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Chapter 8Markowitz Portfolio Theory8.1Expected Returns and CovarianceThe main question in portfolio theory is the following:Given an initial capital V (0), and opportunities (buy or sell) in N securities forinvestment, how would you allocate the capital so that the return on the portfoliois optimal in certain way?More specifically, what we are looking for is a collection of weights:w1 , w2 , . . . , wNwith w1 w2 w3 · · · wN 1, and wi V (0) invested in security i for i 1, 2, . . . , N , such that the return of the portfolio is optimal in certain sense. Ourcommon sense suggests that you will need to take more risk if you seek highexpected returns. On the other hand, investors are always risk averse in the sensethat they demand the largest expected return, given the risk level, or the leastpossible risk, given the expected return. This leads to the concept of efficientportfolios. An efficient portfolio is a portfolio having simultaneously the smallestpossible risk for its given level of expected return and the largest possible expectedreturn for its given level of risk. The collection of all efficient portfolios is calledthe efficient frontier.Given the weights w1 , . . . , wN , the number of shares to invest in security i isni wi V (0)Si (0)(8.1)and the value of the portfolio at T isV (T ) NXni Si (T )i 1so the return of the portfolio over [0, T ] isR[0,T ] NXi 11wi Ri(8.2)
where Ri (Si (T ) Si (0))/Si (0) is the return of security i. We can see that thereturn of the portfolio is a linear combination of the returns of individual securities.The study of the portfolio return is therefore a study of various linear combinationsof a collection of random variables.Sometimes it is more convenient to use the log return: Si (T )(8.3)ri logSi (0)which has the useful property: suppose 0 t0 t1 t2 · · · tm T , the logreturn over [0, T ] S(T )S(t1 ) S(t2 )S(tm )r log log···· r(1) r(2) · · · r(m)S(0)S(t0 ) S(t1 )S(tm 1 )where r(k) is the log return over (tk 1 , tk ). It should be pointed out that for shortperiod of time, S(t t)r(t, t t) log log (R(t, t t) 1) R(t, t t) (8.4)S(t)So quite often we do not distinguish between these two returns, as long as the timeperiod is short.8.2Assumptions of Markowitz TheoryBefore we begin the discussion on the Markowitz theory, we state some assumptionsfor the market: Investors are rational. The supply and demand equilibrium is instantly achieved. There are no arbitrage opportunities. Access to information is available to all participants. Price moves are efficient. The market is liquid. There is no transaction cost. There are no taxes. Everyone has the same opportunity of borrowing and lending.2
We also establish some facts about the return vector R. There is a financialimplication relating to the concept of linear independence: there is no redundantsecurity in the portfolio, that is, there is no security in the portfolio that has areturn as a linear combination of returns from other securities in the portfolio.The consequence of this is that the covariance matrix V is invertible. Since V isalways semi-positive definite, the additional property that V is invertible impliesthat V is positive definite.How do we describe the portfolio risk? We should consider the variance of thereturn of the portfolio, which consists of weighted sum of the securities variancesand the weighted sum of securities covariances, in the form of wT V w, where w (w1 , w2 , . . . , wN )T is the weight vector.8.3Two-Security Portfolio TheoryAssumptions: two securities with µ1 6 µ2 , σ1 σ2 and correlation 1 ρ 1,and we use µ1σ12ρσ1 σ2µ , V (8.5)µ2ρσ1 σ2σ22We can calculate the portfolio return and varianceµP w1 µ1 w2 µ2 µT w(8.6)σP2 w12 σ12 w22 σ22 2w1 w2 ρσ1 σ2 wT V w(8.7)Suppose we target a portfolio with portfolio return µP , the minimization problemis to minimize σP subject to wT e 1, where e (1, 1)T and wT µ µP . Thisproblem is quite easy to solve as w can be determined from the two constraintsfirst: µP µ2wµ(8.8), wµ w 1 wµµ1 µ2which leads to the portfolio variance satisfyingσP2 Aµ2P BµP C(8.9)for some constants A, B, and C. It can be shown that A 0 and C 0, therefore ifwe trace all these pairs (σP , µP ), we will find a hyperbola. With this information wewill try to determine the efficient frontier from this hyperbola. This is a hyperbolawith opening to the right, and we can find the turning point, which correspondsto the absolute minimum for σP :s(σ22 ρσ1 σ2 )µ1 (σ12 ρσ1 σ2 )µ2detCσG ,µ (8.10)Gσ12 σ22 2ρσ1 σ2σ12 σ22 2ρσ1 σ2In figure 8.1, we use an example where µ1 0.1, µ2 0.15, σ1 0.2, σ2 0.4,and ρ 0.5, and plot the Markowitz efficient frontier. The minimum-varianceportfolio has µG 0.1 and σG 0.2 which is the portfolio with w1 1 andw2 0.We can make the following remarks:3
µ0.240.16global minimumvariance portfolio0.0800.080.160.240.320.40.48σ-0.08Figure 8.1: Markowitz Efficient Frontier for Two Security Portfolios The efficient frontier consists of the upper branch of the hyperbola, includingthe turning point. We can in general reduce risk through diversification if we have reliable correlation information. Riskless portfolios exist only in the case ρ 1. In the case ρ 0, we can create portfolios with0 σG min{σ1 , σ2 }and this portfolio requires no short selling (0 wG 1).8.4Efficient Frontier for N -Securities with ShortSellingAssumptions: σi 0 for i 1, . . . , N . Expected returns µi ’s are distinct, and risk levels σi ’s are distinct. Unlimited short sales are allowed.Given w1 , w2 , . . . , wN , we can express the portfolio return and risk as µP (w) wT µ, σP (w) wT V w4
The following will be needed in the calculations:A eT V 1 e 0B µT V 1 eC µT V 1 µ 0AC B 2 0The minimization problem we want to solve is1 Tw Vw2subject to wT e 1 and wT µ µminimize(8.11)(8.12)We cannot determine w from these two constraints like in the 2-security case.In fact, there will be infinitely many portfolios satisfying these two constraints.To solve this problem, we use the Langrange multiplier approach. The Lagrangefunction isL(w, λ) f (w) λ1 (1 wT e) λ2 (µ wT µ) f (w) λT h(w)where λ1λ2λ ,h 1 wT eµ wT µ(8.13) (8.14)To solve the Lagrange multiplier problem, we differentiate the Lagrange functionwith respect to each component of w and λ. The derivatives with respect to wyields L 0 wj w λ1 V 1 e λ2 V 1 µ(8.15) L 0 λ11 wT e 0(8.16) L 0 λ2µ wT µ 0(8.17)V w λ1 e λ2 µThe other two derivatives giveandUsing Eq.(8.15) in these two equations, we obtain the following 2 2 system forλ1 and λ2 :(eT V 1 e)λ1 (µT V 1 e)λ2 1(8.18)(eT V 1 µ)λ1 (µT V 1 µ)λ2 1(8.19)C µBµA B, λ2 2AC BAC B 2(8.20)which givesλ1 5
Therefore the solution for w is the minimum-variance portfolio weight vector C µBµA B 1wµ V e V 1 µ(8.21)AC B 2AC B 2The efficient frontier is determined from the hyperbolaσP2 (µ) Aµ2 2Bµ CAC B 2(8.22)Similar to the 2-security case, the turning point gives the global minimum-varianceportfolio: V 1 e1 B, wG (8.23)(σG , µG ) ,AA AThe set FP,N (σP (w), µP (w)) : wT e 1is called the feasible set, where each point corresponds to a portfolio with theconstraints met.In the case N 2, the feasible set is just a one-parameter set that is the hyperbolaitself. In the cases N 3, it is represented by the region enclosed and includingthe hyperbola show in the following graph.8.5Efficient Frontier N -Securities without ShortSellingIf short selling is not allowed, the weight vector has additional constraints so thespace of weights become WN w : wT e 1, wi 0, i 1, . . . , NThe optimization problems with this nonlinear constraint are much more difficult than the original optimization problem with linear constraints, and in mostsituations some nonlinear programming methods are needed to solve this problem.8.6Mutual Fund TheoremIn Eq.(8.20), we wonder if there are points (σ, µ) on the efficient frontier thatcorrespond to particular λ1 and λ2 values. In particular, we ask what happenswhen λ1 0 or λ2 0. In the first case, we haveµ CV 1 µC2 µD , w wD , σ 2 2 σ DBBB6
For reasons that become clear later, we call this portfolio the diversified portfolioso the solution above is denoted by a subscript D. In the second case when weconsider the portfolio that corresponds to λ2 0, we haveBV 1 eµ µG , w wG ,AA2σ 2 σGthat means that λ2 2 corresponds to the global minimum variance portfolio.Are these two portfolios so special that we can obtain any other portfolio on thefrontier as a linear combination of these two portfolios? A straightforward calculation shows that given any required expected return µ, we can findaµ A(C µB)AC B 2such that the portfolio with the following weight vectorwµ aµ wG (1 aµ )wDwill have the desired expected return µ. This is called a mutual fund theorem orseparation theorem.µ0.24λ 1 00.16λ 2 00.0800.080.160.240.320.40.48σ-0.08Figure 8.2: Two Mutual Fund PortfoliosAs it turns out, this can be achieved with any two portfolios on the frontier so themore general mutual fund theorem states: Any minimum variance portfolio w canbe expressed in terms of any two distinct minimum variance portfoliosw s1 w a s2 w bwhere wa 6 wb , and s1 and s2 satisfying s1 s2 1 can be calculated by certainformula similar to the formula for aµ .7
The significance of the mutual fund theorem is that if you want to construct aportfolio with a required µ, you don’t have to build from scratch to pick all theindividual securities according to the weight wµ . All you have to do is to invest intwo minimum variance portfolios (mutual funds) with the weight vectors w1 andw2 corresponding to two points on the frontier, and calculate the allocation s1 ands2 similar to aµ . In another word, you would let people do the work for you.8
Bibliography[1] A. O. Petters and X. Dong, An Introduction to Mathematical Finance andApplications. Springer Undergraduate Texts in Mathematics and Technology,2016.9
Chapter 8 Markowitz Portfolio Theory 8.1 Expected Returns and Covariance The main question in portfolio theory is the following: G
