Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 4

Matrix Algebra

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrix algebra generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. Matrix algebra collects the various partial derivatives of a single function with respect to many variables, and of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations, such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. Calculations in portfolio theory, financial economics, and financial econometrics rely on the use of matrix algebra because of the need to manipulate large data inputs.

• Matrix algebra is used for optimal portfolio selection.
• Matrix algebra is useful for computing expected return of a portfolio that contains many assets.
• Matrix algebra is useful for computing the variance (or risk) of a portfolio that contains many assets.
• Optimal portfolio weights are calculated by maximizing the risk-adjusted return of a portfolio or by maximizing expected utility of a risk-averse investor. For either case, matrix algebra is useful for determining optimal asset allocation.
• Matrix algebra is used in financial risk management.
• A matrix is used to describe the outcomes or payoff of an investment or venture.
• Matrix algebra is used for computing value-at-risk and ...