Abstract

In this paper we study both market risks and non-market risks, without complete markets assumption, and discuss methods of measurement of these risks. We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties ‘coherent’. We examine the measures of risk provided and the related actions required by SPAN, by the SEC/NASD rules and by quantile based methods. We demonstrate the universality of scenario-based methods for providing coherent measures. We offer suggestions concerning the SEC method. We also suggest a method to repair the failure of subadditivity of quantile-based methods.

Introduction

We provide in this paper a definition of risks (market risks as well as nonmarket risks) and present and justify a unified framework for the analysis, construction and implementation of measures of risk. We do not assume completeness of markets. These measures of risk can be used as (extra) capital requirements, to regulate the risk assumed by market participants, traders, insurance underwriters, as well as to allocate existing capital.

For these purposes, we:

1. (1) Define ‘acceptable’ future random net worths (see Section 2.1) and provide a set of axioms about the set of acceptable future net worths (Section 2.2);

2. (2) Define the measure of risk of an unacceptable position once a reference, ‘prudent’, investment instrument has been specified, as the minimum extra capital (see Section 2.3) which, invested in the reference instrument, makes the future value of the modified position become acceptable;

3. […]