An Analytic Derivation Of The Efficient Portfolio Frontier
An Analytic Derivation of the Efficient Portfolio FrontierAuthor(s): Robert C. MertonReviewed work(s):Source: The Journal of Financial and Quantitative Analysis, Vol. 7, No. 4 (Sep., 1972), pp.1851-1872Published by: University of Washington School of Business AdministrationStable URL: http://www.jstor.org/stable/2329621 .Accessed: 11/04/2012 12:12Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at ms.jspJSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.University of Washington School of Business Administration is collaborating with JSTOR to digitize, preserveand extend access to The Journal of Financial and Quantitative Analysis.http://www.jstor.org
ANALYSISJOURNALOF FINANCIALAND QUANTITATIVESeptember 1972AN ANALYTICDERIVATIONOF THE EFFICIENT PORTFOLIOFRONTIERRobert C. Merton*I.explicitly,isshown thatunder certainSuppose theresecuritydenotedby E.,securitydenotedby a.,by aidenoted2 'j .im, and if we furthertionof the otherand the vestedsecurity 0, i 1,as a linearmatrixportfoliosas the locusfor a prescribedand jthon the ith2can be representedof allof a portfoliothe ithrisky,assumedis definedm securitiesof the valuebetweenon the ithreturnthe expectedthe variance-covarianceThe frontierhave the smallestpercentageareno securitythenfor deriv-Are RiskySecuritiesof the returnm securitiesassume thatQ [a i. ]. is nonsingular.from thesebe constructedfolioswithof returnsthe covarianceIt isincorrect.securitiesare m riskythe separationtechniquegraphicalSet When AllPortfolioThe EfficientII.isfrontierportfolioing the efficientare verified.of a mutual fund theorem.the classicconditions,are derivedcharacteristics,from thesein termsresultsfrontiersfor theseand proved in the contextstatedfor more than e most importanttheorem,the efficientpaper,and the characteristicsHowever,qualitativeapproach has been to displayIn thisof graphs.2in the literature.at lengththe generalassets,1frontierportfolioefficientof the mean-variance,The characteristicshave been discussedIntroduction.combina-of returns,which canof feasibleexpectedreturn.in the ithsecurity,portLet*Massachusetts Institute of TechnoZogy. I thank M. SchoZes, S. Myers,Aid from the NationaZ ScienceG. Pogue, and a referee for helpfuZ suggestions.Foundation is gratefuZZy acknowZedged.Seeworks of Markowitz [10] and Tobin [17 and 18].1See the classicalThe limitedreferences.Sharpe [16] for a modern treatment and additionalin Borch [2], Feldsteinassumption is discussedof the mean-variancevaliditySamuelson [15] has shown that mean[5], Hakansson [6], and Samuelson [14].Further, Mertonfor "compact" distributions.variance is a good approximationis valid in intertype analysis[11 and 12] has shown that mean-varianceand assetproblems when trading takes place continuouslytemporal portfolioprice changes are continuous.to thisfor example,2Exceptionsmodels:see,graphicalFama [4],of equilibriumare the discussionsanalysisBlack [1], Mossin [13], and Lintner [8 and 9].1851
1,.,describedm, and as a definitionalresult,as the setthatof portfoliosE x. 1.Then, the frontierthe constrainedsatisfycan beminimizationproblem,mmin2(1)tosubjectx2 mEm1E 1where a2 isequalto E.3the variance21 110 xmXC1i(3c)0 1 -on Q.the x'sthatEx-(4)Xk Rwhere the vij3Itallowed.(1) can be rewrittenare defined mxE ]The standardyy 2[l[returnas,1ifirst-orderi l Ei 1- 2 .ii-expectedconditionsfor am,x.E.,satisfySystem (3) is linearwithon the frontierEm Elmxx Y [E 11[E1 1 i j cxiiwhere y1 and y2 are the multipliers.criticalpoint are,Further,Xiof the portfolio(3a)ijx.E.Using Lagrange multipliers,min {(2) xa1ij(3) minimizein the x'sl 1v .jj yk j2E vas the elementsis assumed that borrowingHence, the only constraintCa2and are unique by the assumptionwe have from (3a) thatand hence,l kjk 1,of the inverse*.,Mof the variance-and short-sellingof all securitiesison the xi is that they sum to unity.1852
[v.]i.e.,matrix,covariancek 1, .,(4) by Ek and summing overMultiplyingm, we have thatZlXkEk (5)I1 kEk(4)and summingoverk 1,l(6)ZmE1'k yl1l1 kjk'1 vJEthatm, we have.,yl1v jEjE y Yk jk1 Ey2y vkjEEjj1E1 vkj*Define:mZmv1 1 kjjE.; jAC From (3b),Zm m v jEjE1 1 kj j k;Bm Em v1 1 kj(3c),(5),and (6),we have a simplesystemlinearfor y1 and y2sE By1 Ay2(7)1Ay1 Cy2 EZE and thatvEY v11 1 kjj Ejkj k(7) for y1 and y2, we find thatwhere we noteSolving that ZB O and C 0.4(CE - A)(8)D1(B - AE)D'2where D - BC - A2 0.54Q is a nonsingularand positivedefinite.variance-covarianceIt followsdirectlyfor all j and k, and B and C are quadratic(unless all Ei 0).are strictlypositive25Because Q22for y1 and y2 from (8) intoWe can now substituteis positive2B C - 2A B A B B(BC - Adefinite,0) BD.matrixissymmetricis also.Hence, vk vjkkj-lforms of Q which means that theythat ZBut B 0,1853and, therefore,(4)QIll1ijvhence(AE - B)(AE. -B)ijD 0.
to solvewithfor the proportionsexpectedreturnof each riskyheldassetE Z1 vkMultiply(CE(B - AE )A) Elvk.-k 1k(3a)(10)and sum fromby xiZEmof a,i 1xjj2 Jijij 1x1iEiFrom the definition(3b),a2(10)impliesy1E y 2.for Y1 and Y2 from (8) into (11),of a frontieras a functionportfoliowe writevarianceof its2a the frontierfirstand secondstrictlyconvex functionda2i.e.,-E 0dEof2 [CE of the minimum-varianceportfolioa 1/C are the expectedxk to be thein the kth asset,thenDefineportfolio.investedm1xkC k 1,1854where0from (9),(14)of theA(12) where E - A/C andof the minimum-varianceExaminationA]-when E2C D2dE2I is a graph ofasto E shows that a2 is a(12) with respect-0proportionis a parabola.spacedareturn,for theDdaand varianceexpectedof E with a unique minimum point(13)Figurethe equation- 2AE BCEin mean-variancederivatives.m Substituting(12),m to derive.,1and (3c),(11)Thus,portfolioE; namely,x(9)returnin the frontier.,m*
FIGURE IMEAN-VARIANCE PORTFOLIO FRONTIER:RISKY ASSETS ONLYE1855E
It is usualFrom (12)plane.of the mean-varianceinsteadda(CE - A)dEDod2cda,,-1 1dEE2tionisportfolioThe efficientalongof the frontierthe frontier Vof the frontierCa.in Figureof feasible(the setII,withstartingis1E- V(CE for the efficient C/portfolioE ED(C2- 1)frontier 1 / DC(a21856is2)portfoliosdeviation)standardThe equationand moving to the northeast.The equationare the asymptotesfor a givenreturn(17)(18)-Efrontierportfolioexpectedhave the.largestportfolioform with E on the ordinatein the standardE -partFigureportfolio.devia-II graphsare(16)linedof E, and the minimum standardfunctionThe broken linesand a on the abscissa.whose equationsO.the same as the minimum-variancewhich is a hyperbola,the frontierwe have thatand (13),Dc3Daconvexa is a strictlyplanedeviation(CE - 2AE B)/Da /(15)From (15),in the mean-standardthe frontierto presentthatis the heavy-the minimum-variancefor E as a functionof a
FIGURE IIMEAN-STANDARDDEVIATION PORTFOLIO FRONTIER:RISKY ASSETS ONLY1857
III.A Mutual Fund Theorem6satisfyingTheorem I. Given m in choosingor from tfoliosfrom justand the proportionsof the individualk 1., 1vk.(B - AE.)/Dk 1, Egk m.,k-portfolio,.A)/D,withIf we definem.,k 1,hk)gk andto be ablethe proportionsof preferences1 hk.m expected1.of riskytheirassets(or equivalently,returns,m.to constructof the two funds chosen by the investorsecuritiesain theasindividualstwo funds,fund must be independentasset,(CE.definitions,by theirBecause we want all(as describedof the frontiervkcompactlyxkon theof two specificcombinationfor any individualin the kthk --jNote that)mportfolio.E, invested(9) can be rewrittenby a linearthe proportion(9) describes any portfolioto show thatportfolioan optimal(19)thensuchfrom among the originalportfoliosis sufficientitbe an efficientreturnbetweencan be attainedfrontierEquationm assets,from thesetheretwo funds.To prove Theorem I,efficientconstructedII,so as to maximizewho choose their portfoliosindividuals,dependent only on the mean and variance of their portfolios,be indifferentassetsfunds")portfoltios("mutualof Sectionthe conditionsvariances,optimalheld by eachindependentof E)must be independentand covariances.Letin the kth asset,of the firstfund's value investedandak be the proportionlet bkthe proportionproportion ofof thethe secondsecond fund'sfund's valuevalue investedinvestedin the kkth assetinbkbebe thelettheorems, seeof mutual fund or "separation"6For a general discussionto Theorem I is proved in MertonA theorem similar[3].Cass and Stiglitzwhen asset returns are lognormallydecisionsportfolio[11] for intertemporalOne advantage offunctions.have concave utilityand investorsdistributed,of the separationtheorem over the classicalthe mutual fund descriptionis extended to an intertemporalcomes when the analysisinterpretationgraphical("funds") are required tomore than two portfolioswhere mal portfoliosspan the space of investors'as a mutual fundeach portfolioIn this case, it is quite natural to interpretis impossible.descriptionto the investor while graphicalproviding a "service"See, for example, Merton [12].1858
(ZT ak zn bk 1),and ak and bk must satisfyxk Egk hk Xak (1(21)withE.returnexpectedEthe conditionand Ethat,7Allsolutions(6 # 0) thatwhere 6 and a are constantstwo funds,k 1,X)bk,"mix" of the funds thatwhere X is the particularfolio-respectively.,i,the efficientgenerateshave X 6E - ato (21) willdepend on the expectedSubstitutingak and bk be independentportof thereturnsfor X in (21) and imposingof E, we have thatak and bk mustsatisfy(22)6(akgkhk bk-1 0, (22) can be solvedFor 6- bk)k 1,(ak -bk) .,m.for ak and bk to giveak bk gk/6(23)bk hk ag k/6 Factorsthe vectorsatisfya and b are two linearlyspace of frontier(23) willmust be frontierbe calleda setboth funds holdingsEBecauseEEb1 k k'bax.of basisalthoughin1b Ek k'vectorsthatTwo portfoliosportfolios.form a basisdeterminedmhEkE hgZElkTwo such portfoliosby theirHence,expectedmklforwhose holdingsthey need not be efficient.are 0),and Efromreturns.1, we have thatk kE (l aa6(24)EbAlternatively,frontierk 1,given values6for Ea and Eb the constants6 and a can be written7Two funds with proportionsak and bk which satisfythe efficientas a subset,portfolios,includingwith elements ak and b , k(20).xk, where the xk satisfy8a and b are m-vectorsm-vectorwith elements1859(21) willportfolios. 1,.,generatem.x isallan
as(E(25)(Ea Eb)EbOl-(E)baone basissetX, thatand urnsdepend on 6 and a.and of the second2aabEmabE11'bthe two portfoliosithe varianceof the first, and[C 2(aC - A6)]/D62(23) as follows:i1 - 1 Zg.h.a31ijij riiijijb bA5 22% bm(26) and (28),a b ahow theare frontierboth portfolios2Ac6 B6 )/D6Jabs we usethe covariance,(24) describesEquation2b:fund,CTa are theportfoliosof basisBecauseb2 (CaZ(29)or wealthpreferencescan be co-mbined to determine(24) and (12)(27)Usingof individualand covariances.variances,(26)(28)programthe investmentfollowof a setcharacteristicsThe essentialTo find0) and then,knowledge"managers" canThe funds'Hence Theorem I is proved.distribution.portfolios,fund, 2a#(din (23) withoutprescribedoftheof the two funds to determineportfolio.optimal6 and a arbitrarilychoosetransformationsand as can be seen in (25)basis,need onlyof preferences.Thus, the investorknow the means, variances,mix,to nonsingularanotherintoof portfoliosare independentvaluestheirfor 6 and a correspondvaluesDifferent[C a2Z11ggaijij](C 2D6D5we can finduncorrelatedthose(i.e.,of 6 and a that willcombinationsa ab 0).For 6Ca2 B62 2Aa6 Ca - A1860 0.0;makeaab 0 when
(29)isan equationbe an equationfor a conicfor an ellipseIf we restrictand becausesection,(seeFigureboth portfoliosA2BC -D 0, it must-III).to be efficient9and take the conventionthta2a b that a2 b then Ea Eb E A/C, and from (25), 6 must be positiveanda Ad/C.One could show that the line a Ad/C is tangent to the ellipseatthe point(portfolios 0, 6 0) as drawn in Figureathatare uncorrelatedIII.do not exist.From (38), 2 0, which implies that all efficientab-b 2 2correlated.Further, aab ab if and only ifaEb A/Cfrom (24),E which impliesthatTherefore,we have thatportfoliosa Ad/C.two efficientare positivelyIf a Ad/C, then,the portfolioheld by the fund with2proportionsb is the minimum-variance portfoliowith ab 1/C and bk Xk.Because 1/C is the smallestvariance of any feasibleportfolio)it must be that,for efficientbeaabwillthesmallest when one of the portfoliosportfolios, is the minimum-variancefund willportfolio.have the characteristics(30)In thiscase,the portfolioof the otherthat1 ACEa2C1aakakD62Emdkj , I kj'kk 1 . M where 6 is arbitrary.There does not appear to be a "natural"it willbe usefulchoiceto know the characteristicsfor the valueof the frontierof 6.portfolioHowever,whichsatisfiesdE(31)E-RdEfor some givenIf we choosevalueof R.6 such thatFrom (15))the portfolioda alongthe frontierwith proportionsequalsDa/(CE - A).a satisfies(31),then(32a)6 C(A - CR)D9Although the paper does not impose equilibriummarket clearingit is misleadingto allow as one of the mutual funds a portfoliothatwould hold as his optimal portfolio.1861conditions,no investor
FIGURE IIIELLIPSE AMONGFRONTIERPORTFOLIOSZERO-CORRELATIONa11862
Em1(32b)(32b)a1 a kE A/C,If R then 6 0 and the portfolio6 O, and the portfoliocannot be satisfiedThe implicationsIV.by keepingof theseIn thisfeasible1"''2Noticethatassetsmmin {21(33)the casethe constraintiZ1jof E and G.valuesxii Is Risklessto includeassetsa risklessII,asset,assetwiththe frontierthe problem:by solvingR - EEsection.the availableway to (2) in Section ijin which all(31)and equationand adding a (m 1)stas beforeX.X. y1If R E, thenin the followingthe analysisis determinedportfoliosmeSet When One of the AssetsIn an analogousR.with finitebe discussedwe extendthe same m riskyreturnwillanalyzedsection,.,0is efficient.portfolioPortfoliosections 1,If R E, 6 0inefficient.resultsThe Efficienta guaranteedof allisby any frontierThe previousare risky.Vk (E. - R)kjj,k(A -RC)X (E- R)]}.1 does not appear in (33) becausewe haveX are unconthe xi,.xXtheucn1 i i.e.,.,xmstrained by virtue of the fact that xm 1 can be always chosen such that 1 is satisfied.the analyticsThis substitutionnot Qnly simplifiesE1 derived later ininto some resultsof solving(33) but also provides insightexplicitlysubstitutedforx 1 1 -M lthe paper.The first-order(34a)derivedconditionsEljiij1 from (33) are-X(Ei1-i 1,R).m,O E - R - E X (E - R).ili(34b)In a fashionsimilarto the previoussection,we derivethe equationforthe frontier,IE-(35)which is drawn in Figureportfoliosas a functionRI cvCR - 2AR B,IV, and the proportionsof riskyfor the frontierassetsof E are(E - R) E'1 vk (E. - R)(36)Xk2J -k 1,CR -2AR B1863.,m.
FIGURE IVMEAN-STANDARDDEVIATION PORTFOLIO FRONTIEREJV16Rc e c1864
and the efficientconvex),locusthe efficientSincein cf, allof the frontierpartwithportfolioof the efficientholdingsare perfectlyportfoliosefficient(inefficient)the lowerwhere E R.of the frontierportionthatof the riskysalesshortrepresentsisis linearand (36),From (35)correlated.locusnot strictly(althoughis convexIV, the frontierin FigureAs picturedthe same O.allBecauseforward to show thatis riskless;assetsfund holdsthe risklessonlyassets(i.e.,k 1).funds,Given m assetswith returnassetone containingsuch thatallwillThe proof only risky0 and bk 0,of Theorem II followswho chooseX 6(E - R) 1 - a,xkifv(E-R)/(CR(A - RC)/(CR(6#(E -R)0),theirportfoliosasset,so as toof theirfrom among theportfoliosif R E.and onlyto provingthe approachmutualonly the risklessonly on the mean and variancebetweenII and aof efficient10a unique pairtwo funds,1kjof Sectionthe conditionsin choosingor from theseE1k(37)am ,and the otherassetsdependentbe indifferentUk E EDefineexistsindividuals,functionsm l assetsoriginalsection,satisfyingR, thereonly riskyrisk-aversemaximize utilityportfolios,of the previousin the notationfund containsand the othersecurityonesuch thatm).Theorem II.risklesson the frontier.portfoliosthe two mutual funds can be chosennamely,straight-than Theorem I when one of thewants a theorem strongerone usuallyisin which one of the securitiesin the caseany two distinctselectingby simplyis riskless,However,Theorem I holdsitcorrelated,are perfectlyportfoliosefficientTheorem I.- 2AR B),Ifthen- 2AR B).then we have thatuk Xak (1 - X) bk 6(E - R)(ak - bk) (1 k 1,.,)(abkbk)m,here are strongerportfoliosonly efficientloThe reasons for considering9. Given that one of the funds holds only thethan those given in footnotethe aggregate demand for a mutual fund with portfolioproportionsrisklessasset,whichwould have to be negative,part of the frontieralong the inefficientof the mutual fund theorem.if not the mathematics,the spirit,violates1865k
and(38) 1 -x(E - R)(ARC)/(CR- 1-Z1 ak and(39)2AR B)bM l) tl -a)R)(a m lwheream1have that-1 - E11bk. bm l(am lm lfor ak and bk, we(37)Solving bak 'Uk/6bk (a1)-k 1,uk/6 m*.*,and(40)Now requiretheonlyrisklessaccomplished(A - RC)/6(CRI bTl-b 11(cl--- RC)/6(CR1)(Aone of the fundsthatasset 0,bk(i.e.,a 1.by choosing2- 2AR B).b) hold(say the one with proportionsk 1,If it2AR B)-is alsom and b 1 1) which.,thatrequiredthe otherisfundhold only risky assets 0), then from (43), 6 - (A - RC)/(CR - 2AR I(i.e.,a, ,and as can be seen in (40),Note that if R A/C, 6 0, which is not allowed,in thissincecase,b aMl bm 1 1.for all k 1, .,which means thatpositionsisfund holdsthe two mutual funds are differenti 0 for some k.m and ais zero.whose net valueinefficientefficient.the "risky"From (39))a hedged portfolioHowever,E akIf R A/C, then 6 0, and the portfolioIf R A/C, then 6 0, and the portfolioEa R).When R A/C, the compositionof the efficientrisky portfoliois(i.e.,(41)akkEm v(E,(A1 kjR?of long and short-isR).k m.(A -RC)Thus, Theorem II is proved.The traditionalthe assetsapproachexists.in Figurethe efficientis to graph the efficientis risklessand then to draw a lineillustratedto findingfrom the interceptV.Then one couldSuppose thatchoosetangentthe pointfrontierfrontierfor riskyto the efficientassetsonly,frontier(E*, Cr*) as drawn in Figureone mutual fund to be the riskless1866when one ofassetasVand the
FIGUREVRELATIONSHIPBETWEENEFFICIENT FRONTIERS:E RERisky Assets Only/E/ II RL I1867
to be (E*, C*) which containsotherTheorem II,two such mutual funds existV).in FigurecasestandardWhen Rlines(withcurvefrontierassetfrontiercan one constructthe entirelinesassetsThe intuitiveonly.1of a risklessintroductionforindividualshow thatFigureGiven thatin(41)(i.e.,k 1,xk ak,folio"),the fundamentalmarket line,thatthea portfoliowithwhen one coulddirectlyE1(Ek(E--equilibrium1 v.(E.ijj-(E*, c*),"M" denotesassetmust be the same as"forpricingthemarketmodel,port-theas follows:- RC),-RC)andin the literature11There seems to be a tendencyto draw graphs withR E and an upper tangency (e.g.,Fama [4, p. 26] and Jensen [7, p. 174)In Sharpe [16, Chapter 4], the figuresappear to have R E and a doubZetangency.1868theHence, we have as- RC), from (44)R)) CTk/(AiR)E r viaik/(AR)/(Awithsolutioncase withm1 i *.aVJij'k 1(to ruleis no reasondeviation.of the capitalm M-M1thereR E.m where.,resultcan be derived byfor riskyis that within the market portfoliothe proportionskaMincluded)resultthe only possibleand standardfor equilibriumsecurity(42)securitywas not possibleas a generalV isreturnexpectedconditionUnder no conditionto selectselection,portfolioout R E, one can easilya necessaryto the hyperbolicof the frontierfor thisthisVI andis a lower tangencyE, there is possibleassets;homogeneous expectations,market portfolio'sin Figuresassets.among riskyAlthoughitand thebe).the riskless(withexplanationamounts of riskynet nonpositiveonly choosefrontierand(31) and (32),above the upper asymptote.asset,returntheE and a, and the frontierto the upper and lower partsdrawing tangentexpectedthey shouldWhen Ronly.liesand the efficientwithare the asymptotesincluded)assetsfor riskyBut, byif R E A/C (as isin equationsfor finiteis no tangency only.when R E are displayedsolutionsgraphicalof the factby virtueassetsand onlyin (41)(asto thoseE. therethe risklessVII.ifE* and a*, was derivedare identicalThe properfor riskythe ris on the efficient(E*, c*)thatassetsonly risky
FIGUREVIEFFICIENT FRONTIERS:RELATIONSHIPBETWEENE RERisky Assets Only-R E:IIII - .c1869
FIGURE VIIEFFICIENT FRONTIERS:RELATIONSHIPBETWEENE REisky Assets OnlyRE t-I\II I0I IO-u 0-a81870
(43)xmxi aM'imMm M E1x. Eliminating(A - RC) by combiningwhich is the security-(EM - R)/(A(42) and (43),Ek - R cT2(44)(Ei(EM-R)/(ARC),fromn (42)- RC)we derivek 1,R),-.,mmarket line.REFERENCES[1][2]Black, F. "Capital Market Equilibriumcoming in Journal of Business.Borch,with RestrictedK. "A Note on Uncertaintyand Indifference36, January 1969.Borrowing."Curves."Forth-Review ofEconomic Studies,(3]Cass, D., and J. Stiglitz."The Structure of Investor PreferencesandAsset Returns, and Separabilityin PortfolioA ContributionAllocation:to the Pure Theory of Mutual Funds." JournaZ of Economic Theory, 2, June1970.[4]Fama, E. "Risk, Return, and Equilibrium."79, January-February1971.[5]in the Theory of LiquidityFeldstein,M. S. "Mean-Variance AnalysisPreferenceand PortfolioSelection."Review of Economic Studies,36,January 1969.[6]Hakansson, N. H. "Capital Growth and the Mean-Variance Approach to Portfolio Selection."JournaZ of FinanciaZ and Quantitative AnaZysis, January 197[7]Jensen, M. "Risk, the Pricing of Capital Assets,and the EvaluationInvestment Portfolios."JournaZ of Business, Vol. 42, April 1969.[8]J. "The Valuation of Risk Assets and the Selectionof RiskyLintner,in Stock PortfoliosInvestmentsand Capital Budgets." Review of Economicsand Statistics,XLVIII, February 1965.[9]. "SecurityPrices,Risk,JournaZ of PoZiticaZ Economy,and MaximalGainsoffrom Diversification."JournaZ of Finance, March 1968.[10]Markowitz, H. PortfoZio SeZection: EfficientNew York: John Wiley and Sons, 1959.1871Diversificationof Investment.
[11]Rules in a Continuous-TimeMerton! R.C. "Optimum Consumption and PortfolioModel." JournaZ of Economic Theory, Vol. 3, December 1971.[12]. "An IntertemporalCapitalAssetin a CapitalAssetPricingModel."Forthcomingin Econometrica.[13]Mossin, J. "EquilibriumOctober 1966.[14]Pays."Samuelson, P.A. "General Proof that DiversificationFinanciaZ and QuantitativeAnaZysis, Vol. II, March 1967.[15]. "The FundamentalApproximationin Terms of Means, Variances,37, October 1970.Studies,Theory and CapitaZ Markets.Sharpe, W. PortfoZioBook Company, 1970.[17]Preferenceas BehaviorTobin, J. "LiquidityEconomic Studies,Vol. 25, February 1958. "The rnaZ ofPortfolioAnalysisand Higher Moments." Review of Economic[16][18]Market."Portfolioby F.H. Hahn and F.P.R.1872New York: McGraw-HillToward Risk."Selection."Brechling.Review ofThe TheoryofInterestNew York: Macmillan,
AN ANALYTIC DERIVATION OF THE EFFICIENT PORTFOLIO FRONTIER Robert C. Merton* I. Introduction The characteristics of the mean-variance, efficient portfolio frontier have been discussed at length in the literature. 1 However, for more than three assets, the general appr
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Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. 3 Crawford M., Marsh D. The driving force : food in human evolution and the future.
