Efficient Portfolios with no
short-sale restriction
MGT 4850
Spring 2009
University of Lethbridge
Portfolio return
• One period return => ET 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 ΓTS 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 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.
Covariance of two portfolios
• Expected return of portfolio X is a column
vector Ex (Nx1)
• Expected return of portfolio Y is a column
vector Ey (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 = XTS 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(r3) – c = Sz
• This solution produces x:
z= S-1{ E(r) – c }
x= {x1,….. Xn }
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)
Finding Envelope Portfolios
12%
10%
Portfolio mean return
x, the tangency portfolio
given c
8%
6%
c
Zero beta portfolio
4%
2%
0%
0%
10%
20%
30%
40%
50%
Portfolio standard deviation
60%
70%
80%
90%
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(r1 )-c= z1σ11 + z2σ12 + z3σ13 + z4σ14
E(r2 )-c= z1σ21 + z2σ22 + z3σ23 + z4σ24
E(r3 )-c= z1σ31 + z2σ32 + z3σ33 + z4σ34
E(r4 )-c= z1σ41 + z2σ42 + z3σ43 + z4σ44
• 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={x1,….. xN}
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