Modern Portfolio Theory Modern Portfolio Theory History of MPT - - PowerPoint PPT Presentation

Modern Portfolio Theory Modern Portfolio Theory

History of MPT History of MPT

► ►1952 Horowitz

1952 Horowitz

► ►CAPM (Capital Asset Pricing Model)

CAPM (Capital Asset Pricing Model) 1965 Sharpe, 1965 Sharpe, Lintner Lintner, , Mossin Mossin

► ►APT (Arbitrage Pricing Theory) 1976 Ross

APT (Arbitrage Pricing Theory) 1976 Ross

What is a portfolio? What is a portfolio?

► ► Italian word

Italian word

► ► Portfolio weights indicate the fraction of the

Portfolio weights indicate the fraction of the portfolio total value held in each asset portfolio total value held in each asset

► ►

= (value held in the = (value held in the i i-

• th

th asset)/(total portfolio asset)/(total portfolio value) value)

► ► By definition portfolio weights must sum to one:

By definition portfolio weights must sum to one:

1 2 1

1

n n

x x x x

−

+ + + + = …

i

x

Data needed for Portfolio Calculation Data needed for Portfolio Calculation

► ► 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

Covariances of returns for all pairs of assets I

• f returns for all pairs of assets I

and j : and j :

( )

i

Var r

( )

i

E r

( )

,

i j

Cov r r

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

►Compute them from knowledge of the

probability distribution of returns (population parameters)

►Estimate them from historical sample data

using statistical techniques (sample statistics)

Examples Examples

Market Economy Market Economy Probability Probability Return Return Normal Normal environment environment

1:3 1:3 10% 10%

Growth Growth

1:3 1:3 30% 30%

Recession Recession

1:3 1:3

• 10%

10%

( ) ( ) ( ) ( )

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

The portfolio’ ’s expected return is a weighted sum s expected return is a weighted sum

• f the expected returns of assets 1 and 2.
• f the expected returns of assets 1 and 2.

( ) ( ) ( ) ( ) ( )

1 1 2 2 1 1 2 2 v

E r E r E r w E r w E r

w w

= + = +

( ) ( )

( )

( )

( ) ( ) ( ) ( )

[ ]

( ) ( ) ( )

2 2 2 2 2 1 1 1 2 2 2 2 2 2 1 2 1 2 1 2 2 2 1 1 2 2 1 2 1 2

( ) ( )

• 2
• 2

,

v v v

Var 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

The variance is the square-

• weighted sum of

weighted sum of the variances plus twice the cross the variances plus twice the cross-

• weighted

weighted covariance. covariance.

► ► If

If

( ) ( ) ( )

1 2 1,2 1 2

, , ,

v v v v

Cov r r E r Var r μ σ ρ σ σ = = =

then then

1 1 2 2 2 2 2 2 2 1 1 2 2 1 2 1,2 1 2

2

v v

w w w w w w μ μ μ σ σ σ ρ σ σ = + = + +

Where is the corellation

1,2

ρ

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 form

also the expected returns can be write in form

• f vector
• f vector

► ► and let C the covariance matrix

and let C the covariance matrix where where 1

T

uw =

[ ]

1 2

, , ,

n

m μ μ μ = …

1,1 1, ,1 , n n n n

c c C c c ⎛ ⎞ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ …

• ,

( , )

i j i j

c 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:

1

C − ∃

T v

mw μ =

2 T v

wCw σ =

Proof Proof

( )

T v v i i i i i i

E r E w r w q mw μ μ ⎛ ⎞ = = = = ⎜ ⎟ ⎝ ⎠

∑ ∑

( )

2 , , v v i i i i j j i i j T i j i j i j

Var r Var w r Cov w r w r w w c wCw σ ⎛ ⎞ ⎛ ⎞ = = = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = =

∑ ∑ ∑ ∑

Correlation Correlation

, 2 2 2 2 2 2 1 1 2 2 , ,

2

n n v n n i j i i j j i j

w w w w w σ σ σ σ ρ σ σ = + + + +∑ …

, , 2 2 2 2 2 2 1 1 2 2 ,

1 , , 2

i j n n v n n i i j j i j

if i j i n w w w w w ρ σ σ σ σ σ σ = ∀ ∈ = + + + +∑ …

1 1 2 2 v n n v gen

w w w σ σ σ σ σ = + + + = …

Correlation(2) Correlation(2)

► ► An equally

An equally-

• weighted portfolio of

weighted portfolio of n n assets: assets:

1

1 1 1 1

i i gen v gen gen n v gen gen

w n n n n n σ σ σ σ σ σ σ σ = = ⎛ ⎞ ⎛ ⎞ = + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = = …

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)

,

1

i j i i gen

if w n ρ σ σ = = =

2 2 1

1 1

v gen gen n

n n σ σ σ ⎛ ⎞ ⎛ ⎞ = + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ …

2

1

v gen v gen

n n n n σ σ σ σ ⎛ ⎞ = ⇒ = ⎜ ⎟ ⎝ ⎠

Diversification Diversification

Diversification(2) Diversification(2)

If we have 3 element in our portfolio than the variance

• f 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, the

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

► ► The diversification effect is the reduction in

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

► The size of the diversification effect depends on

the degree of correlation

Optimal portfolio selection Optimal portfolio selection

► ► How to choose a portfolio?

How to choose a portfolio?

► ► Minimize risk of a given expected return? Or

Minimize 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 j i j

Minimize w w σ σ = ∑∑

( ) ( )

1 1

1 1 2

n i i n i i v i

subject to w w r μ

= =

= =

∑ ∑

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

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

Solving optimal portfolios Solving optimal portfolios

► ►The locus of all frontier portfolios in the

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

portfolio frontier

► ►The upper part of the portfolio frontier gives

The upper part of the portfolio frontier gives

efficient frontier efficient frontier portfolios

portfolios

► ►Minimal variance portfolio

Minimal variance portfolio

Portfolio frontier with two assets Portfolio frontier with two assets

► ► Let and let and

Let and let and

► ► Then

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

1 2

r r >

1

w w =

2

1 w w = −

( )

1 2

1

v

wr w r μ = + −

( ) ( )

2 2 2 2 1 2 1,2

1 2 1

v

w w w w σ σ σ σ = + − + −

2 1 2 v

r w r r μ − = −

v

μ w

Minimal variance portfolio Minimal variance portfolio

► ► We use and

We use and

► ► Lagrange function

Lagrange function

1

T

uw =

2 T v

wCw σ =

( )

,

T T

L w wCw uw λ λ = −

1

2 2

T

wC u w uC u λ λ

−

− = ⇒ =

1

1 2

T

uC u λ

−

⇒ =

1

2

T

uC u λ

−

⇒ =

1 1 T

uC w uC u

− −

⇒ =

Minimal variance curve Minimal variance curve

1 1 1 1 1 1 1 1 1 1 T T T T v v T T T T

mC m uC m uC uC u mC u mC w uC u mC m uC m mC u μ μ

− − − − − − − − − −

⎡ ⎤ ⎡ ⎤ − + − ⎣ ⎦ ⎣ ⎦ = −

Where

T v

mu mw =

Some examples in MATLAB Some 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 μ σ ρ ρ μ σ ρ ρ μ σ ρ ρ = = = = = = = = = = = =

[ ]

0.2 0.13 0.17 [1 1 1] m u = =

1

C −

We calculate C and

[ ]

1 1

0.091 0.0679 1.0232

T

uC w uC u

− −

= = −

1.578 8.614 0.845 2.769 1.422 11.384

v v v

w μ μ μ ⎡ ⎤ ≅ − − − + ⎣ ⎦

Using MATLAB Using MATLAB

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

► ►Frontcon

Frontcon function; with this function we

function; with this function we can calculate some efficient portfolio can calculate some efficient portfolio

► ►[

[ pkock pkock, , preturn preturn, , pweigths pweigths]= ]= = = frontcon

frontcon(

(returns,Cov,n,preturn,limits,gro returns,Cov,n,preturn,limits,gro up,grouplimits up,grouplimits) )

Examples in MATLAB Examples in MATLAB

► ► pkock

pkock – – covariances covariances of the returned portfolios

• f the returned portfolios

► ► preturn

preturn – – returns of the returned portfolios returns of the returned portfolios

► ► pweighs

pweighs – – weighs of the returned portfolios weighs of the returned portfolios

► ► returns

returns – – the stocks return the stocks return

► ► cov

cov – – covariance matrix covariance matrix

► ► n

n – – number of portfolios number of portfolios

► ► group, group limits

group, group limits – – min and max weigh min and max weigh

► ► Other functions:

Other functions: portalloc

portalloc, , portopt portopt