Efficient Portfolios without short sales MGT 4850 Spring 2007 University of Lethbridge

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.

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

Simultaneous Equations Solve simultaneously for x and y: x + y=10 x − y=2 CAPM with risk free asset – max slope for the tangent portfolio Black’s zero beta CAPM –finding graphically zero beta portfolio

Calculating the efficient frontier Only four risky assets

Short sales allowed from ch. 9

Find two efficient portfolios The product of the inverse S matrix and vector of returns will serve as a starting point to calculate weights – each entry of the vector is divided by the sum of all entries Second portfolio is found in the same way but the inverse S is multiplied by the vector of returns minus a constant.

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

Efficient Portfolio no short sales Using Solver as discussed in previous class Solver and VBA to built the efficient frontier