Modern Portfolio Modern Portfolio TheoryTheory

History of MPTHistory of MPT

►1952 Horowitz1952 Horowitz►CAPM (Capital Asset Pricing Model) CAPM (Capital Asset Pricing Model)

1965 Sharpe, Lintner, Mossin1965 Sharpe, Lintner, Mossin►APT (Arbitrage Pricing Theory) 1976 RossAPT (Arbitrage Pricing Theory) 1976 Ross

What is a portfolio?What is a portfolio?

► Italian wordItalian word► Portfolio weights indicate the fraction of Portfolio weights indicate the fraction of

the portfolio total value held in each assetthe portfolio total value held in each asset► = (value held in the i-th asset)/(total = (value held in the i-th asset)/(total

portfolio value)portfolio value)► By definition portfolio weights must sum to By definition portfolio weights must sum to

one:one:

1 2 1 1n nx x x x

ix

Data needed for Portfolio Data needed for Portfolio CalculationCalculation

► Expected returns for asset i : Expected returns for asset i :

► Variances of return for all assets i : Variances of return for all assets i :

► Covariances of returns for all pairs of Covariances of returns for all pairs of assets Iassets I

and j : and j :

iVar r

iE r

,i jCov r r

Where do we obtain this Where do we obtain this data?data?

►Compute them from knowledge of the probability distribution of returns (population parameters)

►Estimate them from historical sample data using statistical techniques (sample statistics)

ExamplesExamplesMarket Market

EconomyEconomy ProbabilityProbability ReturnReturn

Normal Normal environmentenvironment 1:31:3 10%10%

GrowthGrowth 1:31:3 30%30%

RecessionRecession 1:31:3 -10%-10%

1 3 0,30 1 3 0,10 1 3 0,10 0,10E r

2 2 2

2 2 2

( ) 1 3 0,30 0,10 1 3 0,10 0,10 1 3 0,10 0,10

1 3 0,20 1 3 0,0 1 3 0,20 0,0267

Var r

Portfolio of two assets(1)Portfolio of two assets(1)

► The portfolio’s expected return is a weighted The portfolio’s expected return is a weighted sum of the expected returns of assets 1 and 2. sum of the expected returns of assets 1 and 2.

1 1 2 2 1 1 2 2vE r E r E r w E r w E rw w

22 2 2 2

1 1 1

22 2

2 2 2 1 2 1 2 1 2

2 2

1 1 2 2 1 2 1 2

( ) ( ) -

- 2 -

2 ,

v v vVar r E r E r w E r E r

w E r E r w E rr E r E r

w Var r w Var r w Cov r r

w

w

Portfolio of two assets(2) Portfolio of two assets(2)

► The variance is the square-weighted The variance is the square-weighted sum of the variances plus twice the sum of the variances plus twice the cross-weighted covariance.cross-weighted covariance.

► If If

1 21,2

1 2

,, ,v v v v

Cov r rE r Var r

thenthen

1 1 2 2

2 2 2 2 21 1 2 2 1 2 1,2 1 22

v

v

w w

w w w w

Where is the corellation

1,2

Portfolio of Multiple Assets(1)Portfolio of Multiple Assets(1)

►We can write weights in form of matrix We can write weights in form of matrix

► also the expected returns can be write in also the expected returns can be write in form of vector form of vector

► and let C the covariance matrix and let C the covariance matrix

where where

1 Tuw

1 2, , , nm

1,1 1,

,1 ,

n

n n n

c c

C

c c

, ( , )i j i jc Cov r r

Portfolio of Multiple Assets(2)Portfolio of Multiple Assets(2)

► Because C is symmetric then Because C is symmetric then

► Then the expected return is equal with:Then the expected return is equal with:

►Variance of returns is equal with:

1C

Tv mw

2 Tv wCw

ProofProof

Tv v i i i i

i i

E r E w r w q mw

2

,,

v v i i i i j ji i j

Ti j i j

i j

Var r Var w r Cov w r w r

w w c wCw

CorrelationCorrelation

,2 2 2 2 2 21 1 2 2 ,

,

2n n

v n n i j i i j ji j

w w w w w

,

,2 2 2 2 2 21 1 2 2

,

1 , ,

2

i j

n n

v n n i i j ji j

if i j i n

w w w w w

1 1 2 2v n n v genw w w

Correlation(2)Correlation(2)

► An equally-weighted portfolio of An equally-weighted portfolio of n n assets:assets:

1

1

1 1

1

i i gen

v gen genn

v gen gen

wn

n n

nn

If the correlation is equal with 1 then between i and j is linear connection;if i grow then j grow to and growth rate is the same

Correlation(3)Correlation(3)

,

10i j i i genif w

n

2 2

1

1 1v gen gen

nn n

21

v gen v gen

nnn n

DiversificationDiversification

Diversification(2)Diversification(2)

If we have 3 element in our portfolio than the variance of portfolio is much lower

Diversification(3)Diversification(3)

► Reducing risk with this technique is called Reducing risk with this technique is called diversification diversification

►Generally the more different the assets are, Generally the more different the assets are, the greater the diversification.the greater the diversification.

► The diversification effect is the reduction in The diversification effect is the reduction in portfolio standard deviation, compared with portfolio standard deviation, compared with a simple linear combination of the standard a simple linear combination of the standard deviations, that comes from holding two or deviations, that comes from holding two or more assets in the portfoliomore assets in the portfolio

►The size of the diversification effect depends on the degree of correlation

Optimal portfolio selectionOptimal portfolio selection

►How to choose a portfolio?How to choose a portfolio?

► Minimize risk of a given expected return? OrMinimize risk of a given expected return? Or► Maximize expected return for a given risk.Maximize expected return for a given risk.

2,

n n

v i j i ji j

Minimize w w

1

1

1 1

2

n

ii

n

i i vi

subject to w

w r

Optimal portfolio selection Optimal portfolio selection (2)(2)

Solving optimal portfolios Solving optimal portfolios “graphically” “graphically”

Solving optimal portfoliosSolving optimal portfolios

►The locus of all frontier portfolios in The locus of all frontier portfolios in the plane is called the plane is called portfolio frontierportfolio frontier

►The upper part of the portfolio frontier The upper part of the portfolio frontier gives gives efficient frontierefficient frontier portfolios portfolios

►Minimal variance portfolio Minimal variance portfolio

Portfolio frontier with two Portfolio frontier with two assetsassets

► Let and let and Let and let and ► ThenThen

For a given there is a unique that For a given there is a unique that determines the portfolio with expected determines the portfolio with expected return return

1 2r r1w w 2 1w w

1 21v wr w r

2 2 2 21 2 1,21 2 1v w w w w

2

1 2

v rw

r r

v w

Minimal variance portfolioMinimal variance portfolio

►We use and We use and

► Lagrange function Lagrange function

1 Tuw 2 Tv wCw

, T TL w wCw uw

12 02

TwC u w uCu

112

TuC u

1

2TuC u

1

1 T

uCw

uC u

Minimal variance curve Minimal variance curve 1 1 1 1 1 1

1 1 1 1

T T T Tv v

T T T T

mC m uC m uC uC u mC u mCw

uC u mC m uC m mC u

WhereT

vmu mw

Some examples in MATLABSome examples in MATLAB

1 1 1,2 2,1

2 2 2,3 3,2

3 3 1,3 3,1

0.2 0.25 0.3

0.13 0.28 0.0

0.17 0.20 0.15

0.2 0.13 0.17

[1 1 1]

m

u

1C We calculate C and

1

10.091 0.0679 1.0232

T

uCw

uC u

1.578 8.614 0.845 2.769 1.422 11.384v v vw

Using MATLABUsing MATLAB

Examples in MATLAB(2)Examples in MATLAB(2)

►Frontcon Frontcon function; with this function function; with this function we can calculate some efficient we can calculate some efficient portfolioportfolio

►[[pkock, preturn, pweigthspkock, preturn, pweigths]=]===frontconfrontcon((returns,Cov,n,preturn,limitreturns,Cov,n,preturn,limits,group,grouplimitss,group,grouplimits))

Examples in MATLABExamples in MATLAB

► pkock – pkock – covariances of the returned covariances of the returned portfolios portfolios

► preturn – preturn – returns of the returned portfoliosreturns of the returned portfolios► pweighs – pweighs – weighs of the returned portfoliosweighs of the returned portfolios► returns – returns – the stocks returnthe stocks return ► cov – cov – covariance matrixcovariance matrix ► n – n – number of portfoliosnumber of portfolios► group, group limits – group, group limits – min and max weighmin and max weigh►Other functions: Other functions: portalloc, portoptportalloc, portopt