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Calculating the Variance –Covariance matrix . MGT 4850 Spring 2008 University of Lethbridge. Efficient Portfolios. Efficient frontier Black (1972) – convex combination of any two efficient portfolios, e.g. if we have two efficient portfolios we can find the whole efficient frontier.

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Calculating the Variance –Covariance matrix

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Calculating the Variance –Covariance matrix

MGT 4850

Spring 2008

University of Lethbridge

Efficient Portfolios

• Efficient frontier

• Black (1972) – convex combination of any two efficient portfolios, e.g. if we have two efficient portfolios we can find the whole efficient frontier.

• Minimize portfolio variance, subject to defined return and sum of weights equal 1.

Transpose and Multiplication

• Weights - column vector Γ (row vector ΓT)

• Returns - column vector E (row vector ET)

• Portfolio return ET Γ

• 25 stocks portfolio varianceΓTSΓ

ΓT(1x25)*S(25x25)* Γ(25x1)

• To calculate portfolio variance we need the variance/covariance matrix S.

variance/covariance matrix

• Using Excess Returns

• Return data for variance-covariance p. 151

• Excess return matrix R and its transpose RT for the calculation of S matrix

• RT R/10 → S (p. 153-154).

VBA (skip for now)

Function VarCovar(rng As Range) As Variant

Dim i As Integer

Dim j As Integer

Dim numCols As Integer

numCols = rng.Columns.Count

Dim matrix() As Double

ReDim matrix(numCols - 1, numCols - 1)

For i = 1 To numCols

For j = 1 To numCols

matrix(i - 1, j - 1) = Application.WorksheetFunction.Covar(rng.Columns(i), rng.Columns(j))

Next j

Next i

VarCovar = matrix

End Function

variance/covariance matrix

• Offset Function → returns a reference to a range that is a given number of rows and columns for a given reference