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 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 (1x 25 )*S( 25 x25)* 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
A T for the calculation of S matrix
• A T A/(M1) → S (p. 292).

Excess Returns
(fixing the row cell A$23)

A T A/(M1) → 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.
1 T

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

Single Index Model