On the Role of the Growth Optimal Portfolio in Finance


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1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN
2 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen 1 January 19, 2005 Absrac. The paper discusses various roles ha he growh opimal porfolio GOP) plays in finance. For he case of a coninuous marke we show how he GOP can be inerpreed as a fundamenal building block in financial marke modeling, porfolio opimizaion, coningen claim pricing and risk measuremen. On he basis of a porfolio selecion heorem, opimal porfolios are derived. These allocae funds ino he GOP and he savings accoun. A risk aversion coefficien is inroduced, conrolling he amoun invesed in he savings accoun, which allows o characerize porfolio sraegies ha maximize expeced uiliies. Naural condiions are formulaed under which he GOP appears as he marke porfolio. A derivaion of he ineremporal capial asse pricing model is given wihou relying on Markovianiy, equilibrium argumens or uiliy funcions. Fair coningen claim pricing, wih he GOP as numeraire porfolio, is shown o generalize risk neural and acuarial pricing. Finally, he GOP is described in various ways as he bes performing porfolio Mahemaics Subjec Classificaion: primary 90A12; secondary 60G30, 62P20. JEL Classificaion: G10, G13 Key words and phrases: Growh opimal porfolio, porfolio opimizaion, marke porfolio, fair pricing, risk aversion coefficien. 1 Universiy of Technology Sydney, School of Finance & Economics and Deparmen of Mahemaical Sciences, PO Box 123, Broadway, NSW, 2007, Ausralia
3 1 Inroducion There exiss an increasing lieraure on he growh opimal porfolio GOP), which is he porfolio ha maximizes expeced uiliy from erminal wealh. I was originally discovered in Kelly 1956) and Laané 1959) and laer sudied and applied in Breiman 1961), Long 1990) and by many oher auhors. The curren paper highlighs some aspecs of he cenral role ha he GOP plays in finance. The Nobel prize winning work by Markowiz 1952, 1959) on singleperiod meanvariance porfolio selecion has provided he foundaion for modern porfolio heory. In he conex of dynamic invesmen planning in a coninuous ime seing he meanvariance approach has received lile aenion. The lieraure has focussed more on invesors seeking o maximize expeced uiliy, which is a deparure from he meanvariance approach. In pracice, few if any invesors know heir uiliy funcion. Furhermore, uiliy funcions ha are widely used for mahemaical convenience may no necessarily come close o an adequae descripion of an invesor s view on expeced reurn owards risk. Wide spread in he lieraure is he use of he ArrowPra absolue risk aversion coefficien, see Luenberger 1997). In he curren paper a risk aversion coefficien is inroduced in a coninuous ime seing, which allows o achieve in a unified framework he main objecives of uiliy maximizaion and meanvariance porfolio selecion. I will be shown ha he growh opimal porfolio GOP) plays a cenral role in his conex and more generally in finance. Under naural assumpions he marke porfolio of invesable wealh is shown o form a proxy of he GOP. This permis he derivaion of he capial asse pricing model CAPM), as developed by Sharpe 1964), Linner 1965), Mossin 1966) and Meron 1973a), wihou relying on Markovianiy, expeced uiliy maximizaion and equilibrium argumens. Similarly, he Markowiz efficien fronier, see Markowiz 1959), and he Sharpe raio, see Sharpe 1964), can be direcly derived in he given coninuous ime framework wih he GOP as benchmark. The mehod employed has similariies o ha proposed in Fleming & Sein 2004), which uses sochasic opimal conrol o derive cerain characerisics for an economy. The GOP plays a naural role as numeraire porfolio, see Long 1990), in he pricing of coningen claims wih he real world probabiliy measure as pricing measure. The fair pricing concep of he benchmark approach generalizes he risk neural pricing mehod of he arbirage pricing heory, see Black & Scholes 1973), Meron 1973b), Ross 1976), Harrison & Kreps 1979) and Harrison & Pliska 1981). The exisence of an equivalen risk neural pricing measure is no required. The prescribed benchmark approach, wih he GOP as is cenral building block, widens he range of models ha can be applied in finance. In Secion 2 he paper inroduces a coninuous benchmark model. Secion 3 sudies various aspecs of porfolio opimizaion and inroduces a risk aversion coefficien. I also discusses uiliy maximizaion, he ineremporal CAPM and 2
4 he measuremen of general and specific marke risk. In Secion 4 he pricing of coningen claims is performed wihou measure ransformaion. Finally, in Secion 5 he GOP is shown in various ways o be he bes performing porfolio. 2 Coninuous Benchmark Model 2.1 Primary Securiy Accouns For an illusraion of he cenral role of he GOP in finance le us consider a coninuous financial marke model. Here A denoes he marke informaion ha is available a ime [0, T ], T 0, ). More precisely, he marke is modeled on a filered probabiliy space Ω, A T, A, P ) wih filraion A = A ) [0,T ], saisfying he usual condiions, see Karazas & Shreve 1991). We consider d coninuous sources of raded uncerainy, which are modeled by d independen sandard Wiener processes W 1, W 2,..., W d. The d sources of raded uncerainy are securiized by d + 1 primary securiy accouns, which are ypically cumdividend share accouns of respecive companies or bonds. For j {0, 1,..., d} we denoe by S j) he value of he jh primary securiy accoun a ime, when expressed in unis of he domesic currency. We emphasize ha all dividends or ineres paymens are reinvesed. To provide a model for coninuous marke dynamics we assume ha he price S j) a ime for he jh primary securiy accoun is he unique srong soluion of he SDE ) ds j) = S j) a j d + b j,k 2.1) dw k for all [0, T ] wih S j) 0 > 0 and j {0, 1,..., d}. Here we require he appreciaion rae processes a j = {a j, [0, T ]} and volailiy processes b j,k =, [0, T ]} o saisfy usual measurabiliy and inegrabiliy condiions for all j {0, 1,..., d} and k {1, 2,..., d}. We assume he exisence of a savings accoun S 0) = {S 0), [0, T ]}, wih { } = exp r s ds, 2.2) {b j,k S 0) for all k {1, 2,..., d} and he shor rae 0 b 0,k = 0 2.3) a 0 = r 2.4) for [0, T ]. Noe ha he appreciaion raes, volailiies and shor rae can depend on sources of uncerainy oher han hose given by he Wiener processes W 1, W 2,..., W d. This means ha he marke is no assumed o be complee. 3
5 By inroducing he appreciaion rae vecor a = a 1, a 2,..., a d ), he uni vecor 1 = 1, 1,..., 1) and he marke price for risk vecor θ = θ 1, θ 2,..., θ d ) i is reasonable o suppose ha he equaion b θ = [a r 1] 2.5) has a unique soluion. This is guaraneed, for insance, by he following naural assumpion. Assumpion 2.1 The volailiy marix b = [b j,k ] d j, [0, T ]. is inverible for each We direcly obain by 2.5) he marke price for risk vecor in he form θ = b 1 [a r 1] 2.6) for [0, T ]. Wihou loss of generaliy, his allows us o rewrie he SDE 2.1) for he jh primary securiy accoun S j) in he form ds j) = S j) [ ] ) r d + b j,k θ k d + dw k 2.7) for [0, T ], j {0, 1,..., d}. For simpliciy, we have chosen a coninuous marke model. However, noe ha he majoriy of he resuls ha will be presened can be generalized o jump diffusion markes. 2.2 Porfolios A sochasic process δ = {δ = δ 0,..., δ d ), [0, T ]} is called a sraegy if δ is predicable and Sinegrable wih respec o he vecor process S = {S = S 0), S 1),..., S d) ), [0, T ]} of primary securiy accouns. Here δ j is he number of unis of he jh primary securiy accoun ha is held a ime [0, T ] and j {0, 1,..., d}. A negaive δ j means ha one is shor δ j unis of he jh primary securiy accoun. For a sraegy δ we denoe by S δ) he value of he corresponding porfolio or wealh process a ime, ha is S δ) = j=0 δ j S j) 2.8) for [0, T ]. A sraegy δ, or corresponding porfolio S δ), is said o be selffinancing if ds δ) = δ j ds j) 2.9) j=0 4
6 for [0, T ]. This means ha no inflow or ouflow of funds akes place for a corresponding selffinancing porfolio and all changes in value are due o gains from rade in he primary securiy accouns. Since we deal only wih selffinancing sraegies and porfolios we will herefore omi he phrase selffinancing. For a given sraegy δ le π j δ, denoe he jh fracion or proporion of he value of a corresponding sricly posiive porfolio S δ) ha is invesed a ime in he jh primary securiy accoun, ha is π j δ, = S j) δj 2.10) S δ) for [0, T ] and j {0, 1,..., d}. Noe ha he fracions always sum o uniy such ha π j δ, = ) j=0 for all [0, T ]. For a given sricly posiive porfolio process S δ) = {S δ), [0, T ]} we obain from 2.9), 2.7) and 2.10) he SDE for [0, T ]. ds δ) = S δ) r d + j=1 π j δ, bj,k ) ) θ k d + dw k 2.12) 2.3 Growh Opimal Porfolio We now idenify he cenral objec of our sudy, he growh opimal porfolio GOP). From 2.12) i follows by applicaion of he Iô formula ha he logarihm of a sricly posiive porfolio S δ) saisfies he SDE d lns δ) ) = g δ d + π j δ, bj,k dw k 2.13) wih growh rae g δ = r + j=1 π j δ, bj,k j=1 θ k 1 2 j=1 π j δ, bj,k ) 2.14) for [0, T ]. The GOP arises naurally from maximizing he growh rae for all sricly posiive porfolios. Definiion 2.2 A GOP is a sricly posiive porfolio process S δ ), [0, T ]} wih maximum growh rae g δ, such ha {S δ ) = g δ g δ 2.15) for all [0, T ] and all sricly posiive porfolio processes S δ). 5
7 Noe ha he above definiion of a GOP is given in a pahwise sense and does no use any condiional expecaion or uiliy funcion. However, i is wellknown ha he porfolio which maximizes expeced logarihmic uiliy from erminal wealh is he GOP, see Karazas & Shreve 1998). Noe ha a GOP is unique up o he choice of is iniial value. To idenify a GOP we have o maximize he growh rae, which is of quadraic form, wih respec o he fracions given in 2.14). For each [0, T ] his opimizaion problem is known o have a unique soluion ha can be idenified by he firs order condiions ) 0 = b j,k θ k πδ, l b l,k 2.16) for each [0, T ] and j {1, 2,..., d}. The soluion π δ, = π 1 δ,,..., πd δ, ) saisfies he equaion π δ, b = θ 2.17) for [0, T ]. Since by Assumpion 2.1 he volailiy marix b is inverible for each [0, T ], we can explicily solve 2.17). Thus, a porfolio S δ ) exiss ha achieves he maximum growh rae. This porfolio is a GOP, which is characerized by he vecor of opimal fracions π δ, = ) b 1 θ 2.18) for all [0, T ]. The GOP value S δ ) SDE [ ds δ ) = S δ ) for [0, T ], where S δ ) 0 > 0. r + l=1 a ime saisfies by 2.18) and 2.12) he ] ) θ k ) 2 d + θ k dw k 2.19) We call a coninuous ime financial marke model of he above ype a coninuous benchmark model. We impose exremely weak condiions on a coninuous benchmark model. The inveribiliy of he volailiy marix in Assumpion 2.1 secures he exisence of he GOP, bu is no necessary as long as a soluion o 2.5) exiss. The exisence of an equivalen risk neural measure, as needed under he risk neural approach, is no required. Therefore, in comparison wih he risk neural approach, he assumpions of he above benchmark model are less demanding. In he above coninuous benchmark model he shor rae, volailiy and marke price for risk processes are very flexible, which provides subsanial freedom in modeling. They may depend on nonraded sources of uncerainy. For insance, hey do no require Markovianiy of he dynamics. This generaliy does no impac on he srucure of he soluion of he above maximizaion problem, which idenifies locally in ime he opimal fracions of he GOP. In he following we will selec similarly locally in ime he fracions of a more general class of porfolios. 6
8 3 Porfolio Opimizaion 3.1 Porfolio Selecion Theorem Wihin his secion we invesigae wha sraegy a raional invesor should naurally selec o opimize he evoluion of his or her wealh. Le us consider he discouned porfolio value S δ) = Sδ) S 0). 3.1) By 2.12) and applicaion of he Iô formula we obain from 3.1) he SDE wih diffusion coefficiens δ) d S = S δ) π δ, b {θ d + dw } = ψ δ, {θ d + dw }, 3.2) ψ δ, = ψ 1 δ,,..., ψ d δ,) = S δ) π δ, b 3.3) for [0, T ]. According o 3.3) he discouned porfolio process S δ) has he discouned drif α δ = ψ δ, θ 3.4) and he aggregae diffusion coefficien for [0, T ]. γ δ = ψ δ, ψ δ, 3.5) Le us now idenify a class of porfolios ha we will call opimal. More precisely, a discouned porfolio will be called opimal if i exhibis for each ime insan he larges discouned drif in comparison o all oher porfolios wih he same aggregae diffusion coefficien. Definiion 3.1 A sricly posiive porfolio process S δ) is called opimal if for all [0, T ] and all porfolios S δ) wih a given aggregae diffusion coefficien value γ δ = γ δ 3.6) one has α δ α δ. 3.7) This opimaliy crierion is defined locally in ime, avoids he inroducion of a uiliy funcion and a ime horizon. As we will see, i enables us o model in a flexible manner he effecs ha expeced uiliy maximizaion aims o achieve. 7
9 As will become clear below, by choosing an opimal discouned porfolio he invesor firsly decides on he level of risk, hen he or she selecs he porfolio wih maximum discouned drif from wihin he class of all such discouned porfolios. Therefore, everyhing else being equal he invesor prefers always more raher han less. This behavior of an invesor is also ermed nonsaiaion. Noe, for an opimal porfolio he iniial value is a flexible parameer. By he same mehods ha we use below one can show ha for zero marke prices for risk all porfolios are opimal. To avoid such unrealisic cases, we make he following assumpions. Assumpion 3.2 The oal marke price for risk θ = θ θ > 0 3.8) is almos surely sricly posiive and he fracion π 0 δ, = 1 θ b ) of wealh invesed by he GOP in he savings accoun does no equal one for all [0, T ]. The following porfolio selecion heorem reveals he general srucure of opimal porfolios. Theorem 3.3 The family of opimal porfolios S δ) is a ime [0, T ] parameerized by he fracion πδ, 0 invesed in he savings accoun. The discouned value a ime of an opimal porfolio saisfies he SDE S δ) δ) d S = S δ) 1 π 0 δ, 1 π 0 δ, θ θ d + dw ) 3.10) and he opimal fracions are of he form for [0, T ]. π δ, = 1 π0 δ, 1 π 0 δ, ) b 1 θ 3.11) The fac ha he soluion o he given consrained opimizaion problem depends on πδ, 0 is ineresing and no eviden ahead. As we will see, mahemaically he above resuls require no more han mulivariae calculus and a basic undersanding of Iô calculus. The proof of Theorem 3.3 is given in Appendix A. I uses similar argumens as ha given in Plaen 2002) or Khanna & Kulldorff 1999), who reaed he special case wih deerminisic coefficiens. By 3.11) and 2.18) 8
10 i follows ha any invesor who prefers an opimal porfolio follows a sraegy where a each ime a fracion of wealh is invesed in he GOP and he res is held in he savings accoun. I is very imporan o noe ha an opimal porfolio reains a free parameer, which can be specified as he fracion of wealh o be held in he savings accoun. This is a consequence of our definiion of opimaliy, where we maximize he enire discouned drif of a porfolio while keeping he aggregae diffusion coefficien value fixed. The GOP can be inerpreed as a muual fund since i is par of all opimal porfolios. Each opimal porfolio can be obained by a, in general, randomly changing combinaion of he muual fund and he savings accoun. This shows ha he GOP plays a naural role in porfolio selecion. Noe ha we obained he above form of opimal fracions in 3.11) wihou exploiing any Markovianiy, expeced uiliy or equilibrium argumen. If one wans o ener uiliy ino he modeling, hen i would have o deermine he raio of he value of risky asses in he opimal porfolio o he oal opimal porfolio value. Mainly due o is generaliy he above muual fund or porfolio selecion heorem is differen o mos oher porfolio selecion heorems presened in he lieraure, see, for insance, Meron 1971) or Karazas & Shreve 1998). As we will see, in some cases expeced uiliy maximizaion can be embedded in he framework of opimal porfolios. 3.2 Risk Aversion Coefficien For an opimal porfolio S δ) one can say ha he fracion πδ, 0 invesed in he savings accoun reflecs he risk aversion of he invesor. Similar o he ArrowPra absolue risk aversion coefficien, see Luenberger 1997), we can now inroduce for an opimal porfolio S δ) he risk aversion coefficien J δ, S δ) ) = 1 π0 δ, 1 π 0 δ, 3.12) a ime [0, T ]. According o Theorem 3.3, an invesor who forms an opimal porfolio invess some fracion of his or her wealh ino he GOP and allocaes he remaining par o he savings accoun. For insance, if all of he invesor s wealh is invesed in he savings accoun, hen he or she is compleely risk averse and he risk aversion coefficien is by 3.12) no finie. Noe ha he risk aversion coefficien depends no only on ime bu also on he level of discouned wealh ha is invesed. For an opimal porfolio S δ) he vecor of fracions in he risky asses can be wrien by 3.11) and 3.12) in he form π δ, = J δ, 9 π δ, S δ) ) 3.13)
11 for [0, T ]. In he special case when an invesor maximizes he growh rae of an opimal porfolio, hen he or she obains a GOP and has by 3.12) a risk aversion coefficien of value one, ha is, for all [0, T ]. J δ, S δ ) ) = ) The risk aversion coefficien appears o be useful as a flexible parameer process for modeling he evoluion of he risk aversion of an invesor over ime. One can le his parameer process depend on a wide range of facors, no only he invesor s level of wealh. For insance, one can make i dependen on changing life circumsances, governmen policies, axaion, herd behavior or oher social or financial facors. The above proposed risk aversion coefficien provides subsanial flexibiliy for modeling and can be easily applied o mulifacor models. In he following we will consider an example wih uiliy funcions. These are usually difficul o handle compuaionally in models wih more han wo facors. 3.3 Uiliy Maximizaion Le us illusrae a sandard siuaion in a Markovian framework, where he risk aversion coefficien 3.12) relaes o expeced uiliy maximizaion. In he simples case one may assume ha b and θ are deerminisic. Consider a sricly increasing and sricly concave, wice differeniable uiliy funcion U ) defined on 0, ) for all [0, T ]. We fix he planning horizon a he deerminisic erminal ime T and endow an invesor wih he discouned invesable wealh x > 0 a ime [0, T ]. By V we denoe he se of fracion processes π δ = {π δ,s, s [, T ]} of nonnegaive porfolios for he invesmen in risky primary securiy accouns. The invesor is assumed o form a nonnegaive porfolio S δ) wih π δ V. The expeced uiliy E U ) δ) S T ) Sδ) = x is maximized wih respec o he choice of differen fracion processes of nonnegaive porfolios wih S = x. Le us assume ha we are given he value δ) funcion ) δ) u, x) = sup E U S T ) Sδ) = x 3.15) π δ V of his problem for, x) [0, T ] [0, ). As we will see, he value funcion u, x) can be inerpreed as a kind of uiliy funcion iself. For a wide class of uiliy funcions and dynamics i has been shown, see, for insance Krylov 1980), Cox & Huang 1989), Fleming & Soner 1992), Korn 1997) and Karazas & Shreve 1998), ha he parial derivaives u, u and 2 u are defined and coninuous on x x 2 [0, T ) 0, ). Furhermore, on he basis of he, so called, Bellman principle, see Fleming & Soner 1992), i has been demonsraed for a wide range of diffusion 10
12 dynamics ha he value funcion u, ) saisfies he HamilonJacobiBellman equaion u, x) + sup π u, x) δ, b θ x + 1 ) π δ V x 2 π δ, b b π δ, x 2 2 u, x) = ) x 2 for [0, T ), x 0, ) wih erminal condiion ut, x) = Ux) 3.17) for x 0, ). To achieve in 3.16) a genuine maximum i is necessary ha he fracions saisfy he firs order condiions πδ, l b l,k θ k u, x) x + 1 ) 2 πδ, l b l,k x 2 2 u, x) = 0 x 2 x 2 π j δ, l=1 3.18) for all [0, T ] and j {1, 2,..., d}. This leads o a soluion ha saisfies, when expressed in vecor and marix form, he vecor equaion b x l=1 [ ] u, x) θ + b π δ, x 2 u, x) x x 2 = 0,..., 0). 3.19) If one subsiues he fracions of an opimal porfolio, see 3.11) and 3.12), δ) ino 3.19), hen one noes for he case J δ, S ) 0 ha he condiion 3.19) becomes [ ] u, x) x 2 u, x) b x θ + = 0,..., 0). 3.20) x J δ, x) x 2 I follows ha an opimal porfolio maximizes he expeced uiliy in 3.15), where he risk aversion coefficien is of he form J δ, x) = x 2 u,x) x 2 u,x). 3.21) x This paricular form of he risk aversion coefficien resembles he ArrowPra absolue risk aversion coefficien, see Luenberger 1997), when inerpreing u, ) as a uiliy funcion iself. In he ypical case when u, x) is sricly increasing and sricly concave in x, he risk aversion coefficien is sricly posiive. By 3.21) and 3.16) we ge for he value funcion he firs order parial differenial equaion PDE) u, x) + 1 θ 2 x u, x) = ) 2 J δ, x) x for [0, T ], x 0, ). Alernaively, we have by 3.22) and 3.21) he PDE u, x) 1 2 ) 2 θ x 2 u, x) = ) J δ, x) x 2 11
13 for [0, T ], x 0, ). Noe ha one can generalize his example o include producion and consumpion, as well as ime dependence and ime inegrals over discouned uiliy funcions. The maximizaion of a discouned and inegraed expeced uiliy for an infinie ime horizon can be similarly reaed. To illusrae more direcly he link beween a given uiliy funcion and he corresponding risk aversion coefficien, consider he classical Meron problem when using power uiliy. In Meron 1971) an explici soluion for asse price dynamics under deerminisic marke prices for risk, shor rae and volailiies, when using power uiliy Ux) = 1 γ xγ, was given wih u, { T δ) S ) = exp θ s 2 γ 2 1 γ) ds } ) γ Sδ) 3.24) for [0, T ], γ 0, 1). The resuling vecor of opimal fracions akes he form π δ, = ) b 1 1 θ 1 γ. 3.25) One noes by 3.11) for an opimal porfolio S δ) he risk aversion coefficien in his case equals he consan J δ, S δ) ) = 1 γ 3.26) for all [0, T ]. This means ha power uiliy yields a consan risk aversion coefficien. One noes from 3.26) ha for γ 0 he risk aversion coefficien approaches one. This corresponds o he choice of he GOP as opimal porfolio. Indeed, one can show for logarihmic uiliy Ux) = lnx) ha he value funcion is u, ) T δ) S ) = ln Sδ) + θ s 2 ds 3.27) 2 for [0, T ]. The corresponding vecor of opimal fracions is of he form π δ, = ) b 1 θ 3.28) for [0, T ], as expeced, see 2.18). On he oher hand, for γ 1 he risk aversion coefficien approaches infiniy, which corresponds o he choice of he savings accoun as opimal porfolio. I is clear from 3.21), when considering explicily given or numerically obained value funcions, hen one can direcly deermine he corresponding risk aversion coefficien in dependence on he given ime and acual value of he discouned porfolio. As shown in Fleming & Soner 1992), Karazas & Shreve 1998) and Korn 1997) i is, in general, difficul o compue he corresponding value funcion 12
14 for a given uiliy funcion and specific marke dynamics. The above proposed concep of opimal porfolios wih risk aversion coefficien provides a mehod of circumvening hese problems. I allows he modeling of opimal porfolios for differen invesors wih a wide range of risk aversion coefficiens. Obviously, he conceps of an opimal porfolio and a risk aversion coefficien remove in invesmen planning he necessiy o fix a paricular ime horizon and a paricular uiliy funcion. In more general mulifacor porfolio selecion problems han he sandard siuaion described above, i is pracically impossible o obain even numerically a soluion of he expeced uiliy maximizaion problem because mulidimensional parial differenial equaions are involved. However, i is sill sraighforward in such a case o deermine he class of opimal porfolios and selec one by characerizing he risk aversion coefficien process. 3.4 Marke Porfolio as Opimal Porfolio The oal invesable wealh of he lh invesor is denoed by S δ l), l {1, 2,..., n}. The marke porfolio S δ +) of invesable wealh is hen he sum of he oal invesable wealh processes of all invesors. In he simples case an invesor may paricipae in a pension fund which could be expeced o form an opimal porfolio. Le us assume ha each invesor holds an opimal porfolio. Assumpion 3.4 Each invesor forms an opimal porfolio wih his or her oal invesable wealh. Therefore, we ge from 3.10) for he discouned value S δ +) of he marke porfolio a ime he SDE n ) d S δ +) = d = = n l=1 n l=1 l=1 S δ l) Sδ l ) S δ l) 1 π 0 δ l, 1 π 0 δ, ) δl 0 ) θ θ d + dw ) θ θ 1 πδ 0 d + dw ), = S δ +) 1 π 0 δ +, 1 π 0 δ, θ θ d + dw ) 3.29) for [0, T ]. One obains from 3.29), Definiion 3.1 and he SDE 3.10) he following resul. 13
15 Corollary 3.5 The marke porfolio of invesable wealh is an opimal porfolio. I forms a GOP if and only if πδ 0 +, = πδ 0, 3.30) for all [0, T ]. By 3.12) he risk aversion coefficien of he marke porfolio equals J δ+, S δ +) ) = 1 π0 δ, 1 πδ 0 +, 3.31) for all [0, T ]. Noe ha if he marke porfolio is no he GOP, hen i is sill always a combinaion of he GOP and he savings accoun. Le us now discuss consequences of he hypoheical assumpion ha he moneary auhoriies aim o maximize he growh rae of he marke porfolio of invesable wealh. One could poenially argue ha hey may be able o achieve his goal by adjusing he shor rae or equivalenly he shor erm money supply. Obviously, he maximizaion of he growh rae of he marke porfolio is equivalen o he selecion of he GOP as marke porfolio of invesable wealh. This refers o he choice of a risk aversion coefficien wih consan value one, see 3.14). This hypoheical assumpion would lead direcly o he observabiliy of he GOP in form of he marke porfolio of invesable wealh. We remark, ha in Plaen 2004a) i is assumed ha he savings accoun is in ne zero supply, which makes he marke porfolio equal o he GOP if πδ 0, = 0 for all [0, T ]. Alernaively, he above indicaed proximiy of marke porfolio and GOP resuls in an asympoic sense via a limi heorem derived in Plaen 2004b, 2004c), where sequences of benchmark models for an increasing number d of risky primary securiy accouns have been sudied. I has been shown ha sequences of globally diversified porfolios wih fracions ha vanish sufficienly fas as d, converge asympoically owards he GOP. Therefore, if he marke porfolio of invesable wealh can be considered o be diversified, hen one has a robusness propery in he sense ha all diversified porfolios, including he GOP, are asympoically very similar. 3.5 Marke Prices for Risk and Risk Premia In relaion 2.5) we inroduced he marke price for risk vecor θ = θ 1, θ 2,..., θ d ), which is given by 2.6). In pracice, he esimaion of he appreciaion rae vecor a, as a parameer process in he drif of a diffusion process, is no realisic. Simple saisical analysis of he esimaes reveal ha here is probably no enough marke daa available, covering a sufficienly long ime period, ha 14
16 allows he direc esimaion of parameers in he appreciaion rae wih a reasonable confidence level. However, by using equaion 2.18) for he fracions of he GOP one can express he marke price for risk vecor in he form θ = b π δ, 3.32) for all imes [0, T ]. Tha is, by 2.10) we obain indirecly he kh marke price for risk as an average of he kh volailiies of he primary securiy accouns weighed by he values invesed in he GOP, ha is θ k = d j=0 bj,k δ j ) S j) d i=0 δi ) S i) 3.33) for [0, T ] and k {1, 2,..., d}. By 3.32) and 3.33) one noes ha marke prices for risk are averages of volailiies. By 2.6) one obains he risk premium vecor p) = a r 1 = b θ of risky primary securiy accouns by using he represenaion p) = b b π δ, 3.34) for all [0, T ]. Consequenly, risk premia are averages of producs of volailiies. By he inerpreaion of he GOP as marke porfolio of invesable wealh one gains an alernaive access o he esimaion of risk premia and marke prices for risk. In his case one can observe also he fracions of primary securiy accouns from he known marke capializaion. Addiionally, one can esimae he volailiy marix from frequenly observed primary securiy accoun daa, which is no a simple ask. However, i appears o be simpler han esimaing appreciaion raes, growh raes or risk premia. For his purpose i is reasonable o use, for insance, quadraic variaions of logarihms of securiy prices. Since hese quaniies can be direcly inferred from he daa of a relaively shor observaion period, one can obain a reasonable esimae for he marke price for risk from esimaed quaniies on he righ hand side of equaion 3.32). This also indicaes a poenial roue for esimaing risk premia, see 3.34), which provides a new mehod for sudying he equiy premium, see Mehra & Presco 1985). 3.6 Risk Measuremen Imporan regulaory requiremens for risk measuremen, see Basle 1996), ask for using a broadly based index, as discussed in Plaen & Sahl 2003), for he measuremen of marke risk. As we will see, he GOP is again he financial quaniy ha, when used as reference uni or benchmark, provides a ransparen and well srucured descripion. Le us now inerpre he GOP as a broadly diversified index, which is obained by a linear combinaion of he marke porfolio and he savings accoun. Then 15
17 he volailiy of he GOP θ = d θ k ) 2, 3.35) which equals he oal marke price for risk, models in a naural way he general marke risk, also ermed sysemaic risk. Le us refer o values ha are expressed in unis of he GOP, as benchmarked values. The jh benchmarked primary securiy accoun process Ŝj) = {Ŝj), [0, T ]}, wih Ŝ j) = Sj) S δ ), 3.36) saisfies, by 2.7), 2.19) and applicaion of he Iô formula, he drifless SDE dŝj) = Ŝj) σ j,k dw k 3.37) for [0, T ] and j {0, 1,..., d}. Here, he j, k)h specific volailiy σ j,k = θ k b j,k 3.38) of he benchmarked jh primary securiy accoun Ŝj) arises from an applicaion of he Iô formula and measures he jh specific marke risk a ime [0, T ] wih respec o he kh Wiener process W k for k {1, 2,..., d} and j {0, 1,..., d}. The above separaion of general and specific marke risk is a naural feaure of he benchmark approach. For he modeling of all primary securiy accouns i is only necessary o specify he volailiy processes σ j,k for all j {0, 1,..., d} and k {1, 2,..., d} ogeher wih he shor rae process r and appropriae iniial values. To see his, noe ha by 3.36) we obain he GOP from he benchmarked savings accoun Ŝ0) and he savings accoun S 0) as he raio S δ ) = S0) Ŝ 0) 3.39) for all [0, T ]. From he jh benchmarked primary securiy accoun Ŝj) and he GOP S δ ) we can hen derive by 3.36) he value of he jh primary securiy accoun in he form S j) for [0, T ] and j {1, 2,..., d}. = Ŝj) S δ ) 3.40) In he case when one models he marke from he perspecive of he domesic currency, as is he case in his paper, he general marke risk is refleced by he marke price for risk processes σ 0,k = θ k, k {1, 2,..., d}. If he firs primary securiy accoun S 1) were a foreign savings accoun, hen he volailiy processes σ 1,k, k {1, 2,..., d}, would provide he marke prices for risk for his foreign currency denominaion. 16
18 3.7 Capial Asse Pricing Model Sharpe 1964), Linner 1965), Mossin 1966) and Meron 1973a) developed he seminal capial asse pricing model CAPM) as an equilibrium model. As we will see, one does no need an equilibrium or expeced uiliy argumen o obain he core saemen of he CAPM. To demonsrae his, le us inroduce he risk premium p δ ) a ime [0, T ] of a porfolio S δ) as he expeced excess reurn, which means by 2.12) and 2.19) ha for [0, T ]. p δ ) = j=1 π j δ, bj,k θ k 3.41) Le X, Y denoe he covariaion of wo processes X and Y a ime, which is defined as he limi in probabiliy of he sum of he producs of he incremens of X and Y based on a ime discreizaion wih vanishing sep size, see Karazas & Shreve 1991). The sysemaic risk parameer β δ ) of a porfolio S δ), he bea, can hen be defined as d d β δ ) = lnsδ) ), lns δ +) ) 3.42) d d lnsδ +) ), lns δ +) ) δ for [0, T ], where S +) denoes he marke porfolio of invesable wealh. Under Assumpion 3.4 we have shown ha he marke porfolio is an opimal porfolio. Therefore, i follows by 3.42), 3.29), 3.12) and 2.12) ha β δ ) = d d j=1 πj δ, bj,k θ k ) 1 π 0 2 δ+, θ 2 1 π 0 δ +, 1 π 0 δ, 1 π 0 δ, = J δ, S δ +) ) θ 2 j=1 This yields by 3.41) and 3.29) he following conclusion. π j δ, bj,k θ k. 3.43) Corollary 3.6 for [0, T ]. The sysemaic risk parameer of a porfolio S δ) has he form β δ ) = p δ) p δ+ ) 3.44) The porfolio bea in 3.44) is exacly in he form ha he CAPM heoreically suggess. This leads direcly o he core saemen of he CAPM wihou relying on any equilibrium or expeced uiliy argumen. 17
19 3.8 Efficien Fronier and Efficien Growh Rae Le us now derive he Markowiz efficien fronier, see Markowiz 1952, 1959), in he given coninuous benchmark model. For an opimal porfolio S δ) i follows by 3.10) ha is aggregae squared volailiy equals ) 2 1 π 0 b δ ) 2 δ, = θ 1 πδ 0 2 =, θ 2 ) ) δ) J δ, S ) and is risk premium akes he form p δ ) = 1 π0 δ, 1 π 0 δ, θ 2 = b δ ) θ 3.46) for [0, T ], see 3.41). Obviously, we have p δ ) 0, such ha by 3.10), 3.45) and 3.46) he appreciaion rae a δ ) for an opimal porfolio can be wrien as a funcion of is squared volailiy b δ ) 2 in he form a δ ) = r + p δ ) = r + b δ ) θ = r + b δ ) 2 θ 3.47) for [0, T ]. This funcion can be inerpreed as he wellknown Markowiz efficien fronier. More precisely, each opimal porfolio S δ) has an appreciaion rae ha is locaed a he efficien fronier a δ ) given in 3.47). Noe ha he efficien fronier moves sochasically up and down over ime in dependence on he shor rae r. Is slopes change also over ime according o he, generally, sochasic oal marke price for risk θ. For a given ime insan [0, T ] he Figure 1 shows he efficien fronier in dependence on he squared volailiy Figure 1: Efficien fronier. b δ ) 2 of he opimal porfolios, where r = 0.05 and θ 2 = This graph also includes he angen wih slope 1 2 a he poin b δ) 2 = θ 2, which corresponds o he squared volailiy of he GOP. The second equaion in 3.47) describes he 18
20 Figure 2: Efficien growh raes. wellknown capial marke line in dependence on he porfolio volailiy b δ ), see Luenberger 1997). Here he slope is direcly he marke price for risk. For illusraion, we plo in Figure 2 for given [0, T ] he growh raes of opimal porfolios S δ) in dependence on heir squared volailiy b δ ) 2. One could call hese growh raes he efficien growh raes, which have he form g δ ) = r + b δ ) 2 θ 1 2 b δ) 2, 3.48) see 2.14), 3.11) and 3.45). One noes ha for b δ ) 2 = θ 2 he growh raes achieve heir maximum, which yields he growh rae of he GOP. For b δ ) = 2 θ he efficien growh rae equals he shor rae. As we will see laer, he GOP is he bes performing porfolio under various crieria, in paricular, for long erm invesors, see Luenberger 1997). 3.9 Sharpe Raio Anoher imporan quaniy in modern porfolio heory is he Sharpe raio, see Sharpe 1964). For any porfolio S δ) he Sharpe raio s δ ) a ime is defined as he risk premium p δ ) over he aggregae volailiy b δ ), ha is, s δ ) = p δ) b δ ) 3.49) d as long as b δ ) = d j=1 πj δ, bj,k ) 2 > 0, [0, T ]. I follows by 3.3) 3.5) ha he Sharpe raio a ime equals he raio s δ ) = αδ γ δ 3.50) 19
21 of he discouned drif over he aggregae diffusion coefficien. This simple bu imporan observaion allows us o employ Theorem 3.3, which idenifies he maximum value of α δ for given γ δ when choosing an opimal porfolio wih a given risk aversion coefficien. Due o he srucure of he discouned porfolio drif and aggregae diffusion coefficien for any opimal porfolio S δ), as given in 3.10), he Sharpe raio 3.50) equals by 3.3), 3.5) and 3.11) for all [0, T ] he oal marke price for risk, ha is s δ ) = θ. 3.51) Noe ha he equaliy follows already direcly from A.6) in he Appendix. More generally, by 3.50) and Theorem 3.3 one can draw he following conclusion. Corollary 3.7 For any risky porfolio S δ) is Sharpe raio s δ ) is never greaer han he marke price for risk, ha is for all [0, T ]. s δ ) θ 3.52) This resul is highly relevan for modern porfolio opimizaion. I limis he achievable Sharpe raio for any risky porfolio. I is similar o he maximum growh rae aained by he GOP, see 2.14) 2.15), which limis he achievable growh raes of all sricly posiive porfolios. The opimal porfolios are basically hose wih maximum Sharpe raio. The mehod employed for he derivaion of Corollary 3.7 is similar o ha described in Fleming & Sein 2004). If one asks for he porfolio ha has he larges Sharpe raio and also he larges growh rae, hen one obains he GOP. As demonsraed in his secion he GOP is a very useful ool for he ineremporal generalizaion of he MarkowizTobinSharpe saic mean variance porfolio analysis. 4 Fair Pricing 4.1 Benchmarked Porfolios Similar o 3.36) we define for any porfolio S δ) is benchmarked value Ŝ δ) = Sδ) S δ ) 4.1) for [0, T ]. By applicaion of he Iô formula ogeher wih 2.12) and 2.19) we obain for Ŝδ) he drifless SDE dŝδ) = δ j) Ŝ j) σ j,k dw k 4.2) j=0 20
22 for [0, T ]. Thus, by 4.2) any benchmarked nonnegaive porfolio Ŝδ) is an A, P )local maringale. As shown in Rogers & Williams 2000), any nonnegaive local maringale is a supermaringale. Thus, we obain he following imporan resul. All nonnegaive benchmarked porfolios are A, P )supermar Corollary 4.1 ingales. In he following we draw some conclusions from his saemen, which do no require any major assumpions. 4.2 Arbirage In developed economies here exiss he legal concep of limied liabiliy. This means ha any invesor mus own a sricly posiive porfolio of oal invesable wealh. As in 3.29), he marke porfolio of invesable wealh can herefore be decomposed ino he oal invesable wealh processes of all invesors. If he oal invesable wealh process of an invesor reaches he level zero or becomes negaive, hen he or she mus declare bankrupcy, respecively. This is a crucial propery of a marke ha is modeled by he following noion of noarbirage, see Plaen 2002). Definiion 4.2 A nonnegaive porfolio S δ) is said o permi arbirage if for S δ) 0 = 0, almos surely, we have P S δ) T > 0) > ) Thus, in he case of arbirage here exiss a nonnegaive porfolio process, which generaes from zero iniial capial, sricly posiive wealh wih sricly posiive probabiliy. Using he supermaringale propery of benchmarked nonnegaive porfolios, described in Corollary 4.1, we proof he following resul. Corollary 4.3 A coninuous benchmark model does no permi arbirage. Proof: Consider a nonnegaive porfolio process S δ), where we have S δ) T S δ) 0 = 0 4.4) almos surely. obain By he supermaringale propery of Ŝ δ), see Corollary 4.1, we Ŝδ) ) Ŝδ) ) E T = E T A 0 Ŝδ) 0 = 0, 4.5) 21
23 almos surely. Due o 4.4) and 4.5) he nonnegaive benchmarked value Ŝδ) T canno be sricly greaer han zero wih any sricly posiive probabiliy, ha is Ŝδ) ) P T > 0 = 0. Thus, i follows P Ŝδ) T > 0) = 0 and he inequaliy 4.3) canno hold, which proves by Definiion 4.2 wih 4.3) ha here is no arbirage. 4.3 Fair Coningen Claim Pricing The described class of benchmark models is more general han he coninuous marke models admied under he sandard risk neural approach as presened in Karazas & Shreve 1998) or Björk 1998). The main difference is ha an equivalen risk neural maringale measure need no exis. Therefore, he sandard risk neural pricing mehodology is, in general, no applicable. A consisen pricing mehodology is needed ha generalizes risk neural pricing. Following Plaen 2002) one can use he fair pricing concep. We call a value process fair if is benchmarked value forms an A, P )maringale. Noe ha a value process need no o be a porfolio process. An A τ measurable random variable H τ wih ) Hτ E <, 4.6) S δ ) τ which pays he amoun H τ a some sopping ime τ, is called a coningen claim. For a given coningen claim he corresponding fair price U Hτ ) a ime [0, τ] is uniquely deermined by he fair pricing formula ) U Hτ ) = S δ ) E Hτ S δ ) τ A. 4.7) Noe ha he above fair pricing formula relies on he GOP, which is used as numeraire porfolio, see Long 1990). Noe ha no all benchmarked porfolios form A, P )maringales. By using he GOP as numeraire and he real world probabiliy measure as pricing measure, he fair pricing concep provides a consrucive and simple way of valuing coningen claims. For fair pricing in pracice i is advanageous o have a good proxy for he GOP as numeraire. Assumpions which ensure ha he marke porfolio of invesable wealh is a good proxy for he GOP were described in Secion 3.4. By modeling and calibraing he dynamics of he GOP i is possible o compue he real world expecaion in 4.7). 22
24 4.4 Risk Neural and Acuarial Pricing Le us illusrae ha he risk neural pricing mehodology appears as a paricular case of fair pricing. In a coninuous benchmark model a presumed risk neural probabiliy measure P θ has he RadonNikodym derivaive Λ T = dp θ wih Λ = dp {Λ, [0, T ]}, where Λ = Sδ ) 0 S 0) = Ŝ0) 4.8) S δ ) S 0) 0 Ŝ 0) 0 for [0, T ] and S 0) is he savings accoun process, see Karazas & Shreve 1998). By 4.8) we can rewrie he fair pricing formula 4.7) as ) S δ ) S τ 0) S 0) ) U Hτ ) = E H S δ ) τ S 0) S τ 0) τ A Λ τ S 0) ) = E H τ A Λ S τ 0) for [0, T ]. If he RadonNikodym derivaive process Λ is an A, P )maringale, hen his relaion leads by he Girsanov Theorem o he risk neural pricing formula S 0) ) U Hτ ) = E θ H S τ 0) τ A 4.9) for [0, τ]. Here E θ denoes expecaion wih respec o P θ and he savings accoun S 0) is he numeraire. However, in a coninuous benchmark model he RadonNikodym derivaive process Λ may only be a sric A, P )local maringale. Therefore, we emphasize ha such a model may no admi an equivalen risk neural maringale measure P θ. However, fair derivaive prices can be always compued as condiional expecaions direcly under he real world probabiliy measure p using he GOP S δ ) as numeraire. This means, by relaxing he assumpion on he exisence of an equivalen risk neural maringale measure, one can choose in a benchmark framework from a wider range of models han available under he sandard risk neural approach. Addiionally, we menion ha in he case when a coningen claim H T, which maures a a fixed dae T, is independen of he GOP S δ ) T, he fair pricing formula 4.7) yields he acuarial pricing formula U HT ) = P, T ) E ) H T A 4.10) for [0, T ]. Here P, T ) denoes he corresponding fair zero coupon bond price S δ ) ) P, T ) = E S δ ) A 4.11) T 23
25 a ime [0, T ]. Thus, he fair pricing concep generalizes no only sandard risk neural pricing bu covers also he classical acuarial pricing. Noe ha he ineres rae can be sochasic in 4.10) and 4.11), which is ofen no considered in he acuarial pricing lieraure. 5 The GOP as Bes Performing Porfolio 5.1 Ouperforming Growh Rae and Expeced Reurn The GOP can be considered o be he bes performing porfolio in various ways. In he following, we describe some mahemaical manifesaions of his fac. By relaion 2.15) in Definiion 2.2 i follows for any sricly posiive porfolio process S δ) ha a any ime he growh rae g δ of he GOP is never smaller han ha of any oher sricly posiive porfolio. This yields a firs characerizaion, which shows ha he GOP ouperforms he growh rae of all oher porfolios. From Corollary 4.1 we know ha any sricly posiive benchmarked porfolio Ŝδ) forms an A, P )supermaringale, which means ha he expeced reurn Ŝδ) +h E ) Ŝδ) Ŝ δ) A 0 5.1) of a benchmarked porfolio over any ime period [, +h] [0, T ] wih h > 0 is always nonposiive. This yields a second characerizaion of ouperformance, where he GOP, when used as benchmark, does no allow any nonnegaive benchmarked porfolio o generae expeced reurns greaer han zero. 5.2 Sysemaic Ouperformance wih Posiive Probabiliy For an invesor i is of ineres o know wheher or no i is possible o sysemaically ouperform he GOP wih some sricly posiive probabiliy by any oher porfolio over any ime period. To make his hird characerizaion mahemaically precise we inroduce he following definiion. Definiion 5.1 A sricly posiive porfolio S δ) is said o sysemaically ouperform wih posiive probabiliy anoher sricly posiive porfolio S δ) if for some sopping imes τ [0, T ] and σ [τ, T ] wih S δ) τ = S δ) τ 5.2) and S δ) σ S δ) σ, 5.3) 24
26 almos surely, i holds P S δ) σ ) > S σ δ) Aτ > ) This means in he sense of Delbaen & Schachermayer 1995) ha he GOP is a, so called, maximal elemen, which is imporan in he arbirage pricing heory. According o he above definiion, if a nonnegaive porfolio sysemaically ouperforms wih posiive probabiliy he GOP, hen i can generae over some ime period wealh ha is sricly greaer han ha accrued via he GOP wih some sricly posiive probabiliy. We can now prove he following resul. Corollary 5.2 No nonnegaive porfolio sysemaically ouperforms wih posiive probabiliy he GOP. Proof: wih benchmarked value Consider a benchmarked, nonnegaive porfolio Ŝδ) = {Ŝδ), [0, T ]} Ŝ δ) τ = 1 5.5) a a sopping ime τ [0, T ], almos surely, and assume for a laer sopping ime σ [τ, T ] he inequaliy Ŝ σ δ) 1, 5.6) almos surely. By he supermaringale propery of Ŝδ), provided by Corollary 4.1, he opional sampling heorem and propery 5.5) i follows ha 0 E Ŝδ) σ ) Ŝδ) Ŝδ) τ Aτ = E σ 1 ) Aτ ) Obviously, due o 5.7) and 5.6), he benchmarked value Ŝδ) σ canno be sricly greaer han Ŝδ) τ = 1 wih any sricly posiive probabiliy. Thus, i follows by 5.6) almos surely ha Ŝδ) σ = 1, which means ha S σ δ) = S σ δ ). This proves by Definiion 5.1 he Corollary Ouperforming he Long Term Growh Rae Le us define for a sricly posiive porfolio S δ) is long erm growh rae g δ as he almos sure limi ) a.s. 1 g δ = lim sup T T ln S δ) T. 5.8) S δ) 0 Noe ha his pahwise defined quaniy does no involve any expecaion. In he special case wih consan shor rae r and consan oal marke price for risk θ he GOP has by he law of large numbers he long erm growh rae g δ = r + θ 2 2. The following resul provides he ousanding propery of he GOP ha afer sufficien long ime is rajecory aains pahwise a value no less han ha of 25
27 any oher sricly posiive porfolio. This is a fourh characerizaion of ouperformance, which expresses a mos desirable feaure of a porfolio for a long erm invesor. Since i can be argued ha moneary auhoriies have a naural ineres in a long erm, pahwise ouperformance of all sricly porfolios by he marke porfolio of invesable wealh, he following heorem suppors he earlier discussed assumpion ha moneary auhoriies aim o maximize he growh of he oal invesable wealh in he marke. Theorem 5.3 The GOP S δ ) has almos surely he greaes long erm growh rae compared wih all oher sricly posiive porfolios S δ), ha is almos surely. g δ g δ, 5.9) Proof: Similar o Karazas & Shreve 1998) we consider a sricly posiive porfolio S δ) wih S δ) 0 = S δ ) ) By Corollary 4.1 he sricly posiive benchmarked porfolio Ŝδ) is an A, P ) supermaringale. As a supermaringale i saisfies by 5.10) he inequaliy exp{ε k} P sup Ŝ δ) > exp{ε k} ) Ŝδ) ) A 0 E k A 0 Ŝδ) 0 = ) k < for all k {1, 2,...} and ε 0, 1), see, for insance, Ellio 1982). For fixed ε 0, 1) one finds P sup ln k < Ŝδ) ) > ε k ) A 0 exp{ ε k} <. 5.12) Now, he BorelCanelli lemma implies he exisence of a random variable K ε such ha ) ln ε k ε sup T k Ŝδ) for all k K ε and k, almos surely. Thus, one has almos surely 1 ) ln Ŝδ) T ε T for all k K ε, and herefore, 1 T ln lim sup T S δ) S δ) 0 ) lim sup T ) 1 T ln S δ ) T S δ ) 0 + ε, 5.13) almos surely. Noing ha relaion 5.13) holds for all ε 0, 1) i follows by 5.8) he inequaliy 5.9). 26
28 Conclusion The paper shows ha he growh opimal porfolio GOP) plays a cenral role in finance. I has been demonsraed via a porfolio selecion heorem ha invesors, who maximize he drif of discouned porfolios wih comparable levels of aggregae diffusion coefficiens, form opimal porfolios which are linear combinaions of he GOP and he savings accoun. The free parameer in he family of opimal porfolios is he risk aversion coefficien which can be exploied for modeling. The risk aversion coefficien conrols he fracion of wealh invesed in he savings accoun. I is illusraed ha a sandard expeced uiliy maximizaion problem leads o an opimal porfolio sraegy, which can be characerized via a corresponding risk aversion coefficien process. By assuming ha invesors prefer opimal porfolios, he marke porfolio of invesable wealh is shown o be an opimal porfolio and key resuls relaed o he capial asse pricing model, he Sharpe raio and he Markowiz efficien fronier are easily derived. The GOP can be used in coningen claim pricing as naural numeraire ogeher wih he real world probabiliy as pricing measure, generalizing risk neural and acuarial pricing. Finally, i has been demonsraed in various ways how he GOP is he bes performing porfolio. A Appendix Proof of he Porfolio Selecion Theorem 3.3 Similarly as in Plaen 2002) le us fix a ime [0, T ] and a consan C > 0. We opimize hen he drif 3.4) α δ ) = d ψk δ ) θk under he consrain 3.6), where ψδ k )) 2 = γ δ = C. A.1) Using he Lagrange muliplier λ we maximize he funcion ) ) Gθ 1,..., θ d, ψδ, 1..., ψδ d, C, λ) = ψδ k θ k + λ C ψ k 2 δ. A.2) For ψ 1 δ,..., ψd δ o be opimal, i is necessary ha he firs order condiions Gθ 1,..., θ d, ψδ 1,..., ψd δ, C, λ) ψδ k = θ k λ 2 ψ k δ = 0 A.3) are saisfied for all k {1, 2,..., d}. This means ha for an opimal sraegy we mus have ψδ k = θk A.4) 2 λ 27
29 for all k {1, 2,..., d}. Using he consrain A.1) we ge from A.4) C = ) ) 2 ψ k 2 θ δ =. A.5) 2 λ We obain by A.4) and A.5) he relaion ψ k δ = C θ θk A.6) for k {1, 2,..., d}. From 3.3) and A.6) i follows for [0, T ] ha π j δ, = 1 S δ) ψδ k 1 j,k ) b = γδ S δ) θ θ k 1 j,k b A.7) for all j {1, 2,..., d}. By 2.11) we hen ge π 0 δ, = 1 π j δ, = 1 j=1 γδ 1 π 0 S δ) δ,) θ A.8) for [0, T ]. Then i follows by A.7) ha γ δ = 1 π0 δ, 1 π 0 δ, S δ) θ A.9) for all [0, T ]. Wih 3.3), A.6) and A.9) we hen obain 3.10). By 3.2), A.6) and A.9) we ge 3.11). Acknowledgemen The auhor likes o express his hanks o Moren Chrisensen, Wendell Fleming, Hardy Hulley, Shane Miller and Jerome Sein for valuable commens on he paper. References Basle 1996). Amendmen o he Capial Accord o Incorporae Marke Risks. Basle Commiee on Banking and Supervision, Basle, Swizerland. Björk, T. 1998). Arbirage Theory in Coninuous Time. Oxford Universiy Press. Black, F. & M. Scholes 1973). The pricing of opions and corporae liabiliies. J. Poliical Economy 81,
30 Breiman, L. 1961). Opimal gambling sysems for favorable games. In Proceedings of he Fourh Berkeley Symposium on Mahemaical Saisics and Probabiliy, Volume I, pp Cox, J. C. & C. F. Huang 1989). Opimum consumpion and porfolio policies when asse prices follow a diffusion process. J. Economic Theory 49, Delbaen, F. & W. Schachermayer 1995). The noarbirage propery under a change of numeraire. Sochasics Sochasics Rep. 53, Ellio, R. J. 1982). Sochasic Calculus and Applicaions. Springer. Fleming, W. H. & H. M. Soner 1992). Conrolled Markov Processes and Viscosiy Soluions. Springer. Fleming, W. H. & J. L. Sein 2004). Sochasic opimal conrol, inernaional finance and deb. J. Banking and Finance. o appear). Harrison, J. M. & D. M. Kreps 1979). Maringale and arbirage in muliperiod securiies markes. J. Economic Theory 20, Harrison, J. M. & S. R. Pliska 1981). Maringales and sochasic inegrals in he heory of coninuous rading. Sochasic Process. Appl. 113), Karazas, I. & S. E. Shreve 1991). Brownian Moion and Sochasic Calculus 2nd ed.). Springer. Karazas, I. & S. E. Shreve 1998). Mehods of Mahemaical Finance, Volume 39 of Appl. Mah. Springer. Kelly, J. R. 1956). A new inerpreaion of informaion rae. Bell Sys. Techn. J. 35, Khanna, A. & M. Kulldorff 1999). A generalizaion of he muual fund heorem. Finance Soch. 32), Korn, R. 1997). Opimal Porfolios. World Scienific. Krylov, N. V. 1980). Conrolled Diffusion Processes, Volume 14 of Appl. Mah. Springer. Laané, H. 1959). Crieria for choice among risky venures. J. Poliical Economy 67, Linner, J. 1965). The valuaion of risk asses and he selecion of risky invesmens in sock porfolios and capial budges. Rev. Econom. Sais. 47, Long, J. B. 1990). The numeraire porfolio. J. Financial Economics 26, Luenberger, D. G. 1997). Invesmen Science. Oxford Universiy Press, New York. Markowiz, H. 1952). Porfolio selecion. J. Finance VII1), Markowiz, H. 1959). Porfolio Selecion: Efficien Diversificaion of Invesmen. Wiley, New York. 29
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