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Introduction to Excel Array Functions This document is intended to introduce the use of Excel to perform basic matrix algebra operations. Sumproduct Definition For matrices A and B, a 11 a 21 A a r1 a 12 a 1c b11 b a 22 a 2 c and B 21 b a r 2 a rc i1 Sumproduct(A, B) b12 b1 j b22 b2 j bi 2 bij a11b11 a12b12 a13b13 arc bij Notes: r = i and c = j. In other words, the two matrices must have the same number of rows as each other and the same number of columns as each other. They do not need to be square matrices (where r = c and i = j). Algebra geeks sometimes call this operation the “dot product” (to distinguish it from matrix multiplication — described later in this document), and symbolize it as A B . Example 7 4 6 9 3 8 and A B 10 12 1 5 2 11 Sumproduct(A, B) a11b11 a12b12 a13b13 arc bij 7 9 4 3 6 8 111 208 Excel Method 1 2 3 4 5 6 7 8 Applied Regression Analysis A 7 5 B 4 2 C 6 11 9 10 3 12 8 1 208 D =SUMPRODUCT(A1:C2,A4:C5) 2 Prof. Juran Transpose Definition For matrix A, a 11 a 21 A a r1 a 11 a 12 T A a 1c a 12 a 1c a 22 a 2 c a r 2 a rc a 21 a r 1 a 22 a r 2 a 2 c a rc Notes: If A is an r x c matrix, then and AT must be a c x r matrix. A does not need to be a square matrix. Example 7 4 6 A 5 2 11 7 5 A 4 2 6 11 T Applied Regression Analysis 3 Prof. Juran Excel Method There is an Excel function for this purpose, called TRANSPOSE. This function is one of a special class of functions called array functions. In contrast with most other Excel functions, array functions have two important differences: They are entered into ranges of cells, not single cells You enter them by pressing Shift+Ctrl+Enter, not just Enter 1 2 3 4 5 6 A 7 5 B 4 2 C 6 11 D E Using the spreadsheet above as an example, we start by selecting the entire range A4:C6. Then type into the formula bar =TRANSPOSE(A1:C2) Press Shift+Ctrl+Enter, and curly brackets will appear round the formula (you can’t type them in). 1 2 3 4 5 6 Applied Regression Analysis A 7 5 B 4 2 7 4 6 5 2 11 C 6 11 D E =TRANSPOSE(A1:C2) 4 Prof. Juran Multiplication Definition For matrices A and B, a 11 a 21 A a r1 a 12 a 1c b11 b a 22 a 2 c and B 21 b a r 2 a rc i1 a 11b11 a 12 b21 a 1c bi 1 a b a b a b 21 11 22 21 2c i1 AB a b a b a b rc i 1 r 1 11 r 2 21 b12 b1 j b22 b2 j bi 2 bij a 11b12 a 12 b22 a 1c bi 2 a 21b12 a 22 b22 a 2 c bi 2 a r 1b12 a r 2 b22 a rc bi 2 a 11b1 j a 12 b2 j a 1c bij a 21b1 j a 22 b2 j a 2 c bij a r 1b1 j a r 2 b2 j a rc bij Notes: It is conventional to describe the shape of a matrix by listing the number of rows first, and the number of columns second. Matrix A above is an r x c matrix, and matrix B is an i x j matrix. In this operation, it is necessary for c = i. However it is not necessary for r = j. In other words, B must have the same number of rows as A has columns, but it is not necessary for B to have the same number of columns as A has rows. The product AB will always be an r x j matrix. Example 9 10 7 4 6 A and B 3 12 5 2 11 8 1 AB 7 9 4 3 68 59 2 3 118 7 10 4 12 61 510 2 12 111 123 124 139 85 Applied Regression Analysis 5 Prof. Juran Excel Method 1 2 3 4 5 6 7 8 9 A 7 5 B 4 2 9 3 8 10 12 1 123 139 124 85 C 6 11 D E =MMULT(A1:C2,A4:B6) Remember: Select the entire range A8:B9 before typing the formula. Press Shift+Ctrl+Enter. You can also get the same results using SUMPRODUCT: 1 2 3 4 5 6 7 8 9 10 11 12 A 7 5 B 4 2 9 3 8 10 12 1 C 6 11 D E F 9 10 G 3 12 H I 8 1 =TRANSPOSE(A4:B6) =SUMPRODUCT(A1:C1,E1:G1 ) Using SUMPRODUCTs =SUMPRODUCT(A1:C1,E2:G2) 123 124 139 85 =SUMPRODUCT(A2:C2,E2:G2) =SUMPRODUCT(A2:C2,E1:G1) Applied Regression Analysis 6 Prof. Juran Inverse Definition First, define a square matrix Ij as a matrix with j rows and j columns, completely filled with zeroes, except for ones on the diagonal: 1 0 0 0 1 0 I 0 0 1 This special matrix is called the identity matrix. Now, for a square matrix A with j rows and j columns, there may exist a matrix called A-inverse (symbolized A 1 ) such that: AA1 Ij Note: Not all square matrices can be inverted, a fact that has implications for regression analysis. Example 123 124 If A 139 85 0.01254 0.01829 Then A 1 0.02050 0.01814 Because AA1 123 0.01254 124 0.02050 139 0.01254 85 0.02050 1230.01829 124 0.01814 139 0.01829 85 0.01814 1 0 0 1 I2 Applied Regression Analysis 7 Prof. Juran Excel Method 1 2 3 4 5 6 A B 123 139 124 85 -0.01254 0.01829 0.02050 -0.01814 C D check 1 0 E 0 1 F G H =MMULT(A2:B3,A5:B6) =MINVERSE(A2:B3) Remember: Select the entire range A5:B6 before typing the formula. Press Shift+Ctrl+Enter. Applied Regression Analysis 8 Prof. Juran Appendix: Application to Financial Portfolios Here is a spreadsheet model of a three-stock portfolio optimization problem: A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Mean return Variance of return StDev of return B Stock 1 0.140 0.200 0.447 C Stock 2 0.110 0.080 0.283 D Stock 3 0.100 0.180 0.424 E F Stock 1 1.00 0.80 0.70 Stock 2 0.80 1.00 0.90 Stock 3 0.70 0.90 1.00 Stock 1 0.333 Stock 2 0.667 Stock 3 0.000 Actual 0.120 Required =SUMPRODUCT(B2:D2,B14:D14) >= 0.120 Correlations Stock 1 Stock 2 Stock 3 G H I J Stock 1 0.2000 0.1012 0.1328 Stock 2 0.1012 0.0800 0.1080 Stock 3 0.1328 0.1080 0.1800 Covariances Stock 1 Stock 2 Stock 3 Investment decision Fractions to invest Total 1 = Required 1 Expected portfolio return Portfolio variance Portfolio stdev 0.103 0.321 =MMULT(B14:D14,MMULT(H8:J10,TRANSPOSE(B14:D14))) =SQRT(B20) In cell B18 the SUMPRODUCT function is used to calculate the expected return on the portfolio. The expected return is a function of (a) the expected returns on the three stocks and (b) the portfolio weights (fractions to invest). To be explicit: SUMPRODUCT(B2:D2,B14:D14) =B2*B14+C2*C14+D2*D14 1x1 2 x2 3 x3 =0.14*0.333+0.11*0.667+0.10*0.000 =0.120 The portfolio weights are decision variables in this problem; if these change, then of course the expected return on the portfolio would also change. In cell B20, the MMULT and TRANSPOSE functions are combined to calculate the variance of the portfolio. For notational purposes, let’s define two matrices: A x1 x2 x3 0.333 0.667 0.000 COV11 COV12 COV13 0.2000 0.1012 0.1328 B COV21 COV22 COV23 0.1012 0.0800 0.1080 COV31 COV32 COV33 0.1328 0.1080 0.1800 Applied Regression Analysis 9 Prof. Juran Recall that: The covariance of a random variable with itself is its variance. For example, COV11 12 The covariance of two random variables is independent of order. For example, COV13 COV31 Now the explicit calculation: MMULT(B14:D14,MMULT(H8:J10,TRANSPOSE(B14:D14))) ABAT x1 x1 (1) (2) x2 COV11 COV12 COV13 x1 x3 COV21 COV22 COV23 x2 COV COV COV 31 32 33 x3 (3) x2 COV11x1 COV12 x2 COV13 x3 x3 COV21x1 COV22 x2 COV23 x3 COV x COV x COV x 31 1 32 2 33 3 x1COV11x1 x1COV12 x2 x1COV13 x3 (4) x2COV21x1 x2COV22 x2 x2COV23 x3 x3COV31x1 X 3COV32 x2 x3COV33 x3 x1COV11x1 x2COV22 x2 x3COV33 x3 (5) 12 x12 22 x22 32 x32 2 x1 x2COV12 2 x2 x3COV23 2 x1 x3COV13 (6) 0.447 2 0.3332 0.28320.667 2 0.424 2 0.0002 (7) x1COV12 x2 x2COV21x1 x2COV23 x3 x3COV32 x2 x1COV13 x3 x3COV31x1 20.3330.667 0.1012 20.667 0.0000.1080 20.3330.0000.1328 0.10275 (8) This calculation is usually presented in the form of step (6) in introductory statistics classes, so as to avoid frightening people with the more general matrix notation. Applied Regression Analysis 10 Prof. Juran