Modern Portfolio Theory

Transcription

1 Modern Portfolio Theory

2 History of MPT 1952 Horowitz CAPM (Capital Asset Pricing Model) 1965 Sharpe, Lintner, Mossin APT (Arbitrage Pricing Theory) 1976 Ross

3 What is a portfolio? Italian word Portfolio weights indicate the fraction of the portfolio total value held in each asset x i = (value held in the i-th asset)/(total portfolio value) By definition portfolio weights must sum to one: x x x x = n 1 n

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

5 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)

6 Examples Market Economy Probability Return Normal environment 1:3 10% Growth 1:3 30% Recession 1:3-10% E( r ) = 13( 0,30) + 13( 0,10) + 13( 0,10) = 0,10 ( ) ( ) ( ) Var( r ) = 1 3 0,30 0, ,10 0, ,10 0,10 = ( ) ( ) ( ) = 13 0, , ,20 = 0,0267

7 Portfolio of two assets(1) The portfolio s s expected return is a weighted sum of the expected returns of assets 1 and 2. ( ) = ( ) + ( ) = ( ) + ( ) E r E r E r we r w E r v w w ( 2) 2 2 ( 2 ) ( ) Var( r ) = E r + E( r ) = w E r -E r + v v v ( ) ( ) 2-2 w ( )- ( ) ( ) [ ] + w E r E r + w E rr E r E r = ( ) ( ) 2 (, ) = w Var r + w Var r + w Cov r r w 2

8 Portfolio of two assets(2) The variance is the square-weighted sum of the variances plus twice the cross-weighted covariance. If Cov ( r ) ( ) ( ) 1, r2 μv = E rv, σv = Var rv, ρ1,2 = σσ then μ = wμ + w μ v σ = w σ + w σ + 2ww ρ σ σ v ,2 1 2 Where ρ 1,2 is the corellation

9 Portfolio of Multiple Assets(1) We can write weights in form of matrix 1 T = uw also the expected returns can be write in form of vector m = μ μ μ [ ],,, n 1 2 and let C the covariance matrix where ci, j = Cov( ri, rj ) C c1,1 c1, n = cn,1 c n, n

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

11 σ μ Proof = E r = E wr = wq = mw ( ) μ T v v i i i i i i = Var ( r ) = Var w r = Cov w r w r = 2 v v i i i i j j i i j = wwc = i, j i j i, j wcw T

12 Correlation nn, v = w1 1 + w w + n n 2 i, jwi iwj j i, j σ σ σ σ ρ σ σ if ρ = 1 i, j i, n i, j nn, v = w1 1 + w w + n n 2wi iwj j i, j σ σ σ σ σ σ σ = wσ + w σ + + w σ = σ v n n v gen

13 Correlation(2) An equally-weighted portfolio of n assets: 1 wi = σi = σgen n 1 1 σv = σgen + + σgen n 1 n 1 σv = n σ σ gen = gen n n 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

14 Correlation(3) 1 if ρ = 0 w = σ = σ n i, j i i gen 2 2 σ 1 σ 1 σ = + + v gen gen n 1 n n 1 n σ n = σ σ = σ n n 2 v gen v gen

15 Diversification

16 Diversification(2) If we have 3 element in our portfolio than the variance of portfolio is much lower

17 Diversification(3) Reducing risk with this technique is called diversification Generally the more different the assets are, the greater the diversification. The diversification effect is the reduction in portfolio standard deviation, compared with a simple linear combination of the standard deviations, that comes from holding two or more assets in the portfolio The size of the diversification effect depends on the degree of correlation

18 Optimal portfolio selection How to choose a portfolio? Minimize risk of a given expected return? Or Maximize expected return for a given risk. Minimize σ n n 2 v wiw jσ i, j i j = subject to () n 1 w = 1 i= 1 i ( 2) n i= 1 wr i i = μ v

19 Optimal portfolio selection (2)

20 Solving optimal portfolios graphically

21 Solving optimal portfolios The locus of all frontier portfolios in the plane is called portfolio frontier The upper part of the portfolio frontier gives efficient frontier portfolios Minimal variance portfolio

22 Portfolio frontier with two assets Let r1 > r2 and let and Then μ = + w1 ( 1 ) v wr w r 1 2 = w w2 1 = w σ ( 1 ) 2 ( 1 ) v = w σ + w σ + w w σ ,2 For a given μ there is a unique w that v determines the portfolio with expected return μv r2 w = r r 1 2

23 Minimal variance portfolio 1 = uw We use and T σ = wcw 2 T v Lagrange function (, ) L w λ = wcw λuw T T λ 2wC λu = 0 w = uc 2 1 = λ 2 1 T uc u λ = T uc u u T w = uc uc u 1 1 T

24 Minimal variance curve w = mc m μ uc m uc + μ uc u mc u mc 1 T 1 T 1 T 1 T uc u mc m uc m mc u 1 T 1 T 1 1 T 1 T 1 v v Where muv = mw T

25 Some examples in MATLAB μ = 0.2 σ = 0.25 ρ = ρ = ,2 2,1 μ = 0.13 σ = 0.28 ρ = ρ = ,3 3,2 μ = 0.17 σ = 0.20 ρ = ρ = ,3 3,1 m = u = [ ] [1 1 1] We calculate C and C 1 w 1 uc = = uc 1 u T [ ] w μv μv μ v

26 Using MATLAB

27 Examples in MATLAB(2) Frontcon function; with this function we can calculate some efficient portfolio [pkock, preturn, pweigths]= =frontcon(returns,cov,n,preturn,limits,groreturns,cov,n,preturn,limits,gro up,grouplimits)

28 Examples in MATLAB pkock covariances of the returned portfolios preturn returns of the returned portfolios pweighs weighs of the returned portfolios returns the stocks return cov covariance matrix n number of portfolios group, group limits min and max weigh Other functions: portalloc, portopt