quant_review - FIN 275 - Investment Analysis and Global...

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Unformatted text preview: FIN 275 - Investment Analysis and Global Portfolio Management Quantitative Review Prof. Gergana Jostova 1 Working with Matrices Most financial applications involve working with a series of asset (stock, bond, portfolio) returns or prices over a period of time. Such a combination of cross-sectional and time-series data is called a panel and is most easily analyzed with matrices. A matrix is a rectangular array of elements arranged in horizontal rows and vertical columns , and usually enclosed in brackets. For our purposes, the elements of a matrix will always be real numbers, but in general they can also be functions or complex numbers. A matrix whose elements are numbers is called a constant matrix. 1 2 3 4 Matrices are designated by boldface uppercase letters. A general matrix A having r rows and c columns may be written A = a 11 a 12 ... a 1 c a 21 a 22 ... a 2 c ... ... ... ... a r 1 a r 2 ... a rc where the elements of the matrix are double subscripted to denote location. By convention, the row index precedes the column index, thus a 25 represents the element of A appearing the second row and fifth column, while a 31 represents the element appearing in the third row and first column. A matrix A may also be denoted as [ a ij ], where a ij denotes the general element of A appearing in the i th row and j th column. A matrix having r rows and c columns has order (or size) r c . The order of the matrix is very important for matrix multiplications. The transpose of a matrix A , denoted as A T or A , is obtained by converting the rows of A into columns of A T one at a time in sequence. If A has order m n , then A T has order n m . Be careful when multiplying matrices whether the matrix or its transpose should be used. In finance, we use matrices to combine the historic return series of several stocks into one struc- ture, for example: R = r Jan,ibm r Jan,msft r Jan,lu r Feb,ibm r Feb,msft r Feb,lu r Mar,ibm r Mar,msft r Mar,lu r Apr,ibm r Apr,msft r Apr,lu r May,ibm r May,msft r May,lu r Jun,ibm r Jun,msft r Jun,lu r Jul,ibm r Jul,msft r Jul,lu Square matrices are generally used to summarize variances and covariances among multiple assets:...
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quant_review - FIN 275 - Investment Analysis and Global...

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