Efficient Portfolios with no short-sale restriction MGT 4850 Spring 2009 University of Lethbridge

Portfolio return One period return => E T Γ Matrix of 60 monthly returns for 30 industry portfolios (60x30) Column vector of equal weights (30x1) We get 60 period returns of a portfolio of equally weighted industries (column vector)

Market risk Calculate covariance of portfolio returns with market return Calculate variance of market returns Beta of the protfolio

Copy versus Functions Transpose of a vector or matrix created with the function changes with change in the origin, e.g. portfolio variance Γ T S Γ will recalculate correctly if we change weights in the original vector of weights. Another way to avoid this error is to check “validate” when we copy and “paste special” - “transpose”

Overview CAPM and the risk-free asset –CAPM with risk free asset –Black’s (1972) zero beta CAPM The objective is to learn how to calculate: –Efficient Portfolios –Efficient Frontier

Notation Weights – a column vector Γ (Nx1); it’s transpose Γ T is a row vector (1xN) Returns - column vector E (Nx1); it’s transpose E T is a row vector (1xN) Portfolio return E T Γ or Γ T E 25 stocks portfolio variance Γ T S Γ Γ T (1x25)*S(25x25)* Γ(25x1) To calculate portfolio variance we need the variance/covariance matrix S.

Covariance of two portfolios Expected return of portfolio X is a column vector E x (Nx1) Expected return of portfolio Y is a column vector E y (Nx1) (note you have the same number of returns, whether the portfolio have the same number of assets or not) Variance-covariance matrix S (NxN) Covariance x,y = X T S Y

Theorems on Efficient Portfolio Solve simultaneously for x and y: x + y=10 x − y=2 Arbitrary chosen constant c:

Portfolio on the envelope Vector z solves the system of simultaneous linear equations: E(r 3 ) – c = Sz This solution produces x: z= S -1 { E(r) – c } x= {x 1, ….. X n }

Calculating the efficient frontier Only four risky assets

Find two efficient portfolios Minimum Variance Market portfolio Use proposition two to establish the whole envelope CML SML

Zero beta CAPM Black (1972)

Notation R is column vector of expected returns S var/cov matrix c – arbitrary constatnt z – vector that solves the system of linear equations R-c = Sz Solving for z needs inverse matrix of S (S -1 )

Simultaneous equations E(r 1 )-c= z 1 σ 11 + z 2 σ 12 + z 3 σ 13 + z 4 σ 14 E(r 2 )-c= z 1 σ 21 + z 2 σ 22 + z 3 σ 23 + z 4 σ 24 E(r 3 )-c= z 1 σ 31 + z 2 σ 32 + z 3 σ 33 + z 4 σ 34 E(r 4 )-c= z 1 σ 41 + z 2 σ 42 + z 3 σ 43 + z 4 σ 44 The vector z assigns proportions to each asset. Find the weights as a proportion of the sum.

The Solution is an envelope portfolio Vector z is: z= S -1 {R-c} Vector z solves for the weights x x={x 1,….. x N }

Calculating two envelope portfolios (p.268) Choose arbitrary a constant; solve for 0 constant also: Weight vector is calculated from z by dividing each entry of z by the sum of all entries of the z vector.

Weights portfolio X c=0

Weights portfolio Y c=0.04