Market Risk Analysis Volume II: Practical Financial Econometrics

II.2

Principal Component Analysis

II.2.1 INTRODUCTION

This chapter introduces the statistical factor models that are based on principal component analysis (PCA) and that are commonly applied to model the returns on portfolios and the profit and loss (P&L) of cash flows. Such models may also be applied to assess portfolio risks and hence to provide the risk adjusted performance measures that are used to rank investments.¹

Statistical factor models for portfolios are based on factors that have no economic or financial interpretation. A principal component representation for the percentage return on each asset in the investor's universe is derived from an eigenvector analysis of a very large covariance matrix, based on the returns on all the assets in the portfolio. Each principal component represents the percentage return on a statistical risk factor and, by choosing the number of principal components in the representation for each asset return, the investor can adjust the asset's specific risk. Then optimal portfolios are constructed by adjusting the weights to match the systematic risk, systematic return and specific risk characteristics desired by the investor.

Factor models for portfolios of interest rate sensitive instruments such as bonds, floating rate notes, forward rate agreements and swaps assume the portfolio has already been mapped to a fixed set of risk factors which are standard vertices along one or more yield curves.² In this case a PCA may be based on a covariance ...