Chapter 1 Portfolio Theory with Matrix Algebra

1 Chapter 1 Portfolio Theory with MatrixAlgebraUpdated: August 7, 2013 When working with large portfolios, the Algebra of representing portfolioexpected returns and variances becomes cumbersome. The use of Matrix (lin-ear) Algebra can greatly simplify many of the computations. Matrix algebraformulations are also very useful when it comes time to do actual computa-tions on the computer. The Matrix Algebra formulas are easy to translateinto Matrix programming languages like R. Popular spreadsheet programslike Microsoft Excel, which are the workhorse programs of manyfinancialhouses, can also handle basic Matrix calculations. All of this makes it worth-while to become familiar with Matrix techniques for Portfolio Portfolios with Three Risky AssetsConsider a three asset Portfolio problem with assets denoted and Let ( = )denote the return on asset and assume that the constantexpected return (CER) model holds: ( 2 )cov( )= Example 1 Three asset example data12 CHAPTER1 PORTFOLIOTHEORYWITHMATRIXALGEBRAS tock Pair (i,j) (A,B) (A,C) (B,C).

2 Three asset example gives example data on monthly means, variances and covariancesfor the continuously compounded returns on Microsoft, Nordstrom and Star-bucks (assets A, B and C) based on sample statistics computed over thefive-year period January, 1995 through January, and (risk-return trade-offs) are shown in Figure Clearly, Microsoftprovides the best risk-return trade-offand Nordstrom provides with worst. Let denote the share of wealth invested in asset ( = ) andassume that all wealth is invested in the three assets so that + + =1 The Portfolio return, is the random variable = + + ( )The subscript indicates that the Portfolio is constructed using the x-weights and Theexpectedreturnontheportfoliois = [ ]= + + ( )and the variance of the Portfolio return is 2 =var( )( )= 2 2 + 2 2 + 2 2 +2 +2 +2 Notice that variance of the Portfolio return depends on three variance termsand six covariance terms.

3 Hence, with three assets there are twice as manycovariance terms than variance terms contributing to Portfolio variance. Evenwith three assets, the Algebra representing the Portfolio characteristics ( )- ( ) is cumbersome. We can greatly simplify the Portfolio Algebra usingmatrix example data is also analyized in the Excel spreadsheet PORTFOLIOS with THREE RISKY p pMSFTNORDSBUXGLOBAL MINE1E2 Figure : Risk-return tradeoffs among three asset portfolios. The portfo-lio labeled E1 is the efficient Portfolio with the same expected return asMicrosoft; the Portfolio labeled E2 is the efficient Portfolio with the sameexpected return as Starbux. The Portfolio labeled GLOBAL MIN is the min-imum variance Portfolio consisting of Microsoft, Nordstrom and Starbucks, Portfolio Characteristics Using Matrix NotationDefine the following3 1column vectors containing the asset returns andportfolio weightsR= x= In Matrix notation we can lump multiple returns in a single vector which wedenote byR Since each of the elements inRis a random variable we callRarandom vector.

4 The probability distribution of the random vectorRis4 CHAPTER1 PORTFOLIOTHEORYWITHMATRIXALGEBRA simply the joint distribution of the elements are jointly normally distributed and this joint distribution is com-pletely characterized by the means, variances and covariances of the can easily express these values using Matrix notation as follows. The3 1vector of Portfolio expected values is [R]= = [ ] [ ] [ ] = = and the3 3covariance Matrix of returns isvar(R)= var( )cov( )cov( )cov( )var( )cov( )cov( )cov( )var( ) = 2 2 2 = Notice that the covariance Matrix is symmetric (elements offthe diago-nal are equal so that = 0,where 0denotes the transpose of )sincecov( )=cov( ) cov( )=cov( )andcov( )=cov( )

5 Example 2 Example return data using Matrix notationUsing the example data in Table we have = = 0 04270 00150 0285 = 0 0100 0 0018 0 00110 0018 0 0109 0 00260 0011 0 0026 0 0199 PORTFOLIOS with THREE RISKY ASSETS5In R, these values are created using> <- c("MSFT", "NORD", "SBUX")> = c( , , )> names( ) = > = Matrix (c( , , ,+ , , ,+ , , ),+ nrow=3, ncol=3)> dimnames( ) = list( , )> NORD > NORD SBUXMSFT The return on the Portfolio using Matrix notation is =x0R=( ) = + + Similarly, the expected return on the Portfolio is = [x0R]=x0 [R]=x0 =( ) = + + The variance of the Portfolio is 2 =var(x0R)=x0 x=( ) 2 2 2 = 2 2 + 2 2 + 2 2 +2 +2 +2 6 CHAPTER1 PORTFOLIOTHEORYWITHMATRIXALGEBRAThe condition that the Portfolio weights sum to one can be expressed asx01=( ) 111 = + + =1 where1is a3 1vector with each element equal to another Portfolio with weightsy=( )0 The return onthis Portfolio is =y0R= + + Later on we will need to compute the covariance between the return on port-folioxand the return on portfolioy cov( ) Using Matrix Algebra ,this covariance can be computed as =cov( )

6 =cov(x0R y0R)=x0 y=( ) 2 2 2 = 2 + 2 + 2 +( + ) +( + ) +( + ) Example 3 Portfolio computations in RConsider an equally weighted Portfolio with = = =1 3 Thisportfolio has return =x0 Rwherex=(1 3 1 3 1 3)0 Using R, theportfolio mean and variance are> = rep(1,3)/3> names( ) = > = crossprod( , )> = t( )%*% *% > = sqrt( )> [,1] PORTFOLIOS with THREE RISKY ASSETS7[1,] > [,1][1,] , consider another Portfolio with weight vectory=( )0=(0 8 0 4 0 2)0and return =y0R Thecovariancebetween and is> = c( , , )> names( ) = > = t( )%*% *% > [,1][1,] Finding the Global Minimum Variance PortfolioThe global minimum variance portfoliom=( )0for the threeasset case solves the constrained minimization problemmin 2 = 2 2 + 2 2 + 2 2 ( )+2 +2 +2 + + =1 The Lagrangian for this problem is ( )= 2 2 + 2 2 + 2 2 +2 +2 +2 + ( + + 1) and thefirst order conditions (FOCs) for a minimum are0= =2 2 +2 +2 + ( )0= =2 2 +2 +2 + 0= =2 2 +2 +2 + 0= = + + 1 8 CHAPTER1 PORTFOLIOTHEORYWITHMATRIXALGEBRAThe FOCs ( ) gives four linear equations in four unknowns which can besolved tofind the global minimum variance Portfolio weights and.

7 Using Matrix notation, the problem ( ) can be concisely expressed asminm 2 =m0 ( )The four linear equation describing thefirst order conditions ( ) has thematrix representation 2 2 2 2 12 2 2 2 12 2 2 2 11110 = 0001 or, more concisely, 2 1100 m = 01 ( )The system ( ) is of the formA z =b whereA = 2 1100 z = m andb= 01 The solution forz is thenz =A 1 b ( )Thefirst three elements ofz are the Portfolio weightsm=( )0for the global minimum variance Portfolio with expected return =m0 and variance 2 =m0 m Example 4 Global minimum variance Portfolio for example dataUsing the data in Table 1, we can use R to compute the global minimumvariance Portfolio weights from ( ) as PORTFOLIOS with THREE RISKY ASSETS9> = cbind(2* , rep(1, 3))> = c(rep(1, 3), 0)> = rbind( , )> = c(rep(0, 3), 1)> = solve( )%*% > = [1.]

8 3,1]> NORD , the global minimum variance Portfolio has Portfolio weights =0 4411 =0 3656and =0 1933 and is given by the vectorm=(0 4411 0 3656 0 1933)0 ( )The expected return on this Portfolio , =m0 is> = (crossprod( , ))> [1] Portfolio variance, 2 =m0 m and standard deviation, are> = (t( )%*% *% )> = sqrt( )> [1] > [1] Figure , this Portfolio is labeled global min . Alternative derivation of global minimum variance portfolioThefirst order conditions ( ) from the optimization problem ( ) can beexpressed in Matrix notation as0(3 1)= (m ) m=2 m+ 1 ( )0(1 1)= (m ) =m01 1 ( )10 CHAPTER1 PORTFOLIOTHEORYWITHMATRIXALGEBRAU sing ( ),first solve form:m= 12 11 Next, multiply both sides by10and use ( ) to solve for :1=10m= 12 10 11 = 2 110 11 Finally, substitute the value for back into ( ) to solve form:m= 12( 2)110 11 11= 1110 11 ( )Example 5 Finding global minimum variance Portfolio for example dataUsing the data in Table 1, we can use R to compute the global minimumvariance Portfolio weights from ( ) as follows.

9 > = rep(1, 3)> = solve( )> = *% > = ((t( )%*% *% ))> = > [,1]MSFT NORD Finding Efficient PortfoliosThe investment opportunity set is the set of Portfolio expected return, and Portfolio standard deviation, values for all possible portfolios whoseweights sum to one As in the two risky asset case, this set can be describedin a graph with on the vertical axis and on the horizontal axis. Withtwo assets, the investment opportunity set in ( ) space lies on a curve(one side of a hyperbola). with three or more assets, the investment oppor-tunity set in ( ) space is described by set of values whose general PORTFOLIOS with THREE RISKY ASSETS11is complicated and depends crucially on the covariance terms However,we do not have to fully characterize the entire investment opportunity we assume that investors choose portfolios to maximize expected returnsubject to a target level of risk, or, equivalently, to minimize risk subject toa target expected return, then we can simplify the asset allocation problemby only concentrating on the set of efficient portfolios.

10 These portfolios lieon the boundary of the investment opportunity set above the global mini-mum variance Portfolio . This is the framework originally developed by HarryMarkowitz, the father of Portfolio Theory and winner of the Nobel Prize Markowitz, we assume that investors wish tofind portfoliosthat have the best expected return-risk trade-off. Markowitz characterizedthese efficient portfolios in two equivalent ways. In thefirst way, investorsseek tofind portfolios that maximize Portfolio expected return for a givenlevel of risk as measured by Portfolio variance. Let 2 0denote a target levelof risk. Then Harry Markowitz characterized the constrained maximizationproblem tofind an eff