Optimal Dynamic Proportional and Excess of Loss Reinsurance under Dependent Risks

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1 Modern Economy, 6, 7, Pulsed Onlne June 6 n ScRes. ttp:// ttp://dx.do.org/.436/me Optmal Dynamc Proportonal Excess of Loss Rensurance under Dependent Rsks Crstna Goso, Ester C. Lar, Marna Ravera Department of Economcs Busness Studes, Unversty of Genoa, Genoa, Italy Receved May 6; accepted June 6; pulsed 5 June 6 Copyrgt 6 y autors Scentfc Researc Pulsng Inc. Ts work s lcensed under te Creatve Commons Attruton Internatonal Lcense (CC BY. ttp://creatvecommons.org/lcenses/y/4./ Astract In ts paper, we study an optmal rensurance strategy comnng a proportonal an excess of loss rensurance. We refer to a collectve rsk teory model wt two classes of dependent rsks; partcularly, te clam numer of te two classes of nsurance usness as a varate Posson dstruton. In ts contest, our am s to maxmze te expected utlty of te termnal wealt. Usng te control tecnque, we wrte te Hamlton-Jaco-Bellman equaton, n te specal case of te only excess of loss rensurance, we otan te optmal strategy n a closed form, te correspondng value functon. Keywords Rensurance, Proportonal Rensurance, Excess of Loss Rensurance, Hamlton-Jaco-Bellman Equaton. Introducton In te last two decades te optmal rensurance prolem as ad an mportant mpact n te actuaral lterature. Several autors ave studed ts prolem wt dfferent purposes referrng to dfferent surplus processes. Startng from te classcal model were te process of te total clam amount as a Posson compound dstruton or follows a dffuson process, te adjustment coeffcent, or te expected utlty of te termnal wealt are een optmzed (see, for example, [] []. Wt smlar optmzaton ams, a more realstc model as een often consdered, wt two or more dependent classes of nsurance usness. Smlar approaces are, for example: n [3] were te excess of loss nsurance s consdered te adjustment coeffcent or te expected utlty of te termnal wealt are maxmzed, n [4] n [5] were te expected utlty of te termnal weal ts maxmzed, n [6] were te adjustment coeffcent s maxmzed. Ts paper consders two classes of nsurance usness, dependent troug te numer of How to cte ts paper: Goso, C., Lar, E.C. Ravera, M. (6 Optmal Dynamc Proportonal Excess of Loss Rensurance under Dependent Rsks. Modern Economy, 7, ttp://dx.do.org/.436/me

2 clams, consders te proportonal te excess of loss rensurances. Te paper s organzed as follows: n Secton te assumptons te model are explaned, n Sectons 3 4, te prolem s presented; susequently te Hamlton-Jaco-Bellman (HJB equaton s gven dscussed n some partcular cases. In Secton 5, te prolem wt te only excess of loss rensurance s solved; te optmal strategy te correspondng value functon are otaned.. Te Model We consder te fnte tme orzon [, T], < T < a model n wc two dependent rsks are nvolved. In partcular, we assume two classes of nsurance usness, eng te clam numer processes correlated. Te arrval clam processes are { N, =, ; t [, T] } we assume tat tese processes are Posson processes defned as follows: N = Q + Q, =,, t [, T], ( were Q, Q Q are Posson rom varales wt postve parameters θ, θ θ respectvely. We furtermore assume tat Xj, j =,,, are te rom varales clam sze of te rsks =,,, were we assume tat X j X j ave te same dstruton functons F F wt F (x =, for x, expected value E X j = µ < +, =,. We also assume tat te moment generatng functons: M ( r, =,, j =,,, = =, are mutually ndependent, ndepen- exst. As usually stated, te rom varales { Xj,,, j,, } dent of { N, =, ; t [, T] }. Let S ( t, t [, T],, te made assumptons, te process {( N, N ; t [, T] } X j =, te aggregate clams amounts for te two classes of nsurance rsk. Because of, as a varate Posson dstruton S S t are correlated y θ resultng: N S t = Xj, =, ( j= We consder te rom varales X, =,, dentcally dstruted to X, j =,,, respectvely. We assume tat te rom varales X are upper lmted or tat F ( s lms + s =. We denote y c, =,, te premum rate, calculated y te expected value prncple, ncludng a safety loadng coeffcent η : c = + η θ + θ µ, η >, =, (3 We assume tat te prncpal nsurer can mplement ot a proportonal an excess of loss rensurance referred to ot classes of nsurance rsks, wt te respectve retenton levels a (,, =, for te proportonal rensurance, retenton lmts (, +, =,, for te excess of loss rensurance. We terefore denote y {( a, a,,, t } any admssle control strategy tat, for smplcty, we denote y ( a,. Te rensurer, ecause of te proportonal rensurance, would pay ( a X for eac clam of -type; owever, ecause of te furter excess of loss rensurance, e pays: tat s: a X f ax a X + ax = X f ax, =, ( a X f X a Z = X f X a We assume tat all te premums are pad usng te expected value prncple. Terefore, te rensurance premum rate at tme t s, for eac class of rsk: j (4 76

3 [ ] P a, = + γ θ + θ E Z, γ > η, =,, were we ave assumed te safety loadng coeffcents γ > η, tat s: P(, ( a ( a = + γ θ + θ µ a ( F d. s s (5 Terefore, after te rensurances, te premum rate for te nsurer s: P: = c P a, = a F s d s. = = 3. Te Prolem ( a θ θ µ η γ γ θ θ ( (6 We assume tat te nsurer can coose, for every tme tt, [, T], te a ( t,,, to te oservale nformaton aout te nsurance rsk processes up to tme t. Ts means tat t = accordng a t, =,, are te control parameters tat allow us to consder te followng set of admssle strateges: {(,,, (, ; (,,,, [, ]} A= a t a t t t = a a t t + t T Te man goal for te nsurer s to coose an optmal rensurance strategy tat maxmze te expected exponental utlty of termnal wealt. To solve ts prolem, we wll use a dynamc programmng approac. After te rensurance, rememerng (4, referrng to te j-t clam of type, te nsurer pays X j ax f X a X j = f X. a Hence, te total clam amount carged to te nsurer at tme t, referred to te -type clam s: N S t = X, =, j= It follows tat te surplus process R( t controlled y te rensurance strateges, evolves over te tme as follows: dr = ( ( ( a θ + θ µ η γ + + γ θ + θ a ( F d d s s t = ds t + S t = Pdt d S t + S t. We recall tat te process {( N,, ; t [, T] } = as a varate Posson dstruton wt statonary ncrements; terefore, usng results n [3] [5], t results tat: ( + d + ( + d + S t t S t t S t S t j ( θ θ θ wt proalty + + dt mn ( ax, : = Y wt proalty θdt = mn ( ax, : = Y wt proalty θdt Y + Y wt proalty θdt We assume an nsurer s utlty functon u R Ru C ( R ue defned as follows: :,, wt u > u <. In partcular, let e x u x =, >, x R. (9 (7 (8 77

4 Te nsurer looks for an optmal control strategy so as to maxmze te expected utlty of te termnal surplus under te ntal condton regardng te x state at tme t. We consder te followng value functon: wt te oundary condton = ( = [ V tx, sup EuRT Rt x, t, T, a, A u( x 4. Te Infntesmal Generator te HJB Equaton ( V T, x =. ( We are ale to fnd te nfntesmal generator for te process R( t for te functonvv. Ts allows us to wrte te HJB equaton; we prove te followng teorem:, Teorem. Let V e defned y ( let V C ([, T] R. Terefore, V satsfes te followng HJB equaton: + + V( tx t P V( tx x θ a a, A V( tx as F ( s sup,,, d a a a (, d θ (, d d + θ V tx as F s+ V tx as az F s F z, a F θv tx θ V( tx, az df ( z a F θv tx, a a V ( t, x as d F ( s + θ + θ V( tx, F F ( θ + θ + θ V( tx, = a a Proof. We derve te followng nfntesmal generator LV ( t, x for te process V. Te procedure s smlar to tat used n [7] [8]. Rememerng (7 (8, t results: were we ave: LV t ( t, x = lm E( V ( t+, R( t+ V ( t, R R = x. LV t t x V t x P V t x + lm EV ( t+, x+ P Y V( t, x θ + lm EV ( t+, x+ P Y V( t, x θ + lm EV ( t+, x+ P Y Y V( t, x θ, (, = lm ( +, + (, ( ( θ + θ + θ lm V ( t+, x+ P V ( t, x ( ( θ + θ + θ V t+, x+ P V( tx, + P + P = + + V( tx, V( tx, = + P ; t x lm P + P : = t ( R t for te functon 78

5 terefore we fnd: lm EV ( t+, x+ P Y V ( t, x θ a = θ V( tx, as d (, (,,, ; F s+ θv tx F θv tx = a lm EV ( t+, x+ P Y Y V ( t, x θ = θ a a V( t, x as az df ( s df ( z + a V t x as F F s a (, d + a V tx az F F z a (, d + V( tx, F F V( tx, a a (, V( tx, V tx LV(, a t tx = + P+ θ V tx, as df s t x a a a (, d θ (, d d + θ V tx as F s+ V tx as az F s F z, a F θv tx θ V( tx, az df ( z a F θ V( tx, + θ a a (, d V tx a F s s θ V( tx, F F ( V( tx, ; a a θ θ θ te Equaton ( s terefore fulflled y V. As we specfed efore, we assume te utlty functon (9, nspred y [] [4] [8]-[], we look for a soluton of te prolem (, wt te condton ( of te form: wt QT =. We note tat: Terefore, ( can e wrtten as follows: ( x Qt V( tx, = e, (, V tx t (, V tx x (, = Q tv tx (, = V tx ( = V t, x k V t, x e k 79

6 sup, e d θ a as V( tx Q P θ a, A F( s + a a z e d a a a s a z e d d + θ F z F s F z e a + a z F θ + θ e df ( z a (3 as + F θe + θ a e d a F + θ e F F + ( θ + θ + θ =. a a Oservng tat: ( s e df s = e F + a e F sd s, =,, a a as a as ( t follows tat Equaton (3, dvdng y V( tx, > rememerng (6, can e wrtten as follows: sup Q t a F s ds + ( θ a, ( θ µ η a A γ γ θ θ ( = a as a az θ ( θ θ θ ( + a e F s ds + a e F z dz a as ( e d a a z θ aa F s s e ( F z dz =. In te partcular case were = +, =,, we otan te case study regardng te only proportonal rensurance, as analyzed n [4] [8]; under ts assumpton, Equaton (6, rememerng (3 te assumptons made aout te rom varales X, gves: Terefore, (3 ecomes: ( ( θ θ ( γ µ P = c + + a. sup, = = V( tx Q, ( c ( θ a A θ ( γ ( a µ as + az as az F ( s + + F ( z + F ( z F ( z } θ e d θ e d θ e d d + θ + θ + θ =. (5 It s ovously tat Equaton (5 s te same equaton found n [8]; furtermore Equaton (5 dvded y V( tx, < concdes wt Equaton (3.4 of [] wt r = δ ( q, q calculated y means of te expected value prncple. In te partcular case were a =, =,, we ave te only excess of loss rensurance case. Under ts assumpton, Equaton (3, wrtten on te form (4, s:, Q t F s s = + θ θ µ η A γ γ θ θ ( sup d s z ( θ θ ( ( θ θ ( s z θ e ( F ( s ds e ( F ( z dz }. + e F s ds + e F z dz = (4 (6 7

7 In te followng secton we consder ts case. 5. Te Excess of Loss Rensurance Case We face te prolem (6, wt condton (, tat s QT =. We wrte (6 as follows sup g (,, A wt condtons: We ave: (, = (7, =, (8 QT =. (9 ( g z ( ( ( ( ( ( = F θ + θ + γ e θ + θ + θ e F z dz ( (, ( g s ( ( ( ( ( ( = F θ + θ + γ e θ + θ + θ e F s ds ( g s negatve defned. Te relatve proof can e mmedately otaned usng results n [3]. We can terefore searc te soluton of te prolem (7, wt condtons (8, lookng for te solutons of te followng system: from wc we deduce tat, at te ponts were te gradent of g s zero, te Hessan matrx of (, tat s, lettng ξ = λ ( F( (, g = λ, =, λ =, =,, =,, rememerng ( (, we look for te soluton of te system: ( F z z ( F s s z ( s ( θ + θ + γ e θ + θ + θ e d = ξ θ + θ + γ e θ + θ + θ e d = ξ ( ξ =, =,, =,. Te solutons can e of te followng four knds: We oserve tat te soluton tat s mpossle. ( I ( II ( III ( IV = = = > > = > >. ( I = = does not exst. Indeed, te system ( gves: θ + θ γ θ + θ γ 7

8 Accordng to results n [3], we ave: ( II = an d >. Te system ( can e wrtten as: from wc, f: we ave: + e F z dz z ( θ θ γ θ ( ( θ + θ ( + γ e = ξ = > ln ( + γ z ( θ + θ γ θ ( e F z d, z (3 ( + γ ln =, = (III > =. In a smlar way to (II, f: ln ( + γ s ( θ + θ γ θ ( we ave: ln ( + γ =, = ( IV > an d >. Te system ( gves: e F s d, s (4 z ( θ + θ ( + γ = e ( θ + θ + θ e ( F ( z dz s ( θ + θ ( + γ = e ( θ + θ + θ e ( F ( s ds (5 ξ = ξ =, =,. In [3], t s proved tat under te assumpton tat ot (3 (4 are not satsfed, tat s te optmal strategy (, ln ( + γ z ( θ + θ γ > θ ( e F z dz (6 ln( + γ s ( θ + θ γ > θ (, fulfllng (5, exsts wt We oserve tat, from (8, eng true also (7 snce e F s d, s (7 < < ln ( + γ. (8 fulflled te system (5, t results tat: < < ln ( + γ. (9 Fnally, we recall tat (3 (4 are ncompatle (see [3]. We are so ale to fnd te value functon, susttutng te optmal strategy n (7, tat s n (6, otan- QT =. If (3 s fulflled, we ave: ng Q( t wt te condton 7

9 : ln( + γ ( e z ( d Q t = Q t = θ ; + θ µ γ η θ + θ + γ F z z T t = f (4 s fulflled, t results: (3 : ln( + γ ( e s ( d Q t = Q t = θ ; + θ µ γ η θ + θ + γ F s s T t = (3 f (6 (7 are at te same tme fulflled, we ave: : = 3 = ( θ + θ µ ( γ η Q t Q t = s ( θ θ ( γ ( + + e F s ds z ( θ θ ( γ ( + + e F z dz s z + θ e ( F ( s ds e ( F ( z d z ( T t. Te results otaned n ts secton are collected wtn te followng teorem. Teorem. Te optmal strategy (, concernng a wole excess of loss rensurance te correspondng value functon are te followng: f (3 t s were Q ( t s gven y (3; f t s were Q ( t s gven y (3; f ln ( + γ z ( θ + θ γ θ ( e F z dz q =, q = ln ( + γ V( tx, = e ln ( x Q ( + γ s ( θ + θ γ θ ( q e F s ds = ln ( + γ, q = V( tx, = e ln ( x Q ( + γ z ( θ + θ γ > θ ( e F z dz 73

10 t s were Q 3 t s gven y (3. ln ( + γ s ( θ + θ γ > θ ( e F s ds < q < ln +, < < ln + ( γ q ( γ V( tx, = e ( x Q 3 Acknowledgements We tank te Edtor te Referees for ter comments. References [] Lang, Z. Guo, Y. ( Optmal Comnng Quota-Sare Excess of Loss Rensurance to Maxmze te Expected Utlty. Journal of Appled Matematcs Computng (JAMC, 36, -5. ttp://dx.do.org/.7/s [] Scmdl, H. ( Optmal Proportonal Rensurance Polces n a Dynamyc Settng. Scnavan Actuaral Journal,, ttp://dx.do.org/.8/ [3] Centeno, M.L. (5 Dependent Rsks Excess of Loss Rensurance. Insurance: Matematcs Economcs, 37, ttp://dx.do.org/.6/j.nsmateco.4.. [4] Lang, Z. Yuen, K.C. (6 Optmal Dynamc Rensurance wt Dependent Rsks: Varance Premum Prncple. Scnavan Actuaral Journal,, ttp://dx.do.org/.8/ [5] Yuen, K.C., Lang, Z. Zou, M. (5 Optmal Proportonal Rensurance wt Common Sock Dependence. Insurance: Matematcs Economcs, 4, -3. ttp://dx.do.org/.6/j.nsmateco [6] Hu, F. Yuen, K.C. ( Optmal Proportonal Rensurance under Dependent Rsks. Journal of Systems Scence Complexty, 5, 7. ttp://dx.do.org/.7/s x [7] Cetn, U. (4 An Introducton to Markov Processes Ter Applcatons n Matematcal Economcs. Lecture Notes. Department of Statstc, London Scool of Economcs Poltcal Scence, London. [8] Goso, C., Lar, E.C. Ravera, M. (5 Optmal Proportonal Rensurance n a Bvarate Rsk Model. Modern Economy, 6, ttp://dx.do.org/.436/me [9] Ln, X. L, Y. ( Optmal Rensurance Investment for a Jump Dffuson Rsk Process under te CEV Model. Nort Amercan Actuaral Journal, 5, 3, ttp://dx.do.org/.8/ [] Jonson, N.L., Kotz, S. Balakrsnan, N. (997 Dscrete Multvarate Dstrutons. Wley & Sons, New York. 74