Dynamic Asse iabiliy Managemen under Model Uncerainy Ferenc Horvah, Frank de Jong, and Bas J.M. Werker Ciy Universiy of Hong Kong Tilburg Universiy January 13, 218 Absrac We analyze a dynamic Asse iabiliy Managemen problem wih model uncerainy in a complee marke. The fund manager acs in he bes ineres of he pension holders by maximizing he expeced uiliy derived from he erminal funding raio. We solve he robus muli-period Asse iabiliy Managemen problem in closed form, and idenify wo consiuens of he opimal porfolio: he myopic demand, and he liabiliy hedge demand. We find ha even hough he invesmen opporuniy se is sochasic, he invesor does no have ineremporal hedging demand. We also find ha model uncerainy induces a more conservaive invesmen policy regardless of he risk aiude of he fund manager, i.e., a robus invesmen sraegy corresponds o risk exposures which provide a much sronger liabiliy hedge. JE classificaion: C61, G11, G12 Keywords: asse liabiliy managemen, liabiliy-driven invesmen, robusness, uncerainy, ambiguiy Corresponding auhor. Ciy Universiy of Hong Kong, Deparmen of Economics and Finance, 83 Ta Chee Avenue, Kowloon Tong, Kowloon, Hong Kong. Tel.: +852 3442 763. E-mail address: f.horvah@ciyu.edu.hk
1 Inroducion Financial insiuions, such as pension funds and insurance companies, are exposed o several sources of risk hrough heir asses and heir liabiliies. During heir decision-making process, hey simulaneously consider he poenial effecs of heir decisions on boh he asse and he liabiliy side of heir balance shee, hence he erm Asse iabiliy Managemen. However, unless he decision maker he financial insiuion) 1 knows he rue model he daa-generaing process driving he asse and liabiliy values) precisely, i faces no only risk, bu also uncerainy. Disregarding uncerainy can lead o subopimal invesmen and Asse iabiliy Managemen decisions, hus financial insiuions wan o make decisions ha no only work well when he underlying model for he sae variables holds exacly, bu also perform reasonably well if here is some form of model misspecificaion Maenhou 24)). In he lieraure hese decisions are called robus decisions. Alhough he uiliy loss resuling from model misspecificaion can be subsanial Branger and Hansis 212)), he majoriy of he lieraure sill assumes perfec knowledge of he underlying model on he decision maker s side. Our aim wih his paper is o fill his gap in he dynamic Asse iabiliy Managemen lieraure. Our model feaures a complee financial marke wih sochasic ineres raes governed by an N-facor Gaussian affine erm srucure model. The fund manager solves a dynamic Asse iabiliy Managemen problem under model uncerainy. Using he maringale mehod of Cox and Huang 1989), we provide he opimal erminal wealh, he leas-favorable physical probabiliy measure, and he opimal invesmen policy in closed form. We find ha he opimal porfolio weighs consis of wo componens: he myopic demand and he liabiliy hedge demand, bu nowihsanding he sochasic invesmen opporuniy se, he fund manager does no have an ineremporal hedging demand componen. We hen use 42 years of U.S. daa o calibrae our model. We show ha robusness induces a more conservaive invesmen policy: a robus fund manager s opimal risk exposures are closer 1 In his paper we focus on pension funds, bu our resuls can be inerpreed in a more general sense, and hey are valid for any financial insiuion which has o make Asse iabiliy Managemen decisions. 1
o he liabiliy risk exposures, hence reducing he speculaive demand and increasing he liabiliy-hedging demand. In parallel wih his, a robus fund manager invess less in he consan mauriy bond fund wih a relaively shor mauriy in our numerical example 1 year) and also in he sock marke index han an oherwise idenical non-robus fund manager. The porfolio weigh of he consan mauriy bond fund wih mauriy equal o he invesmen horizon increases due o is srong liabiliy hedge effec, and hus effecively reduces he exposure o he facor-specific risk sources. Our paper relaes o he lieraure on Asse iabiliy Managemen. The modern Asse iabiliy Managemen lieraure daes back o eibowiz 1987), who inroduces he concep of he surplus funcion he excess of he plan s asse value over he value of is liabiliies). Based on his noion, Sharpe and Tin 199) exend he Mean-Variance porfolio allocaion model of Markowiz 1952) o an Asse iabiliy Managemen model. The basic idea is ha insead of asse reurns, he invesor cares abou he surplus reurns, where surplus means he value of asses minus he value of liabiliies. Sharpe and Tin 199) find ha he opimal porfolio consiss of wo componens: a speculaive porfolio and a liabiliy-hedge porfolio. Moreover, only he speculaive porfolio depends on he invesor s preferences, he liabiliy-hedge porfolio is he same for each invesor. More recenly, Hoevenaars, Molenaar, Schoman, and Seenkamp 28) exend he muli-period porfolio selecion model of Campbell and Viceira 25) ino a muli-period Asse iabiliy Managemen model. They confirm he finding of Sharpe and Tin 199) ha he opimal porfolio consiss of wo pars: a speculaive porfolio and a liabiliy hedge porfolio. Insead of maximizing a subjecive uiliy funcion, Shen, Pelsser, and Schoman 214) assume ha he fund manager minimizes he expeced shorfall i.e., he expeced amoun by which he value of liabiliies exceeds he value of asses) a he erminal dae. This assumpion emphasizes ha he fund manager acs in he bes ineres of he sponsoring firm, bu does no consider he ineres of he pension holders. In conras o his, van Binsbergen and Brand 216) assume ha he objecive funcion is a sum of wo pars: he firs par expresses he posiive) uiliy of he pension holders, while he second par represens he negaive) uiliy of he 2
sponsoring firm. To be more concree, pension holders derive posiive) uiliy from a high funding raio a he erminal dae, while he sponsoring firm derives negaive) uiliy from having o provide addiional conribuions o he fund in order o keep he funding raio above one hroughou is life cycle. This model of van Binsbergen and Brand 216) ness he non-robus version of he model of Shen, Pelsser, and Schoman 214), if he weigh of he uiliy funcion of he pension holders is se o zero. Our paper also relaes o he lieraure on robus dynamic asse allocaion. Anderson, Hansen, and Sargen 23) in heir seminal paper develop a framework for dynamic asse allocaion models, which allows he invesor o accoun for being uncerain abou he physical probabiliy measure. Wihin his framework, Maenhou 24) provides an analyical and homoheic soluion o he robus version of he Meron problem. Maenhou 26) exends his model o incorporae a sochasic invesmen opporuniy se. Branger, arsen, and Munk 213) solve a robus dynamic sock-cash allocaion problem including reurn predicabiliy, while Munk and Rubsov 214) also allow for ambiguiy abou he inflaion process. Horvah, de Jong, and Werker 216) provide a non-recursive formulaion of he problem of Maenhou 24), and also exend i o models feauring ineres rae risk. The paper is organized as follows. Secion 2 inroduces our model, i.e., he financial marke and he fund manager s objecive funcion. Moreover, Secion 2 also provides he analyical soluion of he robus dynamic Asse iabiliy Managemen problem. In Secion 3 we calibrae our model o 42 years of U.S. marke daa using Maximum ikelihood and he Kalman filer. In Secion 4 we link he level of uncerainy aversion o he heory of Deecion Error Probabiliies. In Secion 5 we quaniaively analyze he effecs of model uncerainy on he opimal Asse iabiliy Managemen decision. Secion 6 concludes. 3
2 Robus Asse iabiliy Managemen Problem Our model feaures a complee financial marke and a robus fund manager. By robusness we mean ha he fund manager is uncerain 2 abou he underlying model. To be more precise, we assume ha she is uncerain abou he physical probabiliy measure. She has a base measure B in mind, which she hinks o be he mos reasonable probabiliy measure. Bu she is uncerain abou wheher he base measure is indeed he rue measure or no, so she considers oher probabiliy measures as well. We call hese alernaive measures and denoe hem by U. We provide he exac relaionship beween B and U, and also he resricions on he se of U measures under consideraion in Secion 2.3. 2.1 Financial marke We consider pension funds which have access o a complee, arbirage-free financial marke consising of a money-marke accoun, N consan mauriy bond funds, and a sock marke index. The shor rae is assumed o be affine in an N-dimensional facor F, i.e., r = A + ι F, 1) where ι denoes a column vecor of ones. The facor F follows an Ornsein-Uhlenbeck process under he base measure, i.e., df = κf µ F )d + σ F dw B F,. 2) Here κ is an N N diagonal marix wih he mean reversion parameers in is diagonal; µ F is an N-dimensional column vecor conaining he long-erm means of he facors under he base measure B; σ F is an N N lower riangular marix, wih sricly posiive elemens in 2 The erms uncerainy and ambiguiy have slighly differen meanings in he behavioral finance lieraure. In he robus asse pricing and robus asse allocaion lieraure, however, hey are used inerchangeably. Since our paper primarily belongs o his laer branch of he lieraure, we do no differeniae beween he meaning of uncerainy and ambiguiy, and use he wo words inerchangeably. 4
is diagonal; and WF, B is an N-dimensional column vecor of independen sandard Wiener processes under B. The sock marke index can be correlaed wih he facor F, i.e., [ ds = S r + σ F,Sλ ] ) F + σ N+1,S λ N+1 d + S σ F,SdW F, B + σ N+1,S dwn+1, B, 3) where λ F and λ N+1 are he marke prices of risk corresponding o he base measure B, σ F,S is an N-dimensional column vecor governing he covariance beween sock and bond reurns, σ N+1,S is a sricly posiive consan, and W B N+1, is a sandard Wiener process under he base measure B, which is independen of WF, B. The liabiliy of he pension fund is assumed o evolve according o d = r + σ F,λ ) ) F + σ N+1, λ N+1 d + σ F,dW F, B + σ N+1, dwn+1, B, 4) where σ F, is an N-dimensional column vecor, and σ N+1, is a scalar. To simplify noaion, we denoe W B F, and W B N+1, joinly as W B = W B F, W B N+1,, 5) λ F and λ N+1 joinly as σ F,S and σ N+1,S joinly as and σ F, and σ N+1, joinly as λ = σ S = σ = λ F λ N+1 σ F,S σ N+1,S σ F, σ N+1,, 6), 7). 8) 5
2.2 The liabiliy-risk-neural measure Our fund manager, as we describe in more deail in Secion 2.3, is opimizing over he erminal funding raio. Therefore, o faciliae he problem solving process, hroughou he paper we use he liabiliy value as numeraire. Since he financial marke is complee and free of arbirage opporuniies, here exiss a unique probabiliy measure under which he value of any raded asse scaled by he value of liabiliy is a maringale. e X be he value of any raded asse, wih dx = X [ r + σ Xλ ] d + X σ XdW B. 9) Applying Io s lemma, we find he dynamics of he asse price scaled by he value of he liabiliy as Defining d ) X = ) X σ X σ ) λ σ ) d + dw B ) X σ X σ ) dw B. 1) = dw λ d, 11) wih λ = λ σ, 12) he dynamics of he asse X scaled by he liabiliy value can be rewrien as d ) X = ) X σ X σ ) dw. 13) Then 12), ogeher wih 11), uniquely deermines he relaionship beween he liabiliyrisk-neural measure and he base measure B. 6
2.3 Preferences, beliefs, and problem formulaion We consider a pension fund manager who acs in he bes ineres of he pension holders. She is risk-averse, and she has CRRA preferences over he erminal funding raio. The fund manager wans o maximize her expeced uiliy, bu she is uncerain abou he physical probabiliy measure under which he expecaion is supposed o be calculaed. She has a base measure B) in mind, bu she considers oher, alernaive probabiliy measures U) as well. We assume ha he invesor knows which evens will happen wih probabiliy one and wih probabiliy zero, i.e., she considers only alernaive probabiliy measures which are equivalen o he base measure. We now formalize he relaionship beween he base measure B and he alernaive measure U as dw U = dw B u)d, 14) where W B and W U are N +1)-dimensional sandard Wiener processes under he measures B and U, respecively. Similarly o idenifying λ as he N + 1)-dimensional vecor of prices of risks of he base measure B, we can idenify u) as he N + 1)-dimensional vecor of prices of risks of U. 3 We assume ha λ is consan, while u) is assumed o be a deerminisic funcion of ime. 4 We now formalize he robus opimizaion problem of he fund manager. Her invesmen horizon is T, she has a uiliy funcion wih a consan relaive risk aversion of γ > 1 over he erminal funding raio, 5 and a subjecive discoun rae of δ >. Her uncerainy-olerance is deermined by he parameer Υ, which is allowed o be sochasic. 3 Throughou he paper we assume us) 2 ds <. 4 We could allow λ o be a deerminisic funcion of ime wihou much change in our conclusions, bu i would resul in more complex expressions due o ime-inegrals involving λ ). Thus, since for our purposes a consan λ suffices, we hroughou ake λ o be consan. 5 The case γ = 1 corresponds o he fund manager having log-uiliy. All of our resuls can be shown o hold for he log-uiliy case as well. 7
Problem 1. Given iniial funding raio A /, find an opimal pair {A T, U} for he robus uiliy maximizaion problem ) A V = inf sup U A T E U exp δt ) + AT T ) 1 γ 1 γ [ Υ s exp δs) EU log du ) ] } db s ds, 15) s subjec o he budge consrain ) E AT = A. 16) T The formulaion of Problem 1 follows he logic of he Maringale Mehod of Cox and Huang 1989): he fund manager opimizes over he erminal wealh A T. 6 The firs par of he objecive funcion in Problem 1 expresses ha he fund manager derives uiliy from he erminal funding raio. The second par is a penaly erm, which assures ha he invesor will use a pessimisic, bu reasonable physical probabiliy measure o calculae her expeced uiliy. This penaly erm in line wih Anderson, Hansen, and Sargen 23) is he inegral of he discouned ime-derivaive of he Kullback-eibler divergence also known as he relaive enropy) beween he base measure B and he alernaive measure U, muliplied by he fund manager s uncerainy-olerance parameer Υ s. Inuiively, his penaly erm is high if he alernaive measure U and he base measure B are very differen from each oher, and low if hey are similar o each oher. If U and B coincide, he penaly erm is zero. Using Girsanov s heorem, we can express he Kullback-eibler divergence as E U [ log ) ] du db = [ 1 EU 2 us) 2 ds us)dw U s = 1 2 u) 2. 17) ] To insure homoheiciy of he soluion, i.e., ha he opimal porfolio weighs do no 6 Acually, he fund manager opimizes over he erminal funding raio A T / T. However, as we describe i in more deail in Secion 2.1, he liabiliy process is assumed o be exogenous, hence choosing an opimal erminal funding raio A T / T is equivalen o choosing only an opimal erminal wealh A T. 8
depend on he acual funding raio, we following Maenhou 24) express he manager s uncerainy-olerance parameer as Υ = exp δ) 1 γ ) A V, 18) θ where V A / ) is he value funcion of he fund manager a ime, i.e., ) A V = inf sup U A T E U exp δt ) + AT T ) 1 γ 1 γ [ Υ s exp δs) EU log du ) ] } db s ds, 19) s subjec o he budge consrain E AT T ) = A. 2) Subsiuing 18) ino he value funcion 15), and also making use of 17), we can rewrie Problem 1 in a form which has a non-recursive goal funcion. This is saed in he following heorem, he proof of which is provided in he Appendix. Theorem 1. If he fund manager s uncerainy-olerance parameer Υ akes he form 18), hen he value funcion in Problem 1 is equivalen o ) A V = inf sup U A T E U 1 γ exp 2θ ) ) 1 γ A T u) 2 T d δt 1 γ, 21) subjec o he budge consrain ) E AT = A. 22) T As noed by Horvah, de Jong, and Werker 216), he expression in 21) provides an alernaive inerpreaion of robusness: he goal funcion of a robus fund manager is equivalen o he goal funcion of a more impaien 7 non-robus fund manager. Besides 7 By a fund manager being more impaien, we mean ha her subjecive discoun rae is higher. 9
increasing he subjecive discoun rae, he oher effec of robusness is a change in he physical probabiliy measure from B o U. 2.4 Opimal Terminal Funding Raio To solve he robus dynamic AM problem, we apply he maringale mehod developed by Cox and Huang 1989), and adaped o robus problems by Horvah, de Jong, and Werker 216)). The nex heorem which we prove in he Appendix provides he opimal erminal wealh and he leas-favorable disorions. Theorem 2. The soluion o Problem 1 under 18) is given by  T = T A [ exp 1 γ E exp [ 1 γ wih he leas-favorable disorion λ + û ) ) σ d + 1 γ λ + û )) σ d + 1 γ λ + û ) ) dw ] λ + û )) dw ], 23) û ) = θ γ + θ λ. 24) Using he maringale mehod o solve he robus dynamic AM problem has he advanage of providing insigh ino he opimizaion process of he fund manager. The form of he opimal erminal wealh 23) suggess ha he decision process of he fund manager can be separaed ino wo pars. Firs, as a saring poin, she wans o obain a perfec hedge for he liabiliies a ime T, i.e., she wans a erminal wealh equal o T. Then, she modifies his erminal wealh based on her preferences o achieve he opimal erminal wealh. The leas-favorable disorion of he fund manager differs in wo imporan aspecs from he leas-favorable disorion of an oherwise idenical invesor who opimizes over her erminal wealh, insead of he erminal funding raio see Horvah, de Jong, and Werker 216)). Firs, he leas-favorable disorion of he fund manager is independen of ime, while he leas-favorable disorion of an invesor deriving uiliy from erminal wealh 1
conains a ime-dependen componen. This ime-dependen componen is presen due o he ineremporal hedging poenial of he consan mauriy bond funds, and i resuls in he invesor having a less severe disorion. However as we show in Secion 2.5, deriving uiliy from he erminal funding raio insead of he erminal wealh, he fund manager does no have an ineremporal hedging demand for he consan mauriy bond funds. So, inuiively, regardless of how far away from he end of her invesmen horizon he fund manager is, she will disor her base measure o he same exen. Anoher aspec in which he leas-favorable disorion of he fund manager differs from he leas-favorable disorion of an oherwise idenical invesor deriving uiliy from erminal wealh is ha he marke price of risk in which he leas-favorable disorion is affine corresponds o he liabiliy value as numeraire, insead of o he money marke accoun. 8 Inuiively, his means ha he magniude of he disorion is reduced due o he fund manager deriving uiliy from he erminal funding raio insead of he erminal wealh. This is rue for boh û F and for û N+1. Because he liabiliy process behaves very similarly o a zero-coupon bond wih approximaely 15 years of mauriy, we expec he elemens of σ F, o be negaive. Since he firs N elemens of λ are also negaive, and he difference beween he marke price of risk corresponding o o he money marke accoun as numeraire and he marke price of risk corresponding o he liabiliy as numeraire is σ, deriving uiliy from he erminal funding raio insead of he erminal wealh reduces he posiive) elemens of û F. The same logic applies o û N+1. The marke price of risk using he money marke accoun as numeraire, i.e., λ N+1 is posiive, and inuiion suggess σ,n+1 is also posiive, herefore, opimizing over he erminal funding raio insead of he erminal wealh will reduce he magniude of he negaive) û N+1. If he fund manager is no uncerainy averse a all, her θ parameer is equal o zero and her leas-favorable disorion reduces o zero as well. In oher words, she will use her base probabiliy measure B o evaluae her expeced uiliy. A he oher exreme, if her 8 Tha is, he fund manager s leas-favorable disorion is affine in λ, while he leas-favorable disorion of an oherwise idenical invesor who derives uiliy from erminal wealh is affine in λ, i.e., he marke price of risk of he base measure B over he risk-neural measure wih he money marke accoun as numeraire. 11
uncerainy aversion i.e., her θ parameer) is infiniy, she uses he globally-leas-favorable disorion ũ = λ. 25) Opimizing over he erminal wealh, he globally-leas-favorable disorion would be equal o he marke price of risk using he money marke accoun as numeraire, and he invesor would consider he scenario when she receives no compensaion above he risk-free rae for bearing any risk. For a fund manager opimizing over he funding raio, however, his scenario would sill be of value in he sense ha she would sill be willing o bear some risk, due o is hedging poenial. 9 To achieve he leas-favorable disorion, he fund manager has o correc for his and hence her leas-favorable disorion becomes he marke price of risk of he measure over he base measure B. 2.5 Opimal Porfolio Sraegy Since our financial marke is complee, here exiss a unique invesmen process which enables he fund manager o achieve he opimal erminal wealh 23). We provide he opimal risk exposure process corresponding o his opimal invesmen policy in Corollary 1, and he opimal invesmen process iself in Corollary 2. Boh proofs are provided in he Appendix. Corollary 1. Under he condiions of Theorem 2, he opimal invesmen is a coninuous re-balancing sraegy where he exposures o he N+1 risk sources as a fracion of wealh are ˆΠ = 1 γ + θ λ + σ 26) = 1 γ + θ λ + 1 1 ) σ. 27) γ + θ 9 We would like o emphasize here ha his hedging poenial refers o he liabiliy hedge, i.e., by being exposed o some risk in he above-menioned scenario he invesor can achieve a lower volailiy of her erminal funding raio han by invesing everyhing in he money marke accoun. 12
The form of he opimal exposure o he risk sources in 26) also reflecs he separaion of he invesmen decision ino wo pars which we described in Secion 2.4, i.e., he fund manager firs achieves a perfec hedge of he liabiliy second par of 26)), hen she modifies her exposure according o her preferences firs par of 26)). If he correlaion beween he asse reurns and he liabiliy reurn is zero i.e., if σ = ), hen he opimal exposure o he risk sources is equal o he scaled marke price of risk, i.e., λ / γ + θ), which in ha case coincides wih λ/ γ + θ). 1 In he nex corollary we provide he unique opimal invesmen process, wih noaion B τ) = [B τ 1 ) ι;... ; B τ N ) ι], 28) where τ j denoes he mauriy of bond fund j, and B ) is defined as B) = I exp { κ}) κ 1. 29) Corollary 2. Under he condiions of Theorem 2, he opimal invesmen is a coninuous re-balancing sraegy where he fracion of wealh invesed in he consan mauriy bond funds is ˆπ B, = 1 γ + θ B τ) 1 σ F ) 1 λ F λ ) N+1 σ F,S σ N+1,S + 1 γ θ B τ) 1 σ F γ + θ ) 1 σ F, σ N+1, σ N+1,S σ F,S ), 3) and he fracion of wealh invesed in he sock marke index is λ N+1 ˆπ S, = γ + θ) σ N+1,S 1 γ θ) σ N+1,. 31) γ + θ) σ N+1,S 1 We wan o sress ha his does no mean ha he opimal decision for an invesor opimizing over erminal wealh only insead of over he erminal funding raio) is equal o λ / γ + θ) or λ/ γ + θ). The reason of he difference is ha even if he liabiliy process is a consan i.e., σ = ), he fund manager sill hedges agains i, and he liabiliy hedge demand is equal o he negaive of he ineremporal hedge demand. Thus, he wo laer demand componens of he fund manager cancel ou. In conras wih his, if he invesor opimizes over her erminal wealh only, she sill has a non-zero ineremporal hedging demand. 13
In line wih Sharpe and Tin 199) and Hoevenaars, Molenaar, Schoman, and Seenkamp 28), we find ha he opimal porfolio consiss of wo pars: a speculaive porfolio firs line of 3) and firs par of 31)), and a liabiliy hedge porfolio second line of 3) and second par of 31)). The source of he liabiliy hedge demand is he covariance beween he asse reurns and he liabiliy reurns. The higher he covariance beween he bond fund reurns and he liabiliy reurns i.e., he lower he elemens of σ F, ), he higher he opimal porfolio weigh of he consan mauriy bond funds. Also: he higher he covariance beween he sock marke index reurn and he liabiliy reurn i.e., he higher σ N+1, ), he higher he opimal porfolio weigh of he sock marke index. Inuiively, a higher covariance beween he reurn of an asse and he liabiliy induces a higher opimal invesmen in ha paricular asse, because a higher covariance provides a higher hedging poenial and herefore makes he asse more desirable. The second erms wihin he brackes in boh he firs and he second line of 3) are correcion erms o he speculaive consan mauriy bond fund demand and he liabiliy hedge consan mauriy bond fund demand, respecively. These wo correcion erms arise due o he covariance beween he bond reurns and he sock marke index reurn. The higher his covariance i.e., he lower he elemens of σ F,S ), he lower he correcion erm o boh he speculaive bond demand and he liabiliy hedge bond demand. The inuiion of his is ha a higher covariance beween he consan mauriy bond fund reurns and he sock marke index reurn resuls in he same invesmen in he sock marke index providing a higher exposure o he N facors, and hence o reain he opimal exposure o hese facors, he consan mauriy bond funds should have lower porfolio weighs han wih zero covariance. We find ha he opimal asse allocaion is deermined by he sum of he risk-aversion parameer and he uncerainy-aversion parameer, i.e., by γ + θ. This is in line wih, e.g., Maenhou 24), Maenhou 26), and Horvah, de Jong, and Werker 216). Inuiively, a robus fund manager behaves he same way as a non-robus, bu more risk-averse fund manager. 14
Table 1. Parameer esimaes and sandard errors Esimaed parameers and sandard errors using Maximum ikelihood. We observed four poins weekly on he U.S. zero-coupon, coninuously compounded yield curve, corresponding o mauriies of 3 monhs, 1 year, 5 years and 1 years; and he oal reurn index of Daasream s US-DS Marke. The observaion period is from 5 January 1973 o 29 January 216. Esimaed parameer Sandard error ˆκ 1.763.24 ˆκ 2.37.18 Â.862.13 ˆλ F,1.178.1528 ˆλ F,2.5899.1528 ˆλ N+1.318.1528 ˆσ F,11.28.9 ˆσ F,21.24.12 ˆσ F,22.155.3 ˆσ F S,1.35.38 ˆσ F S,2.121.35 ˆσ N+1.1659.25 3 Model Calibraion The wo-facor version of our model for he financial marke is idenical o he model of Horvah, de Jong, and Werker 216). Hence, we direcly adap he esimaes herein for our model parameers. For compleeness, we briefly recall he esimaion mehodology followed by Horvah, de Jong, and Werker 216). The model is calibraed o U.S. marke daa using he Kalman filer and Maximum ikelihood. The daa consis of weekly observaions of he 3-monh, 1-year, 5-year, and 1-year poins of he yield curve, and Daasream s U.S. Sock Marke Index. The observaion period is from 1 January 1973 o 29 January 216. The saring values of he filered facors are equal o heir long-erm means. The parameer esimaes can be found in Table 1. All model parameers are esimaed wih small sandard errors, he only excepion being he marke price of risk. This confirms he validiy of our model seup, namely, ha he fund manager is uncerain abou he physical probabiliy measure, which ogeher wih her considering only equivalen probabiliy measures is equivalen o saying ha she is 15
uncerain abou he marke price of risk. As a proxy for he liabiliy process, we follow van Binsbergen and Brand 216) and use he price of a zero-coupon bond. Inuiively, we hink of he liabiliy as a rolled-over asse wih consan duraion. As of he duraion iself, we use 15 years, which is approximaely he average duraion of U.S. pension fund liabiliies van Binsbergen and Brand 216)). Then, he volailiy parameers of he liabiliies are σ F, = σ F B 15) ι 32) and σ N+1, =. 33) Using our parameer esimaes in Table 1, he esimaed volailiy vecor of he liabiliy process is [ ] σ =.121.51. 34) 4 Deecion Error Probabiliies In he previous secion we esimaed he model parameers relaed o he financial marke, based on hisorical daa. Calibraing he parameers relaed o he preferences, i.e., he risk-aversion parameer γ and he uncerainy-aversion parameer θ, is less sraighforward. There is no agreemen in he lieraure abou wha he relaive risk aversion of a represenaive invesor precisely is, bu he majoriy of he lieraure considers risk aversion parameers beween 1 and 5 o be reasonable. Several sudies aemp o esimae wha a reasonable risk aversion value is, usually by using consumpion daa or by conducing experimens. Friend and Blume 1975) esimae he relaive risk aversion parameer o be around 2; Weber 1975) and Szpiro 1986) esimae i o be beween abou 1.3 and 1.8; he esimaes of Hansen and Singleon 1982) and Hansen and Singleon 1983) are.68.97 and.26-2.7, respecively; using nondurable consumpion daa, Mankiw 1985) esimaes 16
he relaive risk aversion o be 2.44-5.26, and using durable goods consumpion daa i o be 1.79-3.21; Barsky, Juser, Kimball, and Shapiro 1997) use an experimenal survey o esimae he relaive risk aversion parameer of he subjecs, he mean of which urns ou o be 4.17; while in he sudy of Halek and Eisenhauer 21) he mean relaive risk aversion is 3.7. aer in his secion we vary he risk-aversion parameer beween 1 and 5 o see is effec on he opimal invesmen decision. Calibraing he uncerainy-aversion parameer θ is even more complicaed han he calibraion of he risk aversion. Ever since he seminal paper of Anderson, Hansen, and Sargen 23), he mos puzzling quesions in he robus asse pricing lieraure are relaed o how o quanify uncerainy aversion, and how much uncerainy is reasonable. Anderson, Hansen, and Sargen 23) propose a heory o address hese problems based on he Deecion Error Probabiliies. They assume ha he invesor can observe a sample of hisorical daa, and she performs a likelihood raio es o decide wheher hese daa are generaed by a daa-generaing process corresponding o he base measure B, or by a daageneraing process corresponding o he alernaive measure U. Based on his es, he invesor is assumed o be able o correcly guess he rue physical probabiliy measure in p% of he cases, i.e., she is wrong in 1 p) % of he cases. Making his 1 p) % equal o he probabiliy of making an error based on he likelihood raio es, we can disenangle he risk aversion and he uncerainy aversion. The quesion of wha a reasonable level of 1 p)%, i.e., he Deecion Error Probabiliy, is, is he subjec of an acive line of research. Anderson, Hansen, and Sargen 23) sugges ha Deecion Error Probabiliies beween 1% and 3% are plausible. Now we give he formal definiion of he Deecion Error Probabiliy. Definiion 1. The Deecion Error Probabiliy DEP) is defined as DEP = 1 2 P B log db ) du < + 12 P U log db ) du >. 35) Following he reasoning of Horvah, de Jong, and Werker 216), we can express he 17
Deecion Error Probabiliy in closed form. This is saed in Theorem 3 and in Corollary 3. Theorem 3. Assume ha he fund manager coninuously observes he prices of N consan mauriy bond funds, and he level of he sock marke index. The observaion period lass from H o he momen of observaion,. Then, he deecion error probabiliy of he fund manager for given U is 1 DEP = 1 Φ 2 H u s) 2 ds ), 36) where u ) is defined in 14). Subsiuing he leas-favorable disorion 24) ino 36), we obain he closed-form expression in Corollary 3. Corollary 3. Assume ha he condiions of Theorem 3 hold. Then, he deecion error probabiliy of he fund manager for he leas-favorable U is DEP = 1 Φ θ 2 γ + θ) ) H λ. 37) The Deecion Error Probabiliy used by a fund manager who is no uncerainy-averse a all i.e., whose θ parameer is zero) is.5. Tha is, she migh as well flip a coin o disinguish beween wo probabiliy measures insead of performing a likelihood-raio es on a sample of daa. On he oher hand, a fund manager wih an uncerainy aversion parameer of infiniy uses he lowes possible Deecion Error Probabiliy, which is 1 Φ 1/2) ) H λ. 11 We assume ha he observaion period of he invesor is 42 years, 12 and ha her Deecion Error Probabiliy is 1%. Given ha she has access o a relaively long sample of 11 One migh expec ha he lowes possible Deecion Error Probabiliy is zero, which would mean ha he fund manager knows he physical probabiliy measure precisely. However, as 37) also shows, his is only he case if he lengh of her observaion period is infiniy, i.e., H =. If her observaion period is finie, he limiaion of available daa will always resul in he fund manager no being able o correcly ell apar wo probabiliy measures in 1% of he cases. 12 We use 42 years of marke daa o esimae our model parameers, hus i is a reasonable assumpion ha he fund manager has access o he same lengh of daa. Even hough our observaion frequency is weekly, assuming ha he fund manager can observe daa coninuously does no cause a significan difference. 18
daa, and ha she can observe he prices coninuously, our choice of 1% as he Deecion Error Probabiliy is jusifiable. 5 Policy Evaluaion Now we use our parameer esimaes from Secion 3 o analyze he effecs of robusness on he opimal AM decision, if he fund manager has access o a money marke accoun, o wo consan mauriy bond funds wih 1 and 15 years of mauriies, and o a sock marke index. Her invesmen horizon is 15 years. We show he quaniaive relaionship beween he level of uncerainy aversion and he opimal exposure o he differen sources of risk, and also he opimal porfolio weighs. We find ha regardless of he risk aiude of he fund manager, robusness subsanially changes he magniude of her opimal porfolio weighs. Generally speaking, robusness ranslaes ino making more conservaive AM decisions. More concreely, while invesing significanly less in he sock marke index and he consan mauriy bond fund wih he shorer 1 year) mauriy, he fund manager increases her invesmens in he consan mauriy bond fund wih he same mauriy as her invesmen horizon 15 years) and in he money marke accoun. Figure 1 shows he opimal exposure of he fund manager o he hree risk sources for differen levels of risk aversion and uncerainy aversion. Her uncerainy aversion is measured by he Deecion Error Probabiliy. If she uses a Deecion Error Probabiliy of 5%, hen she is no uncerainy-averse a all, while if she uses he lowes possible level of Deecion Error Probabiliy which in our case is 2.8%), her uncerainy aversion is infiniely high. Figure 2 shows he opimal porfolio weighs, which enable he fund manager o achieve he opimal exposure o he risk sources. Due o he inheren naure of affine erm srucure models, our fund manager akes a high shor posiion in he money marke accoun, and she uses his money o obain a highly leveraged long posiion in he 1-year consan mauriy bond fund. 13 In Table 2 we provide he numerical values of he opimal 13 If here are no consrains on he posiion which he fund manager can ake in he differen asses, i is a common finding in he lieraure ha she akes exremely large shor and longe posiions o achieve he 19
exposures and he opimal porfolio weighs for differen levels of risk aversion of robus and non-robus fund managers. A non-robus fund manager applies a Deecion Error Probabiliy of 5%, while a robus fund manager assumes a Deecion Error Probabiliy of 1%. We find ha he fund manager regardless of her risk-aversion and uncerainy-aversion always chooses a negaive exposure o he firs wo risk sources. This is inuiive, since hese wo risk sources have negaive marke prices of risk. Bu i is less sraighforward why an infiniely risk-averse or an infiniely uncerainy-averse fund manager decides o ake a sricly negaive exposure o hese risk sources, insead of oping for zero exposure. Given an infiniely high risk or uncerainy aversion, he fund manager s opimal AM decision is o obain a perfec hedge for he liabiliies. Since he liabiliies are a linear combinaion of he firs wo facors, she will expose herself o hese facors o an exen which is equal o he exposure of he liabiliies o hese facors. This can be shown analyically by aking he limi of he righ-hand-side of 26) as θ. Were he fund manager opimizing over he erminal wealh insead of he erminal funding raio, her opimal exposure o he firs wo facors would be also sricly negaive, bu for a differen reason: in his case her goal would be o achieve an exposure equal o ha of a zero-coupon bond wih he mauriy of her invesmen horizon, hus eliminaing risk and uncerainy oally, since she will receive he face value of he zero-coupon bond a he end of her invesmen horizon for sure. ooking a her decision from a differen angle: her myopic demand for he consan mauriy bond funds would be zero, and he enire oal sricly posiive) demand would be due o he ineremporal hedging demand. In our case, when he fund manager opimizes over he erminal funding raio, and she is eiher infiniely risk-averse or infiniely uncerainy-averse, her myopic demand for he consan mauriy bond funds is zero, and her oal demand is due o he liabiliy hedge demand. The exposure of an infiniely risk-averse or infiniely uncerainy-averse fund manager o he sock-marke-index-specific source of risk is zero, because we assumed ha he liabiliy opimal risk exposures. See, e.g., Brennan and Xia 22), Figure 4 and Figure 6. 2
Figure 1. Opimal exposure o he risk sources Opimal exposure of he fund manager o he risk sources as a funcion of he Deecion Error Probabiliy DEP), for differen levels of relaive risk aversion. We use our parameer esimaes in Table 1, and assume ha he liabiliy value is always equal o a zero-coupon bond wih 15 years of mauriy. The fund manager s invesmen horizon is 15 years. A DEP of 5% corresponds o a non-uncerainy-averse fund manager, while a DEP of 2.8% corresponds o a fund manager wih infiniely high uncerainy aversion. value is always equal o he value of a zero-coupon bond wih 15 years of mauriy, and he value of such a bond is no influenced by he sock-marke-index-specific risk source. In absence of his assumpion, he opimal exposure of an infiniely risk-averse or an infiniely 21
Figure 2. Opimal porfolio weighs Opimal porfolio weighs as a funcion of he Deecion Error Probabiliy DEP), for differen levels of relaive risk aversion. We use our parameer esimaes in Table 1, and assume ha he liabiliy value is always equal o a zero-coupon bond wih 15 years of mauriy. The fund manager s invesmen horizon is 15 years. A DEP of 5% corresponds o a nonuncerainy-averse fund manager, while a DEP of 2.8% corresponds o a fund manager wih infiniely high uncerainy aversion. 22
Table 2. Opimal risk exposures and porfolio weighs Opimal exposures and porfolio weighs for differen levels of risk aversion of robus and non-robus fund managers. The non-robus exposures and porfolio weighs correspond o a Deecion Error Probabiliy of 5%, while heir robus counerpars assume a Deecion Error Probabiliy of 1%. Opimal exposures and porfolio weighs are calculaed using he assumpions of Secion 3. Opimal porfolio weighs and exposures γ = 1 γ = 3 γ = 5 Non-robus Robus Non-robus Robus Non-robus Robus Π 1 -.17 -.14 -.14 -.13 -.13 -.12 Π 2 -.59 -.25 -.23 -.12 -.16 -.9 Π 3.32.12.11.4.6.2 π B1) 43% 1494% 1343% 498% 86% 299% π B15) 54% 83% 85% 94% 91% 97% π S 192% 71% 64% 24% 38% 14% π MMA -4176% -1548% -1392% -516% -835% -31% uncerainy-averse fund manager would be σ N+1,, due o he reasoning in he previous paragraph. 14 If he fund manager were opimizing over he erminal wealh insead of he erminal funding raio, her opimal exposure o he sock-marke-index-specific source of risk would be zero even wihou our previous assumpion abou he liabiliies, because her oal demand would be equal o he ineremporal hedging demand, and he sock-markespecific risk source canno be hedged ineremporally. We also find ha boh a higher risk aversion and a higher uncerainy aversion resul in a lower opimal exposure in absolue value o he risk sources, and in order o achieve his lower exposure he fund manager has lower opimal porfolio weigh again, in absolue value) in he consan mauriy bond fund wih 1 year of mauriy and in he sock marke index. Her opimal porfolio weigh for he consan mauriy bond fund wih 15 years of mauriy is, on he oher hand, an increasing funcion of boh risk aversion and uncerainy aversion, due o is srong liabiliy hedge poenial. 15 14 The sock-marke-index-specific risk source affecs he liabiliy of a pension fund if, e.g., he pension payou is linked o he indusry wage level, and he indusry wage level is affeced by he sock-markeindex-specific risk source via, e.g., performance-dependen wage schemes. 15 The fac ha he effecs of risk aversion and uncerainy aversion have he same sign can direcly be deduced from he risk-aversion parameer and he uncerainy-aversion parameer appearing only as a sum in he opimal porfolio weighs in 3) and 31). 23
Accouning for uncerainy aversion has a subsanial effec on boh he opimal exposures o he risk sources and he opimal porfolio weighs. We find ha he opimal exposures in absolue value) o each of he risk sources are a decreasing funcion of he level of robusness. I.e., he more uncerainy-averse he fund manager, he less exposure she finds opimal o each risk source. The decrease in he opimal exposure is especially subsanial in he case of he sock-marke-specific risk source more han 6%) and he facor-specific risk source wih a higher absolue value of) marke price of risk more han 4%), while i is less significan in he case of he facor-specific risk source wih a lower absolue value of) marke price of risk. The inuiion behind his is ha as he uncerainy aversion of he fund manager increases and approaches infiniy, he opimal exposures approach he volailiy loadings of he liabiliy, concreely -.121, -.51, and. The opimal exposure of a non-robus fund manager wih log-uiliy i.e., γ = 1) o he firs risk source is -.17, hence here is no much scope for reducion in he magniude of his exposure. 16 In conras wih his, he exposure of a non-robus fund manager wih log-uiliy o he second and hird risk sources is relaively higher in magniude -.59 and.32) due o heir higher marke price of risk in absolue erms). Moreover, he magniudes of liabiliy exposures o he second and hird risk sources are lower han ha of he firs risk source -.51 and, respecively), herefore here is more scope for reducion in he opimal exposure as he uncerainy aversion increases. The lower exposure levels due o robusness ranslae o a lower demand for he consan mauriy bond fund wih 1 year of mauriy more han 62% decrease) and for he sock-marke-index also more han 62%). The demand for he consan mauriy bond fund wih a mauriy equal o he invesmen horizon of he fund manager, however, increases wih he level of robusness, due o is liabiliy hedging poenial. 16 I.e., even if her level of robusness is infiniy, her opimal exposure would sill be -.121, which is equal o he liabiliy exposure o his risk source. 24
6 Conclusion We have shown ha model uncerainy has significan effecs on Asse iabiliy Managemen decisions. A fund manager who derives uiliy from he erminal funding raio and who accouns for model uncerainy does no necessarily have an ineremporal hedging demand componen, even hough he invesmen opporuniy se is sochasic. Robusness subsanially changes he opimal exposures o he risk sources: as he level of uncerainy aversion increases, he opimal exposures approach he liabiliy exposures o he respecive risk sources. In he case of a wo-facor affine erm srucure model and an addiional, sock-marke-specific risk source, opimal exposures can change by more han 6% due o robusness. These changes in he risk exposures ranslae ino subsanial changes in he opimal porfolio weighs as well: while a robus fund manager invess less in he consan mauriy bond fund wih a relaively shor mauriy and also in he sock marke index, she increases her invesmen in he consan mauriy bond fund wih a mauriy equal o her invesmen horizon o make use of is liabiliy hedge poenial. In our model we assume ha he fund manager acs in he bes ineres of he pension holders, and she does no consider he ineres of he pension fund sponsors. Exending he model o include he negaive uiliy derived by he pension fund sponsors from having o conribue o he fund can provide furher insigh ino he effecs of model uncerainy on more complex Asse iabiliy Managemen decisions. We also assume ha he fund manager s uncerainy olerance parameer is linear in he value funcion, hence he soluion of our robus dynamic Asse iabiliy Managemen problem is homoheic, and i can be obained in closed form. There is, however, an acive and curren debae in he lieraure wheher his funcional form of he uncerainy olerance is jusifiable. Solving our robus dynamic Asse iabiliy Managen problem wih a differenly formulaed uncerainy olerance parameer is anoher fruiful line of furher research. 25
Appendix Proof of Theorem 1. Subsiuing 17) and 18) ino 15), he value funcion a ime saisfies V [ A ) ] { [ A ) ] 1 γ = E U T 1 γ) u s) 2 [ ) ] } A exp δt ) + V s ds 1 γ 2θ s { [ A ) ] 1 γ } { = E U T exp δt ) + E U 1 γ) u s) 2 [ ) ] } A V s ds 1 γ 2θ s 1 γ) u s) 2 [ V A ) ] s s ds, 38) 2θ where ) A and U denoe he opimal erminal wealh and leas-favorable physical measure, T respecively. Inroduce he square-inegrable maringales, under U, M 1, = E U M 2, = E U { [ A ) T ] 1 γ } exp δt ), 39) 1 γ { 1 γ) u s) 2 [ ) ] } A V s ds. 4) 2θ s The maringale represenaion heorem see, e.g., Karazas and Shreve 1991), pp. 182, Theorem 3.4.15) saes ha here exis square-inegrable sochasic processes Z 1, and Z 2, such ha { [ A ) T ] 1 γ } M 1, = E U exp δt ) + Z 1 γ 1,sdW s U, 41) { M 2, = E U 1 γ) u s) 2 [ ) ] } A V s ds + Z 2θ 2,sdWs U. 42) Subsiuing in 38), we can express he dynamics of he value funcion as [ ) ] [ ) ] A 1 γ) u ) 2 A dv = V d + Z 1, + Z 2, ) dw U. 43) 2θ s 26
[ This linear backward sochasic differenial equaion wih he erminal condiion V A ) ] T T = exp δt ) [ ) ] A 1 γ T /1 γ)) has an explici paricular soluion see, e.g., Pham 29), pp. 141-142). The unique soluion o 43) is given by [ ) ] A Γ V = E U { [ A ) Γ T exp δt ) T 1 γ ] 1 γ }, 44) where Γ solves he linear differenial equaion 1 γ) u ) 2 dγ = Γ d; Γ = 1, 45) 2θ i.e., Γ = exp 1 γ) u s) 2 2θ ) ds. 46) Subsiuing ino 44), we obain he closed-form soluion of he value funcion as V X ) = E U { exp 1 γ) u s) 2 ds δt 2θ ) } X 1 γ T, 47) 1 γ wih [ ) A ] and U represening he opimal funding raio and he leas-favorable physical probabiliy measure. As a resul, we obain 21). Proof of Theorem 2. The firs sep of he opimizaion is o deermine he opimal erminal wealh, given he budge consrain. In order o deermine he opimal erminal wealh, we 27
form he agrangian from 21) and 22). This agrangian is 1 γ A ) = inf sup U EU exp 2θ A T [ y E AT = inf sup U A T T E ) A ]} ) ) 1 γ A T u) 2 T d δt 1 γ ) [ du 1 γ T ] ) 1 γ A T exp u) 2 T d δt d T 2θ 1 γ [ ) y E AT A ]}, 48) T where y is he agrange-muliplier. Now we solve he inner opimizaion, aken U as given. The firs-order condiion for he opimal erminal funding raio, denoed by ÂT / T, is  T T = du ) 1 { γ d T exp 1 γ y 1 γ [ 1 γ 2θ ]}. 49) T u) 2 d δt Afer subsiuing he opimal erminal funding raio ino he budge consrain, we obain he agrangian as y 1 γ = A { E du ) 1 { [ γ d T exp 1 1 γ ]}}. 5) T γ 2θ u) 2 d δt Togeher wih he Radon-Nikodym derivaive ) du = exp d { + λ + u s)) dw s [ ) λ + u s) σ 1 λ + σ + u s) 2 σ 2)] } ds, 51) 2 we subsiue he agrangian back ino 49) o deermine he opimal erminal funding raio 28
as  T T = A [ exp 1 γ E exp [ 1 γ û ) + λ ) σ d + 1 γ û ) + λ ) σ d + 1 γ û ) + λ ) dw ] û ) + λ ) dw ]. 52) Muliplying boh sides by T, we obain 23), and his proves he firs par of Theorem 2. Now we solve he ouer opimizaion problem. Subsiuing he opimal erminal wealh back ino he value funcion, we obain A ) 1 γ V A ) = 1 γ exp 1 γ exp 2θ û ) + λ ) σ d + 1 2γ [ λ + σ + u ) 2 σ 2] ) d u) 2 d δt 1 2 ) û ) + λ 2 d. 53) Now we can wrie down he firs-order condiion for u ) and we obain u ) = θ γ + θ λ, 54) which is indeed he same as 24). This complees he proof. Proof of Corollary 1. The opimal wealh process can be wrien as  = E ) ÂT. 55) T Subsiuing he opimal erminal wealh 23) ino 55), he opimal wealh a ime becomes  = A [ 1 exp γ [ E exp 1 γ û s) + λ ) dw s exp 1 2γ 2 + 1 γ û s) + λ 2 ds û ) + λ ) dw + 1 γ ] û s) + λ) σ ds ) û ) + λ ) σ d ]. 56) Subsiuing he soluion of he sochasic differenial equaion 4) for, he opimal wealh 29
a ime becomes { Â =A exp + + + 1 γ 1 ) ) û F + λ F + σ F, + σ F B s) ι dwf,s γ 1 ) ) û N+1 + λ N+1 + σ N+1, dwn+1,s + γ A )) + ι µ F σ F σ + ι B ) F û s) + λ ) } σ ds E exp [ 1 γ ι B s) σ F σ ds µ F σ F σ )) + 1 2 σ 2 exp 1 2γ 2 ) û s) + λ 2 ds û ) + λ ) dw + 1 γ û ) + λ ) σ d 57) ]. Applying Io s lemma, he opimal wealh dynamics can be expressed as dâ =...d + Â γ û + λ + γσ ) dw. 58) Subsiuing he leas-favorable disorion for û, we obain dâ =...d + Â ) 1 γ + θ λ + σ dw. 59) From 59) he opimal risk exposures follow direcly. This complees he proof. Proof of Corollary 2. Using he porfolio weighs π B, and π S,, he opimal wealh dynamics can be expressed as dâ =...d + Â π B,B σ F dwf, [ ] ) + π S, σ F,S ; σ N+1,S dw. 6) Moreover, he opimal wealh dynamics can equivalenly be wrien as 59). Then, due o he maringale represenaion heorem we can wrie down a sysem of N+1 equaions by making he exposures o he N+1 risk sources in 59) and 6) equal o each oher. Solving his equaion sysem, we indeed obain he opimal porfolio weighs 3) and 31). This complees he proof. 3
Proof of Theorem 3. From 11) and 14) he Radon-Nikodym derivaive db/du follows direcly. Subsiuing his ino Definiion 1 and evaluaing he expecaions, he closed-form soluion in 36) is obained. Proof of Corollary 3. Subsiuing he leas-favorable disorions 24) ino 36), 37) is immediaely be obained. 31
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