Financial and Actuarial Mathematics
TU Wien, Austria

 2nd European Actuarial Journal (EAJ) Conference & Educational Workshop TU Vienna, September 8-12, 2014

# Abstracts of Posters

### ... in alphabetical Order

Poster Presentation, Section: Risk Management and Solvency II
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## CLARAMUNT BIELSA M. Mercè

Dept. Matemàtica Econòmica, Financera i actuarial, Facultat d'Economia i Empresa, Universitat de Barcelona, Spain

### Optimal stop-loss reinsurance under the maximization of the joint survival probability

The stop-loss reinsurance stands out among reinsurance contracts in the insurance market. It presents an interesting property: it is optimal if the criterion of minimizing the variance of the cost of the insurer is used. We analyse this contract in one period from the point of view of the insurer and the reinsurer. The optimal stop-loss contract is obtained if the criterion used is the maximization of the joint survival probability. We consider two different optimization problems.

In the first one, the reinsurance premium is fixed and so are the initial values of the reserves of the insurer and the reinsurer. In addition, the parameters of the reinsurance maximize the joint survival probability.

It is usually considered that the reinsurance premium is a function of the parameters of the stop-loss reinsurance and the total cost. In that instance, the reinsurer would apply for the calculation of the premium some of the usual criteria, for instance, the expected value, variance and standard deviation principles. We adopt as a criterion for the calculation of the reinsurer’s premiums the maximization of the joint survival probability, given as fixed both the values of the parameters of the reinsurance contract and the initial values of the reserves of the insurer and the reinsurer. Then, in the second optimization problem, the joint survival probability is considered to be a function of the reinsurance premium.

Joint work with Anna Castañer (Universitat de Barcelona).

Poster Presentation, Section: Life and Pension Insurance Mathematics
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## FRAGKOS Nikos

Department of Statistics, Athens University of Economics and Business, Greece

### The interplay between social security pension and saving plans: The economic value of tax incentives

We assume that individuals would be creating Individual Pension Plan accounts as a complementary investment plan to their Social Security.
To make people get interested in such investment one way is for the Government to give tax incentives:
Assume there are people at different tax brackets as a function of their income.
Suppose also that there will be investment accounts similar to the American IRA’s (or different depending on the country) sold in banks, investment houses or perhaps Private pension Companies like in Turkey.
The system will work as follows:
The citizen will be investing a predetermined amount to the personal account every month, which will be invested in financial portfolios of his choice.
(The amount will have a minimum value so that the proceeds will have some economic value at the end of the accumulation period. There will also be an upper value.)
The yearly amount saved/investment will be tax-deductible.
We may also assume that the investment returns will be tax free.
This will create a second benefit to the participant but at the same time it will give rise to a shortfall (will create a decrease) of government revenues.
The proposal of this study is to create a mechanism so that the government will “sort of” get back what they have given throughout the accumulation period through the tax incentives.

Joint work with Irini Dimitriyadis (Bahcesehir University Istanbul, Turkey).

Poster Presentation, Section: Risk Management and Solvency II
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## HEINY Johannes

University of Copenhagen, Denmark

### Asymptotic theory for large sample covariance matrices

In risk management an appropriate assessment of the dependence structure of multivariate data plays a crucial role for the trustworthiness of the obtained results. The case of heavy-tailed components is of particular interest.

We consider asymptotic properties of sample covariance matrices of such time series, where both the dimension and the sample size tend to infinity simultaneously.

Joint work with Thomas Mikosch.

Poster Presentation, Section: Non-Life Insurance Mathematics
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## KLEINERT Florian

University of Manchester, UK

### A formula for the ruin probability in finite time driven by Lévy processes

We study the problem of ruin probabilities with finite horizon driven by a Lévy processes. As it is well known typically no closed form solution for such problems exist and hence numerical methods are required. To this end we introduce a new algorithm, which is based on Carr's 'Canadisation' technique (Carr (1998)). This means we are working on a stochastic time grid. Following this, we introduce an explicit formula for the ruin probability in finite time driven by a huge class of Lévy processes. Hereby, the formula is viable for any Lévy process whose law at an independent, exponentially distributed time consists of a (possibly infinite) mixture of exponentials. This includes Compound Poisson processes, Brownian motion plus (hyper)exponential jumps, but also the recently introduced rich class of so-called meromorphic Lévy processes (Kyprianou et al. (2012)). We provide error bounds, illustrate the results with some numerics and compare them to cases where explicit formulas for the ruin probability on a deterministic time grid (Brownian Motion, Compound Poisson processes with exponential distributed claims) are known.

This is joint work with Kees van Schaik (University of Manchester).

Poster Presentation, Section: Non-Life Insurance Mathematics
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## KRAUS Daniel

Applied Statistics, Technische Universität München, Germany

### Using R vine copulas to explain dependencies of health care costs

Using R vine copulas in order to explain the dependencies of insurance data has become very popular. The reasons for this are manifold. High dimensional multivariate data sets with complex dependence structures (e.g. high tail dependence) can be modeled using only bivariate copulas by a pair copula construction (Aas et al (2009)). This procedure results in flexible model fitting with results that are easy to interpret.

The data set we will consider contains the total costs and deductibles of ambulant, stationary and dental treatments for the years 2005 to 2007 for a large number of insured individuals. Further, for each person specific and demographic variables such as age, gender and ZIP code are provided.

Using 9-dimensional R vine copulas we will investigate the dependence structure of the three different cost types between the three years. The likelihood of the resulting model is compared to models using several truncated R vine copulas with clustered tree structures, for example grouping the variables by years or categories.

Finally, we fit a model containing only the costs from 2005 and 2006 and use it to predict the health care costs of the year 2007 and compare the predictions with the actual results.

Reference:
[1] Aas, K., C. Czado, A. Frigessi, and H. Bakken. 2009. Pair-copula constructions of multiple dependence. Insurance Mathematics and Economics 44: 182-198.

Poster Presentation, Section: Non-Life Insurance Mathematics
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## MACCI Claudio

University of Rome Tor Vergata, Italy

### Asymptotic results for empirical means of independent geometric distributed random variables, and applications to weak records

We consider independent geometric distributed random variables which satisfy suitable hypotheses (in particular they can have different geometric distributions). We study large and moderate deviations for their empirical means. Moreover, motivated by the interest of weak records in insurance, we also present asymptotic results for sequences of weak records of i.i.d. discrete random variables. In fact it is known that these weak records can be expressed in terms of sums of independent geometric distributed random variables.

Joint work with Barbara Pacchiarotti.

Poster Presentation, Section: Mathematical Finance with Applications in Insurance
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## MAKOGIN Vitalii

Taras Shevchenko University of Kyiv, Ukraine

### Example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet

In the classical Black-Scholes pricing model the randomness of the stock price is due to Brownian motion. It had been suggested that one should replace the standard Brownian motion by a fractional Brownian motion. In the present talk we consider fractional Brownian sheet (multiparameter process). This Gaussian self-similar random field is an extension of a fractional Brownian motion.

We consider the fields which are self-similar with respect to every coordinate with individual index. Such fields are used to call anisotropic and in the Brownian case they usually are called as Brownian sheets. The investigation of self-similar random fields was caused by the evidence of the self-similarity property of phenomena in climatology and environmental sciences. But it is known that investigations of problems in climatology and financial mathematics often use the same stochastic models. So, applications of self-similar fields in finance and actuarial science are expected.

It is known that a fractional Brownian motion is a self-similar process with stationary increments. So, fractional Brownian motion is unique in the sense that the class of all fractional Brownian motions coincides with that of all Gaussian self-similar processes with stationary increments.

The fractional Brownian sheet has stationary rectangular increments. The properties of fractional Brownian sheet and fractional Brownian motion seem to be quite similar. The aim of this talk is an answer to the following question:
Is a fractional Brownian sheet unique Gaussian self-similar field with stationary rectangular increments?
The answer is no and we present an example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet.

We prove some properties of covariance function for self-similar fields with rectangular increments. Using Lamperti transformation we obtain necessary and sufficient conditions on covariance function of stationary field for the corresponding self-similar field to have stationary rectangular increments.

Joint work with Yuliya Mishura (Taras Shevchenko University of Kyiv).

References:
[1] Genton, M.G., Perrin, O., Taqqu, M.S.: Self-similarity and Lamperti transformation for random fields. Stochastic Models 23, 397-411 (2007).
[2] Makogin, V.I., Mishura, Yu.S.,: Strong limit theorems for anisotropic self-similar fields. Modern Stochastics: Theory and Application 1, 1-22 (2014).
[3] Makogin, V., Mishura, Yu.: Example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet. Preprint, (2014), arXiv:1403.1215.

Poster Presentation, Section: Non-Life Insurance Mathematics
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## MARTÍNEZ-MERINO Luisa Isabel

Department of Statistics and Operation Research, University of Cádiz, Spain

### A multivariate extension of the stop-loss order with applications in insurance

In insurance, an important reason for quantifying losses in the tail of distributions is to compare risks and, for this purpose, stochastic orders can be used. One of the most popular stochastic orders among univariate risks is the stop-loss order (also named the increasing convex order). In this work, we suggest a generalization of the stop-loss order to the multivariate setting to compare portfolios of risks. This new stochastic order is closely related to the multivariate risk measures recently introduced by Cousin and Di Bernardino (2013, 2014). In particular, we extend a stop-loss order preserving property for these measures from the case of Archimedean copulas (as stablished by Hürlimann, 2014) to the case of conditional increasing copulas.

Joint work with Miguel A. Sordo and Alfonso Suarez-Llorens.

Poster Presentation, Section: Non-Life Insurance Mathematics
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## NI Weihong

Institute for Actuarial and Financial Mathematics, Department of Mathematical Sciences, University of Liverpool, UK

### Ruin Probabilities with Dependence on the Number of Claims Within Fixed Window Time

We analyse the ruin probability for the Cramér renewal risk process with consideration of an inter-arrival time depending on a number of claims that have come within past fixed time-window. This adjusted model could be explained through the construction of a regenerative process whose properties are employed in further analysis. Asymptotic results of ruin probabilities for different regimes of the claim distributions will be examined and discussed followed by explanatory examples.

Joint work with Corina Constantinescu and Zbigniew Palmowski.

Poster Presentation, Section: Economics of Insurance
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## PETER Richard

Institute for Risk Management and Insurance, LMU Munich, Germany

### Risk management and saving: income effects and background risk

We study the interplay of intertemporal risk management and saving decisions. We define risk management broadly by allowing the activity to influence the severity of loss, the probability of loss or both simultaneously. Due to the similar cost-benefit structure of risk management and saving decisions a substitution effect arises whose implications we analyze for changes in income and background risk. Typically, the direct effects for risk management and saving move in the same direction but because of substitution net effects become a priori ambiguous. We resolve this ambiguity by deriving necessary and sufficient conditions. Our paper cautions against the use of single-instrument models as spurious results will emerge.

Joint work with Annette Hofmann (HSBA).

Keywords: risk management, saving, income effects, background risk, substitution

Poster Presentation, Section: Mathematical Finance with Applications in Insurance
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## RIBAS Carmen

Departament de Matemàtica Econòmica, Financera i Actuarial, University of Barcelona, Spain

### An investment-consumption model with life insurance and time-inconsistent preferences

Life insurance and life settlements are studied in an investment-consumption model in a continuous time setting. First, a consumption and portfolio rules problem with a life insurance is described. The problem is studied for the cases of CRRA and CARA utility functions. Special attention is devoted to the effects of changing the instantaneous discount rate of time preference. The model is described for different (deterministic) discount functions. Time-consistent equilibria are derived for models with time-inconsistent preferences. Next, life settlements are introduced for the problem when there is just one risk-free asset. For the model with life settlements, the problem of finding the optimal moment for selling the contract is analysed.

Joint work with Jesus Marin-Solano.

Poster Presentation, Section: Non-Life Insurance Mathematics
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## STØVE Bård

Department of Mathematics, University of Bergen, Norway

### Recognizing and visualizing copulas: an approach using local Gaussian approximation

In this paper we examine the relationship between a newly developed local dependence measure, the local Gaussian correlation, and standard copula theory. We are able to describe characteristics of the dependence structure in different copula models in terms of the local Gaussian correlation. Further, we construct a goodness-of-fit test for bivariate copula models. An essential ingredient of this test is the use of a canonical local Gaussian correlation and Gaussian pseudo-observations which make the test independent of the margins, so that it is a genuine test of the copula structure. A Monte Carlo study reveals that the test performs very well compared to a commonly used alternative test. We also propose two types of diagnostic plots which can be used to investigate the cause of a rejected null. Finally, our methods are applied to a "classical" insurance data set.

This is joint work with Geir Drage Berentsen, Dag Tjøstheim and Tommy Nordbø.

Poster Presentation, Section: Life and Pension Insurance Mathematics
Wednesday (Sept. 10, 2014), 17:30 - 19:00, at the Welcome Reception, main lecture hall (FH1)

## VIDAL-MELIÁ Carlos

Department of Financial Economics and Actuarial Science, University of Valencia, Spain

### Integrating retirement and long-term care (LTC) annuities using a notional defined contribution (NDC) framework

With the aim of improving the efficiency of LTC insurance and universalizing its coverage, this paper develops a multistate overlapping generations model (MOLG) that integrates retirement and LTC annuities into a generic NDC framework. The results achieved in the numerical example we present make sense and show an optimal integration of both annuities into the NDC framework. Our model has many practical implications for policymakers because it can be implemented without too much difficulty, it would help to mitigate individual risk, it would universalize LTC coverage with a "fixed" cost, it would make it easier to adapt the system to changing realities, and it would discourage politicians from making promises about future LTC benefits without the necessary funding support. The model is in line with observations made by Murtaugh et al (2001) and Webb (2009) in that the risks of LTC needs and longevity are negatively correlated, and therefore, as Brown & Warshawsky (2013) and Davidoff (2009) point out, the integration of LTC and retirement annuities into the same system may broaden its appeal. Pitacco (2002) also proposed the establishment of an LTC insurance scheme embedded within the retirement pension system as a way of improving the diffusion of LTC insurance cover. Similarly, Barr (2010) gives sound reasons for extending social security to provide mandatory cover for LTC. Finally, the paper by Ventura-Marco & Vidal-Meliá (2014a) shows that the NDC framework could be useful for this purpose.

Joint work with Javier Pla-Porcel (GMS Management Solutions S.L.) and Manuel Ventura-Marco (University of Valencia).

Keywords: NDC, Pay-as-you-go, Retirement, LTC Insurance, Social Security

Please send comments and suggestions to the conference & workshop secretariat e-mail: eaj2014@fam.tuwien.ac.at.

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