research shoud aso anayze situations where this independence assumption is not appropriate. A Proofs Proof of Lemma 4.1. The strict positivity and the martingae property immediatey foow from the definition of ϕ. So there remains the proof of the normaization ϕ 0 = 1. Using the assumptions of Mode 2.1 and the tower property we obtain note that T 0 =, Ω} J 1 ϕ 0 = E ϕ n = E E ϕ n Φ = E exp Iα 1 + α 2Φ Iα 1 + α 2φ Iα 1 + α 2 2 s 2 /2 } = 1. This proves the caim. =0 Proof of Theorem 4.2. Note that we have C i,k i = X i,k i X i,k i 1, therefore it is sufficient to prove the caim for cumuative caims C i,k i. We first condition on the knowedge of the chain adder parameters Φ, Further, ϕ n = 1 ϕ t E ϕ k C i,k i T t = 1 ϕ t E ϕ n C i,k i T t = 1 ϕ t E E ϕ n C i,k i T t, Φ T t. J =1 =1 I exp α 1ξ, } J 1 =0 exp α 2Φ Iα 1 + α 2φ Iα 1 + α 2 2 s 2 2 σ 2 } Iα This means, that conditionay on Φ, the first term in the brackets is the ony random term in ϕ n. Define Hence, for k > t, ϕ Φ t = E ϕ n T t, Φ = J =1 t I J 1 =0 =1 exp E ϕ n C i,k i T t, Φ = E exp α 1ξ, α 1Φ 1 α 2 1σ 2 1/2 } Iα 1 + α 2Φ Iα 1 + α 2φ Iα 1 + α 2 2 s 2 }. 2 ϕ Φ k C i,k i Tt, Φ. For the ast term, note that ϕ Φ t t=0,...,n is a martingae for the fitration T t, Φ t=0,...,n and that the cumuative caim C i,k i = C i,t i k i =t i+1 exp ξ i,} + 1 ony contains terms for accident year i which are conditionay independent given Φ. This impies that, for k > t, E ϕ Φ k We therefore concude that 1 ϕ t C i,k i Tt, Φ E ϕ k C i,k i T t = Ci,t i ϕ t k i 1 = ϕ Φ t C i,t i E ϕ Φ t k i 1 =t i 15 =t i exp Φ + α 1σ 2 + σ 2 /2 } + 1. exp Φ + α 1σ 2 + σ 2 /2 } + 1 T t. A.1

16 There are three important observations that aow to cacuate this ast expression. The first is that E ϕ Φ t Tt = ϕ t which is the tower property for conditiona expectations. The second comes from Theorem 3.1, namey we have posterior independence of the Φ s, conditionay given T t. This impies that expected vaues over the products of Φ can be rewritten as products over expected vaues. The third observation is that in the expected vaue of A.1 we have exacty the same product terms as in ϕ t except for the deveopment periods t i,..., k i 1}. This impies that a terms cance except the ones that beong to these deveopment parameters. If, in addition, we cance a constants and T t-measurabe terms we arrive at 1 ϕ t E ϕ k C i,k i T t k i 1 = C i,t i =t i E exp I t 1α 1 + α 2Φ } exp Φ + α 1σ 2 + σ 2 /2 } + 1 Tt E exp I t 1α 1 + α 2Φ } T t So there remains the cacuation of the terms in the product of the right-hand side of the equaity above. Using Theorem 3.1 we obtain, for t i,..., k i 1}, E exp I t 1α 1 + α 2Φ } exp Φ + α 1σ 2 + σ 2 /2 } + 1 Tt This proves Theorem 4.2. E exp I t 1α 1 + α 2Φ } T t E exp 1 + α2 + I t 1α1Φ} Tt = exp α 1σ 2 + σ 2 /2 } + 1 E exp α 2 + I t 1α 1Φ } T t } } = exp φ t + s t 2 /2 + σ 2 /2 exp α 2 + I t 1α 1s t 2 + α 1σ Proof of Theorem 4.4. We ony prove the caim for the best-estimate reserves because the proof for the riskadusted reserves is competey anaogous. From Coroary 3.2 we see that φ t Therefore we can concentrate on this term. First we study the decouping of φ t use the credibiity formua for this term we obtain φ t = β t with credibiity weight given by ξ t + 1 β t φ = γ t 1 ξ t 1, γ t 1 φ t 1, s 2 is the ony random term in f t. conditionay given T t 1. If we γ t 1 = σ 2 + t. 1s2 This is the we-known iterative update mechanism of credibiity estimators; see for exampe Bühmann-Giser 3, Theorem 9.6. Therefore, conditiona on T t 1, ξ t 1,+1 is the ony random term in f t. Since a these terms beong to different accident years and deveopment periods for t i,..., J 1} we have posterior independence, conditiona on T t 1, which impies, for k > t I, that 2 2 E C i,t i f t f t 1 1 = E E C i,t i f t f t 1 1 = E =t i E C i,t i T t 1 2 =t i E Ts f t T t 1 E f t Iteration of this argument competes the proof. =t i 1 1 Tt 1 Ts = E Tt 1 C i,t i 1 Ts 1 =t i 1 f t 1 f t T s. 16

17 References 1 Artzner, P., Eisee, K Supervisory insurance accounting mathematics for provision - and sovency capita - requirement. Astin Buetin 40/2, Bühmann, H., Debaen, F., Embrechts, P., Shiryaev, A.N On Esscher transforms in discrete finance modes. Astin Buetin 28/2, Bühmann, H., Giser, A A Course in Credibiity Theory and its Appications. Springer. 4 Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R., Laeven, R Risk measurement with equivaent utiity principes. Statistics & Decisions 24/1, Fömer, H, Schied, A Stochastic Finance. 2nd edition. de Gruyter, Studies in Mathematics Fortuin, C.M., Kasteeyn, P.W., Ginibre, J Correation inequaities on some partiay ordered sets. Comm. Math. Phys. 22, Möhr, C Market-consistent vauation of insurance iabiity by cost of capita. Submitted preprint. 8 Sazmann, R., Wüthrich, M.V Cost-of-capita margin for a genera insurance iabiity runoff. Astin Buetin 40/2, Tsanakas, A., Christofides, N Risk exchange with distorted probabiities. Astin Buetin 36/1, Wang, S.S A cass of distortion operators for pricing financia and insurance risks. J. Risk Insurance 67/1, Wüthrich, M.V Runoff of the caims reserving uncertainty in non-ife insurance: a case study. Zavarovaniski horizonti 6/3, Wüthrich, M.V., Bühmann, H., Furrer, H Market-Consistent Actuaria Vauation. 2nd edition. Springer. 13 Wüthrich, M.V., Merz, M Stochastic Caims Reserving Methods in Insurance. Wiey. 17

18 a.y. deveopment year i φ σ s Tabe 3: Cumuative payments Ci, = =0 X i,, i + 17, parameters φ, σ and s. 18

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