Calculating the Variance – Covariance matrix MGT 4850 Spring 2007 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 Γ T ) Returns - column vector E (row vector E T ) Portfolio return E T Γ 25 stocks portfolio variance Γ T S Γ Γ T (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 R T for the calculation of S matrix R T R/10 → S (p ).

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

Single Index Model