COMPUTING AND DATA ANALYSIS WITH EXCEL. Matrix manipulation and systems of linear equations

Transcription

1 COMPUTING AND DATA ANALYSIS WITH EXCEL Matrix manipulation and systems of linear equations

2 Outline 1 Matrices Addition Subtraction Excel functions that return more than one cell Solving systems of linear equations Matrix inversion method Gaussian elimination method Jacobi method Gauss-Seidel method

3 Matrix Commands in Excel Introduction: Excel can perform some useful basic matrix operations namely: Addition & subtraction; Scalar multiplication & division; Some common operations and Excel functions employed: Operation Transpose Matrix multiplication Matrix inverse Determinant of matrix Excel Function TRANSPOSE; MMULT; MINVERSE; MDETERM;

4 Addition, Subtraction and Scalar Multiplication, Etc. To add two named 3 x 2 matrices, A and B: i) Highlight a blank 3 x 2 results area in the spreadsheet. (If the results area is too small, you will get the wrong answer.) ii) Type =A+B in the formula bar and press the CTRL, SHIFT and ENTER keys simultaneously. (Note: A+B above refers to the named range or defined names of matrices A and B) You must use the CTRL, SHIFT,ENTER keys simultaneously if you want to perform a matrix computation. (If you don t do this, you will get an error message or the wrong answer.) If you click on any cell in the result, the formula {=A+B} will be displayed. In Excel, the { } brackets indicate a matrix (array) command.

5 Matrix Transpose Suppose A is a 3 x 2 matrix. The transpose of A, A, will be 2 x 3. Select a 2 x 3 results area, type =TRANSPOSE(A) in the formula bar and press CTRL, SHIFT, ENTER. Matrix transpose using Excel

6 Matrix transpose Step: i) Determine the resultant matrix and select the appropriate number of cells. That is, how many rows by columns are required for the resultant matrix?

7 Matrix Transpose Step: ii) Type in the transpose formula and select the range of cells or named range iii) Remember to simultaneously press Ctrl + shift + enter

8 Matrix Multiplication Suppose A and B are named 3 x 2 and 2 x 3 matrices. Then AB is 3 x 3 and BA is 2 x 2. This illustrates the fact that in general, AB is not equal to BA, even if the matrices are conformable. Select a blank 3 x 3 area for the result AB. Type =MMULT(A,B) in the formula bar and press CTRL, SHIFT, ENTER to generate AB.

9 Matrix multiplication, MMULT The product of two matrices A & B can be calculated using the MMULT function. The result is a matrix with a number of rows equal to the number of rows in matrix A and a number of columns in matrix B. The syntax is: MMULT(array1, array2) where array 1 and 2 are respectively matrix A & B Multiply matrix A (3x2) by matrix B (2x2) Step: i) Define a named range for each of matrix A and B (ref to MOS studyguide) ii) Determine the resultant matrix and select the required Excel cells.

10 Matrix multiplication, MMULT (cont d.) Step: ii) Enter the formula the resultant matrix and select the required excel cells. ii) Remember to simultaneously press Ctrl + Shift + Enter

11 Matrix Determinant, MDETERM The function MDETERM calculates the determinant of a square matrix. This operation results in a single value or scalar. No cell within the range can be either empty or contain text, Excel will return an error message in such cases. Syntax, =MDETERM(array) Determinant of a matrix B is written as: det( B) B

12 Matrix Determinant, MDETERM (conti.) The determinant of matrix B is -7

13 Matrix Inverse, MINVERSE The inverse of a matrix is defined as one which when multiplied by the original matrix will return the identity matrix. The matrix inverse function in Excel has the following syntax, =MINVERSE(array) This returns the elements of the inverse matrix corresponding to the matrix given by the input (array). Example: Find the inverse of the following matrix B B B 3 1 Next, let s do this in excel B (1*5) (4*3) Hence, B or

14 Matrix Inverse, MINVERSE (cont d.) As in the previous operations, select the appropriate number of cells before typing in the matrix inverse formula MINVERSE and as usual simultaneously press Ctrl + Shift + Enter Question: Can we invert matrix C? Why?

15 Excel (Cont d) Natural progression from this course is the use of VBA to automate or speed up the manipulation process. see Benninga, S. (2000), Financial Modelling, MIT Press.

16 Summary If A is singular, then det(a) = 0. Exercise: Check that det(ab) = det(ba) = det(a).det(b), where A and B are square matrices. Highlight the cells of the matrix and right-click your mouse. From the flyer, look for DEFINE NAME. Enter a name for the matrix. You can now use the name of the matrix formula instead of the matrix values. Remember Excel is not case sensitive. In other words, names like futures, Futures and FUTURES are the same in excel world. Choose a location for the matrix (or vector) and enter the elements of the matrix. Highlight the cells of the matrix and choose INSERT NAME DEFINE. Enter a name for the matrix. You can now use the name of the matrix in formulae.

17 Summary Exercise: Choose A and B so that AB exists. Check that (AB)' = B 'A using MMULT (matrix multiplication). What do you think (ABC)' is equal to? Suppose A and B are named 3 x 2 and 2 x 3 matrices. Then AB is 3 x 3 and BA is 2 x 2. This illustrates the fact that, in general, AB is not equal to BA, even if the matrices are conformable. Select a blank 3 x 3 area for the result AB. Type =MMULT(A,B) in the formula bar and press CTRL, SHIFT, ENTER to generate AB. Suppose B is a square 2 x2 matrix. Select a 2 x 2 area for the inverse of B. Type =MINVERSE(B) in the formula bar and press CRTL, SHIFT, ENTER. If B is singular (non-invertible), you will get an error message. Suppose A and B have the same dimension and are both invertible. Show that (AB) -1 = B -1 A -1. What do you think (ABC) -1 is equal to?

18 Solving Systems of Linear Equations (in MS Excel) Simple simultaneous equations in2 variables Solve the simultaneous equations in MS Excel: 3x + 2y =8 2x + 5y =-2 You need 3 steps only: i) Create a matrix of the coefficients and constant ii) Determine the inverse matrix of the coefficients using MINVERSE iii) Multiply the inverse matrix with the matrix of constants, use MMULT

19 Solving Systems of Linear Equations (in MS Excel) Short exercise: Use MS Excel to solve the following simultaneous equations 1) x y z 4 2x 3y z 7 x 2y 3z 6 2) 3a b 3c 8 5a 3b 6c 4 6a 4b c 20 3) 2x y z 3 y z 2 4x y z 12

20 Calculating Variance-Covariance Matrix (From Excess returns) Lab Exercise Steps: i) Download the annual stock price for six companies listed on FTSE 100 ii) From the guide in lecture 2, calculate the average annual return for each company iii) Next calculate the excess return as (each return value less the average annual return for the asset iv) Now, transpose the matrix of excess asset returns v) Calculate the variance-covariance matrix from Mmult(transposed matrix, original excess return matrix) divided by the number of observations per asset