Handbook of Asset and Liability Management: From models to optimal return strategies

34

Statistical Advanced Tools

Qui veut voyager loin ménage sa monture. (Racine)

34.1 EXTREME POINTS

The “80/20 rule” was invented by Vilfredo Pareto by the end of the 19th century. As an economist and as a landowner, he discovered that 80% of the land belonged to 20% of the population. Moreover, he found that 80% of pea production came from 20% of his plantation.

He decided then to apply this “rule” to economy: extreme movements explain 80% of the yields of the portfolios.

During the crash of 1987, the Dow Jones index lost 22.6% and every value of the index was affected. This shows that in extreme market movements, the correlations between assets change radically.

Managers introduced indicators to monitor this extreme correlation moves such as the “Spearman rhô” or the rank statistical correlation. This is indeed the linear correlation between the random variables X and Y considering their statistical laws F:

This correlation can easily be computed and it takes only into account the dependence structure between the two variables.

34.2 COPULAS

Copulas are used more and more often in modern ALM modelling.

We call X₁…X_n a set of n random variables with a joint law F(x₁…x_n) and with marginal laws F_i(x). Then, we define the copula by the following formula:

F(x₁, ..., x_n)= C(F₁(x₁), ..., F_n(x_n))

Or equivalently:

C(u₁, ..., u_n = F(F₁^–1(u₁, ..., F_n^–(u_n))

When there are only two ...