MEASURING PORTFOLIO RISK UNDER PARTIAL DEPENDENCE INFORMATION

• Author: Steven Vanduffel,Carole Bernard,Michel Denuit
• DOI: http://doi.org/10.1111/jori.12165
• Published date: 01 September 2018
• Date: 01 September 2018

©2016 The Journal of Risk and Insurance. Vol.85, No. 3, 843–863 (2018).

DOI: 10.1111/jori.12165

Measuring Portfolio Risk Under Partial

Dependence Information

Carole Bernard

Michel Denuit

Steven Vanduffel

Abstract

The bounds for risk measuresof a portfolio when its components have known

marginal distributions but the dependence among the risks is unknown

are often too wide to be useful in practice. Moreover, availability of addi-

tional dependence information, such as knowledge of some higher-order

moments, makes the problem significantly more difficult. We show that re-

placing knowledge of the marginaldistributions with knowledge of the mean

of the portfolio does not result in significant loss of information when esti-

mating bounds on value-at-risk. These results are used to assess the margin

by which total capital can be underestimated when using the Solvency II or

RBC capital aggregation formulas.

Introduction

Quantifying the risk of a sum of random variables has long been a central problem in

numerous disciplines, including engineering, finance, and insurance. Clearly, portfo-

lios are at the core of the insurance business, as the insurer counts on diversification

effects to control the risk of her entire portfolio. For an insurance portfolio, the as-

sumption of independence among the policies is sometimes realistic, in which case

the insurer can resort to the Central Limit Theorem, to Monte Carlo methods, or to

more accurate methods such as Panjer’s recursion to accurately quantify the maxi-

mum loss in a given period of time at a certain probability level (i.e., the value-at-risk

Carole Bernard is at the Department of Accounting, Law and Finance, Grenoble Ecole

de Management, Grenoble, France. Bernard can be contacted via e-mail: carole.bernard@

uwaterloo.ca. Michel Denuit is at the Institut de Statistique, Biostatistique and Sciences Ac-

tuarielles, Universit´

e Catholique de Louvain, Louvain-la-Neuve, Belgium. Denuit can be

contacted via e-mail: michel.denuit@uclouvain.be. Steven Vanduffel is at the Faculty of Eco-

nomics, Vrije Universiteit Brussel, Elsene, Belgium. Vanduffel can be contacted via e-mail:

steven.vanduffel@vub.ac.be. Carole Bernard acknowledges support from NSERC, the CAE re-

search grant in Waterloo, and the Humboldt Research Foundation. The financial support of

PARC“Stochastic Modelling of Dependence” 2012-17 awarded by the “Communaut ´

e franc¸aise

de Belgique” is gratefully acknowledged by Michel Denuit. Steven Vanduffel acknowledges

support from FWO, BNP Paribas Fortis and the Chair Stewardship of Finance. We thank Jing

Yao for excellent researchassistance.

843

844 The Journal of Risk and Insurance

[VaR]).In the majority of cases, however, the individual risks are influenced by one or

more common factors, such as geography or economic environment, and it is difficult

to specify the joint distribution.

Ultimately, certain assumptions will then be made, resulting in the choice of a multi-

variate model to measure the risk of the portfolio at hand. However, in reality there

are many models that are consistent with the available information. It is then of great

interest (and importance) for stakeholders to assess the robustness of the model, par-

ticularly with respect to the maximum or minimum value for a certain risk measure

that can be justified given available information. Notably,regulators confronted with

scandals such as the subprime crisis and the LIBOR fraud seem to care more and

more about model uncertainty (and complexity). For example, in its discussion paper,

the Basel Committee (2013) insists that one of the desired objectives of a solvency

framework concerns comparability: “Two banks with portfolios having identical risk

profiles apply the framework’s rules and arrive at the same amount of risk-weighted

assets and two banks with different risk profiles should producerisk numbers that are

different proportionally to the differences in risk.” Or, as the financial news agency

Bloomberg Bloomberg reports(Groendahl, 2013): “While firms submit their models to

national regulators for validation, they don’t have to disclose them publicly. Surveys

by the Basel Committee have shown that risk-weightings for the same assets vary

among banks, undermining their credibility.”

There is a rich literature devoted to the risk bounds of a risky portfolio when the

marginal distributions of the components (describing the stand-alone risks) are as-

sumed to be known, but not their interdependence; see, among others, R¨

uschendorf

(1982), Denuit, Genest, and Marceau (1999), Puccetti and R¨

uschendorf (2012a), Wang

and Wang(2011), Peng, Yang,and Wang (2013), Embrechts, Puccetti, and R ¨

uschendorf

(2013), and Bernard, Jiang, and Wang (2014). Specifically, when the portfolio is ho-

mogeneous (all marginal distributions are identical), sharp (i.e., best-possible) risk

bounds can be obtained explicitly. However, the analysis for inhomogeneous portfo-

lios is fairly complicated, and theoretical results in this regard arescarce. Puccetti and

R¨

uschendorf (2012a) and Embrechts, Puccetti, and R¨

uschendorf (2013) recently devel-

oped the rearrangement algorithm (RA) as a practical way to approximate sharp VaR

bounds. While their numerical examples provide evidence that the RA is indeed able

to approximate the bounds accurately, they also show that the gap between worst-case

and best-case VaR numbers is typically very high. Furthermore, the upper bound on

VaR is always larger than the VaR that one would obtain in cases in which all risks

have maximum correlation (comonotonic), a situation that practitioners find difficult

to accept.

The situation described above stresses that information on dependence is crucial in

order to build models that provide trustworthy risk numbers in the sense that upper

and lower bounds for these numbers remain within a reasonablerange. In the presence

of additional dependence information, the literature on risk bounds is more limited.

Embrechts, Puccetti, and R¨

uschendorf (2013) consider the situation in which some of

the bivariate distributions are known, Denuit, Genest, and Marceau (1999) study risk

bounds assuming that the joint distribution of the risks is bounded by some distribu-

tion, and Cheung and Vanduffel(2013) study bounds when the marginal distributions