THIELE CENTRE. Portfolio Management with Stochastic Interest Rates and Inflation Ambiguity. Claus Munk and Alexey Rubtsov

Transcription

1 THIELE CENTRE for applied mahemaics in naural science Porfolio Managemen wih Sochasic Ineres Raes and Inflaion Ambiguiy Claus Munk and Alexey Rubsov Research Repor No. 6 December 1

2

3 Porfolio Managemen wih Sochasic Ineres Raes and Inflaion Ambiguiy Claus Munk 1 and Alexey Rubsov 1 Deparmen of Finance Copenhagen Business School cm.fi@cbs.dk Deparmen of Mahemaics Aarhus Universiy avrubsov@imf.au.dk Absrac We solve a sock-bond-cash porfolio choice problem for a risk- and ambiguiyaverse invesor in a seing where he inflaion rae and ineres raes are sochasic. The expeced inflaion rae is unobservable bu he invesor may learn abou i from realized inflaion and observed sock and bond prices. The invesor is aware ha his model for he observed inflaion is poenially misspecified and he seeks an invesmen sraegy ha maximizes his expeced uiliy from real erminal wealh and is also robus o inflaion model misspecificaion. We solve he corresponding robus Hamilon-Jacobi-Bellman equaion in closed form and derive and illusrae a number of ineresing properies of he soluion. For example ambiguiy aversion affecs he opimal porfolio hrough he correlaion of price level wih he sock index a bond and he expeced inflaion rae. Furhermore unlike oher seings wih model ambiguiy he opimal porfolio weighs are no always decreasing in he degree of ambiguiy aversion. 1 Inroducion Since he seminal work of Meron numerous sudies have been devoed o he opimal porfolio choice of a risk-averse invesor under various assumpions. The vas majoriy of hese papers including Meron s papers make he unrealisic assumpion ha he probabiliy disribuions of all relevan random quaniies are known by he invesor. Following he ideas of Knigh 191 he experimenal sudies of Ellsberg 1961 and Bossaers e al. 1 show ha individuals are no only averse o risk known probabiliy disribuion bu also averse o ambiguiy unknown probabiliy disribuion. In his paper we solve he problem of a riskand ambiguiy-averse invesor who can inves in a sock index a long-erm nominal bond and in shor-erm deposis cash. Ineres raes and he inflaion rae vary sochasically. The invesor does no observe he expeced inflaion rae and is uncerain abou he correc process for he consumer price level. Using he robus conrol approach of Anderson Hansen and Sargen 3 we derive he opimal invesmen sraegy in closed form compare i wih imporan special cases and illusrae he properies of he opimal porfolio by a numerical example. 1

4 Our paper exends he exising lieraure sudying he impac of inflaion on porfolio choice. Campbell and Viceira 1 and Brennan and Xia among ohers specify he price processes of radable asses in nominal erms as well as he inflaion process and derive and sudy opimal dynamic porfolios of invesors wih consan relaive risk aversion. In hese papers all relevan sae variables are assumed observable and he probabiliy disribuions of all processes are assumed known. Bensoussan Keppo and Sehi 7 assume he invesor observes he consumer price index wih noise and hus he inflaion rae is no fully observed bu an esimae can be filered from observed quaniies. This esimae is hen used for deermining he real wealh and real consumpion. Chou Han and Hung 11 assume ha he price level is fully observable bu ha he expeced inflaion rae is unobservable o invesor. All hese papers disregard model uncerainy. We allow for an unobservable expeced inflaion rae and uncerainy abou he relevan consumer price index. Nex we moivae hese wo model feaures. The inflaion rae is he change in he price level of a baske of consumpion goods. Alhough he price level is direcly observable o he invesor i is reasonable o assume ha he drif of he price level process he expeced inflaion rae is no direcly observed from he prices of consumer goods or financial asses nor from publicaions of macroeconomic saisics. Using he Bayesian approach formalized by Lipser and Shiryaev 1 he invesor learns abou he process for he expeced inflaion rae from observaions of he price level he sock price and he ineres rae if he expeced inflaion is correlaed wih hese variables. We also assume ha he invesor is uncerain abou he correc process o use for he observed inflaion process and wans o derive a porfolio sraegy which is robus o a poenial model misspecificaion. This is a reasonable model feaure because he idenificaion of a paricular inflaion process requires a subsanial amoun of daa see Anderson Hansen and Sargen 3 for a discussion of his issue which migh no be available for he invesor. Firs he official consumer price index may reflec a differen composiion of consumpion goods han ha preferred by he invesor and can herefore be inappropriae for he individual invesor. Secondly he composiion of he baske of goods changes over ime wih weighs varying new goods enering and oher goods leaving he index. Thirdly he official consumer price index may no appropriaely reflec changes in he qualiy of he differen goods see he discussion in Griliches 1961 and Prenice and Yin. We focus on uncerainy abou he inflaion process bu we allow uncerainy abou he inflaion process o spill over ino uncerainy abou he expeced nominal reurn on he sock and he bond. Such an effec would be in line wih he discussions of Uppal and Wang 3 and Vardas and Xepapadeas 1 who argue ha when ambiguiy is relaed o economy-wide facors he preference for robusness is he same for all processes in he model. The invesor has a reference model for he observed inflaion process bu he is also aware of he fac ha oher models migh be a beer represenaion of realiy. As a resul he wans o derive invesmen rules ha are robus o he proposed ype of inflaion model misspecificaion and ha perform reasonably well across a se of plausible models. The discrepancy beween he reference model and alernaive models is defined in erms of relaive enropy which serves as a penaly in he opimizaion procedure. This penaly measures he invesor s uncerainy abou

5 he reference model. Following Anderson Hansen and Sargen 3 he opimal porfolio is obained in closed form afer solving he robus Hamilon-Jacobi-Bellman equaion associaed wih our dynamic decision problem. In he opimal porfolio of our model he ambiguiy aversion parameer is muliplied by various combinaions of he correlaion coefficiens beween he sock bond inflaion and expeced inflaion rae. In paricular if he price level process is no correlaed wih he securiies hen he level of uncerainy abou he inflaion model misspecificaion does no influence he opimal porfolio. This sands in conras wih he resuls of Maenhou 4 Flor and Larsen 11 and Branger Larsen and Munk 1 among ohers where he ambiguiy aversion parameer eners he opimal porfolio independenly. In hese papers he agen is uncerain abou he models for radable asses whereas in our model he ambiguiy is abou he inflaion process which implies ha he securiies can only be used as a hedge agains he inflaion uncerainy if he former are correlaed wih he price level process. We show ha he uncerainy abou he inflaion process affecs he invesor s posiions in boh he sock and he bond. This differs from Flor and Larsen 11 where ambiguiy abou he ineres rae process influences only he opimal bond posiion because he bond is a perfec insrumen for hedging agains he ineres rae risk. When he invesor is ambiguous abou he inflaion process his opimal posiions in boh asses are affeced because neiher he sock nor he bond can perfecly hedge agains he inflaion risk. In our model wih ambiguiy abou he inflaion model a more risk-averse invesor does no necessarily have smaller speculaive componens in his opimal porfolio of he sock and a bond. Alhough in models wih no ambiguiy see for example Sørensen 1999 and Munk and Sørensen 4 he speculaive componens decrease as he risk aversion increases his is generally no so when model uncerainy is inroduced see for example Rohschild and Sigliz 1971 Fishburn and Porer 1976 Meyer and Ormison 1985 Hadar and Seo 199 Gollier In our model he behavior of he speculaive porfolios wih respec o he risk aversion depends on numerous parameer values. However in a special case when he price level process is posiively correlaed wih he sock price and he sock price is negaively correlaed wih he bond hen he speculaive componens are indeed decreasing in risk aversion. The opimal invesmen sraegies wih sochasic ineres raes have been sudied in many papers. Sørensen 1999 and Korn and Kraf 1 provide a soluion for an invesor who can inves in he sock index and a bond in a seing wih he Vasicek 1977 erm srucure. Campbell and Viceira 1 and Brennan and Xia analyze he effec of inflaion on he opimal invesmen sraegy. Koijman Nijman and Werker 11 sudy opimal consumpion and porfolio problem aking ino accoun annuiy risk a reiremen. Van Hemer 5 considers morgages as a par of a homeowner s financial porfolio. In Munk and Sørensen 4 he soluion is obained for non-markovian dynamics of he opporuniy se. Munk and Sørensen 1 solve he problem for he invesor wih sochasic labor income. All processes in hese papers are assumed o be known and all parameers are observable. By allowing boh for learning abou he expeced inflaion rae and for price level model uncerainy our paper combines wo srands of he porfolio choice 3

6 lieraure. Gennoe 1986 Brennan 1998 Lakner 1998 and Bjørk Davis and Landén 1 assume ha he expeced raes of reurn on he risky asses are unobserved. As menioned above Bensoussan Keppo and Sehi 7 and Chou Han and Hung 11 invesmen problems wih parial observabiliy of inflaion process parameers has been sudied in oher papers. On he oher hand several papers assume all parameers and variables are observable bu incorporae model uncerainy ino a porfolio choice problem. Maenhou 4 adaps he general robus conrol framework of Anderson Hansen and Sargen 3 o a dynamic porfolio choice problem wih power uiliy. He considers he simple Meron seing wih a single sock and a riskless asse wih consan invesmen opporuniies and assumes ambiguiy abou he expeced rae of reurn on he sock. In an exension Maenhou 6 invesigaes he role of ambiguiy aversion when he expeced sock reurn varies over ime following an Ornsein-Uhlenbeck process. Liu 1 exends ha analysis o Epsein-Zin preferences. Flor and Larsen 11 solve he opimal invesmen problem when he invesor is ambiguous abou he models for he ineres rae and he sock. These papers assume ha all parameers and sae variables are observable. Finally wo recen papers sudy porfolio choice models involving boh unobservabiliy and ambiguiy as we do. Liu 11 considers a model wih a regime-swiching expeced sock reurn wih he curren regime being unobservable. Branger Larsen and Munk 1 exend he model of Maenhou 6 o he case where he expeced sock reurn also has an unobservable componen and he invesor learns abou his componen based on observed sock reurns and he observable componen of he expeced sock reurn. We focus on unobservabiliy and ambiguiy relaed o inflaion insead of he expeced sock reurn. This paper if organized as follows. In Secion we formulae he porfolio choice problem. In Secion 3 we provide he opimal soluion discuss i and compare i wih opimal soluions o oher relevan models. In Secion 4 we analyze he opimal porfolios in a numerical example based on esimaes of all model parameers. In paricular o deermine a reasonable range for he ambiguiy aversion parameer we compue he so-called deecion-error probabiliies. The proofs of some resuls are given in he Appendix. Mahemaical Formulaion Le Ω F P be a complee probabiliy space wih a righ-coninuous filraion {F s } s [T ]. All sochasic processes inroduced below are defined on his probabiliy space. We consider an invesor who can rade in a sock index zero-coupon bonds and a money marke accoun cash. According o he Vasicek 1977 model he nominal shor-erm ineres rae follows an Ornsein-Uhlenbeck process dr = κ r r d σ r db P.1 where κ > is he degree of mean reversion r > is he long-run mean of he ineres rae σ r > is he ineres rae volailiy and B P is a sandard Brownian 4

7 moion. Le q r be he marke price of ineres rae risk which is assumed o be consan. Wih his dynamics for he shor-erm ineres rae he price P of a nominal zero-coupon bond paying one uni of accoun a ime T is given by where he funcions a and b κ are ax = r + σ rq r κ P = e a T b κ T r b κ x = 1 κ 1 e κx. σ r κ x b κ x + σ r 4κ b κx. From Io s lemma he dynamics of he price of such a bond is dp = P r + qd + σ P db P where q = q r σ P is he expeced excess reurn on he bond and σ P = σ r b κ T is he bond price volailiy. 1 Noe ha because ineres raes are driven by a one-facor model an unconsrained invesor would no benefi from rading in more han one bond and he invesor can obain exacly he same uiliy no maer which bond he rades in. The agen can also inves in a sock index wih nominal price S modeled by ds = S r + αd + σ S db S.3 where he posiive consans α and σ S are he expeced excess reurn and volailiy respecively and B S is a sandard Brownian moion. Le X denoe he nominal value of he invesor s porfolio a ime. The evoluion of he porfolio value is dx = r X Θ S Θ P d + Θ S ds S + Θ P dp P = X r + απ S + qπ P d + σ S Π S db S + σ P Π P db P where Θ S and Θ P represen he amouns of wealh invesed in he sock and he bond respecively. Equivalenly Π S and Π P represen he fracions of wealh invesed in he sock and he bond respecively so ha 1 Π S Π P is he fracion of wealh invesed in he bank accoun ha provides a reurn given by he shor-erm ineres rae. Thus he conrol sraegy is represened by Π S Π P. Le Z be he price level of he consumpion good or a baske of consumpion goods. Define he price level process dz = β Z d + σ Z Z db Z.4 1 For simpliciy of noaion we suppress he dependence of σ P on T. 5

8 wih he unobserved drif parameer he expeced inflaion rae given by dβ = λ β β d + σ β db β.5 where λ > is he degree of mean reversion β > is he long-run mean of he expeced inflaion rae σ Z is he price level process volailiy σ β is he volailiy of he expeced inflaion rae and B Z B β are sandard Brownian moions. The Brownian moions B Z B S B P B β are assumed o be correlaed wih he correlaion marix 1 ρ ZS ρ ZP ρ Zβ ρ ZS 1 ρ SP ρ Sβ ρ ZP ρ SP 1 ρ P β. ρ Zβ ρ Sβ ρ P β 1 We assume ha he correlaion coefficiens ake values in he inerval 1 1. The process.4 and Ornsein-Uhlenbeck process.5 are quie common in modeling he price level and he expeced inflaion rae respecively see for example Brennan and Xia and Bensoussan Keppo and Sehi 9. However in Brennan and Xia he expeced inflaion rae is assumed o be observable whereas in Bensoussan Keppo and Sehi 9 he expeced inflaion rae is consan bu he price level is assumed o be unobservable. Since he Brownian moions B Z B S B P B β are assumed o be correlaed he invesor can obain an esimae ˆβ of he unobserved expeced inflaion rae β based on he observed processes S Z r using Bayesian learning. According o Lipser and Shiryaev he Equaions and.5 as observed by he invesor are see Appendix A dz = ˆβ Z d + Z σ Z d ˆB Z.6 ds = S r + αd + σ S ρ ZS d ˆB Z + 1 ρ ZS d ˆB S.7 dr = κ r r d σ r R 3 d ˆB Z + R 4 d ˆB S + R 5 d ˆB P.8 d ˆβ = λ β ˆβ d + A Z σ Z d ˆB Z + A S σ S d ˆB S + A P σ P d ˆB P.9 where ˆB Z ˆB S ˆB P T is an F SZr -adaped Brownian moion wih he filraion F SZr = σ{s τ Z τ r τ τ } and A P = mr 1R 4 R R 3 + σ Z σ β R R 5 R 8 A S = R 1m + σ Z σ β R R 7 σ P σ Z R R 5 σ S σ Z R A Z = σ Zσ β R 6 + m R σz 1 = ρ ZS R = 1 ρ ZS R 3 = ρ ZP R 4 = ρ SP ρ ZS ρ ZP R 1 ρ 5 = ZS 1 R 3 R 4 R 6 = ρ Zβ R 7 = ρ Sβ ρ ZS ρ Zβ 1 ρ ZS R 8 = ρ P β R 3 R 6 R 4 R 7 R 9 = 1 R6 R7 R R 8. 5 Here m is he limi value as of he deerminisic variance given by m = E[β ˆβ F SZr ] and i can be shown ha m = K + K 4 K 1 K3 K 1 6

9 where K 1 = R R 5 + R 1 R 5 + R 1 R 4 R R 3 σ Z R R 5 K = λ + σ Z σ β ρ Zβ R R 5 R 1 R R 5R 7 + R 1 R 4 R R 3 R R 5 R 8 K 3 = σ βρ Zβ + R 7 + R 8 σ β. σ Z R R 5 We assume ha learning was long enough and ake he variance o be equal o m. Equaions.6.9 consiue he reference model of he invesor. As i was menioned in he Inroducion our invesor is uncerain abou he probabiliy disribuion for he observed processes.6.9. In oher words he realizes ha he reference model is only an approximaion of realiy and he wans o consider a se of plausible alernaive models which we now specify. Le e be an F SZr -progressively measurable process valued in R and define he Radon- Nikodým derivaive process [ dp ξ e e = E dp = exp F SZr ] 1 + k S + kp e s ds e s d ˆB s Z + k S d ˆB s S + k P d ˆB s P where k S and k P are consans. According o Girsanov s heorem he process e sds + ˆB Z B Z B S B P = k S e sds + ˆB S k P e sds + ˆB P.1 is a Brownian moion wih respec o probabiliy measure P e. According o his model misspecificaion we rewrie he equaions for he wealh X he price level process Z he shor-erm ineres rae r and he esimae of expeced inflaion ˆβ in he form dx = X r + απ S + qπ P σ S Π S a 1 + σ P Π P a e d + K 1 d B Z + K d B S + K 3 d B P.11 dz = Z ˆβ σ Z e d + σ Z d B Z.1 dr = κ r r a 3 e d σ r R 3 d B Z σ r R 4 d B S σ r R 5 d B P.13 d ˆβ = λ β ˆβ a 4 e d + A Z σ Z d B Z + A S σ S d B S + A P σ P d B P.14 where for simpliciy we inroduced he following noaion a 1 = R 1 + k S R a = R 3 +k S R 4 +k P R 5 a 3 = σ r a a 4 = A Z σ Z +k S A S σ S +k P A P σ P K 1 = σ S Π S X R 1 + σ P Π P X R 3 K = σ S Π S X R + σ P Π P X R 4 and K 3 = σ P Π P X R 5. These equaions The same assumpion was made by Scheinkman and Xiong 3 Dumas Kurshev and Uppal 9 and Branger Larsen and Munk 1. 7

10 represen alernaive models indexed by he process e. The invesor is uncerain abou which model from he se is he rue model and wans o derive robus invesmen rules ha work reasonably well for all hese models. Since he processes in.6.9 are assumed o be correlaed he ambiguiy abou he inflaion migh ranslae ino he inflaion-specific ambiguiy abou he oher processes. The consans k S and k P deermine wheher he price level uncerainy influences he sock price and ineres rae processes. In paricular we have [ ds = S r + α σ S R 1 + k S R e d + σ S R 1 d B Z + R d B ] S. Thus if k S = ρ ZS / 1 ρ ZS hen here is no ambiguiy abou he sock price a 1 =. If in addiion k P = R 1R 4 R R 3 R R 5 hen here is also no uncerainy abou he ineres rae process a =. Any oher values of k S and k P imply ha uncerainy abou he price level spills over ino uncerainy abou he expeced nominal sock reurn and abou he expeced nominal bond reurn. This seing is similar o Uppal and Wang and Vardas and Xepapadeas 1 where he cases wih equal and differen componen perurbaions o a Brownian moion are considered. Here equal perurbaions mean ha ambiguiy is relaed o economy-wide facors and hus he preference for robusness is he same for all processes. We consider an agen wih CRRA consan relaive risk aversion uiliy who wans o derive an invesmen sraegy for he ime inerval [ T ] in order o maximize he expeced uiliy from real erminal wealh X T /Z T. Le us denoe he sae variables by y x z r ˆβ and he opimal invesmen sraegy by Π Π S Π P. Therefore we define he reward funcional realized when choosing an alernaive model specified by e as w e y Π = 1 [ XT 1 γ] 1 γ EPe y.15 Z T and he value funcion as v y = sup inf Π U[T ] e E[T ] [ T w e y Π + Ey Pe e ] s Ψs Y s ds where he parameer γ > γ 1 is he consan relaive risk aversion and T e s Ψs X s Z s r s ˆβ s ds.16 is he penaly erm for deviaing from he reference model. 3 To obain wealhindependen opimal porfolio weighs and also for analyical racabiliy we follow θ Maenhou 4 by assuming ha Ψ y = where θ > is called he 1 γvy ambiguiy aversion parameer. 4 A large value of Ψ corresponds o a small penaly which means ha he invesor is more uncerain abou he model. 3 To simplify he noaion we wrie Π insead of {Π S s Π P s } s [T ] and e insead of {e s } s [T ]. The expecaion operaor wih respec o he probabiliy measure P e is defined as Ey[ ] Pe E Pe [ X = x Z = z r = r ˆβ = ˆβ]. 4 For a criique of his approach see Pahak. 8

11 We define he space U of admissible sraegies {Π s } s [T ] aking values in R as sraegies ha saisfy he following condiions 1. Π : [ T ] Ω R is an F SZr -progressively measurable process;. Under Π for any x he wealh equaion.11 admis a unique srong soluion; 3. The inegrabiliy condiions necessary for he expecaion operaor in.15 o be well defined are saisfied; 4. X a.s. [ T ]. The space E[ T ] is defined o be he space of F SZr -progressively measurable processes e such ha he process.1 is a Radon-Nikodým derivaive. 3 Soluion The problem.16 is difficul o solve direcly. We derive and solve a corresponding highly non-linear second-order parial differenial equaion ha he value funcion v y should saisfy he so-called robus Hamilon-Jacobi-Bellman HJB equaion see Anderson Hansen and Sargen 3. Le π = π S π P be he vecor of fracions of wealh invesed a ime [ T ] in he sock π S and he bond π P hen he corresponding robus HJB equaion is sup inf π R e R { v + z ˆβ σz e v z + x r + απ S + qπ P [σ S π S a 1 + σ P π P a ]e v x + κ r r a 3 e v r + λ β ˆβ a 4 e v ˆβ + 1 zσ Z v zz + σ Z xz σ S π S ρ ZS + σ P π P ρ ZP vzx σ Z σ r zρ ZP v zr + σ ZA Z zv z ˆβ + 1 x σ S π S + σ P π P + σ S σ P π S π P ρ SP vxx σ r x σ P π P + σ S π S ρ SP vxr + σ β x σ S π S ρ Sβ + σ P π P ρ P β vx ˆβ + 1 σ rv rr σ r σ β ρ P β v r ˆβ + 1 AZ σ Z + A S σ S + A P σ P } v ˆβ ˆβ + e =. Ψ We assume ha he value funcion is sufficienly smooh and ha he HJB equaion admis a classical soluion. Proposiion 3.1. The soluion o problem.16 is of he form v x z r ˆβ = 1 x 1 γh r ˆβ. 1 γ z The funcion h r ˆβ is given by h r ˆβ = exp 1 γb λ T ˆβ + 1 γb κ T r + c where b κ is defined in. b λ is defined similarly and he funcion c solves he ordinary differenial equaion B.5 in Appendix B. The wors-case shock is e = θ σ S π S a 1 + σ P π P a σ Z + a 3 b κ T a 4 b λ T 3.1 9

12 and he opimal invesmens in he sock and he bond respecively are where π S spec = 1 σ S K π1 S = πspec S + πinfl S + πs raeb κ T + πunobs S b λt π1 P = πspec P + πinfl P + πp raeb κ T + πunobs P b λt α γ + θa q γρ SP + θa 1 a γσ S γσ P πinfl S = σ Zθa 1 1 γρ ZS γ + θa σ Z θa 1 γρ ZP γρ SP + θa 1 a γσ S K πrae S = θσ ra a 1 a ρ SP π S unobs = 1 γσ S K π P spec = 1 σ P K γσ S K [ θγa 4 a 1 a ρ SP σ β 1 γ γ + θa ρ ] Sβ γρ SP + θa 1 a ρ P β q γ + θa 1 α γρ SP + θa 1 a γσ P γσ S πinfl P = σ Zθa 1 γρ ZP γ + θa 1 σ Z θa 1 1 γρ ZS γρ SP + θa 1 a γσ P K πrae P = θσ ra a a 1 ρ SP + σ rγ 1 γσ P K γσ P πunobs P = 1 [ γθa 4 a a 1 ρ SP γσ P K σ β 1 γ γ + θa 1ρ ] P β γρ SP + θa 1 a ρ Sβ and Proof. See Appendix B. K = γ1 ρ SP + θa 1 + a a 1 a ρ SP. We analyze he porfolio given in Proposiion 3.1 for he case when he invesor s uncerainy abou he price level also means ha he is ambiguous abou he sock price and he ineres rae process namely we assume k S = k P = and hus a 1 = ρ ZS and a = ρ ZP. A similar analysis for he case when he agen is uncerain abou he inflaion only which means ha a 1 = a = follows easily. The opimal wealh allocaion o he available securiies consiss of four componens. Firs we discuss he speculaive componens π S spec and π P spec which involve weighed combinaions of expeced excess reurns on he sock α and he bond q. Similarly o oher models wih sochasic ineres raes Sørensen 1999 Korn and Kraf 1 Flor and Larsen 11 among ohers we also have ha if he expeced excess reurn on he sock increases hen he sock becomes more aracive which corresponds o he increase in π S spec. On he oher hand he increase in he expeced excess reurn on he bond makes he bond more aracive for he invesor and he value of π P spec becomes larger. Anoher difference beween he speculaive 1

13 componens is ha πspec S is consan because q = q r σ P see Secion whereas πspec P is ime-dependen σ P = σ r b T. The erms πinfl S and πp infl represen he hedge agains he inflaion risk. The invesor includes his hedge in he porfolio o proec he real value of his wealh. Ineresingly even if he sock bond is no correlaed wih he inflaion i sill can be used as he hedge if i is correlaed wih he bond sock which in urn is correlaed wih he inflaion. On he oher hand hese erms vanish if he available securiies canno be used o hedge agains he inflaion risk for he sock ρ ZS = ρ ZP = or ρ ZS = ρ SP = and for he bond ρ ZP = ρ ZS = or ρ ZP = ρ SP =. The erms also vanish if he inflaion is locally deerminisic σ Z = and if ρ ZS ρ ZP ρ SP = for he sock and ρ ZP ρ ZS ρ SP = for he bond. 5 This propery of he opimal porfolio is similar o Bensoussan Keppo and Sehi 9 where he opimal porfolio includes he hedge agains inflaion risk if he sock price is correlaed wih he inflaion. If he bond price is correlaed wih he inflaion which is usually he case he uncerainy abou he laer inroduces an exra erm πrae S in π S and an addiional erm in he componen πrae. P These erms vanish if here is no ambiguiy θ = or he ineres rae is locally deerminisic σ r = or ρ ZP ρ ZS ρ SP =. This is in conras wih Flor and Larsen 11 where sock price and ineres rae no inflaion model ambiguiy does no inroduce addiional erms o he sock invesmen. I should also be poined ou ha he influence of hese componens on he opimal porfolio decreases o zero when he invesmen horizon T approaches zero. The erms πunobs S and πp unobs arise from unobservabiliy of he sochasic expeced inflaion rae. These componens appear because he expeced inflaion rae is assumed o be sochasic and unobservable. The erms disappear if he expeced inflaion rae is deerminisic. The presence of erms ha hedge agains changes in unobserved parameers is common for he porfolio choice problems see for example Lakner 1998 Bjørk Davis and Landen 1 Branger Larsen and Munk 1. In conras o Maenhou 6 where he ambiguiy aversion parameer is simply added o he risk aversion parameer he ambiguiy aversion parameer θ in our model is muliplied by various combinaions of he correlaion coefficiens ρ ZS ρ ZP and ρ SP. In paricular if he inflaion is uncorrelaed wih he risky asses ρ ZS = ρ ZP = hen he model uncerainy does no influence he opimal porfolio because he securiies canno be used in hedging agains he inflaion model misspecificaion. The same explanaion holds for he case when a 1 = a = which means ha he price level ambiguiy does no ranslae ino he uncerainy abou he sock price and he ineres rae. Nex we compare he opimal porfolio given in Proposiion 3.1 wih soluions o similar invesmen problems. To make he paper self-conained we briefly describe each model and provide he corresponding soluions. The following porfolios are opimal for he invesor who wans o maximize he expeced uiliy of: erminal wealh Sørensen 1999 Korn and Kraf 1. All variables sock bond and ineres rae are assumed observable wih known dynamics. The 5 The hedge agains he inflaion risk is also zero if θ = 1 γ bu since empirical sudies suppor γ > 1 and θ has o be posiive his is unlikely o be he case. 11

14 model is implicily saed in real erms as inflaion is no modeled. The opimal porfolio is π S = ασ P qσ S ρ SP γσs σ P 1 ρ SP π P = qσ S ασ P ρ SP γσp σ S1 ρ SP }{{ + γ 1 b T. γ bt }}{{} speculaive hedge The opimal sock invesmen π S is represened by he speculaive componen only. On he oher hand he proporion π P of wealh invesed in he bond consiss of boh a speculaive and an ineres rae hedge componen. erminal wealh wih sock price model ambiguiy Flor and Larsen 11. All variables are observable inflaion is no modeled. The invesor in uncerain abou he drif of he sock price wih associaed ambiguiy aversion parameer θ S. The opimal porfolio weighs in he sock and he bond respecively are π3 S ασ P qσ S ρ SP = γ + θ S σs σ P 1 ρ SP qσ S γρ π3 P SP + γ + θ S1 ρ SP αγσ P ρ SP = γγ + θ S σp σ S1 ρ SP }{{ + γ 1 b T. γ bt }}{{} speculaive hedge The inroduced uncerainy abou he sock price process alers he speculaive componens of π3 S and π3 P. I also follows ha he componen of π3 P ha hedges he ineres rae risk does no change when he uncerainy is inroduced. erminal wealh wih bond price model ambiguiy Flor and Larsen 11. All variables are observable inflaion is no modeled. The invesor in uncerain abou he drif of he bond price wih associaed ambiguiy aversion parameer θ P. Then he opimal invesmen sraegy is π4 S = ασ P qσ S ρ SP γσs σ P 1 ρ SP qσ S γ + θ P ρ π4 P SP + γ1 ρ SP αγ + θ P σ P ρ SP = γγ + θ P σp σ S1 ρ SP }{{ } speculaive + γ 1 b T + θ P. γ + θ P bt γ + θ P }{{} hedge In conras wih he previous model sock price process ambiguiy he uncerainy abou he bond price process influences only he componens of π4 P he opimal wealh allocaion in he bond. The opimal invesmen in he sock π4 S is he same as π S. Comparison of π3 P and π4 P shows ha he hedge componen of π4 P hedges boh he ineres rae risk and he model uncerainy; see Flor and Larsen 11 for a discussion of his issue. 1

15 real erminal wealh wih inflaion model ambiguiy. The invesor is uncerain abou he price level dynamics bu can observe he expeced inflaion rae. This is our model excep ha he expeced inflaion rae in our model is assumed unobservable. The soluion is he same as in Proposiion 3.1 bu wih m = which in urn means ha only componens π S unobs and πp unobs change. 6 This also implies ha hese componens do no depend on he price level process volailiy σ Z. real erminal wealh wih no ambiguiy bu an unobserved sochasic expeced inflaion rae. This is our model wihou ambiguiy. The soluion is he same as in Proposiion 3.1 bu wih θ =. A comparison of he models shows ha differen sources of ambiguiy influence differen componens in he opimal porfolio. Since only bonds are used in hedging he ineres rae risk he ambiguiy abou he sock price process does no influence he hedge compare π P and π3 P. In our model if he radable asses are no correlaed wih he inflaion hey canno be used in hedging agains he inflaion and herefore he uncerainy abou he price level process has no effec on he opimal porfolio. On he oher hand he ambiguiy abou he bond price process adds an exra erm o he bond porfolio no o he sock porfolio and his erm represens he hedge agains he model uncerainy see π4 P. If he uncerainy is abou he price level process hen exra erms appear in boh he sock porfolio and he bond porfolio see π1 S and π1 P. As i is usually he case he speculaive componen of he opimal porfolio decreases as he invesor s risk aversion increases see for example he models in Sørensen 1999 Munk and Sørensen 4 Flor and Larsen 11 among ohers. On he oher hand pessimisic deerioraions in beliefs do no necessarily decrease he demand for he risky asses. 7 Since in our model he invesor chooses he financial sraegy ha is opimal under he wors-case probabiliy disribuion for he inflaion process he speculaive demand for he risky asses does no necessarily decrease when he invesor s risk aversion increases. In paricular if he price level process is misspecified hen dπs spec < and dπp spec < are equivalen o he following wo dγ dγ values being posiive α γ 1 ρ SP q ρ SP σ S σ P q γ 1 ρ SP α ρ SP + θρ ZS γ1 ρ SP + K σ P σ S α + θρ ZP γ1 ρ SP + K ρ ZP q σ S ρ ZS σ P q ρ ZP α ρ ZS σ P σ S respecively where K is defined in Proposiion 3.1. Therefore he behavior of he speculaive porfolios π S spec and π P spec as funcions of he ambiguiy aversion parameer γ depend on parameer values. 6 To obain he opimal porfolio one should append fourh column σ β R 9 T o Λ use σ β R 6 σ β R 7 σ β R 8 σ β R 9 as he fourh row and consider Brownian moion W Z W S W P W β T insead of ˆB Z ˆB S ˆB P T. In he HJB equaion his change is equivalen o seing m = and using σ 44 = σβ. As far as he opimal porfolio is concerned his change is he same as aking he variance m of ˆβ o be zero. 7 See Rohschild and Sigliz 1971 Fishburn and Porer 1976 Meyer and Ormison 1985 Hadar and Seo 199 Gollier 1995 and he references in hese papers. 13

16 In our model assuming ha he Sharpe raio of he sock α σ S is greaer han ha of he bond q σ P he speculaive porfolio πspec S decreases when γ increases if he inflaion is posiively correlaed wih he sock price or more generally if α/σ S q/σ P > ρ ZS ρ ZP. Similarly condiions ha ensure ha he speculaive demand in he bond is a decreasing funcion of he ambiguiy aversion parameer γ can be deduced. 4 Numerical Example Since he opimal robus porfolio and he corresponding wors-case model depend on he preference parameer θ some ools of is esimaion are necessary. 8 We assume ha he invesor has measuremens of S Z ˆβ over some finie ime inerval of lengh N. As suggesed by Anderson Hansen and Sargen 3 he parameer θ should be chosen in such a way ha he approximaing model and he worscase model are sufficienly similar which makes i difficul for he invesor o use a likelihood raio es in choosing eiher model based on he ime series of lengh N. 4.1 Deecion-Error Probabiliies In his secion we follow he procedure suggesed by Maenhou 6 namely we apply Fourier inversion o find he deecion-error probabiliy ε N θ which is hen used o deermine how similar he reference and he wors-case models are. Anderson Hansen and Sargen 3 sugges using θ such ha ε N θ is no less han.1. This choice will make i difficul for he robus invesor o disinguish he wo models saisically. The wors-case model considered by he robus invesor is given by wih e = e where e is defined in 3.1. Define he Radon-Nikodým derivaives Ξ 1 E P[ ] dp e dp F SZr and Ξ E Pe [ ] dp F SZr dp e and consider he logarihm of hese derivaives ξ 1 ln Ξ 1 = e sd ˆB s Z + k S e sd ˆB S s + k P e sd ˆB P s 1 + k S + k P e s ds ξ ln Ξ = e sd ˆB s Z + k S e sd ˆB s S + k P e sd ˆB s P k S + k P e s ds. Based on he sample wih size N he decision maker will discard he reference model misakenly for he wors-case model if ξ 1N >. On he oher hand if he wors-case model is rue hen i will be rejeced erroneously if ξ N > or ξ 1N <. According o his we define he deecion error probabiliy ε N θ = 1 Prξ 1N > P F + 1 Prξ 1N < P e F. 8 Noe ha he ambiguiy aversion parameer depends on he precise model se-up and source of ambiguiy and herefore has o be esimaed on a case-by-case basis. 14

17 I can hen be shown ha see Appendix C N ε N θ = 1 1 erf K where K = 1 e s ds and erfx = x e d. Noe ha we obained a π closed-form expression for he deecion-error probabiliy in conras o Maenhou 6 and Branger Larsen and Munk 1 who rely on numerical echniques of solving differenial equaions. 4. Model Parameers For he numerical analysis of our model we apply he parameers esimaed by Brennan and Xia from a ime series for 5 years. For concreeness we assume ha he bond he invesor rades in a any dae is a zero-coupon bond mauring 1 years laer. This implies a consan bond price volailiy σ P = σ r b κ 1. The assumed parameer values are shown in Table 1. 9 Table 1: Parameer values in our numerical example. σ S σ Z σ β σ r σ P λ β qr κ α ρ SZ ρ Sβ ρ ZP ρ SP ρ P β ρ Zβ.6.54 ± ±.3 In Table we presen he deecion-error probabiliies for differen values of he risk aversion parameer γ and he ambiguiy aversion parameer θ for N = 5 years and T = 1 years. I can be shown ha ε N θ < regardless of he parameer values N so ha he deecion-error probabiliy decreases when he daa sample increases. Furhermore lim ε Nθ =. In he following example we choose γ = 4 and θ = 5. N Wih his choice of θ he deecion-error probabiliy is greaer han.1. 9 Since Brennan and Xia esimae he parameers for real ineres raes we accordingly adjus heir parameers o be applicable in our model. Alhough ρ SZ and ρ Zβ were no esimaed by Brennan and Xia we perform he analysis for heir values equal o.3 and.3. These values of ρ SZ were used in Bensoussan Keppo and Sehi 9. Esimaions of Fama and Schwer 1977 Gulekin 1983 Ferson and Harvey 1991 and Moerman and van Dijk 1 also show ha hese correlaion coefficiens can be quie differen. Similarly we ake ρ ZP o be equal o -.3 because bond prices are negaively correlaed wih inflaion. 15

18 Table : Deecion-error probabiliies ε N θ for differen values of γ and θ for N = 5 years and T = 1 years. θ γ = γ = γ = γ = Opimal Porfolios Nex we analyze he opimal porfolios π1 S π1 P and heir componens. To beer undersand he influence of he unobservabiliy of he expeced inflaion rae and ambiguiy abou he inflaion process he opimal porfolios are compared wih he following special cases provided in Secion 3: he expeced inflaion rae is observed; here is no ambiguiy abou he price level process. We also discuss he influence of he ambiguiy aversion parameer θ and he risk aversion parameer γ on he opimal porfolio. Figure 1 illusraes he opimal porfolios π1 S and π1 P wih he corresponding componens given in Proposiion 3.1. The figure shows ha if he sock price is posiively correlaed wih he inflaion ρ ZS =.3 he invesor should decrease his opimal sock holdings as his invesmen horizon T decreases which is in line wih ypical invesmen advice. On he oher hand if ρ ZS =.3 he sock becomes more aracive for he invesor as T decreases. The mos influenial ime-varying componen in his sock porfolio is πunobs S b λt ha adjuss he opimal porfolio due o unobservabiliy of he sochasic expeced inflaion rae. I is worh poining ou ha he speculaive componen is he larges in he porfolio. The opimal bond posiion is an increasing funcion of he invesmen horizon T boh when ρ ZS =.3 and when ρ ZS =.3. The componens πraeb P κ T and πunobs P b λt of he porfolio significanly adjus he opimal wealh allocaion in he bond. However his influence weakens over ime because as i was poined ou in Secion 3 hese componens decrease o zero as he remaining invesmen horizon shorens. As a resul he opimal bond posiion changes from long o shor as he invesmen horizon becomes smaller. Comparing he opimal sock and bond porfolios we see ha he ime-varying componens have more effec on he bond holdings han on he sock holdings. Ineresingly he speculaive componens for he sock and he bond porfolios are of opposie sign regardless of he correlaion beween he sock and he inflaion. Figure shows he opimal porfolios π1 S and π1 P when he expeced inflaion rae is observed by he invesor. As i was poined ou in Secion 3 only he componens πunobs S and πp unobs are affeced by he change in he assumpion. Compared o he unobservable case he invesor should inves less more in he sock if ρ ZS =.3 ρ ZS =.3. A he same ime he opimal invesmen in he bond is smaller when 16

19 a Opimal sock porfolio and is componens ρ SZ =.3 b Opimal bond porfolio and is componens ρ SZ =.3 c Opimal sock porfolio and is componens ρ SZ =.3 d Opimal bond porfolio and is componens ρ SZ =.3 Figure 1: Opimal wealh allocaion π1 S in he sock green line on he lef plos wih is componens and opimal wealh allocaion π1 P in he bond green line on he righ plos wih is componens. The op plos are for ρ SZ =.3 and he boom plos are for ρ SZ =.3. Red line is he speculaive componen. Blue line is he hedge agains he price level process Z. Dashed blue line represens he model ambiguiy adjusmen. Dashed red line represens he componen ha arises from unobservable sochasic expeced inflaion rae β. 17

20 a Opimal sock porfolio and is componens ρ SZ =.3 b Opimal bond porfolio and is componens ρ SZ =.3 c Opimal sock porfolio and is componens ρ SZ =.3 d Opimal bond porfolio and is componens ρ SZ =.3 Figure : Opimal wealh allocaion π1 S in he sock green line on he lef plos wih is componens and opimal wealh allocaion π1 P in he bond green line on he righ plos wih is componens when he expeced inflaion rae is observable. The op plos are for ρ SZ =.3 and he boom plos are for ρ SZ =.3. Red line is he speculaive componen. Blue line is he hedge agains he price level process Z. Dashed blue line represens he model ambiguiy adjusmen. Dashed red line represens he componen ha arises from sochasic expeced inflaion rae β. he expeced inflaion rae is observed. The res of he analysis of Figure is similar o ha done for Figure 1. The opimal porfolios for he invesor wih no ambiguiy abou he price level process are shown in Figure 3. The behavior of he opimal porfolios and he corresponding componens in Figure 3 is similar o ha in Figure 1 and Figure so ha he above discussion applies. However since he componen π S rae in he opimal sock porfolio becomes zero his porfolio is heavily dominaed by he speculaive componen. The corresponding componen π P rae in he bond porfolio is no zero because i includes he hedge agains he ineres rae risk ha does no vanish when he invesor is ambiguous abou he inflaion. For he ease of exposiion Figure 4 shows he opimal porfolios for an invesor who is ambiguous abou he price level and does no observe he expeced inflaion rae Figure 1; 18

21 a Opimal sock porfolio and is componens ρ SZ =.3 b Opimal bond porfolio and is componens ρ SZ =.3 c Opimal sock porfolio and is componens ρ SZ =.3 d Opimal bond porfolio and is componens ρ SZ =.3 Figure 3: Opimal wealh allocaion π1 S in he sock green line on he lef plos wih is componens and opimal wealh allocaion π1 P in he bond green line on he righ plos wih is componens when here is no ambiguiy abou he inflaion process. The op plos are for ρ SZ =.3 and he boom plos are for ρ SZ =.3. Red line is he speculaive componen. Blue line is he hedge agains he price level process Z. Dashed blue line represens he model ambiguiy adjusmen. Dashed red line represens he componen ha arises from unobservable sochasic expeced inflaion rae β. 19

22 ambiguous abou he price level and observes he expeced inflaion rae Figure ; no ambiguous abou he price level and does no observe he expeced inflaion rae Figure 3. I is clear from he figure ha an ambiguiy-averse invesor who does no observe he expeced inflaion rae invess more in he bond compared o an invesor who eiher observes he expeced inflaion rae or is no ambiguous abou he price level. In his seing he opimal bond invesmen is he smalles when he invesor is no ambiguous abou he price level. Ineresingly his is rue for boh ρ SZ =.3 and ρ SZ =.3. On he oher hand a change in he correlaion ρ SZ also changes he aiude of an ambiguous invesor oward he sock invesmen making him inves less more in he sock over ime when ρ SZ =.3 ρ SZ =.3. However if he invesor is no ambiguous abou he price level process hen he decreases his sock holdings wih ime for boh values of he correlaion. Figure 5 illusraes how he opimal porfolios π1 S and π1 P depend on he ambiguiy aversion parameer θ. As one can see from he figure he more he invesor is ambiguiy-averse he more he invess in he bond. On he oher hand higher values of he ambiguiy aversion parameer lead o higher lower values of he sock invesmen if ρ ZS =.3 ρ ZS =.3. Noe ha some of he componens of he opimal porfolios are increasing in θ and oher componens are decreasing in θ. This is in conras o Maenhou 6 where he ambiguiy aversion parameer is simply added o he risk aversion parameer and hus he opimal porfolio is decreasing in θ. On he oher hand our findings are similar o he model of Flor and Larsen 11 in which he speculaive componen of he bond porfolio decreases in θ if he invesor is ambiguous abou he bond price dynamics only and increases in θ if he invesor is uncerain abou he sock price process only. Figure 6 shows he opimal porfolios π1 S and π1 P as funcions of he risk aversion parameer γ. As discussed in Secion 3 he impac of γ on he speculaive componens of he opimal porfolios depend on parameer values. I follows from he figure ha he same applies for he oal opimal porfolios. The more risk-averse invesor invess less in he sock and more in he bond if he correlaion beween he sock price and inflaion is ρ ZS =.3. On he oher hand if ρ ZS =.9 hen he more risk-averse invesor invess more in he sock.

23 a Opimal sock porfolios ρ SZ =.3 b Opimal bond porfolios ρ SZ =.3 c Opimal sock porfolios ρ SZ =.3 d Opimal bond porfolios ρ SZ =.3 Figure 4: Opimal wealh allocaion π1 S in he sock lef plos and opimal wealh allocaion π1 P in he bond righ plos under differen assumpions on he expeced inflaion rae and ambiguiy. The op plos are for ρ SZ =.3 and he boom plos are for ρ SZ =.3. Green line is he opimal porfolio when he expeced inflaion rae is unobserved and here is ambiguiy abou he price level process. Red line is he opimal porfolio when he expeced inflaion rae is observable and here is ambiguiy abou he price level process. Blue line is he opimal porfolio when he expeced inflaion rae is unobservable and here is no ambiguiy abou he price level process. 1

24 a Opimal sock porfolio and is componens ρ SZ =.3 b Opimal bond porfolio and is componens ρ SZ =.3 c Opimal sock porfolio and is componensρ SZ =.3 d Opimal bond porfolio and is componens ρ SZ =.3 Figure 5: Opimal wealh allocaion π1 S in he sock green line on he lef plos wih is componens and opimal wealh allocaion π1 P in he bond green line on he righ plos wih is componens as funcions of he ambiguiy aversion parameer θ. The value of he invesmen horizon T is 1 years. The op plos are for ρ SZ =.3 and he boom plos are for ρ SZ =.3. Red line is he speculaive componen. Blue line is he hedge agains he price level process Z. Dashed blue line represens he model ambiguiy adjusmen. Dashed red line represens he componen ha arises from unobservable sochasic expeced inflaion rae β.

25 a Opimal sock and opimal bond porfolios ρ SZ =.3 b Opimal sock and opimal bond porfolios ρ SZ =.9 Figure 6: Opimal wealh allocaion π1 S in he sock green line and opimal wealh allocaion π1 P in he bond red line as funcions of he risk aversion parameer γ. The value of he invesmen horizon T is 1 years. The lef plo is for ρ SZ =.3 and he righ plo is for ρ SZ =.9. 5 Conclusion In his paper we solve he problem of opimal porfolio choice under he assumpions ha he invesor is ambiguous abou he price level process and ha he expeced inflaion rae is unobservable in a seing wih sochasic ineres raes. The opimal wealh allocaion in he sock index and a zero-coupon bond is obained in closed form. We show ha he influence of he ambiguiy aversion parameer on he opimal porfolio depends on he correlaion beween he sae variables. The uncerainy abou he price level process influences he opimal posiions in boh he sock index and he bond. We also show ha when here is ambiguiy abou he model for he inflaion process he more risk-averse invesor does no necessarily inves less in he speculaive porfolios. The opimal porfolio is illusraed by a numerical example. A Opimal Filering To keep he same noaion as in Lipser and Shiryaev 1 we rewrie he Equaions.3.4 and.5 in he following form dz Z ds S dr = + r + α κ r r + }{{} A dw β + }{{} B 1 1 }{{} A 1 β d σ Z σ S R 1 σ S R σ r R 3 σ r R 4 σ r R 5 } {{ } B dw Z dw S dw P 3

26 and dβ = λ }{{} β + λ }{{} a β d + σ β R 9 }{{} a 1 b 1 β dw + [σ β R 6 } σ β R 7 {{ σ β R 8 ] } b dw Z dw S dw P where W Z W S W P W β T is a sandard Brownian moion relaive o he filraion F and he coefficiens R i i = are defined in such a way ha db Z db S db P db β = 1 R 1 R R 3 R 4 R 5 R 6 R 7 R 8 R 9 dw Z dw S dw P dw β and R 1 = ρ ZS R = R 5 = 1 ρ ZS R 3 = ρ ZP R 4 = ρ SP ρ ZS ρ ZP 1 ρ ZS 1 R 3 R 4 R 6 = ρ Zβ R 7 = ρ Sβ ρ ZS ρ Zβ 1 ρ ZS R 8 = ρ P β R 3 R 6 R 4 R 7 R 9 = 1 R6 R7 R R 8. 5 Assuming ha for a R he condiional disribuion P β a Z S r is Gaussian P-a.s. wih mean ˆβ = E[β Z S r ] and variance m = E[β ˆβ Z S r ] equivalenly he disribuion of β is condiionally Gaussian we have from Theorem 1.6 in Lipser and Shiryaev 1 ha he condiional disribuion P β a F SZr is also Gaussian P-a.s. 1 Therefore applying Theorem 1.7 in Lipser and Shiryaev 1 we have ha he observed expeced inflaion rae ˆβ = E[β F SZr ] saisfies d ˆβ = λ }{{} β + λ ˆβ d }{{} a a { 1 ] } + [σ Z σ β ρ Zβ σ S σ β ρ Sβ σ r σ β ρ P β +m [1 ] }{{}}{{} A T 1 b B σ Z σ S R 1 σ r R 3 σ S R σ r R 4 σ r R 5 1 σ Z σ S R 1 σ S R σ r R 3 σ r R 4 σ r R 5 }{{} dz Z ds S dr r + α κ r r B B 1 } {{ } A + 1 }{{} A 1 ˆβ d 1 I is easy o check ha he assumpions of he heorem are saisfied because enries in all marices are consan. 4 1

27 where m = E[β ˆβ F SZr ] is deerminisic and b B = b 1 B1 T + b B T = [ ] + [σ β R 6 σ β R 7 σ β R 8 ] = [σ Z σ β ρ Zβ σ S σ β ρ Sβ σ r σ β ρ P β ] σ Z σ S R 1 σ r R 3 σ S R σ r R 4 σ r R 5 B B 1 = B 1 B T 1 + B B T 1 = B B T 1 = B T 1 B 1 where B T 1 = 1 σ Z R 1 σ Z R R 1 R 4 R R 3 σ Z R R 5 1 σ S R R 4 σ S R R 5 1 σ rr 5. We firs evaluae b B + m A T 1 which yields ] b B + m A T 1 = [σ Z σ β ρ Zβ + m σ S σ β ρ Sβ σ r σ β ρ P β Now we muliply vecor b B + m A T 1 by marix B T 1 o obain b B + m A T 1 B T 1 [ = σ β ρ Zβ + m m R 1 + σ Z σ β R R 7 m R 1 R 4 R R 3 + σ Z σ β R R 5 R ] 8 σ Z σ Z R σ Z R R 5 = [σ Z A Z σ S A S σ P A P ] where A P = m R 1 R 4 R R 3 + σ Z σ β R R 5 R 8 σ P σ Z R R 5 A S = R 1m + σ Z σ β R R 7 σ S σ Z R A Z = σ Zσ β ρ Zβ + m. σz The following vecor defines a Brownian moion see Lipser and Shiryaev 1 Vol. p.35 relaive o filraion F SZr 1 σ Z dz Z 1 σ S R 1 σ S R ds S r + α + ˆβ d σ r R 3 σ r R 4 σ r R 5 dr κ r r 1 σ Z dz = R Z 1 1 ˆβ d σ Z R σ S R σ S db S R 1 R 4 R R 3 σ Z R R 5 R 4 σ S R R 5 1 σ r db P = 1 dz σ Z Z R 1 dz σ Z R Z R 1 R 4 R R 3 dz σ Z R R 5 Z σ rr 5 ˆβ d ˆβ d + 1 R db S ˆβ d R 4 R R 5 db S + 1 R 5 db P 5 = d ˆB Z d ˆB S d ˆB P A.1

28 where ˆB Z ˆB S ˆB P T is he Brownian moion. From his equaliy we easily obain ha db S = R d ˆB S + R 1 d ˆB Z db P = R 5 d ˆB P + R 4 d ˆB S + R 3 d ˆB Z. Thus we have d ˆβ = λ β ˆβ d + A Z σ Z d ˆB Z + A S σ S d ˆB S + A P σ P d ˆB P. Therefore he filered equaions can be wrien in erms of he Brownian moion A.1 as dz = Z ˆβ d + σ Z d ˆB Z ds = S r + αd + σ S R 1 d ˆB Z + R d ˆB S dr = κ r r d σ r R 5 d ˆB P + R 4 d ˆB S + R 3 d ˆB Z. B Robus HJB Equaion We rewrie equaions for wealh X price level process Z shor-erm ineres rae r and drif ˆβ in marix form dz Z ˆβ Z σ Z dx dr = X r + απ S + qπ P κ r r d + K 1 K K 3 d ˆB Z σ r R 3 σ r R 4 σ r R 5 d ˆB S d ˆβ λ β ˆβ A Z σ Z A S σ S A P σ d ˆB P P We inroduce perurbaions o his sysem by adding a drif e sds1 k S k P T o he Brownian moion ˆB Z ˆB S ˆB P T. The resuling vecor B Z B S B P T is a Brownian moion under probabiliy measure P e. The perurbed sysem of equaions is dz dx dr d ˆβ = Z ˆβ σ Z e X r + απ S + qπ P K 1+k S K +k P K 3 X e d κ r r + σ r R 3 + k S R 4 + k P R 5 e λ β ˆβ A Z σ Z + k S A S σ S + k P A P σ P e }{{} M Z σ Z + K 1 K K 3 d B Z σ r R 3 σ r R 4 σ r R 5 d B S. A Z σ Z A S σ S A P σ d B P P }{{} Λ According o Anderson Hansen and Sargen 3 we evaluae he symmeric marix σ 11 σ 1 σ 13 σ 14 Σ = ΛΛ T = σ 1 σ σ 3 σ 4 σ 31 σ 3 σ 33 σ 34 σ 41 σ 4 σ 43 σ 44 6

29 where we defined σ 11 = Z σ Z σ = X σs Π S + σ P Π P + σ S σ P Π S Π P ρ SP σ 33 = σ r σ 44 = σ Z A Z + σ S A S + σ P A P σ 1 = σ Z X Z σs Π S R 1 + σ P Π P R 3 σ 13 = σ Z σ r Z R 3 σ 14 = σ Z A ZZ σ 34 = σ r σ β ρ P β σ 3 = σ r X σp Π P +σ S Π S ρ SP σ 4 = σ β X σs Π S ρ Sβ + σ P Π P ρ P β. We denoe he Hessian and he gradien of he value funcion v wih respec o sae variables z x r and ˆβ respecively as v x i x j v zz v zx v zr v z ˆβ v zx v xx v xr v x ˆβ v zr v xr v rr v r ˆβ v z ˆβ v x ˆβ v r ˆβ v ˆβ ˆβ v x i Le π = π S π P be he vecor of fracions of wealh invesed a ime [ T ] in he sock π S and he bond π P hen according o Anderson Hansen and Sargen 3 he robus HJB equaion is In paricular sup inf π R e R v v + sup inf M T + 1 π R e R x race Σ i v x i x j v z v x v r v ˆβ. + e =. Ψ { v + z ˆβ σz e v z + x r + απ S + qπ P [σ S π S a 1 + σ P π P a ]e v x + κ r r a 3 e v r + λ β ˆβ a 4 e v ˆβ + 1 zσ Z v zz + σ Z xz σ S π S n 1 + σ P π P n vzx σ Z σ r zn v zr + σ ZA Z zv z ˆβ + 1 x σ S π S + σ P π P + σ S σ P π S π P n 4 vxx B.1 σ r x σ P π P + σ S π S n 4 vxr + σ β x σ S π S n 5 + σ P π P n 6 vx ˆβ + 1 σ rv rr σ r σ β n 6 v r ˆβ + 1 AZ σ Z + A S σ S + A P σ P } v ˆβ ˆβ + e = Ψ where n 1 = ρ ZS n = ρ ZP n 3 = ρ Zβ n 4 = ρ SP n 5 = ρ Sβ and n 6 = ρ P β. To find he infimum over e we ake he derivaive wih respec o e and se i equal o zero. d zσ Z ev z xσ S π S a 1 + σ P π P a ev x a 3 ev r a 4 ev de ˆβ + e =. Ψ The value e ha gives he infimum is e = Ψ zσ Z v z + xσ S π S a 1 + σ P π P a v x + a 3 v r + a 4 v ˆβ. To simplify he noaion we use φ = 1 γ. Le us look for a soluion in he form v z x r ˆβ = φ 1 x φh z r ˆβ and assume ha Ψ = θ x φ. h z Plugging hese funcions ino e we obain e = θ σ S π S a 1 + σ P π P a σ Z + a 3 h r φ h + a 4 φ h ˆβ. h 7