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Introduction to Computational Finance and

Financial Econometrics

Portfolio Theory with Matrix Algebra

Eric Zivot

Spring 2015

Eric Zivot (Copyright©2015)Portfolio Theory1 / 54

Outline

1Portfolios with Three Risky Assets

Portfolio characteristics using matrix notation

Finding the global minimum variance portfolio

Finding efficient portfolios

Computing the efficient frontier

Mutual fund separation theorem again

Eric Zivot (Copyright©2015)Portfolio Theory2 / 54

Example

Example: Three risky assets

LetRi(i=A,B,C) denote the return on assetiand assume thatRi follows CER model: R i≂iid N(μi,σ2i) cov(Ri,Rj) =σij

Portfolio "x":

x i= share of wealth in asseti x

A+xB+xC= 1

Portfolio return:

R p,x=xARA+xBRB+xCRC.Eric Zivot (Copyright©2015)Portfolio Theory3 / 54

Example cont.

Stocki μiσiPair (i,j)σijA (Microsoft) 0.0427 0.1000 (A,B) 0.0018

B (Nordstrom) 0.0015 0.1044 (A,C) 0.0011

C (Starbucks) 0.0285 0.1411 (B,C) 0.0026Three asset example data.

In matrix algebra, we have:

A B C) (0.0427

0.0015

0.0285)

2AσABσAC

ABσ2BσBC

ACσBCσ2C)

((0.1000)20.0018 0.0011

0.0018 (0.1044)20.0026

0.0011 0.0026 (0.1411)2)

)Eric Zivot (Copyright©2015)Portfolio Theory4 / 54

Outline

1Portfolios with Three Risky Assets

Portfolio characteristics using matrix notation

Finding the global minimum variance portfolio

Finding efficient portfolios

Computing the efficient frontier

Mutual fund separation theorem again

Eric Zivot (Copyright©2015)Portfolio Theory5 / 54

Matrix Algebra Representation

R=( (R A R B R C) A B C) ),1=( (1 1 1) x=( (x A x B x C)

2AσABσAC

ABσ2BσBC

ACσBCσ2C)

Portfolio weights sum to 1:

x ?1= (xAxBxC)( (1 1 1) =x1+x2+x3= 1Eric Zivot (Copyright©2015)Portfolio Theory6 / 54

Portfolio return

R p,x=x?R= (xAxBxC)( (R A R B R C) =xARA+xBRB+xCRC

Portfolio expected return:

p,x=x?μ= (xAxBxX)( A B C) =xAμA+xBμB+xCμCEric Zivot (Copyright©2015)Portfolio Theory7 / 54

Computational tools

R formula:

t(x.vec)%*%mu.vec crossprod(x.vec, mu.vec)

Excel formula:

MMULT(transpose(xvec),muvec)

--Eric Zivot (Copyright©2015)Portfolio Theory8 / 54

Portfolio variance

2p,x=x?Σx

xAxBxC)(

2AσABσAC

ABσ2BσBC

ACσBCσ2C)

(x A x B x C) =x2Aσ2A+x2Bσ2B+x2Cσ2C + 2xAxBσAB+ 2xAxCσAC+ 2xBxCσBC

Portfolio distribution:

R p,x≂N(μp,x,σ2p,x)Eric Zivot (Copyright©2015)Portfolio Theory9 / 54

Computational tools

R formulas:

t(x.vec)%*%sigma.mat%*%x.vec

Excel formulas:

MMULT(TRANSPOSE(xvec),MMULT(sigma,xvec))

MMULT(MMULT(TRANSPOSE(xvec),sigma),xvec)

--Eric Zivot (Copyright©2015)Portfolio Theory10 / 54

Covariance Between 2 Portfolio Returns

2 portfolios:

x=( (x A x B x C) ),y=( (y A y B y C) x ?1= 1,y?1= 1

Portfolio returns:

R p,x=x?R R p,y=y?R

Covariance:

cov(Rp,x,Rp,y) =x?Σy =y?ΣxEric Zivot (Copyright©2015)Portfolio Theory11 / 54

Computational tools

R formula:

t(x.vec)%*%sigma.mat%*%y.vec

Excel formula:

MMULT(TRANSPOSE(xvec),MMULT(sigma,yvec))

MMULT(TRANSPOSE(yvec),MMULT(sigma,xvec))

--Eric Zivot (Copyright©2015)Portfolio Theory12 / 54

Derivatives of Simple Matrix Functions

LetAbe ann×nsymmetric matrix, and letxandybe ann×1 vectors. Then, ∂∂xn×1x ?y=( ((∂∂x 1x?y ∂∂x nx?y) ))=y,(1) ∂∂xn×1x ?Ax=( ((∂∂x 1x?Ax ∂∂x nx?Ax) ))= 2Ax.(2)Eric Zivot (Copyright©2015)Portfolio Theory13 / 54

Outline

1Portfolios with Three Risky Assets

Portfolio characteristics using matrix notation

Finding the global minimum variance portfolio

Finding efficient portfolios

Computing the efficient frontier

Mutual fund separation theorem again

Eric Zivot (Copyright©2015)Portfolio Theory14 / 54

Computing Global Minimum Variance Portfolio

Problem: Find the portfoliom= (mA,mB,mC)?that solves: min mA,mB,mCσ2p,m=m?Σms.t.m?1= 11Analytic solution using matrix algebra

2Numerical Solution in Excel Using the Solver (see

3firmExample.xls)

Eric Zivot (Copyright©2015)Portfolio Theory15 / 54

Analytic solution using matrix algebra

The Lagrangian is:

L(m,λ) =m?Σm+λ(m?1-1)

First order conditions (use matrix derivative results): 0 (3×1)=∂L(m,λ)∂m=∂m?Σm∂m+∂∂mλ(m?1-1) = 2·Σm+λ1 0 (1×1)=∂L(m,λ)∂λquotesdbs_dbs17.pdfusesText_23