Introduction to Financial
Modeling
MGT 4850
Spring 2008
University of Lethbridge
Topics
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The power of Numbers
Quantitative Finance
Risk and Return
Asset Pricing
Risk Management and Hedging
Volatility Models
Matrix Algebra
MATRIX ALGEBRA
• Definition
– Row vector
– Column vector
Matrix Addition and Scalar Multiplication
• Definition: Two matrices A = [aij] and B = [bij
] are said to be equal if Equality of
these matrices have the same size, and for each
index pair (i, j), aij = bij , Matrices
that is, corresponding entries of A and B are
equal.
Matrix Addition and Subtraction
• Let A = [aij] and B = [bij] be m × n matrices.
Then the sum of the matrices, denoted by A +
B, is the m × n matrix defined by the formula A
+ B = [aij + bij ] .
• The negative of the matrix A, denoted by −A,
is defined by the formula −A = [−aij ] .
• The difference of A and B, denoted by A−B, is
defined by the formula A − B = [aij − bij ] .
Scalar Multiplication
• Let A = [aij] be an m × n matrix and c a
scalar. Then the product of the scalar c
with the matrix A, denoted by cA, is
defined by the formula Scalar cA = [caij ] .
Linear Combinations
• A linear combination of the matrices A1,A2,
. . . , An is an expression of the form c1A1
+ c2A2 + ・ ・ ・ + cnAn
Laws of Arithmetic
• Let A,B,C be matrices of the same size m × n, 0 the m
× n zero
• matrix, and c and d scalars.
• (1) (Closure Law) A + B is an m × n matrix.
• (2) (Associative Law) (A + B) + C = A + (B + C)
• (3) (Commutative Law) A + B = B + A
• (4) (Identity Law) A + 0 = A
• (5) (Inverse Law) A + (−A) = 0
• (6) (Closure Law) cA is an m × n matrix.
Laws of Arithmetic (II)
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(7) (Associative Law) c(dA) = (cd)A
(8) (Distributive Law) (c + d)A = cA + dA
(9) (Distributive Law) c(A + B) = cA + cB
(10) (Monoidal Law) 1A = A
Matrix Multiplication