Efficient Portfolios without short sales

MGT 4850

Spring 2007

University of Lethbridge

Notation

• Weights – a column vector Γ (Nx1); it’s transpose ΓT is a row vector (1xN)

• Returns - column vector E (Nx1); it’s transpose ET is a row vector (1xN)

• Portfolio return ET Γ or ΓT E• 25 stocks portfolio variance ΓTS Γ ΓT(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