Portfolio Efficiency: Traditional Mean-Variance Analysis versus Linear Programming

Transcription

1 Porfolio Efficiency: Tradiional Mean-Variance Analysis versus Linear Programming Seve Eli Ahiabu Universiy of Torono Spring 003 Please send commens o Sephen.ahiabu@uorono.ca I hank Prof. Adonis Yachew for his commens and suggesions in his projec. All remaining shorcomings remain enirely mine.

2 Absrac So srong is he influence of Markoviz [95] on modern finance ha porfolio selecion asks and efficiency ess are dominaed by one definiion of meanvariance efficiency. No much regard is paid o he fac ha sandard meanvariance uiliy funcions saisfy he necessary and sufficien condiions of expeced uiliy heory if and only if reurn disribuions are ellipical. In his paper, I explore efficiency implicaions of porfolios using he mainsream meanvariance mehodology and compare my resuls o a new es approach which assumes only nonsaiaion and concaviy of he uiliy funcions; similar o Arrow [97]. The resuls are ineresing and sugges ha beer diversificaion is required of he oherwise popular porfolio, he value-weighed S&P 500. Keywords: MV-efficiency, HL-efficiency, Spanning, Inersecion I Inroducion The conceps of Spanning and Inersecion are major bedrocks of modern finance in general and porfolio selecion heory in paricular. Given marke compleeness, he marke porfolio fronier is said o span all asse reurns in he mean-variance sense. A porfolio A wih n + N asses is said o span a narrower porfolio B of n asses if he efficien froniers of he wo porfolios coincide. In fac, an invesor holding he efficien allocaion B does no benefi, mean-variance wise, by adding any of he exra N asses in he broader porfolio A. Thus here is no benefi from furher expansion or conracion of he curren porfolio. Spanning occurs if no invesor benefis from any diversificaion moves irrespecive of heir degree of risk aversion. If however he efficien froniers inersec, hen here is exacly one uiliy funcion for which an invesor does benefi. Only invesors wih ha uiliy funcion will be opimizing if hey fail o add asses from he broader porfolio. Alernaively, here exiss one coefficien of risk aversion for which diversificaion benefis does no occur. Inersecion hus ess wheher or no we can fine any raional agen wih any conceivable degree of risk aversion, however absurd as ofen suggesed in he lieraure on he equiy premium puzzle, who migh no benefi from diversificaion. Spanning hus implies inersecion bu he reverse is no he case. Asses ha are spanned can be ignored for he

3 purpose of porfolio selecion. For diversificaion purposes herefore, i is a sandard ask for an invesor o es for poenial benefis from exending porfolio conen o a broader specrum eiher for higher reurn, o economize on risk or in general, o achieve some preferred reurn disribuion. Mean-variance analysis is a popular ool for analyzing porfolio efficiency. The procedure ypically involves maximizing a specified mean-variance uiliy funcion subjec some consrains including feasibiliy. Since Markoviz [95, 959] and Tobin [958], mean-variance analysis has been dominaed by one definiion of efficiency. According o hese auhors, a porfolio is efficien if here exiss no oher porfolio wihin he feasible se ha is characerized eiher by a higher expeced reurn wih no worse volailiy or by a lower volailiy plus no worse mean reurn. In his paper, I refer o his approach which is he mos popular as he sandard or mainsream mean-variance heory. I is imporan however o sress ha sandard mean-variance analysis is only a subse o he broader framework of Sochasic Dominance and is no necessarily consisen wih expeced uiliy heory (see Arrow [95]). I is only consisen if he reurn disribuion is ellipical, which is hardly he case wih finance daa. In paricular, an invesor may prefer an asse wih lower mean reurn plus higher volailiy if ha asse has more preferred disribuional properies including skewness, kurosis and persisence compared o he benchmark. Tha is, sandard Markoviz-ype definiion of meanvariance efficiency is a special case which is paricularly no ideal given he naure of financial daa. Hanoch and Levy [970] provide an alernaive definiion of mean-variance efficiency which is more consisen and free of disribuional assumpions. In heir framework, a porfolio is efficien if here exiss a monoone and concave (uiliy) funcion ha raionalizes ha porfolio. Pos [00] uses his definiion in his proposed linear programming es for spanning and inersecion using second order sochasic dominance. Thus, by using second order sochasic dominance, his es is mindful only of wo of mos imporan properies of expeced uiliy heory, non-saiaion and risk Oher approached may include minimizing variance subjec o a minimum desirable reurn. 3

4 aversion. Furher, by using linear programming, he es avoids he compuaional burden of quadraic soluions which characerized mainsream mean-variance analysis and has he addiional flexibiliy o incorporae logical exensions such as ransacion coss. In his paper, I evaluae porfolio efficiency and mean-variance spanning and inersecion via he resricive sandard approach iniially using he simple Sharpe Raios and nex wih he popular full characerizaion. I hen repea he ask using he Hanoch- Levy definiion of efficiency and he compare he resuls of boh branches. The res of he paper is organized as follows. The nex secion reviews he lieraure on porfolio efficiency and mean-variance spanning and inersecion. Secion III presens he ess considered in his paper as well as mehods and algorihm employed. In secion IV, I presen he daa and he main resuls. I also discuss briefly he implicaions. Secion V concludes. II Lieraure Review The lieraure on Spanning and Inersecion is vas and he inenion here is no o give a complee overview bu merely o briefly recap he main branches and o indicae o an ineresed reader as o where o look. Applicaions of sandard mean-variance analysis (MVA) abound. DeSanis [995] and Cumby and Glen [990] employ MVA o quesion wheher US-invesors can benefi from inernaional diversificaion. Taking he viewpoin of a US invesor who iniially only invess in he US, hese auhors sudy he quesion wheher hey can enhance he mean-variance characerisics of heir porfolio by also invesing in oher developed markes. DeSanis [994], Bekaer and Urias [996], Errunza, Hogan and Hung [998], and DeRoon, Nijman and Werker [00] invesigae mean-variance porfolio advanages o he US invesor who holds asses in he developed markes in he US, Japan and Europe by invesing in emerging markes. Glen and Jorion [993] ake he argumen furher by invesigaing wheher mean-variance opimizing invesors wih welldiversified inernaional porfolios should add currency fuures o heir porfolios. Tha is, Sharpe [97] 4

5 should hey hedge he currency risk ha arises from posiions aken in cross border socks and bonds? Some auhors have explored spanning and inersecion in join formulaions involving mean-variance froniers and volailiy bounds in wha has come o be ermed dualiy ess. Ferson, Foerser, and Keim [993], DeSanis [994], Ferson [995] and Bekaer and Urias [996] demonsrae ha he hypohesis of mean-variance spanning and inersecion can be reformulaed in erms of he volailiy bounds similar o hose by Hansen and Jagannahan [99]. In heir framework, he quesion is wheher he se of addiional asses conain informaion abou he volailiy of he sochasic discoun facor or pricing kernel ha is no already presen in he curren porfolio. A mean-variance improvemen in his case occurs if he diversificaion ino emerging markes for insance provides igher volailiy bounds on he sochasic discoun facor han reurns from he developed markes only. Bansal and Lehmann (997) provide a bound on he mean of he logarihm of he pricing kernel, using growh opimal porfolios. Balduzzi and Kallal (997) show how addiional knowledge abou risk premia may lead o sharper bounds on he volailiy of he discoun facor and Balduzzi and Roboi (000) use he minimum variance discoun facor o esimae risk premia associaed wih economic risk variables. There is lieraure ha uses condiioning informaion. Finance reurn daa are hardly independenly and idenically disribued (i.i.d.). Cochrane [997] and Bekaer and Urias [996] develop models ha allow he incorporaion of condiional informaion in heir ess. Though heir procedures are inuiive and involve only a rescaling of reurns, a disadvanage of his mehod is ha he dimension of he esimaion and esing problem increases quickly. Harvey [989], Campbell and Viceira [998] and DeRoon, Nijman and Werker [998] show how he problem can be largely circumvened by assuming ha variances and covariances are homoskedasic, while expeced reurns are allowed o vary over ime, alhough his assumpion is largely in conflic wih he empirical evidence regarding ime-varying second momens. Appealing o his simplifying assumpion however, he auhors show ha he condiioning variables can easily be accouned for by using hem as addiional regressors. The resricions for he inersecion and spanning hypoheses hen become similar o he resricions in he sandard case wih i.i.d. 5

6 variables. This way of incorporaing condiional variables also has he addiional advanage ha he regression esimaes indicae he economic circumsances, i.e., for wha values of he condiioning variables, inersecion and spanning can or can no be rejeced as demonsraed in Shanken [990] and Ferson and Schad [996]. Markoviz [95, 959] and Tobin [958] presen quaniaive approaches o porfolio analysis. Their prescripion remains dominan in pracice o dae. Markoviz propose choosing he porfolio ha minimizes variance subjec o a resricion on he mean reurn. These mehodologies involve large quadraic programming soluion rules. To simplify he problem, Yamakazi and Konno [99] presen linear mehods involving mean absolue error analysis (MADA) while Young [998] formalize a maximum (and minimum) reurn approach requiring linear programming. The laer also esablishes he exac relaionship beween his minimax approach and Markoviz s. Mos mean-variance analysis mehodologies (mehods above) consider when he invesmen possibiliy se is given. Kandel and Sambaugh [987] and De Roon, Nijman and Werber [00] propose mean-variance spanning and inersecion ess in he case where IPOs exis in he marke. Though dominaed by sandard mean-variance analysis, i is imporan o sress ha sandard MVA is only a subse of he broader subjec of Sochasic Dominance (SD). The advanages of using his sandard form include is racabiliy, ease of esabiliy (see Huberman and Kandel [987] and De Roon e al. [00]) as well as flexibiliy o allow for logical exensions such as ransacions cos and shor selling consrains. As poined by Bigoelow [993], his sandard definiion of MVA is no necessarily consisen wih expeced uiliy heory; i is consisen only if reurn disribuion is ellipical. Hanoch and Levy [970] make his claim much more inuiive when in heir characerizaion of resuls when differen classes of mean-variance uiliy funcions are assumed. They sugges ha he sronger he resricions assumed on admissible uiliy funcions, he closer one ges o individual complee preference ordering. Therefore, he number of iems in he efficien se is reduced as he condiion for dominance becomes more specialized as in he sandard mean-variance uiliy case. 6

7 Meyer [979] presens necessary and sufficien condiions for esing wheher or no a given porfolio is efficien in he Hanoch-Levy sense. Similar o he case of sandard MVA, he implemenaion of such ess involve quadraic programming which can be compuaionally asking. Pos [00] demonsraes a way o derive necessary and sufficien condiions ha require only linear programming hence reducing he compuaional burden enormously. His es also offers increased flexibiliy including he opporuniy o consider ransacion coss and shor selling consrains. In he nex secion, I recap sandard mean-variance analysis saring wih he Sharpe raio and hen a full characerizaion. Porfolio efficiency ess are highlighed for boh sandard MVA and for he Hanock-Levy ype uiliy due Pos [00]. III Mehods and Procedures The Sharpe raio is perhaps he crudes and quick source of he sandard meanvariance crierion. A porfolio is mean-variance efficien if no alernaive feasible porfolio yields a leas he same mean reurn wih a lower variance or a higher mean reurn wih a wors he same variance. 3 The raio basically is 4 where µ ( ωr) and ( ωr) SR µ σ ( ωr) φ ( ωr) = σ are he mean reurn and sandard deviaion of he porfolio ω. φ is simply daa se available up o dae. I is easy o spo poenial flaws wih he Sharpe raio. For insance, his crierion auomaically ranks risk free asses as he mos efficien wih a raio of posiive infiniy regardless of he disribuion of reurns on risky asses. This is so because no oher asse has zero variance excep he risk free asse. 3 An alernaive measure of porfolio performance similar o he Sharpe Raio is Jensen s alpha following Jensen [968]. 4 Oher measuremens of Sharpe Raio use mean-o-variance raio. This is simply a normalizaion and yields same porfolio ranking as he above formulaion. 7

8 A more rigorous mean variance analysis follows. An invesor is assumed o be faced wih he one period problem of maximizing he indirec uiliy of fuure wealh [ u( )] max E W + ω subjec o he consrains W = Wω r + and ω i = (see Ingersoll + [987]). ω is he vecor of weighs assigned o each of he n asses wihin he porfolio and i n is an n uni vecor. r + is an n vecor of nex period reurns. Uiliy u is assumed o saisfy he usual properies. The agen s problem hus can be rewrien as: [ u( W ω r + )] + η( ω i ) max E ω where η is a Lagrange muliplier. The firs order condiion of he above implies ha [ m + r + ] in E = where m + is he sochasic discoun facor or pricing kernel which is assumed o exis if he law of one price holds ω + m is given as = W u ( W r ) η n n m. As suggesed by he previous secion, and as shown by Ferson, Foerser, and Keim [993], DeSanis [994], Ferson [995] and Bekaer and Urias [996], he concep of mean-variance spanning and inersecion has a dual inerpreaion in erms of volailiy bounds. In his regard, mean-variance spanning means ha he volailiy bound derived from he reurns, + r is he same as he bound derived from ( r r ) minimum variance sochasic discoun facors for, + m r r, ( ) + variance sochasic discoun facors for ( r ), + ;, +. Therefore, he, are also he minimum r, + ;, +, and he asse reurns, + r do no provide informaion abou he necessary volailiy of sochasic discoun facors ha is no already presen in, + r. Using he definiion for covariance cov ( x y ) = E ( x y ) E ( x ) E ( y ) rewrie he above FOC as: E [ r ] [ E ( m )] i [ E ( m )] [ m r ] + = + n + cov + +, we can 5 Subsiuing he sronger assumpion of no arbirage for he law of one price, one can show ha m > 0. The same resul is arrived a when one inerpres he kernel funcion as he ineremporal rae + of subsiuion. 8

9 The opimal porfolio weighs ω can be found from he above if he uiliy funcion and Lagrange muliplier η are known. The problem becomes more racable if one resrics he objecive o meanvariance opimizaion (ha is assuming a mean-variance uiliy funcion). Assume ha a fund iniially has n asses in is porfolio. A porfolio ω is mean-variance efficien if i is chosen o opimizaion n subjec o he consrain i = γ max u = ω µ ω ω () ω ω where µ is an ( ) n vecor of gross reurns. γ is he coefficien of risk aversion and he ( n n) variance covariance marix of reurns. Opimal allocaion requires he assigned weighs o be generaed as: ω = γ ( µ ) In he above, he Lagrange muliplier η can be inerpreed as he zero-bea rae, i.e. he η i n reurn of he porfolio ha is no correlaed wih he opimal porfolio. is ( ) r, The reurn vecor on he enire marke n while r, is ( ) R can be pariioned ino ( r ) r,, (),, where N. We regard r, as he benchmark porfolio and r, he vecor of es asses. If he benchmark porfolio is efficien mean-variance wise, hen we expec efficien porfolios o be of he form: wih ω being ( n ) and ω being [( n + N ) ] we have ( µ η i ) = γ ω n ω ω = (3) 0, a pariioning of which implies µ η in ω = γ µ η in 0 The firs equaion in (4) implies ω = γ ( µ η ) second line, we have:. From equaion (), in he general case (4) i n and subsiuing his ino he ( i N i ) + µ (5) = η n 9

10 If here is only one value of η ( γ ) for which his condiion holds, we say ha here is inersecion. In his case, he wo efficien froniers inersec, and here can exis a raional risk averse invesor (i.e. an invesor wih a specific degree of risk aversion) who has no benefi in erms of sandard mean-variance radeoff from including he exra N asses ino her porfolio. If his condiion holds for every value of η ( γ ), we say ha here is spanning. Thus, spanning implies: µ µ = i n i N = 0 0 Spanning means ha he mean-variance fronier of he n asses compleely coincides wih he mean-variance fronier of he n + N asses and no invesor, irrespecive of degree of risk aversion can benefi from furher diversificaion. Inersecion on he oher hand means ha he wo mean-variance froniers have only one common poin (porfolio). In he wo asse case equaion (5) becomes σ σ µ = η + µ σ (6) σ where subscrips refer o he iniial asse porfolio and he asse for poenial σ addiion. In (6), he raio σ coincidenally is he slope coefficien (bea) from regressing he reurn of asse on ha of asse. The hypohesis ω = 0 can hus be esed by running he regression: r α + u (7), = + β r, and esing he resricion α η( β ) unobserved. Pre-muliplying equaion () by Solving for relaive risk aversion coefficien, =. Again, he Lagrange muliplier is i, we have = i ω = γ i ( µ η i ) γ = i η (8) µ i i This implies ha one can es for inersecion (for a specific invesor wih a given degree of risk aversion γ hence η ) as well as es spanning (for all invesors irrespecive of. 0

11 γ ) using he above mehodology. In oher words, esing inersecion involves choosing γ (hence η ) and esing wheher he condiion α η( β ) = holds. A es for spanning implies his condiion holds for all γ (hence η ). This requires he join hypohesis β = and α = 0. 6 To es for inersecion in his sandard Markoviz-ype uiliy framework, I reparameerize and es = 0 κ in he regression r, η = κ + β ( r, η) + u for several values of γ (hence η ). The es of spanning involve he reparameerized regression r α + u and he join hypohesis λ = β = 0 and, r, = + λ r, α = 0. Hanoch-Levy (HL) approach is criical of he above mainsream MVA due o disribuional assumpions which are inheren in he uiliy funcion specificaion. Since uiliy funcions are no observable, his provides raional for he use of general assumpions such as nonsaiaion and risk aversion. This noion is eviden in heir alernaive definiion of porfolio efficiency. A porfolio opimal relaive o some funcion ω Ω is HL-efficien if i is u U where U is he se of all monoone concave uiliy funcions. This is a much sronger definiion han he sandard of mean-variance definiion of efficiency (see above). In conras o he Sharpe raio, he HL definiion ypically classifies a riskless fund as inefficien, in consisency wih Arrow [97] since socks generally have higher mean reurn over he risk free rae. Thus, a HL-efficien porfolio saisfies min u U T { max{ u( rω) df( r) u( rω ) df ( r) } = min max [ u( r ) ( )] = 0 = ω u rω ω Ω u U ω Ω where F ( r) is he empirical reurn disribuion funcion. The linear programming es by Pos (00) is designed using he above approach. His es quesions wheher or no one can consruc a monoone and concave uiliy u U ha can raionalize he porfolio choice T ω. In oher words, can we find suppor 6 An alernaive es is spanning and inersecion is Jensen s alpha approach, due Jensen [968].

12 lines for a monoone and concave quadraic funcion ha jusifies he diversificaion sraegy eviden in he curren porfolio? If we can, hen we can call such a funcion a poenial uiliy funcion for an opimizing agen wih ha porfolio. The above minimax formulaion has ofen appeared in he lieraure as measures of producion efficiency 7 as well as in he finance lieraure (see Young [998]). In he HL definiion, he class of monoone and concave quadraic uiliy funcions is resriced o: U { u U u( r) = α + α r + α r α 0, α + α } : The firs consrain α 0 endures concaviy. can be seen as he marginal 3 increase/decrease in reurn due o a porfolio reallocaion and hence he second consrain α + α 0 resrics he funcion o be sricly increasing over he enire reurn 3 inerval. If he relevan porfolio is efficien, hen such suppor lines mus exis, and if such suppor lines exis, hen he porfolio mus be efficien. There are wo imporan issues o noe concerning he above saisic. Firs, he saisic does no represen a meaningful performance measure ha can be used o rank differen porfolios wih regards degree of efficiency. Secondly, he suppor lines are used as an insrumen for esing he efficiency of he porfolio raher as an esimae of he uiliy funcion for which he given porfolio is efficien. This is because once a porfolio is found o be efficien, here ypically exiss muliple candidae uiliy funcions ha could equally jusify ha porfolio. The linear programming formulaion requires he es saisic ξ T ( ω ) min θ : ( α + α r ω )( r ω r ) ( ) = Θ = θ, α, α Θ α α R R : α + α 3, T, 3 3 wih {( ) } i + θ 0 i... n + N. The problem hence has hree choice variables θ, α and α 3 and n + N + consrains. The porfolio ω is efficien if and only if ( ω ) = 0 ξ. 8 7 See Debreu (95) and Farrell (957). 8 See Pos [00] for proof.

13 In he above, he necessary condiion follows direc from Kuhn-Turker condiions = ω Ω T T of solving for ω = arg max u( r ω) Tha is o say alernaive porfolios T T = u for u U. Thus he condiion ( r ω )( r ω r ω) 0 ω Ω ω is opimal for he concave monoone se of uiliy funcions only if all ω Ω are enveloped by he angen hyperplane defined by he vecor u ( r ω ) ( u ( r ω ),... u ( r Tω ) α ), we know ha ( ( ), ( ω u rω u r ) is feasible i.e. ( u ( rω ), u ( r ω ) Ω 3. By consrucion (given our choice of α and inequaliy (9) above implies ha necessary condiion for efficiency. ω is efficien only if ξ ( ω ) = 0 Sufficien condiions are esablished by using ( ) Ω,α 3 (9). The, which is he α for he opimal porfolio and using some concave funcion u( W ) α W + α =. If 3W ω is efficien i.e. ξ ( ω ) = 0, hen T ( r ω ) ω = rr u ( rω ) ω r T u = ω Ω = max (0 ) T Jensen s inequaliy and u concave imply ha u T ( r ω) ( ω ) ω = max u r ω Ω = r for T T all ω Ω. Given u ( W ) as above, we have u( r ω) = ( ω ) ω = u r = r. This T T ogeher wih equaion (0 ) gives u( r ) = ( ) = ω u r = ω r rω. Combining his T T wih he Jensen s inequaliy gives u ( r ω ) ω ( ω ) ω = rr u r = r { = 0 when ω = ω } which simply implies he sufficien condiion also being ξ ( ω ) = 0. One observaion abou he es saisic is ha here is always a feasible soluion T α and, in which case = ( r i r ω ) for insance = α 0 3 = T = θ necessarily saisfies he consrain. This refers o he case of a risk neural agen wih linear uiliy and who seeks o maximize only expeced reurn. A second observaion is as follows. Consider an iniial porfolio of jus one sock ( n = ) and se, ω = ω. Thus r i r,, = and hence 3

14 r ω = 0. Immediaely, one ges he resul θ = 0 and he porfolio is efficien. In r i, oher words, if here is only one asse in he invesmen opporuniy se, hen ha single asse makes an efficien porfolio. To es he efficiency hypohesis, Pos (00) develops an alernaive formulaion which he calls he dual formulaion which is very similar ha oulined above. The dual saisic is given as: ψ ( ω ) max{ µ ( ω, ω ) : σ ( ω, ω ) 0} ω Ω where he mean difference beween he curren porfolio T = ω and a poenial feasible T T alernaive is µ ( ω, ω ) = ( r i ω r ω ). ( ω, ω ) = ( r ω )( r ω r ω ), T = σ is he co-movemen measure beween boh porfolios. Again, he derivaion of his is quie similar o ha recapped above (see Pos [00]). The porfolio only if ( ω ) = 0 ψ. i, ω is HL-efficien if and This paper adops he dual formulaion for he ess conduced in he nex secion. The algorihm used is raher simply bu no so fas for a large asse space and high accuracy level. Firs, I decide he degree of precision ha an invesor may consider imporan wih regards porfolio weighs. Tha is, an invesor seeking fairly high accuracy may choose weighs o he decimal of say for ψ. Then I ake he curren porfolio weigh vecor 5 0. Nex, I guess a arbirary posiive value ω and formulae all possible alernaive weigh permuaions o he required accuracy level bearing in mind he n + N i= invesmen consrains = ω and ω 0 i (,... n N ) i i +. Using each hypoheical weigh vecor ω, I evaluae µ ( ω,ω ) and σ ( ω,ω ). If σ ( ω, ω ) 0 and µ ( ω,ω ) is less han he previously saved ψ, I replace ψ o equal he curren µ ( ω,ω ) and also save he curren weigh vecor. Then I ry he nex hypoheical weigh vecor. My final ψ is ha remaining afer he enire ieraion is done. 4

15 IV Daa Descripion and Main Resuls In he curren paper, I use monhly reurn daa on seven of he mos widely wached benchmark porfolios/indexes of sock marke aciviy. The firs is S&P 500 Table : Descripive Saisics: monhly reurn daa in percenages, July 96 o Dec 00. S & P B / L B/ M B/ H S / L S / M S / H Mean SD Minimum Maximum Skewness Kurosis Jarque-Bera Variance-Covariance Marix S & P B / L B/ M B/ H S / L S / M S / H S & P B / L B/ M B/ H S / L S / M S / H Uncondiional Correlaion Marix S & P B / L B/ M B/ H S / L S / M S / H S & P.0 B / L B/ M B/ H S / L S / M S / H Monhly reurn for S&P 500 were rerieved from CRSP Compuserve daabase. These reurns are inclusive of dividends. Reurns on Fama and French porfolios are from he web sie of Kenneh French a hp://mba.uck.darmouh.edu/pages/ faculy/ken.french/. Jarque-Bera [980] provides an LM es for normaliy. In his case, I es he normaliy of reurns. The resul is disribued as Chi-Squared wih degrees of freedom. The es rejecs he normaliy assumpion in all 7 cases. 5

16 value weighed cum-dividend reurn wih a ime range of July 96 o December 00 from CRSP daabase. The res six are Fama and French book-o- marke sored and sizesored benchmark porfolio reurn daa wih he same ime range. 9 Summary descripive saisics are repored in able. Ineresingly, he S&P 500 has one of he lowes mean reurns among he seven porfolios. In compensaion, i does exhibi modes volailiy. The index correlaes highes wih F&F porfolio coded B / L and lowes wih S / L. The porfolio B / L seems o be a raher conservaive porfolio exhibiing he lowes reurn, volailiy, kurosis and negaive skewness. S / H seems o be he mos advenurous porfolio. Jarque-Bera [980] es for normaliy repors srong rejecion of he null in all seven cases. A poenial implicaion is ha sandard mean-variance analysis Markoviz [95] is flawed. Nex, I compare condiional Sharpe raio for he above porfolios. Sharpe Raios repored in figure do no show overwhelming performance advanage for or agains any porfolio since all he graphs seem neck-o-neck, perhaps in excepion of F&F S / L where S&P seems o do beer. However, his is far from being conclusive since his pair as well exhibi he lowes correlaion which suggess a good avenue for risk curbing diversificaion. Furher, as highlighed earlier, he Sharpe raio can be highly uninformaive and even erroneous. For insance, wo risk free asses wih differen reurns will be ranked equally wih a raio of posiive infiniy because boh have zero reurn variance. 9 See hp://mba.uck.darmouh.edu/pages/faculy/ken.french/daa_library.hml 6

17 Figure : Condiional Sharpe Raios, S&P verses F&F benchmark porfolios. These raios sar from he 0 s observaion (from March 943) up o he 98 h (Dec 00). The firs 00 observaions were used as previous informaion (condiioning informaion) o calculae he raio repored a dae 0 and so on. The solid lines are moving (condiional) Sharpe raios for S&P 500 while he broken lines are moving Sharpe raios for he F&F porfolio repored in each ile. 7

18 Tess for inersecion were carried ou for varying degrees of relaive risk aversion using sandard mean-variance uiliy maximizaion as oulined in he previous secion. The quesion is wheher or no a given agen wih specific degree of risk aversion γ (hence η ) will be benefi from diversifying beyond he S&P benchmark and add of he F&F porfolios. Table shows -saisics for : α = η( β ) reparameerized regression r, η = κ + β ( r, η) + u. Table : -saisic for differen degrees of relaive risk aversion. γ H in using he B / L B/ M B/ H S / L S / M S / H As explained in Secion III, in all he above cases, he variable 0 r, r, refers o S&P 500 reurns while refers o he reurns of he F&F porfolio noed a he op of each column. η is derived from γ and he appropriae variance-covariance marix using equaion (8) above. Tha is, in µ γ η =. i i Table above suggess a definie case of non-rejecion of he hypohesis of inersecion for all degrees of risk aversion even as high as 0. For he range of relaive risk aversion considered, agens do no benefi from diversificaion. Agens will no be acing inefficienly by failing o add any F&F porfolio o he benchmark S&P 500. As one considers agens wih higher risk aversion, he inersecion he value of he saisic rises slighly wihin reasonable ranges of γ ye he hypohesis is sill no where close o n n 8

19 he rejecion zone. The firs row of able refers o he risk neural agen who is ineresed only in expeced reurn. One way o inerpre hese resuls is ha from he perspecive of a risk neural agen, he mean reurn of all seven asses are no saisically differen hence no diversificaion suggesed on he firs row. Comparing able o able suggess reasons why diversificaion benefis are no prevalen via he mean-variance approach. Mos of he F&F porfolios have slighly higher reurns han S&P 500 reurn. However, he former have higher volailiy as well as srong correlaion wih he laer. The es for spanning asks wheher or no no agen will benefi from diversificaion, hus irrespecive of he degree of risk aversion. From he es of inersecion above, i is apparen ha spanning (ha no agen benefis) will mos probably no be rejeced eiher given he raher low saisics repored. Table 3 below shows Wald saisics for he join hypohesis β = and α = 0 in each of he six cases. The es is conduced here using he reparameerized regression wih H : λ = β = 0 and α 0. 0 = r +, r, = α + λ r, u Table 3: Wald Saisic irrespecive of degrees of relaive risk aversion. B / L B/ M B/ H S / L S / M S / H Wald Again he variable r, refers o S&P 500 reurns while F&F porfolios. The saisic is derived as ( ) ( δˆ ) ( δˆ ) ( δˆ ) δˆ dh d l dh h h( δˆ ) dδ dδdδ he number of resricions under H. 0 dδ r, disribued χ wih degree of freedom equal Wald saisic wih degrees of freedom has 5% criical value of According o able 3, spanning canno be rejeced in any of he six cases. In words, agens, irrespecive of degree of risk aversion will no benefi from diversificaion beyond he S&P benchmark if he oher available asses are he Fama and French porfolios aop each column. As is obvious from he above ess, mainsream mean-variance analysis depends srongly on disribuional assumpions which are inaccurae as eviden from he 9

20 descripive saisics and he Jarque-Bera es saisics in able. HL-efficiency hus offers an opporuniy o es efficiency of porfolios relying on accepable properies of uiliy funcions being nonsaiaion and risk aversion; a es wih fewer reurn disribuion assumpions. Table 4: Dual formulaion saisic ψ comparing he value-weighed S&P 500 o each F&F porfolio. B / L B/ M B/ H S / L S / M S / H ψ ψ = 0 implies a sand-alone S&P 500 porfolio is efficien and here is no need o diversiy if he only oher asse available is he F&F Porfolio on each column. For insance in he 3 rd column, if asse B/ M is available, hen a porfolio made solely of S&P 500 is sub-opimal. In conras o he sandard mean-variance ess, he dual es saisic above rejecs efficiency of a porfolio made solely of he value weighed S&P 500 when alernaive asses B/ M, B/ H, S / M and S / H are available for diversificaion purposes. I is imporan o cauion again ha values repored in able 4 only compare he S&P o each of he six Fama and French porfolios. The dual saisic ψ by design does no offer an opporuniy o rank hese six Fama and French porfolios based only on informaion conained in he able above. V Conclusion Sharpe [97] has remarked ha if he essence of he porfolio analysis problem can be adequaely capured in a form suiable for linear programming mehods, he prospec for pracical applicaion would be grealy enhanced. In he curren paper, I have demonsraed an example of he growing number of applicaions of linear programming o effecively answer opical quesions in finance which oherwise would have required a large quadraic soluion. According o he sandard mean-variance uiliy funcion approach similar o Markoviz [95], spanning and inersecion analyzes he effec ha he inroducion of addiional asses has on he mean-variance fronier. If he mean-variance fronier of he benchmark asses and he fronier of he benchmark plus he new asses have exacly one poin in common, hen hey inersec. This is wha is ermed inersecion. This means 0

21 ha an agen wih ha mean-variance uiliy funcion is opimizing my holding ha benchmark. For an agen wih ha uiliy funcion, here is no benefi in sandard meanvariance uiliy from adding he new asses. If he mean-variance fronier of he benchmark asses plus he new asses coincides wih he fronier of he benchmark asses only, here is spanning. In his case no sandard mean-variance uiliy opimizing invesor can benefi from adding he new asses o her iniial porfolio of he benchmark asses. The forgoing definiion is only accurae if reurn disribuion is ellipical. Hanoch and Levy [970] provide alernaive definiion for mean-variance efficiency much more consisen wih Arrow s heorem even in a world of non-ellipical reurns. In his paper, I implemened ess of spanning and inersecion for a porfolio made solely of he value-weighed S&P 500 as a benchmark porfolio using boh es approaches: mainsream mean-variance uiliy approach using quadraic opimizaion and a second es designed o saisfy he Hanoch-Levy definiion of porfolio efficiency and make use of linear programming ools. The objecive is o es wheher here exiss spanning and inersecion of six Fama and French porfolios, considered here as asses available for possible diversificaion. While sandard mean-variance ess are unable o rejec he hypohesis of efficiency of he benchmark, he Hanoch-Levy approach rejecs in four of he six cases. There are herefore mean-variance uiliy benefis from exending he asse holding beyond he S&P benchmark. The difference in resuls is accouned for by disribuional assumpions inheren in mainsream mean-variance opimizaion heory.

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