1. Portfolio Statistics  Portfolio Expected Return: E(r p ) = w T r  Portfolio Variance:  2 p = w T  w  Sum of portfolio weights: = w T 1 –where w is a 1xn vector (column) of weights, –r is a 1xn vector of expected return on the n assets, –  is a nxn matrix containing the variances and covariances –1 is a 1xn vector of ones  w T is the transpose of w  Note that the resulting portfolio expected return E(r p ), portfolio variance (  2 p ) and sum of portfolio weights are all scalars.

2. Matrix Functions in Excel

3. Scalar versus Matrix  Scalar is a single cell –As long as the end result is a scalar, the formula will be treated as a scalar even if the intermediate calculations involve matrices.  Matrix is an array (range) –Both vectors and matrices are considered Matrix by Excel –Entering formulas for an array Select the range with the proper dimension Type formula Press Ctrl+Shift+Enter

4. Excel Matrix Functions  SUMPRODUCT(array1,array2..array30) –Arrays must be of the same dimension but can be row or column vectors –This function is useful whenever you need to computed a weighted average –You can get the same result using SUM(array1*array2) entered as an array, i.e. Ctrl+Shift+Enter.  MMULT(x,y) –X and Y must be conforming in dimensions (consistent)  MINVERSE(x) –Returns the inverse of a matrix  MDETERM(x) –Returns the determinant of a matrix

5. Other Related Excel Functions  SUMSQ(array1,array2..)

6. Review Materials for Matrix Algebra  http://www.mathphysics.com/spingarn/lane/ http://www.mathphysics.com/spingarn/lane/  http://planetmath.org/encyclopedia/MatrixOperations.html http://planetmath.org/encyclopedia/MatrixOperations.html