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 variance ΓTS Γ
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