Estimation and allocation of insurance risk capital

203 Pages·2007·1.2 MB·English

Estimation and allocation of
insurance risk capital
by
Hyun Tae Kim
A thesis
presented to the University of Waterloo
in fulfilment of the
thesis requirement for the degree of
Doctor of Philosophy
in
Actuarial Science
Waterloo, Ontario, Canada, 2007
c Hyun Tae Kim 2007
(cid:13)
I hereby declare that I am the sole author of this thesis. This is a true copy of the
thesis, including any required final revisions, as accepted by my examiners.
I understand that my thesis may be made electronically available to the public.
Hyun Tae Kim
ii
Acknowledgements
When properly managed, risks can be opportunities. In my life, looking back, those
risks have turned to be blessings; it just took a fair time to realize that.
First of all I thank my wife Jinsun (Julia) for her love, support, and patience
during my Ph.D. studies. Ever since we got married she has been my stronghold
and the closest friend. Our beloved children, Christopher and Hannah, have been
very nicely behaving; there will be one less student in my family now, but I am
sure they wouldn’t mind. My parents have been my role models ever since I started
understanding things in life; they taught me by examples. My parents-in-law played
a special role in my life as well.
I am deeply indebted to Professor Mary Hardy, my supervisor, who knows ex-
actly how to train and support her apprentice. Her knowledge, insight, passion, and
patience have been my life-time lessons; I could not ask for a better mentor as a stu-
dent. Special thanks go to Professor Harry Panjer, my master program’s supervisor,
who has supported me over various obstacles since I arrived at UW; Professor Gord
Willmot for his support and kindness during my studies; Professor Paul Marriott
generously spent much time with me in reviewing and discussing the thesis material,
which substantially improved not only the quality of the final thesis, but also my
ability to think critically. Professor Tony Wirjanto at Economics went through my
thesis cover to cover and provided valuable suggestions as well as a list of typos in
the thesis; Professor Jed Frees, the external examiner, made valuable suggestions on
various spots in the thesis which I had missed.
I am glad to know Professor David Matthews, the chair of the department, who
has done great things for me. Professor Ken Seng Tan always cheered up my spirit
whenever I felt blue. Professor Rob Brown kindly supported me during my master’s
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program with an exciting project. I also thank graduate officers of the department
for their supports. Students in our department are lucky to have such remarkable
support staff. Mary Lou Dufton, the graduate secretary of the department, knows
my school life better than my wife does, and I owe her cookies and chocolates beyond
numbering. I have had a lot of help from the expertise of Lucy Simpson, Gwen Sharp,
Joan Hatton, Anissia Anniss, and Amy Aldous. I also like to thank Linda Lingard,
who now is retired, for her special care for me during my early days at UW.
There are many students that I am thankful of. Joonghee, So-Yeun, Chang-kee,
Yunhee, Chongsun and other Korean students have shared much of my memories on
the 6th floor of the MC building. Thanks also go to Kai, Johnny, Ali, Bill, George
and Diego for their friendships. My office mates Lilia and Ivan made our office bigger
and brighter than it actually is.
Chapter 5 of thethesis was possible thanks tothe data that GGYprovided forme.
I am very grateful to David Gilliland who allowed me to use the AXIS software and
Wes Leong for his technical support and effort he spared for me. I would also like to
acknowledge theresearch support fromourdepartment, theInstitute for Quantitative
Finance and Insurance, Faculty of Mathematics, Natural Sciences and Engineering
Research Council of Canada, the Society of Actuaries.
Finally, and most of all, I thank the Lord who have made all these things possible
and led me to where I stand today.
“The riddles of God are more satisfying than the solutions of man.”
Gilbert K. Chesterton
Introduction to the Book of Job, 1907
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Abstract
Estimating tail risk measures such as Value at Risk (VaR) and Conditional Tail Ex-
pectation (CTE) is a vital component in financial and actuarial risk management.
The CTE is a preferred risk measure, due to coherence and a widespread acceptance
in actuarial community. In particular we focus on the estimation of the CTE using
both parametric and nonparametric approaches.
In parametric case the conditional tail expectation and variance are analytically
derived for the exponential distribution family and its transformed distributions.
For small i.i.d. samples the exact bootstrap (EB) and the influence function are
used as nonparametric methods in estimating the bias and the the variance of the em-
pirical CTE. In particular, it is shown that the bias is corrected using the bootstrap
for the CTE case. In variance estimation the influence function of the bootstrapped
quantile is derived, and can be used to estimate the variance of any bootstrapped
L-estimator without simulations, including the VaR and the CTE, via the nonpara-
metricdeltamethod. Anindustry modelareprovided byapplying theoreticalfindings
on the bias and the variance of the estimated CTE.
Finally a new capital allocation method is proposed. Inspired by the allocation
of the solvency exchange option by Sherris (2006), this method resembles the CTE
allocationinitsformandproperties, buthasitsownuniquefeatures,suchasmanager-
based decomposition. Through a numerical example the proposed allocation is shown
to fail the no undercut axiom, but we argue that this axiom may not be aligned with
the economic reality.
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Contents
1 Introduction 1
1.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Risk measure as required capital . . . . . . . . . . . . . . . . . . . . . 7
1.4 The Conditional Tail Expectation . . . . . . . . . . . . . . . . . . . . 13
1.5 Modelling Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Capital Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Conditional tail moments of the EF and its related distributions 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 The Exponential Family . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Conditional tail moments of the EF . . . . . . . . . . . . . . . . . . . 29
2.3.1 When generating function exists . . . . . . . . . . . . . . . . . 29
2.3.2 When no generating function exists . . . . . . . . . . . . . . . 36
2.4 Transformed Distributions . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
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3 Bias correction of risk measures using the bootstrap 51
3.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Estimating VaR and CTE using finite sample . . . . . . . . . . . . . 54
3.3 Bias of sample estimates of VaR and CTE . . . . . . . . . . . . . . . 56
3.4 The bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.2 Bootstrapping L-estimators . . . . . . . . . . . . . . . . . . . 63
3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5.2 Estimating the 99% Quantile risk measure . . . . . . . . . . . 71
3.5.3 Estimating the 95% CTE . . . . . . . . . . . . . . . . . . . . 76
3.6 A practical guideline . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Variance estimation of bootstrapped risk measures 85
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 The bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.1 Improvement using recursion . . . . . . . . . . . . . . . . . . . 92
4.3 Nonparametric delta method . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 IF of bootstrapped L-estimator . . . . . . . . . . . . . . . . . . . . . 101
4.5 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5 A simulation study 113
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
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5.3 Simulation backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.5 Alternative model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.6 Model and parameter uncertainty . . . . . . . . . . . . . . . . . . . . 130
5.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 A new capital allocation method 135
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 Value vs. Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2.1 Two measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2.2 Valuation perspective . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.3 Solvency perspective . . . . . . . . . . . . . . . . . . . . . . . 142
6.3 Allocation review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.4 A new allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.5 Properties of the new allocation . . . . . . . . . . . . . . . . . . . . . 154
6.5.1 Fairness of the allocation . . . . . . . . . . . . . . . . . . . . . 155
6.5.2 Further decomposition . . . . . . . . . . . . . . . . . . . . . . 156
6.6 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.7 No undercut axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7 Discussion and future work 171
7.1 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.2 Future ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.2.1 CTE Comparison: parametric vs. nonparametric . . . . . . . 174
7.2.2 The Shifted Power Transform . . . . . . . . . . . . . . . . . . 174
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7.2.3 Bootstrap after transformation . . . . . . . . . . . . . . . . . 176
7.2.4 A semi-parametric method for CTE estimation . . . . . . . . 176
7.2.5 Fitting the GPD for Tail Region . . . . . . . . . . . . . . . . . 177
7.2.6 Allocation via Bootstrap . . . . . . . . . . . . . . . . . . . . . 178
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List of Tables
1.1 US RBC framework for life insurer . . . . . . . . . . . . . . . . . . . 3
1.2 Canada MCCSR framework . . . . . . . . . . . . . . . . . . . . . . . 4
3.1 99% Quantile estimators for the Lognormal example . . . . . . . . . 73
3.2 99% Quantile estimators for the RSLN2 example . . . . . . . . . . . 74
3.3 99% Quantile estimators for the Pareto example . . . . . . . . . . . 75
3.4 95% CTE estimators for the lognormal example . . . . . . . . . . . . 77
3.5 95% CTE estimators for the RSLN-2 example . . . . . . . . . . . . . 78
3.6 95% CTE estimators for the Pareto example . . . . . . . . . . . . . . 78
3.7 Application of MSE test to the three examples, 99% CTE. . . . . . . 83
4.1 Variance of the EB CTE 95% for three examples . . . . . . . . . . . . 110
5.1 Basic statistics based on the full data (Unit: CDN dollar) . . . . . . . 120
5.2 Simulation result of the Seg. fund model . . . . . . . . . . . . . . . . 123
6.1 Ordinary balance sheet of an insurer . . . . . . . . . . . . . . . . . . 140
6.2 Entitlements of shareholders and policy holders at time 1 . . . . . . . 140
6.3 Economic balance sheet of an insurer . . . . . . . . . . . . . . . . . . 141
6.4 Entitlements of shareholders and policy holders at time 1 for line i . . 149
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