Acceped Manuscrip Opimal mean-variance efficiency of a family wih life insurance under inflaion risk Zongxia Liang, Xiaoyang Zhao PII: S0167-6687(15)308-1 DOI: hp://dx.doi.org/10.1016/j.insmaheco.016.09.004 Reference: INSUMA 70 To appear in: Insurance: Mahemaics and Economics Received dae: November 015 Revised dae: Sepember 016 Acceped dae: 9 Sepember 016 Please cie his aricle as: Liang, Z., Zhao, X., Opimal mean-variance efficiency of a family wih life insurance under inflaion risk. Insurance: Mahemaics and Economics (016), hp://dx.doi.org/10.1016/j.insmaheco.016.09.004 This is a PDF file of an unedied manuscrip ha has been acceped for publicaion. As a service o our cusomers we are providing his early version of he manuscrip. The manuscrip will undergo copyediing, ypeseing, and review of he resuling proof before i is published in is final form. Please noe ha during he producion process errors may be discovered which could affec he conen, and all legal disclaimers ha apply o he journal perain.
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY WITH LIFE INSURANCE UNDER INFLATION RISK Zongxia Liang and Xiaoyang Zhao Deparmen of Mahemaical Sciences, Tsinghua Universiy, Beijing 100084, China Absrac. We sudy an opimizaion problem of a family under mean-variance efficiency. The marke consiss of cash, a zerocoupon bond, an inflaion-indexed zero-coupon bond, a sock, life insurance and income-replacemen insurance. The insananeous ineres rae is modeled as he Cox-Ingersoll-Ross (CIR) model, and we use a generalized Black-Scholes model o characerize he sock and labor income. We also ake ino accoun he inflaion risk and consider our problem in he real marke. The goal of he family is o maximize he mean of he surplus wealh a he reiremen or deah of he breadwinner and minimize is variance by finding a porfolio selecion. The efficien fronier and opimal sraegies are derived hrough he dynamic programming mehod and he echnique of solving associaed nonlinear HJB equaions. We also presen a numerical illusraion o explore he impac of economical parameers on he efficien fronier. Submission Classificaions: IE13, IM1, IE43, IE53, IB1. MSC(010): 91G10, 91B16,91E0. JEL classificaion: G11, C61. Keywords: Mean-variance efficiency; Surplus wealh of he family; Income-replacemen Insurance; Lagrange dual mehod; Replicaion of asses; Dynamic programming; Nonlinear HJB equaions. 1. Inroducion While making long-erm financial invesmen decisions, a family faces no only he financial risk bu also he inflaion risk and he moraliy risk for he breadwinner. The inflaion risk can be hedged by holding inflaion-linked securiies such as inflaion-indexed zero-coupon bonds and he moraliy risk for he breadwinner can be hedged by Email: zliang@mah.singhua.edu.cn(z.liang), zhao-xy13@mails.singhua.edu.cn(x.zhao) 1
ZONGXIA LIANG AND XIAOYANG ZHAO enering he insurance marke. Life insurance, for example, is a ypical insrumen o preven he family from financial crises caused by he sudden deah of he breadwinner. Moreover, since more and more people prefer o ge married and have babies a an older age, i is more common ha he younger generaion is no economically independen when he breadwinner reires. As a resul, i is worh considering ha how o allocae family asses o ensure enough money o suppor he family afer he reiremen or deah of he breadwinner unil he economic independence of he nex generaion. There has been los of researches on he demand for he life insurance since he 1960s. Recenly, Pliska and Ye](007) sudy he opimal conrol problem abou life insurance, consumpion and invesmen. They assume ha wage earner s deah is independen of he financial marke, and conver he problem from an uncerain lifeime siuaion o a fixed lifeime siuaion. Nielsen and Seffensen1](008) invesigae he opimal invesmen and life insurances policies under he minimum and maximum consrains. They find ha he wealh consiss of hree pars and wo of hem are similar o European pu opions and call opions. Kwak e al.18](011) look ino an opimal invesmen, life insurance and consumpion decision of a family. They consider he consumpion of children afer he deah of parens and successfully ge an explici soluion hrough he maringale mehod. Kronborg and Seffensen17](013) invesigae he problem of opimal consumpion, invesmen and life insurance wih surrender opion guaranee, where invesors wih a deerminisic labor income can inves in he financial marke and buy life insurance or annuiies, and are assumed o fulfill an American capial guaranee. On he oher hand, since he financial planning of a family ofen lass for a long ime, he inflaion risk should be reaed as an imporan facor ha can affec he overall performance of he plan. Fischer1](1975) sudies he demand for inflaion-indexed bonds by solving he iner-emporal opimizaion problem of households under he inflaion risk. He highlighs he role of indexed bonds and provides a raionale for inroducing an indexed bond marke. Campbell and Viceira7](001), Brennan and Xia5](00) probe he imporance of indexed bonds for he long-erm and conservaive invesmen. Munk e al.0](004) provide he opimal asse allocaion sraegy of a power uiliy invesor who can inves in cash, nominal bonds, and socks in a model ha exhibis mean-revering sock reurns. Gong
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 3 and Li14](006) consider he consumpion-invesmen problem wih indexed bonds and subsisence consumpion consrain. Siu3](011) examines he regime swiching environmen as well as he inflaion risk. Han and Hung15](01) invesigae he DC pension plans and show ha indexed bonds guaranee he inflaion-adjused annuiy. Meanwhile, he mean-variance efficiency is a quaniaive ool which allows people o make allocaion by considering he rade-off beween risk and reurn. Mean-variance problem sems from he breakhrough of Markowiz19](195) and his work has laid down foundaions of modern financial heories. Zhou and Li6](000) ransform he mean-variance problem ino he linear quadraic(lq) problem and many recen lieraures are based on his procedure. The mean-variance efficiency for companies wih liabiliies is discussed in Chiu and Li9](006), Chen e al.8](008) and Xie e al.4](008). Bai and Zhang1](008) sudy he opimal invesmen-reinsurance policies for an insurer under mean-variance crierion hrough he LQ mehod and he dual mehod. Björk, Murgoci and Zhou3](014) consider differen risk aversions in he mean-variance problem and inroduce he equilibrium conrol o derive he efficien fronier. Guan and Liang13](015) sudy he opimizaion problem of he DC pension plan under mean-variance crierion. Therefore, i is pracical and essenial for a family o ake ino accoun all of he financial risk, inflaion risk and moraliy risk when measuring risks hrough mean-variance efficiency. Neverheless, when adding life insurance ino he marke o hedge he moraliy risk, he radiional LQ mehod and he dynamic programming mehod become less useful because of heir sophisicaed conrol echniques and ineviable difficulies for solving he associaed nonlinear HJB equaion. There is no lieraure on he mean-variance problem under boh moraliy risk and inflaion risk a he same ime due o hese difficulies. Thus o incorporae he inflaion risk and he moraliy risk ino a financial decision, we consider a mean-variance problem of a family wih he marke including cash, a zero-coupon bond, an inflaion-indexed zero-coupon bond, a sock, life insurance and incomereplacemen insurance. The breadwinner of he family has an uncerain lifeime and receives labor income unil a fixed ime of reiremen if he(she) is alive. During he working period of he breadwinner, he family purchases life insurance and has an income-replacemen insurance offered by he breadwinner s employer o preven an absence
4 ZONGXIA LIANG AND XIAOYANG ZHAO of labor income caused by he sudden deah of he breadwinner. The objecive of he family is o minimize he variance wih a fixed mean of he surplus wealh. A firs, we apply he Lagrange dual mehod o handle he consrain and ransform he problem o an unconsrained problem. Then we consruc some hypoheical financial insrumens o replicae labor income and simplify he problem. Finally he associaed nonlinear HJB equaion is obained and convered ino six parial differenial equaions relaed o he HJB equaions. The efficien fronier and opimal sraegies are derived and deailed numerical explanaions for efficien fronier are also provided. The res par of his paper is organized as follows: Secion describes our model and esablishes he opimal conrol problem. Secion 3 consrucs a hypoheical financial insrumen o replicae labor income so ha we can ransform he problem o a self-financing problem and derive he soluion hrough he dynamic programming mehod. Secion 4 gives he numerical analysis. Secion 5 concludes his paper. The appendices conain rigorous proofs on main resuls in his paper.. Formulaion We consider a mean-variance opimizaion problem of a family unil a fixed ime T, when he nex generaion of he family is economically independen and can suppor he family. Le T 1 be he breadwinner s reiremen ime and τ be he breadwinner s deah ime. The family wans o hedge hese wo kinds of risks so ha i will have enough money unil T. Le (Ω, F, {F } 0,T ], P) be a filered complee probabiliy space saisfying he usual condiions (cf.16]). All he processes on (Ω, F, P) below are adaped o he filraion {F } 0,T ]..1. The Financial Marke. The financial marke consiss of a risk free asse(i.e., he cash accoun), a risky asse(i.e., a sock), a zero-coupon bond and an inflaion-indexed zero-coupon bond. The price of he risk free asse is ds 0 () S 0 () = r()d, S 0(0) = 1, (.1) where r() is he insananeous nominal ineres rae. I is assumed ha r() follows he CIR process proposed by Cox e al.(cf.10]). dr() = a br()]d σ r r()dwr (), r(0) = r 0. (.)
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 5 Here a, b and σ r are posiive consans saisfying he condiion a > σr, which ensures he insananeous nominal ineres rae r() > 0 for all ime > 0. {W r ()} is a sandard Brownian moion on (Ω, F, {F } 0,T ], P). The CIR model describes he evoluion of ineres raes and is a ype of one facor model. I is an exension of he Vasicek model and can avoid he possibiliy of negaive ineres raes, which is more pracical. Besides, a zero-coupon bond, which delivers a nominal payoff of 1 when mauring a dae T, is available in he financial marke. Le B(, T ) denoe he price of he zero-coupon bond a ime. Wih he marke price of risk W r () being λ r r(), he price of he zero-coupon bond has he form(cf.6]): B(, T ) = exp { h 0 (T ) h 1 (T )r() }, where h 0 (T ) = a e ξ(t ) 1 log{ σ r ξh 1 (T ) } (ξ + b λ rσ r )(T ) ], (e ξ(t ) 1) h 1 (T ) = ξ + (ξ + b λ r σ r )(e ξ(t ) 1), ξ (b λ r σ r ) + σr. The dynamics of B(, T ) is db(, T ) B(, T ) = r()d + σ B(T ) dw r () + λ r r()d ], (.3) where σ B (T ) = h 1 (T )σ r r(). Since i is unrealisic for an individual o find all zero-coupon bonds in he financial marke, we use a rolling bond wih a consan mauriy τ 1 in our model, whose dynamics is db τ 1 B τ 1 = r()d + σ B (τ 1 ) dw r () + λ r r()d ], (.4) where σ B (τ 1 ) = h 1 (τ 1 )σ r r(). B(, T ) and B τ 1 relaion: db(, T ) B(, T ) = h 1(T ) h 1 (τ 1 ) db τ 1 B τ 1 has he following + 1 h 1(T ) ]ds 0 () h 1 (τ 1 ) S 0 (). In addiion o he zero-coupon bond, here is a sock in he financial marke, whose price saisfies ds 1 () S 1 () =r()d+σ s 1 r() λr r()d+dwr () ] +σ s λs d+dw s () ], (.5)
6 ZONGXIA LIANG AND XIAOYANG ZHAO where S 1 (0) = 1, λ s is he marke price of risk wih respec o he sandard Brownian moion W s (), which is independen of W r (). The price of he sock is always influenced by he ineres raes, so in order o aking ino accoun all he feaures of i, we add degrees of freedom o he original model and use a generalized Black-Scholes model o characerize he sock. In addiion, we ake ino accoun he risk of inflaion, and i is assumed ha he nominal price level process is dp () P () = πr()d + σ p 1 r() λr r()d + dwr () ] + σ p λs d + dw s () ] + σ p3 λp d + dw p () ], (.6) where P (0) = p 0. W p (), whose marke price of risk is λ p, is a sandard Brownian moion independen of W r () and W s (). Here we also use he generalized Black-Scholes model o characerize he nominal price level since his model will include all he poenial impacs of differen risks and ensure ha he nominal price level is always posiive. The fourh asse in he marke is an inflaion-indexed zero-coupon bond, whose pay off a is mauring dae T is P (T ). Denoe he price of i a ime by I(, T ), we can ge is explici soluion(cf.11]). I(, T ) = P ()expq 0 (T ) q 1 (T )r()], where q 0 (T ) = a log{ eζ(t ) 1 σr ζq 1 (T ) } (ζ + b λ rσ r + σ p1 σ r )(T )], q 1 (T ) = ζ (e ζ(t ) 1)(1 π) ζ + (ζ + b λ r σ r + σ r σ p1 )(e ζ(t ) 1), (b λ r σ r + σ p1 σ r ) + σ r(1 π). According o he explici form of I(, T ), we have di(, T ) I(, T ) =r()d + σ I 1 (T ) dw r () + λ r r()d ] + σ I dws () + λ s d ] + σ I3 dwp () + λ p d ], (.7) where σ I1 (T ) = (σ p1 + q 1 (T )σ r ) r(), σ I = σ p, σ I3 = σ p3. Similarly, we use a rolling inflaion-indexed bond wih a consan
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 7 mauriy of τ in our model. di τ I τ =r()d + σ I1 (τ ) dw r () + λ r r()d ] + σ I dws () + λ s d ] + σ I3 dwp () + λ p d ], (.8) where σ I1 (τ ) = (σ p1 + q 1 (τ )σ r ) r(), and he relaion beween I(, T ) and I τ is di(, T ) I(, T ) = diτ I τ + q 1(T ) q 1 (τ ) db τ 1 h 1 (τ 1 ) B τ ds ] 0(). 1 S 0 () Since τ 1 and τ are consan, we denoe σ B (τ 1 ) = σ B and σ I1 (τ ) = σ I1... Labor Income and The Insurance Marke. We suppose ha he breadwinner receives a coninuous nonnegaive labor income sream unil ime min{t 1, τ}, which means he family will no ge he income afer he deah or reiremen of he breadwinner. Denoe labor income a ime by D(), and i is driven by he following process: dd() D() =µr()d + σ d 1 r() λr r()d + dwr () ] + σ d λs d + dw s () ] + σ d3 λp d + dw p () ], where D(0) = d 0. This is also indeed a generalized Black-Scholes model, which ensures ha labor income is no negaive. Moreover, he expeced labor income growh rae is allowed o depend on he level of he ineres raes. This assumpion reflecs he influence of ineres rae on labor income: labor income usually increases more frequenly in booming periods(high ineres raes) han in recessions(low ineres raes). Assume ha he breadwinner is alive a ime 0 and has a lifeime τ, which is a posiive random variable defined on he probabiliy space (Ω, F, P) and is independen of he filraion {F } 0,T ], and he(she) has an insananeous deah rae λ() defined as λ() = lim 0 P( τ < + τ ). We define he condiional probabiliy survival funcion by F(s, ) P(τ > s τ > ). Noicing he definiion of {λ()}, i is easy o see F(s, ) = exp { s λ(u)du }. (.9)
8 ZONGXIA LIANG AND XIAOYANG ZHAO Given he wage earner being alive a ime s, denoe he condiional probabiliy densiy for deah a ime s by f(s, ), hen f(s, ) = λ(s)exp { s λ(u)du }. (.10) This insananeous deah rae model was considered by Bowers e al.4](1997). Going back o he insurance marke, here are wo insrumens in our model o hedge he moraliy risk of he breadwinner, whose sudden deah may lead o he loss of labor income. Firsly, he family can purchase life insurance coninuously by paying premiums a rae m() a ime. If he breadwinner dies a ime wih he premium paymen rae m(), hen he life insurance company needs o pay an amoun m() of money o he family. Here η() : 0, T ] R + is a coninuous, η() deerminisic and specified funcion, namely he so called he insurance premium-payou raio. In general, η() > λ(), bu we assume ha he marke is fricionless and η() = λ() for simpliciy. Besides, he income-replacemen insurance is offered by many companies as an employmen benefi, i replaces a person s income when she or he loses he abiliy o work. In general, we should consider he disabiliy risk as well as he moraliy risk and discuss moraliy raes for healhy and disabled individuals separaely. However, for mahemaic simpliciy we assume an income-replacemen insurance is provided as an employmen benefi for he breadwinner and in our paper his or her family can only ge he money afer he breadwinner s deah raher han disabiliy. Tha is, he breadwinner akes ou a fixed porion of his (her) labor income, denoing θ (θ < 1), and pus i ino he employer s fund accoun. Then if he(she) suddenly dies, he family can sill receive he same amoun of income as before from he employer unil his(her) sauory reiremen ime. To be more deailed, he breadwinner receives a labor income wih rae (1 θ)d() insead of D(), and if he(she) suddenly dies, he family will coninue o receive a labor income wih rae (1 θ)d() unil T 1 raher han lose i. 1.3. The Opimizaion Problem. People who have babies a an old age migh face he problem ha heir nex generaion has no been economically independen when hey reire or die. The nominal wealh 1 To ensure ha he expeced discouned oal labor income of he breadwinner when he employer offers he income-replacemen insurance is equal o he case ha he employer does no offer i, we assume ha θ saisfies: E{ T 1 τ D()e 0 r(s)ds d} = E{ T 1 0 0 (1 θ)d()e 0 r(s)ds d}.
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 9 of he family a ime is denoed by X() and he iniial wealh a ime 0 is X 0. The sraegy a ime consiss of four decisions: he premium paymen rae m() 0, he money π B () invesed in he rolling bond, he money π S () invesed in he sock and he money π I () invesed in he rolling inflaion-indexed bond. We denoe he sraegy a ime by α() = ( m(), π B (), π S (), π I () ). So he amoun invesed in he risk free asse is X() π B () π S () π I () and dx() = π B ()X() dbτ 1 B τ 1 + π S ()X() ds 1() S 1 () + π I()X() diτ I τ +(1 π B () π S () π I ())X() ds 0() m()d+ (1 θ)d()d S 0 () = π B ()σ B +π S ()σ s1 r()+πi ()σ I1 ]X() ] dw r ()+λ r r()d +π S ()σ s +π I ()σ I ]X() dw s ()+λ s d ] +π I ()σ I3 X() dw p ()+λ p d ] +r()x()d m()d+(1 θ)d()d. (.11) If he breadwinner dies a ime, hen he oal legacy M() consiss of he family s wealh and he life insurance amoun. Besides, we also need o plus he presen value of oal fuure labor income from he employer afer he breadwinner s deah, i.e., M() = X() + m() λ() + E T1 (1 θ)d(s)e s r(u)du ds F ]. A sraegy α( ) is said o be admissible if : (i) {α()} is progressively measurable w.r.. (Ω, F, {F } 0,T], P); (ii) {α()} ensures { T1 τ E X() (π B ()σ B + π S ()σ s1 r() + πi ()σ I1 ] 0 } +π S ()σ s + π I ()σ I ] + π I ()σ I3 ] )d < + ; (iii) For any given iniial daa ( 0, r 0, p 0, X 0 ) 0, T] (0, + ) 3, he sochasic differenial equaion(.11) has a unique soluion. Denoe by A he se of all admissible sraegies. The family will have no income afer ime T 1 so i has o reserve enough money for living expenses unil ime T. Assuming he consan consumpion floors of he family is C, define A() T T 1 CI(, s)ds,
10 ZONGXIA LIANG AND XIAOYANG ZHAO hen he minimum wealh ha he family should reserve a T 1 τ is A(T 1 τ). Define he surplus wealh of family as he difference beween he oal wealh of i and he minimum wealh i should reserve a ime T 1 τ. Then considering he risk of inflaion, he family wans o minimize he variance of he surplus wealh in he real marke wih he expeced value of i being fixed. Namely, min α( ) A { } VarM R (τ)1 {τ T1 }+X R (T 1 )1 {τ>t1 } A R (T 1 τ) u], subjec o EM R (τ)1 {τ T1 }+X R (T 1 )1 {τ>t1 } A R (T 1 τ)]=u. Here M R, X R and A R sand for he process in he real marke, i.e., M R () = M() P (), X R() = X() P (), A R() = A() P (). (.1) 3. Soluion o he Opimizaion Problem In his secion he Lagrange dual mehod is applied o handle he consrains and our problem urns ou o be an unconsrained problem. Then wih he assumpion of he complee marke and no arbirage, we consruc some hypoheical financial insrumens o replicae he labor income (1 θ)d(), and finally employ he dynamic programming approach o derive he nonlinear HJB equaion from which we ge he opimal feedback conrol. Firsly, wih a Lagrange muliplier β, we define he Lagrange dual problem as { min E M R (τ)1 {τ T1 } + X R (T 1 )1 {τ>t1 } A R (T 1 τ) u] α( ) A } + βm R (τ)1 {τ T1 } + X R (T 1 )1 {τ>t1 } A R (T 1 τ) u]. (3.1) Based on he Lagrange dual heory, he opimal value of original problem (.1) can be obained by maximizing he opimal value of problem (3.1) over β R. Since he deah ime τ is independen of he filraion {F } 0,T ], he following heorem ells us ha he expecaion wih he random ime horizon T 1 τ can be convered ino a deerminisic planning horizon, see ](006) for deails.
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 11 Theorem 3.1. The problem (3.1) can be rewrien as follows: { T1 min E α( ) A 0 f(s, 0)(M R (s) A R (s) u + β) ds + F (T 1, 0)(X R (T 1 ) A R (T 1 ) u + β) } β, (3.) where F (s, ) and f(s, ) are given by (.9) and (.10), respecively. Proof. Since E M R (τ)1 {τ T1 } + X R (T 1 )1 {τ>t1 } A R (T 1 τ) u ] =E (M R (τ) A R (τ) u) 1 {τ T1 } ] + (X R (T 1 ) A R (T 1 ) u) 1 {τ>t1 } =E T 1 f(s, 0)(M R (s) A R (s) u) ds 0 + F (T 1, 0)(X R (T 1 ) A R (T 1 ) u) ] and E M R (τ)1 {τ T1 } + X R (T 1 )1 {τ>t1 } A R (T 1 τ) u ] =E T 1 f(s, 0)(M R (s) A R (s) u)ds 0 + F (T 1, 0)(X R (T 1 ) A R (T 1 ) u) ], we complee he proof. Le β = β u. We now rewrie he problem (3.) in form of dynamic programming. For any sraegy α = (m, π B, π S, π I ), we define { T1 J(, x, r, p, d; α) E,x,r,p,d f(s, ) M R (s) A R (s) + β ] ds + F (T 1, ) X R (T 1 ) A R (T 1 ) + β ] }, τ > (3.3) where E,x,r,p,d ] means he condiional expecaion given he iniial value X() = x, r() = r, P () = p, D() = d a ime and indicaes ha he breadwinner is alive a. Define V (, x, r, p, d) inf α(,x,r,p,d) A() {J(, x, r, p, d; α)}, (3.4) where he definiion of A() is similar o A, i.e., he iniial condiion (X(0), r(0), P (0), D(0)) is replaced by (X(), r(), P (), D()).
1 ZONGXIA LIANG AND XIAOYANG ZHAO 3.1. Transform he problem. Since he exisence of labor income makes he problem non self-financing and difficul o solve, wih he assumpion of he complee marke and no-arbirage, we consruc a hypoheical financial insrumen o replicae i. Suppose here is a hypoheical insrumen in he marke whose pay off a is mauriy ime s is (1 θ)d(s) and denoe is value a ime ( s) by C(, s), hen is explici expression is C(, s) = (1 θ)d()expf 0 (s ) f 1 (s )r()], where f 0 (s ) = a log{ eη(s ) 1 σ r ηf 1 (s ) } (η + b λ rσ r + σ d1 σ r )(s )], (e η(s ) 1)(1 µ) f 1 (s ) = η + (η + b λ r σ r + σ r σ d1 )(e η(s ) 1), η (b λ r σ r + σ d1 σ r ) + σr(1 µ). According o he explici form of C(, s), we have dc(, s) C(, s) =r()d + σ c 1 (s ) dw r () + λ r r()d ] + σ c dws () + λ s d ] + σ c3 dwp () + λ p d ], (3.5) where σ c1 (s ) = (σ d1 + f 1 (s )σ r ) r(), σ c = σ d, σ c3 = σ d3. Define a new process H() by H() T1 C(, s)ds. In fac, H() represens he presen value a ime of fuure labor income from ime o T 1, ha is, H() = E T 1 (1 θ)d(s)e s ] r(u)du ds F. Theorem 3.. The dynamics of H() is given by dh() + (1 θ)d()d H() = r()d + σ h1 () dw r () + λ r r()d ] + σ h dws () + λ s d ] + σ h3 dwp () + λ p d ], (3.6) where σ h1 () = T1 C(, s)σ c1 (s )ds, σ h = σ d, σ h3 = σ d3. H()
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 13 Proof. Using Iô formula o H() we have dh() + (1 θ)d()d T1 { = C(, s) r()d + σ c1 (s ) ] dw r () + λ r r()d (3.7) + σ c dws () + λ s d ] + σ c3 dwp () + λ p d ]} ds. Simplifying (3.7) ends he proof. Similarly, using Iô formula o A() we can give he dynamics of A() Theorem 3.3. The dynamics of A() is given by da() A() =r()d + σ a 1 () dw r () + λ r r()d ] + σ a dws () + λ s d ] + σ a3 dwp () + λ p d ], (3.8) where σ a1 () = C T T 1 I(, s)σ a1 (s )ds, σ a = σ p, σ a3 = σ p3. A() Le Z() = X() + H() A(). I is easy o see from(.11), (3.6) and (3.8) ha he dynamics of Z() is dz() = m()d + Z()r()d + σ h1 ()H() + σ a1 ()A()+ (π B ()σ B +π S ()σ s1 r()+πi ()σ I1 )X() ] (dw r ()+λ r r()d) + σ h H()+σ a A()+(π S ()σ s +π I ()σ I )X() ] (dw s ()+λ s d) + σ h3 H() + σ a3 A() + π I ()σ I3 X() ] (dw p () + λ p d). (3.9) Theorem 3.4. The dynamics of Z R () is given by dz R () = Z R ()(1 π)r()d + Z R () π ] B ()σ B + π S ()σ s1 r() + πi ()σ I1 (dw r () + λ ) r r()d + Z R () π ]( S ()σ s + π I ()σ I dws () + λ s d ) (3.10) + Z R () π I ()σ I3 ( dwp () + λ p d ) m R ()d, where m R () = m() P (), λ r = λ r σ p1, λ s = λ s σ p, λ p = λ p σ p3, and π B (), π S (), π I (), π() are given by (6.6). Proof. See appendix.
14 ZONGXIA LIANG AND XIAOYANG ZHAO Now we conver he problem wih process X R () ino he problem wih Z R () and for any sraegy α = (m R, π B, π S, π I ), define J(, z, r, p; α) E,z,r,p { T1 f(s, ) Z R (s) + m R(s) λ(s) + β ] ds + F (T 1, ) Z R (T 1 ) + β ] τ > }, (3.11) where E,z,r,p ] means he condiional expecaion given he iniial value Z R () = z, r() = r, P () = p a ime. Define { } V (, z, r, p) inf J(, z, r, p; α), (3.1) α(,z,r,p) A() hen V (, z, r, p) = V (, x, r, p, d). For any sraegy α and he corresponding sraegy α, J(, z, r, p; α) is equal o J(, x, r, p, d; α) in (3.3), here z = x+h(0) A(0). A naural necessary assumpion is ha p z > 0, ha is, x + H(0) > A(0). 3.. The HJB equaion. In his subsecion we derive he opimal sraegy hrough he dynamic programming mehod. The class of C 1,,, (0, T 1 ) R (0, + ) (0, + )) C(0, T 1 ] R (0, + ) (0, + )) funcions is denoed simply by C. We can prove he following classical verificaion heorem. Theorem 3.5. Le v(, z, r, p) C, i.e., v(, z, r, p) C 1,,, (0, T 1 ) R (0, + ) (0, + )) C(0, T 1 ] R (0, + ) (0, + )) saisfy for every α( ) A, v λ()v + v z (1 π)rz + πb ()zσ B λr r + π S ()z(σ s1 λr r + σ s λs ) + π I ()z(σ I1 λr r + σp λs + σ p3 λp ) m R () ] + 1 v zzz ( π B ()σ B + π S ()σ s1 r + πi ()σ I1 ) + ( π S ()σ s + π I ()σ p ) + ( π I ()σ p3 ) ] + vr (a br) + 1 v rrσ rr +v p p ( λr σ p1 r+ λ s σ p + λ p σ p3 +σ p 1 r+σ p +σ p 3 +πr ) v pr prσ p1 σ r (3.13) + 1 v ppp ( σ p 1 r + σ p + σ p 3 ) vrz zσ r r ( πb ()σ B + π S ()σ s1 r + π I ()σ I1 ) + vpz zp σ p1 r ( πb ()σ B + π S ()σ s1 r + πi ()σ I1 ) +σ p ( πs ()σ s + π I ()σ p ) +σp3 π I ()σ p3 ] +λ() ( z+ m R () λ() + β ) G(v; α( )) 0,
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 15 for all (, z, r, p) (0, T 1 ] R (0, + ) (0, + )), wih v(t 1, z, r, p) = (z + β), (z, r, p) (R (0, + ) (0, + )). Then v(, z, r, p) V (, z, r, p), α A, (, z, r, p) (0, T 1 ] R (0, + ) (0, + )). Moreover, if here exiss an admissible sraegy α( ) A such ha G(v; α( ))=0 holds for all (, z, r, p) (0, T 1 ] R (0, + ) (0, + )), hen v(, z, r, p) = V (, z, r, p) and his sraegy α( ) is he opimal sraegy. Proof. The proof is sandard and we omi i here, see for example Secion 5.5 5](1999) for deails. The following heorem gives he opimal value funcion and sraegies. Theorem 3.6. The opimal value funcion V (, z, r, p) is in class C and has he following form. Moreover, he sraegies of opimal conrol problem (3.1) is given by V (, z, r, p) = V 1 (, r)z + V 5 (, r)z β + V 3 (, r) β, m R(, z, r, p) = λ()(v 1(, r)z + V 5 (, r) β (z + β)), π B(, z, r, p) = σ s σ p3 σ r r()(v1r (, r)z + V 5r (, r) β) + (σ s1 σ p3 λs r() σs σ p3 λr r() + σs σ I1 λp ] σ s1 σ p λp r())(v1 (, r)z + V 5 (, r) β) ( V 1 (, r)zσ B σ s σ p3 ), π S(, z, r, p) = (σ p λp σ p3 λs )(V 1 (, r)z + V 5 (, r) β) V 1 (, r)zσ s σ p3, π I(, z, r, p) = λ p (V 1 (, r)z + V 5 (, r) β) V 1 (, r)zσ p3, where V 1 (, r), V 5 (, r) and V 3 (, r) are in C 1, (0, T 1 ) (0, + )) C(0, T 1 ] (0, + )) and given by (6.16), (6.7) and (6.31), respecively, λ r = λ r σ p1, λ s = λ s σ p and λ p = λ p σ p3. Proof. See appendix. The value funcion consiss of hree erms, z, z β and β. There is no erm abou p or p since he problem is considered in he real marke. Given he parameers as Table 1, we find ha V 1 (, r), he coefficien of erm z, will decrease as increases unil he reiremen ime T 1, when V 1 (T, r) = 1. This means when he breadwinner faces
16 ZONGXIA LIANG AND XIAOYANG ZHAO reiremen, he conribuion of he wealh z o he value funcion will increase more slowly. I is no srange because when he invesmen period decreases, he family has less ime o inves leading ha i has less profi from he marke. Besides, he coefficien of erm z β, V 5 (, r), is always posiive because he wealh always has a posiive conribuion o he value funcion, and i will decrease o as increases o T 1. All of he hree erms will decrease as r increases, leading he value funcion has a negaive correlaion wih r. I means ha when he ineres rae is greaer, he family has o ake more risk if hey wan o leave he same money as before in he real marke. Insurance. The opimal insurance sraegy is linear wih he wealh z and he parameer β. The coefficien of z is λ()(v 1 (, r) 1), which is posiive. This means when he family has more wealh, i will pu more money in he life insurance marke in case of he sudden deah of he breadwinner. In addiion, if he insananeous deah rae of he breadwinner, λ(), increases, he family will also pu more money in he life insurance marke, since i has o ake more moraliy risk of he breadwinner, and life insurance is a direc insrumen o hedge i. Invesmen in he zero-coupon bond. We rewrie π B (, z, r, p) as π B(, z, r, p) = σ I 1 λp + (σ s 1 λs σ p3 σ s λr σ p3 σ s1 σ p λp ) r() σ B σ p3 σ B σ s σ p3 + V 1r(, r)σ r r() λ r r()v5 (, r) V 1 (, r)σ B V 1 (, r)zσ B + V 5r(, r) βσ r r() V 5(, r) βσ s1 σ p λp r() V 1 (, r)zσ B V 1 (, r)zσ B σ s σ p3 + σ s 1 λs r()v5 (, r) V 1 (, r)zσ B σ s + σ I 1 λp V 5 (, r) V 1 (, r)zσ B σ p3. The firs erm is a fixed proporion of he wealh and we call i a speculaive demand. The second erm is a correcion erm for he correlaion beween he ineres rae and he sock and inflaion risk, we can see ha his erm will disappear if hese hree risks are independen. The hird and he fourh erm reflec he influence of he ineres rae on he invesmen. These wo erms only depend on he behaviour of he ineres rae. The following wo erms are correcion erms for he parameer β. The las wo erms reflecs he influence of he sock and inflaion risk, separaely.
π S OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 17 Invesmen in he sock. The invesmen proporion in he sock, (, z, r, p), can be rewrien as π S(, z, r, p) = (σ p λ p σ p3 λ s ) σ s σ p3 + σ p λ p V 5 (, r) β V 1 (, r)zσ s σ p3 λ sv 5 (, r) β V 1 (, r)zσ s. The firs erm is a speculaive demand, which is only correlaed wih he wealh. The second erm is he correcion erm for he inflaion risk, we can see ha if he inflaion risk is independen of he risk of sock, ha is, σ p = 0, hen his erm will disappear. Thus he correlaion beween he inflaion risk and he risk of sock has a posiive effec on he proporion of invesmen in he sock. This can be explained as follows: if here is a correlaion beween he risk of sock and he inflaion risk, hen he sock can hedge he inflaion risk in some sense and he family will inves more in he sock. The las erm is a resul of comprehensive effec of he ineres rae, he parameer β and he wealh. Besides, he influence of labor income is refleced in equaion (6.6), he relaionship beween π S and π S. π I Invesmen in he inflaion-indexed zero-coupon bond. We rewrie (, z, r, p) as π I(, z, r, p) = λ p σ p3 λ p V 5 (, r) β V 1 (, r)zσ p3. The sign of i depends on he sign of λ p = λ p σ p3. The firs erm is also a speculaive demand and is a fixed proporion of he wealh. The second erm is grealy influenced by he inflaion risk and he ineres rae and is similar o he las erm of π S (, z, r, p). Remark 3.7. Replacing (, z, r, p) wih (, ZR (), r(), p()) in m R (, z, r, p), π B (, z, r, p), π S (, z, r, p) and π I (, z, r, p), we can ge he opimal feedback conrol sraegy m R (, Z R (), r(), p()), π B (, Z R (), r(), p()), π S (, Z R (), r(), p()) and π I (, Z R (), r(), p()). 3.3. The original problem. Since problem (3.1) is equal o problem (3.4), i.e., he opimal values are equal and he relaion of opimal sraegy is given by (6.1) and (6.6), we can ge he opimal value funcion and sraegy of he original problem (3.). The value funcion of problem (3.) is V (0, x, r, p, d) β = V (0, z, r, p) β = V 1 (0, r)( x + H(0) A(0) ) x + H(0) A(0) + V 5 (0, r) (β u) p p + V 3 (0, r)(β u) β
18 ZONGXIA LIANG AND XIAOYANG ZHAO and he opimal sraegy is given by m R(, x, r, p, d) = λ()(v 1(, r) 1)(x + H() A()) + β(v 5 (, r) )], { πb(, x, r, p, d) = π B(, z, r, p)σ p3 σ s σ B σ s σ p3 σ h1 ()+σ s σ p3 σ p1 r()+σs σ I1 σ d3 ] σ s σ I1 σ p3 +σ p3 σ d σ s1 r() σp σ d3 σ s1 r() HR () σ s σ p3 π B (, z, r, p)σ B } ] +σ a1 () σ p1 r() AR () (σ B σ s σ p3 x)+ π B(, z, r, p) q 1(τ ) h 1 (τ 1 ), { πs(, x, r, p, d) = (σ p3 σ s π S(, z, r, p) σ p3 σ d +σ p σ d3 )H R () } σ p3 σ s π S(, z, r, p)a R () σ p3 σ s x + π S(, z, r, p), πi (, x, r, p, d) = ( π I (, z, r, p)σ p 3 σ d3 +σ p3 )H R () π I (, z, r, p)σ p 3 A R () + π I (, z, r, p)+1. σ p3 x Problem (3.) is he Lagrange dual problem of problem (.1) and depends on β. We can solve problem (.1) by maximizing (3.) over all β. The value funcion V (0, x, r, p, d) β is a quadraic funcion of β and exisence of finie maximum value depends on he sign of β. Lemma 3.8. The coefficien of β in V (0, x, r, p, d) β is negaive. Proof. Since he coefficien of β in V (0, x, r, p, d) β is V 3 (0, r) 1 and k(, r) λ() = λ()v 5 (, r) λ() 4 V 5(, r) we have V 3 (0, r) 1 = < σ rrv 5r (, r) 4V 1 (, r) ( λ s + λ p + λ rr)v 5 (, r) 4V 1 (, r) + λ r σ r rv 5 (, r)v 5r (, r) V 1 (, r) λ() = λ() 4 (V 5(, r) ) r(σ rv 5r (, r) λ r V 5 (, r)) 4V 1 (, r) ( λ s + λ p)v 5 (, r) 4V 1 (, r) T1 0 T1 0 < 0, ( e u ]) 0 λ(θ)dθ E,r k(u, q(u)) du + e T 1 0 λ(θ)dθ 1 ( e u 0 λ(θ)dθ λ(u) ) du + e T 1 0 λ(θ)dθ 1 = 0,
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 19 hus his ends he proof. Afer a lile algebra compuaion, we ge he Lagrange muliplier β = V 5(0, r)(x + H(0) A(0)) + upv 3 (0, r). p(v 3 (0, r) 1) Therefore we conclude he following heorem. Theorem 3.9. The opimal value of problem (.1) is Var M R (τ)1 {τ T1 } + X R (T 1 )1 {τ>t1 } A R (T 1 τ) ] = V 3 (0, r) 4(V 3 (0, r) 1) u V 5(0, r) V 3 (0, r) x + H(0) A(0) ] p + (V 3(0, r)v 5 (0, r) 4V 1 (0, r)v 3 (0, r) +4V 1 (0, r)v 3 (0, r) 4V 5 (0, r) ) 4(1 V 3 (0, r))v 3 (0, r) (x + H(0) A(0)) p. We can observe from Theorem 3.9 ha he inersecion of he meanvariance fronier and he mean-axis in he plane is V 5(0,r) x+h(0) A(0) V 3, (0,r) p which is a muliple of he oal wealh in he real marke, x+h(0) A(0). p According o Theorem 3.9, seing u = u σmin = V 5(0,r) x+h(0) A(0), we obain he global minimum variance: p Var minm R (τ)1 {τ T1 } + X R (T 1 )1 {τ>t1 } A R (T 1 τ)] V 3 (0,r) = (V 3(0, r)v 5 (0, r) 4V 1 (0, r)v 3 (0, r) +4V 1 (0, r)v 3 (0, r) 4V 5 (0, r) ) 4(1 V 3 (0, r))v 3 (0, r) (x + H(0) A(0)) p. I is obvious ha raional invesors should selec he expeced wealh more han u σmin. 4. Sensiiviy analysis In his secion, we will discuss some numerical examples o illusrae how some main parameers effec he efficien fronier. The efficien fronier shows how much risk a family has o bear if hey wan o achieve a given expecaion of surplus wealh in he real marke a ime T 1 τ. Obviously, he risk will increase as he expecaion value increases. To simplify he expression in he graph, denoe Var M R (τ)1 {τ T1 } + X R (T 1 )1 {τ>t1 } A R (T 1 τ)] by Var (MXR) and EM R (τ)1 {τ T1 } + X R (T 1 )1 {τ>t1 } A R (T 1 τ)] by E (MXR).Unless
0 ZONGXIA LIANG AND XIAOYANG ZHAO oherwise saed, he values of he parameers are presened in Table 1 below. 3.5 x 104 3 impac of C on he efficien fronier C=0.5 C=1 C=1.5.5 Var * (MXR) 1.5 1 0.5 0 0 50 100 150 00 50 300 350 400 450 500 E * (MXR) Figure 1. How consan consumpion floor C impacs on he efficien fronier Fig.1 illusraes he impac of consan consumpion floor of a family, C, on he efficien fronier. A greaer C sands for a higher consan consumpion floor, which means ha he family needs o reserve more wealh o mainain he same consumpion level from T 1 o T. As we can see, when C increases from 0.5 o 1.5, he minimum wealh ha he family should reserve a T 1 is more. Thus he family has o ake more risk o ge he same value as E (MXR). Besides, i is also demonsraed ha he inersecion of he efficien fronier wih he E (MXR) axis decreases if C increases, which implies ha he family will keep less wealh when C increases assuming i does no wan o ake any risk. Fig. reveals he influence of labor income on he efficien fronier. The increase of µ implies ha here is a closer relaionship beween labor income and he ineres rae. Since r() is posiive in our model, i is also indicaed ha he growh of income is faser. The family will gain more from ime o T 1 τ, which, as shown in Fig., resuls in a lower risk level hey could bear a he same E (MXR) value. For he same reason, i will reserve more wealh when µ increases. Fig.3 shows how he efficien fronier will vary as he inflaion level changes. A greaer π leads o a higher increasing rae of he inflaion rae. E (MXR) consiss of wo pars: he oal wealh of he family and
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 1 Table 1. values of he parameers in our model parameers in ineres rae model value a 0.64 b 0.0 σ r 0.0 λ r 0.0 r 0 0.05 τ 1 0 parameers in sock model value σ s1 0.0 σ s 0.0 λ s 0.0 parameers in inflaion model value π 0.5 σ p1 0.0 σ p 0.0 σ p3 0.0 λ p 0.0 p 0 1 τ 0 parameers in labor income model value µ 0.7 σ d1 0.0 σ d 0.0 σ d3 0.0 d 0 1 oher parameers 0 35 T 1 60 T 75 X 0 1 value λ() 10 10 C 1
ZONGXIA LIANG AND XIAOYANG ZHAO 3.5 x 104 3 impac of on he efficien fronier =0.5 =0.7 =1.5 Var * (MXR) 1.5 1 0.5 0 0 50 100 150 00 50 300 350 400 450 500 E * (MXR) Figure. Influence of labor income on he efficien fronier.5 3 x 104 =0.3 =0.5 =0.7 impac of on he efficien fronier Var * (MXR) 1.5 1 0.5 0 50 100 150 00 50 300 350 400 450 500 E * (MXR) Figure 3. How inflaion level effec on he efficien fronier he minimum wealh he family should reserve a ime T 1 τ. Since we consider he problem in he real marke, and here is a erm I(, s) in he definiion of A(), which eliminaes mos par of he effec of inflaion, i is naural ha inflaion level P () has lile influence on he second par A R (). However, he firs par, namely he oal wealh reserved by he family a ime T 1 τ, is closely relaed o he level of inflaion. As is illusraed in Fig.3, a higher inflaion level makes he family bear more risk a he same level of expecaion. We are also concerned by he influence of reiremen ime T 1 and ime T when he second generaion becomes economically independen.
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 3 3.5 x 104 3 impac of r 0 on he efficien fronier when T 1 =60 and T =75 r 0 =0.03 r 0 =0.05 r 0 =0.07.5 Var * (MXR) 1.5 1 0.5 0 50 100 150 00 50 300 350 400 450 500 E * (MXR) Figure 4. Influence of reiremen ime and ineres rae on he efficien fronier 7 x 104 6 impac of r 0 on he efficien fronier when T 1 =55 and T =75 r 0 =0.03 r 0 =0.05 r 0 =0.07 5 Var * (MXR) 4 3 1 0 0 50 100 150 00 50 300 350 400 450 500 E * (MXR) Figure 5. Influence of reiremen ime and ineres rae on he efficien fronier Besides, since he invesmen plan will las for a long ime, i is essenial o consider he effec of he change of he ineres rae. Fig.4 o Fig.6 show how efficien fronier will change wih he change of hese wo facors. Firsly, when r 0 increases from 0.03 o 0.07, a higher ineres rae will be expeced. Since he expeced reurn from all financial asses is posiively correlaed wih he ineres rae, he family is expeced o have more wealh a ime T 1 τ. Besides, he value of he minimum wealh he family should keep a T 1 τ, namely A(T 1 τ), is negaively correlaed wih he ineres rae. In oher words, as a discouned value,
4 ZONGXIA LIANG AND XIAOYANG ZHAO 4.5 5 x 104 4 impac of r 0 on he efficien fronier when T 1 =60 and T =80 r 0 =0.03 r 0 =0.05 r 0 =0.07 3.5 3 Var * (MXR).5 1.5 1 0.5 0 0 50 100 150 00 50 300 350 400 450 500 E * (MXR) Figure 6. Influence of reiremen ime and ineres rae on he efficien fronier T 1 τ will decrease wih he increase of he ineres rae. As shown in all hree figures, boh facors reduce he family s risk olerance if i wans o ge he same expeced value E (MXR). Secondly, comparing Fig.4 and Fig.5, he reiremen ime in Fig.5 is five years earlier han Fig.4. As a resul, he family will have less ime o accumulae wealh before T 1 τ and he ime beween T 1 τ and T is longer, which makes i more difficul o keep he same amoun of money. We can observe ha no maer how he ineres rae varies, as long as i is a he same level in he wo figures, an earlier reiremen ime will lead o a higher risk level. Thirdly, in conras o Fig.4, Fig.6 is he siuaion where he second generaion becomes economically independen five years laer. This assumpion resuls in more wealh o reserve a T 1 τ, since i akes more ime for he younger generaion o become economically independen. For he reason ha he ime for he family o accumulae wealh is no changed, hey have o ake more risk o ge he same expeced value E (MXR). Besides, observing Fig.5 and Fig.6, we can find ha here will have a more significan influence for a five years earlier reiremen ime compared o a five years delayed ime of economic independence. This means ha when ineres raes are he same, o ge he same expeced value of E (MXR), he family has o ake more risk if i chooses o le he breadwinner reire five years earlier raher han o ge he nex generaion employed five years laer. This reflecs he fac ha he family can sill ge a posiive influence from working despie he negaive effec of is consumpion.
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 5 5. Conclusion We sudy a mean-variance opimizaion problem of a family wih he breadwinners who have babies a an old age. This group of people have o face he problem ha heir nex generaion has no been economically independen when hey reire or die. So he objecive of he family is o minimize he variance wih a fixed mean of he surplus wealh. We ake ino accoun he inflaion risk and sudy he problem in he real marke raher han he nominal marke. In addiion o his, we suppose ha he family can inves in life insurance and incomereplacemen insurance o make up for he economic loss due o he accidenal deah of he breadwinner. This assumpion is pracical and essenial, hough under his assumpion he previous radiional LQ mehod and dynamic programming mehod become less useful because he associaed HJB equaion is nonlinear. To solve his problem, we firsly apply Lagrange dual mehod o handle he consrain. Then we consruc some hypoheical financial insrumens o replicae labor income and simplify he problem. A las, we decompose he associaed nonlinear HJB equaion ino six parial differenial equaions and solve hem. A he end of his paper, we presen how he change of economic siuaion influence he efficien fronier and give some reasonable explanaions. 6. Appendix 6.1. Proof of Theorem 3.4. Noicing ha X R () = X() P () Iô formula for i, we have and using dx R () = X R ()π B ()σ B + π S ()σ s1 r() + πi ()σ I1 σ p1 r()](dwr () + λ r r()d) + X R ()π S ()σ s + π I ()σ I σ p ](dw s () + λ s d) + X R ()π I ()σ I3 σ p3 ](dw p () + λ p d) + (1 π)r()x R ()d m R ()d + X R ()(σp 1 r() + σp + σp 3 ) σ p1 r()(πb ()σ B + π S ()σ s1 r() + πi ()σ I1 ) σ p (π S ()σ s + π I ()σ I ) σ p3 π I ()σ I3 ]d + (1 θ)d R ()d,
6 ZONGXIA LIANG AND XIAOYANG ZHAO where D R () = D(). If we le P () π B () = π B () + q 1(τ ) h 1 (τ 1 ), π S() = π S (), π I () = π I () 1, λ r = λ r σ p1, λ s = λ s σ p, λ p = λ p σ p3, (6.1) and noicing ha σ I1 = (σ p1 + q 1 (τ )σ r ) r(), σ I = σ p, hen σ I3 = σ p3, σ B = h 1 (τ 1 )σ r r(), dx R () = (1 π)r()x R ()d + X R () Similarly, and π B ()σ B + π S ()σ s1 r() + πi ()σ I1 ] (dw r ()+ λ r r()d)+xr () π S ()σ s + π I ()σ I ](dw s ()+ λ s d) + X R () π I ()σ I3 (dw p ()+ λ p d) m R ()d+(1 θ)d R ()d. (6.) da R () A R () = (1 π)r()d+(σ a 1 () σ p1 r())(dwr ()+ λ r r()d) (6.3) dh R () + (1 θ)d R ()d H R () = (1 π)r()d + (σ h1 () σ p1 r())(dwr () + λ r r()d) + (σ h σ p )(dw s () + λ s d) + (σ h3 σ p3 )(dw p () + λ p d). By (6.), (6.3) and (6.4) we have dz R () = m R ()d + (1 π)z R ()r()d + (σ h1 () σ p1 r())hr () + (σ a1 () σ p1 r())ar () (6.4) + ( π B ()σ B + π S ()σ s1 r() + πi ()σ I1 )X R ()](dw r () + λ r r()d) + (σ h σ p )H R () + ( π S ()σ s + π I ()σ I )X R ()](dw s () + λ s d) + (σ h3 σ p3 )H R () + π I ()σ I3 X R ()](dw p () + λ p d). (6.5)
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 7 Seing π B () = { σ s σ p3 σ h1 () σ s σ p3 σ p1 r() σs σ I1 σ d3 +σ s σ I1 σ p3 σ p3 σ d σ s1 r() + σp σ d3 σ s1 r()]hr () + σ s σ p3 σ a1 () σ p1 r()]ar () + σ B σ s σ p3 π B ()X R () } { σ p3 σ s σ B Z R () }, π S () = (σ p 3 σ d σ p σ d3 )H R () + σ p3 σ s π S ()X R (), σ p3 σ s Z R () π I () = (σ d 3 σ p3 )H R () + σ p3 π I ()X R (), σ p3 Z R () we finally ge (3.10). (6.6) 6.. Proof of Theorem 3.6. Differeniaing F ( ) wih respec o α(, z, r, p) and leing is firs order parial derivaives be zero yields he following equaions: m R(, z, r, p) = λ()(v z (z + β)), π B(, z, r, p) = σ s σ p3 σ r rvrz + (σ s1 σ p3 λs r σs σ p3 λr r + σ s σ I1 λp σ s1 σ p λp r)vz + (σ s σ p3 σ I1 σ s σ p3 σ p1 r)vpz ] (Vzz zσ B σ s σ p3 ), π S(, z, r, p) = (σ p λp σ p3 λs )V z V zz zσ s σ p3, (6.7) π I (, z, r, p) = λ p V z + σ p3 V pz V zz zσ p3. Subsiuing (6.7) ino (3.13) we have 0 =V λ()v + ((1 π)r + λ())z + βλ() ] V z λ()v z 4 1 (σr V rz V λ r V z σ p1 pv pz ) r + ( λ s V z + σ p pv pz ) zz + ( λ p V z + σ p3 pv pz ) ] + V r (a br) + 1 V rrσ rr + V p p ( λr σ p1 r + λ s σ p + λ p σ p3 + σ p 1 r + σ p + σ p 3 + πr ) (6.8) V pr prσ p1 σ r + 1 V ppp ( σ p 1 r + σ p + σ p 3 ), wih boundary V (T 1, z, r, p) = (z + β).
8 ZONGXIA LIANG AND XIAOYANG ZHAO Since Z R () is a process in he real marke, we guess ha P () has no influence on he value funcion, ha is, V (, z, r, p) = V (, z, r) and we assume V (, z, r, p) can be wrien in he following form: V (, z, r, p) = V 1 (, r)z + V (, r)z + V 3 (, r) β + V 4 (, r) β + V 5 (, r) βz + V 6 (, r), (6.9) wih boundary V 1 (T 1, r) = V 3 (T 1, r) = 1, V 5 (T 1, r) =, V (T 1, r) = V 4 (T 1, r) = V 6 (T 1, r) = 0. Subsiuing he form in (6.9) ino (6.8) we obain { V 1 + ( π λ r)r + (λ() λ s λ p) ] V 1 λ()v1 + a + (σ r λr b)r ] V 1r + 1 } σ rrv 1rr σ rrv1r z + V + (1 π)rv λ()v 1 V + (a br)v r + 1 σ rrv rr ( λ s + λ p + λ rr)v σ rrv 1r V r + λ r σ r r(v 1 V r + V 1r V )] z V 1 V 1 + V 3 λ()(v 3 V 5 ) λ() 4 V 5 + (a br)v 3r + 1 σ rrv 3rr ( λ s + λ p + λ rr)v5 σ rrv5r + σ r λ r rv 5r V 5 ] β 4V 1 4V 1 V 1 + V 4 λ()(v 4 V ) λ() V V 5 + (a br)v 4r + 1 σ rrv 4rr ( λ s + λ p + λ rr)v V 5 V 1 + σ r λ r r(v r V 5 + V V 5r ) V 1 V 1 σ rrv r V 5r ] β V 1 + V 5 + (1 π)rv 5 + λ()v 1 λ()v 1 V 5 + (a br)v 5r + 1 σ rrv 5rr ( λ s + λ p + λ rr)v 5 σ rrv 1r V 5r + σ r λ r r(v 1r V 5 + V 1 V 5r )] z β V 1 V 1 + V 6 λ()v 6 λ() 4 V + (a br)v 6r + 1 σ rrv 6rr ( λ s + λ p + λ rr)v 4V 1 + λ r σ r rv V r σ rrv ] r = 0. V 1 4V 1
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 9 Leing he coefficiens of z and β be zero we ge he following six simpler equaions. V 1 + ( π λ r)r + (λ() λ s λ p)]v 1 λ()v1 + a + (σ r λr b)r]v 1r + 1 σ rrv 1rr σ rrv1r = 0, (6.10) V 1 V 1 (T 1, r) = 1, V + (1 π)rv λ()v 1 V + (a br)v r + 1 σ rrv rr ( λ s+ λ p+ λ rr)v σ rrv 1r V r r σ r r(v 1 V r +V 1r V ) + λ = 0, V 1 V 1 V (T 1, r) = 0, V 3 λ()(v 3 V 5 ) λ() 4 V 5 + (a br)v 3r + 1 σ rrv 3rr ( λ s + λ p + λ rr)v5 σ rrv5r + σ r λ r rv 5r V 5 = 0, 4V 1 4V 1 V 1 V 3 (T 1, r) = 1, V 4 λ()(v 4 V ) λ() V V 5 +(a br)v 4r + 1 σ rrv 4rr ( λ s+ λ p+ λ rr)v V 5 + σ r λ r r(v r V 5 +V V 5r ) σ rrv r V 5r = 0, V 1 V 1 V 1 V 4 (T 1, r) = 0, V 5 +(1 π)rv 5 +λ()v 1 λ()v 1 V 5 +(a br)v 5r + 1 σ rrv 5rr ( λ s+ λ p+ λ rr)v 5 σ rrv 1r V 5r + σ r λ r r(v 1r V 5 +V 1 V 5r ) = 0, V 1 V 1 V 5 (T 1, r) =, V 6 λ()v 6 λ() 4 V + (a br)v 6r + 1 σ rrv 6rr ( λ s + λ p + λ rr)v + λ r σ r rv V r σ rrvr = 0, 4V 1 V 1 4V 1 V 6 (T 1, r) = 0. (6.11) (6.1) (6.13) (6.14) (6.15) Nex we will solve hese equaions one by one. equaion (6.10). Now we sar wih
30 ZONGXIA LIANG AND XIAOYANG ZHAO Theorem 6.1. There exiss a unique soluion V 1 (, r) C 1, (0, T 1 ) (0, + )) C(0, T 1 ] (0, + )) o he equaion (6.10), and i is given by where V 1 (, r) = h 1 (, T 1, r) + T1 h1 (, s, r)λ(s)ds ] 1, (6.16) h1 (, s, r) = exp{ h 11 (, s) + h 1 (, s)r}, (6.17) and h11 (, s) = a s ( h 1 (u, s) λ(u))du + ( λ s + λ p)(s ), h1 = σ r λ r + b 4bσ r λr + 4(1 π)σ r, h1 (, s) = λ 1 λ exp{ h1 (s )} λ 1 λ λ 1 exp{ h1 (s )} λ, if h1 > 0, σ r λr b σ r λ r b σr + 1 σ r (s )(σ r λ r b) σr h1 σ 4 r an{arcan( σ r λr b h1 ), if h1 = 0, + 1 h1 (s )}] + b σ r λ r, < 0. σr if h1 Proof. Suppose he soluion of (6.10) is of he following form: V 1 (, r) = h 1 (, r) 1 wih boundary h 1 (T 1, r) = 1. Subsiuing i ino (6.10) we have 0 = h 1 ( π λ r)r + (λ() λ s λ p)]h 1 + a + (σ r λr b)r]h 1r + 1 σ rrh 1rr + λ(). (6.18) I is hard o solve his equaion direcly because of he nonlinear erm λ(), so we firsly solve he siuaion wihou λ(). Assume h 1 (, r) is he soluion of equaion 0 = h 1 ( π λ r)r + (λ() λ s λ p)]h 1 + a + (σ r λr b)r]h 1r + 1 σ rrh 1rr. (6.19) Then we suppose h 1 (, r) has he form h1 (, r) = exp{ h 11 () + h 1 ()r}.
OPTIMAL MEAN-VARIANCE EFFICIENCY OF A FAMILY 31 Subsiuing i ino (6.19) and leing he coefficien of r be zero, we ge h 11() + a h 1 () (λ() λ s λ p) = 0, (6.0) h11 (T 1 ) = 0, and h 1() + (σ r λr b) h 1 () + 1 σ r h 1() ( π λ r) = 0, h1 (T 1 ) = 0. (6.1) To solve (6.1), rewrie i as h 1() = 1 σ r h 1() (σ r λr b) h 1 () + ( π λ r). Le h1 = σ r λ r + b 4bσ r λr + 4(1 π)σ r be he discriminan of quadraic equaion If h1 1 σ r h 1() (σ r λr b) h 1 () + ( π λ r) = 0. (6.) > 0, le λ 1 and λ be wo real roos of (6.), which can be expressed as hen λ 1, = b σ r λ r ± h1, σr If h1 = 0, hen h1 () = λ 1λ exp{ h1 (T 1 )} λ 1 λ λ 1 exp{ h1 (T 1 )} λ. h1 () = If < 0, hen h1 h1 h1 () = Thus σ 4 r h1 () = σ r λr b σr + 1 σ r(t 1 )(σ r λr b) σ r λ r b. σr an{arcan( σ r λ r b ) + 1 ] (T h1 1 )} h1 λ 1 λ exp{ h1 (T 1 )} λ 1 λ λ 1 exp{ h1 (T 1 )} λ, if h1 > 0, σ r λr b σr λ r b σr+ 1 σ r(t 1 )(σ r λr b) σr h1 σ 4 r an{arcan( σr λ r b ) h1, if h1 = 0, + 1 h1 (T 1 )}] + b σ r λ r, < 0. σr if h1 + b σ r λ r. σr
3 ZONGXIA LIANG AND XIAOYANG ZHAO We ge he following expression hrough (6.0) h11 () = a T1 ( h 1 (u) λ(u))du + ( λ s + λ p)(t 1 ). (6.3) Now we ge he explici soluion h 1 (, r) of equaion (6.19). To represen h 1 (, r) by h 1 (, r), we le T 1 in h 1 (, r) o be a variable, ha is, assume ha and Then h1 (, s) = h11 (, s) = a λ 1 λ exp{ h1 (s )} λ 1 λ λ 1 exp{ h1 (s )} λ, if h1 > 0, σ r λr b σ r λ r b σr+ 1 σ r(s )(σ r λr b) σr h1 σ 4 r an{arcan( σ r λ r b ) h1, if h1 = 0, + 1 h1 (s )}] + b σr λ r, < 0, σr if h1 s ( h 1 (u, s) λ(u))du + ( λ s + λ p)(s ). h1 (, r) = h 1 (, T 1, r) = exp{ h 11 (, T 1 ) + h 1 (, T 1 )r} is he soluion of (6.19). Wrie (6.19) as h 1 + h 1 = 0. Here for any funcion f(, r), is a differenial operaor defined as f ( π λ r)r+(λ() λ s λ p)]f+a + (σ r λr b)r]f r + 1 σ rrf rr. Thus h 1 (, r) saisfies Le Then h 1 + h 1 + λ() = 0. h 1 (, r) h 1 (, T 1, r) + h 1 (, r) = h 1 (, T 1, r) + and T1 h 1 (, r) = h 1 (, T 1, r) + T1 h1 (, s, r)λ(s)ds. (6.4) h1 (, s, r) λ(s)ds h 1 (,, r)λ(), (6.5) T1 λ(s) h 1 (, s, r)ds. (6.6) Combining (6.5) and (6.6), and noicing ha h 1 (,, r) = 1, we finally have h 1 + h 1 + λ() = 0.