DEPARTEMENT TOEGEPASTE ECONOMISCHE WETENSCHAPPEN

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1 DEPARTEMENT TOEGEPASTE ECONOMISCHE WETENSCHAPPEN ONDERZOEKSRAPPORT NR 9730 GOMONOTONICITY AND MAXIMAL STOP-LOSS PREMIUMS by Ja DHAENE Shau WANG Virgiia R. YOUNG Marc GOOVAERTS Katholieke Uiversiteit Leuve Naamsestraat 69, Leuve

2 ONDERZOEKSRAPPORT NR 9730 COMONOTONICITY AND MAXIMAL STOP-LOSS PREMIUMS by Ja DHAENE Shau WANG Virgiia R. YOUNG Marc GOOVAERTS 0/1997/2376/32

3 CorI1ootoicity ad Maxirllal Stop-Loss Preriurlls* Ja Dhaeet Shau Wag+ Virgiia R. Youg Marc J. Goovaerts' Abstract I this paper, we ivestigate the relatioship betwee comootoicity ad stoploss order. vve prove our mai results by usig a characterizatio of stop-loss order withi the framework of Yaari's (1987) dual theory of choice uder risk. Wag ad Dhaee (1997) explore related problems i the case of bivariate radom variables. vve exted their work to a arbitrary sum of radom variables ad preset several examples illustratig our results. 1 Itroductio The stop-loss trasform is a importat tool for studyig the riskiess of a isurace portfolio. I this paper, we cosider the idividual risk theory model, where the aggregate claims of the portfolio are modeled as the sum of the claims of the idividual risks. We ivestigate the aggregate stop-loss trasform of such a portfolio without makig the usual assumptio of mutual idepedece of the idividual risks. \Vag ad Dhaee (1997) explore related problems i the case of bivariate radom variables. We exted their work to a arbitrary sum of radom variables. To prove results cocerig orderig of risks, oe ofte uses characterizatios of these orderigs withi the framework of expected utility theory, see e.g. Kaas et a1. (1994). We, however, rely o the framework of Yaari's (1987) dual theory of choice uder risk. Our results are easier to obtai i this dual settig. I Sectio 2,,ye provide otatio ad a brief itroductio to Yaari's dual theory of risk. We itroduce a special type of depedecy betwee the idividual risks, the otio 'J. Dhaee ad M.J. Goovaerts would like to thak the fiacial support of Oderzoeksfods KU.Leuve (grat OT/97/6). -Katholieke Uiversiteit Leuve, Uiversiteit va Amsterdam, Uiversiteit Atwerpe. =Uiversity of Waterloo. i Uiversity of Wiscosi-Madiso. '"Katholieke Uiversiteit Leuve, Uiversiteit va Amsterdam. 1

4 of "comootoicity". Loosely speakig, risks are comootoic if they" move i the same directio". I Sectio 3, we cosider stop-loss order. It is well-kow that stop-loss order is the order iduced by all risk-averse decisio makers whose prefereces amog risks obey the axioms of utility theory. We show that the class of decisio makers, whose prefereces obey the axioms of Yaari's dual theory of risk ad who have icreasig cocave distortio fuctios, also iduces stop-loss order. From this characterizatio of stop-loss order, we fid the followig result: If risk Xi is smaller i stop-loss order tha risk Yi, for i = 1,..., TI, ad if the risks Yi are mutually comootoic, the the respective sums of risks are also stop-loss ordered. I Sectio 4, we characterize the stochastic domiace order withi Yaari's theory. I Sectio 5, we cosider the case that the margial distributios of the idividual risks are give. We derive a expressio for the maximal aggregate stop-loss premium i terms of the stop-loss premiums of the idividual risks. Fially, i Sectio 6, we preset several examples to illustrate our results. We remark that Wag ad Youg (1997) further cosider orderig of risks uder Yaari's theory. They exted first ad secod stochastic domiace orderigs to higher orderigs i this dual theory of choice uder risk. 2 Distortio Fuctios ad Comootoicity For a risk X (i.e. a o-egative real valued radom variable with a fiite mea), we deote its cumulative distributio fuctio (cdf) ad its decumulative distributio fuctio (ddf) by Fx ad Sx respectively: Fx (x) = Pr {X :::; ;r:}, Sx(x) = Pr{X > :r}, O:::;;r: < CXl, O:::;;r < CXl. I geeral, both Fx ad Sx are ot oe-to-oe so that we have to be cautious i defiig their iverses. We defie F"yl ad S"yl as follows: if{x: Fx(x) 2: p}, if {x : Sx (x) :::; p}, 0< p:::; L O:::;p<L F;:I(O) = 0, Sx 1 (1) =0. where we adopt the covetio that if q; = CXl. We remark that F.yl is o-decreasig, SXl is o-icreasig ad S;/ (p) = F"y 1 (1 - p). Startig from axioms for prefereces betwee risks. Vo Neuma ad Morgester (1947) developed utility theory. They showed that, vvithi this axiomatic framework, a decisiomaker has a utility fuctio 'U such that he or she prefers risk X to risk Y (or is idifferet betwee them) if ad oly if E (u( - X)) 2: E (u( - Y)). 2

5 Yaari (1987) presets a dual theory of choice uder risk. I this dual theory, the cocept of )) distortio fuctio" emerges. It ca be cosidered as the parallel to the cocept of "utility fuctio" i utility theory. Defiitio 1 A distortio fuctio 9 is a o-decreasig fuctio 9 g(o) = 0 ad g(l) = 1. [0,1] --+ [0,1] with Startig from a axiomatic settig parallel to the oe i utility theory, Yaari shows that there exists a distortio fuctio 9 such that the decisio maker prefers risk X to risk Y (or is idifferet betwee them) if ad oly if Bg(X) :S Hg(Y), where for ay risk X, the "certaity equivalet" Bg (X) is defied as Hg (X) = /= g[sx(x)]dx = /1 Si/(q)dg(q). io.~ We remark that By (X) = E(X) if 9 is the idetity. For a geeral distortio fuctio 9 1 the certaity equivalet Hg (X) ca be iterpreted as a "distorted" expectatio of X. See Wag ad Youg (1997) for a discussio of Yaari's axioms i a isurace cotext. I the followig sectios, we use two special families of distortio fuctios for provig some of our results. I the followig lemma. we derive expressios for the certaity equivalets Hg (X) of these families of distortio fuctios. For a subset A of the real umbers, we use the otatio fa for the idicator fuctio which equals 1 if ::e E A ad 0 otherwise. Lemma 1 (a) Let the distortio fuctio 9 be defied by g( x) = f (x > p), 0 :S x :S I, for a arbitrary, but.fixed, p E [0,1]. The for ay risk X the certaity equi'vazet Hg(X) zs gwe by (b) Let the distortio fuctio 9 be de.fied by g(x) = mi (x/p, 1), 0 :S.r :S I, for a arbitrary, but.fixed, p E (0,1]. The for ay risk X, the certaity equi'vazet Bg(X) zs gwe by Proof. (a) First let 9 be defied by g(x) = f(x > p). As we have for ay :r > 0 that S dx) :S p {::} S)/(p) :S x, we fid g(sx(x)) = { Ix < S;/(p), ' 0, x 2: SX1(p), from which we immediately obtai the expressio for the certaity equivalet. 3

6 (b) Now let 9 be defied by g(x) = mi (x/p, 1). I this case we fid g(sx(x)) = { I S'y(x)/p,.1: < S"~l(p), _,,1: 2 S-X I (p), from which we immediately obtai the desired result. Yaari's axiomatic settig oly differs from the axiomatic settig of expected utility theory by modifyig the idepedece axiom. This modified axiom ca be expressed i terms of "comootoic" risks. Defiitio 2 The r-isks Xl, X 2,... ) X ar-e said to be mutually comootoic if Q,'f/',Ij of the followig equivalet coditios hold: (1) The cdf FX1,x2,oo,xof (XI,X2)...,X) satisfies for all Xl'..., :r; 2 o. (2) Ther-e exists a mdom variable Z ad o-deer-easig fuctios UI,..., 'U o R s'/u:h that (XI,...,X) ~ (UI(Z)"",u(Z)), (3) For- ay uifor-mly distr-ibuted mdom var-iable C o [0, 1], we have that I the defiitio above, the otatio "~,, is used to idicate that the two multivariate radom variables are equal i distributio. The proof for the equivalece of the three coditios is a straightforward geeralizatio of the proof for the bivariate case cosidered i,yag ad Dhaee (1997). The followig theorem states that the certaity equivalet of the sum of mutually comootoic risks is equal to the sum of the certaity equivalets of the differet risks. Theorem 2 lfthe r-isksxi,x2",.,x ar-e mutually eomootoie, the H9(XI + X X) = LHg(Xi)' i=l Proof. A proof for the bivariate case ca be foud i Wag (1996). A geeralisatio to the multivariate case follows immediately by cosiderig the fact if XI,X2"",X are mutually comootoic, the also Xl + X X - l ad X are mutually comootoic. If we restrict to the class of cocave distortio fuctios, the the certaity equivalet is subadditive, which meas that the certaity equi\'alet of a sum of risks is smaller tha or equal to the sum of the certaity equivalets. This property is stated i the followig theol-em. 4

7 Theorem 3 If the distortio fuctio 9 is cocave, the for ay risks Xl, X 2, "', X we have that Hq(XI + X 2 + '" + X) :s; L Hg(Xi)' i=l The theorem above is a straightforward geeralizatio of the bivariate case cosidered i ~Wag ad Dhaee (1997). 3 Stop-Loss Order ad Comootoicity For ay risk X ad ay d 2:: 0, we defie (X - d)+ = max(o, X - d). The stop-loss premium \vith retetio d is the give by E(X - d)+. Defiitio 3 A risk X is said to precede a risk Y i stop-loss order, writte X :S;sl Y, ~f for all retetios d 2:: 0, the stop-loss premium for risk X is smaller tha that for risk Y: E(X - d)+ :s; E(Y - d)+. I the followig theorem, we derive characterizatios of stop-loss order, withi the framework of Yaari's dual theory of choice uder risk. Theorem 4 For ay risks X ad Y, the followig coditio's are equivalet: (1J X :S;sl Y (2) For all cocave distortios fuctios, we have that Hg(X) :s; Hg(Y). (3/ For all distortio fuctios 9 defied by g(x) = mi (x/p, 1), p E (0,1], we have that Hg(X) :s; Hg(Y). Proof. (1) ::::} (2) : Relyig o the fact that stop-loss order is the trasitive closure of the order i dagerousess (Mliller, 1996), ad o the domiated covergece theorem, we oly have to prove that if X ad Yare ordered i dagerousess. writte X :S;D Y, the Hg(X) :s; I-Jg(Y) for all cocave distortio fuctios. Hece. let us assume that X :S;D Y; that is, E(X) < E(Y) ad there exists a real umber c 2:: 0 such that 5 x (:r) > 5 y (x)forall::c<c, 5 x (x) < 5y (x) for all T 2:: c..\"ow let g be a distortio fuctio. As g is o-decreasig, we immediately fid g(5x (:r)) > g(sy(x)) for all.t <C, g(5x (:r)) < g(5y (x)) for all:r >c.

8 Also assume that 9 is cocave i [0,1]. Thus, for each y i [0,1], there exists a lie l(:r) = ayx + b, with l(y) = g(y) ad l(x) ~ g(x) for all x i [0,1]. As l(y) = g(y), we fid that l(x) = ay(x - y) + g(y). Hece, l(x) ~ g(x) ca be writte as g(1;) - g(y) ~ ay(x - y) for all x i [0.1]. As 9 is o-decreasig, we fid that ay ~ 0. Further. ay is a o-icreasig fuctio of ii By substitutig Sx(x) ad Sy(x) for x ad y i the above iequality, we obtai 9 (Sx(x)) - 9 (Sy(x)) ~ as,.-(x) (Sx(:r) - Sy(x)) for all x ~ 0. From the crossig coditio for 9 (Sx(x)) - 9 (Sy(x)) ad the fact that asy(x) IS a odecreasig fuctio of x, we fid 9 (Sx(x)) - 9 (Sy(x)) ~ as, (c) (S\{r) - Sy(:r)) for all x > 0. Takig the itegral over both members of the iequality above leads to where the last iequality holds because E(X) ~ E(Y). Hece.,ve have prove that coditio (1) implies coditio (2). (2) ::::;, (3) : This follows immediately. because g(x) = mi (x/p, 1) defies a cocave distortio fuctio. (3) ::::;, (1) : For ay distortio fuctio 9 defied by g(:r) = mi (x/p, 1), p E (0,1]. we fid from Lemma 1 that 'Hg(X) ~ Hg(Y) is equivalet to l d P S-;/(p) + -1 Sx(x)dx + E(X - d)+ ~ P Syl (p) + 1 Sy(x)dx + E(Y - d)-i' Jd sx ~) 5; ~) for all d ~ O. We have to prove that E(X - d)+ ~ E(Y - d)+ for ay d ~ O. If Sx(d) = 0, the E(X - d)+ = so that E(X - d)+ ~ E(Y - d)+. ::\ow assume that Sx(d) > O. ad let p = Sx(d). Note that i geeral S)/(p) ~ d ad that for S;/(p) ~ 1; ~ d we have that Sx{r) = p. Hece, Hg(X) ~ Hg(Y) ca be rewritte as E(X - d)+ ~ r d 1 (Sy(:r;) - p) dx + E(Y - d)+. J 5;; (p) As Syl(p) ~ X <=> Sy(x) ~ p, we fid that the itegral i the iequality above is always egative, from which it follows that E(X - dh ~ E(Y - d)-t-. As the proof holds for ay d 2': 0, we fid that coditio (1) follows from coditio (3). Withi the framework of expected utility theory. stop-loss order of two risks is equivalet to sayig that oe risk is preferred over the other by all risk-averse decisio makers. From 6

9 the theorem above, we see that we have a similar iterpretatio for stop-loss order withi the framework of Yaari's theory of choice uder risk: Stop-loss order of two risks is equivalet to sayig that oe risk is preferred over the other by all decisio makers who have odecreasig cocave distortio fuctios. See Wag ad Youg (1997) for related results. :'\ote that our Theorem 4 is more geeral tha the correspodig result of Wag ad Youg (1997) because we do ot assume that the distortios are differetiable. It is well-kow that stop-loss order is preserved uder covolutio of mutually idepedet risks, see e.g. Goovaerts et al. (1990). I the followig theorem we cosider the case of mutually comootoic risks. Theorem 5 If Xl,X2, ""X ad Y 1, Y2,..., Y are sequeces of risks 'with Xi S:.SI Yi (i L...,) ad with Y 1,Y2,...,Y mutually comootoic, the L.: Xi S:.SI L.: Yi. i=l i=l Proof. Usig Theorems 2, 3 ad 4 we fid that for ay cocave distortio fuctio 9, i=l i=l \\'hich proves the theorem. ~ote that i the theorem above, we make o assumptio cocerig the depedecy amog the risks Xi' This meas that the theorem is valid for ay depedecy amog these risks. For ay risk X ad ay uiformly distributed radom variable U o [0,1], we have that X ~ Fx 1 (U). From this fact, we obtai the followig corollary to Theorem 5. Corollary 6 For ay radom variable U, uiformly distributed o [0,1]' ad ay risks Xl' X 2,...,X, we have L.: Xi S:.sl L.: Fjy} (U). i=l i=l Aother proof for this corollary. i terms of" supermodular order" 1 ca be foud i M tiller ). :'-Jote that (Xl, X 2,..., X) ad (Fjyj 1 (U), FX21(U),..., Fx~ (U)) have the same margial distributios, while the risks Fx/(U). i = 1,...,71" are mutually comootoic. Hece, Corollary 1 states that, withi the class of all multivariate risk with give margials Xl, X 2,..., X 1 the stop-loss premiums of Xl + X X are maximal if the risks Xi are mutually c0mootoic. 7

10 4 Stochastic Domiace ad Comootoicity I this sectio, \ve first examie whether Theorem 5, which holds for stop-loss order. also holds i the case of stochastic domiace, i.e. if)) S s/' is replaced by )) S sf". Defiitio 4 A risk Y is said to stochastically domiate a risk X, wriue X Sst. Y; ~l the followig coditio holds: Sx(x) S Sy(x) for all x 2: o. Let X I,X2, YI ad Y2 be uiformly distributed radom variables defied o [0,1]. with X 2 = 1- Xl ad Y I - Y2. The we have that Y I ad Y2 are comootoic. Further, Xi Sst Y, (i = 1, 2). After some straightforward calculatios, we fid that FXdx2(x) < FYI +1'2 (x) if 0 S x < l. F Xl + X 2 ( X ) > FYl""I-1'2 ( X ) if x 2: 1. Hece, Xl + X 2 is ot stochastically domiated by}] + Y2 so that Theorem 5 does ot hold i the case of stochastic domiace. However, stochastic domiace implies stop-loss order, so we should have that Xl +X2 Ssl Y I + Y2. This follows ideed from the crossig coditio. Theorem 7 For ay risks X ad Y, the followig coditios are equivalet: (1) X Sst Y. (2) For all distortio fuctios 9 we have that Hg(X) S Hg(Y). (3) SXI(p) S Syl (p) for all p E [0,1]. Proof. (1) =? (2) : Straightforward. (2) =? (3) : Let p E [0,1] ad cosider the distortio fuctio 9 defied by g(;r) = l(x > p). 0 S x S 1. The proof the follows from Lemma 1. (3) =? (1) : For a fixed x 2: 0, let p = Sy(x). From S.~l(p) S Syl(p), we haw that S" (Syl(p)) S P = Sy(x). Note that i geeral, S)-;l(p) S X. As Sx is o-icreasig, we fid SX(x) S Sx (Syl(p)) S Sy(x). As the proof holds for ay x 2: 0, we have prove that coditio (3) implies coditio (1). Withi the framework of utility theory, it is well-kow that stochastic domiace of t,, o risks is equivalet to sayig that oe risk is preferred over the the other by all decisio makers who prefer more to less. From the theorem above, we see that, withi the framework of Yaari's theory of choice uder risk, stochastic domiace of risk Y over risk X holds if ad oly if all decisio makers with (o-decreasig) distortio fuctio prefer risk X. 8

11 Actually, preservatio of stochastic domiace is a axiom i both utility theory ad Yaari's dual theory. Hece, the fact that coditio (1) implies coditio (2) is a direct result of this axlo1. 5 Maximal Stop-Loss Premiums i the Multivariate Case From Corollary 6, we cocluded that i the class of all multivariate risks with give margials (Xl, X 2,..., X ), the stop-loss premiums are maximal if the risks Xi, i = 1,...,, are mutually comootoic. For comootoic risks Xi, the stop-loss premium with retetio d is give by ::\ ow we will derive aother expressio for this upper boud. Theorem 8 Let Xl,... ) X be mutually comootoic risks. The for ay retetio d 2' 0, we have E(XI X - d)+ = LE(Xi - di)+ - i=l [d - SXl (Sx(d))] Sx(d) where X = Xl X ad the di are defied by di = Sx;(Sx(d)). Proof. If Sx(d) = 0, the the iequality trivially holds. ::\ow assume that Sx(d) > O. Let p = Sx(d) ad defie a distortio fuctio g by g(x) = mi (x/p, 1) for 0 :::; x :::; 1. As Xl,'",X are mutually comootoic, we fid from Theorem 2 that Csig Lemma 1 this equality ca be \Hitte as from which we fid because S;/(p) = L~l Sx;(p) for comootoic risks, see Deeberg (1994) or Wag (1996). O the other had, we have that 9

12 Now combie these two equalities to obtai the desired result. From Theorem 8 we see that, apart from a correctio factor, ay stop-loss premium for,. ~ the sum of comootoic risks ca be writte as a sum of stop-loss premiums for the idividual risks ivolved. Note that i geeral we have that SxJ(Sx(d)) < d. However, if S,,{r) > Sx(d) for all x < d, the Sx 1 (Sx(d)) = d, so that i this case E(X X - d)+ = 2: E(Xi - di)+ i=l with the di as defied i Theorem 8. I this case,,, e also have that L~'=l di = d. 6 Examples I this fial sectio, we show by example how to evaluate stop-loss premiums for the sum X = Xl + X X of the mutually comootoic risks Xl, X 2,..., X. We first cosider the case for which all risks have a two-poit distributio ad the three cases for which all risks have cotiuous distributios. Example 1: The Idividual Life lvlodel Assume that each risk Xi, (i = 1,...,) has a two-poit distributio i 0 ad ai > 0 with Pr(Xi = ai) = qi. The ddf of Xi is the give by if 0 ::; x < ai, if x ~ ai, from which we fid S-l C ) = { Oi, if 0 ::; p < qi x, p 0, if qi ::; P ::; l. Without loss of geerality, we assume that the radom variables Xi are ordered such that q1 2:... 2: q Now assume that the risks are comootoic, the we have if 0 ::; p < q 1 if qj-t-l ::; P < qjl if q1 ::; P < 1. Hece, if 0 ::; x < ai, if a aj ::; x < a ajll, if :1: 2: a 10

13 which meas that X is a discrete radom variable with poit-masses i 0, a1, a] + a2, a1 + a2 + a3,.. " a1 + a a For d such that a aj :s: d < a aj+l,we fid if i < j + 1, if i 2- j + 1, so that S-;/(Sx(d)) = S.~1(qj_H),,ve fially fid from Theorem 8 that = L:Si;(Qj+l) = al aj. i=l 'f d > "\", 1 '_ L.,i=l ai This idividual life model is more extesively cosidered i Dhaee ad Goovaerts (1996). Example 2: Expoetial l\iargials Assume that each Xi, (i = 1,"', ) IS distributed accordig to the Expoetial (bi) distributio (bi > 0) with ddf give by SX i (x) = e- x / bi, x > O. For comootoic Xi, the iverse ddf of their sum X is Si 1 (p) = -blp, Sx(x) = e- X / b, x> O. I other words, the comootoic sum of expoetial radom variables is expoetially distributed. Heilma (1986) cosiders the case of = 2. Oe ca easily verify that the stop-loss premium with retetio d is give by Example 3: Pareto lviargials Assume that each Xi (i = 1,".,) is distributed accordig to the Pareto (a, bi) distributio (a, b i > 0) with ddf give by bi ) a ( bi +:r ' x> O. 11

14 For comootoic Xi, the iverse ddf of their sum X is SXl(p) = b (p-l/a -1), i which b = L~'=l bi. Thus, x> O. I other words, the comootoic sum of Pareto radom variables (with idetical first parameter) is a Pareto radom variable. Oe ca easily verify that for ay d 2:: 0 we have that E(X _ d)+ = (_b_)a-l b b+d a_i' a> 1. Example 4: Expoetial-Iverse Gaussia Margials Assume that each Xi, (i = 1....,) is distributed accordig to the expoetial-iverse Gaussia (bi, Ci) distributio (bi, Ci > 0) with ddf give by x> 0, see Hesselager, Wag ad Willmot (1997). I this case the iverse ddf of Xi is fb: SXi (p) = - (I p) - v-l p. 4Ci Ci Thus, for comootoic Xi, the iverse ddf of their sum X is If Sx 1 (p) = - 1 (I p) 2 - -I p. 4c C Sx(x) = e:rp [-2JC (J;y; + b - Jb)], x> O. I other words, the comootoic sum of expoetial-iverse Gaussia radom variables is also a expoetial-iverse Gaussia radom variable. Oe ca easily verify that for ay d 2:: 0 we have that E(X - d)+ ~ exp [-2Ji (.,jd+ b - Jb)] [Jd ~ b + ;c]. 12

15 Refereces Deeberg, D. (1994). No-Additive MeasuTe Ad Itegral, Kluwer Academic Publisher's, Bosto. Dhaec, 1. ad Goovaerts, M. (1996). Depedecy of risks ad stop-loss order, ASTIN Bulleti, 26(2), Dhaee, 1. ad Goovaerts, M. (1997). O the depedecy of Tisks i the idividual lile model, Isurace: Mathematics ('1 Ecoomics, 19, Goo'uaerts, M.J., Kaas, R.,va Heerwaarde, A.E. ad Ba'uwelickx, 1'. (1990). Eflecti'ue actuarial methods, Isurace Series, Vol. 3, North-Hollad. Amsterdam, Ncw YOTk, Oxford, Tokyo. Heilma, W.R. (1986). O the impact of the idepedece of risks o stop-loss prem'illms, Isurace: JvIathematics ad Ecoomics, 5, Hesselager, 0.; Wag, S. ad Willmot, G. (1997). Expoetial ad scalc mixt'utes ad equilibrium distributios, Workig Paper. Kaas, R., va Heerwaarde, A.E., ad Goovaerts, M.J. (1994). Risks, Educatio SeTies 1, CAIRE, Brussels. Orderig of Actuarial Miiller, A. (1996). OrdeTig of Tisks: a comparati'ue study via stop-loss trasfotts, Isu:race: Mathematics ad Ecoomics Miiller, A. (1997). Stop-loss otdet fot pottfolios of depedet risks. Isurace: Mathematics ad Ecoomics, to appeat. vo Neuma, 1., MorgesteT, O. (1947). TheoTY of games ad ecoomzc beha'uiour, secod editio, PTiceto UiveTsity PTess, Priceto, New JeTsey. \Fag, S. (1996). PTemium calculatio by trasformig the layet premium desity, ASTIN Bulleti 26, \/Fag, S. ad Dhaee, 1. (1997). Comootoicity, cottelatio ordet ad ptemium priciples, submitted. l,fag, S. ad Youg, V.R. (1997). OrdeTig Tisks: utility theoty versus YaaTi's dv,al theory of Tisk, submitted. Ycwri, M. E. (1987). The dual theory of choice udet Tisk, Ecoometrica 55,

OEPARTEMENT TOEGEPASTE ECONOMISCHE WETENSCHAPPEN

j. OEPARTEMENT TOEGEPASTE ECONOMISCHE WETENSCHAPPEN ONDERZOEKSRAPPORT NR 9812 A NOTE ON THE STOP-LOSS PRESERVING PROPERTY OF WANG'S PREMIUM PRINCIPLE by C. RIBAS M.J. GOOVAERTS J.DHAENE Katholieke Universiteit