Market Risk Analysis Volume I: Quantitative Methods in Finance

I.2

Essential Linear Algebra for Finance

I.2.1 INTRODUCTION

This chapter aims to equip readers with the tools of matrix algebra that are applied in finance. Starting with basic definitions and notation, we provide a detailed understanding of matrix algebra and its important financial applications. An understanding of matrix algebra is necessary for modelling all types of portfolios. Matrices are used to represent the risk and return on a linear portfolio as a function of the portfolio weights and the returns and covariances of the risk factor returns. Examples include bond portfolios, whose value is expressed as a discounted cash flow with market interest rates as risk factors, and stock portfolios, where returns are represented by linear factor models. Matrices are used to represent the formulae for parameter estimates in any multiple linear regressions and to approximate the returns or changes in price of non-linear portfolios that have several risk factors.

Without the use of matrices the analysis becomes extremely cumbersome. For instance, it is easy to use matrices to solve a set of simultaneous equations in many variables, such as the linear equations that arise when a trader hedges an options portfolio against changes in its risk factors. The covariance matrix and the Cholesky decomposition of this matrix lie at the heart of financial analysis: they are used to simulate correlated returns and to measure portfolio risk. In highly correlated systems, such as returns on futures ...