Testing for Concordance Ordering

THE GRADUATE INSTITUTE OF INTERNATIONAL STUDIES 40, Bd. du Pont d’Arve PO Box, 1211 Geneva 4 Switzerland Tel (++4122) 312 09 61 Fax (++4122) 312 10 26 http: //www.fame.ch E-mail: [email protected]

FAME - International Center for Financial Asset Management and Engineering

ORDERING

Université Catholique de Louvain Olivier SCAILLET

TESTING FOR

CONCORDANCE

HEC Genève and FAME Université de Genève

INTERNATIONAL CENTER FOR FINANCIAL ASSET MANAGEMENT AND ENGINEERING

40 bd du Pont d'Arve • P.O. Box 3 • CH-1211 Geneva 4 • tel +41 22 / 312 0961 • fax +41 22 / 312 1026 http://www.fame.ch • e-mail: [email protected]

R

ESEARCH

P

APER

S

ERIES

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Executive summary

We propose inference tools to analyse the ordering of concordance of random

vectors. Random variables are concordant if they tend to be all large together or

small together. Concordance of random variables conveys the idea of clustering

of large and small events. An ordering of concordance was initially considered

for two random variables but has been extended to the multivariate case. This

ordering corresponds to a natural notion of stochastic dominance between two

distributions functions with fixed marginals. Large and small values will tend to

be more often associated under the distribution which dominates the other one.

Detection of concordant behaviour is especially important in risk management

of large portfolios of insurance contracts or financial assets. In these portfolios

the main risk is the occurrence of many joint default events or simultaneous

downside evolution of prices. An accurate knowledge of concordance between

claims or financial asset prices will help to assess this risk of loss clustering and

there from allows for taking appropriate action to ensure that the risk incurred

by the financial institution remains within its stated risk appetite. Clearly the

presence of concordance affects risk measures and asset allocatio ns resulting

from optimal portfolio selection. Analysis of concordance cannot be neglected

and reveals much of the danger associated to a given position. Some illustrations

of the practical relevance of the concordance order are given in the paper.

Modelling of concordance can be fully parametric or semiparametric. In the first

case specific parametric forms are selected for the dependence structure and the

margins, while in the second case margins are left unspecified. The dependence

structure is expressed by means of a parametric copula function. Once these

models have been estimated a natural concern of a risk manager ought to be:

does the adopted modelling reflect the dependence structure present in the data

safely enough? The aim of this paper is to provide inference tools to answer this

question. These tools will be tests for the ordering of concordance for random

variables. In fact we propose testing procedures for concordance order between

the chosen model and the empirical distribution. This allows for checking

whether the estimated parametric or semiparametric model gives a safe picture

of the association between small and large observed losses.

The analysis in the bivariate case relies on tests for upper and lower quadrant

dominance of the true distribution by a parametric or semiparametric model, i.e.

for a parametric or semiparametric model to give a probability that two variables

are simultaneously small or large at least as great as it would be were they left

unspecified. Tests for its generalisation in higher dimensions, namely joint lower

and upper orthant dominance, are also analysed. The parametric and

semiparametric setting are based on the copula representation for multivariate

distribution (a review of this concept is given in the paper), which allows for

disentangling behaviour of margins and dependence structure. We propose two

types of testing procedures for each setting. The first procedure is based on a

formulation of the dominance concepts in terms of values taken by random

variables, while the second procedure is based on a formulation in terms of

probability levels. For each formulation a distance test and an intersection-union

test for inequality constraints are developed depending on the definition of null

and alternative hypotheses.

An empirical illustration of the use of the testing procedures is given for US

insurance claim data. Indeed often insurance processes involve correlated pairs

of variables. A relevant example is the LOSS and allocated adjustment expenses

(ALAE, in short) on a single claim. ALAE's are type of insurance company

expenses that are specifically attributable to the settlement of individual claims

such as lawyers' fees and claims investigation expenses. We analyse in the paper

a US database made of 1,466 uncensored observed values of the random vector

(LOSS, ALAE). The joint modelling in the parametric setting relies on the

Pareto distribution to model the margins and the Gumbel copula to model the

dependence structure. We keep this copula in the semiparametric setting. The

empirical results in both frameworks suggest that the Gumbel copula reflects the

dependence structure in the data safely enough. This should reassure actuaries in

their use of this copula when computing an insurance premium.

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