(2.1) Markowitz’s mean-variance formulation (2.2) Two …

1 2. mean-variance portfolio theory( ) markowitz s mean-variance formulation ( ) Two-fund theorem( ) Inclusion of the riskfree markowitz mean-variance formulationSuppose there areNrisky assets, whose rates of returns are given by the randomvariablesR1, , RN, whereRn=Sn(1) Sn(0)Sn(0), n= 1,2, , (w1 wN)T, wndenotes the proportion of wealth invested in assetn,withNXn=1wn= 1. The rate of return of the portfolio isRP=NXn= There does not exist any asset that is a combination of other assets in theportfolio, that is, non-existence of redundant = (R1R2 RN) and1= (1 1 1) are linearly independent, otherwiseRPis a constant irrespective of any choice of portfolio first two moments ofRPare P=E[RP] =NXn=1E[wnRn] =NXn=1wn n,where n=Rn,and 2P= var(RP) =NXi=1 NXj=1wiwjcov(Ri, Rj) =NXi=1 NXj=1wi denote the covariance matrix so that 2P=wT example whenn= 2, we have(w1w2) 11 12 21 22!

2 W1w2!=w21 21+w1w2( 12+ 21) +w22 The portfolio risk of return is quantified by 2P. In mean-varianceanalysis, only the first two moments are considered in the port-folio model. Investment theory prior to markowitz consideredthe maximization of Pbut without The measure of risk by variance would place equal weight onthe upside deviations and downside In the mean-variance model, it is assumed that i, iand ijareall portfolioConsider two risky assets with known meansR1andR2, variances 21and 22, of the expected rates of returnsR1andR2, togetherwith the correlation coefficient .Let 1 and be the weights of assets 1 and 2 in this mean:RP= (1 )R1+ R2,0 1 portfolio variance : 2P= (1 )2 21+ 2 (1 ) 1 2+ 2 represent the two assets in a mean-standard deviation diagram(recall: standard deviation = variance )As varies, ( P,RP) traces out a conic curve in the = 1, it is possible to have = 0 for some suitable choice ofweight.

3 In general, putting two assets whose returns are negativelycorrelated has the desirable effect of lowering the portfolio particular, when = 1, P( ; = 1) =q(1 )2 21+ 2 (1 ) 1 2+ 2 22= (1 ) 1+ is the straight line joiningP1( 1,R1) andP2( 2,R2).When = 1, we have P( ; = 1) =q[(1 ) 1 2]2=|(1 ) 1 2|.When is small (close to zero), the corresponding point is close toP1( 1,R1). The lineAP1corresponds to P( ; = 1) = (1 ) 1 pointA(with zero ) corresponds to = 1 1+ quantity (1 ) 1 2remains positive until = 1 1+ > 1 1+ 2, the locus traces out the upper 1< <1, the minimum variance point on the curve thatrepresents various portfolio combinations is determined by 2P = 2(1 ) 21+ 2 22+ 2(1 2 ) 1 2= 0 setgiving = 21 1 2 21 2 1 2+ formulation of markowitz s mean-variance analysisminimize12 NXi=1 NXj=1wiwj ijsubject toNXi=1wiRi= PandNXi=1wi= 1.

4 Given the target expectedrate of return of portfolio P, find the portfolio strategy that mini-mizes form the LagrangianL=12 NXi=1 NXj=1wiwj ij 1 NXi=1wi 1 2 NXi=1wiRi P where 1and 2are Lagrangian then differentiateLwith respect towiand the Lagrangian mul-tipliers, and set the derivative to zero. L wi=NXj=1 ijwj 1 2Ri= 0, i= 1,2, , N.(1) L 1=NXi=1wi 1 = 0;(2) L 2=NXi=1wiRi P= 0.(3)From Eq. (1), the portfolio weight admits solution of the formw = 1( 11+ 2 )(4)where1= (1 1 1)Tand = (R1R2 RN) determine 1and 2, we apply the two constraints1 =1T 1 w = 11T 11+ 21T 1 .(5) P= T 1 w = 1 T 11+ 2 T 1 .(6)Writea=1T 11, b=1T 1 andc= T 1 , we have1 = 1a+ 2band P= 1b+ for 1and 2: 1=c b P and 2=a P b , where =ac that 1and 2have dependence on P, which is the targetmean prescribed in the variance minimization 6=h1, and 1exists.

5 Since is positive definite, soa >0, c >0. By virtue of the Cauchy-Schwarz inequality, > minimum portfolio variance for a given value of Pis given by 2P=w T w =w T ( 1 11+ 2 1 )= 1+ 2 P=a 2P 2b P+c .The set of minimum variance portfolios is represented by a paraboliccurve in the 2P Pplane. The parabolic curve is generated byvarying the value of the parameter , when Pis plotted against P, the set of minimumvariance portfolio is a hyperbolic are the asymptotic values of lim d Pd P?d Pd P=d Pd 2Pd 2Pd P= 2a P 2b2 P= a P bqa 2P 2b P+cso thatlim d Pd P= s P, we obtain 1=c b P and 2=a P b , and the optimalweightw = 1( 11+ 2 ).To find the global minimum variance portfolio , we setd 2Pd P=2a P 2b = 0so that P=b/aand 2P= 1/a. Correspondingly, 1= 1/aand 2= 0. The weight vector that gives the global minimum varianceis found to bewg= 11a= 111T portfolio that corresponds to 1= 0 is obtained when Pis taken to becb.

6 The value of the other Lagrangian multiplier isgiven by 2=a cb b = weight vector of this particular portfolio isw d= 1 b= 1 1T 1 .Also, 2d=a cb 2 2b cb +c = setGivenNrisky assets, we form various portfolios from plot the point ( P,RP) representing the portfolios in the Rdiagram. The collection of these points constitutes the feasible setor feasible a 3-asset portfolio , the various combinations of assets 2and 3 sweep out a curve between them (the particular curve takendepends on the correlation coefficient 12).A combination of assets 2 and 3 (labelled 4) can be combined withasset 1 to form a curve joining 1 and 4. As 4 moves between 2 and3, the curve joining 1 and 4 traces out a solid of feasible regions1. If there are at least 3 risky assets (not perfectly correlatedand with different means), then the feasible set is a solid two-dimensional The feasible region isconvex to the left.

7 That is, given any twopoints in the region, the straight line connecting them doesnotcross the left boundary of the feasible region. This is because theminimum variance curve in the mean-variance plot is a variance set and efficient fundsThe left boundary of a feasible region is called theminimum varianceset. The most left point on the minimum variance set is called theminimum variance point. The portfolios in the minimum varianceset are calledfrontier a given level of risk, only those portfolios on theupper halfof the efficient frontier are desired by investors. They are calledefficient portfoliow is said to be mean-variance efficient if there existsno portfoliowwith P Pand 2P 2P, except itself. That is,you cannot find a portfolio that has a higher return and lower riskthan those for an efficient Two-fund theoremTwo frontier funds (portfolios) can be established so that any fron-tier portfolio can be duplicated, in terms of mean and variance , asa combination of these two.

8 In other words, all investors seekingfrontier portfolios need only invest in combinations of these convex combination (that is, weights are non-negative)of ef-ficient portfolios is an efficient portfolio . Let i 0 be the weightof Fundiwhose rate of return isRif. SinceEhRifi bafor alli, wehavenXi=1 iEhRifi nXi=1 iba= (w11 w1n), 11, 12andw2= (w21 w2n)T, 21, 22are twoknown solutions to the minimum variance formulation with expectedrates of return 1 Pand 2P, ijwj 1 2Ri= 0, i= 1,2, , n(1)nXi=1wiri= P(2)nXi=1wi= 1.(3)It suffices to show that w1+ (1 )w2is a solution correspondsto the expected rate of return 1P+ (1 ) w1+(1 )w2is a legitimate portfolio with weights that sumto Eq. (1) is satisfied by w1+ (1 )w2since the system ofequations is Note thatnXi=1h w1i+ (1 )w2iiRi= nXi=1w1iRi+ (1 )nXi=1w2iRi= 1P+ (1 ) minimum variance portfolio with target mean Pcan be uniquelydecomposed into the sum of two portfoliosw P=Awg+ (1 A)wdwhereA=c b P a minimum- variance portfolio whose solution of the Lagrangianmultipliers are 1and 2, the optimal weight isw P= 1 11+ 2 1 = 1(awg) + 2(bwd).

9 Observe that the sum of weights is 1a+ 2b=ac Pb +b Pa b =ac b2 = set 1a=Aand 2b= 1 A, where 1=c Pb and 2= Pa b .24 Indeed, any two minimum- variance portfolios can be used to substi-tute forwgandwd. Supposewu= (1 u)wg+uwdwv= (1 v)wg+vwdwe then solve forwgandwdin terms ofwuandwv. Thenw P= 1awg+ (1 1a)wd= 1a+v 1v uwu+1 u 1av uwv,where sum of coefficients = , variances and covariances of the rates of return of 5 riskyassets are procedure to find the two funds in the minimum varianceset:1. Set 1= 1 and 2= 0; solve the system of equations5Xj=1 ijv1j= 1, i= 1,2, , actual weightswishould be summed to one. This is doneby normalizingv1k s so that they sum to onew1i=v1iPnj= normalization, this gives the solution towg, where 1=1aand 2= Set 1= 0 and 2= 1; solve the system of equations:5Xj=1 ijv2j=Ri, i= 1,2, , s to normalization, this gives the solution towd, where 1= 0and 2= above procedure avoids the computation ofa=1T 11andb=1T 1.

10 * Note thatw1corresponds to the global minimum variance know that g=b/a; how about d? d= Twd= T 1 b= in expected returns = d g=cb ba= ab> , difference in variances = 2d 2g=cb2 1a= ab2> is the covariance of portfolio returns for any two minimumvariance portfolios?WriteRuP=wTuRandRvP=wTvRwhere R= (R1 RN)T. Recall thatwg= 11aandwd= 1 bso that gd= cov 11aR, 1 bR = 11a T 1 b!=1T 1 ab=1asinceb=1T 1 .31In general,cov(RuP, RvP) = (1 u)(1 v) 2g+uv 2d+ [u(1 v) +v(1 u)] gd=(1 u)(1 v)a+uvcb2+u+v 2uva=1a+uv particular,cov(Rg, RP) =wTg wP=1 1 wPa=1a= var(Rg)for any any Portfoliou, we can find another Portfoliovsuch that thesetwo portfolios are uncorrelated. This can be done by setting1a+uv ab2= mean-variance criterion can be reconciled with the expectedutility approach in either of two ways: (1) using a quadraticutilityfunction, or (2) making the assumption that the random returns arenormal utilityThe quadratic utility function can be defined asU(x) =ax b2x2,wherea >0 andb >0.