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. 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 variance ΓTS Γ 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 295 Excess return matrix A and its transpose AT for the calculation of S matrix AT A/(M-1) → S (p. 292).

Excess Returns (fixing the row cell A$23)

AT A/(M-1) → S (p. 292).

Population vs. Sample

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

Array function 298 (Shift+CTRL+Enter)

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

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

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

Single Index Model