Modern Portfolio Modern Portfolio Theory Theory
Dec 30, 2015
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