Three Essays in Finance and Actuarial Science

140 Pages·2017·1.52 MB·English

by  Regis Luca

Three Essays in Finance and Actuarial Science
Regis Luca
To cite this version:
Regis Luca. Three Essays in Finance and Actuarial Science. Economies et finances. Università degli
studi di Torino, 2011. Français. ￿NNT: ￿. ￿tel-00804585￿
HAL Id: tel-00804585
https://theses.hal.science/tel-00804585
Submitted on 25 Mar 2013
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est
archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents
entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non,
lished or not. The documents may come from émanant des établissements d’enseignement et de
teaching and research institutions in France or recherche français ou étrangers, des laboratoires
abroad, or from public or private research centers. publics ou privés.
Universit`a di Torino
THREE ESSAYS IN FINANCE
AND ACTUARIAL SCIENCE
Luca Regis
Scuola di Dottorato in Economia ”V.Pareto”,
indirizzo in Statistica e Matematica Applicata

Universit`a di Torino
Dipartimento di Statistica e Matematica Applicata
”Diego De Castro”
THREE ESSAYS IN FINANCE
AND ACTUARIAL SCIENCE
Luca Regis
Scuola di Dottorato in Economia ”V.Pareto”,
indirizzo Statistica e Matematica Applicata
Ciclo XXII
Supervisore: Prof. ssa Elisa Luciano
Coordinatore: Prof. Paolo Ghirardato
Acknowledgements
The first and most important thank goes to my supervisor Elisa Luciano, for
her constant and invaluable support throughout all my Ph.D. years, fruitful
discussions and suggestions. I am deeply indebted to Fabrizio Restione, Al-
berto Fasano and the Risk Management Department at Fondiaria-Sai s.p.A.
for providing me with the data I use in the first Part of my dissertation. I
would also like to thank Carmelo Genovese for helpful discussions on the first
Chapter.
The work benefited of comments from participants to the IV MAF Confer-
ence (Ravello, 7-9 April 2010), the IRMC 2010 (Firenze, 4-6 June 2010) and
the XII Workshop on Quantitative Finance (Padova, 27-28 January 2011).
I kindly thank the Fondazione Collegio Carlo Alberto and the Dipartimento
di Statistica e Matematica Applicata ”D. De Castro” and their staffs for fa-
cilities and technical assistance. Finally, a sincere thank goes to my family
and Marta, for supporting me during these years.
Contents
I Bayesian models for stochastic claims reserving 5
1 A Bayesian stochastic reserving model 7
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Deterministic models . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Fisher Lange method . . . . . . . . . . . . . . . . . . . 9
1.2.2 Chain Ladder method . . . . . . . . . . . . . . . . . . 9
1.2.3 TowardsaBayesianperspective: theBornhuetter-Ferguson
method . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Stochastic models for claims reserving . . . . . . . . . . . . . . 11
1.3.1 ODP model and GLM theory . . . . . . . . . . . . . . 12
1.3.2 Obtaining full predictive distributions by simulation . . 13
1.4 Bayesian models and Markov Chain Monte Carlo methods . . 15
1.5 An ODP Bayesian model for claims reserving . . . . . . . . . 18
1.5.1 Link with GLM theory and CL parameters . . . . . . . 21
1.5.2 Multivariate normal priors . . . . . . . . . . . . . . . . 21
1.5.3 Gamma priors . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Applying the model in practice . . . . . . . . . . . . . . . . . 26
1.6.1 Tests of convergence to the prior and to the chain lad-
der estimates . . . . . . . . . . . . . . . . . . . . . . . 26
1.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 Conclusions and further possible model extensions . . . . . . . 33
1.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Bayesian Copulas for Claims Reserving 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 A Bayesian approach for computing Lob’s reserves . . . . . . . 39
2.3 A copula approach to aggregate across LoBs . . . . . . . . . . 40
2.3.1 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.2 Applying copulas to claims reserving . . . . . . . . . . 43
2.4 A Bayesian copula approach . . . . . . . . . . . . . . . . . . . 44
2.4.1 Bayesian Gaussian copula . . . . . . . . . . . . . . . . 45
2
2.5 An application to an Italian Insurance Company . . . . . . . . 46
2.5.1 Estimation of the marginals . . . . . . . . . . . . . . . 46
2.5.2 Estimation of reserves through copulas . . . . . . . . . 47
2.5.3 Estimation of reserves using Bayesian Copulas . . . . . 51
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7 Appendix - Generating n-dimensional copulas . . . . . . . . . 54
2.7.1 The Genest and Rivest approach . . . . . . . . . . . . 54
2.7.2 Generating 4-dimensional Archimedean copulas . . . . 55
II A theoretical model of Capital Structure for Fi-
nancial Conglomerates 61
3 Financial Conglomerates 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3 Financial Groups and their supervision . . . . . . . . . . . . . 67
3.3.1 Financial conglomerates and coinsurance . . . . . . . . 67
3.3.2 The supervision of financial groups . . . . . . . . . . . 68
3.4 Modelling financial conglomerates . . . . . . . . . . . . . . . . 69
3.4.1 Basic set up: the stand alone firm and the HG . . . . . 69
3.4.2 Integrated conglomerates . . . . . . . . . . . . . . . . . 74
3.4.3 Holding/Subsidiary structures . . . . . . . . . . . . . . 76
3.5 Numerical Analysis: HG and IC . . . . . . . . . . . . . . . . 80
3.5.1 The Stand Alone constrained firm case: calibration . . 80
3.5.2 Integrated conglomerates . . . . . . . . . . . . . . . . . 81
3.6 Numerical Analysis: HS . . . . . . . . . . . . . . . . . . . . . 84
3.6.1 Consolidated Supervision . . . . . . . . . . . . . . . . . 84
3.6.2 Regulating subsidiaries . . . . . . . . . . . . . . . . . . 86
3.6.3 HCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6.4 Regulatory arbitrage in units subject to different cap-
ital requirements . . . . . . . . . . . . . . . . . . . . . 89
3.7 Concluding comments . . . . . . . . . . . . . . . . . . . . . . 90
3.8 Appendix A - Stand Alone Maximization Problem . . . . . . . 91
3.9 Appendix B - HS value maximization problem . . . . . . . . . 97
3
4
III Hedging Mortality in Stochastic Mortality Mod-
els 99
4 Hedging Mortality Risk 101
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Cox modelling of mortality risk . . . . . . . . . . . . . . . . . 103
4.2.1 Instantaneous death intensity . . . . . . . . . . . . . . 104
4.2.2 Ornstein-Uhlenbeck and Feller processes . . . . . . . . 105
4.3 Forward death intensities . . . . . . . . . . . . . . . . . . . . . 108
4.4 Financial risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.5 Change of measure and insurance valuations . . . . . . . . . . 111
4.6 HJM restriction on forward death intensities . . . . . . . . . . 115
4.7 Mortality risk hedging . . . . . . . . . . . . . . . . . . . . . . 118
4.7.1 Dynamics and sensitivity of the reserve . . . . . . . . . 118
4.7.2 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.8 Mortality and financial risk hedging . . . . . . . . . . . . . . . 122
4.9 Application to a UK sample . . . . . . . . . . . . . . . . . . . 124
4.9.1 Mortality risk hedging . . . . . . . . . . . . . . . . . . 125
4.9.2 Mortality and financial risk hedging . . . . . . . . . . . 127
4.10 Summary and conclusions . . . . . . . . . . . . . . . . . . . . 129
4.11 Appendix: Cox Processes . . . . . . . . . . . . . . . . . . . . . 129
Part I
Bayesian models for stochastic
claims reserving
5
6
The first part of my Ph.D. dissertation develops a Bayesian stochastic
model for computing the reserves of a non-life insurance company.
The first chapter is the product of my research experience as an intern at
the Risk Management Department of Fondiaria-Sai S.p.A.. I present a short
review of the deterministic and stochastic claims reserving methods currently
appliedinpracticeandIdevelopa(standard)Over-DispersedPoisson(ODP)
BayesianmodelfortheestimationoftheOutstandingLossLiabilities(OLLs)
of a line of business (LoB). I present the model, I illustrate the theoretical
foundations of the MCMC (Markov Chain Monte Carlo) method and the
Metropolis-Hastings algorithm used in order to generate the non-standard
posterior distributions. I apply the model to the Motor Third Party Liabil-
ity LoB of Fondiaria-Sai S.p.A..
The Risk Management Department of the company was already developing a
Bayesian model for stochastic claims reserving when I began my intern. My
contribution to the project consisted in a re-parametrization of the model,
that allowed to adopt unrestricted distributional assumptions for the priors.
The practical implementation of this model, which I describe in Chapter 1,
was my own work. I am grateful to the Head of the Technical Risk Manage-
ment Department Fabrizio Restione and to Carmelo Genovese for giving me
the opportunity to work on such an interesting subject and for their helpful
suggestions.
This chapter is also an introduction to the next one, in which I explore the
problem of computing the prudential reserve level of a multi-line non-life
insurance company. In the second chapter, then, I present a full Bayesian
model for assessing the reserve requirement of multiline Non-Life insurance
companies. The model combines the Bayesian approach for the estimation of
marginal distribution for the single Lines of Business and a Bayesian copula
procedure for their aggregation. First, I consider standard copula aggre-
gation for different copula choices. Second, I present the Bayesian copula
technique. Up to my knowledge, this approach is totally new to stochastic
claims reserving. The model allows to ”mix” own-assessments of dependence
between LoBs at a company level and market wide estimates. I present an
application to an Italian multi-line insurance company and compare the re-
sults obtained aggregating using standard copulas and a Bayesian Gaussian
copula. I am again grateful to Carmelo Genovese who suggested the use of
a time series of loss ratios to estimate standard copula parameters.