5 - Modelling I: Linear dependence

Summary

Introduction

Risk modelling beyond the most elementary requires stochastically dependent variables. The non-linear part of the theory, much needed in insurance, is treated in Chapter 6, and the topic here is linear relationships which are the mainworkhorse for financial risk. Two examples are shownin Figure 5.1. On the left, monthly log-returns on two equity indexes from the NewYork Stock Exchange (NYSE) are scatter plotted for a period of 25 years. They tend to move in the same direction and by related amounts. This is cross-sectional dependence; what happens at the same time influences both simultaneously. The dynamic or longitudinal side is indicated on the right. Equity returns R_0:k accumulated over k months are plotted against k. They start at zero (by definition) and then climb steadily until the investments in 2001 were 10–15 times more valuable than at the beginning. A downturn (only partly shown) then set in.

The first part of this chapter concerns cross-sectional dependence with random vectors X = (X₁, …, X_J). Models for pairs were treated in Section 2.4, and their scatterplots in Figure 2.5 (look them up!) match the real data in Figure 5.1 left fairly well. It is those models that are now being extended to J variables. They play a main role in longitudinal modelling too, where the setup is a random sequence X₁, X₂, … with X_k occurring at time t_k = kh. The value of the time increment h depends on the application. In long-term finance 1 year is often sufficient, yet much (and important) theoretical modelling applies when h → 0. How models on different time scales are related is discussed at the end of the chapter.