Upper stop-loss bounds for sums of possibly dependent risks with given means and variances

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1 Statistics & Probability Letters 57 (00) 33 4 Upper stop-loss bounds for sums of possibly dependent risks with given means and variances Christian Genest a, Etienne Marceau b, Mhamed Mesoui c a Departement de mathematiques et de statistique, Universite Laval, Quebec, Canada GK 7P4 b Ecole d actuariat, Universite Laval, Quebec, Canada GK 7P4 c Departement de mathematiques et d informatique, C.P. 500, Universite duquebec a Trois-Rivieres, Trois-Rivieres (Quebec) Canada G9A 5H7 Received August 000 received in revised form September 00 Abstract Consider non-negative random variables X :::X n whose marginal means and variances are known. The purpose ofthis paper is to compare two dierent strategies for nding an upper bound on the stop-loss premium (X + + X n d)=e{max (0X + + X n d)} that are valid for all retention amounts d 0in the absence ofinformation concerning the type or degree ofdependence between the risks X i. One approach consists ofmaximizing the premium over all possible values ij = corr(x i X j ) 6 i j6 n. As it turns out, however, a better solution exploits results ofdhaene et al. (Schweiz. Aktuarver. Mitt. (000) 99) on the maximality ofcomonotonic risks in the stop-loss order. Explicit calculations and numerical illustrations ofthe proposed bounds are given. c 00 Elsevier Science B.V. All rights reserved. Keywords: Comonotonicity Frechet bounds Stop-loss bounds Stop-loss ordering. Introduction In actuarial mathematics, a risk is typically modeled by a random variable X 0 representing the monetary value, either of(i) a single insurance claim, (ii) the sum ofall claims for an individual contract, or (iii) a portfolio ofcontracts over a given time period. Ofspecial interest is the degree ofdangerousness ofsuch a risk, once the amount d 0 retained by the insured (or the rst insurer) has been factored out. This can be measured, in particular, by the stop-loss premium (X d)=e{max(0x d)} Corresponding author. address: genest@mat.ulaval.ca (C. Genest) /0/$ - see front matter c 00 Elsevier Science B.V. All rights reserved. PII: S (0)00039-

2 34 C. Genest et al. / Statistics & Probability Letters 57 (00) 33 4 for which it is useful to have upper bounds when the distribution of X is not completely determined. Beginning with the work ofbuhlmann et al. (977), there has been a great deal ofliterature on this subject, comprehensively reviewed by Hurlimann (999). For example, when the only available information is E(X )= 0 and var(x )= it is well known (cf., e.g., Theorem of Jansen et al., 986) that for all retention amounts d 0, the best upper bound on (X d) is given by d + if0 6 d (d)= () d + (d ) + if d + : For bounds under more restrictive hypotheses on the claim size distribution (e.g., additional moment conditions, nite range, unimodality, etc.), see De Vylder and Goovaerts (98), Jansen et al. (986) and Hurlimann (996), among others. The purpose ofthis paper is to examine strategies for nding upper bounds on (X d) when X = X + + X n in the presence ofpartial information on the marginal distributions ofthe X i s but no specic knowledge about their dependence structure. This problem arises when X represents an aggregate claim whose components might exhibit stochastic dependence. When the marginals are fully specied, Dhaene and Goovaerts (996), Muller (997), Bauerle and Muller (998) and Wang and Dhaene (998), have shown that the best upper stop-loss bound is obtained when the X i s are comonotonic, i.e., when their underlying copula is the Frechet upper bound. Hurlimann (998) worked on the same problem, assuming that nothing ofthe marginals is known but the means, variances and correlation between two risks X and X that are diatomic, i.e., whose probability mass function is concentrated on two points. Two dierent approaches for bounding the stop-loss premium (X d) are considered in Section when the rst two moments ofthe X i s are given. A natural solution consists ofmaximizing the premium over all possible correlation structures between these risks. Another way to proceed is to exploit the fact, mentioned by Dhaene et al. (000), that the sum of comonotonic risks Y :::Y n constitutes an upper bound on the sum ofthe X i s in the stop-loss order whenever (X i d) 6 (Y i d) for all d 0 and 6 i 6 n. Explicit calculations ofthe resulting upper bounds are described in Section 3 for the special case n =, and the better ofthe two solutions is identied in Section 4. A couple ofillustrations are given in Section 5, and concluding remarks are made in Section 6.. Upper stop-loss bounds for an arbitrary sum of risks Suppose that X :::X n are non-negative random variables with E(X i )= i 0 and var(x i )= i 0 6 i 6 n. In this section, two natural approaches are described for deriving stop-loss bounds

3 C. Genest et al. / Statistics & Probability Letters 57 (00) on X = X + + X n in the absence ofcomplete information concerning the joint distribution of the X i s. First approach. A simple solution consists ofxing temporarily the values of ij = corr(x i X j ) 6 i j6n, so that n n E(X )= = i and var(x )= ()= i + ij i j : i j i= In view of(), one can then assert that (X d) 6 + () (d) for all d 0, where the subscript makes explicit the dependence ofthe bound on the assumed vector ofvalues ofthe ij s. If P represents the set ofall possible such vectors, an upper bound on (X d) that does not depend on the arbitrary choice of is then given by sup + () (d): () P Second approach. An alternative upper bound on (X d) inspired by the work ofdhaene et al. (000) is given by {F (U)+ + Fn (U)d} (3) where U is a uniform random variable on the interval (0 ) and F i = F ii with + if0 6 x 6 + F (x)= + x (x ) + if + x : Indeed, it is easy to check (e.g., using Theorem. on p. ofthe book by Kaas et al., 994) that {Fi (U)d}= + i (d) yields the best upper bound on (X i i d), so that Y i =Fi (U) dominates X i in the stop-loss order. Inequality (3) then follows from Theorem 5 of Dhaene et al. (000), since the Y i s are comonotonic. It is natural to ask which ofquantities () or (3) yields the better bound. This issue is investigated next in the special case where X is the sum oftwo risks. i= 3. Computation of the bounds in the case n = It is shown in the appendix that for arbitrary but xed i and i i=, one has sup + () (d)=+ () (d) 66 for all d 0. Accordingly, bound () on the stop-loss premium on a risk X = X + X reduces to + with = + and = + : (4)

4 36 C. Genest et al. / Statistics & Probability Letters 57 (00) 33 4 To compute bound (3), assume without loss ofgenerality that a + + and use the fact that F (u)= + a 0 if0 6 u 6 + u if u (u ) + (5) to derive the cumulative distribution function of V = F (U)+F (U). Writing F V (v)=p(v 6 v) as P(V 6 v U 6 a )+P(V 6 v a U6a )+P(V 6 v U a ) one can see right away that the rst term equals a, since U 6 a implies F (U)=F (U) 0, and U is uniformly distributed on (0 ). Similarly, the second term is simply P{U 6 F (v)a U6 a } = P[a U6 min{f (v)a }] = max [0 min {F (v)a } a ]: Furthermore, noting that for all a u6, one has F (u)+f (u)=f (u) with and given by (4), the third term can be reexpressed as Letting P{a U6 F (v)} = max{0f (v) a }: b = lim v a F (v)= + and b = F (a )= + b 3 = F (a )= + it follows from (5) that b 6 b 6 b 3, and one may then verify easily that a if0 6 v 6 b F (v) if b v6b F V (v)= a if b v6b 3 F (v) if b 3 v :

5 C. Genest et al. / Statistics & Probability Letters 57 (00) Elementary calculations further yield + (d)+ if0 6 d 6 b (V d)= (b 3 d)( a )+ + (b 3 ) if b 6 d 6 b 3 + (d) if d b 3 where and are as dened in (4). That this bound is continuous in d follows in part from the fact that the middle term may be written alternatively as (b 3 d)( a )+ + (b )+ = and that (b 3 b )( a )= =. 4. Comparison of the bounds when n = In the limiting case a = a, it is easy to see that = = = = =, whence b = b and a = a = a + : Since b 3 is then equal to ( + )=(), it follows that F V F, which implies that bounds () and (3) coincide in this case. When a a, however, bound (3) turns out to be dierent, and uniformly better, than bound (). To prove this, rst observe that the distribution functions F and F V associated with these bounds have the same expected value, namely + (0) = (V 0) =. It is also clear that and a = F V (v) 6 F (v)=a 0 6 v 6 b a = F V (v) F (v) b 6 v 6 b 3 since the two curves are monotone increasing and coincide for all v b 3. Ifit can be shown that F and F V cross once and only once in the interval [b b 3 ], the inequality (V d) 6 + (d) d 0 will then follow as a special case of Theorem 3.a. in Shaked and Shantikumar (994). To this end, take a 6 a as before, and introduce b = F (a)= + whose value is comprised in the interval (b b 3 ]. If b b, then f(v)=f V (v) F (v) is continuous and strictly increasing on [b b ] with f(b )=a a 0 and f(b )=a F (b ) 0, so that there can be only one root. If b b, two cases must be considered, according as F V (b) is strictly smaller or larger than a.

6 38 C. Genest et al. / Statistics & Probability Letters 57 (00) 33 4 When F V (b) a, f(v) = 0 ifand only if ( ) ( ) v v = (6) where (u)=u= u + is strictly monotone and increasing. There is, therefore, one and only one root. When F V (b) a, there is a single root of f on [b b], since F (v) is constant on that interval and hence f is strictly increasing on this domain. Furthermore, any additional root of f in the interval [b b ] must satisfy relation (6), and is therefore unique, provided it exists. If one could nd such a root, however, then one would still have f(v) 0 for all b 6 v 6 b, or else f(b ) would be strictly negative, which is impossible. In this case again, therefore, F and F V cross once and only once, which completes the argument. 5. Examples As an illustration ofthe above result, Fig. displays the upper bound (3) as a function ofthe retention amount d 0 when X and X have mean =75, =50 and variance =6 50 and = , respectively. Since in this case a =a ==3 (V d)= + (d) with = + =35 and = ( + ) = 50. This dashed line may be compared to the solid line corresponding to the exact value of (X d) when X and X are comonotonic Pareto random variables with distribution function ( ) i P(X i 6 x)= =4 = 55 = 450: i + x As indicated by Wang and Dhaene (998), among others, the latter is actually an upper bound on the stop-loss premium when the marginal distributions ofthe two risks are known to have Pareto distributions with these specic parameters but their joint dependence is uncertain Fig.. Upper bound (3) on the stop-loss premium when X and X have mean =75, =50 and variance =6 50 and =45 000, and (i) nothing else is known about their distributions (dashed line) (ii) the marginals are known to be Pareto (solid line).

7 C. Genest et al. / Statistics & Probability Letters 57 (00) Fig.. Top panel: Bounds on the stop-loss premiums on X + X when X and X have mean = 75, = 50 and variance =40 65 and =4 500, and (i) nothing else is known about their distributions and Approach is used (dashed line) (ii) nothing else is known about their distributions and Approach is used (dotted line) (iii) marginals are known to be exponential-inverse Gaussian (solid line). Bottom panel: Blown-up version ofthe rst part ofthe previous graph showing more distinctly the dierence between bounds () and (3). As a second example, the bounds () and (3) are compared in Fig. with the exact value of (X d) when X and X are comonotonic and are distributed as F(x)= exp{ ( x + )} x 0 i.e., follow an exponential-inverse Gaussian distribution, whose mean and variance are given by + 5=4+ + and respectively. (Note the typographical error in the formula for the variance given by Hesselager et al., 998).

8 40 C. Genest et al. / Statistics & Probability Letters 57 (00) 33 4 In Fig., the solid line depicts (X d) when = 75 = 50 =40 65 and = In this case, a 0:570 and a 0:6538, so that (V d) (dotted line) is uniformly smaller than + (d) (dashed line). Here, = + = 35 and =( + ) = 66 8:9. 6. Discussion In the two illustrations provided above, the bound (3) based only on the knowledge ofthe rst two moments ofthe X i s is obviously more conservative than that corresponding to the case where the marginals are totally specied. With additional information concerning higher marginal moments of X and X, however, it should be possible to improve the upper stop-loss bound derived in Section 4 by exploiting, say, results ofjansen et al. (986). Following these authors, one could also use either Approach or to derive an upper stop-loss bound for the sum of two risks with nite support. In the case n, an advantage ofbound (3) over bound () is that it can be more readily computed in explicit form. One may conjecture that it is also tighter, but considering the diculty ofmaximizing + (d) over a multidimensional set P ofpossible correlation vectors, a proof () along the same lines as Section 4 does not seem practical. This issue, as well as the possible optimality ofbound (3), remains to be addressed. Acknowledgements The authors are grateful to an anonymous referee whose comments led to a signicant improvement in this paper s contribution. Partial funding in support of this work was provided by the Natural Sciences and Engineering Research Council ofcanada, by the Fonds pour la formation de chercheurs et l aide a la recherche du Gouvernement du Quebec, and by the Chaire en Assurance Industrielle-Alliance de l Universite Laval. Appendix It must be shown that for arbitrary d 0 + () (d) dened in () in terms of = + and = + + is maximized when =. To this end, rewrite this bound as a function of t = =, viz. (td)=f d (t){d 6 h(t)} + g d (t){d h(t)} where h(t)= ( + t)=, f d (t)= d +t and g d (t)= d ( + ) d + t: If t 0 =( ) = and t =( + ) =, one must then show that one has (td) 6 (t d)for all t [t 0 t ] and any xed retention amount d 0.

9 C. Genest et al. / Statistics & Probability Letters 57 (00) Three cases must be considered, according as d h(t )d6h(t 0 )orh(t 0 ) d h(t ). When d h(t )(td)=g d (t) for all t 0 6 t 6 t, and hence is an increasing function of t, whence (td) 6 (t d). Likewise when d 6 h(t 0 ), (td)=f d (t) for all t 0 6 t 6 t and since the latter is monotone increasing, (td) 6 (t d) once again. Finally, suppose that h(t 0 ) d h(t ) and let t d? = h (d) =d=. Two subcases must be considered, according as t 0 6 t 6 t d? or t? d 6 t 6 t. In the rst subcase, (td)=g d (t) 6 g d (t d?)= f d (t d?)== since g d is monotone increasing. In the second subcase, (td)=f d (t) and this increasing function is maximized at t, so that (td) 6 (t d)=f d (t ) for all t 0 6 t 6 t. This completes the argument. References Bauerle, N., Muller, A., 998. Modeling and comparing dependencies in multivariate risk portfolios. ASTIN Bull. 8, Buhlmann, H., Gagliardi, B., Gerber, H.U., Straub, E., 977. Some inequalities for stop-loss premiums. ASTIN Bull. 9, De Vylder, F., Goovaerts, M.J., 98. Upper and lower bounds on stop-loss premiums in case ofknown expectation and variance ofthe risk variable. Mitt. Verein. Schweiz. Versicherungsmath Dhaene, J., Goovaerts, M.J., 996. Dependency ofrisks and stop-loss order. ASTIN Bull. 6, 0. Dhaene, J., Wang, S., Young, V., Goovaerts, M.J., 000. Comonotonicity and maximal stop-loss premiums. Schweiz. Aktuarver. Mitt Hesselager, O., Wang, S., Willmot, G.E., 998. Exponential and scale mixtures and equilibrium distributions. Scand. Actuar. J., 5 4. Hurlimann, W., 996. Improved analytical bounds for some risk quantities. ASTIN Bull. 6, Hurlimann, W., 998. On best stop-loss bounds for bivariate sums by known marginal means, variances and correlation. Mitt. Verein. Schweiz. Versicherungsmath. 34. Hurlimann, W., 999. Extremal moment methods and stochastic order: applications in actuarial science. Unpublished monograph. Jansen, K., Haezendonck, J., Goovaerts, M.J., 986. Upper bounds on stop-loss premiums in case ofknown moments up to the fourth order. Insurance Math. Econom. 5, Kaas, R., van Heerwaarden, A.E., Goovaerts, M.J., 994. Ordering ofactuarial Risks. Caire, Brussels. Muller, A., 997. Stop-loss order for portfolios of dependent risks. Insurance Math. Econom., 9 3. Shaked, M., Shantikumar, J.G., 994. Stochastic Orders and Their Applications. Academic Press, New York. Wang, S., Dhaene, J., 998. Comonotonicity, correlation order and premium principles. Insurance Math. Econom., 35 4.