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, Lintner Lintner, , Mossin Mossin 1965 Sharpe, ► 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 x ► = (value held in the i i- -th th asset)/(total portfolio asset)/(total portfolio ► = (value held in the i value) value) ► By definition portfolio weights must sum to one: By definition portfolio weights must sum to one: ► + + + + = … x x x x 1 − 1 2 n 1 n

Data needed for Portfolio Calculation Data needed for Portfolio Calculation ( ) E r ► Expected returns for asset i : ► Expected returns for asset i : i ( ) ► ► Variances of return for all assets i : Variances of return for all assets i : Var r i ► ► Covariances Covariances of returns for all pairs of assets I of returns for all pairs of assets I ( ) Cov r r , and j : and j : i j

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 Probability Return Market Economy Probability Return Normal Normal 1:3 10% 1:3 10% environment environment 1:3 30% Growth 1:3 30% Growth 1:3 -10% 10% Recession 1:3 - Recession ( ) ( ) ( ) ( ) E r = + + − = 1 3 0,30 1 3 0,10 1 3 0,10 0,10 ( ) ( ) ( ) Var r = − 2 + − 2 + − − 2 = ( ) 1 3 0,30 0,10 1 3 0,10 0,10 1 3 0,10 0,10 ( ) ( ) ( ) = 2 + 2 + − 2 = 1 3 0,20 1 3 0,0 1 3 0,20 0,0267

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 ► of the expected returns of assets 1 and 2. of the expected returns of assets 1 and 2. ( ) ( ) ( ) ( ) ( ) = + = + w w E r E r E r w E r w E r v 1 1 2 2 1 1 2 2 ( ) ( ) ( ) ⎡ ⎤ = + = + 2 2 2 2 2 Var r ( ) E r E r ( ) w E r - E r ⎣ ⎦ v v v 1 1 1 ( ) [ ] ( ) ( ) ( ) ( ) ⎡ ⎤ + + = 2 2 2 w w E r - E r 2 w E rr - E r E r ⎣ ⎦ 2 2 2 1 2 1 2 1 2 ( ) ( ) ( ) = + + 2 2 w w Var r w Var r 2 w Cov r r , 1 1 2 2 1 2 1 2

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- -weighted weighted the variances plus twice the cross covariance. covariance. ► If If ► ( ) Cov r r , ( ) ( ) μ = σ = ρ = 1 2 , , E r Var r σ σ v v v v 1,2 1 2 ρ then then Where is the 1,2 corellation μ = μ + μ w w v 1 1 2 2 σ = σ + σ + ρ σ σ 2 2 2 2 2 w w 2 w w v 1 1 2 2 1 2 1,2 1 2

Portfolio of Multiple Assets(1) Portfolio of Multiple Assets(1) = T ► ► We can write weights in form of matrix We can write weights in form of matrix 1 uw ► also the expected returns can be write in form ► also the expected returns can be write in form [ ] = μ μ μ … of vector of vector m , , , 1 2 n ⎛ ⎞ … c c 1,1 1, n ⎜ ⎟ ► and let C the covariance matrix and let C the covariance matrix ► = ⎜ � � � C ⎟ ⎜ ⎟ � c c ⎝ ⎠ where where = n ,1 n n , c Cov r r ( , ) , i j i j

Portfolio of Multiple Assets(2) Portfolio of Multiple Assets(2) ∃ C − 1 ► Because C is symmetric then ► Because C is symmetric then ► ► Then the expected return is equal with: Then the expected return is equal with: μ = T mw v ► Variance of returns is equal with: σ = 2 T wCw v

Proof Proof ⎛ ⎞ ∑ ∑ ( ) μ = = = μ = T E r E ⎜ w r ⎟ w q mw v v i i i i ⎝ ⎠ i i ⎛ ⎞ ⎛ ⎞ ∑ ∑ ∑ ( ) σ = = = = 2 ⎜ ⎟ Var r Var ⎜ w r ⎟ Cov w r w r v v i i i i j j ⎝ ⎠ ⎝ ⎠ i i j ∑ = = T w w c wCw i j i j , i j ,

Correlation Correlation n n , + ∑ σ = σ + σ + + σ ρ σ σ … 2 2 2 2 2 2 w w w 2 w w v 1 1 2 2 n n i j , i i j j i j , ρ = ∀ ∈ 1 , , if i j i n , i j n n , + ∑ σ = σ + σ + + σ σ σ … 2 2 2 2 2 2 w w w 2 w w v 1 1 2 2 n n i i j j i j , σ = σ + σ + + σ = σ … w w w v 1 1 2 2 n n v gen

Correlation(2) Correlation(2) ► An equally ► An equally- -weighted portfolio of weighted portfolio of n n assets: assets: If the 1 = σ = σ correlation is w i i gen equal with 1 n then between i ⎛ ⎞ ⎛ ⎞ 1 1 and j is linear σ = σ + + σ … ⎜ ⎟ ⎜ ⎟ connection; v gen gen ⎝ ⎠ ⎝ ⎠ n n if i grow then j 1 n grow to and 1 σ = σ = σ growth rate is n n v gen gen the same

Correlation(3) Correlation(3) 1 ρ = = σ = σ 0 if w i j , i i gen n 2 2 ⎛ ⎞ ⎛ ⎞ 1 1 σ = σ + + σ … ⎜ ⎟ ⎜ ⎟ v ⎝ gen ⎠ ⎝ gen ⎠ n n 1 n 2 ⎛ ⎞ 1 n σ = σ ⇒ σ = σ ⎜ ⎟ n n v gen v gen ⎝ ⎠ n

Diversification Diversification

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, 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. ► = ∑∑ n n n ∑ ( ) σ σ = 2 1 1 Minimize w w subject to w i v i j i j , = i 1 i j n ∑ ( ) = μ 2 w r i i v = i 1

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 portfolio frontier plane is called portfolio frontier plane is called ► The upper part of the portfolio frontier gives ► The upper part of the portfolio frontier gives efficient frontier portfolios efficient frontier portfolios ► Minimal variance portfolio ► Minimal variance portfolio

Portfolio frontier with two assets Portfolio frontier with two assets > = = − r r w w ► ► Let and let and Let and let and w 1 w 1 2 1 2 ( ) μ = + − ► Then Then ► wr 1 w r v 1 2 ( ) ( ) σ = σ + − σ + − σ 2 2 2 2 w 1 w 2 w 1 w 1 2 1,2 v μ For a given there is a unique that w For a given there is a unique that v determines the portfolio with expected return determines the portfolio with expected return μ − r = v 2 w − r r 1 2

Minimal variance portfolio Minimal variance portfolio = σ = 2 T T 1 uw wCw ► We use and ► We use and v ( ) λ = − λ T T L w , wCw uw ► ► Lagrange function Lagrange function λ − − λ = ⇒ = 1 2 wC u 0 w uC T u 2 − λ 1 uC 2 ⇒ = − ⇒ = 1 T ⇒ λ = 1 uC u w − − 2 1 T 1 T uC u uC u

Minimal variance curve Minimal variance curve ⎡ − − μ − ⎤ − + ⎡ μ − − − ⎤ − 1 T 1 T 1 1 T 1 T 1 mC m uC m uC uC u mC u mC ⎣ ⎦ ⎣ ⎦ = v v w − − − − − 1 T 1 T 1 T 1 T uC u mC m uC m mC u = T mu mw Where v