Risk‐Minimizing Reinsurance Protection For Multivariate Risks

• Date: 01 March 2014
• Author: K. C. Cheung,S. C. P. Yam,K. C. J. Sung
• Published date: 01 March 2014
• DOI: http://doi.org/10.1111/j.1539-6975.2012.01501.x

© The Journal of Risk and Insurance, 2014, Vol. 81, No. 1, 219–236

DOI: 10.1111/j.1539-6975.2012.01501.x

219

RISK-MINIMIZING REINSURANCE PROTECTION FOR

MULTIVARIATE RISKS

K. C. Cheung

K. C. J. Sung

S. C. P. Yam

ABSTRACT

In this article, we study the problem of optimal reinsurance policy for

multivariate risks whose quantitative analysis in the realm of general law-

invariant convex risk measures, to the best of our knowledge, is still absent

in the literature. In reality, it is often difficult to determine the actual depen-

dence structureof these risks. Instead of assuming any particular dependence

structure, we propose the minimax optimal reinsurance decision formula-

tion in which the worst case scenario is first identified, then we proceed to

establish that the stop-loss reinsurances are optimal in the sense that they

minimize a general law-invariant convex risk measure of the total retained

risk. By using minimax theorem, explicit form of and sufficient condition for

ordering the optimal deductibles are also obtained.

INTRODUCTION

In spite of frequent substantial financial crises in the recent three decades, risk man-

agement has become the major focus in finance from both theoretical and practi-

cal perspectives. Artzner et al. (1999) proposed an axiomatic approach for coherent

measures of risk that satisfies the properties of monotonicity, positive homogeneity,

translation invariance, and subadditivity. Besides, they also provided a representa-

tion theorem for those risk measures; a similar result, in a different context, has been

also obtained in Huber (1981). Since then, many scholars have made various impor-

tant contributions along this direction. For example, as a generalization of coherent

K. C. Cheung and K. C. J. Sung are at the Department of Statistics, and Actuarial Science, The

University of Hong Kong, Pokfulam Road, Hong Kong. S. C. P.Yam is with the Department of

Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. K. C. Cheung can

be contacted via e-mail: kccg@hku.hk.The authors thank Jan Dhaene, Ludger R¨

uschendorf,

and many seminar and conference participants, for pointing out more accurate references,

and for their supportive comments and inspiring suggestions. K. C. Cheung acknowledges

the financial support of the Research Grants Council of HKSAR (Project No. HKU701409P).

S. C. P. Yam acknowledges financial support from The Hong Kong RGC GRF 404012 with the

project title “Advanced Topics in Multivariate Risk Management in Finance and Insurance.”

S. C. P.Yam also expresses his sincere gratitude to the hospitality of both Hausdorff Center for

Mathematics of the University of Bonn and Mathematisches Forschungsinstitut Oberwolfach

(MFO) in the German Black Forest during the preparation of the present work.

220 THE JOURNAL OF RISK AND INSURANCE

measures of risk, the notion of convex risk measures was introduced in the works

by Frittelli and Rosazza Gianin (2002), F¨

ollmer and Schied (2002, 2004), and Heath

and Ku (2004). As an important example, average value at risk (AV@R) is further

explored in the works by Acerbi and Tasche (2002), Delbaen (2002), and Rockafellar

and Uryasev (2001). If one imposes an additional axiom that the same value will be

assigned to two risky positions with a common distribution, a property known as

law-invariant, representation result was obtained by Kusuoka (2001) in the coherent

case, and by Frittelli and Rosazza Gianin (2005) in the convex case. Further prop-

erties of law-invariant risk measures can also be found in Jouini et al. (2006) and

Song and Yan (2009). Although the study of risk measures in the above works is es-

sentially developed for bounded risks, generalizations to the unbounded risks have

been investigated recently, such as in Filipov´

ıc and Svindland (2012). In addition, as

a special case of law-invariant convex risk measures, distortion risk measures enjoy a

wide application in practice. For a survey of these results and their connections with

comonotonicity, see, for example, Dhaene et al. (2006).

Because of the popularity in both interest in and actual use of various risk measures,

many classical insurance-related problems have been retreated in the risk measure

paradigm. For example, Dhaene et al. (2005) considered the optimal asset allocation

problem, Schied (2007) dealt with portfolio selection problem,and Annaert et al. (2009)

investigated the efficiency of the classical portfolio insurance problem under value

at risk (V@R) or AV@R. Motivated by the seminal works of Arrow (1963) and Borch

(1960), many researchers aim at seeking for optimal designs of (re)insurance contract

so that the risk measure of the retained loss can be minimized. For example, Cai and

Tan(2007), Cai et al. (2008), and Cheung (2010) considered the optimal increasing con-

vex indemnity schedule which minimizes the V@R and the conditional tail expectation

(CTE) of the retained cost. Another interesting study is given by Balb´

as et al. (2009) in

which stop-loss contracts were shown to be optimal under certain scenarios; also see

Balb´

as (2011) for the relevance of stop-loss contracts in optimal reinsurance problems

involving coherent risk measures. Recently, Cheung et al. (Forthcoming) considered

the optimal reinsurance decision problem under general law-invariant risk measures,

including V@R, CTE, and general convex risk measures. Based on real-life experience

and heuristics, truncated stop-loss reinsurance is obviously less welcome by both

reinsurers and buyers because it may cause potential moral hazard or swindle. Un-

like to some existing works that allow truncated stop-loss reinsurance to be optimal,

similar to our previous work in Cheung et al. (Forthcoming), we now only consider a

reinsurance contract to be feasible if it satisfies the two properties, namely: (1) an addi-

tional unit of loss cannot result in more than a unit increment of indemnity claim, and

(2) for any additional loss claim, at least not lesser compensation could be requested.

The present research studies a single-period risk-measure-based optimal reinsurance

decision problem for a basket of ninsurable risks. Although there is some theoretical

work on characterization of measures of multivariate risks, such as Ekeland et al.

(2012) and R ¨

uschendorf (2006, 2012), apart from a few applications and implementa-

tions in the financial context (such as in Kiesel and R ¨

uschendorf, 2008, 2010), similar

consideration in the insurance literature is still rare. To the best of our knowledge,

risk-minimizing reinsurance design in a multivariate setting is still absent in the lit-

erature. Whenever one studies a problem involving more than one risk, a knowledge

in the joint distribution of the multivariate risks is essential. In reality, acquiring the