Utility portfolio optimization with liability and multiple risky assets under the extended CIR model

Transcription

1 Uiliy porfolio opimizaion wih liabiliy and muliple risky asses under he exended CIR model HAO CHANG Tianjin Polyechnic Universiy Deparmen of Mahemaics Bin-shui Wes Road 399, 3387 Tianjin CHINA JI-MEI LU Tianjin Polyechnic Universiy School of Elecrical Engineering and Auomaion Bin-shui Wes Road 399, 3387 Tianjin CHINA Absrac: This paper sudies an asse and liabiliy managemen problem wih exended Cox-Ingersoll-Ross (CIR) ineres rae, where he financial marke is composed of one risk-free asse and muliple risky asses and one zero-coupon bond. We assume ha risk-free ineres rae is driven by exended CIR ineres rae model, while liabiliy is modeled by Brownian moion wih drif and is generally correlaed wih sock price. Firsly, we use sochasic opimal conrol heory o obain Hamilon-Jacobi-Bellman (HJB) equaion for he value funcion and choose power uiliy and exponenial uiliy for our analysis. Secondly, we obain he closed-form soluions o he opimal invesmen sraegies by applying variable change echnique. Finally, a numerical example is presened o analyze he dynamic behavior of he opimal invesmen sraegy and provide some economic implicaions for our resuls. Key Words: CIR ineres rae model; liabiliy process; dynamic porfolio selecion; sochasic opimal conrol; opimal invesmen sraegy; 1 Inroducion Dynamic porfolio selecion problems wih sochasic ineres raes have been paid much more aenions o in recen years. The Vasicek model (1977) and Cox- Ingersoll-Ross(CIR) model (1985) were he mos imporan models which described he erm srucure of ineres rae dynamics. Sanon (1997) presened a nonparameric echnique o esimae he drif and diffusion of he shor rae, and he marke price and he ineres rae risk. Korn and Kraf (21) used sochasic opimal conrol approach o sudy porfolio problems wih sochasic ineres raes and obained he closed-form soluions of he opimal porfolios, and furher presened verificaion heorem of he opimal soluions in he sochasic ineres rae environmens. Deelsra e al.(2, 23) invesigaed an invesmen and consumpion problem in a CIR framework and he opimal policy for an insurer in he presence of a minimum guaranee respecively. Laer, Grasselli (23) sudied an invesmen problem where he ineres rae followed he CIR dynamics and obained he opimal invesmen sraegy in explici form for HARA uiliy. Bielecki e al.(25) focused on risk sensiive porfolio managemen problem wih CIR ineres rae dynamics. Liu (27) assumed ha risk-free ineres rae and he appreciae rae and volailiy erm were funcions of sae variable, which followed Markovian diffusion process, and applied sochasic opimal conrol approach o sudy an invesmen and consumpion problem in he finie horizon. Gao (28) invesigaed a defined conribuion pension funds problem wih affine ineres rae dynamics, which included he Vasicek model and he CIR model, and obained he explici soluion of he opimal invesmen sraegy. Li and Wu (29) was concerned wih a dynamic porfolio selecion problem wih CIR ineres rae and Heson s sochasic volailiy and obained he explici expression of he opimal invesmen policy. Ferland and Waier (21) applied backward sochasic differenial equaion heory o sudy he mean-variance model wih exended CIR ineres rae dynamics and presened he explici expressions of he opimal invesmen sraegy and he efficien fronier. Nowadays, he asse and liabiliy managemen (ALM) problem is of boh heoreical ineres and pracical imporance, and has araced more and more aenions in he las decade in he acuarial science and financial modelling. In recen years, many scholars have sudied ALM problems in differen siuaions. Leippold e al.(24), Yi e al.(28), Chen and Yang (211) and Yao e al.(213) sudied mean-variance ALM problems in a muli-period environmen under differen marke assumpions. Chiu and Li (26), Xie e al.(28) and Zeng and Li E-ISSN: Volume 9, 214

2 (211) invesigaed mean-variance ALM problem in a coninuous-ime framework under differen liabiliy processes. Chen e al.(28), Xie (29) and Li and Shu (211) considered coninuous-ime meanvariance ALM problems in he regime swiching seing; Chiu and Wong (211, 213) focused on dynamic porfolio selecion problems wih coinegraed asses boh in a mean-variance framework and expeced uiliy framework respecively. Laer, Chiu and Wong (212, 213) invesigaed mean-variance ALM problems, where he prices of he asses are coinegraed. Leippold e al.(211) and Yao e al.(213) concerned mean-variance ALM problems wih endogenous liabiliy boh in a muli-period seing and coninuous-ime seing respecively. In he above lieraure menioned, risk-free ineres rae was all kep fixed. Such an assumpion is oo resricive for many financial insiuions. In fac, he ineres rae isn always fixed in he real-world environmens. Hence, considering ALM problems in he sochasic ineres rae environmens for many financial insiuions or invesor is more pracice. To he bes of our knowledge, he ALM problem wih CIR ineres rae dynamics has no been repored in he exising lieraures. In his paper, we assumed ha risk-free ineres rae is driven by he CIR model, and here are muliple risky asses and one zero-coupon bond in he financial marke. The liabiliy process is governed by Brownian moion wih drif and is generally correlaed wih sock price dynamics. We firsly use dynamic programming principle o ge he HJB equaion for he value funcion and invesigae he opimal invesmen sraegies in he power uiliy and exponenial uiliy cases. Due o he sochasic ineres rae model and liabiliy process, hose facors make he HJB equaion more sophisicaed. Forunaely, we inroduce a differenial operaor o ransform he equaion (16) ino (23), which is easily solved direcly. Secondly, we obain he closedform soluions o he opimal invesmen sraegies by applying variable change echnique. Finally, a numerical example is presened o analyze he dynamic behavior of he opimal invesmen sraegy and provide some economic implicaions for our resuls. There are hree main conribuion in his paper: (i) he ALM problem wih CIR ineres rae dynamics is sudied; (ii) he explici expressions of he opimal invesmen sraegies are obained in he power uiliy and exponenial uiliy cases; (iii) a verificaion heorem for he porfolio opimizaion problem wih CIR ineres rae dynamics is provided. The res of his paper is organized as follows. The problem formulaion is presened in Secion 2. In Secion 3, we use dynamic programming principle o obain he HJB equaion and obain he explici expression and geomeric srucure of he opimal invesmen sraegies in he power uiliy and exponenial uiliy cases by applying variable change echnique. Secion 4 presens a numerical example o illusrae he effec of marke parameers on he opimal invesmen sraegies and gives some economic implicaions. Secion 5 concludes he paper. 2 Problem formulaion In his secion we provide a dynamic porfolio selecion problem wih liabiliy process and exended CIR ineres rae. Throughou his paper, we assume ha ( ) represens he ranspose of a vecor or a marix, [, T ] represens he fixed and finie invesmen horizon, (Ω, F, {F } T, P ) represens a given filered complee probabiliy space, where {F } T is a filraion and each F can be inerpreed as he informaion available a ime, and any decision made a ime is based on hose informaion. 2.1 Financial marke Assume ha he financial marke is composed of one risk-free asse and muliple risky asses and one zerocoupon bond. One risk-free asse is inerpreed as cash or bank accoun, whose price a ime is denoed by S (). Then S () evolves according o ds () S () = r()d, S () = 1, (1) where r() is risk-free ineres rae. In his paper, we suppose ha he dynamics of shor rae r() is driven by CIR ineres rae model: dr() = (a br())d σ r r()dwr (), r() = r >, (2) where W r () is a one-dimension sandard Brownian moion defined on (Ω, F, {F } T, P ), a, b and σ r are all posiive real consans and saisfy he condiion: 2a > σ 2 r. I leads o ha r() > for all [, T ]. Muliple risky asses is he socks, whose price of he ih sock a ime is denoed by S i (), i = 1, 2,, n. Then he dynamics of S i () can be described by (referring o Deelsra and Grasselli e al.(23), Ferland and Waier (21)): ds i () n S i () = r()d + σ ij (dw j () + λ j d)+ j=1 E-ISSN: Volume 9, 214

3 σ ir r()(dwr () + λ r r()d), Si () = s i >, (3) where W S () = (W 1 (), W 2 (),, W n ()) is a n dimension independen and sandard Brownian moion defined on (Ω, F, {F } T, P ) and is independen of W r (). In addiion, Σ S = (σ ij ) n n represens he volailiy marix of he sock. Leing Σ r = (σ 1r, σ 2r,, σ nr ), Λ = (λ 1, λ 2,, λ n ), hen Σ r r() represens he volailiy of sock price generaed by he volailiy of ineres rae, λ r r() and Λ can be aken as risk compensaion coefficien vecor generaed by he volailiy risk of he sock and he risk of ineres rae respecively. I implies ha he sock prices are influenced by he shor rae and corresponding marke price of risk. The zero-coupon bond wih mauriy T, and he price of zero-coupon bond a ime is denoed by B(, T ), hen he dynamics of B(, T ) can be represened by db(, T ) B(, T ) = r()d + σ B()(dW r () + λ r r()d), B(T, T ) = 1, (4) where σ B () = σ r ()h() r(), and we have h() = 2(1 e m(t ) ) m (b λ r σ r ) + e m(t ) (m + b λ r σ r ), 2.2 Liabiliy process m = (b λ r σ r ) 2 + 2σ 2 r. (5) Assume ha an invesor is equipped wih an iniial endowmen w > and an iniial liabiliy l > a ime =, hen he ne iniial wealh of an invesor is given by x = w l. Suppose ha he accumulaive liabiliy of he invesor a ime is denoed by L(), hen L() can be described by he following Brownian moion wih drif: dl() = ud + vdw L (), L() = l >, (6) where u and v are he posiive consans, and W L () is a one-dimension sandard Brownian moion on (Ω, F, {F } T, P ). In his paper, we assume ha W L () is correlaed wih sock prices and he correlaion coefficien beween W L () and W i () is denoed by ρ i, i = 1, 2,, n. Leing ρ = (ρ 1, ρ 2,, ρ n ), hen W L () can be expressed as: W L () = ρ W S ()+ 1 ρ 2 WL (), where W L () is a one-dimension sandard Brownian moion on (Ω, F, {F } T, P ) and is independen of W S (). Therefore, liabiliy process L() can be rewrien as dl() = ud + vρ dw S () + v 1 ρ 2 d W L (), 2.3 Wealh process L() = l >. (7) Assume ha he amoun invesed in he ih sock a ime is denoed by π i (), i = 1, 2,, n, and he amoun invesed in he zero-coupon bond is denoed by π B (), hen he amoun invesed in he risk-free asse is given by π () = X() n π i () π B (), i=1 where X() represens he ne wealh of an invesor a ime. Suppose ha he financial marke is fricionless and is allowed o shor-selling he sock. Leing π S () = (π 1 (), π 2 (),, π n ()), hen he ne wealh process under rading sraegy π S () and π B () can be wrien as: dx() = (X() + n i=1 n i=1 π i () π B ()) ds () S () π i () ds i() S i () + π B() db(, T ) B(, T ) dl(). Taking (1), (3), (4) and (7) ino consideraion, we can ge dx() = [r()x() + π S ()(Σ S Λ + Σ r λ r r()) +π B ()σ B ()λ r r() u]d +(π S()Σ S vρ )dw S () v 1 ρ 2 d W L () +(π S()Σ r r() + πb ()σ B ())dw r (), (8) wih he iniial value X() = x. 2.4 Opimizaion crierion Definiion 1(Admissible sraegy). Trading sraegy π S () and π B () are admissible if he following condiions are saisfied: (i) π S () and π B () are progressively measurable; (ii) E{ [(π S ()Σ S vρ ) 2 + (v 1 ρ 2 ) 2 +(π S()Σ r r() + πb ()σ B ()) 2 ]d} < ; (iii) For all iniial condiions (, r, x ), he wealh process X() wih X() = x has a pahwise unique soluion. E-ISSN: Volume 9, 214

4 We denoe he se of all admissible sraegies by Γ, and he opimizaion problem of he invesor in he uiliy framework can be expressed as: Maximize EU(X(T )). (9) π S Γ,π B Γ where U( ) represens uiliy funcion and saisfies he condiion: he firs-order derivaive U(x) > and he second-order derivaive Ü(x) <. 3 The opimal porfolios In his secion, we apply dynamic programming principle and variable change echnique o solve he problem (9) and choose power uiliy and exponenial uiliy for our analysis. We define he value funcion J(, r, x) of he problem (9) as J(, r, x) = sup E[U(X(T )) X() = x, r() = r ] π S Γ,π B Γ wih boundary condiion: J(T, r, x) = U(x). Theorem 1.. Assume ha J(, r, x) is coninuously differeniable wih respec o [, T ], and wice coninuously differeniable wih respecive o (r, x) R R, hen J(, r, x) saisfies he following HJB equaion: sup {J + [rx + π S ()(Σ S Λ + Σ r λ r r) π S Γ,π B Γ +π B ()σ B ()λ r r u]jx [(π S()Σ S vρ ) 2 + (v 1 ρ 2 ) 2 +(π S()Σ r r + πb ()σ B ()) 2 ]J xx +(a br)j r σ2 rrj rr σ r r[π S ()Σ r r + πb ()σ B ()]J xr } =. (1) where J, J r, J rr, J x, J xx, J xr represen he firs-order and second-order parial derivaives of he value funcion wih respec o he variables, r, x. Proof. According o Theorem 3.1 of Lin and Li (211), we assume ha J(, r, x) = sup π S Γ,π B Γ E[J( θ, r( θ), X π S,π B ( θ))]. For any sopping ime θ R,T, where R,T represens he se of all sopping ime valued in [, T ]. Considering he ime θ = +h, for arbirary π S Γ, π B Γ, we have J(, r, x) E[J( + h, r( + h), X π S,π B ( + h))]. Applying Iô s formula beween and + h, we have J( + h, r( + h), X π S,π B ( + h)) = J(, r, x) + +h ( J + lπ S,π B J)(u, r(u), X π S,π B (u))du+ local maringale. where l π S,π B J is defined by +h l π S,π B J = [rx + π S ()(Σ S Λ + Σ r λ r r) +π B ()σ B ()λ r r u]jx [(π S()Σ S vρ ) 2 + (v 1 ρ 2 ) 2 +(π S()Σ r r + πb ()σ B ()) 2 ]J xx +(a br)j r σ2 rrj rr σ r r[π S ()Σ r r + πb ()σ B ()]J xr. Then we obain ( J + lπ S,π B J)(u, r(u), X π S,π B (u))du. Dividing by h and leing h, his leads o J + lπ S,π B J. On he oher hand, suppose ha π S Γ and π B Γ are he opimal invesmen sraegies, hen we have J(, r, x) = E[J( + h, r( + h), X π S,π B ( + h))]. By similar argumens as above, we obain J + lπ S,π B J =. Summarizing he above argumens, we obain ha J(, r, x) should saisfy J + sup π S Γ,π B Γ l π S,π B J =. This complees he proof of Theorem 1. The firs-order maximizing condiions for he opimal sraegy yield: π S () = (Σ S) 1 Λ J x J xx + (Σ S) 1 ρv, (11) π B () = λ r r Λ Σ 1 S Σ r r σ B () + σ r r σ B () Jrx Jx J xx ρ vσ 1 S Σ r r. (12) J xx σ B () E-ISSN: Volume 9, 214

5 Plugging (11) and (12) ino (1), we ge J + rxj x + (Λ ρv u)j x v2 (1 ρ 2 )J xx + (a br)j r σ2 rrj rr 1 2 ( Λ 2 + λ 2 rr) J 2 x J xx +λ r σ r r J xj rx J xx 1 2 σ2 rr J 2 rx J xx =. (13) The following subsecion, we ry our bes o solve he second-order nonlinear parial differenial equaion (13) by conjecuring he srucure of he value funcion. 3.1 Power uiliy Assume ha power uiliy is expressed as U(x) = x, < 1,. Conjecuring a soluion o (13) wih he following form: J(, r, x) = (x g(, r)) f(, r), g(t, r) =, f(t, r) = 1. (14) The parial derivaives of (14) are given by J = (x g) 1 g f + J r = (x g) 1 g r f + J x = (x g) 1 f, (x g) f, (x g) f r, J xx = ( 1)(x g) 2 f, J rx = ( 1)(x g) 2 g r f + (x g) 1 f r, J rr = ( 1)(x g) 2 g 2 rf + (x g) f rr 2(x g) 1 f r g r (x g) 1 g rr f. Puing he above parial derivaives in (13), we ge (x g) [ f + (r 2( 1) ( Λ 2 + λ 2 rr))f +(a br σ2 rrf rr 1 λ rσ r r)f r 2( 1) σ2 rr f r 2 f ] ( 1)(x g) 2 v 2 (1 ρ 2 )f +(x g) 1 [g + (a br + λ r σ r r)g r σ2 rrg rr rg (Λ ρv u) ] =. Eliminaing he dependence on he variable x, we ge he following wo parial differenial equaions under he condiion: 1 ρ 2 =. f + (r +(a br + 2( 1) ( Λ 2 + λ 2 rr))f 1 λ rσ r r)f r σ2 rrf rr 2( 1) σ2 rr f r 2 =, f(t, r) = 1; (15) f g rg + (a br + λ r σ r r)g r σ2 rrg rr + u Λ ρv =, g(t, r) =. (16) Lemma 1. Assume ha he soluion of (15) is of he srucure f(, r) = e D 1()+D 2 ()r wih boundary condiions: D 1 (T ) = and D 2 (T ) =, hen under b he condiion: < 2 (λ r σ r, D b) 2 +2σr 2 1 () and D 2 () are given by (22) and (21), respecively. Proof. Subsiuing f(, r) = e D 1()+D 2 ()r ino (15), we ge { e D 1()+D 2 ()r Ḋ 1 () 2( 1) Λ 2 + ad 2 () +r[ḋ2() + ( 2( 1) λ2 r) +( 1 λ 1 rσ r b)d 2 () 2( 1) σ2 rd2()] 2 Furher, we can ge he following wo equaions: } =. Ḋ 2 () + ( 2( 1) λ2 r) + ( 1 λ rσ r b)d 2 () 1 2( 1) σ2 rd2() 2 =, D 2 (T ) = ; (17) Ḋ 1 () 2( 1) Λ 2 +ad 2 () =, D 1 (T ) =. The equaion (17) can be rewrien as 1 m 1 m 2 (18) 1 1 ( )dd 2 () D 2 () m 1 D 2 () m 2 = σr 2 (T ). (19) 2( 1) E-ISSN: Volume 9, 214

6 where m 1 and m 2 are wo differen roos of he following quadraic equaion: 1 2( 1) σ2 rd2() 2 ( 1 λ rσ r b)d 2 () ( 2( 1) λ2 r) =. b i.e. under he condiion < 2 (λ r σ r, we ge b) 2 +2σr 2 m 1,2 = λ rσ r ( 1)b σ 2 r ( 1)(λr σ r b) ± 2 + ( 1)(2σr 2 b 2 ). σ 2 r Solving he equaion (19), we obain (2) D 2 () = m 1m 2 (1 exp{ σ2 r 2( 1) (m 1 m 2 )(T )}) m 1 m 2 exp{ σ2 r 2( 1) (m 1 m 2 )(T )}. (21) Inegraing he boh sides of (18), we ge D 1 () = a D 2 ()d 2( 1) Λ 2 (T ). (22) Therefore, he proof of Lemma 1 is compleed. Lemma 2. Suppose ha he soluion o (15) is of he srucure g(, r) = (u Λ ρv) ĝ(u, r)du, hen ĝ(, r) saisfies he following equaion: ĝ rĝ + (a br + λ r σ r r)ĝ r σ2 rrĝ rr =, ĝ(t, r) = 1. (23) Proof. We inroduce a differenial operaor as follows. g = rg + (a br + λ r σ r r)g r σ2 rrg rr. Then (16) is rewrien as g + g + u Λ ρv =, g(t, r) =. (24) Considering g(, r) = (u Λ ρv) ĝ(u, r)du, we yield g = (u Λ ρv)ĝ(, r) = (u Λ ρv)( g r = (u Λ ρv) g rr = (u Λ ρv) ĝ(u, r) du ĝ(t, r)), u ĝ(u, r) du, r 2 ĝ(u, r) r 2 du. Furher, we have g = (u Λ ρv) Insering g and g ino (u Λ ρv) In fac, we have ĝ(u, r) u ĝ(u, r)du. ĝ(u, r) [ + ĝ(u, r)]du u (u Λ ρv)(ĝ(t, r) 1) =. + ĝ(u, r) =, ĝ(t, r) = 1. The proof of Lemma 2 is compleed. Lemma 3. Leing ĝ(, r) = e D 3()+D 4 ()r is he soluion o he equaion (23), where boundary condiions are given by D 3 (T ) = and D 4 (T ) =, hen D 3 () and D 4 () are deermined by (28) and (27), respecively. Proof. Puing ĝ(, r) = e D 3()+D 4 ()r in he equaion (23) yields: e D 3()+D 4 ()r {Ḋ3() + ad 4 () +[Ḋ4() 1 + (λ r σ r b)d 4 () σ2 rd 2 4()]r} =. We can decompose he above equaion ino he following wo equaions in order o eliminae he dependence on r: Ḋ 4 () 1 + (λ r σ r b)d 4 () σ2 rd 2 4() =, D 4 () =. (25) Ḋ 3 () + ad 4 () =, D 3 () =. (26) Using he same analysis as he equaions (17) and (18), we derive D 4 () = m 3m 4 (1 exp{ 1 2 σ2 r(m 3 m 4 )(T )}) m 3 m 4 exp{ 1 2 σ2 r(m 3 m 4 )(T )}. (27) T D 3 () = a D 4 ()d. (28) where m 3 and m 4 are given by m 3,4 = b λ rσ r (b λr σ r ) σr 2 ± 2 + 2σr 2 σr 2. Therefore, we complee he proof of Lemma 3. Furher, we have J x = 1 (x g), J xx 1 E-ISSN: Volume 9, 214

7 J rx = g r + 1 J xx 1 (x g)f r f = g r (x g)d 2(). Finally, we obain he opimal invesmen sraegies in he power uiliy case. Proposiion 1. Assume ha uiliy funcion is given by U(x) = x, wih < 1 and, hen under he condiions: < Min{ b 2 (λ r σ r b) 2 +2σ 2 r, 1}, and ρ 2 = 1, he opimal invesmen sraegies of he problem (9) are given by π S() = 1 1 (Σ S) 1 Λ(X() g) + (Σ S) 1 ρv, π B() = 1 1 λr (29) r() Λ Σ 1 S Σ r r() (X() g)+ σ B () σ r r() σ B () ( 1 1 (X() g)d 2() g r ) ρ vσ 1 S Σ r r(). σ B () (3) where g = g(, r) = (u Λ ρv) ĝ(u, r)du, ĝ(, r) = e D 3()+D 4 ()r, D 2 (), D 3 () and D 4 () are given by (21), (28) and (27), respecively. Remark1. If he liabiliy is no considered, i.e. u = v =, hen g = g(, r) =. Therefore, he opimal policy of he problem (9) for power uiliy is reduced o π B() = 1 1 λr π S() = 1 1 (Σ S) 1 ΛX(), r() Λ Σ 1 S Σ r r() X()+ σ B () 1 1 σr r() σ B () X()D 2(). where D 2 () is given by(21). Remark2. If, hen we have D 2 () =. I is all well-known ha power uiliy is degeneraed o logarihm uiliy when. Hence, he opimal policy of he problem (9) is given by πs() = (Σ S) 1 Λ(X() g) + (Σ S) 1 ρv. πb() = λ r r() Λ Σ 1 S Σ r r() (X() g)+ σ B () σ r r() σ B () ( g r ) ρ vσ 1 S Σ r r(). σ B () where g = g(, r) = (u Λ ρv) ĝ(u, r)du, ĝ(, r) = e D 3()+D 4 ()r, D 3 () and D 4 () are given by (28) and (27), respecively. 3.2 Exponenial uiliy Assume ha exponenial uiliy is expressed as U(x) = 1 β e βx, β >. Furher, we suppose ha he soluion o (13) is of he following srucure J(, r, x) = 1 exp{ βk(, r)(x L(, r))+m(, r)}. β (31) wih boundary condiions: K(T, r) = 1, L(T, r) =, M(T, r) =. Then we have J = J( βxk + βk L + βkl + M ), J x = J( βk), J xx = J( βk) 2, J r = J( βxk r + βk r L + βkl r + M r ), J rx = J( βk)( βxk r +βk r L+βKL r +M r + K r K ), J rr = J( βxk r + βk r L + βkl r + M r ) 2 +J( βxk rr + βk rr L + 2βK r L r + βkl rr + M rr ). (32) Puing (32) ino (13), afer some calculaions, we obain +βk J { βx[k + rk + (a br + λ r σ r r)k r σ2 rrk rr σrr 2 K2 r K ] [ L K (K + (a br + λ r σ r r)k r σ2 rrk rr σrr 2 K2 r K ) + L + (a br + λ r σ r r)l r σ2 rrl rr + u Λ ρv + 1 ] 2 v2 (1 ρ 2 )βk +M + (a br + λ r σ r r σrr 2 K r K )M r σ2 rrm rr 1 2 ( Λ 2 + λ 2 rr + σrr( 2 K r K )2 2λ r σ r r K } r K ) =. (33) Eliminaing he dependence on he variable x, we can decompose (33) ino he following hree parial differenial equaions: K + rk + (a br + λ r σ r r)k r σ2 rrk rr σ 2 rr K2 r K =, K(T, r) = 1; (34) E-ISSN: Volume 9, 214

8 [ L βk K (K + (a br + λ r σ r r)k r σ2 rrk rr σ 2 rr K2 r K ) + L + u Λ ρv σ2 rrl rr + (a br + λ r σ r r)l r + 1 ] 2 v2 (1 ρ 2 )βk =, L(T, r) = ; (35) M + (a br + λ r σ r r σ 2 rr K r K )M r σ2 rrm rr 1 2 ( Λ 2 + λ 2 rr + σrr( 2 K r K )2 2λ r σ r r K r K ) =. (36) Lemma 4. Assume ha he soluion o (34) is of he form: K(, r) = e D 5()+D 6 ()r, wih boundary condiions given by D 5 (T ) = and D 6 (T ) =, hen D 6 () and D 5 () are deermined by (39) and (4), respecively. Proof. Plugging K(, r) = e D 5()+D 6 ()r ino (34) yields e D 5()+D 6 ()r {Ḋ5() + ad 6 () +[Ḋ6() (λ r σ r b)d 6 () 1 2 σ2 rd 2 6()]r} =. Then we have Ḋ 6 ()+1+(λ r σ r b)d 6 () 1 2 σ2 rd 2 6() =, (37) Ḋ 5 () + ad 6 () =. (38) Solving (37) and (38), we ge D 6 () = m 5m 6 (1 exp{ 1 2 σ2 r(m 5 m 6 )(T )}) m 5 m 6 exp{ 1 2 σ2 r(m 5 m 6 )(T )}, (39) T D 5 () = a D 6 ()d. (4) where m 5 and m 6 are given by m 5,6 = λ rσ r b (b λr σ r ) σr 2 ± 2 + 2σr 2 σr 2. The proof is compleed. Lemma 5. Suppose ha he soluion o (35) is L(, r) = (u Λ ρv) ˆL(u, r)du and ˆL(, r) = e D 7()+D 8 ()r, wih boundary condiions: D 7 (T ) = and D 8 (T ) =, hen under he condiion ρ 2 = 1, we obain D 8 () = D 4 () and D 7 () = D 3 (). Proof. Taking (34) and he condiion ρ 2 = 1 ino consideraions, we ge L rl + (a br + λ r σ r r)l r σ2 rrl rr + u Λ ρv =, L(T, r) =. (41) Applying he same approach as (16) and referring o he proof of Lemma 2 and Lemma 3, we can easily ge he resuls of Lemma 5. Lemma 6. Assume ha he soluion o (36) is expressed as M(, r) = D 9 ()+D 1 ()r, wih boundary condiions given by D 9 (T ) = and D 1 (T ) =, hen D 1 () and D 9 () are deermined by (45) and (46), respecively. Proof. Inroducing M(, r) = D 9 () + D 1 ()r in he equaion (36), we derive Ḋ 9 () + ad 1 () 1 2 Λ 2 +r[ḋ1() + (λ r σ r b σ 2 rd 6 ())D 1 () 1 2 (λ r σ r D 6 ()) 2 ] =. (42) Comparing he coefficiens on he boh sides of (42), we ge Ḋ 1 () + (λ r σ r b σ 2 rd 6 ())D 1 () 1 2 (λ r σ r D 6 ()) 2 =, D 1 (T ) = ; (43) Ḋ 9 ()+ad 1 () 1 2 Λ 2 =, D 9 (T ) =. (44) Solving (43) and (44) yields: D 1 () = 1 2 e (λ rσ r b σ 2 rd 6 ())d (λ r σ r D 6 ()) 2 e (λ rσ r b σrd 2 6 ())d d, (45) D 9 () = a D 1 ()d 1 2 Λ 2 (T ). (46) The proof of Lemma 6 is compleed. Furher, applying (32) and he resuls of Lemma 4, Lemma 5 and Lemma 6, we have J x = 1 J xx βk, J rx J xx = K r K x K r K L L r M r βk K r βk 2 = D 6 ()(x L) L r 1 βk (D 1() + D 6 ()). Finally, we can summarize he opimal invesmen sraegies of he problem (9) for exponenial uiliy maximizaion in he following Proposiion 2. E-ISSN: Volume 9, 214

9 Proposiion 2. If uiliy funcion is given by U(x) = 1 β e βx, β >, hen under he condiion ρ 2 = 1, he opimal policies of he problem (9) are πs() = 1 βk (Σ S) 1 Λ + (Σ S) 1 ρv, (47) πb() = 1 βk λr r() Λ Σ 1 S Σ r r() σ B () + σ r r() [D 6 ()(X() L) L r σ B () 1 βk (D 1() + D 6 ())] ρ vσ 1 S Σ r r(). (48) σ B () where K = K(, r) = e D 5()+D 6 ()r, L = L(, r) = (u Λ ρv) ˆL(u, r)du, ˆL(, r) = e D 7 ()+D 8 ()r, D 5 (), D 6 (), D 7 (), D 8 () and D 1 () are given by Lemma 4-Lemma 6. Remark3. If here is no liabiliy, i.e. u = v =, and i leads o L = L(, r) =. Therefore, he opimal policies wih CIR ineres rae dynamics for exponenial uiliy ae given by π B() = 1 βk λr σ r r() σ B () πs() = 1 βk (Σ S) 1 Λ, r() Λ Σ 1 S Σ r r() + σ B () [D 6 ()X() 1 βk (D 1() + D 6 ())]. where K = K(, r) = e D 5()+D 6 ()r, in addiion, D 5 (), D 6 () and D 1 () are given by (4), (39), (45), respecively. 4 Numerical analysis This secion provides a numerical example o illusrae he impac of marke parameers on he opimal invesmen sraegy. Assume ha he financial marke is composed of one risk-free asse and wo risky asses and one zero-coupon bond. Throughou his secion, unless oherwise saed, he basic parameers are given by a =.35, b =.4, σ r =.1, r() =.5, Λ = (.4,.6), Σ r = (.16,.32), λ r =.16, u = 2, v = 3, ρ ( = (.6,.8) ), =, T = 2, X() = 1, Σ S =. Noice ha in he following figures, he amoun invesed in he bank accoun is denoed by he hick line, i.e. π (), and he sum of he amoun invesed in he socks is given by he dashed line, i.e. 2 i=1 π i (), and he amoun invesed in he zero-coupon bond is represened by he orange line, i.e. πb (). 4.1 Sensiiviy analysis in he power uiliy case Under power uiliy, assume ha he risky aversion facor is given by =.2. I can be seen from every gragh of Fig.1-Fig.6 ha how marke parameers affec he opimal invesmen sraegy. Some resuls are summarized as follows. (a1) 2 i=1 π i () is no sensiive o he parameer b, and πb () increases wih respec o (w.r.) he value of b, and π () decreases w.r. he parameer b. I is shown from he equaion (1) ha he expeced value of ineres rae will decrease wih he increasing value of b. I means ha he he bigger he value of b becomes, he smaller amoun an invesor invess he risk-free asse and he bigger amoun in he zero-coupon bond. (a2) 2 i=1 π i () is no sensiive o he parameer σ r as well, and πb () decreases in he value of σ r, and π () is increasing wih he parameer σ r. Noice ha when he value of σ r is increasing, he volailiy of ineres rae is increasing as well. So, i ells us ha an invesor should inves he less amoun in he zero-coupon bond and inves he more amoun in he risk-free asse. (a3) 2 i=1 π i () and π B () are all decreasing wih he increasing value of he parameer u, and π () increases w.r. he parameer u. Noing ha he bigger he value of u is, he bigger he expeced value of liabiliy is. I displays ha an invesor should inves he less amoun of wealh in he risky asses and inves he more amoun in he risk-free asse. (a4) 2 i=1 π i () and π B () increases w.r. he parameer v, and π () decreases w.r. he paremeer v. In fac, he parameer v represens he volailiy of liabiliy. Hence, he larger he value of v is, he more risk he liabiliy resuls ino. I illusaes ha an invesor should inves he more amoun of wealh in he risky asses and invess he less amoun in he risk-free asse in order o hedge he risk of liabiliy. (a5) 2 i=1 π i () keeps fixed almos wih he invesmen horizon T, while πb () decreases w.r. he parameer T and π () increases w.r. he value of T. I shows ha πb () is decreasing and π () is increasing when he value of T is increasing. In fac, he bigger he value of T is, he bigger he value of T is. Hence, i indicaes ha as ime elapses, an invesor should inves he more amoun of wealh in he zero-coupon bond and keeps he less amoun in he risk-free asse. (a6) 2 i=1 π i () and π B () increases w.r. risk aversion facor and π () decreases w.r. he value of. Under power uiliy, he risk aversion coefficien of an invesor is denoed by 1. Therefore, when he value of is increasing, he risk aversion agree of invesor is decreasing. I leads o ha an invesor would E-ISSN: Volume 9, 214

10 inves he more amoun of wealh in he risky asses and inves he less amoun in he risk-free asse. 4.2 Sensiiviy analysis in he exponenial uiliy case Under exponenial uiliy, suppose ha β =.1 and λ r =.1, he oher marke parameers keep fixed. Fig.7-Fig.12 illusrae ha how marke parameers impac on he opimal invesmen sraegy. In addiion, he following conclusions are drawn. (b1) 2 i=1 π i () is no sensiive o he parameer b, while πb () increases wih respec o b and π () decreases wih respec o b. This is o say, he bigger he value of b is, he more amoun of wealh an invesor invess in he zero-coupon bond and he less amoun in he risk-free asse. (b2) 2 i=1 π i () keeps fixed almos w.r. he parameer σ r, in he meanime, πb () decreases w.r. he parameer b and π () increases w.r. he value of b. I implies ha as he value of b becomes larger, an invesor should inves he less amoun of wealh in he zero-coupon bond and inves he more amoun in he risk-free asse. (b3) 2 i=1 π i () is fixed in he value of λ r, while πb () is decreasing wih he value of λ r and π () is increasing. I indicaes ha when he value of λ r become larger, an invesor should inves he less amoun of wealh in he zero-coupon bond and inves he more amoun in he risk-free asse. (b4) 2 i=1 π i () and π B () increases wih respec o he parameer v, while π () decreases w.r. he value of v. I ells us ha when he volailiy of liabiliy is increasing, he invesor should inves he more amoun of wealh in he risky asses in order o hedge he risk resuled from he liabiliy. (b5) 2 i=1 π i () and π B () is decreasing in he value of T, while π () increases w.r. he parameer T. I implies ha as ime elapses, an invesor would ake more risk and inves more money in he risky asses. (b6) 2 i=1 π i () and π B () decreases wih he value of β, while π () increases w.r. he parameer β. I is well-known ha he risk aversion coefficien in he exponenial uiliy case is given by β. Therefore, he bigger he value of β is, he more risk aversion an invesor is faced wih. This is he reason why an invesor would inves less money in he risky asses and inves more money in he risk-free asse. 5 Conclusions In his paper, we have sudied an asse and liabiliy managemen problem wih CIR ineres rae dynam- opimal invesmen sraegy sock bank accoun Figure 1: The impac of b on he opimal policies wih power preference opimal invesmen sraegy b Σr sock bank accoun Figure 2: The impac of σ r on he opimal policies wih power preference opimal invesmen sraegy u sock bank accoun Figure 3: The impac of u on he opimal policies wih power preference E-ISSN: Volume 9, 214

11 sock opimal invesmen sraegy sock bank accoun opimal invesmen sraegy bank accoun v b Figure 4: The impac of v on he opimal policies wih power preference Figure 7: The impac of b on he opimal policies wih exponenial preference 6 15 opimal invesmen sraegy sock bank accoun opimal invesmen sraegy 1 5 sock bank accoun T Σr Figure 5: The impac of T on he opimal policies wih power preference Figure 8: The impac of σ r on he opimal policies wih exponenial preference 1 opimal invesmen sraegy sock bank accoun opimal invesmen sraegy sock bank accoun Η Λr Figure 6: The impac of on he opimal policies wih power preference Figure 9: The impac of λ r on he opimal policies wih exponenial preference E-ISSN: Volume 9, 214

12 opimal invesmen sraegy sock bank accoun Figure 1: The impac of v on he opimal policies wih exponenial preference opimal invesmen sraegy v T sock bank accoun Figure 11: The impac of T on he opimal policies wih exponenial preference opimal invesmen sraegy sock bank accoun Figure 12: The impac of β on he opimal policies wih exponenial preference Β ics. The financial marke is composed of one risk-free asse and muliple risky asses and one zero-coupon bond. The liabiliy process is assumed o be driven by Brownian moion wih drif and be generally correlaed wih sock price dynamics, while he price processes of socks and zero-coupon are affeced by ineres rae dynamics. The closed-form soluions o he opimal invesmen sraegies for power uiliy and exponenial uiliy are obained. We also presen a numerical example o analyze he influence of marke parameers on he opimal invesmen sraegies and provide some economic implicaions. Some imporan resuls are found: (i) he opimal invesmen sraegy wih liabiliy and sochasic ineres rae is more sophisicaed han ha wih sochasic ineres rae only; (ii) he opimal policies for power and exponenial uiliy have opposie rend wih respec o risk aversion facor; (iii) he opimal policies for power and exponenial uiliy almos have he same rend in he value of he parameer b, σ r, v, T. In fuure research on he asse and liabiliy managemen problem, i would be ineresing o exend our model o he following wo aspecs. On he one hand, we can consider he ALM problem wih sochasic ineres rae dynamics in he mean-variance framework and aim o obain he closed-form soluion o he opimal policy and efficien fronier. On he oher hand, we can also consider he ALM problem wih oher sochasic dynamics in he HARA framework and expec o achieve he explici expression of he opimal policy, for example, Heson s sochasic volailiy and CEV model and so on. In hose siuaions, i may be difficul o guess he form of he value funcion, and i needs us o explore new mehodology. Acknowledgemens: The research was suppored by he Humaniies and Social Science Research Youh Foundaion of Minisry of Educaion (No:11YJC796), he Higher School Science and Technology Developmen Foundaion of Tianjin (No: 21821). References: [1] O.A. Vasicek, An equilibrium characerizaion of he erm srucure, Journal of Financial Economics. 5, 1977, pp [2] J.C. Cox, J.E. Ingersoll, S.A.Ross, A heory of he erm srucure of ineres raes, Economerica. 53, 1985, pp [3] R. Sanon, A nonparameric model of erm srucure dynamics and he marke price of ineres rae risk, The Journal of Finance. 52, 1997, pp E-ISSN: Volume 9, 214

13 [4] R. Korn, H. Kraf, A sochasic conrol approach o porfolio problems wih sochasic ineres raes, SIAM Journal of Conrol and opimizaion. 4, 21, pp [5] G. Deelsra, M. Grasselli, P.F. Koehl, Opimal invesmen sraegies in a CIR framework, Journal of Applied Probabiliy. 37, 2, pp [6] G. Deelsra, M. Grasselli, P.F. Koehl, Opimal invesmen sraegies in he presence of a minimum guaranee, Insurance: Mahemaics and Economics. 33, 23, pp [7] M. Grasselli, A sabiliy resul for he HARA class wih sochasic ineres raes, Insurance: Mahemaics and Economics. 33, 23, pp [8] T.R. Bielecki, S. Pliska, S.J. Sheu, Risk Sensiive Porfolio Managemen wih Cox-Ingersoll- Ross Ineres Raes:The HJB Equaion, SIAM Journal on Conrol and Opimizaion. 44, 25, pp [9] J. Liu, Porfolio selecion in sochasic environmens, The Review of Financial Sudies. 2, 27, pp [1] J.W. Gao, Sochasic opimal conrol of DC pension funds, Insurance: Mahemaics and Economics. 42, 28, pp [11] J.Z. Li, R. Wu, Opimal invesmen problem wih sochasic ineres rae and sochasic volailiy: maximizing a power uiliy, Applied Sochasic Models in Business and Indusry. 25, 29, pp [12] R. Ferland, F. Waier, Mean variance efficiency wih exended CIR ineres raes, Applied Sochasic Models in Business and Indusry. 26, 21, pp [13] M. Leippold, F. Trojani, P.A. Vanini, geomeric approach o muli-period mean-variance opimizaion of asses and liabiliies, Journal of Economics Dynamics and Conrol. 28, 24, pp [14] L. Yi, Z.F. Li, D. Li, Muli-period porfolio selecion for asse liabiliy managemen wih uncerain invesmen horizon, Journal of Indusrial and Managemen Opimizaion. 4, 28, pp [15] P. Chen, H.L. Yang, Markowiz s mean variance asse liabiliy managemen wih regime swiching: a muli-period model, Applied Mahemaical Finance. 18, 211, pp [16] H.X. Yao, Y. Zeng, S.M. Chen, Muli-period mean variance asse liabiliy managemen wih unconrolled cash flow and uncerain imehorizon,economic Modelling. 3, 213, pp [17] M.C. Chiu, D. Li, Asse and liabiliy managemen under a coninuous-ime mean-variance opimizaion framework, Insurance: Mahemaics and Economics. 39, 26, pp [18] S.X. Xie, Z.F. Li, S.Y. Wang, Coninuous-ime porfolio selecion wih liabiliy: mean-variance model and sochasic LQ approach, Insurance: Mahemaics and Economics. 42, 28, pp [19] Y. Zeng, Z.F. Li, Asse liabiliy managemen under benchmark and mean variance crieria in a jump diffusion marke, Journal of Sysems Science and Complexiy. 24, 211, pp [2] P, Chen, H.L. Yang, G. Yin, Markowiz s mean variance asse liabiliy managemen wih regime swiching: a coninuous-ime model, Insurance: Mahemaics and Economics. 43, 28, pp [21] S.X. Xie, Coninuous-ime mean variance porfolio selecion wih liabiliy and regime swiching, Insurance: Mahemaics and Economics. 45, 29, pp [22] Z.C. Li, H.S. Shu, Opimal porfolio selecion wih liabiliy managemen and Markov swiching under consrained variance, Compuers and Mahemaics wih Applicaions. 61, 211, pp [23] M.C. Chiu, H.Y. Wong, Mean variance porfolio selecion of coinegraed asses, Journal of Economic Dynamics and Conrol. 35, 211, pp [24] M.C. Chiu, H.Y. Wong, Opimal invesmen for insurer wih coinegraed asses: CRRA uiliy, Insurance: Mahemaics and Economics. 52, 213, pp [25] M.C. Chiu, H.Y. Wong, Mean-variance asseliabiliy managemen: coinegraed asses and insurance liabiliies, European Journal of Operaional Research. 223, 212, pp [26] M.C. Chiu, H.Y. Wong, Mean-variance principle of managing coinegraed risky asses and random liabiliies, Operaions Research Leers. 41, 213, pp [27] M. Leippold, F. Trojani, P. Vanini, Muli-period mean variance efficien porfolios wih endogenous liabiliies, Quaniaive Finance. 11, 211, pp [28] H.X. Yao, Y.Z. Lai, Y. Li, Coninuous-ime mean variance asse liabiliy managemen wih endogenous liabiliies, Insurance: Mahemaics and Economics. 52, 213, pp E-ISSN: Volume 9, 214

14 [29] X. Lin, Y.F. Li, Opimal reinsurance and invesmen for a jump diffusion risk process under he CEV Model, Norh American Acuarial Journal. 15, 211, pp E-ISSN: Volume 9, 214