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Calculating the Variance –Covariance matrix

Calculating the Variance –Covariance matrix. MGT 4850 Spring 2009 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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  1. Calculating the Variance –Covariance matrix MGT 4850 Spring 2009 University of Lethbridge

  2. 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.

  3. 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.

  4. variance/covariance matrix • Using Excess Returns • Return data for variance-covariance 295 • Excess return matrix A and its transpose AT for the calculation of S matrix • AT A/(M-1) → S (p. 292).

  5. Excess Returns (fixing the row cell A$23)

  6. AT A/(M-1) → S (p. 292).

  7. Population vs. Sample

  8. VBA (optional) 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

  9. Array function 298 (Shift+CTRL+Enter)

  10. variance/covariance matrix 299 • Offset Function → returns a reference to a range that is a given number of rows and columns for a given reference

  11. Minvarprt=1.S-1/1. S-1.1T

  12. Eff P=S-1[E(r)-c]/Sum{S-1[E(r)-c]}

  13. Single Index Model

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