**Unformatted text preview: **Springer Proceedings in Mathematics & Statistics Jaime A. Londoño
José Garrido
Monique Jeanblanc
Editors Actuarial
Sciences
and Quantitative
Finance
ICASQF2016, Cartagena, Colombia,
June 2016 Springer Proceedings in Mathematics & Statistics
Volume 214 Springer Proceedings in Mathematics & Statistics
This book series features volumes composed of selected contributions from
workshops and conferences in all areas of current research in mathematics and
statistics, including operation research and optimization. In addition to an overall
evaluation of the interest, scientific quality, and timeliness of each proposal at the
hands of the publisher, individual contributions are all refereed to the high quality
standards of leading journals in the field. Thus, this series provides the research
community with well-edited, authoritative reports on developments in the most
exciting areas of mathematical and statistical research today. More information about this series at Jaime A. Londoño • José Garrido
Monique Jeanblanc
Editors Actuarial Sciences and
Quantitative Finance
ICASQF2016, Cartagena, Colombia,
June 2016 123 Editors
Jaime A. Londoño
Departamento de Matemáticas y Estadí-stica
Universidad Nacional de Colombia
Manizales, Caldas, Colombia José Garrido
Department of Mathematics and Statistics
Concordia University
Montréal, QC, Canada Department of Mathematics
National University of Colombia
Bogota, Bogota, Colombia
Monique Jeanblanc
LaMME, Batiment IBGBI
Université d’Evry Val D’Essone
Evry Cedex, Essonne, France ISSN 2194-1009
ISSN 2194-1017 (electronic)
Springer Proceedings in Mathematics & Statistics
ISBN 978-3-319-66534-4
ISBN 978-3-319-66536-8 (eBook)
Library of Congress Control Number: 2017955004
Mathematics Subject Classification (2010): 62P05, 91B30, 91G20, 91G80
© Springer International Publishing AG 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, express or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.
Printed on acid-free paper
This Springer imprint is published by Springer Nature
The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface The chapters in this volume of the Springer Proceedings in Mathematics and
Statistics entitled “Actuarial Sciences and Quantitative Finance: ICASQF2016,
Cartagena, Colombia, June 2016” are from selected papers presented at the Second
International Congress on Actuarial Science and Quantitative Finance, which took
place in Cartagena from June 15 to 18, 2016. The conference was organized
jointly by the Universidad Nacional de Colombia, Universidad de Cartagena,
Universidad del Rosario, Universidad Externado de Colombia, Universidad de los
Andes, ENSIIE/Université Evry Val d’Essonne, and ADDACTIS Latina. It also
received support from Universidad Industrial de Santander, Ambassade de France en
Colombie, and ICETEX. The conference took place in the Claustro de San Agustín
and Casa Museo Arte y Cultura la Presentación in the walled city of Cartagena.
This congress was the second edition of a series of events to be organized every
other year, with the objective of becoming a reference in actuarial science and
quantitative finance in Colombia, the Andean region (Peru, Colombia, Venezuela,
Ecuador, and Bolivia), and the Caribbean. The congress had participation from
researchers, students, and practitioners from different parts of the world. This
second edition helped enhance the relations between the academic and industrial
actuarial and financial communities in North America, Europe, and other regions of
the world.
The emphasis of the event was equally distributed between actuarial sciences and
quantitative finance and covered a variety of topics such as Statistical Techniques in
Finance and Actuarial Science, Portfolio Management, Derivative Valuation, Risk
Theory, Life and Pension Insurance Mathematics, Non-life Insurance Mathematics,
and Economics of Insurance.
The event consisted of plenary sessions with invited speakers in the areas of
actuarial science and quantitative finance, oral sessions of contributed talks on
these topics, as well as short courses taught by some of the invited speakers and
poster sessions. The list of invited speakers reflects the broad variety of topics:
Nicole El Karoui (Self-Exciting Process in Finance and Insurance for Credit
Risk and Longevity Risk Modeling in Heterogeneous Portfolios), Julien Guyon v vi Preface (Path-Dependent Volatility), Christian Hipp (Stochastic Control for Insurance: New
Problems and Methods), Jean Jacod (Estimation of Volatility in Presence of High
Activity Jumps and Noise), Glenn Meyers (Aggressive Backtesting of Stochastic
Loss Reserve Models—Where It Leads Us), Michael Sherris (To Borrow or Insure?
Long-Term Care Costs and the Impact of Housing), Qihe Tang (Mitigating Extreme
Risks Through Securitization), and Fernando Zapatero (Riding the Bubble with
Convex Incentives). Topics for short courses included the following: The New
Post-crisis Landscape of Derivatives and Fixed Income Activity Under Regulatory
Constraints on Credit Risk, Liquidity Risk, and Counterparty Risk (Nicole El
Karoui); Stochastic Control for Insurers: What Can We Learn from Finance, and
What Are the Differences? (Christian Hipp); High-Frequency Statistics in Finance
(Jean Jacod); and Using Bayesian MCMC Models for Stochastic Loss Reserving
(Glenn Meyers).
Additionally, researchers and students presented oral contributions and posters.
There were 30 contributed oral presentations, 26 invited oral contributions, and ten
poster presentations. We received 85 contributions and 34 invited contributions. The
selection process was the result of careful deliberations, and 54 oral contributed
presentations of the 85 submissions and 20 posters were accepted. Authors came
from different corners of the world and countries of origin including Australia,
Brazil, Canada, Chile, Colombia, Egypt, France, Germany, Italy, Jamaica, Mexico,
Spain, Switzerland, the United Kindom, Uruguay, and the United States. The
number of contributions along with the total number of 279 registered participants
shows the steady growth of the congress and its consolidation as the main event of
the area in the Andean region and the Caribbean.
The congress put the emphasis on enhancing relations between industry and
academia providing a day to address problems arising from the financial and
insurance industries. As a matter of fact, topics and speakers themselves came from
these sectors. The congress provided practitioners a platform to present and discuss
with academics and students different approaches in addressing problems arising
from the industries in the region.
The current proceedings are based on invitations to selected oral contributions
and selected contributions presented by the invited speakers. All contributions were
subject to an additional review process. The spectrum of the eight papers published
here reflects the diverse nature of the presentations: there are five papers on actuarial
sciences and three papers on quantitative finance.
Special thanks go to the members of the organizing committee, which included
Javier Aparicio (Colombia, ADDACTIS Latina), Prof. Sergio Andrés Cabrales
(Colombia, Universidad de los Andes), Prof. Carlos Alberto Castro (Colombia,
Universidad del Rosario), Prof. Margaret Johanna Garzón (Colombia, Universidad
Nacional de Colombia, Bogotá), Prof. Sandra Gutiérrez (Colombia, Universidad
de Cartagena), Prof. Jaime A. Londoño (Colombia, Universidad Nacional de
Colombia, Bogotá), Prof. Sergio Pulido (France, ENSIIE/Université Evry Val
d’Essonne), Prof. Javier Sandoval (Colombia, Universidad Externado de Colombia), Preface vii and Prof. Arunachalam Viswanathan (Colombia, Universidad Nacional de Colombia, Bogotá). Finally, we would like to thank all the conference participants who
made this event a great success.
Manizales, Colombia
Montréal, QC, Canada
Evry Cedex, France
May 2017 Jaime A. Londoño
José Garrido
Monique Jeanblanc Contents Part I Actuarial Sciences
Robust Paradigm Applied to Parameter Reduction in Actuarial
Triangle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gary Venter 3 Unlocking Reserve Assumptions Using Retrospective Analysis. . . . . . . . . . . . .
Jeyaraj Vadiveloo, Gao Niu, Emiliano A. Valdez, and Guojun Gan 25 Spatial Statistical Tools to Assess Mortality Differences in Europe . . . . . . . .
Patricia Carracedo and Ana Debón 49 Stochastic Control for Insurance: Models, Strategies, and Numerics . . . . . .
Christian Hipp 75 Stochastic Control for Insurance: New Problems and Methods . . . . . . . . . . . . 115
Christian Hipp
Part II Quantitative Finance
Bermudan Option Valuation Under State-Dependent Models . . . . . . . . . . . . . . 127
Anastasia Borovykh, Andrea Pascucci, and Cornelis W. Oosterlee
Option-Implied Objective Measures of Market Risk with Leverage . . . . . . . 139
Matthias Leiss and Heinrich H. Nax
The Sustainable Black-Scholes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Yannick Armenti, Stéphane Crépey, and Chao Zhou
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 ix Part I Actuarial Sciences Robust Paradigm Applied to Parameter
Reduction in Actuarial Triangle Models
Gary Venter Abstract The recognition that models are approximations used to illuminate
features of more complex processes brings a challenge to standard statistical testing,
which assumes the data is generated from the model. Out-of-sample tests are
a response. In my view this is a fundamental change in statistics that renders
both classical and Bayesian approaches outmoded, and I am calling it the “robust
paradigm” to signify this change. In this context, models need to be robust to
samples that are never fully representative of the process. Actuarial models of
loss development and mortality triangles are often over-parameterized, and formal
parameter-reduction methods are applied to them here within the context of the
robust paradigm.
Keywords Loss reserving • Mortality • Bayesian shrinkage • MCMC 1 Introduction
Section 2 discusses model testing under the robust paradigm, including out-ofsample tests and counting the effective number of parameters. Section 3 introduces
parameter-reduction methods including Bayesian versions. Section 4 reviews actuarial triangle modeling based on discrete parameters by row, column, etc., and
how parameter-reduction can be used for them. Section 5 gives a mortality model
example, while Sect. 6 illustrates examples in loss reserving. Section 7 concludes. 2 Model Testing Within the Robust Paradigm
Both Bayesian and classical statistics typically assume that the data being used to
estimate a model has been generated by the process that the model specifies. In
many, perhaps most, financial models this is not the case. The data is known to come
G. Venter ()
University of New South Wales, Sydney, NSW, Australia
e-mail: [email protected]
© Springer International Publishing AG 2017
J.A. Londoño et al. (eds.), Actuarial Sciences and Quantitative Finance, Springer
Proceedings in Mathematics & Statistics 214, 3 4 G. Venter from a more complex process and the model is a hopefully useful but simplified
representation of that process. Goodness-of-fit measures that assume that the data
has been generated from the sample are often not so reliable in this situation, and
out-of-sample tests of some sort or another are preferred. These can help address
how well the model might work on data that was generated from a different aspect of
the process. I have coined the term “robust paradigm” to refer to statistical methods
useful when the data does not come from the model.
Much statistics today is based on pragmatic approaches that keep the utility of
the model for its intended application in mind, and regularly deviate from both pure
Bayesian and pure classical paradigms. That in itself does not mean that they are
dealing with data that does not come from the models. In fact, even out-of-sample
testing may be done purely to address issues of sample bias in the parameters, even
assuming that the data did come from the model. But simplified models for complex
processes are common and pragmatic approaches are used to test them. This is what
is included in the robust paradigm.
When models are simplified descriptions of more complex processes, you can
never be confident that new data from the same process will be consistent with
the model. In fact with financial data, it is not unusual for new data to show
somewhat different patterns from those seen previously. However, if the model is
robust to a degree of data change, it may still work fairly well in this context. More
parsimonious models often hold up better when data is changing like that. Out-ofsample testing methods are used to test for such robustness.
A typical ad hoc approach is the rotating 45 ths method: the data is divided, perhaps
randomly, into five subsets, and the model is fit to every group of four of these five.
Then the fits are tested on the omitted populations, for example by computing the
negative loglikelihood (NLL). Competing models can be compared on how well
they do on the omitted values.
A well-regarded out-of-sample test is leave one out, or “loo.” This fits the model
many times, leaving out one data point at a time. Then the fit is tested at each omitted
point to compare alternative models. The drawback is in doing so many fits for each
model.
In Bayesian estimation, particularly in Markov Chain Monte Carlo (MCMC),
there is a shortcut to loo. The estimation produces many sample parameter sets from
the posterior distribution of the parameters. By giving more weight to the parameter
sets that fit poorly at a given data observation, an approximation to the parameters
that would be obtained without that observation can be made. This idea is not new,
but such approximations have been unstable.
A recent advance, called Pareto smoothed importance sampling, appears to have
largely solved the instability problem. A package to do this, called loo, is available
with the Stan package for MCMC. It can be used with MCMC estimation not
done in Stan as well. It allows comparison of the NLL of the omitted points
across models. This modestly increases the estimation time, but is a substantial
improvement over multiple re-estimation. Having such a tool available makes loo
likely to become a standard out-of-sample fitting test. Robust Paradigm Applied to Parameter Reduction in Actuarial Triangle Models 5 This is a direct method to test models for too many parameters. Over-fitted
models will not perform well out of sample. If the parameters do better out of
sample, they are worth it. Classical methods for adjusting for over-parameterization,
like penalized likelihood, are more artificial by comparison, and never have become
completely standardized. In classical nonlinear models, counting the effective
number of parameters is also a bit complex. 2.1 Counting Parameters
In nonlinear models it is not always apparent how many degrees of freedom are
being used up by the parameter estimation. One degree of freedom per parameter is
not always realistic, as the form of the model may constrict the ability of parameters
to pull the fitted values towards the actual values.
A method that seems to work well within this context is the generalized degrees
of freedom method of Ye (1998). Key to this is the derivative of a fitted point from
a model with respect to the actual point. That is the degree to which the fitted point
will change in response to a change in the actual point. Unfortunately this usually
has to be estimated numerically for each data point.
The generalized degrees of freedom of a model fit to a data set is then the sum
across all the data points of the derivatives of the fitted points with respect to the
actual points, done one at a time. In a linear model this is just the number of
parameters. It seems to be a reasonable representation of the degrees of freedom
used up by a model fit, and so can be used like the number of parameters is used in
linear models to adjust goodness-of-fit measures, like NLL. A method of counting
the effective number of parameters is also built into the loo package. 3 Introduction to Parameter Reduction Methodology
Two currently popular parameter reduction methodologies are:
• Linear mixed models (LMM), or in the GLM environment GLMM
• Lasso—Least Absolute Shrinkage and Selection Operator 3.1 Linear Mixed Models
LMM starts by dividing the explanatory variables from a regression model into two
types: fixed effects and random effects. The parameters of the random effects are to
be shrunk towards zero, based perhaps on there being some question about whether
or not these parameters should be taken at face value. See for example Lindstrom
and Bates (1990) for a discussion in a more typical statistical context. 6 G. Venter Suppose you are doing a regression to estimate the contribution of various factors
to accident frequency of driver/vehicle combinations. You might make color of
car a random effect, thinking that probably most colors would not affect accident
frequency, but a few might, and even for those you would want the evidence to be
fairly strong. Then all the parameters for the car color random effects would be
shrunk towards or to zero, in line with this skepticism but with an openness to being
convinced.
This could be looked at as an analysis of the residuals. Suppose you have done the
regression without car color but suspect some colors might be important. You could
divide the residuals into groups by car color. Many of these groups of residuals
might average to zero, but a few could have positive or negative mean—some of
those by chance, however. In LMM you give color i parameter bi and specify that
bi is normal with mean zero and variance di 2 , where 2 is the regression variance
and di is a variance parameter for color i. LMM packages like in SAS, Matlab, R,
etc. generally allow a wide choice of covariance matrices for these variances, but
we will mainly describe the base case, where all of them are independent.
The di ’s are also parameters to be estimated. A color with consistently high
residuals is believably a real effect, and it would be estimated with a fairly high
di to allow bi to be away from zero. The bi ’s are usually assumed to be independent
of the residuals. LMM simultaneously maximizes the probability of the bi ’s, P.b/,
and the conditional probability of the observations given b, P.yjb/, by maximizing
the joint likelihood P.y; b/ D P.yjb/P.b/.
For a bi parameter to get further...

View
Full Document