Markowitz Mean-Variance Portfolio Theory

1 Markowitz Mean-Variance Portfolio Return RatesAn investment instrument that can be bought and sold is often calledanasset. Suppose we purchase an asset forx0dollars on one date andthen later sell it forx1dollars. We call the ratioR=x1x0thereturnon the asset. Therate of returnon the asset is given byr=x1 x0x0=R ,x1=Rx0andx1= (1 +r) it is possible to sell and asset that we do not own. Thisis calledshort selling. It works somewhat as follows. Suppose you wishto short (or short sell) a particular stockXXX. You begin by askingyour stock broker if their firm is holdingXXXin the total pool ofstocks owned by all of their customers. If the brokerage does hold (ormanage) some of stockXXX, you can ask them sell any number ofstockXXXup to the number that they hold. This sale is creditedagainst your account as a debt equal to the number of stockXXXthey sell on your behalf. That is, your debt is not denominated indollars, but rather in the number of stockXXXthat you are shorting( your account isshortby the given number of stockXXX).

2 Onyour account asset sheet, this short sale appears as a negative numberassociated with the shorted asset. Remember, this negative numberis not denominated in dollars, but rather in the number of stocks ,or assets, shorted. Due to the sale of stockXXXyou have receivedx0dollars. Eventually, you must ask the brokerage to buy the samenumber of stockXXXback as you originally asked them to sell andreturn this stock to the pool of assets that they are holding for theircustomers. On the date at which you return stockXXXyou ask yourbroker to re-purchase it at its current going value ofx1dollars andreturn it to the brokerage s asset pool. Ifx1< x0, then you have madea profit on this transaction; otherwise, you have a loss. The return andrate of return on this transaction are given byR= x1 x0=x1x0andr=( x1) ( x0) x0=x1 x0x0,respectively. Short selling can be very risky, and many brokerage firmsdo not allow it. Nonetheless, it can be us now consider constructing a Portfolio consisting have an initial budget ofx0dollars that we wish to assign tothese assets.

3 The amount that we assign to assetiisx0i=wix0fori= 1,2, .. , n, wherewiis a weighting factor for asseti. We al-low the weights to take negative values, and when negative it meansthat the asset is being shorted in our Portfolio . To preserve the budgetconstraint we require that the weights sum to 1: ni=1wi= 1. That is,the sum of the investments =n i=1wix0=x0n i=1wi= that by shorting some stocks we open up more funds for thepurchase of other stocks, because when we short a stock we receive thedollar value of that stock today and we can turn around and re-investthose dollars elsewhere with the purchase of other the return on asseti, then the total receipts from ourportfolio isx1=n i=1 Riwix0=x0n i=1 Riwi,and so the total return from the Portfolio isR=n i= addition, we have that the rate of return from assetiisri=Ri 1,i= 1,2, .. , n. Hence the rate of return on the Portfolio isr=R 1 = (n i=1 Riwi) (n i=1wi) =n i=1(Ri 1)wi=n i= Basics of Markowitz Mean-Variance PortfolioTheoryIn the Markowitz Mean-Variance Portfolio Theory , one models the rateof returns on assets as random variables.

4 The goal is then to choose theportfolio weighting factorsoptimally. In the context of the Markowitztheory an optimal set of weights is one in which the Portfolio achievesan acceptable baseline expected rate of return with the variance of the rate of return of an instrument is taken as asurrogate for its the random variable associated with the rate of return forasseti, fori= 1,2, .. , n, and define the random vectorz= .Set i=E(ri), m = ( 1, 2, .. , n)T, and cov(z) = . Ifw=(w1, w2, .. , wn)Tis a set of weights associated with a Portfolio , thenthe rate of return of this portfolior= ni=1riwiis also a randomvariable with mean mTwand variancewT w. If bis the acceptablebaseline expected rate of return, then in the Markowitz Theory an opti-mal Portfolio is any Portfolio solving the following quadratic program:Mminimize12wT wsubject to mTw b,and eTw= 1,where e always denotes the vector of ones, , each of the componentsof e is the number 1. The KKT conditions for this quadratic programare0 = w m e(1) b mTw,eTw= 1,0 (2) T(mTw b) = 0(3)for some , IR.

5 Since the covariance matrix is symmetric andpositive definite, we know that if (w, , ) is any triple satisfying theKKT conditions thenwis necessarily a solution toM. Indeed, it iseasily shown that ifMis feasible, then a solution toMmust alwaysexist and so a KKT triple can always be found IRm n, B IRm t, E IRs n, F IRs t, M IRt n, Q IRn nandH IRt twithQandHsymmetric,and letr IRm,andh assume that the symmetricmatrix Q=[Q MTM H]is positive semi-definite. If the following quadratic program is feasible,then it has finite optimal value and a solution attaining this optimalvalue exists:minimize12[uTQu+ 2vTM u+vTHv]subject toAu+Bv rEu+F v=h0 u . any square root of the matrix Q, and setx= (uT, vT) =IRm+ IRs IRn+. Then the constraint region for the QP canbe written asF={x IRn IRt:T x }, whereT= A BE FI0 .By assumptionF 6= . Making the change of variabley=Sxin theQP yields the QPminimize12 y 2subject toy SF={Sx:x F}.Since the setFis a closed nonempty polyhedral convex set, so is thesetSF.

6 Hence the solution yto this QP is given by the point closestto the origin in the closed setSFwhich must exist since this set isclosed and nonempty. Therefore, the solution set to the original QP isnonempty and is given by{x: y=Sx, x F}. Assume that is nonsingular and that wis a solution toM( wexists by Proposition ). We consider two cases. b<mT w: In this case, the complementarity condition (3) im-plies = 0. Hence the KKT conditions reduce to the two equa-tions 0 = w e and eT w= 1. Multiplying the first throughby 1yields w= 1e. Multiplying this equation throughby e and using the fact that eT w= 1 gives = (eT 1e) , w= (eT 1e) 1 is important to note that this value ofwgives the smallestpossible variance over all portfolios since it solves the problemMmin-var: minimize12wT wsubject to eTw= , the return associated with the least variance so-lution is min-var=mT 1eeT denote the set of weights associated with the minimum vari-ance solution wbywmin-varas observe that is the minimum variance weightswmin-varare feasible forM, that is, if mTwmin-var b, thenwmin-varmust be the solution toMsince it the solution to the prob-lemMmin-var.

7 Therefore, when solvingMone first computeswmin-varand checks to see if the inequality mTwmin-var b5holds. If it does hold, thenwmin-varsolvesMand no furtherwork is required. If it does not hold then you know that theconstraint mTw= bat the solution toM. b= mT w: Multiplying (1) through by 1gives(4) w= 1m + this formula for wand (2), we get the two equations b= mT 1m + mT 1e1 = mT 1e + eT 1e,or equivalently, the 2 2 matrix equation(5)[mT 1m mT 1emT 1e eT 1e]( )=( b1).Properties of positive definite matrices can be used to show thatthe matrixT=[mT 1m mT 1emT 1e eT 1e]= [m e]T 1[m e]is always positive semi-definite There are a number of waysto see this. The simplest is to exploit the factored formT=[m e]T 1[m e]. But a simple test is to check that0< = (mT 1m)(eT 1e) (mT 1e) will always be the case whenever the vectors m and e arelinearly = 0, then it must be the case thatm= efor some IR. In this case, if b/ 6= 1, then the problemMisnecessarily infeasible.

8 If b/ = 1, thenwmin-varsolvesMwhichwould have been detected by first computingwmin-varand thenchecking its feasibility >0, the system (5) can be solved to give = eTvand = mTv,wherev= 1 1( be m).Plugging these values into (4) gives the optimal solutionw= 1eeT 1e+ [ 1meT 1m 1eeT 1e]= (1 ) 1eeT 1e+ 1meT 1m= (1 )wmin-var+ wmk,6wherewmk= 1meT 1mand = b(mT 1e)(eT 1e) (mT 1e)2 .Observe that the optimal set of weights is a linear combina-tion of the two sets of weightswmin-varandwmk, both of whichsatisfy the constraint eTw= 1. We have labeled weightswmkas themarketweights since they incorporate all of the marketinformation on the assets under us now recap the solution proceedure PROCEEDUREC heck FeasibilityFirst check feasibility. For this we need only check to see if m is parallelto e. If it is, then m = efor some IR. In this case the problemis infeasible if b> . If m = eand b , then compute 1andevaluate the minimum variance weightswmin-var. These weights Minimum Variance SolutionIfMis feasible, then compute 1and the minimum variance solutionwmin-var= 1eeT mTwmin-var b, thenwmin-varsolves the Two Portfolio SolutionIf the problem is feasible andwmin-varis not the solution, then computethe market weightswmk= 1meT 1m,and form the vectorv=wmk solution toMis then of the formw=wmin-var+ (wmk wmin-var) =wmin-var+ determine use the identity mTw= bto get = b is remarkable that every solution to the Markowitz problemMcanbe represented as a linear combination of only two portfolios.

9 Thesebeing the the minimum variance Portfolio with weightswmin-varand ourmarketportfolio with weightswmk. In the next section we will showthat this is a general principal regardless of whether is invertible Efficient Frontier and the Two-Fund TheoremIn practice, one would like to have a better understanding of thereturn-risk trade-off since we want to both maximize return while min-imizing risk. An alternative strategy is to try to balance these twoobjectives in a single objective function. One way to do this is to solvethe QPM minimize12wT w mTwsubject to eTw= that for >0 the term mTwtries to push mTwupwards tocounter balance the downward pull of the term12wT w. The upwardpush on mTwincreases as is increased. The KKT conditions for theQPM are0 = w m e(6)eTw= 1.(7)Note that these conditions are quite similar to the conditions (1)-(3)except that we no longer require that mTw= b. However, if wsolvesM and we set b= mT w, then the solution sets ofM andMcoincide!

10 Proceeding as in the case ofMunder the assumption that isinvertible, we find that the solution to (6)-(7) is given by =1 mT 1eeT 1eandw = 1eeT 1e+ [ 1mmT 1e 1eeT 1e]= (1 ) 1eeT 1e+ 1mmT 1e= (1 )wmin-var+ wmk,wherewmin-varandwmkare as defined in the previous section and = (mT 1e).8 This gives a value for bof b= mTw = min-var+ eT 1e,where = (eT 1e)(mT 1m) (mT 1e) = 0, we get, as expected, min-var, while as we seethat b if 0< = (eT 1e)(mT 1m) (mT 1e)2. That is, if0< , the solution toM traces out all possible solutions toMforall possible values of bas moves from 0 to + . Thus, in order tocompletely understand the risk-return trade-off, we need only graphthe curve( (wT w ), r ) = ( (var(r )), E(r )),wherer =wT r ,as varies from 0 to + . This is very reminiscent of a mean-standarddeviation curve! Indeed, we will see that it is exactly this concept! Inthe context of Markowitz Mean-Variance Portfolio Theory , this is calledtheefficiency curveorefficient frontier.