OPTIMAL TRADE CREDIT REINSURANCE PROGRAMS WITH SOLVENCY REQUIREMENTS

Transcription

1 OPTIMAL TRADE CREDIT REINSURANCE PROGRAMS WITH SOLVENCY REQUIREMENTS Lca Passalacqa Dipartimento di Scienze Attariali e Finanziarie La Sapienza, Università di Roma Viale Regina Elena, Roma, Italy lca.passalacqa@niroma1.it ABSTRACT In this paper we stdy the design of the optimal strategy of reinsrance for a particlar line of heavy-tailed non-life type of insrance, namely trade credit insrance. Optimality is achieved by maximizing shareholders expected rate of retrn over a reisrance program composed of a proportional qota share treaty and a complex mlti-layer excess-of-loss treaty with mltiple reinstatements. The work participates to the debate on insrance firm risk management within the Solvency II framework by considering, in particlar, the constraint imposed by the solvency capital reqirement and by assming, both for the insrer and the reinsrer, a pricing principle based on a cost-of-capital approach. De to the natre of this type of insrance, we se a loss model that takes into accont correlations between defalts. To cope with the complex payoff of the non-proportional treaty, the comptation of the loss distribtion is performed within a Monte Carlo framework eqipped with an additional importance sampling variance redction techniqe. Finally, we discss nmerical reslts obtained for a realistic credit insrance portfolio. KEYWORDS Solvency, reinsrance, excess-of-loss, credit insrance.

2 1 Introdction In this paper we stdy the design of the optimal strategy of reinsrance for a particlar line of heavy-tailed non-life type of insrance, namely trade credit insrance. The reinsrance program is formed by the combination of a simple proportional treaty (qota share reinsrance) and a complex mlti-layer excess-of-loss reinsrance treaty with mltiple reinstatements. This combination is often present in the practice of credit insrance as an alternative to the combination of a proportional treaty and a stop-loss treaty. While proportional reinsrance is generally conceived to redce the capital reqirement to levels jdged acceptable by the insrance firm management, the second type of treaties (either excess-ofloss or stop-loss) is designed to protect the insrer against extreme ( catastrophic ) losses. In this work, similarly to [18] and [29], the optimality of the reinsrance program will be inspired to the principle of economic capital allocation from the shareholders point of view, taking into accont the constraint imposed by the solvency reqirement. Differently from past works, a new cost-of-capital inspired pricing principle will be adopted. The investigation is inspired by the debate on insrance firm risk management, and in particlar by the gidelines of the solvency capital reqirements assessed by the Solvency II framework [6]. Trade credit insrance protects sellers from the risk of a byer nonpayment, either de to commercial or political risks. Given the natre of the risks involved, the measrement of risk capital reqires the se of a model acconting for the presence of correlations between defalts, that prodce heavy tails in the loss distribtion. In this sense modeling isses are similar to those involved in the development of the Basel II framework for the adeqacy of banking capital reqirements [2]. Among the large variety of credit risk models acconting for correlations in defalt, either explicitly or implicitly (for a review common references are e.g. [9] and [26]) the model sed in or analysis belongs the class of the so-called factor models, i.e. those models based on conditional independence between defalts. One of the main advantages of the model is that it allows the calclation of the distribtion of the insrance company aggregate loss in a fast semi-analytical way by nmerical inversion of the probability generating fnction (PGF) of the loss distribtion. When the complex payoff of excess-of-loss contracts is taken into accont the semianalytical tractability of the model is spoiled, so that the comptation of the loss distribtion has to be performed within a Monte Carlo framework. In principle this is not a problem, bt there cold be comptational isses when the confidence levels reqired in the valation are large, as those sed in the calclation of risk capitals and solvency margins. For example in the Solvency II framework the solvency capital reqirement shall correspond to the 99.5% confidence level Vale-at-Risk over one year period (see [6], art. 100), while it is common practice for insrance companies to se a confidence level defined of the basis of their credit reptation, which may easily reqire vales of the confidence level larger than 99.9%. To decrease compting time we improve on plain Monte Carlo reslts by the se of an importance sampling techniqe, originally proposed by Glasserman and Li [21], tailored for the application of the model. Timing performance are relevant in the process of 1

3 optimizing the large nmber of parameters of the reinsrance treaties, since the soltion of the optimization problem will also be fond nmerically, with a well established [25, 28] random search techniqe known as simlated annealing. In the existing literatre the valation of excess-of-loss reinsrance treaties has been considered by several other athors, in particlar by Sndt [27] and, more recently, by Mata [19], Wahlin and Paris [30] and Hürlimann [13]. However, all these athors have adopted an approach based on the collective model of risk theory (see e.g. [16] p. 45) which is based on a series of hypotheses the most important being the assmption that claims shold be independent and identically distribted. The rest of paper is organized as follows: in the first section we introdce trade credit insrance and the comptation of the aggregate loss gross of reinsrance; we then discss the reinsrance contracts, we formlate the optimization problem, describe the model sed for the loss comptation and finally provide a realistic nmerical example. 2 Loss distribtion for trade insrance In the following we consider a trade credit insrance company over a single period [0, T ]. In order to follow Solvency II indications and withot loss of generality, we will later assme T = 1 year. At time t = 0 the company holds a portfolio P of polices protecting the insred parties against the risk of losses they may sffer de to defalt of contracts with deferred payments. The natre of the event determining the defalt is contractally specified and can encompass commercial events as well as political events or natral disasters. Ths, the company is exposed to the defalt of the N cstomers, hereafter byers, of the insred parties, where N is typically very large. Trade credit companies generally offer coverage either for single byer or for the whole set of byers trading with the insred party dring the coverage period, even if they are still nknown at the moment the policy is issed. Since insrance conditions (premim rate, evental riders sch as benefit participation, etc.) have to be contractally specified ex-ante, trade insrance companies maintain the right to cap dynamically the amont granted in case of the defalt of each byer to a maximm vale, M i (t), with t [0, T ] and i = 1,..., N. Therefore, nlike a conventional loan or bond, for which there is a fixed face amont that cannot be changed once the loan has been made or the bond prchased, credit insrance is based on limits of coverage that might be varied as risk perception changes. For the same reason one expects the vale a single name trade insrance policy to be smaller than the otherwise financially eqivalent credit defalt swap contract. We define E i (t) as the exposre of the company towards the defalt of the i-th byer, by sbtracting from M i (t) the fraction of the loss retained by the insred. In practice, this is done by mltiplying M i (t) by the (contractally specified) coverage fraction ρ i E i (t) := ρ i M i (t). (1) Typical vales of ρ i ranges from 80% to 90% and generally depend on the geographical area and economic sector to which the byer belongs. 2

4 For simplicity sake we shall frther assme that the byers referring to one given policy are distinct from those referring to any other policy. If this were not the case than E i (t) shold have been nderstood as the aggregate exposre at time t. Moreover, we assme that defalt is an absorbing state for the reference period [0, T ], so that a byer cannot defalt more than once. Let Y i be the byers defalt indicators over the reference time period and τ i the corresponding defalt time, that is 1 if τ i T ; Y i = 1I {τi T } = (2) 0 otherwise; and let c i ( 0, E i (τ i ) ] be a random variable representing the claim amont arising from the insolvency of byer i for whom there was a pre-defalt maximm credit allowance E i (τ i ); the total loss sffered by the company over the reference period is then N L = L[0, T ] = c i Y i. (3) i=1 As pointed ot by the Credit & Srety PML Working Grop [5] while for loans and bonds it is possible to define a loss severity rate, a similar qantity does not exist in trade credit insrance since the exposre is changed dynamically. Possibly, a reference indicator might be provided by the random variable c i /E i (τi 1), with time measred in years. However, the are no pblicly available statistics for c i /E i (τi 1) or similar ratios. On the Eropean market, typical vales for the loss severity ranges between 55% and 70% (see e.g. [31], Exhibit 15). Later on we shall assme that c i = E i (0) i = 1,... N. (4) Let now F (x), x 0 be the distribtion fnction of L defined in the probability space (Ω, P, F). In insrance practice companies often employ internal models for which there is no closed-form expression for F (x), that therefore has to be determined in a nmerical way. Similarly, we shall se in or nmerical example a realistic loss model able to accont for the granlarity of the insred portfolio P. This model will later be described in section 5. From F (x) it is then possible to compte the statistics of the distribtion sed as indicators of risk. These encompass moment-based variables, sch as a) the expected loss EL = E[L]; b) the variance V (L) = V[L]; c) the standard deviation SD(L) = V[L] 1/2 ; and qantile-based variables sch as d) the Vale-at-Risk VaR α at the confidence level α, with α (0, 1), VaR α = inf{l R, P [ L > l ] 1 α} = inf{l R, P [ L l ] α}; (5) 3

5 e) the expected shortfall at confidence level α (ES α ), defined as in [10], ES α = E[L L VaR α ] = 1 1 α E[L 1I {L VaR α }], (6) f ) the generalized expected shortfall at confidence level α (GES α ) GES α = 1 { ( )} E[L 1I {L VaRα}] + VaR α (L) 1 α P[L VaR α (L)], (7) 1 α which is the most poplar coherent risk measre in the sense of Artzner et al. [1]; g) the risk capital at confidence level α RC α (L) = VaR α (L) EL(L). (8) Remark. Notice that VaR α (L) is simply the α-qantile of the loss distribtion VaR α (L) = F (α) := inf{l R + : F L (l) α} (9) where for continos distribtions the generalized inverse fnction F (x) (see, e.g., [11]) coincides with the ordinary inverse F 1 (x). Remark. As already stated, VaR α (L) is not a coherent measre of risk; in particlar it is not sb-additive, so that the relation VaR α (L 1 + L 2 ) VaR α (L 1 ) + VaR α (L 2 ) is not atomatically satisfied for all distribtion of aggregate loss L = L 1 +L 2 (L 1, L 2 > 0). As a conseqence, also the risk capital RC α (L) is not sb-additive. 3 Reinsrance Classical reinsrance can be classified into two types: the proportional and the non-proportional. These two types are often combined in what is defined a reinsrance program. Real reinsrance programs might be fairly complex. In the following we shall stdy a program where the proportional treaty is extremely simple (pre qota share ) while the non-proportional part is more similar to market practice. 3.1 Proportional reinsrance In this case all the characteristics of each risk (exposres, claims, premia, etc.) are proportionally shared between insrer and reinsrer, with the fraction of cession per risk eqal to 4

6 λ i, (λ i [0, 1]) and conseqentially a retention per risk eqal to 1 λ i. In market practice, there exist two forms of proportional insrance: a strictly proportional type of qota share reinsrance for which λ i = λ (i = 1,... N), and a srpls type of reinsrance where the claims are redistribted in sch a way that λ i = max{c i c i, 0} 0 if c i < c i = c i c i c i if c i > c i c i where c i > 0 is the so-called line amont of retained loss by the cedant. In is also common practice of reinsrers to deviate from the proportionality rle by applying a sliding scale commission, that is by increasing the reinsrance premim with increasing sffered losses. Noticeably, the problem of the optimal program for proportional reinsrance was addressed by de Finetti [7] already in 1940 sing a mean-variance approach. 3.2 Non-proportional reinsrance In this case the insrer seeks cover for claims jdged too high. Non-proportional forms of insrance inclde (see, e.g., [20]) 1. stop-loss reinsrance where the reinsred will retain aggregate losses p to a maximm amont C, so that the net loss is L sl = L max{l D, 0} = min{l, D} where D > 0 is called the retention level of the ceding company; 2. largest claims reinsrance where the reinsrer garantees, at time t = 0, that the k largest claims originated by the portfolio P will be covered; 3. excess-of-loss reinsrance where the reinsred retains only a fraction of each claim according to N L xl = L Y i max{c i d, N 0} = Y i min{c i, d} i=1 i=1 where d > 0 is called in different insrance branches the priority or the dedctible. 3.3 The reinsrance program In this work we consider a trade credit reinsrance program composed by a qota share treaty followed by a mlti-layer excess-of-loss treaty with mltiple reinstatements. The latter are treaties formed by J protection layers, each defined by the limit m j, the dedctible l j, and by a nmber of reinstatements r j (j = 1,... J). It is cstomary to refer to the 5

7 layers with the short notation m j xs l j. For each claim amont c i the layer j will provide a coverage Z (j) i defined as } {max{c i l j, 0}, m j Z (j) i = min, (10) as far as the aggregate loss X j = n i=1 Z (j) i is smaller than (r j + 1)m j. The contract might also inclde an aggregate dedctible L j, so that the effective coverage C j provided by each layer is } {max{x j L j, 0}, (r j + 1) m j C j = min. (11) For a fixed nmber J of adjacent layers, i.e. layers for which l j+1 = l j +m j (j = 1,... J 1), the parameters characterizing the treaty can be groped in a (2J +1)-dimensional vector ν of components ν = {l 1, m 1,... m J, r 1,... r J } The loss W sffered by the re insrer and the loss L XL sffered by the cedant are respectively W = J j=1 C j, L XL = (12) n i=1 c i Y i W. Remark: the layer are reinstated in the sense that every time a claim hits a layer there is an extra premim charged to the cedant at a pre-determined rate ϕ j, sally pro rata to the claim size. If Π (j) 0 is the initial premim paid by the cedant for the layer j, the extra premim Π (j) i is given by Π (j) i Z (j) i = ϕ j Π (j) 0 i = 1,... n. (13) m j Typical vales of the rates ϕ j are 100 % and 50%, while for ϕ j = 0 one speaks of free reinstatements. When r j 1 the total premim paid by the cedant to the reinsrer for the j-th layer is therefore a random variable with C (k) j ( Π (j) = Π (j) ϕ j m j = min r j k=1 {max{x j L j k m j, 0}, m j } ) C (k) j, (14). (15) Remark: Notice that the non-proportional form of reinsrance can be mapped to financial credit derivatives (see, e.g. [3]). In this sense stop-loss reinsrance can be exactly replicated 6

8 by writing a call option on the loss generated by portfolio P while largest-claim reinsrance can be replicated by a basket of first-to-defalt and n-to-defalt swaps. Similarly, the payoff strctre of the excess-of-loss treaties considered here finds a conterpart in the strctre of collateralized debt obligation (CDO s), albeit for the absence of the reinstatement mechanism in the latter. 4 Optimization strategy Similarly to the works of [18] and [29] we consider the problem of optimizing the reinsrance program as a problem of efficient allocation of the shareholders capital. Ths, in or approach reinsrance is not expected to minimize rin probability, bt only to maintain risks below a target level, defined at least by the solvency reqirement. Differently from the above mentioned works we consider a pricing principle based on the cost of capital, nder which it will trn ot that it might be difficlt for the cedant to obtain a benefit from reinsrance in terms of expected rate of retrn on the economic capital of the shareholder. The rate of retrn the shareholders expect on their capital depends both on the net operating profit (after taxes) and the constraints fixed by the Reglators. Ths, we need to specify a liability generating portfolio, a loss model, a reinsrance program, and a pricing principle both for the insrer and the reinsrer. On the contrary, we shall neglect taxes and other frictional effects sch as doble taxation, financial distress or agency costs. We shall frther assme that at time t = 0 the shareholders of the insrance company provide a initial capital that is invested in risk-free assets with matrity T. Ths at time t = T the srpls U T is U T = v + Π T L (16) where v = v(0, T ) is the risk-free discont factor over the period [0, T ], Π T is the vale at time T of the aggregate premim and L is the aggregate loss. From eq. (16) it immediately follows that the rin probability is given by P[U T < 0]. In credit trade insrance premia are sally linked to the trnover of the insred; ths the vale at time t of the aggregate premim Π t is in general a random variable p to time T. For simplicity we shall assme that Π t is deterministic, that all payments are made in advance, and that the insrance company invest the premim in risk-free assets with matrity T. We frther define Π := Π 0 = v Π T as the vale of the aggregate premim at time t = 0. Later, we shall frther assme that also the reinsrance premia are deterministic, so that the only sorce of incertitde is the loss generated by the portfolio P. The Solvency II capital reqirement implies that P[( T + Π T L) > 0] α (17) with α = Similarly, a strategic reptational reqirement wold imply that P[( T + Π T L) > 0] α R (18) 7

9 where α R (0, 1) is a confidence level fixed on the basis of the insrer target reptation. Therefore we can assme P[( T + Π T L) > 0] α = max{α R, α} (19) and define, the minimm capital, as the vale of that satisfies and the excess of initial capital δ as P[( T + Π T L) > 0] = α (20) δ = (21) From eq. (19) it follows immediately that vale of the minimm capital at time T, T, shold be T = VaR α (L) Π T (22) so that the vale of the minimm capital provided by the shareholders is ] = v [VaR α (L) Π T (23) Remark. Strictly speaking, nothing prevents VaR α (L) to be smaller than Π T, so we make the additional assmption that > 0. Remark. Notice that nder the hypothesis that δ > 0 the probability β() the company is solvent is larger than α β() := P[( T + Π T L) > 0] > P[( T + Π T L) > 0] = α (24) The expected rate of retrn ρ() for the capital is (assming limited liability) ρ() = E[max { T + Π T L, 0}] For continos distribtions the first term in (25) can be rewritten E[( T + Π T L) 1I {L Lβ() }] = 1 β() ( T + Π T ) 0 = 1 β() ( T + Π T ) Lβ() = 1 β() ( T + Π T ) E[L] + 0 x df (x) ± ( L 1 β() 1 (25) Lβ() x df (x) = L β() ) ES β() x df (x) = 8

10 ) where L β() = F (β() 1. As a conseqence eq. (25) becomes ρ() = 1 β() ( T + Π T ) E[L] + ( 1 β() ) ES β() 1 (26) Since in practice β() α 99.5%, we shall assme β() 1, so that ρ() ρ () := 1 δ T + RC α (L) 1 (27) with δ T = T T. The rate ρ () can be decomposed as if the shareholders had invested in a risk-free asset and a risky fnd with expected retrn i P ρ () = 1 δ v + RC α(l) + δ = i δ + RC α(l) 1 (28) i P = RC α(l) 1 = RC α (L) ] 1 (29) v [VaR α (L) Π T A well established link between the expected rate of retrn and economic capital is provided by the se of the EVA R, or Economic Vale Added (EVA R is a registered trademark of the US conslting firm Stern Stewart). In its simple form: EVA R = NP h (30) where NP is net operating profit after tax (adjsted for varios acconting items), h (the hrdle rate) is the weighted average cost of capital (WACC) reqired by the shareholders, and is economic capital. The hrdle rate h 0 that makes EVA R vanish is h 0 = ρ () i. Ths, for the minimm capital reqirement one has that the expected vale of EVA R is: E() = Π T E[L] h ( VaR α (L) Π T ) (31) For reason that will be clear in the following we prefer the se of the expected rate of retrn to that of EVA R in the statement of the optimization problem. We shall now discss how different pricing principles affect the vales of the expected rate of retrn ρ (). In particlar, we consider two principles of the form Π T = E[L]+δΠ T (L), namely 1. (PP1) a traditional expected vale principle where θ > 0 is a constant, and Π (1) T = (1 + θ) E[L] (32) 9

11 2. (PP2) a Cost-of-Capital (CoC) inspired principle ( ) Π (2) T = E[L] + (µ i) VaR α (L) E[L] = E[L] + (µ i) RC α (L) where i = i(0, T ) is the risk-free interest rate over the period [0, T ] and µ i is an exogenos constant having the dimensions of a rate of retrn over the same time period. Under the first principle and [ ] = v T = v RC α (L) θ E[L] ρ () = i δ + i RC α(l) + θ E[L] RC α (L) θ E[L] (33) (34) i P = i P (ζ) = i ζ + θ ζ θ ζ = RC α(l) E[L] [ E = h v E[L] θ 1 + v h ] ζ v h Notice that i P (ζ) is monotonically decreasing in ζ. For typical vales of i = 5%, θ = 1/3, α = 99.5% the vale i P (ζ) that wold be obtained with a log-normal distribtion of mean 1 and variance 2 is obtained for ζ 7.6, and ths i P (ζ ) 10%. Under the first principle the shareholder is likely to prefer portfolios with a low vale of ζ ( low tails ) and a high vale of E[L]. Under the second principle the sitation is qite different. In fact, one has that (35) (36) = v T = v [ 1 (µ i) ] RC α = [ 1 v µ ] RC α (37) and ρ () = i δ + v µ 1 v µ (38) i P (µ) = v µ 1 v µ µ (39) [ ( )] E() = RC α (L) (µ i) h 1 v µ (40) Eqations (39) and (40) imply that shareholders expected retrn is identical for portfolios having different loss distribtions bt the same risk capital RC α (L). Moreover, since the expected vale of EVA R increases with increasing risk capital, shareholders wold prefer heavy tails since the risk is properly remnerated by the premim paid by the insred. 10

12 Remark. Eq. (35) implies that if θ = 0 then i P (ζ) = i for any loss distribtion; similarly (39) implies that for µ = i then i P (i) = i for any loss distribtion. Remark. It is to verify that the two coples of eqations, respectively eqs. (35) and (36), eqs. (39) and (40), correctly satisfy the relation h 0 = i P i. We now consider the effects of reinsrance. For simplicity we assme that the expected vale of the reinsrance premim in t = T is Π R T and that the reinsred loss relief in t = T is L R. The shareholders expected retrn is then ρ() = E[max { T + Π T L Π R T + L R, 0}] Under strict proportional reinsrance only, with a ceded fraction λ (0, 1), 1 (41) Π R T = λ Π T LR = λ L (42) while the cedant retains a fraction (1 λ) of the premim and of the loss. It easy is to check that the new minimm capital reqirement QS T is eqal to (1 λ) T ; moreover ρ () can be written as ρ QS λ () = i δ + λ + i P (1 λ) = ρ () (i P i) (43) where the sperscript QS indicates that a qota share treaty has been applied and the sbscript λ refers to the parameter of the treaty. Similarly, when excess-of-loss reinsrance after qota share is applied eq. (41) is changed into [ ] E max{ T + (1 λ)π T (1 λ)l Π XL T (ν) + L XL ν, 0} ρ() = 1 (44) where Π XL T (ν) and L XL ν are respectively the premim for the excess-of-loss treaty and the corresponding loss relief for a given set of excess-of-loss reinsrance treaty parameters ν. The total loss sffered by the insrer is now L λ,ν = (1 λ) L L XL ν (45) The new minimm reqirement and expected rate of retrn are respectively XL T (λ, ν) = VaR α ( L λ,ν) (1 λ)π T + Π XL T (ν) (46) ρ XL λ,ν() = i δ + XL T + i XL P (λ, ν) XL T (47) i XL P (λ, ν) = RC α( L λ,ν) 1 (48) v XL T 11

13 4.1 A first optimality problem We shall now adopt as first optimality criterion the maximization of the expected rate of retrn ρ XL P (λ, ν) with respect to the set of parameters {λ, ν} nder the constraint XL T (λ, ν) = κ T, where κ (0, 1) is a target redction factor of the initial minimm capital reqirement. Ths, we define problem PB1 as PB1 sbject to: max λ,ν ρ XL λ,ν() XL T (λ, ν) = κ T (49) Remark. The fact that the released capital is left in the investment portfolio and contribtes to the expected rate of retrn ρ XL λ,ν() at the risk free rate i acts as a barrier against the possibility of fll reinsrance. The soltion of PB1 depends on a) the choice of a pricing principle for the cedant and the reinsrer and b) of a loss model. If, following the indications of the Solvency II framework, we se the pricing principle PP2 for both the cedant and the reinsrer, we then have what we define as problem PB1a. In this case eq. (46) and eq. (48) become XL T ( ) ( ) (λ, ν) = RC α ( L λ,ν) (1 λ)(µ i)rc α L + (µr i)rc γ L XL ν (50) i XL P (λ, ν) = RC α ( L λ,ν) v [ RC α ( L λ,ν) (1 λ)(µ i)rc α ( L ) + (µr i)rc γ ( L XL ν )] 1 (51) where µ R > i and γ (α, 1) are constants fixed by the reinsrer on the basis of its risk aversion and target reptation. In the next section we shall introdce the loss model. Before that, notice that i XL P (λ, ν) > i P iif ( ) ( ) (1 λ)(µ i)rc α L (µr i)rc γ L XL ν > (µ i)rcα ( L λ,ν) ( ) (1 λ)rc α L > RCα ( L λ,ν) + µ R i µ i RC ( ) γ L XL ν RC α (1 λ)l > RC α (1 λ)l L XL ν + µ R i µ i RC ( ) γ L XL ν (52) Eq. (52) shows that in the case µ R = µ and γ = α there cold be an increase in the expected rate of retrn if the risk capital were not sb-additive, that althogh possible is qite nlike. Later on we shall nmerically investigate PB1a in this setting. 12

14 Differently, if we chose principle PP2 for the cedant and PP1 for the reinsrer we obtain problem PB1b. In this case i XL P (λ, ν) > i P iif ( ) (1 λ)(µ i)rc α L θr E [ ] L XL ν > (µ i)rcα ( L λ,ν) ( ) (1 λ)rc α L > RCα ( L λ,ν) + θ ] R µ i E[ L XL ν RC α (1 λ)l > RC α (1 λ)l L XL ν + θ ] R µ i E[ L XL ν (53) RC α (1 λ)l > RC α (1 λ)l L XL ν + θ R µ i E [ ] L XL ν RC α ( L XL ν ( ) )RC α L XL ν where θ R is the risk loading parameter of the reinsrer. In this case it is possible for the cedant to increase the expected rate of retrn by choosing ν in order to have a heavy tailed distribtion of L XL ν. Later we shall nmerically investigate PB1b in this setting by assming θ R = Π T /E[L] A second optimality problem Whenever it is not possible to enhance the expected rate of retrn throgh reinsrance, it is still significant to investigate if there is a trade off between a redction in the expected rate of retrn and the redction in risk, where for the latter we have to specify a risk measre, e.g. the standard deviation of the loss distribtion. We can ths define a second problem PB2 min λ,ν SD( L λ,ν) sbject to: XL T (λ, ν) = κ T ρ XL λ,ν() = constant (54) This problem is very similar to that solved in the pioneering work of de Finetti [7], except for the presence of non-proportional reinsrance, the solvency constraint, the correlations between risks and possibly for the se of a qantile-based pricing principle. However, PB2 is a not as promising as it cold seem at first glance. In fact, whilst the constraint on the capital reqirement (which is essentially a constraint on the qantile of the loss distribtion) is not fixing the standard deviation of the distribtion, it is nevertheless very likely to restrict strongly the accessible range of vales. Later on we shall nmerically investigate this isse by determining the smallest possible vale of the standard deviation 13

15 by solving problem PB2b defined as PB2b min λ,ν sbject to: SD( L λ,ν) XL T (λ, ν) = κ T (55) 5 The loss model The loss model adopted in or stdy is based on the CreditRisk + methodology [4] and is detailed in [23]. It is a credit portfolio loss model developed for financial applications that implements comptational techniqes derived from actarial loss models and is characterized by the fact that the distribtion of the aggregate loss can be obtained via a semi-analytical procedre based on a recrrence eqation, or alternatively on nmerical procedres sch as the fast Forier transform. Extensions of the original model by several athors are collected in [22]. In the model defalts are driven by stochastic common risk factors, that are independent Gamma-distribted random variables s = (s 1,..., s K ), with mean 1 and variances σ = (σ1, 2..., σk). 2 Conditional on these random variables, each defalt indicator Y i is assmed to be Poisson distribted with mean p i p i = p i (s) = p i ( ωi0 + ω i1 s ω ik s K ), i = 1,... N, (56) for some positive coefficients (weights) ω i0,..., ω ik for which K k=0 ω ik = 1, and a positive parameter p i, corresponding to the nconditional mean p i = E[p i (s)]. The choice of the Poisson distribtion, the so-called Poisson approximation, is done in the sake of the analytical tractability of the model and correspond to an approximation of a Bernolli random variable. In the model the Y i may be viewed as a Poisson random variable with a random mean. Nmerical effects of this approximation are discssed in [22] p. 289, and in [23]. Differently, claim amonts c i are assmed to be deterministic. As already stated, the cedant loss, gross of reinsrance, is then given by N L = c i Y i (57) i=1 where N is the nmber of exposres in the cedant portfolio. The risk factor are interpreted as representing different market segments, or sectors in the model jargon. Using Gamma random variables for the risk drivers allows calclation of the distribtion of L throgh nmerical inversion of its probability generating fnction, as shown in [4]. Moreover it trns ot that the distribtion of the nmber of defalts in the portfolio is eqal to the distribtion of a sm of independent negative Binomial random variables. 14

16 5.1 Monte Carlo simlation The analytical tractability of the model is lost when the complex payoff of mlti-layer excess-of-loss treaties with fixed reinstatements have to be considered. In this case to compte the loss distribtion one can revert to Monte Carlo techniqes. Monte Carlo simlation withot importance sampling is particlarly simple. In each replication, it is sfficient to generate the common risk factors s k from the distribtions Γ(y; α k, β k ), k = 1,..., K, with α k = 1 σ 2 k β k = σ 2 k k = 1,..., K. (58) In this way each s k has mean 1 and variance σk. 2 Then one has to generate the vale of the defalt indicators Y i from a Poisson distribtion with mean p i, where p i is calclated according to eq. (56). From the Y i s and the exposres c i the portfolio loss is obtained by sing eq. (57). The se of importance sampling is slightly more complex. Or implementation of importance sampling is based on an exponential twisting approach, originally proposed by Glasserman and Li [21] and flly described in [24]. For the specific case of the CreditRisk + setting, exponential twisting is sed to distort first the defalt indicators Y i (i = 1,... N) and sccessively the risk factors s k (k = 1,... K). Clearly better sampling in the region of large losses is achieved when defalt probabilities are increased. This is done by sing a one-parameter family of the form p i (θ) = p i e θc i p i e θc i + (1 pi ) = p i e θci 1 + p i ( e θc i 1 ), (59) which is ineffective for θ = 0 and ensres that the tilted defalted probabilities p i (θ) are monotonically increasing with θ > 0. The second step of the importance sampling procedre is composed of a similar twist on the s k by some twisting parameters τ k. For a particlar choice of the τ k parameters (see [24]) the likelihood ratio for this two-step change of distribtion depends only on θ and is given by where L = exp { θ L + ψ L (θ) } (60) N ( ψ L (θ) = p i ω i0 e θc i 1 ) ( K N ( α k log 1 β k p i ω ik e θc i 1 )) (61) i=1 k=1 i=1 At this point, nder the θ measre, e θl+ψ L(θ) 1I {L>x} is an nbiased estimator of P[L > x]. To achieve efficient importance sampling θ is chosen to minimize the pper bond on the second moment of this estimator M 2 (θ, x) = E θ [ e 2θL+2ψ L (θ) 1I {L>x} ] P[L > x] 2 e 2θL+2ψ L(θ) P[L > x] 2 (62) 15

17 Exposre Exposre Figre 1: Distribtion of the exposres (in millions of crrency nits) in the portfolio sed for the nmerical evalation. The left plot is in linear-logarithmic scale and shows exposres smaller than one million; the right plot is in logarithmic-logarithmic scale to show the behavior of the high side of the tail. that is by (nmerically) solving the eqation ψ L(θ x ) = x. (63) Eq. (63) can be reinterpreted by saying that the change of measre that optimizes the measrement of P[L > x] is that particlar change which makes x to be the expected vale of the loss nder the distorted measre. Practical se of this importance sampling techniqe reqires to provide the vale of x. In [24] it is shown that an initial gess of x obtained with the PGF inversion techniqe is sfficient to improve in accracy by a factor of abot 5 with respect to plain Monte Carlo. 6 A nmerical evalation We consider a realistic portfolio of polices with N = distinct byers, over a risk horizon of T = 1 year. The inpt data for the calclation of the loss distribtion are the exposres of the byers, from which we determine the claim size assming the hypothesis of eq. (4), i.e. c i = E i (0) (i = 1,... N), the byers nconditional defalt probabilities p i, the nmber of sectors K, the parameters α k, β k (k = 1,... K) of the Gamma distribtions of the risk drivers and the weights ω i,k (i = 1,... N, k = 0,... K). For this analysis we assme an economy with a single macroeconomic sector (K = 1) and an idiosyncratic component of 50 % for all byers (examples for other cases are given in [24]), so that ω i,0 = 0.5 and ω i,1 = 0.5 for i = 1,... N. The credit qality of the byers are compted sing the vales reported by Dietsch and Petey [8] which in trn are based on a very large data sets provided by the trade credit insrance companies Coface and Creditreform. Defalt probabilities are assigned to each 16

18 byer on the basis of an internal indicator of risk class and an indicator of trnover size sing the vales reported in table 3, p. 779 of [8]. Following the analysis done in [23], we assme the coefficient of variation cv i of each byer cv i = V[p i(s)] 1/2, i = 1,... N, (64) E[p i (s)] to be 1.5. The variances of the conditional defalt probabilities V[p i (s)] can be dedced from the above eqation and the knowledge of the defalt probabilities, and sed to determine the parameters of the Gamma distribtion of the risk driver. In a more general framework, see e.g. [22] pag. 249, a mltivariate analysis is reqired to identify the nmber of sectors, the Gamma parameters and the byers weights. The composition of the portfolio is smmarized in tables 1 and 2. The total exposre of the portfolio is abot 9000 M crrency nits. The fraction of large firms is 0.04% (22 exposres larger than 40 M) with the largest exposre being abot 230 M. The distribtion of exposres is reported in linear-log and log-log scale in fig. 1. The calclation of the loss distribtion gross of reinsrance is performed sing three methods: the nmerical inversion of the PGF with the standard Panjer algorithm [4], the plain Monte Carlo method and the Monte Carlo method with importance sampling. The semi-analytical method provides a benchmark for the other two. Moreover, it provides the vale of x = VaR 99.5%, which is sed as inpt for the optimization of the importance sampling techniqe. The calclations are implemented in standard ANSI C code complied with gcc in a Unix environment. Random nmber generation is performed sing the algorithms of the GSL libraries [12]. The reslts obtained in this way are smmarized in Table 4. In general there is a remarkable agreement between the three methods. As discssed in [24] importance sampling increase the accracy by a factor of abot 5. The shape of loss distribtion is shown in fig. 2; notice that it is a heavy tailed distribtion typical of credit risks. The agreement between the nmerical techniqes is frther shown in fig. 3 where the distribtion obtained with the PGF techniqe and the importance sampling techniqe are compared. The figre is in log-scale to allow a better comparison on the right tail. The reinsrance program implemented in the evalation is composed of a fixed nmber J = 3 of consective layers. The parameters vector is then ν = (λ, l 1, m 1, m 2, m 3, r 1, r 2, r 3 ) (65) where r i [0, 5] for i = 1, 2, 3; no other bond are applied. Tables 5, 6 and 7 report the reslts for the problems PB1a, PB2b and PB1b respectively. In the first case we have assmed a one-year risk-free rate of 4% and following Solvency II indications a cost of capital 6% higher than the risk-free rate, ths µ = µ R = 10%. We have also assmed eqal confidence levels for the determination of the risk capital of the cedant and the reinsrer, α = γ = 99.5%. For the second problem we have assmed that θ R 25% by asking θ R to solve the eqation Π T = (1 + θ R )E[L], which in principle wold hold for the cedant; in practice we have assmed that the cedant and the reinsrer had the same risk loading θ. For the last case we have sed the same assmptions of the first case. 17

19 All the three optimization problems have been solved nmerically sing the simlated annealing techniqe [17], in the fast adaptive version [14] provided by Ingberg [15]. Simlated annealing is theoretically well nderstood [25, 28] random-search optimization techniqe which is particlarly sited to solve integer-valed non-convex problems. For each iteration of the search procedre the loss distribtion is compted sing 1000 simlations with importance sampling. The reslts reported in Table 5 confirm that, when the cedant and the insrer se the same pricing principle PP2 with an identical cost of capital, adding an excess-of-loss treaty to a qota share one does not bring a benefit to the insrer in terms of expected rate of retrn since the same coverage cold have obtained with a different pre qota share treaty. Differently, Table 7 shows that a roghly 2% average increase in the expected rate of retrn can be obtained by the insrer if the reinsrer ses pricing principle PP1. This reslt is in qalitative agreement with what reported in the literatre, albeit for different loss models. Finally, Table 6 shows that once the minimal capital reqirement has been fixed, there is very little room for a redction of the standard deviation of the net loss distribtion. As a conseqence, in this case, the efficient frontier in the plane expected rate of retrn vs loss standard deviation compressed in a very small range, with possible improvements of less then 10%. 18

20 References [1] P. ARTZNER, F. DELBAEN, J. EBER, D. HEATH (1999): Coherent measres of risk, Mathematical Finance, 9, n. 3, pp [2] BASEL COMMITTEE ON BANKING SUPERVISION (2006): International Convergence of Capital Measrement and Capital Standards: A Revised Framework Comprehensive Version, available at [3] R. BRUYERE, IT ET AL. (2006): Credit derivatives and strctred credit, John Wiley & Sons Ltd. [4] CREDIT SUISSE FINANCIAL PRODUCTS (1997): CreditRisk + : a credit risk management framework, London. [5] CREDIT & SURETY PML WORKING GROUP, [6] COMMISSION OF THE EUROPEAN COMMUNITIES (2007): Proposal for a directive of the Eropean Parliament and of the Concil on the taking-p and prsit of the bsiness of Insrance and Reinsrance - Solvency II, COD/2007/0143, available at [7] B. DE FINETTI (1940): Il problema dei pieni, Giornale dell Istitto Italiano degli Attari, XI, pp [8] M. DIETSCH, J. PETEY (2004): Shold SME exposres be treated as retail or corporate exposres? A comparative analysis of defalt probabilities and asset correlations in French and German SMEs., Jornal of Banking & Finance, 28, pp [9] D. DUFFIE, J. K. SINGLETON (2003): Credit Risk, Princeton University Press, Princeton. [10] R. FREY, A. J. MCNEIL (2002): VaR and expected shortfall in portfolios of dependent credit risks: conceptal and practical insights, Jornal of Banking & Finance, 26, pp [11] A. J. MNEIL, R. FREY, P. EMBRECHTS (2005): Qantitative risk management, Princeton series in finance, Princeton [12] M. GALASI et. al., GNU scientific library reference manal (2nd Ed.), ISBN: [13] W. HÜRLIMANN (2005): Excess of loss reinsrance with reinstatements revisited, Astin Blletin, 35, 1, pp [14] L. INGBER (1898): Very fast simlated re-annealing, Mathematical Compter Modelling, 12, pp

21 [15] The ASA code is available at [16] R. KAAS et. al. (2001): Modern actarial risk theory, Klwer Academic Pblishers, Boston. [17] S. KIRKPATRICK, C. D. GELATT JR, M. P. VECCHI Optimization by simlated annealing, Science, 4598, [18] Y. KRVAVYCHA, M. SHERRIS, (2006): Enhancing insrer vale throgh reinsrance optimization, Insrance: Mathematics and Economics, 38, pp [19] A. J. MATA (2000): Pricing Excess of loss reinsrance with reinstatements, Astin Blletin,30, n. 2, pp [20] T. MIKOSCH (2004): Non-life insrance mathematics, Springer-Verlag, Berlin. [21] P. GLASSERMAN, J. LI (2003): Importance Sampling for a mixed Poisson model of portfolio credit risk, Proceedings of the 2003 Winter Simlation Conference, S.Chick et al. eds. [22] M. GUNDGLACH, F. LEHRBASS (eds.) (2004): CreditRisk + in the banking indstry, Springer Finance, Berlin [23] L. PASSALACQUA (2006): A pricing model for credit insrance, Giornale dell Istitto Italiano degli Attari, LXIX, pp [24] L. PASSALACQUA (2007): Measring effects of excess-of-loss reinsrance on credit insrance risk capital, Giornale dell Istitto Italiano degli Attari, LXX, pp [25] C. P. ROBERT, G. CASELLA (2004): Monte Carlo Statistical Methods, 2 nd ed., Springer Science, New York. [26] P. J. SCHÖNBUCHER (2003): Credit Derivatives Pricing Models: Models, Pricing and Implementation, John Wiley & Sons Ltd. [27] B. SUNDT (1991): On excess of loss reinsrance with reinstatements, Blletin of the Swiss Association of Actaries, 1, pp [28] P.J.M. VAN LAARHOVEN (1988): Simlated Annealing: Theory and Applications, Klwer Academic Pblisher, Dordrecth, The Netherlands [29] R. VERLAAK, J. BEIRLANT (2003): Optimal reinsrance programs: an optimal combination of several reinsrance protections on a heterogeneos insrance portfolio, Insrance: Mathematics and Economics, 33, pp [30] J. F. WALHIN, J. PARIS (2000): The effects of excess-of-loss reinsrance with reinstatements on the cedant s portfolio, Blätter der Detschen Gesellschaft für Versicherngsmathematik, XXIV, pp

22 [31] A. ZAZZARELLI et. al (2007): Eropean Corporate Defalt and Recovery Rates, , Moody s Special Comment 21

23 Size (trnover in M) firms % Exposre firms % 1 (0,15 to 1) < 1 M (1-7) M (7-40) M > 40 M Total Table 1: The size distribtion for firms in the portfolio sed in the nmerical evalation. Size Risk classes (from low risk class 1 to high risk class 8) Total Table 2: The risk distribtion for firms in the portfolio sed in the nmerical evalation. Risk classes Size classes size 1 size 2 size 3 <1M 1-7M 7-40M 1 (low) (high) Total Table 3: Average annal defalt probabilities (in %) sed in the nmerical evalation. 50% idiosyncratic component PGF inversion plain MC IS MC millions % millions % millions % EL RC 99.5% Table 4: Expected loss and risk capital at the α = 99.5% confidence level gross of reinsrance with N s = simlations. Vales are expressed both in millions of crrency nits and as fraction of the total exposre. 22

24 Probability 6 x Loss Probability Loss Figre 2: Loss distribtion (in millions of crrency nits) gross of reinsrance for K = 1 and 50% idiosyncratic risk IS MC withot reins. IS MC with reins. PGF inversion withot reins Loss Figre 3: Loss distribtion (in millions of crrency nits) with and withot reinsrance for K = 1 and 50% idiosyncratic risk. Importance sampling MC reslts gross of reinsrance are plotted with triangles; they are in good agreement with the continos line that is obtained with PGF inversion. An example of importance sampling MC reslt after reinsrance is also plotted with empty sqared shaped symbols; the reinsrance parameters are the same as in [24]. 23

25 Reqired redction κ parameters λ l m m m r r r gross Expected loss Standard dev Risk Capital Expected Shortfall net after qota share only Expected loss Standard dev Risk Capital Expected Shortfall after qota share and excess-of-loss Expected loss Standard dev Risk Capital Expected Shortfall excess-of-loss only Expected loss Standard dev Risk Capital Expected Shortfall reslts Expected Premim Expected srpls Minimm capital reqirement Expected rate of retrn (%) comparison with pre qota share Pre QS retrn (%) Expected retrn redction (%) St. Dev. redction (%) ES redction (%) Table 5: Reslts for PB1a. The parameters are obtained by maximizing the expected rate of retrn nder a constraint on the fraction of capital reqirement redction. 24

26 Reqired redction κ parameters λ l m m m r r r gross distribtion Expected loss Standard dev Risk Capital Expected Shortfall distribtion after qota share Expected loss Standard dev Risk Capital Expected Shortfall distribtion after qota share and excess-of-loss Expected loss Standard dev Risk Capital Expected Shortfall excess-of-loss only Expected loss Standard dev Risk Capital Expected Shortfall reslts Expected Premim Expected srpls Minimm capital reqirement Expected rate of retrn (%) comparison with pre qota share Pre QS retrn (%) Expected retrn redction (%) St. Dev. redction (%) ES redction (%) Table 6: Reslts for PB2b. The parameters are obtained by minimizing the standard deviation of the net loss nder a constraint on the fraction of capital reqirement redction. 25

27 Reqired redction κ parameters λ l m m m r r r gross distribtion Expected loss Standard dev Risk Capital Expected Shortfall distribtion after qota share Expected loss Standard dev Risk Capital Expected Shortfall distribtion after qota share and excess-of-loss Expected loss Standard dev Risk Capital Expected Shortfall excess-of-loss only Expected loss Standard dev Risk Capital Expected Shortfall reslts Expected Premim Expected srpls Minimm capital reqirement Expected rate of retrn (%) comparison with pre qota share Pre QS retrn (%) Expected retrn redction (%) St. Dev. redction (%) ES redction (%) Table 7: Reslts for PB1b. The parameters are obtained by maximizing the expected rate of retrn of the net loss nder a constraint on the fraction of capital reqirement redction. The reinsrer is assmed to se pricing principle PP1 instead of PP2. 26