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