Optimal investment problem for an insurer with dependent risks under the constant elasticity of variance (CEV) model

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1 Opimal invesmen problem for an insurer wih dependen riss under he consan elasiciy of variance (CEV) model YAJIE WANG Tianjin Universiy School of Science 92, Weijin Road, Tianjin CHINA XIMIN RONG Tianjin Universiy School of Science 92, Weijin Road, Tianjin CHINA HUI ZHAO Tianjin Universiy School of Science 92. Weijin Road, Tianjin CHINA zhaohui Absrac: In his paper, we consider he opimal invesmen problem for an insurer who has n dependen lines of business. The surplus process of he insurer is described by a n-dimensional compound Poisson ris process. Moreover, he insurer is allowed o inves in a ris-free asse and a risy asse whose price process follows he consan elasiciy of variance (CEV) model. The invesmen objecive is maximizing he expeced uiliy of he insurer s erminal wealh.applying dynamic programming approach, we esablish he corresponding Hamilon- Jacobi-Bellman (HJB) equaion. Opimal invesmen sraegy is obained explicily for exponenial uiliy. Finally, we provide a numerical example o analyze he effecs of parameers on he opimal sraegy. Key Words: Opimal invesmen, Dependen riss, Consan elasiciy of variance (CEV) model, Uiliy maximizaion, Hamilon-Jacobi-Bellman (HJB) equaion 1 Inroducion Recenly, opimal invesmen problem for an insurer has been sudied in many lieraures. For diffusion ris model, Browne [1] invesigaed he problem for maximizing he uiliy of erminal wealh and minimizing he probabiliy of ruin. Hipp and Plum [2] assumed he insurer can inves in a risy asse and obained he explici sraegy for ruin probabiliy minimizaion under compound Poisson ris model.laer Liu and Yang [3] exended he research of Hipp and Plum [2] by adding a ris-free asse ino he model of Hipp and Plum [2].Yang and Zhang [4] proposed he jump-diffusion ris process and obained he opimal sraegy for ruin probabiliy minimizaion. Wang e al. [5] applied he maringale approach o invesigae he opimal invesmen problem for an insurer. Bai and Guo [6] supposed ha he insurer can inves in muliple risy asses and obained he opimal sraegy o maximize he uiliy of erminal wealh. Under he objecive of minimizing he ruin probabiliy, Li e al. [7] sudied he opimal invesmen problem for boh he insurer and he reinsurer when he insurer can purchase proporional reinsurance. Besides, some more researches abou he opimal invesmen problems in differen conexs have been exensively sudied, see, Chang and Lu [8],Chang e al. [9],Li and Liu [10],and references herein. corresponding auhor In he above-menioned lieraure, hey generally assume ha he insurance company only has one business line. Bu in pracice, many insurance companies have wo or more differen lines of business, for insance, auo insurance, hird pary insurance, casualy insurance, endowmen insurance, and so on. Wha s more, he n lines of insurance business usually have a relaion of dependence. The classical example of dependen riss is naural disasers, such as an earhquae, yphoon or sunami, where usually cause a leas wo inds of claims such as deah claims, medical claims, ec. Currenly, some researchers began o sudy opimal reinsurance problem wih lines of business.in he saic seing, Ceneno [11] sudied he opimal excess-of-loss reinsurance sraegy for wo dependen classes of insurance riss. By using maringale cenral limi heorem, Bai e al.[12] firs derived a wo-dimensional diffusion approximaion for he wo-dimensional compound Poisson reserve ris process and sudied an opimal excess-of-loss reinsurance problem for he approximaed diffusion model. Liang and Yuen [13] considered he opimal proporional reinsurance problem wih wo dependen riss under he variance premium principle. Yuen e al.[14] exended he research of Liang and Yuen [13] o he case wih he reinsurance premium calculaed under he expeced value principle and he model wih muliple dependen classes of insurance business, closed- E-ISSN: Volume 16, 2017

2 form sraegies are derived for boh compound Poisson ris model and he diffusion approximaion ris model. In his paper,we consider invesmen problem for an insurer wih dependen riss. The insurer is allowed o inves in a ris-free asse and a risy asse. Furhermore, he price process of he risy asse follows he consan elasiciy of variance (CEV) model,which is a naural exension of geomeric Brownian moion (GBM).The CEV model was proposed by Cox and Ross [15]. A firs, he CEV model was usually used in opion pricing research,see e.g., Becers [16],Davydov and Linesy [17],Jones [18]. Recenly, he CEV model has been inroduced o opimal invesmen research by Darius [19].For he porfolio s- elecion problem, Li e al.[20] considered he opimal invesmen problem wih axes, dividends and ransacion coss under he consan elasiciy of variance (CEV) model and obained he soluions for he logarihmic, exponenial and quadraic uiliy funcions. For he opimal reinsurance and invesmen problem, Gu e al.[21] invesigaed he opimal reinsuranceinvesmen problem wih a Brownian moion ris process under he CEV model, and opimal sraegies are derived for insurers wih CRRA or CARA uiliy. For jump-diffusion ris process, Liang e al.[22] adoped he CEV model for sudying proporional reinsurance problem. Besides, here are some oher lieraures s- udied opimal reinsurance-invesmen problems under he CEV model, see, Lin and Li [23],Li e al.[24] among ohers. For he opimal invesmen problem for he DC pension fund, Xiao e al.[25] began o apply he CEV model o invesigae he pension fund invesmen problem and derived he opimal sraegy for logarihm uiliy funcion using he echnologies of Legendre ransform and dual heory. Gao [26], [27] exended he wor of Xiao e al.[25] by solving he opimal soluions for CRRA and CARA uiliy funcions. These papers moivae us o consider he opimal invesmen problem for an insurer wih n dependen lines of business. The surplus process of he insurer is described by a n-dimensional compound Poisson ris process, and he insurer is allowed o inves in a ris-free asse and a risy asse whose price process follows he CEV model. The objecive of he insurer is o maximize he expeced uiliy of his/her erminal wealh. By applying dynamic programming approach, we esablish he Hamilon-Jacobi-Bellman (HJB) equaion associaed wih he opimal problem and ransform i ino a parial differenial equaion. Under some given assumpions, explici soluions o he problem of expeced CARA uiliy maximizaion are derived. Furhermore, we presen a numerical example o analyze he effecs of parameers on he opimal sraegy. This paper is organized as follows. In secion 2, we inroduce he n-dimensional ris model. Secion 3 derives he opimal sraegies o maximize he uiliy of erminal wealh. In secion 4,numerical examples are carried ou o analyze he effecs of parameers on he opimal sraegies. Finally, we give conclusions in Secion 5. 2 Problem formulaion In his paper, we consider an insurer who has n dependen classes of insurance business,such as healh insurance/casualy insurance/hird pary insurance, and so on. The aggregaed claims up o ime in he ih line of business are denoed by N i ()+N() j=1 X ij,and c i sands for he premium rae of he ih line. Then he surplus process of he insurer follows: R() = x + c i N i ()+N() j=1 X ij, (1) where {N i ()} and {N()} are n + 1 independen Poisson processes wih inensiies λ i and λ for i = 1, 2,, n. The claim sizes {X ij, j = 1, 2, } are i.i.d posiive random variables wih common disribuion funcion for i = 1, 2,, n, and {X ij, j = 1, 2, } are independen of {N i ()} and {N()}.Besides,{X ij, j = 1, 2, } are independen of {X j, j = 1, 2, } for i, i, = 1, 2,, n. Denoe E(X i ) = a i and V ar(x i ) = b i for i = 1, 2,, n. Suppose he premium is calculaed according o he expeced value principle, i.e., c i = (1 + η i )(λ i + λ)a i, (2) where η i is he posiive safey loading in line i. According o Bai e al.[12],he approximaion ris model of (1) is where dr () = µ i d + γ i dw (i) (), (3) µ i = (λ + λ i )η i a i, γ i = ((λ + λ i )E[X 2 i ]) 1 2. (4) W (i) () are sandard Brownian moions. Furhermore, he correlaion coefficien of W (i) () and W () () are denoed by ρ i for i, i, = 1, 2,, n, and ρ i = λ γ i γ E(X i )E(X ) = λ γ i γ a i a. (5) E-ISSN: Volume 16, 2017

3 We assume ha he insurer is allowed o inves in a ris-free asse and a risy asse. The price of risfree asse is given by ds 0 () = S 0 ()d, (6) where > 0 is he ineres rae, and he price of risy asse is described by he CEV model: ds() = r 1 S()d + (S()) β+1 dw (0) (), (7) where r 1 > is he appreciaion rae of he risy asse, {W (0) ()} is a sandard Brownian moion defined on a complee probabiliy space (Ω, F, P ).β 0 represens he elasiciy coefficien and (S()) β sands for he insananeous volailiy of risy asse. The correlaion coefficien of W (0) () and W (i) () are denoed by ρ i0 for i = 1, 2,, n. Le π() be he money amoun invesed in he risy asse a ime by he insurer, Then X() π() is he money amoun invesed in he ris-free asse, where X() is he wealh of he insurer a ime. For a rading sraegy π(),he wealh process X() is given by: dx() = π() ds() S() + (X() π())ds 0() S 0 () +dr () = [(r 1 )π() + X() + n µ i]d +π()(s()) β dw (0) () + n γ idw (i) (), X(0) = x. (8) If π() is F -progressively measurable and T 0 (π())2 dx <,π() is called an admissible sraegy. Denoe Π he se of all admissible sraegies. Suppose ha he insurer has a uiliy funcion u( ) which is sricly concave and coninuously differeniable on (, + ). The insurer aims o maximize he expeced uiliy of erminal wealh, i.e., max E[u(X(T ))]. (9) π Π 3 Soluion o opimal invesmen problem for an insurer wih dependen ris In his secion, we ry o find he explici soluions for opimizaion problem (9) for exponenial uiliy by using dynamic programming approach. 3.1 General framewor The value funcion is defined as H(, s, x) = sup H π (, s, x), 0 < < T, (10) π Π H π (, s, x) = E[u(X(T )) S() = s, X() = x]. (11) According o Fleming and Soner [28], if H and H, H x, H xx, H s, H ss, H sx is coninuous, H saisfies he following Hamilon-Jacobi-Bellman(HJB) equaion: H + r 1 sh s + ( x + n µ i )H x ( n +2 i,j=1,i j γi 2 ρ ij γ i γ j )H xx s 2β+2 H ss +s β+1 ( n γ i ρ i0 )H sx + sup π { 1 2 π2 2 s 2β H xx Le A = = +[(r 1 )H x + ( n γ i ρ i0 )s β H xx + 2 s 2β+1 H sx ]π} = 0. γ i ρ i0, B = γi γi i,j=1,i j λa i a j, i,j=1,i j ρ ij γ i γ j and differeniaing wih respec o π in (12),we have (12) π = (r 1 )H x + As β H xx + 2 s 2β+1 H sx 2 s 2β. H xx (13) Subsiuing (13) ino (12),we ransform (12) ino a parial differenial equaion, H + r 1 sh s + ( x + n µ i )H x BH xx s 2β+2 H ss + s β+1 AH sx [(r 1 )H x + As β H xx + 2 s 2β+1 H sx ] s 2β H xx = 0. (14) Nex, we solve opimal problem (9) for common consan absolue ris aversion (CARA) uiliy funcion, i.e., u(x) = 1 q e qx, q > 0. (15) To solve (14),we ry o conjecure a soluion in he following form V (, s, x) = 1 q e[ q(d()x+g(,s))], (16) E-ISSN: Volume 16, 2017

4 wih boundary condiion given by (16) gives d(t ) = 1, g(t, s) = 0. V = q(d ()x + g )V, V s = qg s V, V ss = (q 2 g 2 s qg ss )V, V x = qd()v, V xx = q 2 d() 2 V, V xs = q 2 d()g s V. Plugging V, V s, V ss, V x, V xx, V xs ino (14),afer simplificaion,we have ( qd () q d())x qg q sg s qd()( n µ i ) q2 d 2 ()[B A 2 ] qs 2β+2 g ss (r 1 ) s 2β + qd()(r 1 )A s β = 0. (17) can be spli ino wo equaions and (17) qd () q d() = 0, (18) qg q sg s qd()( n µ i ) q2 d() 2 [B A 2 ] qs 2β+2 g ss (r 1 ) s 2β + qd()(r 1 )A s β = 0. From (18) and d(t ) = 1, we derive To solve (19),le hen (19) d() = e (T ). (20) g(, s) = m(, y), y = s 2β, (21) g = m, g s = 2βs 2β 1 m y, g ss = 2β(2β + 1)s 2β 2 m y + 4β 2 s 4β 2 m yy. Inroducing hese derivaives ino (19),we obain q[m + d()( n µ i ) 2β ym y ] + q2 d 2 () (B A 2 ) 2 q[β(2β + 1)m y 2 +2β 2 ym yy ] y(r 1 ) qd()(r 1 )Ay 1 2 = 0. (22) 3.2 The case of ρ i0 = 0 In his secion, we suppose ha he financial mare and ris model are independen, i.e., W (0) () is independen wih W (i) (),i.e.,ρ i0 = 0,which implies ha A = 0. Under he assumpion ha A = 0,(22) becomes: q[m + d()( n µ i ) 2β ym y ] + q2 d 2 () B 2 q[β(2β + 1)m y 2 +2β 2 ym yy ] y(r 1 ) 2 = 0. (23) We ry o find a soluion o (23) wih he following form: m(, y) = P () + Q()y, (24) wih he boundary condiion Q(T ) = 0 and P (T ) = 0. Then m = P + Q y, m y = Q(), m yy = 0. Plugging m, m y, m yy ino (23),we obain: ( qq + 2βq Q (r 1 ) )y qp qd()( n µ i) + q2 d 2 () B 2 2 qqβ(2β + 1) = 0. Decomposing (25) ino wo equaions, we have qq + 2βq Q (r 1 ) 2 qp qd()( n µ i ) + q2 d 2 () B 2 2 qqβ(2β + 1) = 0. (25) 2 2 = 0, (26) (27) Taing he boundary condiion Q(T ) = 0 and P (T ) = 0 ino accoun, we find he soluion o (26) and (27) are P () = Q() = (r 1 ) 2 2q 2 1 e 2βr0( T ) 2β, (28) µ i e 2(T ) 1 2 [1 e (T ) ] + qb 2 + (2β + 1)(r 1 ) 2 e 2βr0( T ) 1 4q 2β (2β + 1)(r 1 ) 2 ( T ). 4q (29) The following heorem summarizes he above derivaion. E-ISSN: Volume 16, 2017

5 Theorem 1. Suppose ha he financial mare and ris model are independen, i.e.,ρ i0 = 0,he opimal sraegy for problem (9) under he CARA uiliy funcion is given by π1 = e( T ) 2s 2β 2 q [2(r 1 )+ (r 1 ) 2 (1 e 2β( T ) )]. A soluion o HJB equaion (23) is given by where (30) V (, s, x) = 1 q e[ q(d()x+g(,s))], (31) d() = e (T ), g(, s) = P () + Q()s 2β, P () = Q() = (r 1 ) 2 1 e 2βr0( T ) 2q 2, (32) 2β µ i [1 e (T ) ] + qb 2 + (2β + 1)(r 1 ) 2 e 2βr0( T ) 1 4q 2β (2β + 1)(r 1 ) 2 ( T ). 4q e 2(T ) 1 2 (33) Proof: From (13), (16), (21), (24) and (28), we have π 1 = (r 1 )V x + 2 s 2β+1 V sx 2 s 2β V xx = (r 1 )( qd()v ) + 2 s 2β+1 q 2 d()g s V 2 s 2β q 2 d 2 ()V = (r 1 ) + 2 s 2β+1 qg s 2 s 2β qd() = e( T ) 2s 2β 2 q [2(r 1 ) + (r 1 ) 2 (1 e 2β( T ) )]. According o (16), (20), (21) and (24),we derive V (, s, x) as given in (31). Theorem 2. Le V (, s, x) be a soluion o (23),hen he value funcion is H(, s, x) = V (, s, x).for he wealh process X() associaed wih an admissible s- raegy π(),we have E[U(X())] V (0, s, x). In paricular, for π1 () given by Theorem 1 and he corresponding wealh process X1 (), E[U(X 1())] = V (0, s, x). Theorem 2 verifies ha he value funcion coincides wih he soluion o HJB equaion (23) given in Theorem 1 and indicaes ha he sraegy given in Theorem 1 is opimal for problem (9).The prove of Theorem 2 is similar as Zhao e al.[29]. Remar 3. From Theorem 1,we find ha he claim processes have no effec on he opimal sraegy. Under he assumpion ha W (0) () is independen wih W (i) (),i.e.,ρ i0 = 0,he financial mare and ris model are independen, hen he claim process is independen wih he financial mare. In realiy, he impac of claim process of insurer on he volailiy of he financial mare is very small. 3.3 The case of β = 0 Suppose ha β = 0, hen he CEV model reduces o he GBM model, and HJB equaion (12) reduces o H + sup π {[(r 1 )π + x + n µ i]h x [2 π 2 + B + 2Aπ]H xx } = 0. (34) Again,we can differeniae wih respec o π in he following formula: [r 1 )π+ x+ we have µ i ]H x [2 π 2 +B+2Aπ]H xx. π 2 = (r 1 ) 2 Plugging (35) ino (34) H x H xx A. (35) H + ( x + n µ i A(r 1 ) )H x (B A2 )H xx (r 1 ) 2 Hx = 0. H xx (36) In order o solve (34),we ry o find he soluion o (36) in he following srucure: Ṽ (, x) = 1 q exp{ qxe(t ) 1 2 (r 1 ) 2 (T ) + h(t )}, (37) and he boundary condiion Then we have h(t ) = 0. Ṽ x = Ṽ qe(t ), Ṽ xx = q 2 Ṽ e 2(T ), Ṽ = Ṽ [xqe (T ) (r 1 ) 2 h, (T )]. E-ISSN: Volume 16, 2017

6 Inroducing Ṽx, Ṽxx ino π2,we ge π 2 = r 1 q 2 e (T ) A. puing Ṽx, Ṽxx, Ṽ ino (36),we can ge h (T ) = qe (T ) ( n µ i A(r 1 ) ) q2 (B A 2 )e 2(T ), wih he boundary condiion h(t ) = 0 and afer inegraing, we derive h(t ) = 1 2 q2 (B A 2 ) e2 (T ) 1 2 q e(t ) 1 ( n r µ i A(r 1 ) ). 0 (38) According o he above derivaion, we have he following heorem. Theorem 4. In he case ha he risy asse s price follows he GBM model, he opimal sraegy and he corresponding value funcion are as follows: π 2 = r 1 q 2 e (T ) A, (39) Ṽ (, x) = 1 q exp{ qxe(t ) 1 2 (r 1 ) 2 +h(t )}, (40) where h(t ) = 1 2 q2 (B A 2 ) e2 (T ) 1 2 q e(t ) 1 ( n r µ i A(r 1 ) ), 0 A = = γ i ρ i0, B = γi γi i,j=1,i j γ i = ((λ + λ i )E[X 2 i ]) 1 2. λa i a j, Proof: (35) and (37) implies ha π 2 = (r 1 ) 2 = (r 1 ) 2 Ṽ x Ṽ xx A i,j=1,i j Ṽ qer0(t ) Ṽ q 2 e 2(T ) A. = r 1 q 2 e (T ) A. ρ ij γ i γ j From (37) and (38), we derive Ṽ (, x) as given in (40). The following heorem verifies ha he value funcion coincides wih he soluion o HJB equaion (34) given in Theorem 4 and indicaes ha he sraegy given in Theorem 4 is opimal for problem (9). Theorem 5. Ṽ (, x) is he soluion of (34), hen he value funcion is H(, x) = Ṽ (, x). For he wealh process X() associaed wih an admissible sraegy π(),we have E[U(X())] Ṽ (0, x). In paricular, for π2 () given by Theorem 4 and he corresponding wealh process X2 (), E[U(X2())] = Ṽ (0, x). Remar 6. If here is only one business line in he ris model, i.e., i = 1,(39) reduces o he opimal sraegy derived by Yang and Zhang [4]. From (39), we find ha he opimal invesmen s- raegy under he GBM model depends no only on he ime, ineres rae, ris aversion coefficien, appreciaion rae and volailiy of he risy asse, bu also on couning processes and he correlaion coefficiens beween risy asse and claim processes. We can also find his propery in Secion 4 numerical examples. Remar 7. Suppose here are wo dependen lines of business, he impac of he claim processes on he opimal invesmen sraegy is given as follows. (1)If ρ 10 > (a2 2 + b 2) (λ + λ 1 )(a b 1) (a b 1) (λ + λ 2 )(a b 2) ρ 20,he opimal invesmen sraegy decreases as λ increases. (2)If ρ 10 < (a2 2 + b 2) (λ + λ 1 )(a b 1) (a b 1) (λ + λ 2 )(a b 2) ρ 20,he opimal invesmen sraegy increases as λ increases. Proof:For i = 2,he opimal invesmen sraegy is π2 = r 1 q 2 e (T (λ + ) λ1 )(a b 1)ρ 10 (λ + λ2 )(a b 2)ρ 20. Differeniaing π2 wih λ, π2 λ = ρ 10 (a b 1) 2 (λ + λ 1 )(a b 1) ρ 20 (a b 2) 2 (λ + λ 2 )(a b 2). E-ISSN: Volume 16, 2017

7 * * * WSEAS TRANSACTIONS on MATHEMATICS when ρ 10 > (a2 2 + b 2) (λ + λ 1 )(a b 1) (a b 1) (λ + λ 2 )(a b 2) ρ 20, π 2 λ > 0, while ρ 10 < (a2 2 + b 2) (λ + λ 1 )(a b 1) (a b 1) (λ + λ 2 )(a b 2) ρ 20, π2 λ < 0. Remar 7 shows he impac of he couning processes and he correlaion coefficiens beween risy asse and he claim processes on he opimal invesmen sraegy for i = 2.The idea and echnique shown here are sill useful for i > 2. 1 () =0.02 =0.03 =0.04 =0.05 The effec of ineres rae on he opimal invesmen sraegy * 1 () Numerical analysis In his secion, we provide some numerical simulaions o illusrae our resuls. Throughou he numerical analysis, according o Li e al. [20],he basic parameers are given by: = 0.03, r 1 =0.12, =16.16, T =10, β=-0.12, s=67, q=0.05, α 1 = 2, α 2 = 2, λ = 3, λ 1 = 1, λ 2 = 5, ρ 10 = 0.5, ρ 20 = Numerical analysis for he case of ρ i0 = 0 Figure 2: Sensiiviy of he opimal sraegy w.r. 1 () q=0.1 q=0.3 q=0.5 q=0.7 The effec of ris aversion coefficien on he opimal invesmen sraegy * 1 () The effec of appreciaion rae on he opimal invesmen sraegy () r 1 =0.12 r 1 =0.15 r 1 =0.20 r 1 =0.25 Figure 3: Sensiiviy of he opimal sraegy w.r. q () From Figure 3, we find ha he ris aversion coefficien q exers a negaive effec on he opimal sraegy. The insurer is ris averse and hey will inves less in risy asse as he ris aversion coefficien becomes higher. Figure 1: Sensiiviy of he opimal sraegy w.r. r The effec of insananeous volailiy on he opimal invesmen sraegy * 1 () s = s = s = s = Figure 1 shows ha he amoun invesed in risy asse increases wih he appreciaion rae of risy asse r 1.I is because ha as r 1 increases, he insurer will ge more profis from risy asse. Therefore, he insurer would lie o pu more money in he risy asse o gain more profis. In Figure 2, we plo he impac of ineres rae on he opimal sraegy. The opimal sraegy decreases wih.as increases, he ris-free asse is more aracive, he insurer will inves more money in he ris-free asse. Thus, he money invesed in he risy asse becomes less. 1 () Figure 4: s(0) β 0.01 Sensiiviy of he opimal sraegy w.r. E-ISSN: Volume 16, 2017

8 WSEAS TRANSACTIONS on MATHEMATICS In Figure.4, he opimal invesmen sraegy is a decreasing funcion of he insananeous volailiy rae.as s(0) β increases, he volailiy of risy asse becomes bigger. Thus, i is no appropriae o carry ou large-scale invesmen on risy asse. In order o reduce he impac of he volailiy, he insurer will reduce invesmen in he risy asse invesmen The effec of common shoc inensiy on he opimal invesmen sraegy 4.2 Numerical analysis for he case of β = 0 According o Liang and Yuen [13], we assume ha here are wo business lines and he claim sizes X 1j and X 2j are exponenially disribued wih parameers α 1 and α 2,respecively.Then a 1 = 1 α 1, a 2 = 1 2(λ+λ1 ) 2(λ+λ2 ) α 2. α 2, γ 1 = α 1, γ 2 = According o (4) and (5), he correlaion coefficien of he wo lines of business saisfies ρ 12 = λ 2(λ + λ1 ) 2(λ + λ 2 ). The effec of common shoc inensiy on he correlaion coefficien corresponding o he wo lines Figure 6: Sensiiviy of he opimal sraegy w.r. λ ha ρ 10 = 1,ρ 20 = 1,he underwriing ris of line i and invesmen ris has negaive correlaion. In fac,π2 decreases wih λ when ρ i0 > 0 for i = 1, 2 and increases wih λ when ρ i0 < 0. From Figure 6, we find ha in he case ρ 10 = 0.1,ρ 20 = 0.9,he posiive correlaion ρ 20 = 0.9 plays a significan roles on π2,hen π 2 decreases wih λ while when ρ 10 = 0.1,ρ 20 = 0.9,he negaive correlaion ρ 20 = 0.9 plays a significan roles on π2,hus,π 2 is an increasing funcion of λ The effec of appreciaion rae on he opimal invesmen sraegy * () 2 r 1 =0.12 r 1 =0.15 r 1 =0.20 r 1 =0.25 Figure 5: Sensiiviy of he correlaion coefficien corresponding o he wo business lines w.r. λ Figure 5 shows he effec of common shoc inensiy λ on he correlaion coefficien corresponding o he wo dependen business lines. The correlaion coefficien ρ 12 is an increasing funcion of he common shoc inensiy λ. The correlaion beween line 1 and line 2 increases as λ increases. Figure 6 shows he effec of common shoc inensiy λ on he opimal invesmen sraegy π 2. From Figure 6, we can see ha when ρ 10 = 1, ρ 20 = 1, he opimal invesmen sraegy decreases wih λ. Under he assumpion ha ρ 10 = 1,ρ 20 = 1, he correlaion beween he underwriing ris of line i for i = 1, 2 and invesmen ris is perfec posiive correlaed. As λ increases, he underwriing ris becomes larger. In order o reduce overall ris, he insurer will pu less money in he risy asse. On he conrary, in he case Figure 7: Sensiiviy of he opimal sraegy w.r. r 1 Figures 7-10 show he effecs of he appreciaion rae of risy asse r 1,he ineres rae,he volailiy of he risy asse and he ris aversion coefficien q on he opimal sraegy respecively for he case ha he risy asse s price follows he GBM model. As shown in hese figures, we find ha he effecs are similar o hose under he ρ i0 = 0 cases. E-ISSN: Volume 16, 2017

9 WSEAS TRANSACTIONS on MATHEMATICS The effec of ineres rae on he opimal invesmen sraegy * () 2 =0.02 =0.03 =0.04 = Figure 8: Sensiiviy of he opimal sraegy w.r = = = = The effec of volailiy on he opimal invesmen sraegy () 0 Opimal invesmen problem for an insurer has been around in many lieraures. In his paper,we s- udy a more general opimal invesmen problem for an insurer. We consider an insurer who has n dependen classes of insurance business, and adop he consan elasiciy of variance (CEV) model o describe he dynamic of he risy asse s price process. By applying dynamic programming approach, we esablish he corresponding Hamilon-Jacobi-Bellman (HJB) e- quaion. For he objecive of maximizing he expeced uiliy of erminal wealh, we obain explici soluions for he exponenial uiliy funcions under some given assumpions. Finally, a numerical simulaion is presened o analyze he properies of he opimal invesmen sraegy. Some ineresing resuls are found: (1) Under he case of ρ i0 = 0,he financial mare and ris model are independen, we find ha he claim processes have no effec on he opimal sraegy. In pracice, he impac of claim process of insurer on he volailiy of he financial mare is very small. (2) In he case of β = 0,he CEV model reduces o he GB- M model, if ρ i0 0,he opimal invesmen sraegy depends on couning processes and he correlaion coefficiens beween risy asse and claim processes. (3) From he numerical simulaion, we find ha for boh he case of ρ i0 = 0 and he case of β = 0,he appreciaion rae of risy asse has a posiive effec on he opimal sraegies, while ineres rae, volailiy of he risy asse and ris aversion coefficien exer negaive effecs on he opimal sraegies. Under he GBM model, he correlaion corresponding o wo lines of business increases as he common shoc inensiy λ increases. Figure 9: Sensiiviy of he opimal sraegy w.r The effec of ris aversion coefficien on he opimal invesmen sraegy * () 2 q=0.1 q=0.3 q=0.5 q= Figure 10: Sensiiviy of he opimal sraegy w.r. q 5 Conclusion References: [1] S. Browne, Opimal invesmen policies for a firm wih a random ris process: exponenial uiliy and minimizing he probabiliy of ruin, Mahemaics of Operaions Research, Vol.20, No.4, 1995, pp [2] C. Hipp, M. Plum, Opimal invesmen for insurers, Insurance: Mahemaics and Economics, Vol.27, No.2, 2000, pp [3] C. Liu, H. Yang, Opimal invesmen for an insurer o minimize is probabiliy of ruin, Norh American Acuarial Journal, Vol.8, No.3, 2004, pp [4] H. Yang, L. Zhang, Opimal invesmen for insurer wih jump-diffusion ris process, Insurance: Mahemaics and Economics, Vol.37, No.3, 2005, pp [5] Z. Wang, J. Xia, L. Zhang, Opimal invesmen for an insurer: The maringale approach, Insurance: Mahemaics and Economics, Vol.40, No.2, 2007, pp [6] L. Bai, J. Guo, Opimal proporional reinsurance and invesmen wih muliple risy asses and E-ISSN: Volume 16, 2017

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