On the computation of the aggregate claims distribution in the individual life model with bivariate dependencies

Transcription

1 On the computation of the aggregate claims distribution in the individual life model with bivariate dependencies Carme Ribas, Jesús Marín-Solano and Antonio Alegre July 12, 2002 Departament de Matemàtica Econòmica, Financera i Actuarial Universitat de Barcelona Av. Diagonal, Barcelona, Spain Abstract In this paper, we focus on the computation of the aggregate claims distribution in the individual life model when the portfolio is composed of independent pairs of dependent risks. First, we prove that the bivariate probabilities associated to each couple under any intermediate positive (negative) dependency hypothesis about their mortality can always be written as a convex linear combination between the independent and the comonotonic (mutually exclusive) ones. Considering this structure for the bivariate probabilities, we then obtain two recursive schemes for computing the distribution of the aggregate claims of the portfolio. These recursions greatly facilitate the computation of the aggregate claims distribution of a life insurance (sub)portfolio exclusively composed of dependent couples and ease the interpretation of the impact of the dependence on the associated stop-loss premiums. Numerical results are given to demonstrate the applicability and efficiency of the method. Keywords: Individual life model, aggregate claims distribution, (in)dependent risks, comonotonic risks, mutually exclusive risks, stop-loss order. Corresponding author. Telephone/Fax adress: cribas@eco.ub.es 1

2 1 Introduction Approximating the distribution of the aggregate amount of claims incurred in an insurance portfolio over a fixed reference period has been one of the main topics in the actuarial literature during the past two decades. In the individual life model, exact calculations are easily obtained since 1986, when De Pril derived an exact recursive scheme for computing this distribution. An efficient reformulation of this algorithm is given in Waldmann (1994). Dhaene and Vanderbroek (1995), proposed a new recursive scheme for the exact computation of the aggregate claims distribution in the individual risk model with arbitrary positive claim amounts. This algorithm contains the Waldmann (1994) recursion as a particular case. The individual life model has been considered as the one closest to the real situation of the total claims of a life insurance portfolio. It only makes the nearly inevitable assumption of independence of the lifelenghts of insured persons in the portfolio. Many clinical studies, however, have demonstrated positive dependence of paired lives such as a husband and wife. These dependences materially increase the values of the stop-loss premiums for the aggregated loss (for numerical illustrations the interested reader is referred, e.g., to Dhaene and Goovaerts (1997)). It is well known that the stop-loss order is the order followed by any risk averse decision maker and then, the simplifying hypothesis of independence constitute a real financial danger for the company, in the sense that most of their decisions are based on the aggregate claims distribution. Since the beginning of the 1990 s, several papers have treated dependency between risks. Among them, Dhaene and Goovaerts (1996, 1997), Müller (1997), Baüerle and Müller (1998), Wang and Dhaene (1998), Hu and Wu (1999), Dhaene and Denuit (1999), Dhaene et al. (2000), Frostig (2001) and Cossete et al. (2002) study different ordering between two portfolios of dependent risks in the individual risk model. For a portfolio composed of independent pairs of dependent risks, Dhaene and Goovaerts (1996) derived the distribution of the riskiest and safest risks, i.e., the ones generating the largest and smallest stop-loss premiums for the aggregate claim. In another work (1997), they treated multivariate dependencies in the individual life model and found the distribution that maximizes the aggregate claims in stop-loss order, which is also the riskiest one. Their results were generalized in Hu and Wu (1999), where the safest distribution is found. The generalization of these bounds to the individual risk model can be found in Müller (1997) and Dhaene et al. (2000) for the riskiest distribution and in Dhaene and Denuit (1999) for the safest one. 2

3 Within the framework of the individual life model, simple expressions for computing the riskiest and safest distribution for the aggregate claims of a life insurance portfolio with multivariate dependencies are derived in Dhaene and Goovaerts (1997) and Hu and Wu (1999), respectively. The riskiest one follows by assuming that the multivariate distribution corresponding to the dependent risks in the portfolio is given by the Fréchet upper bound (in this case, the risks are said to be mutually comonotonic). On the contrary, the safest one corresponds to the case of dependent risks with the Fréchet lower bound as multivariate distribution when the conditions assuring that the Fréchet lower bound is a proper distribution function are fulfilled (in this case, the risks are said to be mutually exclusive). Hence, in any life insurance portfolio we can consider all kind of dependencies and compute easily the riskiest and safest distributions in the stop-loss order sense. These bound distributions will help us to give an idea of the degree of underestimation of the real risk of a portfolio with dependent risks when computing the distribution of the aggregate claims under the traditional hypothesis of independence. Unfortunately, they turn out to be useless as a measure of the risk of the portfolio since the dependencies between the individual risks will be, in most real situations, nearest to the independency than to these extreme dependency relations. In this paper, we deal with bivariate intermediate positive (negative) dependency relations in the individual life model. Considering a life insurance (sub)portfolio exclusively composed of dependent couples, we prove, in section 3, that the bivariate probabilities associated to each couple under any intermediate positive (negative) dependency hypothesis about their mortality can always be written as a convex linear combination between the independent and the comonotonic (mutually exclusive) ones. Considering such form for the bivariate distributions and assuming that the bivariate dependency relations are completely known, two recursive schemes are derived in section 4 for computing exactly the aggregate claims distribution. The first one is a two stage recursion formula that follows as a particular case of Dhaene and Vanderbroek (1995) algorithm. The second, is a one stage recursion formula that is only valid in case that both policies in each couple have the same amount at risk. These recursions greatly facilitate the computation of the aggregate claims distribution for a life insurance portfolio with bivariate dependencies and also ease the interpretation of the impact of the dependence on the associated stop-loss premiums. Furthermore, when the dependence structure between the members of each couple is not exactly known (as will occur in practice), safe approximations for the aggregate claims distribution are easily obtained with our proposal. Some numerical results are summarized in section 5. 3

4 2 Preliminaries Let us firstly recall the classical individual risk model where all the policies are assumed to be mutually independent. Consider a portfolio of n independent policies which produce at most one claim within a certain reference period (e.g., one year). Suppose that the probability of producing a claim in the reference period and the associated claim amount distributions are given for each policy. The aggregate claims amount of the portfolio, S, is the sum of the n individual claims, X 1,,X n, produced by the policies. We will suppose that S has a finite mean, that the probability of no claims is strictly positive and that the individual claim amounts are positive and integer multiples of some convenient monetary unit, so that one can consider S as being definedonthe non-negative integers. Let the portfolio be classified into a b classes, as displayed in table 1 Claim amount distribution Mortality rate f 1 (x) f 2 (x) f k (x) f a (x) q 1 q 2 q i n k i q b Table 1. Classification of the Portfolio where f k (x) : Conditional distribution of the claim amounts for a policy in column k, given that a claim has occurred, k =1, 2,,a and x =1, 2,,m i. q i : Probability that a policy in row i produces a claim, i =1, 2,,b. We will assume in the sequel that 0 <q i 1 2, for all i. n k i : Number of policies in row i and column k. m = bp P a n k i k=1 i=1 : Maximum possible amount of aggregate claims. 4

5 Each individual policy in the portfolio implies the payment,in case of claim,of an amount given by f k (x), and then, it constitute an individual risk X k i ³ Pr Xi k = x = 1 q i if x =0, with probability function (pf) q i f k (x) if x =1, 2,,m i. (1) Hence, the aggregate claims amount of the portfolio, can be rewritten as S = S = X X n, k=1 i=1 ³ Xi k + + Xi k. {z } n k i times Let p S (s) =Pr(S = s),s=0, 1,,m.The following recursion scheme for computing p S (s) is stated and proved in Dhaene and Varderbroek (1995), giving an optimal way to calculate exactly the aggregate claims distribution. Theorem 1 For the individual model the probabilities p S (s) can be computed by p S (0) = ay by (1 q i ) nk i (2) k=1 i=1 sp S (s) = k=1 i=1 where the coefficients vi k (s) are given by vi k (s) = q Xm i ³ i f k (x) xp(s x) vi k (s x) 1 q i and vi k (s) =0elsewhere. x=1 n k i vi k (s) for s =1, 2,,m, (3) for s =1, 2,,m (4) The individual life model can be obtained from the individual model considered above by choosing f k (x) =δ k,x for k =1, 2,,a, where δ k,x is the Kronecker delta with 1 if k = x, δ k,x = 0 otherwise. In this case, the pf in (1) are given two-point distributions in 0 and k ³ Pr Xi k = x = 1 q i if x =0, q i if x = k. (5) 5

6 Each individual risk in the portfolio can either produce no claim or a fixed positive claim amount k, k =1, 2,,a, during the reference period considered. The claim probability is the probability that the policyholder i dies during this period, i =1, 2,,b, which can be obtained from the mortality tables. Given the individual risk distributions in (5) several recursive methods can be found in the actuarial literature for computing exactly the aggregate claims distribution. The first one is due to De Pril (1986) and generalizes a formula of White and Greville (1959) for the claim numbers distribution. Waldmann (1994) obtains an efficient reformulation of this algorithm by rearrangement of the recursive scheme of De Pril (1986). This recursion follows from Dhaene and Vanderbroek (1995) algorithm in theorem 1 by choosing f k (x) =δ k,x. In the sequel the assumption of mutually independence is relaxed. More precisely, we will assume that the portfolio consists of independent pairs of dependent risks. Hence, the number of contracts in the portfolio, n, isassumedtobeanevennon-negativeinteger. Therefore, we will group risks in table 1 in pairs by columns into a 2 (b+1)b 2 classes Amount at risk Mortality rate (1, 1) (1,a) (2, 1) (k, l) (a, a) (q 1,q 1 ) (q 1,q b ) (q 2,q 2 ) (q i,q j ) (q b,q b ) n kl Table 2. Life insurance portfolio with coupled risks where (k, l) :Amount at risk for the individual policies in each couple, k=1, 2,,a; l =1, 2,,a. (q i,q j ):Mortality rate in the exposure period for the individual policies in each couple, i= 1, 2,,b; j = i,,b. 6

7 n kl : Number of couples with mortality rate (q i,q j ) andamountatrisk(k, l). Further, let X kl = X k i + X l j (6) with the marginal distributions of the X k i and X l j given by (5). Let =Pr(X i =0,X j =0) p k0 =Pr(X i = k, X j =0) p 0l =Pr(X i =0,X j = l) =Pr(X i = k, X j = l) denote the non-zero bivariate probabilities corresponding to Xi k,xl j. Observe that each couple in the portfolio constitute a risk X kl as defined in (6) with pf defined by Pr X kl = x = if x =0, p k0 if x = k, p 0l if x = l, if x = k + l and then, the aggregate claims amount of the portfolio is (7) (8) S = ³ k=1 l=1 i=1 j=i X kl + + X kl {z } n kl times (9) with X kl mutually independent. In this case, the distribution of the aggregate claims is no longer uniquely determined by the pf (5) of the Xi k. Intuitively, it is clear that for a positively (negatively) correlated risk portfolio, assuming the independence hypothesis and computing the pf of S with any of the recursive schemes quoted above will lead to underestimate (overestimate) the associated stop-loss premiums. This fact has been proved in terms of stop-loss order. Definition 2 A risk X is said to precede a risk Y in stop-loss order (written X sl Y ), oralso X is less risky than Y, if their stop-loss premiums are ordered uniformly: E (X d) + E (Y d) + for all retentions d 0. 7

8 Let and S = S 0 = ³ k=1 l=1 i=1 j=i k=1 l=1 i=1 j=i X kl + + X kl {z } n kl times ³ Y kl + + Y kl {z } n kl times denote the total claim amount of the portfolio under two different hypothesis about the dependency relation between the members of each couple.; i.e., the individual risks Xi k and Yi k are identically distributed ( Xi k d = Yi k, for i =1,,b; k =1,,a) and the only difference between both distributions will be given by the bivariate probabilities in (7) that define the sums of the X kl and Y kl bounds for the riskiness of S. (10) (11) in (8) The following theorem is due to Dhaene and Goovaerts (1997) and give Theorem 3 Let S, S 0 be as defined in (10) and (11) and let r denote the Pearson s correlation coefficient. Then we have that r Xi k,xl j r Yi k,yl j, (i =1,,b; j = i,,b; k, l =1,,a) implies S sl S 0 Results in this theorem indicate that the aggregate risk of the portfolio is monotonically increasing with respect to the correlation coefficient between the individual risks in each couple. The expression corresponding to the correlation coefficient of any of the couples in the portfolio, Xi k,xl j, is r Xi k,xj l = q iq j p qi (1 q i ) q j (1 q j ) and, consequently, the aggregate risk of the portfolio, measured in terms of stop-loss order, increases when the probabilities increase. Bounds for these probabilities follow immediately from Fréchet bounds which we define next. (12) Definition 4 Let F 1,F 2 be univariate cumulative distribution functions (cdf s, in short). The Fréchet lower bound is a joint cdf W with margins F 1,F 2 defined by W (x, y) =max{f 1 (x)+f 2 (y) 1, 0} x, y R. 8

9 The Fréchet upper bound is a joint cdf M with margins F 1,F 2 defined by M (x, y) =min{f 1 (x),f 2 (y)} x, y R. These two extreme bound distributions have been frequently considered in the recent actuarial literature. When the bivariate cdf associated to any pair of risks (X 1,X 2 ) is given by the Fréchet upper bound, the risks X 1 and X 2 are said to be mutually comonotonic. On the contrary, when the bivariate cdf associated to this pair is given by the Fréchet lower bound, the risks X 1 and X 2 are said to be mutually exclusive. Within the framework of the individual life model, applications of the notion of comonotonicity can be found in Dhaene and Goovaerts (1997). The case of mutually exclusive risks has been considered in Hu and Wu (1999). In the sequel, we will add the label to the nomenclature corresponding to each couple in the portfolio in order to indicate that the comonotonicity assumption is established. The label will indicate that the risks in each couple are assumed to be mutually exclusive. It is easy to prove that for the individual risks we are considering here, µ i.e., those with marginal pf definedby(5),therisksxi k and Xj l are mutually comonotonic, Xi k, Xj l, if and only if =min(q i,q j ). This means that the death of the younger one in the couple (the one with higher survival probability) implies the death of the older one. µ By symmetry, the risks Xi k and Xj l are mutually exclusive, Xi k, Xj l, ifandonlyif =0, i.e.,if the death of both policyholders can never occur in the reference period considered. It follows immediately that pkl, (i =1,,b; j = i,,b; k, l =1,,a) and then by combining the results in theorem 3 with the expression in (12), we can conclude that S sl S sl S. These bound distributions for the aggregate claims in stop-loss ordering can be improved by considering that the portfolio only contains either positively or negatively correlated risks. If 9

10 r Xi k,xl j 0, (i =1,,b; j = i,,b; k, l =1,,a), then, (i =1,,b; j = i,,b; k, l =1,,a), (13) where the label indicates that the risks Xi k and Xj l are assumed to be mutually independent. Hence, for a portfolio exclusively composed of positively dependent couples S sl S sl S. (14) From a similar reasoning, follows that if the portfolio is exclusively composed of negatively dependent couples, i.e., r Xi k,xl j 0, (i =1,,b; j = i,,b; k, l =1,,a), S sl S sl S. (15) Bounds in (14) and (15) were firstly obtained by Dhaene and Goovaerts (1997) and Hu and Wu (1999), respectively. They indicate that if the risks in the portfolio are positively (negatively) dependent, taking the traditional independence assumption will always lead to underestimate (overestimate) the stop-loss premiums of the portfolio under consideration. Remark that the pf of S and S can be immediately obtained. Indeed, considering such two extreme dependence relations, the sum distributions in (8) follow from the individual pf given in (5) and no additional information about the bivariate dependency relations is needed. Unfortunately, they turn out to be useless for most practical situations. In contrast to the independency assumption, assuming comonotonicity (mutually exclusivity) turns out to be an extremely prudent (riskiness) strategy. The main dependency relations that we could find in any real life insurance portfolio will be closer to the independence than to the comonotonicity (mutually exclusivity) and this last hypothesis can never been accepted for different purposes than giving the best upper (lower) bound if the only information available consist of the individual risk distributions. 3 Intermediate types of dependence In view of previous results, it is clear that more realistic approximations to the aggregate claims distribution than the ones stated in (14) and (15) are only possible in the measure we can introduce some additional information concerning the way of dependency of the components of each 10

11 couple in the portfolio. Restricting ourselves to a portfolio exclusively composed of positively (negatively) dependent couples, we investigate in this section the nature of this information. Let us consider one of the couples in the portfolio Xi k,xl j and assume that the individual risks Xi k and Xj l correspond to a x i-year-old and x j -year-old policyholders. We will denote by T xi and T xj their remaining lifetimes (or time-until-death random variables). Theorem 5 For any i =1,,b; j = i,,b; k, l =1,,a, r Xi k,xj l = r I (T xi t),i T xj t, (16) where I denotes the indicator function and t the reference period in years. Proof. It follows immediately that the expression for r I (T xi t),i T xj t is the one givenin(12)sincepr (T xi t) =q i, Pr (T xi t) =q j and Pr T xi t, T xj t =. Results in theorem 5 indicate that the correlation coefficient corresponding to the risks in each couple depend on the correlation between the bivariate remaining lifetimes for the reference period considered. We prove next that these last quantities, define completely the bivariate probabilities in (7). Case 1. r Xi k,xl j 0 (i =1,,b; j = i,,b; k, l =1,,a) Theorem 6 For any i =1,,b; j = i,,b; the real value for s in [0, 1] such that r I (T xi t),i T xj t µ µ µ T T = s r I t xi,i t xj (17) defines the following relations valid for any k, l =1,,a, = s +(1 s ), p k0 = s p k0 +(1 s ) p k0, p 0l = s p 0l +(1 s ) p 0l, = s +(1 s ). (18) Proof. Note first that the inequalities in (13) assure the existence of the s in [0, 1] that defines (17). Since T xi d = T xi for i =1,,b, by combining (12), (16) and (17), we have that 11

12 Ã! = s which achieves the proof of the firstrelationin(18). relations also hold, note that! p k0 = q i = s Ãq i à +(1 s ) q i Inordertoprovethattheremaining!! Ã! p 0l = q j = s Ãq j +(1 s ) q j = s p k0 +(1 s ) p k0, = s p 0l +(1 s ) p 0l,! Ã! = 1 q i q j + = s Ã1 q i q j + +(1 s ) 1 q i q j + = s +(1 s ). Case 2. r Xi k,xl j 0 (i =1,,b; j = i,,b; k, l =1,,a) Theorem 7 For any i =1,,b; j = i,,b; the real value for s in [0, 1] such that r I (T xi t),i T xj t µ µ µ T T = s r I t xi,i t xj (19) defines the following relations valid for any k, l =1,,a, = s +(1 s ), p k0 = s p k0 +(1 s ) p k0, p 0l = s p 0l +(1 s ) p 0l, = s +(1 s ). (20) Proof. Similar to the proof of theorem 6. From results in theorem 6 (theorem 7), we can conclude that the values of the s that define the relations in (17), (19), turn out to be the key quantity for the exact knowledge of the bivariate probabilities in (7) associated to each couple in the portfolio. Given these values, the bivariate probabilities result, in each point, as a convex linear combination between the corresponding bivariate probabilities when the hypothesis with respect to the dependency relations of the risks 12

13 in each pair are independency and comonotonicity (mutually exclusivity), respectively. Once the probabilities in (7) are given, the sum distributions in (8) are known. Then, the distribution of the aggregate claims of the portfolio can be exactly computed with any of the recursive schemes presented in next section. Nevertheless, at this point, we want to remark that the importance of the results in this paper consist in giving improved upper (lower) bounds for the riskiness of the portfolio but not in obtaining exact results for the aggregate claims distribution. Indeed, exact values are only possible in the measure the information available allows us to compute exactly the joint distributions of the T xi,t xj,i=1,,b; j = i,,b, and this will never occur in practice, except in case of comonotonic (mutually exclusive) risks. Considering any intermediate type of dependence, e.g., a life insurance (sub)portfolio exclusively composed of married couples, the available information consists at most of a bivariate data set containing the ages at death of some couples. In this case, the joint distributions of the T xi,t xj can only be approximations, which can be obtained by using, e.g., the methodology proposed by Genest and Rivest (1993) for identifying a copula form in empirical applications as has been done in Denuit et al. (2001) were a data set with the ages of death of 533 married couples was available. Once the joint distributions of the T xi,t xj are approximated, the approximative values for the s that define the relations in (17) or (19) (depending on the sign of the dependencies in the portfolio) can be obtained. On what concerns to these values, notice that when the risks in the couples are positively dependent, results in theorems 3 and 6 indicate that the coefficients s are monotonically increasing with respect to the aggregate risk of the portfolio, measured in terms of stop-loss order. Hence, our proposal consists in giving approximative values for the coefficients s in (17) combining two criteria: they have to be smaller enough to describe reality, but, they also have to be larger enough to guarantee they will never be exceeded. For safety reasons, the latter criteria has to prevail in our approximations and then, the upper bound obtained turns out to be sharper than the resultant when the comonotonicity hypothesis is assumed for the risks in each couple. On the contrary, when the risks in the couples are negatively dependent, results in theorems 3 and 7 indicate that the coefficients s are monotonically decreasing with respect to the aggregate risk of the portfolio. In this case, the approximative values for the coefficients s in (19) have to be higher enough to describe reality, but also smaller enough to guarantee safety. The lower bound obtained this way turns out to be sharper than the resultant when the risks of each couple are assumed to be mutually exclusive. 13

14 4 Exact recursive procedures In this section, we assume that the coefficients s [0, 1], i =1,,b; j = i,,b, in (17), (19), are exactly known, so that the probabilities in (18), (20), can be exactly computed. Two recursive schemes for computing exactly the probabilities p S (s) corresponding to the aggregate claims amount of the portfolio are next derived. Theorem 8 For the aggregate claims amount of the portfolio defined in (9), the probabilities p S (s) can be computed by p S (0) = ay ay by by k=1 l=1 i=1 j=i 1 p 00 n kl sp S (s) = where the coefficients v kl h v kl (s) = 1 p k0 and v kl (s) =0elsewhere. k=1 l=1 i=1 j=i (s) are given by ³ kp(s k) v kl n kl v kl (s) for s =1, 2,,m, ³ (s k) + p 0l lp(s l) v kl (s l) ³ + (k + l) p (s (k + l)) v kl (s (k + l)) i for s =1, 2,,m Proof. Observe that the pf of the X kl in (8) can be rewritten as Pr X kl if x =0, = x = 1 f kl (x) if x =1,,a, where the conditional claim amount distributions are defined by p k0 /(1 p00 ) if x = k, f kl (x) = p 0l /(1 p00 ) if x = l, /(1 p00 ) if x = k + l and f kl (x) =0elsewhere. Then, the recursion immediately follows from Dhaene and Varderbroek (1995) recursion in theorem 1. In next theorem, we assume that the individual policies in each couple have the same amount at risk, i.e., each couple in the portfolio has an amount at risk given by (k, k), k =1,,a, so that, table 2 consists of a (b+1)b 2 classes, meanwhile the P b i=1 nk i intable1areassumedto be non-negative integers. In next theorem we prove that, in this special case, the probabilities p S (s) can be computed by a one stage recursion formula. 14

15 Theorem 9 Let µ a (i, j, k, t, r) = 1 t ³ p k0 + p 0k tr ³ p kk r and A (k, t, r) =(1) t+1 k i=1 j=i Then, for s =1, 2,,m, the following recursion holds: sp S (s)= min(a,s) X k=1 X [ k] s l=1 lx t=] l+1 2 [ p S (0) = t1 lt ay by k=1 i=1 j=i n k a (i, j, k, t, r). by A (k, t, l t)+2 p 00 n kk (21) Xl1 t=] l 2[ t1 lt1 A (k, t, l t) p S (s lk) where [x] denote the greatest integer less than or equal to x and ]x[ denote the greatest integer larger than or equal to x. (22) Proof The probability generating function of the aggregate claims is given by G S (u) = = mx p S (s) u s = s=0 ay k=1 i=1 j=i by by k=1 i=1 j=i h + ay by by k=1 i=1 j=i ³ p k0 + p 0k nkk G X k (u) Putting u =0in (23) leads to (21). In order to proof (22), take µ ln G S (u) = n kk ln +ln 1+ By expanding we get ln G S (u) = k=1 i=1 j=i n kk Taking the derivative of both sides of (24) " X ln (1) t+1 + t t=1 µ p k0 u k + p kk u 2ki n kk µµ p k0 + p0k + p0k u k + pkk u k + pkk u 2k u 2k t # (23) (24) G 0 S (u) = G S (u) k=1 i=1 j=i X µ (1) t+1 t=1 1 ³ k k p k0 + p 0k +2p kk u n kk t ³ p k0 + p 0k + p kk u k t1 u kt1 15

16 and by applying the binomial theorem lead to Hence, Let Since and G 0 S (u) = G S (u) G 0 S (u) =G S (u) v (u) = r=0 k=1 i=1 j=i ³ k k p k0 + p 0k +2p kk u n kk Xt1 t1 ³ r p k0 + p 0k tr1 ³ p kk u k r X Xt1 k=1 t=1 r=0 X Xt1 k=1 t=1 r=0 min(a,s) X v s1) (0) = (s 1)! k=1/ s =[ s k k] 1 X µ (1) t+1 t=1 1 t u kt1 t1 h r A (k, t, r) u k(t+r)1 +2A (k, t, r +1)u k(t+r+1)1i (25) t1 h r A (k, t, r) u k(t+r)1 +2A (k, t, r +1)u k(t+r+1)1i G s1) S (0) = (s 1)! p S (s 1) s kx t=] s+k 2k [ ³ t t1 s A k, t, k s t +2 k s k 1 X t=] s 2k[ ³ t1 t1 s k A k, t, k s t Taking, according to Leibnitz s formula, the derivative of order s 1 of both sides of (25) and setting u =0leads to min(a,l) sx X sp S (s) = l=1 k=1/ l =[ l k k] l kx t=] l+k 2k [ µ t1 l k t A k, t, l k t +2 l k 1 X t=] l 2k[ µ t1 l k t1 A k, t, l k t p S (s l) and by rearranging terms the recursion (22) follows. Remark that, although the apparently limitation of the conditions stated in theorem 9, the recursion obtained this way will be useful for many real situations. Indeed, in any life insurance portfolios a large number of bivariate dependencies presumably arise from married couples and, in most cases, both contracts will have the same amount at risk. 1 k V (s), where V (s) = k N / 1 k min (a, s) s = ª s k k 16

17 5 Numerical example In this section, we illustrate previous results by a numerical example. Remark that only positive dependencies are considered since they will constitute the greater part of dependencies in any real portfolio. Nevertheless, numerical results for negatively dependent risks follow easily by a similar procedure. We consider as starting point, the portfolio discussed in Kornya (1983) which is represented in the following table. Claim amount distribution Mortality rate Table 3. Kornya s portfolio The portfolio consists of 322 risks. We suppose that it contains 52 positively dependent couples which are presented in table 4, meanwhile the restating 218 risks in table 3 are mutually independent. Claim amount distribution Mortality rate ( , ) ( , ) ( , ) ( , ) ( , ) (1, 1) (2, 2) (3, 3) (4, 4) (5, 5) Table 4. Positively dependent couples in Kornya s portfolio For simplicity reasons, we assume that all the couples in table 4 arise from the same kind of dependency relations (e.g., married couples) so that s = s for i =1,, 5; j = i,, 5. 17

18 Moreover, observe that the amounts at risk for the two risks in each couple have been assumed to be equal, so that the aggregate claims distribution follow either by applying the recursions in theorem 8 and 9. We give in Fig. 1a and b, the exact values of the cumulative distribution functions F S and the stop-loss premiums E (S d) for four portfolios. They differ in the value attributed to the parameter s (that defines the relations in (17)) which determines the degree of dependence between the risks of each couple. We have chosen values of 0, 0.15, 0.25 and 1 for s, where s =0corresponds to the independence hypothesis for all the risks in the portfolio, meanwhile s =1corresponds to the comonotonicity hypothesis for both risks in each couple. In table 5, we give E (S) and Var(S) for each portfolio. It is clear from Fig. 1a and b that the dependency significantly influences both F S (s) and E (S d). In fact, we observe that, for any fixed retention level d, the value of the stop-loss premiums increases with s, the degree of dependency between the members of each couple. Hence, sharper upper bounds than the comonotonic one are easily obtained in case some information concerning the bivariate remaining lifetimesisavailable. s E(S) Var(S) Independence s = s = Comonotonicity Table 5 18

19 (a) F1 s F2 s F3 s 0.6 F4 s s 30 (b) sl1 d 3 sl2 d sl3 d sl4 d d 20 Independence s = 0.15 s = 0.25 Comonotonicity Fig. 1.(a). Cumulative distribution functions F S (s) and (b) stop-loss premiums E (S d). 19

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