Efficient Portfolios with no short-sale restriction

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Efficient Portfolios with no short-sale
restriction

• MGT 4850
• Spring 2009
• University of Lethbridge

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Portfolio return

• One period return gt ET G
• 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)

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Market risk

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

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Copy versus Functions

• Transpose of a vector or matrix created with the
  function changes with change in the origin, e.g.
  portfolio variance GTS G 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

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Overview

• CAPM and the risk-free asset
• CAPM with risk free asset
• Blacks (1972) zero beta CAPM
• The objective is to learn how to calculate
• Efficient Portfolios
• Efficient Frontier

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Notation

• Weights a column vector G (Nx1) its transpose
  GT is a row vector (1xN)
• Returns - column vector E (Nx1) its transpose
  ET is a row vector (1xN)
• Portfolio return ET G or GT E
• 25 stocks portfolio variance GTS G
• GT(1x25)S(25x25) G(25x1)
• To calculate portfolio variance we need the
  variance/covariance matrix S.

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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

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Theorems on Efficient Portfolio

• Solve simultaneously for x and y
• x y10 
• x - y2
• Arbitrary chosen constant c

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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

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Calculating the efficient frontier

• Only four risky assets

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Find two efficient portfolios

• Minimum Variance
• Market portfolio
• Use proposition two to establish the whole
  envelope
• CML
• SML

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Zero beta CAPM Black (1972)
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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)

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Simultaneous equations

• E(r1 )-c z1s11 z2s12 z3s13 z4s14
• E(r2 )-c z1s21 z2s22 z3s23 z4s24
• E(r3 )-c z1s31 z2s32 z3s33 z4s34
• E(r4 )-c z1s41 z2s42 z3s43 z4s44
• The vector z assigns proportions to each asset.
  Find the weights as a proportion of the sum.

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The Solution is an envelope portfolio

• Vector z is
• z S-1 R-c
• Vector z solves for the weights x
• xx1,.. xN

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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.

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Weights portfolio X c0
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Weights portfolio Y c0.04