46 Chapter 5 The Mathematics of Diversification

Chapter The Mathematics of Diversification    ρ21 Σ = ( ρij ) =  ρ31     ρ N1  For i, j = 1, , N ρ12 ρ13 ρ23 ρ32 ρ1N ρ2 N ρ3 N   ρN2 ρN3    ÷ ÷ ÷ ÷ ÷ ÷  Introduction ◆ The reason for portfolio theory mathematics: • To show why diversification is a good idea • To show why diversification makes sense logically Introduction (cont’d) ◆ Harry Markowitz’s efficient portfolios: • Those portfolios providing the maximum return for their level of risk • Those portfolios providing the minimum risk for a certain level of return Introduction ◆ A portfolio’s performance is the result of the performance of its components • The return realized on a portfolio is a linear combination of the returns on the individual investments • The variance of the portfolio is not a linear combination of component variances Return ◆ The expected return of a portfolio is a weighted average of the expected returns of the components: n %  E ( R% p ) = ∑  xi E ( Ri )  i =1 where xi = proportion of portfolio invested in security i and n ∑x i =1 i =1 Variance ◆ ◆ ◆ ◆ ◆ Introduction Two-security case Minimum variance portfolio Correlation and risk reduction The n-security case Introduction ◆ Understanding portfolio variance is the essence of understanding the mathematics of diversification • The variance of a linear combination of random variables is not a weighted average of the component variances Introduction (cont’d) ◆ For an n-security portfolio, the portfolio variance is: n n σ = ∑∑ xi x j ρijσ iσ j p i =1 j =1 where xi = proportion of total investment in Security i ρij = correlation coefficient between Security i and Security j Two-Security Case ◆ For a two-security portfolio containing Stock A and Stock B, the variance is: σ = x σ + x σ + x A xB ρ ABσ Aσ B p A A B B Two Security Case (cont’d) Example Assume the following statistics for Stock A and Stock B: Stock A Stock B Expected return 015 020 Variance 050 060 Standard deviation 224 245 Weight 40% 60% Correlation coefficient 50 10 Global Minimum Variance Portfolio ◆ In a similar fashion, we can solve for the global minimum variance portfolio: µ* 1' V µ = 1'V −1 −1 σ = * (1'V 1) −1 −1 with w* = 1'V V −1 −1 The global minimum variance portfolio is the efficient frontier portfolio that displays the absolute minimum variance 31 Another Way to Derive the MeanVariance Efficient Portfolio Frontier ◆ Make use of the following property: if two portfolios lie on the efficient frontier, any linear combination of these portfolios will also lie on the frontier Therefore, just find two meanvariance efficient portfolios, and compute/plot the mean and standard deviation of various linear combinations of these portfolios 32 33 34 Some Excel Tips ◆ To give a name to an array (i.e., to name a matrix or a vector): • Highlight the array (the numbers defining the matrix) • Click on ‘Insert’, then ‘Name’, and finally ‘Define’ and type in the desired name 35 Excel Tips (Cont’d) ◆ To compute the inverse of a matrix previously named (as an example) “V”: • Type the following formula: ‘=minverse(V)’ and click ENTER • Re-select the cell where you just entered the formula, and highlight a larger area/array of the size that you predict the inverse matrix will take • Press F2, then CTRL + SHIFT + ENTER 36 Excel Tips (end) ◆ To multiply two matrices named “V” and “W”: • Type the following formula: ‘=mmult(V,W)’ and click ENTER • Re-select the cell where you just entered the formula, and highlight a larger area/array of the size that you predict the product matrix will take • Press F2, then CTRL + SHIFT + ENTER 37 Single-Index Model Computational Advantages ◆ The single-index model compares all securities to a single benchmark • An alternative to comparing a security to each of the others • By observing how two independent securities behave relative to a third value, we learn something about how the securities are likely to behave relative to each other 38 Computational Advantages (cont’d) ◆ A single index drastically reduces the number of computations needed to determine portfolio variance • A security’s beta is an example: % COV ( R% i , Rm ) βi = σ m2 where R% = return on the market index m σ m2 = variance of the market returns R% i = return on Security i 39 Portfolio Statistics With the Single-Index Model ◆ Beta of a portfolio: n β p = ∑ xi β i ◆ i =1 Variance of a portfolio: σ 2p = β p2σ m2 + σ ep2 ≈ β p2σ m2 40 Proof Ri = R f + βi ( Rm − R f ) + ei n n n R p = ∑ xi Ri =R f + ∑ xi β i ( Rm − R f ) + ∑ xi ei i =1 i =1 i =1 43 23 βp n ep n n R p = R f + ∑ xi β i Rm − ∑ xi β i R f + ∑ xi ei i =1 i =1 43 43 1i =12 βp βp ep   n n   σ p2 =  ∑ xi β i  σ m2 + ∑ xi2σ ie2 = β p2σ m2 + σ ep2 ≈ β p2σ m2 i =1 i =1 1 43   β p  41 Portfolio Statistics With the Single-Index Model (cont’d) ◆ Variance of a portfolio component: σ = β σ +σ i ◆ i m ei Covariance of two portfolio components: σ AB = β A β Bσ m2 42 Proof Ri = R f + βi Rm − βi R f + ei σ i2 = βi2σ m2 + σ ei2 σ A, B = Cov( RA , RB ) = Cov( R f + β A Rm − β A R f + eA , R f + β B Rm − β B R f + eB ) σ A, B = Cov( β A Rm + eA , β B Rm + eB ) σ A, B = Cov( β A Rm , β B Rm ) + Cov(eA , β B Rm ) + Cov ( β A Rm , eB ) + Cov(eA , eB ) σ A, B = β A β B Cov( Rm , Rm ) = β A β Bσ m2 43 Multi-Index Model ◆ A multi-index model considers independent variables other than the performance of an overall market index • Of particular interest are industry effects – Factors associated with a particular line of business – E.g., the performance of grocery stores vs steel companies in a recession 44 Multi-Index Model (cont’d) ◆ The general form of a multi-index model: % % % % R% i = + β im I m + β i1 I1 + β i I + + β in I n where = constant I% = return on the market index m I% j = return on an industry index β ij = Security i's beta for industry index j β im = Security i's market beta R% i = return on Security i 45 ... portfolio variance is the essence of understanding the mathematics of diversification • The variance of a linear combination of random variables is not a weighted average of the component variances... linear combination of the returns on the individual investments • The variance of the portfolio is not a linear combination of component variances Return ◆ The expected return of a portfolio is... for their level of risk • Those portfolios providing the minimum risk for a certain level of return Introduction ◆ A portfolio’s performance is the result of the performance of its components