THE CARLO ALBERTO NOTEBOOKS


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1 THE CARLO ALBERTO NOTEBOOKS Meanvaiance inefficiency of CRRA and CARA utility functions fo potfolio selection in defined contibution pension schemes Woking Pape No. 108 Mach 2009 Revised, Septembe 2009) Elena Vigna
2 Meanvaiance inefficiency of CRRA and CARA utility functions fo potfolio selection in defined contibution pension schemes Elena Vigna This vesion: Septembe 2009 Fist daft: Mach 2009) Abstact We conside the potfolio selection poblem in the accumulation phase of a defined contibution pension scheme in continuous time, and compae the meanvaiance and the expected utility maximization appoaches. Using the embedding technique pioneeed by Zhou and Li 2000) we fist find the efficient fontie of potfolios in the BlackScholes financial maket. Then, using standad stochastic optimal contol we find the optimal potfolios deived via expected utility fo popula utility functions. As a main esult, we pove that the optimal potfolios deived with the CARA and CRRA utility functions ae not meanvaiance efficient. As a coollay, we pove that this holds also in the standad potfolio selection poblem. We povide a natual measue of inefficiency based on the diffeence between optimal potfolio vaiance and minimal vaiance, and we show its dependence on isk avesion, Shape atio of the isky asset, time hoizon, initial wealth and contibution ate. Numeical examples illustate the extent of inefficiency of CARA and CRRA utility functions in defined contibution pension schemes. Keywods. Meanvaiance appoach, efficient fontie, expected utility maximization, defined contibution pension scheme, potfolio selection, isk avesion, Shape atio. JEL classification: C61, D81, G11, G23. 1 Intoduction 1.1 The poblem The cisis of intenational Pay As You Go public pension systems is focing govenments of most counties to dastically cut pension benefits of futue geneations and to encouage the development of fully funded pension schemes. It is wellknown that the efoms undetaken in most industialized counties give a pefeence towads defined contibution DC) plans athe than defined benefit DB) plans. Thus, defined contibution pension schemes will play a cucial ole in the social pension systems and financial advisos of DC plans will be needing flexible decision making tools to be appopiately tailoed to a membe s needs, to help he making optimal and conscious choices. Given that the membe of a defined contibution pension scheme has some feedom in choosing the Dipatimento di Statistica e Matematica Applicata and CeRP, Collegio Calo Albeto, Univesity of Toino. elena.vigna at econ.unito.it. I am gateful to Paolo Ghiadato and Massimo Mainacci fo useful advice, and to Xun Yu Zhou fo useful comments. I especially thank Bjane Højgaad and Luis Viceia fo fuitful discussions. All emaining eos ae mine.
3 investment allocation of he fund in the accumulation phase, she has to solve a potfolio selection poblem. Taditionally, the usual way to deal with it has been maximization of expected utility of final wealth. In this pape, we show that the optimal potfolios deived with CARA and CRRA utility functions ae not meanvaiance efficient. In fact, we pove that the vaiance of the optimal potfolios is not the minimal vaiance. As a bypoduct, we popose a natual measue of inefficiency, based on the diffeence between optimal potfolio vaiance and minimal vaiance. We show that in geneal inefficiency inceases with time hoizon and Shape atio of the isky asset and deceases with isk avesion. We also pove that the amount of wealth invested in the isky asset at any time is stictly positive. This means that shotselling of the isky asset is pevented by the adoption of meanvaiance efficient stategies in continuoustime. This esult, that is a desiable featue fo pension funds management, is standad in the singlepeiod famewok and is poven to hold tue also in the continuoustime setting. 1.2 Review of the liteatue The liteatue on the accumulation phase of defined contibution pension schemes is full of examples of optimal investment stategies esulting fom expected utility maximization. See fo instance Battocchio and Menoncin 2004), Boulie, Huang and Taillad 2001), Cains, Blake and Dowd 2006), Deelsta, Gasselli and Koehl 2003), Devolde, Bosch Pincep and Dominguez Fabian 2003), Di Giacinto, Gozzi and Fedeico 2009b), Gao 2008), Habeman and Vigna 2002), Xiao, Zhai and Qin 2007). Consistently with the economics and financial liteatue, the most widely used utility function exhibits constant elative isk avesion CRRA); i.e., the powe o logaithmic utility function, see, e.g., Boulie et al. 2001), Cains et al. 2006), Deelsta et al. 2003), Devolde et al. 2003), Gao 2008), Xiao et al. 2007). Some papes use the utility function that exhibits constant absolute isk avesion CARA); i.e., the exponential utility function, see, e.g., Battocchio and Menoncin 2004), Devolde et al. 2003). Finally, Di Giacinto et al. 2009b) use a geneal fom of utility function that includes as a special case a modified vesion of the powe utility function, and Habeman and Vigna 2002) minimize expected loss using a quadatic loss function, a common appoach in pension schemes optimization. In the context of DC pension funds the poblem of finding the optimal investment stategy that is meanvaiance efficient, i.e. minimizes the vaiance of the final fund given a cetain level of expected value of the fund has not been epoted in published aticles. This is not supising and is mainly due to the fact that the exact and igouous multipeiod and continuoustime vesions of the meanvaiance poblem have been poduced only quite ecently. The main eason of this delay in solving such a elevant poblem, since Makowitz 1952) and Makowitz 1959), lies in the difficulty inheent in the extension fom singlepeiod to multipeiod o continuoustime famewok. In the potfolio selection liteatue the poblem of finding the minimum vaiance tading stategy in continuoustime has been solved by Richadson 1989) via the matingale appoach. The same appoach has been used also by BajeuxBesnainou and Potait 1998) in a moe geneal famewok. They also find the dynamic efficient fontie and compae it to the static singlepeiod one. Regading the use of stochastic contol theoy to solve a meanvaiance optimization poblem, a eal beakthough was intoduced by Li and Ng 2000) in a discetetime multipeiod famewok and Zhou and Li 2000) in a continuoustime model. They show how to tansfom the difficult poblem into a tactable one, by embedding the oiginal poblem into a stochastic lineaquadatic contol poblem, that can then be solved though standad methods. These seminal papes have been followed by a numbe of extensions; see, fo instance, Bielecky, Jin, Pliska and Zhou 2005) and efeences theein. In the context of DC pension schemes, the techniques of Zhou and Li 2000) have been used by Højgaad and Vigna 2007). They find the efficient fontie fo the twoasset case as well as fo the n+1 2
4 asset case and show that the tagetbased appoach, based on the minimization of a quadatic loss function, can be fomulated as a meanvaiance optimization poblem, which is an expected esult. Meanvaiance and expected utility ae two diffeent appoaches fo dealing with potfolio selection. It is wellknown that in the singlepeiod famewok the meanvaiance appoach and expected utility optimization coincide if eithe the utility function is quadatic o the asset etuns ae nomal. Futhemoe, in the continuoustime famewok when pices ae lognomal thee is consistency between optimal choices and meanvaiance efficiency at instantaneous level see Meton 1971) and also Campbell and Viceia 2002)). Howeve, this does not imply that an optimal policy should emain efficient also afte two consecutive instants o, moe in geneal, on a time inteval geate than the instantaneous one. In fact, in geneal it does not. In pevious financial liteatue, the lack of efficiency of optimal policies in continuoustime was noted fo instance by some empiical woks that compae meanvaiance efficient potfolios with expected utility optimal potfolios and find that thee ae indeed diffeences between those potfolios. Among these, Hakansson 1971), Gaue 1981) and Gaue and Hakansson 1993). Related wok can be found in Zhou 2003). The aim of this pape is to compae the two leading altenatives fo potfolio selection in DC pension schemes, i.e. expected utility and meanvaiance optimization. Although the fact that a myopically efficient policy is not necessaily efficient ove the entie peiod is not a new esult and has been aleady mentioned see e.g. BajeuxBesnainou and Potait 1998)), this pape poves it diectly and focuses on the moe inteesting aspect of extent of inefficiency wheneve popula utility functions ae used. In paticula, we pove inefficiency of final potfolios on a given time peiod 0, T ) when CRRA and CARA utility functions ae maximized. We intoduce a natual measue of inefficiency of optimal potfolio deived with those utility functions, and find the intuitive esults that in geneal inefficiency inceases with time hoizon T and Shape atio of the isky asset and deceases with isk avesion. This dawback becomes paticulaly elevant in applications to pension funds, given the longtem hoizon involved and the fact that investos should cae moe about behaving efficiently on the entie time hoizon athe than in each single instant. Finally, we pove that the amount invested in the isky asset with a meanvaiance efficient stategy is stictly positive. In othe wods, shotselling is pevented a pioi with adoption of efficient stategies. This esult, togethe with the ecent esult by Chiu and Zhou 2009) that an efficient potfolio must have a nonzeo but not necessaily positive allocation to the iskless asset, sheds futhe light on the composition of efficient potfolios. We end with a numeical example, aimed at showing, in the context of a DC pension scheme, the extent of inefficiency of optimal potfolios deived with CRRA and CARA utility functions with typical isk avesion coefficients. 1.3 Agenda of the pape The emainde of the pape is oganized as follows. To impove eadability of the pape, almost all poofs and deivations of intemediate esults ae elegated in the Appendix. In section 2, we intoduce the model. In section 3, we epot the meanvaiance optimization poblem solved by Højgaad and Vigna 2007). In section 4, we outline the expected utility optimization appoach and solve the optimization poblem with the CARA and CRRA utility functions. In section 5, we state and pove a theoem that shows that the optimal potfolios deived in section 4 ae not efficient in the meanvaiance setting. In section 6, we define a measue of inefficiency, based on the diffeence between the vaiance of the CARA and CRRA optimal inefficient potfolios and the coesponding minimal vaiance and analyze the dependence of inefficiency on isk avesion paamete, Shape atio of isky asset, time hoizon, initial wealth and contibution ate. In section 7, we show that the tagetbased appoach is meanvaiance efficient. In section 8, we epot a numeical example, 3
5 aimed at showing the extent of inefficiency by adopting popula utility functions in a DC pension plan. Section 9 concludes and outlines futhe eseach. 2 The model A membe of a defined contibution pension scheme is faced with the poblem of how to invest optimally the fund at he disposal and the futue contibutions to be paid in the fund. The financial maket available fo he potfolio allocation poblem is the BlackScholes model see e.g. Bjök 1998)). This consists of two assets, a iskless one, whose pice Bt) follows the dynamics: dbt) = Bt)dt, 1) whee > 0, and a isky asset, whose pice dynamics St) follows a geometic Bownian motion with dift λ > 0 and diffusion σ > 0: dst) = λst)dt + σst)dw t), 2) whee W t) is a standad Bownian motion defined on a complete filteed pobability space Ω, F, {F t }, P), with F t = σ{w s) : s t}. The constant contibution ate payed in the unit time in the fund is c 0. The popotion of potfolio invested in the isky asset at time t is denoted by yt). The fund at time t, Xt), gows accoding to the following SDE: dxt) = {Xt)[yt)λ ) + ] + c}dt + Xt)yt)σdW t) X0) = x ) The amount x 0 is the initial fund paid in the membe s account, which can also be null, if the membe has just joined the scheme with no tansfe value fom anothe fund. The membe entes the plan at time 0 and contibutes fo T yeas, afte which she eties and withdaws all the money o convets it into annuity). The tempoal hoizon T is supposed to be fixed, e.g. T can be 20, 30 yeas, depending on the membe s age at enty. The membe of the pension plan has to choose the citeion fo he potfolio selection poblem. She esticts he attention to the two leading appoaches, meanvaiance appoach and expected utility maximization. 3 The meanvaiance appoach In this section, we assume that the individual chooses the meanvaiance appoach fo he potfolio selection poblem. She then pusues the two conflicting objectives of maximum expected final wealth togethe with minimum vaiance of final wealth, namely she seeks to minimize the vecto [ EXT )), V axt ))]. Definition 1 An investment stategy y ) is said to be admissible if y ) L 2 F 0, T ; R). 4
6 Definition 2 The meanvaiance optimization poblem is defined as Minimize J 1 y )), J 2 y ))) EXT )), V axt ))) subject to { y ) admissible X ), y ) satisfy 3). An admissible stategy y ) is called an efficient stategy if thee exists no admissible stategy y ) such that J 1 y )) J 1 y )) J 2 y )) J 2 y )), 5) and at least one of the inequalities holds stictly. In this case, the point J 1 y )), J 2 y ))) R 2 is called an efficient point and the set of all efficient points is called the efficient fontie. Poblem 4) is equivalent to min[ EXT )) + αv axt ))], 6) y ) whee α > 0. Notice that α is a measue of the isk avesion of the individual. Zhou and Li 2000) show that poblem 6) is equivalent to Minimize Jy )), α, µ) E[αXT ) 2 µxt )], 4) subject to { y ) admissible X ), y ) satisfy 3). 7) whee µ = 1 + 2αEXT )). 8) In solving poblem 7) we follow the appoach pesented in Zhou and Li 2000). The deivation of the solution of this LQ contol poblem is fully contained in Højgaad and Vigna 2007), so we efe the inteested eade to this pape fo details and hee epot only the solution. The optimal investment allocation at time t, given that the fund is x, is given by whee yt, x) = λ σ 2 x [ x δe T t) + c ] 1 e T t) ), 9) δ = µ 2α. 10) The evolution of the fund unde optimal contol Xt) is easily obtained: [ ] dxt) = β 2 )Xt) + e T t) β 2 δ + β2 c ) + c β2 c ) dt+ [ ] + βxt) + e T t) βδ + βc ) βc dw t), whee β := λ σ, 12) is the Shape atio of the isky asset. By application of Ito s lemma to 11), we obtain the SDE that govens the evolution of X 2 t): dx 2 t) = [2 β 2 )X 2 t) + 2cXt) + β 2 δ + c )e T t) c )2 ]dt+ 2β{X 2 t) [δ + c )e T t) c ]Xt) + c }dw t). 13) 5 11)
7 If we take expectations on both sides of 11) and 13), we find that the expected value of the optimal fund and the expected value of its squae follow the linea ODE s: dext)) = [ β 2 )EXt)) + e T t) β 2 δ + c ) + c β2 c )]dt EX0)) = x 0 14) and dex 2 t)) = [2 β 2 )EX 2 t)) + 2cEXt)) + β 2 δ + c )e T t) c ) 2]dt EX 2 0)) = x 2 0. By solving the ODE s we find that the expected value of the fund unde optimal contol at time t is EXt)) = x 0 + c ) e β2 )t + δ + c ) e T t) δ + c ) e T t) β2t c, 16) and the expected value of the squae of the fund unde optimal contol at time t is: EX 2 t)) = x 0 + c ) 2 e β 2 2)t δ + c ) 2 e 2T t) β 2t 2c ) δ + c e T t) + 2c ) δ + c e T t) β 2t 2c x0 + c ) e β 2 )t + δ + c ) 2 e 2T t) + c2 At teminal time T we have: and EX 2 T )) = EXT )) = 2. 15) 17) x 0 + c ) ) e β2 )T + δ 1 e β2 T c e β2t, 18) x 0 + c ) ) 2 e β 2 2)T + δ 2 1 e β2 T 2c x 0 + c ) e β2 )T + c2 2 e β2t. 19) We now define an impotant quantity, that will play a special ole in the est of the pape: x 0 := x 0 e T + c et 1). 20) The meaning of x 0 is clea: it is the fund that would be available at time T investing initial fund and contibutions in the iskless asset. The expected optimal final fund can be ewitten in tems of α, β and x 0 : EXT )) = x 0 + eβ2t 1 21) 2α It is easy to see that the expected optimal final fund is the sum of the fund that one would get investing the whole potfolio always in the iskless asset plus a tem, eβ2t 1 2α that depends both on the goodness of the isky asset w..t. the iskless one and on the weight given to the minimization of the vaiance. Thus, the highe the Shape atio of the isky asset, β, the highe the expected optimal final wealth, eveything else being equal; the highe the membe s isk avesion, α, the lowe its mean. These ae intuitive esults. Using 21), 10) and 8), it is possible to wite yt, x) in this way: ) yt, x) = β x x 0 e t + c ) σx et 1) e T t)+β2 T. 22) 2α The amount xyt, x) invested in the isky asset at time t is popotional to the diffeence between the fund x at time t and the fund that would be available at time t investing always only in the iskless asset, minus a tem that depends on β 2, α and the time to etiement. The highe the weight α given to the minimization of the vaiance, the lowe the amount invested in the isky asset, and 6
8 vice vesa, which is an obvious esult. Evidently, α is a measue of isk avesion of the individual: the highe α the highe he isk avesion. It is clea that a necessay and sufficient condition fo the fund to be invested at any time t in the iskless asset is α = + : the exteme) stategy of investing the whole potfolio in the iskless asset is optimal if and only if the isk avesion is infinite. Using 21) and 22) one can expess the optimal investment stategy in tems of the expected final wealth in the following way: [ ] yt, x) = λ σ 2 x E[XT )]e T t) c ) x 1 e T t) e T t) ). 23) 2α The intepetation is that the amount xyt, x) invested in the isky asset at time t is popotional to the diffeence between the fund x at time t and the amount that would be sufficient to guaantee the achievement of the expected value by adoption of the iskless stategy until etiement, minus a tem that depends on α and the time to etiement. In ealistic situations, when the minimization of the vaiance plays a ole in the investo s decisions, expessions 18) and 19) allow one to choose he own pofile isk/ewad. In fact, as in classical meanvaiance analysis, it is possible to expess the vaiance  o the standad deviation  of the final fund in tems of its mean. The subjective choice of the pofile isk/ewad becomes easie if one is given the efficient fontie of feasible potfolios. It can be shown see Appendix A) that the vaiance of the final wealth is e β2 T e β2t 1 2α V axt )) = 1 e β2 T ) 2 = eβ2t 1 4α 2, 24) The vaiance is inceasing if the Shape atio inceases, which is an expected esult: in this case the investment in the isky asset is heavie, leading to highe vaiance. Obviously, the highe the isk avesion α, the lowe the vaiance of the final fund, which is null if and only if α = + : in this case, the potfolio is entiely invested in the iskfee asset and XT ) = EXT )) = x 0. The efficient fontie of potfolios is see Appendix A): ) EXT )) = x 0 + e β2t 1 σxt )). 25) Expectedly, the efficient fontie in the meanstandad deviation diagam is a staight line with slope e β2t 1 which is called pice of isk see Luenbege 1998)): it indicates by how much the mean of the final fund inceases if the volatility of the final fund inceases by one unit. When c = 0, the efficient fontie coincides with that found by Richadson 1989), BajeuxBesnainou and Potait 1998) and Zhou and Li 2000) fo selffinancing potfolios. 4 The expected utility appoach: optimal potfolios fo CARA and CRRA utility functions 4.1 The expected utility maximization poblem In this section, we assume that the individual solves he potfolio selection poblem with the expected utility maximization appoach. Theefoe, he aim is now find the optimal investment stategy ove 7
9 time that maximizes the expected value of final wealth. She then wants to solve Maximize Jy ))) E[UXT ))], subject to { y ) admissible X ), y ) satisfy 3). 26) Poblem 26) is a standad optimization poblem that can be dealt with via classical contol theoy. We efe the inteest eade to classical texts such as Yong and Zhou 1999), Øksendal 1998), Bjök 1998) and contain ouselves to a bief desciption of the basic steps to follow. One fist defines a moe geneal pefomance function Jy ); t, x) = E x [UXT ))], 27) whee E x [ ] = E[ Xt) = x], then defines the optimal value function as the supemum of the pefomance citeion among admissible contols, V t, x) := sup Jy ); t, x). 28) y ) Then, applying a fundamental theoem of stochastic contol theoy, wites the HamiltonJacobi Bellman HJB) equation that the value function associated to this poblem must satisfy: [ V sup y t with bounday condition + xyλ ) + ) + c) V x + 1 ] 2 x2 σ 2 y 2 2 V x 2 = 0, 29) V T, x) = Ux). 30) Then, she wites the optimal contol associated to the poblem, as a function of patial deivatives of the value function: y t, x) = λ V x σ 2, 31) x V xx whee V x = V x and V xx = 2 V x 2, plugs 31) into the HJB equation to find the nonlinea PDE V t + x + c)v x 1 2 β2 V 2 x V xx = 0, 32) with bounday condition 30). By solving the PDE 30)32) one etieves the optimal contol via 31). The usual way to solve the nonlinea PDE 32) is by guessing the fom of the solution exploiting the natual similaity with the utility function selected. The guess technique woks well with the utility functions consideed in this pape, namely the exponential, the logaithmic and the powe utility function. In each of the thee cases, we epot the expected value and the vaiance of teminal wealth. All deivations ae in Appendix B. 4.2 CARA: Exponential utility function Conside the exponential utility function Ux) = 1 k e kx, 33) 8
10 with constant) AowPatt coefficient of absolute isk avesion equal to ARAx) = U x) U x) = k. It can be shown see Appendix B.1) that the expected final wealth is EX T )) = and the vaiance of the final fund is x 0 + c ) e T c + β2 T k = x 0 + β2 T k, 34) V ax T )) = EX T )) 2 ) E 2 X T )) = β2 T k 2. 35) 4.3 CRRA: Logaithmic utility function Conside the logaithmic utility function The constant) AowPatt coefficient of elative isk avesion is Ux) = ln x. 36) RRAx) = U x) U x) x = 1. It can be shown see Appendix B.2) that the expected final wealth is and the vaiance of the final fund is EX T )) = e AT x 0 + c 1 e T )) = x 0 e β2t, 37) V ax T )) = e KT e 2AT )x 0 + c 1 e T )) 2 = EX T ))) 2 e β2t 1), 38) whee A = + β 2, K = 2 + 3β 2. 39) 4.4 CRRA: Powe utility function Conside the powe utility function Ux) = xγ γ, with γ < 1 and constant) AowPatt coefficient of elative isk avesion equal to RRAx) = U x) U x) x = 1 γ. It can be shown see Appendix B.3) that the expected final wealth is and the vaiance of the final fund is EX T )) = e AT x 0 + c 1 e T )) = x 0 e β2 T 1 γ, 40) V ax T )) = e KT e 2AT )x 0 + c 1 e T )) 2 = e β 2 T 1 γ) 2 1)EX T ))) 2, 41) whee A and K ae given by 39). 9
11 5 Meanvaiance vesus expected utility The aim of this section is to pove that the optimal potfolios deived via maximization of expected utility of final wealth with the utility functions that exhibit constant absolute isk avesion and constant elative isk avesion ae not efficient in tems of meanvaiance. Befoe showing the pocedue, it is convenient to ecall some pevious esults. In section 3 we have shown that a membe of a defined contibution pension scheme wanting to solve the following meanvaiance poblem min[ EXT )) + αv axt ))] 42) y ) whee α > 0 measues he isk avesion, should invest optimally in such a way as to obtain a final fund that has the following mean: and the following vaiance: EXT )) = x 0 + eβ2t 1, 43) 2α V axt )) = eβ2t 1 4α 2. 44) In othe wods, fo this poblem thee exists no potfolio that has a final mean equal to 43) with a vaiance stictly lowe than 44). Equivalently, thee exists no potfolio that has a final vaiance equal to 44) with a mean stictly geate than 43). In ode to pove that the utility function U poduces optimal potfolios that ae not efficient, one can poceed along the following steps: 1. Deive the expectation and vaiance of final wealth unde optimal contol associated to the poblem of maximization of UXT )), EXU T )) and V ax U T )). 2. Then, pove eithe a) EXU T )) = EXT )) V ax U T )) > V axt )); o b) V axu T )) = V axt )) EX U T )) < EXT )). As a bypoduct, eithe the diffeence o the diffeence V ax UT )) V axt )) > 0 EXT )) EX UT )) > 0 quantifies the degee of inefficiency of the utility function U. In the poof of Theoem 3 we will follow pocedue a). We ae now eady to state and pove the main esult of this pape. Theoem 3 Assume that the financial maket and the wealth equation ae as descibed in section 2. Assume that the potfolio selection poblem is solved via maximization of the expected utility of final wealth at time T, with pefeences descibed by the utility function Ux), as in section 4.1. Then, the couple V axu T )), EX U T ))) associated to the final wealth unde optimal contol X U T ) is 10
12 not meanvaiance efficient in the following cases: i) Ux) = 1 k e kx ; ii) Ux) = ln x; iii) Ux) = xγ γ. Poof. See Appendix C. 5.1 The special case c = 0: the usual potfolio selection poblem It is athe clea fom the pevious analysis, that the inequalities still hold in the thee cases when c = 0. In this case, fo the poblem to be not tivial it must be x 0 > 0. Theefoe we find that in typical potfolio selection analysis in continuous time, in a standad Black & Scholes financial maket the expected utility maximization citeion with CARA and CRRA utility functions leads to an optimal potfolio that is not meanvaiance efficient. We can summaize this esult in the following coollay. Coollay 4 Assume that an investo wants to invest a wealth of x 0 > 0 fo the time hoizon T > 0 in a financial maket as in section 2 and wealth equation 3) with c = 0. Assume that she maximizes expected utility of final wealth at time T. Then, the couple V axu T )), EX U T ))) associated to the final wealth unde optimal contol XU T ) is not meanvaiance efficient in the following cases: i) Ux) = 1 k e kx ; ii) Ux) = ln x; iii) Ux) = xγ γ. Poof. The poof is obvious, by setting c = 0 in the poof of Theoem 3) and obseving that inequalities 113), 120) and 124), still hold. 6 Measue of inefficiency The esult poven in the pevious section is not supising. In fact, by definition the vaiance of a potfolio on the efficient fontie is lowe o equal than the vaiance of any othe potfolio with the same mean. Howeve, what can be inteesting is the extent of inefficiency of a potfolio found with the expected utility maximization appoach, and its dependence on time hoizon, isk avesion and financial maket. This seems elevant fo applicative puposes, consideing the fact that EU appoach with CARA and CRRA utility functions is widely used in the potfolio selection liteatue, also fo longtem investment such as pension funds. One may well ague that if the individual s pefeences ae epesented by, say, the powe utility function, then she is not meanvaiance optimize, and she does not cae not to be. This is fai. Howeve, we obseve thee things. Fist, it is evidently difficult fo an agent to specify he own utility function and the coesponding isk avesion paamete. On the contay, it is elatively easy to eason in tems of tagets to each. This was obseved also by Kahneman and Tvesky 1979) in thei classical pape on Pospect Theoy. As will be shown late, the tagetbased appoach is meanvaiance efficient. Second, fo most individuals it is athe immediate to undestand the meanvaiance citeion. It is indeed enough to show them two distibutions of final wealth with same mean but diffeent vaiances: in the context of pension funds, given that the final wealth efes to etiement saving, most wokes would pobably choose the distibution with lowe vaiance. Thid, the meanvaiance citeion is still the most used citeion to 11
13 value and compae investment funds pefomances. We theefoe believe that evey futhe step in undestanding the meanvaiance appoach should be encouaged in the context of pension schemes. In this section, we define a measue of meanvaiance inefficiency fo a potfolio. As mentioned in section 5, the inefficiency of an optimal potfolio can be natually measued by the diffeence between its vaiance and the coesponding minimal vaiance. We theefoe define the Vaiance Inefficiency Measue as V IMX UT )) := V ax UT )) V axt )). 45) In each of the thee cases consideed, we analyze the dependence of the inefficiency measue on the elevant paametes of the poblem, namely the isk avesion of the membe, the Shape atio β, the time hoizon T, the initial wealth x 0 and the contibution ate c. We will pefom the analysis of V IMXU T )) fo the thee cases sepaately. 6.1 Exponential utility function When we have V IMX T )) = eβ2t 1) 2αk Ux) = 1 k e kx, 1 k ) = β2 T 2α k 2 ) 1 β2 T e β2t. 46) 1 So that 1. The inefficiency is a deceasing function of the absolute isk avesion coefficient ARA = k. 2. The inefficiency is an inceasing function both of the Shape atio β and the time hoizon T. 3. The inefficiency does not depend on the initial fund x 0 and on the contibution ate c. Let us make some comments on the exteme cases in which the two potfolios coincide and the inefficiency 46) is null. It is athe obvious that fo k + the optimal potfolio is the iskless one, with mean x 0 and zeo vaiance, since the investo has infinite isk avesion. At the same time, due to 112), also α + and the efficient potfolio is the iskless one. Similaly, it is obvious that the diffeence in 46) is null also in the case e β2t = 1. In fact, this is possible if eithe β = 0 o T = 0. In both cases, we have that the optimal potfolio is invested entiely in the iskless asset and the final deteministic potfolio at time T 0 is x Logaithmic utility function When we have So that Ux) = ln x, V IMX T )) = x 2 0e β2t 1) 2 e β2t + 1). 47) 1. The inefficiency is an inceasing function both of the Shape atio β and the time hoizon T. 12
14 2. The inefficiency is an inceasing function of both the initial fund x 0 0 and the contibution ate c 0. Given that x 0 > 0 fo the poblem not to be tivial, the diffeence in 47) is null if and only if e β2t = 1. As obseved ealie, this is possible if eithe β = 0 o T = 0. In both cases, we have that the optimal potfolio is invested entiely in the iskless asset and the final potfolio at time T 0 is x Powe utility function When we have V IMX T )) = With the change of vaiables: we have: x 2 0 e β2t 1 Ux) = xγ γ, e 2β 2 T 1 γ) e β2 T 1 γ) 2 b := e β2 T 1)e β2t 1) e β2 T 1 γ 1) 2 ). 48) a := 1 1 γ, V IMX T )) = V IMa, b) = x2 0 b 1 ba2 +2a+1 b a2 +2a b 2a+1 + 2b a 1). 49) Fom the poof of Theoem 3), point iii), it is athe clea that What is moe difficult to pove is that V IM a V IM b > 0. 50) > 0 51) fo all values of a > 0, so that is still an open poblem. We ae able to pove it fo b > 2 a > a whee a 0.45 is the positive oot of 2a 3 + 4a 2 1 = 0. These values imply RRA < 1 a 2.22 and β 2 T > ln Ou conjectue is that 51) holds fo all possible values of a > 0, b > 1, but this is still to be poved. In conclusion, we have 1. The inefficiency is a deceasing function of the elative isk avesion coefficient RRA = 1 γ. 2. The inefficiency is an inceasing function both of the Shape atio β and the time hoizon T, if β 2 T > ln 2 and a < a, whee a > 0 solves 2a 3 + 4a 2 1 = The inefficiency is an inceasing function of both the initial fund x 0 > 0 and the contibution ate c > 0. One can see that fo γ the optimal potfolio is the iskless one, with mean x 0 and zeo vaiance, since the investo has infinite isk avesion. At the same time, due to 123), also the efficient potfolio will be the iskless one. Theefoe in this case, the diffeence in 48) is null. Similaly, one can see that the diffeence in 48) is null also in the case e β2t = 1. In fact, this is possible if eithe β = 0 o T = 0. In both cases, we have that the optimal potfolio is invested entiely in the iskless asset and the final deteministic potfolio at time T 0 is x 0. 13
15 7 Quadatic loss function: the tagetbased appoach In this section, we show the expected esult that in the famewok outlined in section 2 the quadatic utilityloss function is consistent with the meanvaiance appoach. Most of the esults of this section can be found in Højgaad and Vigna 2007). Since in this pape the focus is on potfolio choices in defined contibution pension schemes, we will conside a modified vesion of the simple quadatic utility function, consideing a tagetbased appoach induced by a quadatic loss function. Optimization of quadatic loss o utility function is a typical appoach in pension schemes. Examples of this appoach can be found fo instance in Boulie, Michel and Wisnia 1996), Boulie, Tussant and Floens 1995), Cains 2000), Habeman and Sung 1994) fo defined benefit pension funds, in Habeman and Vigna 2002), Gead, Habeman and Vigna 2004), Gead, Højgaad and Vigna 2010) fo defined contibution pension schemes. Højgaad and Vigna 2007) conside the poblem of a membe of a DC pension scheme who chooses a taget value at etiement F and chooses the optimal investment stategy that minimizes 1 E [ XT ) F ) 2]. In these cicumstances, we shall say that the membe solves the potfolio selection poblem with the tagetbased TB) appoach. Fo the poblem to be financially inteesting, the final taget F should be chosen big enough, i.e. such that F > x 0. 52) We can see fom Gead et al. 2004) that the optimal investment stategy fo the TB appoach is given on the following fom 2 whee y tb t, x) = λ σ 2 x Gt)), 53) x Gt) = F e T t) c 1 e T t) ), 54) and y tb t, x) = λ σ 2 x [ x F e T t) c )] 1 e T t) ). 55) Let us notice that the function Gt) epesents a sot of taget level fo the fund at time t: should the fund Xt) each Gt) at some point of time t < T, then the final taget F could be achieved by adoption of the iskless stategy until etiement. Howeve, as will be shown, the achievement of Gt) and theefoe the sue achievement of the taget, is pevented unde optimal contol by the constuction of the solution. In ode to pove efficiency of the TB appoach, we now need to set the expected value of final wealth unde optimal contol equal to that of the mean vaiance appoach. To calculate the 1 In a moe geneal model, pesented in Gead et al. 2004), the individual chooses a taget function F t) so as to minimize [ T ] E e ϱt ε 1Xt) F t)) 2 dt + ε 2e ϱt XT ) F T )) 2. 0 Hee, fo consistent compaisons we eliminate the unning cost and select only the teminal wealth poblem. 2 Notice that Gead et al. 2004) conside the decumulation phase of a DC scheme. The diffeence in the wealth equation is that in that case thee ae peiodic withdawals fom the fund wheeas hee we have peiodic inflows into the fund. Fomally the equations ae identical if one sets b 0 = c. 14
16 expected value of the final fund in the TB appoach we let X t) denote the optimal wealth function fo this case. Then in Gead et al. 2004) it can be seen that X t) satisfies the following SDE: dx t) = [Gt) + c + β 2 )Gt) X t))]dt + βgt) X t))dw t). 56) As in pevious wok, let us define the pocess Then Zt) = Gt) X t). 57) dzt) = G t)dt dx t) = β 2 )Zt)dt βzt)dw t), 58) whee in the last equality we have applied 54) and 56). We can see that the pocess Zt) follows a geometic Bownian motion and is given by: Zt) = Z0)e 3 2 β2 )t βw t). 59) Noting that one has Thus GT ) = F, ZT ) = F X T ). EX T )) = F EZT )) = F G0) x 0 )e β2 )T = e β2t x e β2t )F. 60) The expected final fund tuns out to be a weighted aveage of the taget and of the fund that one would have by investing fully in the iskless asset. Futhemoe, it is staightfowad to see that in the TB appoach the final taget cannot be eached. In fact, fom 59), one can see that Zt) > 0 fo t T if Z0) > 0. Let us notice that this holds, due to 52). In fact, Z0) = G0) x 0 = F e T c 1 e T ) x 0 = e T F x 0 ) > 0. 61) Theefoe, the final fund is always lowe than the taget. This esult is not new. A simila esult was aleady found by Gead et al. 2004) and by Gead, Habeman and Vigna 2006) in the decumulation phase of a DC scheme: with a diffeent fomulation of the optimization poblem and including a unning cost, in both woks they find that thee is a natual timevaying taget that acts as a sot of safety level fo the needs of the pensione and that cannot be eached unde optimal contol. Peviously, in a diffeent context, a simila esult was found by Bowne 1997): in a poblem whee the aim is to maximize the pobability of hitting a cetain uppe bounday befoe uin, when optimal contol is applied the safety level the minimum level of fund that guaantees fixed consumption by investing the whole potfolio in the iskless asset) can neve be eached. We ae now eady to state and pove a theoem that shows that the tagetbased appoach is meanvaiance efficient and that each point on the efficient fontie coesponds to the optimal solution of a TB optimization poblem. Theoem 5 Assume that the financial maket and the wealth equation ae as descibed in section 2. Assume that the potfolio selection poblem is solved via minimization of expected loss of final wealth at time T, with pefeences descibed by the loss function Lx). Then, i) the couple V ax L T )), EX L T ))) associated to the final wealth unde optimal contol X L T ) is meanvaiance efficient if Lx) = F x) 2 ; ii) each point V axt )), EXT ))) on the efficient fontie as outlined in section 3 is the solution of an expected loss minimization poblem with loss function Lx) = F x) 2. 15
17 Poof. See Appendix D. Thus, evey solution to a tagetbased optimization poblem coesponds to a point on the efficient fontie, and each point of the efficient fontie can be found by solving a tagetbased optimization poblem. The onetoone coespondence between points of the efficient fontie and tagetbased optimization poblems is given by the following elationship between the paamete α of the meanvaiance appoach and the value of final taget of the tagetbased appoach: whee we have used 60) and 157). α = e β2 T 2F x 0 ), 62) The fact that the tagetbased appoach is a paticula case of the meanvaiance appoach should put an end to the citicism of the quadatic utility function, that penalizes deviations above the taget as well as deviations below it. The intuitive motivation fo suppoting such a utility function in DC schemes: The choice of tying to achieve a taget and no moe than this has the effect of a natual limitation on the oveall level of isk fo the potfolio: once the taget is eached, thee is no eason fo futhe exposue to isk and theefoe any suplus becomes undesiable finds hee full justification in a igouous setting. We notice that a simila esult was mentioned, without poof, by Bielecky et al. 2005). They noticed, howeve, that the potfolio s expected etun would be unclea to detemine a pioi. In contast, hee we povide the exact expected etun and vaiance of the optimal potfolio via optimization of the quadatic loss function. We ae thus able to detemine completely the exact point on the efficient fontie of potfolios. A final emak about an intinsec featue of the optimal efficient investment stategies. Fom 53) we can see that anothe diect consequence of the positivity of Zt) is the fact that unde the tagetbased appoach the amount invested in the isky asset unde optimal contol is always positive. Obviously, this is the case also fo the meanvaiance appoach. This leads us to the fomulation of the following coollay. Coollay 6 Conside the financial maket and the wealth equation as in section 2. Conside the efficient fontie of feasible potfolios, as outlined in section 3. Then, the optimal amount invested in the isky asset at any time 0 t < T is stictly positive. Poof. This follows fom 159), 57), 59) and 61). This is a desiable popety, given that the constained potfolio poblem has not been solved yet fo the tagetbased appoach. In fact, this natual featue allows to educe the bilateal constained potfolio poblem in the noboowing constaint poblem, given that the noshot selling comes with no cost fo the natue of the poblem. Solving the noshot selling constained poblem with the tagetbased appoach in the decumulation phase of a defined contibution pension scheme is subject of ongoing eseach see Di Giacinto, Fedeico, Gozzi and Vigna 2009a)). 8 Numeical application 8.1 The efficient fontie In this section, with a numeical example we intend to illustate the extent of inefficiency of optimal potfolios fo DC pension schemes wheneve CARA and CRRA utility functions ae used to solve 16
18 the potfolio selection poblem. We will do this by compaing optimal inefficient potfolios with the coesponding meanvaiance efficient one. Fo illustative puposes, we will also epot esults fo the lifestyle stategy see e.g. Cains et al. 2006)), widely used by DC pension plans in UK. In the lifestyle stategy the fund is invested fully in the isky asset until 10 yeas pio to etiement, and then is gadually switched into the iskless asset by switching 10% of the potfolio fom isky to iskless asset each yea. The paametes fo asset etuns ae as in Højgaad and Vigna 2007), i.e. = 0.03, λ = 0.08, σ = 0.15, c = 0.1, x 0 = 1, T = 20. Theefoe, the Shape atio is β = 0.33 and the fund achievable unde the iskless stategy is x 0 = The compaison will be done fo each of the thee inefficient utility functions, consideing appopiate values fo the isk avesion displayed. It is fa beyond the scope of this pape to discuss the choice of appopiate values fo the paametes of absolute and elative isk avesion fo the exponential and the powe utility function. Howeve, we notice that while thee seems to be oveall ageement acoss the liteatue egading typical values of the RRA coefficient, this is not the case fo the choice of the ARA coefficient. In addition, thee seems to be little evidence of constant absolute isk avesion displayed by investos see fo instance, Guiso and Paiella 2008)). The value of ARA = 20 used by Battocchio and Menoncin 2004) is not appopiate in this context, because it would imply an α value of aound 37, with implied final taget F = 4.67, too much close to the basic value achievable with the iskless stategy, x 0 = Theefoe, such high values of k, used also elsewhee in the liteatue see fo instance Joion 1985)) have to be consideed too high in this model with this time hoizon. On the othe hand, Guiso and Paiella 2008) suggest that the aveage absolute isk avesion should ange aound 0.02, a too low value fo this context, implying a final taget of F = 129, clealy uneasonable. We have then decided to test diffeent levels of isk avesion fo the powe case, as in many pevious woks of this kind. We will be consideing RRA=1 logaithmic utility), RRA=2 and RRA=5. Howeve, in each case we will epot the coesponding esults also fo the exponential utility function, as implied by the choice of the elative isk avesion. The choice of RRA = 2 is motivated by the evident consensus in the liteatue egading constant elative isk avesion coefficient of about 2. See, fo instance Schlechte 2007), who sets a minimum bound of aound 1.92 with no savings, and of 2.42 in the pesence of savings. Moe specifically, egading active membes of pension schemes Canessa and Doich 2008) in a ecent suvey epoted an oveall aveage of elative isk avesion of about 1.81, depending on the age of the goup unde investigation. In paticula, the RRA coefficient of the goup unde study vaies between 1.59 and 1.88 fo younge membes, and between 2.21 and 2.25 fo olde ones. The choice of RRA = 5, motivated by the impotance of showing esults elative to highe isk avesion, is in line with simila choices fo DC pension plans membes see Cains et al. 2006)) and is consistent with the choice of the final taget opeated by Højgaad and Vigna 2007). Not least, RRA=5 gives an expected final fund vey simila to that empiical found by application of the lifestyle stategy see late) and theefoe allows consistent compaisons. We have then the following thee cases: low isk avesion: RRA = 1, that is the logaithmic utility function, implies α = , which in tun leads to F = 46.66; this coesponds to k = in the exponential model; medium isk avesion: RRA = 2 implies α = 0.44, which in tun leads to F = 14.99; this coesponds to k = 0.24 in the exponential model; high isk avesion: RRA = 5 implies α = 1.61, which in tun leads to F = 7.43; this coesponds to k = 0.87 in the exponential model. Table 1 epots fo each value of the RRA coefficient the coesponding α value, the coesponding taget in the TB appoach, the coesponding coefficient of absolute isk avesion k, mean and 17
19 vaiance of the efficient potfolio, vaiance of the optimal potfolio fo powe and exponential utility function and VIM Vaiance Inefficiency Measue) fo both utility functions. Clealy, when RRA=1 the powe degeneates in the logaithmic utility function. Remak 1 Notice that in the fifth column the MV efficient expected value EXT )) coincides with the expected value associated to the powe and exponential optimal potfolios, EX T )). In the label we epot only EXT )) fo space constaints. RRA Taget ARA MV efficient MV efficient Powe Exponential Powe Exponential 1 γ α F k EXT )) Va XT )) Va X T )) Va X T )) V IM V IM Table 1. It is evident the extent of inefficiency when the isk avesion is too low. Namely, the VIM in the logaithmic case is and when ARA = 0.06 VIM in the exponential case is 463. Moe in geneal, one can obseve that the inefficiency deceases when RRA and ARA incease. This comes diectly fom esults shown in section 6. We also obseve the inteesting featue that in evey choice fo the elative isk avesion coefficient, the inefficiency poduced by the exponential utility function is lowe than that of the powe utility. Figues 1, 2 and 3 epot in the usual standad deviation/mean diagam the efficient fontie and the optimal potfolios in the cases RRA=1, 2, 5, espectively. In Figue 3, we have added also the empiical potfolio obtained via adoption of the lifestyle stategy. In ode to find it, we have caied out 1000 Monte Calo simulations with discetization done on a weekly basis and in each scenaio have applied the lifestyle stategy descibed befoe and obtained the final wealth. We have then plotted the point with coodinates equal to standad deviation and mean of the distibution of final wealth ove the 1000 scenaios. Noticing that the mean of the final wealth fo the lifestyle is 7.31, we have plotted it only in Figue 3, that epots esults fo RRA=5, with mean equal to Fo completeness of exposition, Table 2 epots fo each isk avesion coefficient, the standad deviation of each optimal potfolio, that is the xcoodinate of the optimal point in the Figues, the ycoodinate being the mean, epoted in the last column. As befoe, hee Remak 1 applies. RRA MV Powe Exponential MV efficient 1 γ σxt )) σx T )) σx T )) EXT )) Table 2. 18
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