Chapter 1
Portfolio Theory with Matrix
AlgebraUpdated: August 7, 2013
When working with large portfolios, the algebra of representing portfolio
expected returns and variances becomes cumbersome. The use of matrix (linear) algebra can greatly simplify many of the computations. Matrix algebra
formulations are also very useful when it comes time to do actual computations on the computer. The matrix algebra formulas are easy to translate
into matrix programming languages like R. Popular spreadsheet programs
like Microsoft Excel, which are the workhorse programs of many financial
houses, can also handle basic matrix calculations. All of this makes it worthwhile to become familiar with matrix techniques for portfolio calculations.

1.1

Portfolios with Three Risky Assets

Consider a three asset portfolio problem with assets denoted   and  Let
 ( =   ) denote the return on asset  and assume that the constant
expected return (CER) model holds: ∼  (   2 )cov(   ) =   
Example 1 Three asset example data
1

2CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
Stock  Pair (i,j)  

A

0.0427 0.1000 (A,B)

0.0018B

0.0015 0.1044 (A,C)

0.0011

C

0.0285 0.1411 (B,C)

0.0026

Table 1.1: Three asset example data.
Table 1.1 gives example data on monthly means, variances and covariances
for the continuously compounded returns on Microsoft, Nordstrom and Starbucks (assets A, B and C) based on sample statistics computed over the
five-year period January, 1995 through January, 20001 . The values of and   (risk-return trade-oﬀs) are shown in Figure 1.1. Clearly, Microsoft
provides the best risk-return trade-oﬀ and 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 =   +   +   

(1.1)

The subscript “” indicates that the portfolio is constructed using the xweights    and   The expected return on the portfolio is = [ ] =   +   +   

(1.2)

and the variance of the portfolio return is 2 = var( )(1.3)= 2  2 + 2  2 + 2  2 + 2    + 2    + 2    
Notice that variance of the portfolio return depends on three variance terms
and six covariance terms. Hence, with three assets there are twice as many
covariance terms than variance terms contributing to portfolio variance. Even
with three assets, the algebra representing the portfolio characteristics (1.1)- (1.3) is cumbersome. We can greatly simplify the portfolio algebra using
matrix notation.
1

This example data is also analyized in the Excel spreadsheet 3firmExample.xls.

3

0.05

0.06

1.1 PORTFOLIOS WITH THREE RISKY ASSETS

0.03

p

0.04

E1

MSFT

E2

SBUX

0.01

0.02

GLOBAL MIN

0.00

NORD

0.00

0.05

0.10

0.15

0.20

p

Figure 1.1: Risk-return tradeoﬀs among three asset portfolios. The portfolio labeled “E1” is the eﬃcient portfolio with the same expected return as
Microsoft; the portfolio labeled “E2” is the eﬃcient portfolio with the same
expected return as Starbux. The portfolio labeled GLOBAL MIN is the minimum variance portfolio consisting of Microsoft, Nordstrom and Starbucks,
respectively.

1.1.1

Portfolio Characteristics Using Matrix Notation

Define the following 3 × 1 column vectors containing the asset returns andportfolio weights
⎛
⎛
⎞
⎞


⎜ ⎟
⎜ ⎟
⎜
⎜
⎟
⎟
R = ⎜  ⎟  x = ⎜  ⎟ 
⎝
⎝
⎠
⎠



In matrix notation we can lump multiple returns in a single vector which wedenote by R Since each of the elements in R is a random variable we callR a random vector. The probability distribution of the random vector R is

4CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
simply the joint distribution of the elements of R. In the CER model all
returns are jointly normally distributed and this joint distribution is completely characterized by the means, variances and covariances of the returns.
We can easily express these values using matrix notation as follows. The
3 × 1 vector of portfolio expected values is
⎡⎛
⎞⎤ ⎛
⎞
⎞ ⎛

[ ]

⎢⎜
⎟⎥ ⎜
⎟
⎟ ⎜
⎢⎜
⎟⎥ ⎜
⎟
⎟ ⎜
[R] =  ⎢⎜  ⎟⎥ = ⎜ [ ] ⎟ = ⎜  ⎟ = μ
⎣⎝
⎠⎦ ⎝
⎠
⎠ ⎝
[ ]

and the 3 × 3 covariance matrix of returns is⎛⎞var( ) cov(   ) cov(   )⎜⎟⎜⎟var(R) = ⎜ cov(   ) var( ) cov(   ) ⎟⎝⎠cov(   ) cov(   ) var( )
⎛
⎞
2
 

⎜    ⎟⎜⎟= ⎜    2   ⎟ = Σ⎝⎠2     

Notice that the covariance matrix is symmetric (elements oﬀ the diagonal are equal so that Σ = Σ0 , where Σ0 denotes the transpose of Σ) sincecov(   ) = cov(   ) cov(   ) = cov(   ) and cov(   ) =cov(   )
Example 2 Example return data using matrix notation
Using the example data in Table 1.1 we have⎞⎞ ⎛⎛00427⎟⎜ ⎟ ⎜⎟⎟ ⎜⎜μ = ⎜  ⎟ = ⎜ 00015 ⎟ ⎠⎠ ⎝⎝00285⎞⎛
00100 00018 00011⎟⎜⎟⎜
Σ = ⎜ 00018 00109 00026 ⎟ ⎠⎝
00011 00026 00199

5

1.1 PORTFOLIOS WITH THREE RISKY ASSETSIn R, these values are created using> asset.names <- c("MSFT", "NORD", "SBUX")> mu.vec = c(0.0427, 0.0015, 0.0285)> names(mu.vec) = asset.names> sigma.mat = matrix(c(0.0100, 0.0018, 0.0011,+
0.0018, 0.0109, 0.0026,+
0.0011, 0.0026, 0.0199),+
nrow=3, ncol=3)> dimnames(sigma.mat) = list(asset.names, asset.names)> mu.vec
MSFT
NORD
SBUX
0.0427 0.0015 0.0285> sigma.mat
MSFT
NORD
SBUX
MSFT 0.0100 0.0018 0.0011
NORD 0.0018 0.0109 0.0026
SBUX 0.0011 0.0026 0.0199¥
The return on the portfolio using matrix notation is
⎛
⎞

⎜ ⎟
⎜
⎟
 = x0 R = (     ) · ⎜  ⎟ =   +   +   ⎝⎠

Similarly, the expected return on the portfolio is
⎛

⎞


⎜ ⎟
⎜
⎟
 = [x0 R] = x0 [R] = x0 μ = (     )·⎜  ⎟ =   +  +  ⎝⎠The variance of the portfolio is

⎛

 2

⎞⎛

⎞


   ⎜⎟⎜  ⎟⎜⎜⎟⎟
 2 = var(x0 R) = x0 Σx = (     ) · ⎜    2   ⎟ ⎜  ⎟⎝⎠⎝⎠2
     = 2  2 + 2  2 + 2  2 + 2    + 2    + 2    

6CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
The condition that the portfolio weights sum to one can be expressed as
⎛ ⎞
1
⎜ ⎟
⎟
⎜
x0 1 = (     ) · ⎜ 1 ⎟ =  +  +  = 1
⎝ ⎠1

where 1 is a 3 × 1 vector with each element equal to 1.
Consider another portfolio with weights y = (     )0  The return on
this portfolio is = y0 R =   +   +   
Later on we will need to compute the covariance between the return on portfolio x and the return on portfolio y cov(   ) Using matrix algebra,
this covariance can be computed as  = cov(   ) = cov(x0 R y0 R)
⎛
⎞⎛
⎞

 2    
⎜
⎟⎜
⎟
⎜
⎟⎜
⎟
= x0 Σy = (     ) · ⎜    2   ⎟ ⎜  ⎟⎝⎠⎝⎠2
     =    2 +    2 +    2+(  +   )  + (  +   )  + (  +   )  

Example 3 Portfolio computations in R
Consider an equally weighted portfolio with  =  =  = 13 This
portfolio has return  = x0 R where x = (13 13 13)0  Using R, theportfolio mean and variance are> x.vec = rep(1,3)/3> names(x.vec) = asset.names> mu.p.x = crossprod(x.vec,mu.vec)> sig2.p.x = t(x.vec)%*%sigma.mat%*%x.vec> sig.p.x = sqrt(sig2.p.x)> mu.p.x[,1]1.1 PORTFOLIOS WITH THREE RISKY ASSETS

7[1,] 0.02423> sig.p.x[,1][1,] 0.07587
Next, consider another portfolio with weight vector y = (     )0 = (08
04 −02)0 and return  = y0 R The covariance between  and  is> y.vec = c(0.8, 0.4, -0.2)> names(x.vec) = asset.names> sig.xy = t(x.vec)%*%sigma.mat%*%y.vec> sig.xy[,1][1,] 0.003907
¥

1.1.2

Finding the Global Minimum Variance Portfolio

The global minimum variance portfolio m = (     )0 for the three
asset case solves the constrained minimization problemmin   2 = 2  2 + 2  2 + 2  2

(1.4)

+2    + 2    + 2   
s.t.  +  +  = 1The Lagrangian for this problem is(      ) = 2  2 + 2  2 + 2  2+2    + 2    + 2   
+( +  +  − 1)
and the first order conditions (FOCs) for a minimum are
= 2  2 + 2   + 2   + 


0 =
= 2  2 + 2   + 2   + 
= 2  2 + 2   + 2   + 
0 =


0 =
=  +  +  − 1

0 =

(1.5)

8CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRAThe FOCs (1.5) gives four linear equations in four unknowns which can besolved to find the global minimum variance portfolio weights    and .
Using matrix notation, the problem (1.4) can be concisely expressed asmin  2 = m0 Σm s.t. m0 1 = 1

(1.6)

m

The four linear equation describing the first order conditions (1.5) has thematrix representation
⎞⎛
⎞ ⎛ ⎞
⎛
2

0
2  2  2  1⎟⎜  ⎟ ⎜ ⎟
⎜
⎟
⎜
⎜
⎟
⎟
⎜
⎜ 2  2 2 2  1 ⎟ ⎜  ⎟ ⎜ 0 ⎟⎟⎜⎟ = ⎜ ⎟
⎜
⎟⎜
⎟ ⎜ ⎟
⎜
⎜ 2  2  2 2 1 ⎟ ⎜  ⎟ ⎜ 0 ⎟
⎠⎝
⎠ ⎝ ⎠
⎝
1
1
1
1 0

or, more concisely,

⎛
⎝

2Σ 1
0

1 0

The system (1.7) is of the form

⎞⎛
⎠⎝

m


⎞

⎛

⎠=⎝

0
1

⎞

⎠

(1.7)

A z = b
where

⎛

A = ⎝

2Σ 1
0

1 0

The solution for z is then⎞⎛⎠  z = ⎝

m


⎞

⎛

⎠ and b = ⎝

0
1

⎞

⎠

z = A−1
 b

(1.8)

The first three elements of z are the portfolio weights m = (     )0
for the global minimum variance portfolio with expected return  = m0 μ
and variance  2 = m0 Σm
Example 4 Global minimum variance portfolio for example data
Using the data in Table 1, we can use R to compute the global minimum
variance portfolio weights from (1.8) as follows:9

1.1 PORTFOLIOS WITH THREE RISKY ASSETS> top.mat = cbind(2*sigma.mat, rep(1, 3))> bot.vec = c(rep(1, 3), 0)> Am.mat = rbind(top.mat, bot.vec)> b.vec = c(rep(0, 3), 1)> z.m.mat = solve(Am.mat)%*%b.vec> m.vec = z.m.mat[1:3,1]> m.vec
MSFT
NORD
SBUX
0.4411 0.3656 0.1933

Hence, the global minimum variance portfolio has portfolio weights   =04411  = 03656 and  = 01933 and is given by the vectorm = (04411 03656 01933)0 (1.9)

The expected return on this portfolio,  = m0 μ is> mu.gmin = as.numeric(crossprod(m.vec, mu.vec))> mu.gmin[1] 0.02489
The portfolio variance,  2 = m0 Σm and standard deviation,    are> sig2.gmin = as.numeric(t(m.vec)%*%sigma.mat%*%m.vec)> sig.gmin = sqrt(sig2.gmin)> sig2.gmin[1] 0.005282> sig.gmin[1] 0.07268
In Figure 1.1, this portfolio is labeled “global min”.¥
Alternative derivation of global minimum variance portfolio
The first order conditions (1.5) from the optimization problem (1.6) can be
expressed in matrix notation as(m )= 2 · Σm+ · 1
(3×1)
m
(m )0 == m0 1 − 1
(1×1)

0

=

(1.10)(1.11)

10CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
Using (1.10), first solve for m :1
m = − · Σ−1 12
Next, multiply both sides by 10 and use (1.11) to solve for :1
1 = 10 m = − · 10 Σ−1 1
21⇒  = −2 · 0 −1 1Σ 1Finally, substitute the value for  back into (1.10) to solve for m:
1
1
Σ−1 1
m = − (−2) 0 −1 Σ−1 1 = 0 −1 
2
1Σ 1
1Σ 1

(1.12)

Example 5 Finding global minimum variance portfolio for example data
Using the data in Table 1, we can use R to compute the global minimum
variance portfolio weights from (1.12) as follows:> one.vec = rep(1, 3)> sigma.inv.mat = solve(sigma.mat)> top.mat = sigma.inv.mat%*%one.vec> bot.val = as.numeric((t(one.vec)%*%sigma.inv.mat%*%one.vec))> m.mat = top.mat/bot.val> m.mat[,1]
MSFT
NORD
SBUX
0.4411 0.3656 0.1933
¥

1.1.3

Finding Eﬃcient Portfolios

The investment opportunity set is the set of portfolio expected return,  and portfolio standard deviation,    values for all possible portfolios whose
weights 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. With
two assets, the investment opportunity set in (    )− space lies on a curve(one side of a hyperbola). With three or more assets, the investment opportunity set in (    )− space is described by set of values whose general shape1.1 PORTFOLIOS WITH THREE RISKY ASSETS

11

is complicated and depends crucially on the covariance terms    However,
we do not have to fully characterize the entire investment opportunity set.
If 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 problem
by only concentrating on the set of eﬃcient portfolios. These portfolios lie
on the boundary of the investment opportunity set above the global minimum variance portfolio. This is the framework originally developed by Harry
Markowitz, the father of portfolio theory and winner of the Nobel Prize ineconomics.
Following Markowitz, we assume that investors wish to find portfoliosthat have the best expected return-risk trade-oﬀ. Markowitz characterizedthese eﬃcient portfolios in two equivalent ways. In the first way, investors
seek to find portfolios that maximize portfolio expected return for a given
level of risk as measured by portfolio variance. Let  20 denote a target levelof risk. Then Harry Markowitz characterized the constrained maximization
problem to find an eﬃcient portfolio as
max  = x0 μ s.t.
x

(1.13)

 2 = x0 Σx =  20 and x0 1 = 1Markowitz showed that the investor’s problem of maximizing portfolio expected return subject to a target level of risk has an equivalent dual representation in which the investor minimizes the risk of the portfolio (as measured
by portfolio variance) subject to a target expected return level. Let 0 denote a target expected return level. Then the dual problem is the constrained
minimization problem
min  2 = x0 Σx s.t.
x

(1.14)

 = x0 μ = 0  and x0 1 = 1To find eﬃcient portfolios of risky assets in practice, the dual problem (1.14)
is most often solved. This is partially due to computational conveniences andpartly due to investors being more willing to specify target expected returns
rather than target risk levels. The eﬃcient portfolio frontier is a graph of 
versus   values for the set of eﬃcient portfolios generated by solving (1.14)
for all possible target expected return levels 0 above the expected returnon the global minimum variance portfolio Just as in the two asset case, the12CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
resulting eﬃcient frontier will resemble one side of an hyperbola and is often
called the “Markowitz bullet”.
To solve the constrained minimization problem (1.14), first form the Lagrangian function
( 1  2 ) = x0 Σx + 1 (x0 μ−0 ) + 2 (x0 1 − 1)
Because there are two constraints (x0 μ = 0 and x0 1 = 1) there are two
Lagrange multipliers 1 and 2  The FOCs for a minimum are the linear
equations(x 1  2 )= 2Σx + 1 μ + 2 1 = 0x(x 1  2 )= x0 μ − 0 = 01(x 1  2 )= x0 1 − 1 = 02(1.15)(1.16)(1.17)

These FOCs consist of five linear equations in five unknowns (      1  2 )
We can represent the system of linear equations using matrix algebra as⎛⎞⎛ ⎞ ⎛⎞
0x
2Σ μ 1⎜⎟⎜ ⎟ ⎜⎟⎜ 0⎟⎜ ⎟ ⎜⎟⎜ μ 0 0 ⎟ ⎜ 1 ⎟ = ⎜ 0 ⎟ ⎝⎠⎝ ⎠ ⎝
⎠
0
2
1
1 0 0
or

Az = b0 
where

⎛

⎞

⎛

⎞

⎛

⎞

x
0
2Σ μ 1
⎜ ⎟
⎜
⎜
⎟
⎟
⎜ ⎟
⎜
⎜ 0
⎟
⎟
A = ⎜ μ 0 0 ⎟  z = ⎜ 1 ⎟ and b0 = ⎜ 0 ⎟ 
⎝ ⎠
⎝
⎝
⎠
⎠
10 0 0
2
1

The solution for z is thenz = A−1 b0 

(1.18)

The first three elements of z are the portfolio weights x = (     )0 for
the minimum variance portfolio with expected return  = 0 . If 0 isgreater than or equal to the expected return on the global minimum variance
portfolio then x is an eﬃcient portfolio.

1.1 PORTFOLIOS WITH THREE RISKY ASSETS

13

Example 6 Eﬃcient portfolio with the same expected return as Microsoft
Using the data in Table 1, consider finding a minimum variance portfolio with the same expected return as Microsoft. This will be an eﬃcient
portfolio because   = 00427   = 002489 Call this portfolio
x = (      )0  That is, consider solving (1.14) with target expected return 0 =   = 00427 using (1.18) The R calculations tocreate the matrix A and the vectors z and b  are:> top.mat = cbind(2*sigma.mat, mu.vec, rep(1, 3))> mid.vec = c(mu.vec, 0, 0)> bot.vec = c(rep(1, 3), 0, 0)> A.mat = rbind(top.mat, mid.vec, bot.vec)> bmsft.vec = c(rep(0, 3), mu.vec["MSFT"], 1)
and the R code to solve for x using (1.18) is:> z.mat = solve(A.mat)%*%bmsft.vec> x.vec = z.mat[1:3,]> x.vec
MSFT
NORD
SBUX
0.82745 -0.09075 0.26329
The eﬃcient portfolio with the same expected return as Microsoft has portfolio weights   = 082745  = −009075 and  = 026329 and isgiven by the vectorx = (082745 −009075 026329)0 (1.19)The expected return on this portfolio,  = x0 μ is equal to the target
return   :> mu.px = as.numeric(crossprod(x.vec, mu.vec))> mu.px[1] 0.0427
The portfolio variance,  2 = x0 Σx and standard deviation,    are> sig2.px = as.numeric(t(x.vec)%*%sigma.mat%*%x.vec)> sig.px = sqrt(sig2.px)> sig2.px[1] 0.0084> sig.px[1] 0.09166

14CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
and are smaller than the corresponding values for Microsoft (see Table 1).
This eﬃcient portfolio is labeled “E1” in Figure 1.1. ¥Example 7 Eﬃcient portfolio with the same expected return as Microsoft
To find a minimum variance portfolio y = (      )0 with thesame expected return as Starbucks we use (1.18) with b = (0   1)0 :> bsbux.vec = c(rep(0, 3), mu.vec["SBUX"], 1)> z.mat = solve(Ax.mat)%*%bsbux.vec> y.vec = z.mat[1:3,]> y.vec
MSFT
NORD
SBUX
0.5194 0.2732 0.2075
The portfolioy = (05194 02732 02075)0 (1.20)

is an eﬃcient portfolio because  = 00285   = 002489 The portfolio expected return and standard deviation are:> mu.py = as.numeric(crossprod(y.vec, mu.vec))> sig2.py = as.numeric(t(y.vec)%*%sigma.mat%*%y.vec)> sig.py = sqrt(sig2.py)> mu.py[1] 0.0285> sig.py[1] 0.07355
This eﬃcient portfolio is labeled “E2” in Figure 1.1. It has the same expected
return as SBUX but a smaller standard deviation.
The covariance and correlation values between the portfolio returns  =x0 R and  = y0 R are given by:> sigma.xy = as.numeric(t(x.vec)%*%sigma.mat%*%y.vec)> rho.xy = sigma.xy/(sig.px*sig.py)> sigma.xy[1] 0.005914> rho.xy[1] 0.8772
This covariance will be used later on when constructing the frontier of eﬃcient
portfolios.
¥

1.1 PORTFOLIOS WITH THREE RISKY ASSETS

15

Alternative derivation of eﬃcient portfolio
Consider the first order conditions (1.15)-(1.17) from the optimization problem (1.14). First, use (1.15) to solve for the 3 × 1 vector x:11
x = − 1 Σ−1 μ − 2 Σ−1 1(1.21)
2
2.
Define the 3 × 2 matrix M = [μ .. 1] and the 2 × 1 vector λ = (1  2 )0  Then
we can rewrite (1.21) in matrix form as1
x = − Σ−1 Mλ
2(1.22)

Next, to find the values for 1 and 2 , pre-multiply (1.21) by μ0 and use(1.16) to give11
0 = μ0 x = − 1 μ0 Σ−1 μ − 2 μ0 Σ−1 1
2
2(1.23)

Similarly, pre-multiply (1.21) by 10 and use (1.17) to give1
1
1 = 10 x = − 1 10 Σ−1 μ − 2 10 Σ−1 1
2
2(1.24)

Now, we have two linear equations (1.23) and (1.24) involving 1 and 2
which we can write in matrix notation as⎞⎛ ⎞ ⎛ ⎞
⎛
0 −1
0 −1
1 μ Σ μ μ Σ 1 ⎠ ⎝ 1 ⎠ ⎝ 0 ⎠− ⎝
(1.25)=
2 μ0 Σ−1 1 10 Σ−1 121
Define

⎛
⎝

0

−1

0

−1

0

−1

0

−1

μΣ μ μΣ 1
μΣ 1 1Σ 1

so that we can rewrite (1.25) as⎞⎠ = M0 Σ−1 M = B⎛μ̃0 = ⎝

1
− Bλ = μ̃0 
2

0
1

⎞

⎠
(1.26)

16CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
The solution for λ = (1  2 )0 is thenλ = −2B−1 μ̃0 (1.27)

Substituting (1.27) back into (1.22) gives an explicit expression for the eﬃcient portfolio weight vector x:¡¢11
x = − Σ−1 Mλ = − Σ−1 M −2B−1 μ̃0 = Σ−1 MB−1 μ̃0 
22(1.28)

Example 8 Alternative solution for eﬃcient portfolio with the same expected return as Microsoft
The R code to compute the eﬃcient portfolio with the same expected
return as Microsoft using (1.28) is:> M.mat = cbind(mu.vec, one.vec)> B.mat = t(M.mat)%*%solve(sigma.mat)%*%M.mat> mu.tilde.msft = c(mu.vec["MSFT"], 1)> x.vec.2 = solve(sigma.mat)%*%M.mat%*%solve(B.mat)%*%mu.tilde.msft> x.vec.2[,1]
MSFT 0.82745
NORD -0.09075
SBUX 0.26329
¥

1.1.4

Computing the Eﬃcient Frontier

The analytic expression for a minimum variance portfolio (1.28) can be used
to show that any minimum variance portfolio can be created as a convex
combination of any two minimum variance portfolios with diﬀerent target
expected returns. If the expected return on the resulting portfolio is greaterthan the expected on the global minimum variance portfolio, then the portfolio is an eﬃcient frontier portfolio. Otherwise, the portfolio is an ineﬃcient
frontier portfolio. As a result, to compute the portfolio frontier in (   )space (Markowitz bullet) we only need to find two eﬃcient portfolios. The
remaining frontier portfolios can then be expressed as convex combinationsof these two portfolios. The following proposition describes the process forthe three risky asset case using matrix algebra.

17

1.1 PORTFOLIOS WITH THREE RISKY ASSETS
Proposition 9 Creating a frontier portfolio from two eﬃcient portfolios

Let x = (     )0 and y = (     )0 be any two minimum variance
portfolios with diﬀerent target expected returns x0 μ = 0 6= y0 μ = 1 .
That is, portfolio x solves
min  2 = x0 Σx  x0 μ = 0 and x0 1 = 1xand portfolio y solves
min  2 = y0 Σy y0 μ = 1 and y0 1 = 1yLet  be any constant and define the portfolio z as a linear combination of
portfolios x and y :z =  · x + (1 − ) · y⎞⎛ + (1 − )
⎟
⎜ 
⎟
⎜
= ⎜  + (1 − ) ⎟ ⎠⎝ + (1 − )(1.29)Then(a) The portfolio z is a minimum variance portfolio with expected return andvariance given by = z0 μ =  ·  + (1 − ) ·   2

= z Σz = α2  2 + (1 − )2  2 + 2(1 − ) 
0(1.30)(1.31)

where 2 = x0 Σx  2 = y0 Σy  = x0 Σy(b) If  ≥   where  is the expected return on the global minimum
variance portfolio, then portfolio z is an eﬃcient portfolio. Otherwise, z isan ineﬃcient frontier portfolio.
The proof of (a) follows directly from applying (1.28) to portfolios x andy:x = Σ−1 MB−1 μ̃ 
y = Σ−1 MB−1 μ̃ 18CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
where μ̃ = (  1)0 and μ̃ = (  1)0  Then for portfolio zz =  · x + (1 − ) · y=  · Σ−1 MB−1 μ̃ + (1 − ) · Σ−1 MB−1 μ̃= Σ−1 MB−1 ( · μ̃ + (1 − ) · μ̃ )= Σ−1 MB−1 μ̃ 

where μ̃ =  · μ̃ + (1 − ) · μ̃ = (  1)0 
Example 10 Creating an arbitrary frontier portfolio from two eﬃcient portfolios
Consider the data in Table 1 and the previously computed eﬃcient portfolios(1.19) and (1.20) and let  = 05 From (1.29), the frontier portfolio z isconstructed usingz =  · x + (1 − ) · y⎞⎛⎞⎛
05194082745
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
= 05 · ⎜ −009075 ⎟ + 05 · ⎜ 02732 ⎟⎠⎝⎠⎝
02075026329⎞ ⎛⎛⎞(05)(082745)(05)(05194)
⎟ ⎜
⎜
⎟
⎟ ⎜
⎜
⎟
= ⎜ (05)(−009075) ⎟ + ⎜ (05)(02732) ⎟⎠ ⎝⎝⎠(05)(026329)(05)(02075)
⎞ ⎛
⎛
⎞

06734⎟ ⎜ ⎟⎜⎟ ⎜⎜⎟= ⎜ 00912 ⎟ = ⎜  ⎟ ⎠ ⎝⎝⎠02354


In R, the new frontier portfolio is computed using> a = 0.5
> z.vec = a*x.vec + (1-a)*y.vec> z.vec
MSFT
NORD
SBUX
0.6734 0.0912 0.2354

1.1 PORTFOLIOS WITH THREE RISKY ASSETS

19

Using  = z0 μ and  2 = z0 Σz, the expected return, variance and standard
deviation of this portfolio are> mu.pz = as.numeric(crossprod(z.vec, mu.vec))> sig2.pz = as.numeric(t(z.vec)%*%sigma.mat%*%z.vec)> sig.pz = sqrt(sig2.pz)> mu.pz[1] 0.0356> sig2.pz[1] 0.00641> sig.pz[1] 0.08006
Equivalently, using  =  +(1−) and  2 = 2  2 +(1−)2  2 +2(1 − ) the expected return, variance and standard deviation of thisportfolio are> mu.pz = a*mu.px + (1-a)*mu.py> sig.xy = as.numeric(t(x.vec)%*%sigma.mat%*%y.vec)> sig2.pz = a^2 * sig2.px + (1-a)^2 * sig2.py + 2*a*(1-a)*sig.xy> sig.pz = sqrt(sig2.pz)> mu.pz[1] 0.0356> sig2.pz[1] 0.00641> sig.pz[1] 0.08006
Because  = 00356   = 002489 the frontier portfolio z is an eﬃcient
portfolio. The three eﬃcient portfolios x y and z are illustrated in Figure
1.2 and are labeled “E1”, “E2” and “E3”, respectively.¥
Example 11 Creating a frontier portfolio with a given expected return fromtwo eﬃcient portfolios
Given the two eﬃcient portfolios (1.19) and (1.20) with target expected returns equal to the expected returns on Microsoft and Starbucks, respectively,
consider creating a frontier portfolio with target expected return equal to the
expected return on Nordstrom. Then =  + (1 − ) =  = 00015

0.05

0.06

20CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA

0.04

E1

MSFT

0.03

p

E3
E2

SBUX

0.01

0.02

GLOBAL MIN

0.00

IE4

0.00

0.05

0.10

NORD

0.15

0.20

p

Figure 1.2: Three eﬃcient portfolios of Microsoft, Nordstrom and Starbucks.
and we can solve for  using= − 00015 − 00285= −1901= − 00427 − 00285

Using R, the weights in this frontier portfolio are> a.nord = (mu.vec["NORD"] - mu.py)/(mu.px - mu.py)> z.nord = a.nord*x.vec + (1 - a.nord)*y.vec> z.nord
MSFT
NORD
SBUX
-0.06637 0.96509 0.10128
The expected return, variance and standard deviation on this portfolio are> mu.pz.nord = a.nord*mu.px + (1-a.nord)*mu.py> sig2.pz.nord = a.nord^2 * sig2.px + (1-a.nord)^2 * sig2.py ++2*a.nord*(1-a.nord)*sigma.xy

1.1 PORTFOLIOS WITH THREE RISKY ASSETS

21> sig.pz.nord = sqrt(sig2.pz.nord)> mu.pz.nord
NORD
0.0015> sig2.pz.nord
NORD
0.01066> sig.pz.nord
NORD
0.1033
Because  = 00015   = 002489 the frontier portfolio z is an ineﬃcient frontier portfolio. This portfolio is labeled “IE4” in Figure 1.2.¥
The eﬃcient frontier of portfolios, i.e., those frontier portfolios with expected return greater than the expected return on the global minimum variance portfolio, can be conveniently created using (1.29) with two specificeﬃcient portfolios. The first eﬃcient portfolio is the global minimum variance portfolio (1.4). The second eﬃcient portfolio is the eﬃcient portfolio
whose target expected return is equal to the highest expected return among
all of the assets under consideration. The steps for constructing the eﬃcient
frontier are:
1. Compute the global minimum variance portfolio m by solving (1.6),
and compute  = m0 μ and  2 = m0 Σm.
2. Compute the eﬃcient portfolio x by with target expected return equal
to the maximum expected return of the assets under consideration.
That is, solve (1.14) with 0 = max{1  2  3 } and compute  =x0 μ and  2 = x0 Σx.
3. Compute cov(   ) =   = m0 Σx
4. Create an initial grid of  values {1 09     −09 −1} compute thefrontier portfolios z using (1.29), and compute their expected returns
and variances using (1.29), (1.30) and (1.31), respectively.
5. Plot  against   and adjust the grid of  values to create a nice
plot.

0.05

0.06

22CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA

0.03

p

0.04

MSFT

SBUX

0.01

0.02

GLOBAL MIN

0.00

NORD

0.00

0.05

0.10

0.15

p

Figure 1.3: Eﬃcient frontier of three risky assets.
Example 12 Compute and plot the eﬃcient frontier of risky assets (Markowitz
bullet)To compute and plot the eﬃcient frontier from the three risky assets in Table
1.1 in R use> a = seq(from=1, to=-1, by=-0.1)> n.a = length(a)> z.mat = matrix(0, n.a, 3)> mu.z = rep(0, n.a)
> sig2.z = rep(0, n.a)
> sig.mx = t(m)%*%sigma.mat%*%x.vec> for (i in 1:n.a) {+z.mat[i, ] = a[i]*m + (1-a[i])*x.vec+
mu.z[i] = a[i]*mu.gmin + (1-a[i])*mu.px+
sig2.z[i] = a[i]^2 * sig2.gmin + (1-a[i])^2 * sig2.px ++2*a[i]*(1-a[i])*sig.mx

1.1 PORTFOLIOS WITH THREE RISKY ASSETS

23+ }> plot(sqrt(sig2.z), mu.z, type="b", ylim=c(0, 0.06), xlim=c(0, 0.17),+
pch=16, col="blue", ylab=expression(mu[p]),+xlab=expression(sigma[p]))> text(sig.gmin, mu.gmin, labels="Global min", pos=4)> text(sd.vec, mu.vec, labels=asset.names, pos=4)
The variables z.mat, mu.z and sig2.z contain the weights, expected returns
and variances, respectively, of the eﬃcient frontier portfolios for a grid of values between 1 and −1 The resulting eﬃcient frontier is illustrated in
Figure 1.3. ¥1.1.5

Eﬃcient Portfolios of Three Risky Assets and a
Risk-Free Asset

In the previous chapter, we showed that eﬃcient portfolios of two risky assets
and a single risk-free (T-Bill) asset are portfolios consisting of the highest
Sharpe ratio portfolio (tangency portfolio) and the T-Bill. With three ormore risky assets and a T-Bill the same result holds.
Computing the Tangency Portfolio
The tangency portfolio is the portfolio of risky assets that has the highest
Sharpe ratio. The tangency portfolio, denoted t = (      )0  solves
the constrained maximization problem
maxtt0 μ − (t0 Σt)1
2= − s.t. t0 1 = 1
 

1

where  = t0 μ and   = (t0 Σt) 2  The Lagrangian for this problem is1(t ) = (t0 μ −  ) (t0 Σt)− 2 + (t0 1 − 1)
Using the chain rule, the first order conditions are1
(t )= μ(t0 Σt)− 2 − (t0 μ −  ) (t0 Σt)−32 Σt + 1 = 0t(t )= t0 1 − 1 = 024CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
After much tedious algebra, it can be shown that the solution for t has anice simple expression:t=Σ−1 (μ −  · 1)
10 Σ−1 (μ −  · 1)

(1.32)The location of the tangency portfolio, and the sign of the Sharpe ratio,
depends on the relationship between the risk-free rate  and the expected
return on the global minimum variance portfolio   If     whichis the usual case, then the tangency portfolio with have a positive Sharpe
ratio. If     which could occur when stock prices are falling and theeconomy is in a recession, then the tangency portfolio will have a negative
Sharpe slope. In this case, eﬃcient portfolios involve shorting the tangency
portfolio and investing the proceeds in T-Bills.
Example 13 Computing the tangency portfolio
Suppose  = 0005 To compute the tangency portfolio (1.32) in R for thethree risky asses in Table use> rf = 0.005> sigma.inv.mat = solve(sigma.mat)> one.vec = rep(1, 3)> mu.minus.rf = mu.vec - rf*one.vec> top.mat = sigma.inv.mat%*%mu.minus.rf> bot.val = as.numeric(t(one.vec)%*%top.mat)> t.vec = top.mat[,1]/bot.val> t.vec
MSFT
NORD
SBUX
1.0268 -0.3263 0.2994
The tangency portfolio has weights   = 10268  = −03263 and = 02994 and is given by the vectort = (10268 −03263 02994)0 (1.33)

Notice that Nordstrom, which has the lowest mean return, is sold shortis the tangency portfolio. The expected return on the tangency portfolio, = t0 μ is1.1 PORTFOLIOS WITH THREE RISKY ASSETS

25> mu.t = as.numeric(crossprod(t.vec, mu.vec))> mu.t[1] 0.05189
The portfolio variance,  2 = t0 Σt and standard deviation,    are> sig2.t = as.numeric(t(t.vec)%*%sigma.mat%*%t.vec)> sig.t = sqrt(sig2.t)> sig2.t[1] 0.01245> sig.t[1] 0.1116
Because  = 0005   = 002489 the tangency portfolio has a positive
Sharpe’s ratio/slope given by = − 
005189 − 0005= 04202= 
01116¥
Alternative Derivation of the Tangency Portfolio
Consider forming portfolios of three risky assets with return vector R andT-bills (risk-free asset) with constant return  . Let  denote the vector ofrisky asset weights and let  denote the safe asset weight and assume thatx0 1 +  = 1 so that all wealth is allocated to these assets. The portfolio
return is = x0 R +   = x0 R + (1 − x0 1) =  + x0 (R −  · 1)
The portfolio excess return is −  = x0 (R −  · 1)

(1.34)The expected portfolio return excess return (risk premium) and portfolio
variance are −  = x0 (μ −  · 1)
 2 = x0 Σx

(1.35)(1.36)

26CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRAFor notational simplicity, define R̃ = R− ·1 μ̃ = μ− ·1 ̃ =  − and ̃ =  −   Then (1.34) and (1.35) can be re-expressed as̃ = x0 R̃̃ = x0 μ̃(1.37)(1.38)

To find the minimum variance portfolio of risky assets and a risk freeasset that achieves the target excess return ̃0 = 0 −  we solve theminimization problem
min  2 = x0 Σx s.t. ̃ = ̃0 
xNote that x0 1 = 1 is not a constraint because wealth need not all be allocatedto the risky assets; some wealth may be held in the riskless asset. The
Lagrangian is(x ) = x0 Σx+(x0 μ̃ − ̃0 )

The first order conditions for a minimum are(x )= 2Σx + μ̃ = 0x(x )= x0 μ̃ − ̃0 = 0(1.39)(1.40)

Using the first equation (1.39), we can solve for x in terms of  :1
x = − Σ−1 μ̃
2(1.41)The second equation (1.40) implies that x0 μ̃ = μ̃0 x = ̃0  Then premultiplying (1.41) by μ̃0 gives1
μ̃0 x = − μ̃0 Σ−1 μ̃ = ̃0 
2which we can use to solve for  :=−2̃0
μ̃0 Σ−1 μ̃Plugging (1.42) into (1.41) then gives the solution for x :¶µ2̃01
Σ−1 μ̃
1 −1
Σ−1 μ̃ = ̃0 · 0 −1 
− 0 −1x = − Σ μ̃ = −22μ̃ Σ μ̃μ̃ Σ μ̃(1.42)(1.43)

1.1 PORTFOLIOS WITH THREE RISKY ASSETS

27

The solution for  is then 1 − x0 1
Now, the tangency portfolio t is 100% invested in risky assets so that0t 1 = 10 t = 1 Using (1.43), the tangency portfolio satisfies
10 t = ̃ ·which implies that10 Σ−1 μ̃= 1
μ̃0 Σ−1 μ̃

μ̃0 Σ−1 μ̃̃ = 0 −1 
1 Σ μ̃

(1.44)

Plugging (1.44) back into (1.43) then gives an explicit solution for t :µ 0 −1 ¶Σ−1 μ̃
Σ−1 μ̃
μ̃ Σ μ̃=t =10 Σ−1 μ̃ μ̃0 Σ−1 μ̃
10 Σ−1 μ̃Σ−1 (μ −  · 1)
= 0 −11 Σ (μ −  · 1)
which is the result (1.32) we got from finding the portfolio of risky assetsthat has the maximum Sharpe ratio.

1.1.6

Mutual Fund Separation Theorem Again

When there is a risk-free asset (T-bill) available, the eﬃcient frontier ofT-bills and risky assets consists of portfolios of T-bills and the tangency
portfolio. The expected return and standard deviation values of any sucheﬃcient portfolio are given by =  +  ( −  )
  =    (1.45)(1.46)

where  represents the fraction of wealth invested in the tangency portfolio(1 −  represents the fraction of wealth invested in T-Bills), and  = t0 μand   = (t0 Σt)12 are the expected return and standard deviation on thetangency portfolio, respectively. Recall, this result is known as the mutualfund separation theorem. The tangency portfolio can be considered as amutual fund of the risky assets, where the shares of the assets in the mutualfund are determined by the tangency portfolio weights, and the T-bill canbe considered as a mutual fund of risk-free assets. The expected return-risk

28CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
trade-oﬀ of these portfolios is given by the line connecting the risk-free rate
to the tangency point on the eﬃcient frontier of risky asset only portfolios.
Which combination of the tangency portfolio and the T-bill an investor will
choose depends on the investor’s risk preferences. If the investor is very risk
averse and prefers portfolios with very low volatility, then she will choose a
combination with very little weight in the tangency portfolio and a lot ofweight in the T-bill. This will produce a portfolio with an expected returnclose to the risk-free rate and a variance that is close to zero. If the investor
can tolerate a large amount of volatility, then she will prefer a portfolio witha high expected return regardless of volatility. This portfolio may involve
borrowing at the risk-free rate (leveraging) and investing the proceeds in thetangency portfolio to achieve a high expected return.
Example 14 Eﬃcient portfolios of three risky assets and T-bills chosen byrisk averse and risk tolerant investors
Consider the tangency portfolio computed from the example data in Table
1.1 with  = 0005 This portfolio is> t.vec
MSFT
NORD
1.0268 -0.3263> mu.t[1] 0.05189> sig.t[1] 0.1116

SBUX
0.2994The eﬃcient portfolios of T-Bills and the tangency portfolio is illustrated in
Figure .
We want to compute an eﬃcient portfolio that would be preferred bya highly risk averse investor, and a portfolio that would be preferred by ahighly risk tolerant investor. A highly risk averse investor might have a low
volatility (risk) target for his eﬃcient portfolio. For example, suppose thevolatility target is   = 002 or 2% Using (1.46) and solving for  , theweights in the tangency portfolio and the T-Bill are> x.t.02 = 0.02/sig.t> x.t.02[1] 0.1792> 1-x.t.02[1] 0.8208

1.1 PORTFOLIOS WITH THREE RISKY ASSETS

29

In this eﬃcient portfolio, the weights in the risky assets are proportional to
the weights in the tangency portfolio> x.t.02*t.vec
MSFT
NORD
0.18405 -0.05848

SBUX
0.05367

The expected return and volatility values of this portfolio are> mu.t.02 = x.t.02*mu.t + (1-x.t.02)*rf> sig.t.02 = x.t.02*sig.t> mu.t.02[1] 0.01340> sig.t.02[1] 0.02
These values are illustrated in Figure as the portfolio labeled “E1”.
A highly risk tolerant investor might have a high expected return target
for his eﬃcient portfolio. For example, suppose the expected return targetis  = 007 or 7% Using (1.45) and solving for the  , the weights in thetangency portfolio and the T-Bill are> x.t.07 = (0.07 - rf)/(mu.t - rf)> x.t.07[1] 1.386> 1-x.t.07[1] -0.3862
Notice that this portfolio involves borrowing at the T-Bill rate (leveraging)and investing the proceeds in the tangency portfolio. In this eﬃcient portfolio, the weights in the risky assets are> x.t.07*t.vec
MSFT
NORD
1.4234 -0.4523

SBUX
0.4151The expected return and volatility values of this portfolio are> mu.t.07 = x.t.07*mu.t + (1-x.t.07)*rf> sig.t.07 = x.t.07*sig.t> mu.t.07

0.08

30CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA

0.06

E2

MSFT

0.04

p

tangency

SBUX
0.02

Global min

0.00

E1

NORD

0.00

0.05

0.10

0.15

p

Figure 1.4: Eﬃcient portfolios of three risky assets. The portfolio labeled“E1” is chosen by a risk averse investor with a target volatility of 0.02. The
portfolio “E2” is chosen by a risk tolerant investor with a target expected
return of 0.07.
[1] 0.07> sig.t.07[1] 0.1547
In order to achieve the target expected return of 7%, the investor must tolerate a 15.47% volatility. These values are illustrated in Figure as the portfolio
labeled “E2”.
¥

1.2

Portfolio Analysis Functions in R

The script file portfolio.r contains a few R functions for computing Markowitz
mean-variance eﬃcient portfolios allowing for short sales using matrix alge-1.2 PORTFOLIO ANALYSIS FUNCTIONS IN R
Function

Description

getPortfolio

create portfolio object

31

globalMin.portfolio compute global minimum variance portfolio
efficient.portfolio compute minimum variance portfolio subject to target return
tangency.portfolio

compute tangency portfolio

efficient.frontiercompute eﬃcient frontier of risky assets

Table 1.2: R functions for computing mean-variance eﬃcient portfolios
bra computations2 . These functions allow for the easy computation of the
global minimum variance portfolio, an eﬃcient portfolio with a given target
expected return, the tangency portfolio, and the eﬃcient frontier. These
functions are summarized in Table 1.2.
The following examples illustrate the use of the functions in Table 1.2
using the example data in Table 1.1. We first construct the input data:> asset.names = c("MSFT", "NORD", "SBUX")> er = c(0.0427, 0.0015, 0.0285)> names(er) = asset.names> covmat = matrix(c(0.0100, 0.0018, 0.0011,+
0.0018, 0.0109, 0.0026,+
0.0011, 0.0026, 0.0199),+
nrow=3, ncol=3)> rk.free = 0.005> dimnames(covmat) = list(asset.names, asset.names)
To specify a portfolio object, you need an expected return vector and covariance matrix for the assets under consideration as well as a vector of portfolio
weights. To create an equally weighted portfolio use> ew = rep(1,3)/3> equalWeight.portfolio = getPortfolio(er=er,cov.mat=covmat,weights=ew)
2

To use the functions in this script file, load them into R with the source()function. For example, if portfolio.r is located in C:\mydata run the command
source(“C:/mydata/portfolio.r”) at the beginnin of your R session. Then the functions in the file will be available for your use.

32CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA> class(equalWeight.portfolio)[1] "portfolio"
Portfolio objects have the following components> names(equalWeight.portfolio)[1] "call"
"er"
"sd"

"weights"

There are print(), summary() and plot() methods for portfolio objects.
The print() method gives> equalWeight.portfolio
Call:
getPortfolio(er = er, cov.mat = covmat, weights = ew)
Portfolio expected return:Portfolio standard deviation:
Portfolio weights:
MSFT
NORD
SBUX
0.3333 0.3333 0.3333

0.024230.07587

The plot() method shows a bar chart of the portfolio weights> plot(equalWeight.portfolio)
The global minimum variance portfolio (allowing for short sales) m solves
the optimization problem
min m0 Σm s.t. m0 1 = 1m

To compute this portfolio use the function globalMin.portfolio()> gmin.port <- globalMin.portfolio(er, covmat)> attributes(gmin.port)$names[1] "call"
"er"
"sd"
"weights"$class[1] "portfolio"> gmin.port
Call:1.2 PORTFOLIO ANALYSIS FUNCTIONS IN R

33

0.20
0.15
0.00

0.05

0.10

Weight0.25

0.30

Portfolio Weights

MSFT

NORD

SBUX

AssetsFigure 1.5:
globalMin.portfolio(er = er, cov.mat = covmat)
Portfolio expected return:Portfolio standard deviation:
Portfolio weights:
MSFT
NORD
SBUX
0.4411 0.3656 0.1933

0.02489184
0.07267607

A mean-variance eﬃcient portfolio x that achieves the target expected
return 0 solves the optimization problem
min x0 Σx s.t. x0 1 = 1 and x0 μ = 0 
To compute this portfolio for the target expected return 0 = [  ] =004275 use the efficient.portfolio() function> target.return <- er[1]> e.port.msft <- efficient.portfolio(er, covmat, target.return)> e.port.msft

34CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA
Call:
efficient.portfolio(er = er, cov.mat = covmat,
target.return = target.return)Portfolio expected return:0.0427
Portfolio standard deviation: 0.091656
Portfolio weights:
MSFT
NORD
SBUX
0.8275 -0.0907 0.2633
The tangency portfolio t is the portfolio of risky assets with the highest
Sharpe’s slope and solves the optimization problem
maxtt0 μ − s.t. t0 1 = 1
(t0 Σt)12

where  denotes the risk-free rate. To compute this portfolio with  = 0005
use the tangency.portfolio() function> tan.port <- tangency.portfolio(er, covmat, rk.free)> tan.port
Call:tangency.portfolio(er = er, cov.mat = covmat, risk.free = rk.free)
Portfolio expected return:Portfolio standard deviation:
Portfolio weights:
MSFT
NORD
SBUX
1.0268 -0.3263 0.2994

0.05188967
0.1115816

The the set of eﬃcient portfolios of risky assets can be computed as aconvex combination of any two eﬃcient portfolios. It is convenient to use theglobal minimum variance portfolio as one portfolio and an eﬃcient portfoliowith target expected return equal to the maximum expected return of theassets under consideration as the other portfolio. Call these portfolios m
and x, respectively. For any number  another eﬃcient portfolio can becomputed asz = m + (1 − )x

The function efficient.frontier() constructs the set of eﬃcient portfolios
using this method for a collection of  values. For example, to compute 20
eﬃcient portfolios for values of  between −2 and 15 use

1.2 PORTFOLIO ANALYSIS FUNCTIONS IN R

35

> ef <- efficient.frontier(er, covmat, alpha.min=-2,+alpha.max=1.5, nport=20)> attributes(ef)$names[1] "call"
"er"
"sd"
"weights"$class[1] "Markowitz"
> ef
Call:
efficient.frontier(er = er, cov.mat = covmat, nport = 20, alpha.min = -2,alpha.max = 1.5)
Frontier portfolios’ expected returns and standard deviations
port 1 port 2 port 3 port 4 port 5 port 6 port 7
ER 0.0783 0.0750 0.0718 0.0685 0.0652 0.0619 0.0586
SD 0.1826 0.1732 0.1640 0.1548 0.1458 0.1370 0.1284
port 8 port 9 port 10 port 11 port 12 port 13 port 14
ER 0.0554 0.0521 0.0488 0.0455 0.0422
0.039 0.0357
SD 0.1200 0.1120 0.1044 0.0973 0.0908
0.085 0.0802
port 15 port 16 port 17 port 18 port 19 port 20
ER 0.0324 0.0291 0.0258 0.0225 0.0193 0.0160
SD 0.0764 0.0739 0.0727 0.0730 0.0748 0.0779
Use the summary() method to show the weights of these portfolios. Use
the plot() method to plot the eﬃcient frontier> plot(ef)The resulting plot is shown in Figure 1.6.
To create a plot of the eﬃcient frontier showing the original assets andthe tangency portfolio use> plot(ef, plot.assets=T)> points(gmin.port$sd, gmin.port$er, col="blue")> points(tan.port$sd, tan.port$er, col="red")> sr.tan = (tan.port$er - rk.free)/tan.port$sd> abline(a=rk.free, b=sr.tan)
The resulting plot is shown in Figure 1.7.

36CHAPTER 1 PORTFOLIO THEORY WITH MATRIX ALGEBRA

0.04
0.00

0.02

Portfolio ER

0.06

0.08

Efficient Frontier

0.00

0.05

0.10

0.15

Portfolio SD

Figure 1.6: Plot method for Markowitz object.

1.3

References

Cochrane, J. (2008). Asset Pricing, Second Edition.
Constantinidex, G.M., and Malliaris, A.G. (1995). “Portfolio Theory”,
Chapter 1 in R. Jarrow et al., Eds., Handbooks in OR & MS, Vol. 9, Elsevier.
Ingersoll, Jr., J.E. (1987). Theory of Financial Decision Making, Rowman
and Littlefield, Totowa, NJ.
Markowitz, H. (1959). Portfolio Selection: Eﬃcient Diversification of
Investments, Wiley, New York.

37

1.3 REFERENCES

MSFT

0.04

Portfolio ER

0.06

0.08

Efficient Frontier

0.00

0.02

SBUX

NORD
0.00

0.05

0.10

0.15

Portfolio SD

Figure 1.7: Eﬃcient frontier for three firm example.