38th International ASTIN Colloquium Manchester 13-16 July, 2008 … · 2018. 5. 22. · Addressing...
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38th International ASTIN Colloquium Manchester 13-16 July, 2008 Abstracts
Transcript of 38th International ASTIN Colloquium Manchester 13-16 July, 2008 … · 2018. 5. 22. · Addressing...
38th International ASTIN Colloquium
Manchester13-16 July, 2008
Abstracts
1
Dynamic Financial Analysis of the Minimum Capital
Requirement. Empirical Analysis of Polish Non-life Insurance
Companies
Wojciech Bijak
Institute of Econometrics Warsaw School of Economics
Al. Niepodległości 164, 02-554 Warsaw, Poland [email protected]
Tel. +48 22 564 92 56 Fax. +48 22 564 86 17
Abstract
The Dynamic Financial Analysis (DFA) is a useful tool to forecast assets, liabilities, full financial statements (financial position) and probability of ruin of an insurance company. DFA can be used by regulators and supervisors of insurance markets to test different solutions on capital requirements especially standard formulas of Solvency Capital Requirement (SCR) and Minimum Capital Requirement (MCR). In the European Union in a framework of Solvency II project the Committee of European Insurance and Occupational Pension Supervisors (CEIOPS) identified the MCR as one of the critical issues from the political and technical points of view.
The article describes the role of DFA in the decision process of adoption of the standard MCR formula on the example of Polish non-life insurance companies. The paper presents an interpretation of Margin over Liabilities (MoL) approach to MCR calculation.
The study is based on empirical examples concerning non-life insurance companies operating in Poland in the period 1995-2006.
Key words: DFA, Solvency II, Solvency Margin, Minimum Capital Requirement.
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Capital Allocation by Percentile Layer Neil M. Bodoff, FCAS, MAAA
Willis Re Inc.
1 World Financial Center
200 Liberty Street, Third Floor
New York, NY 10281
Abstract
The goal of this paper is to describe a new approach to capital allocation; the catalyst for this new approach is a new formulation of the meaning of holding Value at Risk (VaR) capital. This new formulation expresses the firm’s total capital as the sum of many granular pieces of capital, or “percentile layers of capital”. As a result, one must allocate capital separately on each layer and perform the capital allocation across all layers. The resulting capital allocation procedure, “capital allocation by percentile layer”, exhibits several salient features. First, it allocates capital to all losses, rather than allocating capital only to extreme losses in the tail of the distribution. Second, despite allocating capital to this broad range of loss events, the proposed procedure does not allocate in proportion to average loss; rather, it allocates disproportionate capital to severe losses. Third, it allocates capital by relying neither upon esoteric parameters nor upon elusive risk preferences. Ultimately, on the practical plane, capital allocation by percentile layer produces allocations that are different from many other methods. Concomitantly, on the theoretical plane, capital allocation by percentile layer leads to new continuous formulas for risk load and utility. Keywords. Capital Allocation; Percentile Layer of Capital; Value at Risk; Enterprise Risk Management; Risk Load; Risk Adjusted Profitability.
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Addressing Credit and Basis Risk Arising From Hedging Weather-related Risk with Weather Derivatives
Patrick L Brockett
Red McCombs School of Business University of Texas at Austin
Linda L. Golden Red McCombs School of Business
University of Texas at Austin [email protected]
Charles C Yang
School of Finance, Insurance, and Economics Florida Atlantic University
Hong Zou Department of Finance and Insurance
Lingnan University, Hong Kong Abstract
Weather derivatives are a relatively recent innovation, but according to the Chicago Mercantile Exchange (CME), are the fastest growing derivative market today. They are used to manage weather related risks which previously were exclusively handled by insurance products, and represent the latest product in the continuing convergence of financial and insurance research and markets. A weather derivative pays-off based upon the realization of an underlying weather index, much like stock index based derivatives pays-off based upon a realization of the underlying stock index. The pay-off from a weather derivative can offset losses generated by adverse weather conditions. The use of these derivatives creates new risks, however, depending upon whether an exchange traded derivative or an over-the-counter (OTC) derivative is used. This paper examines the effectiveness of using a basis derivative strategy in conjunction with an exchange traded weather derivative to mitigate credit risk inherent in OTC transactions, and basis risk inherent in exchange-traded transactions. We examine the effectiveness of this strategy for summer and winter seasons, with both linear and nonlinear hedging instruments. Finally, we compare the effectiveness obtained using the CME and Risk Management Solutions, Inc. (RMS) weather indices. Results show that hedging methods are significantly more effective for winter than for summer, for both the CME and RMS weather indices, and for both linear and nonlinear basis derivative instruments. It is also found that the RMS regional weather indices are more effective than the CME weather indices for creating a basis hedging strategy, and that the effectiveness weather risk management can vary significantly by region of the country. Keywords: credit (default) risk, basis risk, weather risk management, hedging
effectiveness
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Return Attribution Analysis of the UK Insurance Portfolios
George A Christodoulakis1
Manchester Business School, University of Manchester
Emmanuel C Mamatzakis2
Department of Economics, University of Macedonia
January 2008
Abstract
We examine the attribution of premium growth rates for the five main insurance sectors
of the United Kingdom for the period 1969-2005; in particular, Property, Motor,
Pecuniary, Health & Accident, and Liability. In each sector, the growth rates of aggregate
insurance premiums are viewed as portfolio returns which we attribute to a number of
factors such as losses, expenses, their expectations and uncertainty as well as market
power, using the Sharpe (1988, 1992) Style Analysis. Our estimation method differs from
the standard least squares practice which does not provide confidence intervals for style
betas and adopts a Bayesian approach, resulting in a robust estimate of the entire
empirical distribution of each beta coefficients over the long run for the full sample. We
also perform a dynamic analysis of robust estimation for a rolling window of seven
overlapping samples. Our empirical findings show that there are some main differences
across industries as far as the shares attributed to the underlying factors in the long run.
Rolling regressions assist us to identify the variability of these shares over time, but also
across industries.
Keywords: Insurance Premiums, Monte Carlo Integration, Non-Negativity Constraints,
Return Attribution, Sharpe Style Analysis
1 Corresponding author, MBS Crawford House, Booth Street West, Manchester M15 6PB, UK,
[email protected] 2 Egnatia Avenue, Thessaloniki, Greece, [email protected]
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A Loss Reserving Method forIncomplete Claim Data
Rene DahmsBaloise, Aeschengraben 21, CH-4002 Basel
April 3, 2008
Abstract
A stochastic model of an additive loss reserving method based on the assumptionthat the claim reserves are good measures for the remaining exposure is presented.This model combines a projection of payments and a projection of correspondingreported amounts such that both leads to the same ultimate. In addition, thepresented method even works for some kind of incomplete triangles.
We will state estimators for the total necessary reserves and estimators for thecorresponding standard error. Moreover, we will discuss an example based on motorliability data, where we distinguish between property damage and bodily injuryclaims and only have proper data to do so for the last six accounting years.
Keywords: Stochastic Reserving, Loss Development, Mean SquaredError of Prediction.
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The Economic Consequences of a Global Energy Crisis
Anthony Day – Cyber Associates Ltd1
Abstract Modern society is dependent on the availability of growing supplies of energy. This paper looks initially at energy security in the UK: the change from an energy exporter to an energy importer, and the pressures on the electricity supply and distribution infrastructure which are likely to lead to expensive power failures.
At the global level, the oil price has risen dramatically since the start of 2008 and is now three to four times the long-term trend value for the 20th century – this in spite of recession in the US, which traditionally depresses demand. Peak Oil theory predicts an unavoidable decline in supply. The wide range of alternative or renewable fuels will be unable to plug the gap. In the face of energy shortages the future of globalisation and growth economics must be in doubt.
Keywords
Peak Oil, primary energy, coal, gas, renewables, recession, growth, investment returns, business continuity.
1 www.anthony-day.com ; www.susbiz.biz
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Catastrophe Risk Simulation
Measurement and Transfer of Catastrophic Risks. A Simulation
Analysis
Enrique de Alba
Actuarial Science Department, ITAM Department of Statistics and Actuarial Science , U. of Waterloo
e-mail: [email protected]
Jesús Zúñiga Dirección de Actuaría Corporativa, GNP
e-mail: [email protected]
Marco A. Ramírez Corzo e-mail: [email protected]
Abstract
When analyzing catastrophic risk, traditional measures for evaluating risk, such as the probable maximum loss (PML), value at risk (VaR), Tail VAR (TVaR), and others, can become practically impossible to obtain analytically in certain types of insurance, such as earthquake. Given the available information it can be very difficult for an insurer to measure this risk. The transfer of risk in this type of insurance is usually done through reinsurance schemes that can be of diverse types that can greatly reduce the extreme tail of the cedant’s loss distribution. This effect can be assessed mathematically. The PML is defined in terms of a very extreme quantile. Also, under standard operating conditions, insurers use several “layers” of non proportional reinsurance that will be combined with some type of proportional reinsurance. The resulting reinsurance structures will then be very complicated to analyze and to evaluate their mitigation or transfer effects analytically it may be necessary to use alternative approaches, such as Monte Carlo simulation methods. This is what we do in this paper in order to measure the effect of a complex reinsurance treaty on the risk profile of an insurance company. We compute the pure risk premium, PML as well as a host of results: impact on the insured portfolio, risk transfer effect of reinsurance programs, proportion of times reinsurance is exhausted, percentage of years it was necessary to contract reinstatements, etc. Since the estimators of quantiles are known to be biased, we explore the alternative of using an Extreme Value approach to complement the analysis. KEY WORDS: Quantile, Extreme Value, Monte Carlo Methods, PML, VAR, Reinsurance.
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Stochastic modelling of catastrophe risks in DFA models
Dorothea Diers∗
Abstract
Negative developments on the capital markets at the beginning of the millennium along with
the increase in natural catastrophes and terrorist attacks have substantially altered the risk
situation of the insurance industry. Insurance companies have reacted to the altered prevailing
conditions with a paradigm shift in corporate strategy developing from classical turnover
orientation to value- and risk-based management. Companies will only be able to assess the
level of risk capital and moreover the complete distribution of results according to corporate
risk structure with the help of high-quality internal models – DFA models – matched as closely
as possible to the risk situation of the individual company. Measuring and evaluating
catastrophe risk has come to be a very important issue, as a substantial share of the company’s
entire risk capital is committed to natural catastrophes. Whether or not internal models can be
applied depends largely on adequate catastrophe risk modelling. The following study aims to
present two actual approaches in modelling loss due to natural catastrophes taking storms as
an example. Both models use results from natural risks models. The first method is based on
processing complete event loss tables, while the second mathematical statistical approach uses
information from certain return periods. Both methods will be compared using example data,
and their advantages and disadvantages will be pointed out as applicable to value- and risk-
based management. Finally, the study will calculate risk capital, and test the impact of
strategies on risk capital requirement.
Keywords: Internal models, catastrophe risk, storm risk, natural risk models, event loss
tables, mathematical statistical models, risk capital, value- and risk-based management
∗
Provinzial NordWest Holding AG, Münster, Germany, [email protected]
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1
The Actuary’s Toolkit: A View from EMB
P.D. England
EMB Consultancy LLP, Saddlers Court, 64-74 East Street, Epsom, KT17 1HB. [email protected]
http://emb.co.uk
ABSTRACT
In this short paper, some of EMB’s software tools that have been developed to help solve business problems are described, from the perspective of the new and emerging statistical and risk management techniques that have been incorporated. In addition, the challenges of developing actuarial software, and actuarial education, are discussed.
KEYWORDS Bayesian, Bootstrap, Demand Modelling, Dynamic Financial Analysis, Generalised Linear Models, Internal Capital Models, Markov Chain Monte Carlo, Parameter Uncertainty, Predictive Modelling, Price Optimisation, Pricing, Simulation, Spatial Smoothing, Stochastic Reserving.
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ABOUT THE UNCERTAINTY OF PAST INFLATION
A MATHEMATICAL ANSWER TO WHY WE DON’T USE THE DATA OF 80 YEARS AGO
MICHAEL FACKLER
freelance actuary Klopstockstr. 8-IX
80804 Munich Germany
ABSTRACT Insurance data, before being used in statistics, often are adjusted for inflation, which is routinely done by application of a suitable inflation index. Whether or not the index exactly matches the inflation of the business remains somewhat uncertain. A model is proposed for this uncertainty, interpreting the gap between the “true” inflation and the applied index series as a random variable. For a special case the mean squared error of the sample mean is calculated. The result is quite different from the traditional model without inflation uncertainty but it looks very much as actuaries would intuitively say it should.
KEYWORDS
Inflation, index series, sample mean, mean squared error, AR(1), Wilkie model
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Reinsurance Contract Valuation When theLiabilities are of Fractional Brownian Motion
type
N. E. Frangos∗, A. N. YannacopoulosDepartment of Statistics,
Athens University of Economics and Business,and
S. D. Vrontos,Department of Statistics and Insurance Science,
University of Piraeus,Greece
Abstract
In this paper we study the valuation of reinsurance contracts for lia-bilities exhibiting long range dependence modelled by fractional Brownianmotion. We examine both aggregated excess and proportional reinsurancecontracts and we model the contract as an Asian type option. Specificallyusing fractional Ito calculus and ideas from option pricing theory we derivea partial differential equation the solution of which provides the value of thereinsurance policy. An analytical solution is found for this equation and theresults obtained by this approach are compared with the results obtained byMonte-Carlo simulation.
∗Corresponding author, email: [email protected]
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Towards an understanding approach of the Insurance Linked
Securities Market
Mathieu Gatumel∗ Dominique Guegan†
March 31, 2008
Abstract: The paper aims to present the insurance linked securities market behaviour, that
has changed a lot the past three years, both in terms of structure and in terms of ceded risks.
After having introduced some stylized facts characterizing the insurance linked securities we
capture their market price of risk, following the methodologies of Wang (2004), Lane (2000) and
Fermat Capital Management (2005). A dynamical study of the insurance linked securities is
also provided in order to understand the elements driving the spreads: the consequences of the
catastrophic events, the seasonality and the diversification effects between some different risks
are highlighted.
Keywords: insurance linked securities, cat. bonds, market price of risk.
JEL classification: G10, G12, G14
∗Corresponding author. Axa, 9 avenue de Messine, 75008 Paris and Paris School of Economics, CES-MSE,Universite Paris 1 Pantheon Sorbonne, 106/112 Boulevard de l’Hopital, 75647 Paris Cedex 13. Email: [email protected] and [email protected], Tel : +33 1 40 75 57 51.
†Paris School of Economics, CES-MSE, Universite Paris 1 Pantheon Sorbonne, 106/112 Boulevard de l’Hopital,75647 Paris Cedex 13. E-mail: [email protected]., Tel : +33 1 44 07 82 98.
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Climate change and its impact on
building water damage
Ola HaugXeni K. DimakosJofrid F. VardalMagne Aldrin
Norwegian Computing CenterP.O.Box 114, Blindern
N-0314 OSLO
E-mail address [email protected]
Abstract
The insurance industry, like other parts of the financial sector, isvulnerable to climate change. Life as well as non-life products areaffected, and knowledge of future loss levels is valuable. Risk andpremium calculations may be updated accordingly, and dedicated loss-preventive measures can be communicated to customers and regulators.
We have established statistical claims models for the coherence be-tween externally inflicted water damage to private buildings and se-lected meteorological variables. Based on such models and downscaledclimate predictions from the Hadley centre HadAM3H climate model,the estimated loss level of a future scenario period (2071-2100) is com-pared to that of a recent control period (1961-1990). On a nationalscale, payments increase by 15% and 20% under two different CO2
emissions scenarios, but there is substantial regional variability. Ofthe uncertainties inherently involved in such predictions, only the er-ror due to model fit is quantifiable.
Key words: Water damage, buildings, meteorological observations, climatemodel data, Generalized Linear Models, claims models, prediction.
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ON THE NON-LIFE SOLVENCY II MODEL
Werner Hürlimann IRIS integrated risk management ag
Bederstrasse 1 P.O. BOX, CH-8027 Zürich
E-mail : [email protected] URL : www.geocities.com/hurlimann53
Abstract We present a simple actuarial rationale for the non-life economic capital formula proposed for Solvency II in QIS3. On the statistical side we improve on the methodology in QIS3 by defining company specific estimators for all quantities of interest including premium risk and reserve risk volatilities as well as correlation coefficients at the granularity level of lines of business. We develop the non-life Solvency II economic capital formula applying both the value-at-risk (VaR) and conditional value-at-risk (CVaR) risk measures under a log-normal distribution of the portfolio combined ratio, which is defined as the ratio of incurred claims inclusive “run-off” to the premium and reserve volume. We determine confidence levels under which both methods yield approximately identical practical results. Moreover, we point out that economic capital modeling should neither be restricted to a log-normal distribution assumption nor to the VaR and CVaR risk measures and refer to the actuarial literature for various extensions on this. The portfolio combined ratio is decomposed in a weighted sum of the premium risk ratio and the reserve risk ratio as suggested in QIS3. Based on the basic portfolio risk ratio model summarized in the Appendix, we propose simple weighted estimators for all volatilities and correlation coefficients of interest. A numerical example illustrates the use of the proposed estimators.
Key words Non-life Solvency II, premium risk, reserve risk, value-at-risk, conditional value-at-risk,
weighted estimators
16
Transform approach for operational risk modelling: VaR and TCE
Jiwook JangDepartment of Actuarial Studies
Division of Economic and Financial StudiesMacquarie University, Sydney 2109, Australia
Genyuan FuPricewaterhouseCoopers Center
202 Hubin RoadShanghai 200021, People’s Republic of China
This version: April 1, 2007
Abstract
To quantify the aggregate losses from operational risk, we employ actuarial risk model, i.e.we consider compound Cox model of operational risk to deal with stochastic nature of itsfrequency rate in reality. A shot noise process is used for this purpose. A compound Poissonmodel is also considered as its counterpart for the case that operational loss frequency rateis deterministic. As the loss amounts arising due to mismanagement of operational risks areextremes in practice, we assume the loss sizes are Loggamma, Fréchet and truncated Gumbel.We also use an exponential distribution for the case of non-extreme losses. Employing lossdistribution approach, we derive the analytical/explicit forms of the Laplace transform of thedistribution of aggregate operational losses. The Value at Risk (VaR) and tail conditionalexpectation (TCE, also known as TailVaR) are used to evaluate the operational risk capitalcharge. Fast Fourier transform is used to approximate VaR and TCE numerically and the fig-ures of the distributions of aggregate operational losses are provided. Numerical comparisonsof VaRs and TCEs obtained using two compound processes are also made respectively.
Keywords: Operational risk; total loss; the compound Poisson/Cox process; shot noiseprocess; loss distribution; VaR; tail conditional expectation (TCE); Fast Fourier transform.
1
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Extensions of the Wang transform for the pricing of insurance and financial risks
Extensions of the Wang transform for the pricing ofinsurance and financial risks ∗
Masaaki Kijimaa, b and Yukio Muromachia
[email protected] [email protected]
a Graduate School of Social Sciences, Tokyo Metropolitan University,
1-1 Minami-Ohsawa, Hachiohji, Tokyo 192-0397, Japan.
b Daiwa Securities Chair, Graduate School of Economics, Kyoto University,
Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan.
(First Version: April 11, 2008)
Abstract. It is well known that the Wang transform [15] for the pricing of fi-
nancial and insurance risks is derived from Buhlmann’s economic premium prin-
ciple [1]. Recently, some extensions of the Wang transform are proposed, for
example, a multivariate extension by Kijima [5] and a extended class of proba-
bility transforms by Kijima and Muromachi [9]. In this article we describe the
essence of these recent extensions of the Wang transforms and show some special
examples related to the Student’s t distributions.
Keywords: Buhlmann’s economic premium principle, Wang transform, Gaus-
sian copula, non-central t distribution
∗Send all correspondence to Yukio Muromachi, Graduate School of Social Sciences, Tokyo Metropolitan
University, 1-1 Minami-Ohsawa, Hachiohji, Tokyo 192-0397, Japan. Email:[email protected].
1
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Optimal Insurance Coverage of a DurableConsumption Good with a Premium Loading
in a Continuous Time Economy
Masaaki Kijima∗ and Teruyoshi Suzuki†
Abstract
This article analyzes the optimal deductible level of insurance on durable consump-tion goods with a positive premium loading in a continuous-time economy. Assumingfinancial assets and durable consumption goods can be traded without transactioncosts, we provide an explicit solution for the optimal insurance coverage of durableconsumption goods together with optimal trading strategies for the amount of thedurable consumption goods and financial assets. Using the solution, we show thatan increase in premium loading decreases both demand for insured assets and theirinsurance coverage. We also show that an increase in premium loadings can affectoptimal investment strategies through the effects on the optimal amount of durableconsumption goods held. Moreover we show that there exist unique parameter valuesof the loss process such as when no insurance is optimal. Numerical examples help usunderstand how the risk of financial investment and the risk aversion measure affectsan agent’s optimal insurance coverage.
Keywords: Insurance, deductible, durable consumption goods, optimal consumptionand investment
∗Tokyo Metropolitan University, Daiwa Securities Group Chair: Kyoto University†Corresponding author: Graduate School of Economics Hokkaido University, NISHI 7, KITA 9, KITA-KU
SAPPORO 060-0809, JAPAN, e-mail:[email protected]
1
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Solvency II versus IFRS: Cost of Capital Implications for Insurance Firms*
Paul J M Klumpes1 and Kathryn Morgan2
Abstract Solvency II is being developed by the European Union in response to a desire harmonise insurance supervision and to effect greater consistency across the measurement of assets and liabililities of insurance companies across the European Union. Also it will improve the link between minimum solvency and the risks borne by EU insurance firms. Solvency II will require all insurers to formally assess their risks to assess what capital they need, which will be subject to regular monitoring and reporting to regulators. The resulting increased public disclosures are still in development, and may conflict with existing and proposed international financial reporting IFRS requirements for insurance firms, although the current QIS4 exercise is heavily reliant on IFRS, as far as that has developed for insurers. This paper identifies the key differences in measurement, risk margin, entity and reporting for insurance firms and considers their implications for incorporating these risks into cost of capital estimates. This paper discusses international diversity in accounting for insurance contracts as reported under various Generally Accepted Accounting Principles in the UK. Relative to statutory-based accounting principles, UK GAAPs currently allow insurance firms to match income to expenses over the term of an insurance contract in order to provide a more ‘realistic’ basis for reporting to shareholders. However UK GAAP does not employ a coherent and consistent view of how to measure the fair value of a insurance firm’s business. The International Accounting Standards Board (IASB) has tentatively concluded that fair value should be used in accounting for insurance contracts. This paper discusses how existing UK GAAPs differ from statutory accounting principles (SAP), IFRS (proposed fair value) and Solvency II. We use the analysis from this discussion to present new evidence on the cost of equity capital by line of business for for a sample of UK general insurance firms. To do so we obtain beta estimates and then use the full-information industry beta (FIB) methodology to decompose the cost of capital by line. We obtain full information beta estimates using the standard one-factor CAPM model, accounting-based estimates and extend the FIB methodology to incorporate the Fama-French three factor cost of capital model. Contrary to the results of prior research, the analysis suggests that the cost of capital for firms using the Fama-French model are significantly higher than estimates based on the CAPM. In addition, we find evidence of significant differences in the cost of capital across business lines, indicating that the use of a single company-wide cost of capital is generally not appropriate. Finally, we find that cost of capital estimates are significantly understated when applied using GAAP or SAP relative to either IFRS or Solvency II definitions of profit and book value.
* We appreciate comments provided on an earlier draft by participants at an accounting workshop at the University of Warwick. 1 Tanaka Business School, South Kensington Campus, London SW7 2AZ, United Kingdom, Tel: +44 2078237685 Fax: +44 207 8237685 Email: [email protected] 2 UK Actuarial Profession.
20
Most elegant Premium Formulas for the most
general Drop Down Excess of Loss Cover
Erhard Kremer
University of Hamburg
Department of Mathematics
Bundesstr.55
20146 Hamburg
Germany
Abstract
The most general version of the Drop Down Excess-of-Loss Cover is considered. For
that an elegant net-premium theory is derived, giving most elegant net-premium
formulas. The general results are also specialized to three most important special
treaties. The advantage of the new results (to the former ones of Kremer (2005a))
is that certain summations are finite instead of being infinite.
keywords: Net premium, drop down excess-of-loss cover
1
21
Give Credit where Credit is due:Operational Risk goes Bayesian
Dominik D. Lambrigger1 Pavel V. Shevchenko2 Mario V. Wüthrich1
Abstract
Under the new regulatory standards Basel II and Solvency 2, many financialinstitutions adopt a Loss Distribution Approach (LDA) to estimate the opera-tional risk capital charge.
Such an approach requires the combination of internal and external datawith expert opinion in an adequate manner. In this article we present a con-sistent and unified way how this task can be fulfilled. The simultaneous con-sideration of the three different sources of information is done in a Bayesianinference model.
The main idea is to start with external market data which determines aprior estimate. This prior estimate is then modified according to internal ob-servations and expert opinion leading to a posterior estimate. Risk measures asfor instance Value-at-Risk and Expected Shortfall may then easily be inferredfrom this posterior knowledge.
Keywords: Advanced Measurement Approach, Basel II, Bayesian Inference, Loss Dis-tribution Approach, Operational Risk, Quantitative Risk Management, Solvency 2.
1ETH Zurich, Department of Mathematics, 8092 Zurich, Switzerland.2CSIRO Mathematical and Information Sciences, Sydney, Locked Bag 17, North Ryde, NSW,
1670, Australia; e-mail: [email protected].
22
Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims
Huijuan Liu Allianz Global Corporate & Specialty
27 Leadenhall Street, London [email protected]
Richard Verrall
Faculty of Actuarial Science and Insurance Cass Business School, City University, London
Abstract This paper presents a new bootstrap approach to the estimation of the prediction distributions of reserves produced by the Munich chain ladder (MCL) model. The MCL model was introduced by Quarg and Mack (2002) and takes into account of both paid and incurred claims information. In order to produce a bootstrap distribution, this papers addresses the application of bootstrapping methods to dependent data, with the consequence that correlations between the data sets is considered. Numerical examples are provided to illustrate the algorithm and the prediction errors are compared for the new bootstrapping method applied to MCL and a more standard bootstrapping method applied to the chain-ladder technique.
Keywords
Bootstrap, Munich chain ladder, Correlation, Simulation
23
Dynamic Financial Analysis: An Untrodden Path of Catrisk
and Solvency II
Chitro Majumdar1
Extended Abstract
DFA – Dynamic Financial Analysis grew in late 1990s out of practical need rather than
academic research with Casualty Actuarial Society (CAS) being the pioneer of this
genesis with specific needs of insurance sector in mind. In the last few years, non life
insurance corporations in western world have experienced, among other things, pricing
cycles accompanied by volatile insurance profits and increasing catastrophe losses
contrasted by well performing capital markets.
Here are the DFA solutions of several challenges in (re)insurance/ capital market world:
1. DFA (Dynamic Financial Analysis): DFA offering enables the creation of a comprehensive framework to manage Enterprise Risk. DFA in the capital budgeting
decision process of a company is launching a new invention and predicting the impact of
the strategic decision on the balance sheet in a horizon of few years. The P&C Insurance
Risk for the Purpose of Solvency Economic Capital Requirement will be solved by a
multi-line risk factor model where we generate scenarios and apply risk measures to
quantifying inter-dependencies among risks. DFA gives strategy for Enterprise Risk
Management in order to avoid undesirable outcomes which could be disastrous. Hence,
we focus on the downside and on the tail of the profit and loss of distribution is a critical
part of ERM. However, higher returns may require taking more risk. I am limiting the use
of DFA and ERM analytics to scrutinizing the distribution's tail leads to effectively
ignoring valuable information embedded in other parts of the distribution with more
comprehensive type of dependence structure on results by comparing linear and nonlinear
dependencies & copulas. This helps organization to improve their risk control.
2. This has severely affected shareholder value as well as solvency position of many
non-life companies amidst rigid regulatory constraints, hyper competition and changing
risk landscape. The net result capital drains into more profitable market segments.
3. In comparison to the earlier static environment of the insurance market where there
was no particular need for sophisticated analytics: actuarial analysis was carried out on
the
underwriting side – without linkage to the investment side of the company, which was
analyzed separately. Quantitative analysis amounted to modelling a group of isolated
silos, without taking a holistic view. This completely ignored the interplay & interrelation
amongst
1
CHITRO MAJUMDAR: Oracle Financial Services, Level 25, 40 Bank Street, Canary Wharf, London 14
5NR, UK; Email: [email protected].
24
the various risk factors in evaluating a strategic decision and hence in the process ignored
the major economic factors driving shareholder value. DFA overcomes this drawback by
joint
stochastic simulation of risk factors affecting a company. Risk factors typically include
economic risks (e.g. inflation), underwriting risks, asset risks (e.g. stock market return)
and
business risks (e.g. underwriting cycles). We see the DFA to be a more generic tool that
extends the conventional boundaries of the insurance industry. The outcome of a DFA
analysis depends on the stakeholders and objectives of the analysis.
4. DFA tries to facilitate and help justify or explain strategic management decisions with
respect to Strategic asset allocation, Capital allocation & budgeting, performance
measurement, market strategies, business mix, pricing decisions to even pro-forma
analysis. The key success factor in designing an efficient DFA system is not only in the
choice of proper econometric models but to identify the key random variables affecting
asset and liability cash flows and their interrelationship.
5. The factors are then modularized depending on the business and simulated
stochastically under different scenarios before simulating their combined effect on the
company’s bottom line.
6. A properly designed DFA tool hence in that can be used for management decision
support for different strategic alternatives and also as a benchmark for performance. The
strategic assumptions made during the strategic planning process can also be simulated or
changed easily in due course of time for strategic adjustments and control.
7. Commercially available DFA products are usually tailored for specific industry sectors
and hardly have any flexibility, which makes it an analogous tool to Scenario testing.
� We see DFA as a tool for management decision support, which culminates in
predicted financial statements and the distribution of important financial indicators on the
strategic horizon.
� As a concrete example of our vision we site the use of DFA in the capital budgeting
decision process of a company launching a new product and predicting the impact of the
strategic decision on the balance sheet in a horizon of 2-3 years.
� To identify the few factors that will affect the asset liability cash flow are demand
uncertainty, sales volatility, credit risk, volatility in the price of raw materials cost of
capital to name a few.
� Each of these random variables can be stochastically simulated either based on the
distribution of retrospective data or under strategic assumptions. When simulated in a
combined way the future cash flows can be predicted which in return would dictate the
capital requirements in the future.
25
� Depending on the capital structure of the company and simulated interest rate in the
capital market the final earnings volatility of the company can be predicted to identify the
return and associated risks.
� We first decompose the historical vintage performance data into a maturation function
of months-on-books, a quality function of vintage origination date, and an exogenous
function of calendar date. In a second step, the exogenous function is modeled with
macroeconomic data or factors representing portfolio management impacts.
� Stress tests are performed by extrapolating the exogenous function using externally
provided scenarios for extreme macroeconomic events.
� Given concerns about exposure data quality and about model inaccuracy, it is unclear
if the insurers are properly measuring the uncertainty associated with extreme outcomes.
Consideration of that uncertainty leads directly to the determination of the capital
required to support operations. The combination of sophisticated mathematical
measurement of extreme outcomes and dynamic financial analysis (DFA) could lead to a
stronger and more transparent industry.
� The present market is very interested in catastrophe measurement and management
and is planning to begin the collection of relevant data from the regulated insurers. The
expectation is that analysis of the data will lead to a regulatory process involving modern
mathematical techniques and DFA; in particular it is worthwhile to consider the
integration of certain copula modeling techniques in a DFA framework and to conduct
numerical validation tests within a simulation study.
� Possible approaches could include the catastrophic events via Markov Jump Diffusion
Models. For catastrophic events, the assumption that catastrophe claims occur in terms of
the Poisson process is inadequate as these processes rely on the assumption of constant
intensity. Markov Modulated Poisson process can be used to model more realistically the
arrival process for catastrophic events.
� Under the process, the underlying state is governed by a homogenous Markov chain,
in which Markov jump diffusion models can be used to derive pricing formulas and
hedging formulas for catastrophe insurance products, e.g. futures, options, & catastrophe
bonds.
� The methodology could then show how catastrophe insurance product prices are
related to the jump rate of catastrophe events, standard deviation of jump size, and mean
of jump size. Such techniques can also be modified to apply to financial catastrophes and
other extreme events.
� We will find built-in support for most DFA modelling needs: Portfolio risk
management and modelling, Economic capital estimation and allocation, Evaluation of
ceded reinsurance and other management strategies, Cash flow analysis, financial
statement modelling and projections.
26
� The approach is demonstrated in an analysis of the US Mortgage markets.
� “Product”- following should be the terms:
A. Liability Curves, Changes there in due to- a. new cadent / deal, b. New instruments, c.
New Events;
B. Asset Distributions- a. Current Market Senaior
C. Ratings / Solvancy II- a. Cash Flow, b. Simulations: Asset / Liability Balance
Keywords include: Dual-time Dynamics, Nonlinear Dynamics, Time Series Analysis,
Portfolio Forecasting, Scenario-based Forecasting, Retail Lending, Stress Testing,
Macroeconomic Scenarios
Acknowledgement: Michael Smith, Alex Bushel, Sayaji Hande, Liz Thrussell
27
Quantile Based VaR (Value at Risk) to Appraise an Insurance/Reinsurance Business’s ERM-Economic Risk Capital
Chitro Majumdar*
Abstract An urbane econometric theory has been concerned with devising robust and analytically sound techniques for assessing the risk of an asset portfolio. The works of some of the greats of the profession have been used to develop a number of risk measures for quantifying different categories of risks. For an example, volatility measures the uncertainty of returns, beta measures how much an individual asset is likely to move with the general market and Value at Risk, which is a recent innovation, measures the maximum loss (in the probabilistic sense) that is likely to be occurred in the immediate future. Ironically, almost all risk measures developed yet are applicable to a symmetric loss distribution. VaR is not an exception to that tendency. Unfortunately, in the real world, skewness of the loss distribution is not an uncommon feature. Especially for an emerging market (also for developed market), where equity prices are observed to remain in the high volatile state for a reasonably longer time period, the loss distribution is expected to have a skewed nature. In such a situation, a VaR based approach would naturally provide an exuberant / or conservative measure for the economic risk capital of an insurance/ reinsurance business. The firms disperse the risk-capital tradeoff, so that the whole organization makes decisions optimizing the tradeoff. And there will be an issue while measuring the operational risk from ERM (Enterprise Risk Management) perspective. It is important for firms to look beyond the VaR measure for an insurance company’s asset portfolio allocations which could estimate at the probability level corresponding to a default threshold. Then the ERM creates value by optimizing (to measures of tail risk either) the present value of financial distress costs. Moreover we will compare our results with the application of DFA portfolio risk management and modelling, economic capital estimation and allocation, evaluation of ceded reinsurance and other management strategies, cash flow analysis, financial statement modelling and projections. We have introduced an alternative to VaR, advantages of which are akin to the conventional VaR model. However, the new measure would not be viable to the serious limitation of the VaR model. Key words: VaR, ERM Modelling, DFA (Dynamic Financial Analysis) * Chitro Majumdar: Practice leader -Credit derivative, DFA modelling, Oracle Financial Services, Level 25, 40 Bank Street, Canary Wharf, London E14 5NR; [email protected], [email protected], Phone: +44 207 531 4400
28
ZONE-ADAPTIVE CONTROL STRATEGY FOR
A MULTIPERIODIC MODEL OF RISK
Vsevolod K. Malinovskii
Finance Academy
25468, Leningradskiy prosp., 49, Moscow, Russia
E-mail: [email protected]
Abstract
In this paper intended to illustrate the adaptive control approach in insurance, a zone-adaptive control strategy harmonizing the requirements of principles of solvency and equity isconsidered in the simplistic framework of diffusion multiperiodic risk model. The adjacent worksby the author set the similar adaptive control strategies in more realistic Poisson-exponentialmultiperiodic risk model. The room for further generalizations is large. In particular, it is therisk theory insight into the problem of asset–liability and solvency adaptive management ininsurance under deficient information. The latter means that the intensities of the successiveannual claim arrival processes are the random variables which comply with a certain scenario.
Keywords: Multiperiodic insurance process, equity, solvency, zone-adaptive control, diffu-sion risk model
29
Hurricanes in the North Atlantic, should insurance pricing be based on long-term averages?
Trevor Maynard1
Abstract
The key question this note attempts to discuss is “do long-term average hurricane landfall rates adequately reflect current risk levels?” In statistical parlance we could rephrase this as “is the hurricane landfall timeseries stationary?”. It is important to realise the difference between genesis frequency (the number of storms actually arising each year) and US landfall frequency (note this excludes landfalling storms that fall outside of the US). Insurers are interested, primarily, in landfalling storms. This note will present a simple argument to suggest that we should assume the landfall series is not stationary; and should adjust long-term averages accordingly.
1 Trevor Maynard, Manager, Emerging Risks, Lloyds Exposure Management, 1 Lime Street, London, EC3M 7HA, UK. Tel: +44 (0)20 7327 6141, E-mail: [email protected]
30
Modelling the Claims Development Result for
Solvency Purposes
Michael Merz∗, Mario V. Wuthrich†
Abstract
We assume that the claims liability process satisfies the distribution-free chain-
ladder model assumptions. For claims reserving at time I we predict the total
ultimate claim with the information available at time I and, similarly, at time I + 1
we predict the same total ultimate claim with the (updated) information available
at time I + 1. The claims development result at time I + 1 for accounting year
(I, I+1] is then defined to be the difference between these two successive predictions
for the total ultimate claim. In [4, 8] we have analyzed this claims develpment
result and we have quantified its prediction uncertainty. Here, we simplify and
illustrate the results obtained in [4, 8]. We emphasize that these results have direct
consequences for solvency considerations and were (under the new risk-adjusted
solvency regulation) already implemented in industry.
Keywords: Stochastic Claims Reserving, Chain Ladder Method, Claims Development Result,
Loss Experience, Incurred Losses Prior Accident Years, Solvency, Mean Square Error of Predic-
tion.
References:
[4] Merz, M., Wuthrich, M.V. (2007). Prediction error of the expected claims development
result in the chain ladder method. Bulletin of Swiss Association of Actuaries, 1, 117-137.
[8] Wuthrich, M.V., Merz, M., Lysenko, N. (2008). Uncertainty in the claims development
result in the chain ladder method. Accepted for publication in Scand. Act. J.
∗University Tubingen, Faculty of Economics, D-72074 Tubingen, Germany†ETH Zurich, Department of Mathematics, CH-8092 Zurich, Switzerland
1
31
Tax-Deductible Pre-Event Catastrophe Loss Reserves: The Case of Florida1,2
By
Andreas Milidonis3, Ph.D. Martin F. Grace Ph.D., JD
James S. Kemper Professor of Risk Management Department of Risk Management & Insurance, Robinson College of Business, Georgia State University, Atlanta, GA, 30303, USA. Tel: +1 404 413-7469; Email: [email protected]
Accounting and Finance Division, Manchester Business School, University of Manchester, Crawford House, Booth Street East, Manchester, M13 9PL, U.K. Tel: +44(0)161-275-3936 Email: [email protected]
August 21, 2007
Abstract After Hurricane Andrew the U.S. Congress entertained proposals to allow insurers to
employ tax-deferred loss reserves. Interest was strong at first, but as the events receded interest waned. However, after the most recent severe hurricane seasons the proposals are again being discussed. In this paper we examine the institution of catastrophic loss reserves in a stylized model of insurance provisions. First, we find that the benefits of the tax-deferred loss reserves depend on the actuarial assumptions regarding the expected loss distribution. Second, we make the first attempt at estimating the change in consumer behavior and the social welfare implications for permitting tax deferred loss reserves. In sum, we find under specific circumstances there are large welfare gains for allowing the tax deferral of reserves.
Keywords Tax-Deferred Loss Reserve; Catastrophe Financing and Pricing; Extreme Value Theory; Mixture model; Insurance; Reinsurance; 1 This paper will appear ASTIN Bulletin volume 38.1. 2 We would like to thank Sam Cox for suggestions on actuarial methodology. We would also like to thank Richard Derrig, Richard Phillips and Tyler Leverty for comments and assistance on an earlier version of the paper and further acknowledge the prompt and valuable help of Mr. Edward N. Trevelyan of the U.S. Census Bureau. Finally, we would like to thank an anonymous referee for his/her insightful comments. All errors or omissions, however, are the authors’ responsibility. 3 Contact Author.
32
An Actuarial Model of Cross Subsidization in Price-Regulated Insurance Markets under Moral Hazard
Andreas Milidonis, Ph.D. Accounting & Finance Division,
Manchester Business School, The University of Manchester,
Crawford House, Booth Street East, Manchester, M13 9PL, UK. Tel: +44(0) 161 275 3936. Fax: +44(0) 161 275 4023.
Email: [email protected]
Abstract
In a market where consumers can observe their risk characteristics, price regulation may spur moral hazard on behalf of high-risk consumers. Such moral hazard could aggravate cross subsidization from low-risk to high-risk consumers. We construct an inter-temporal model of cross-subsidization from low-risk to high-risk consumers in a price-regulated insurance industry. We analyze the process by which low-risk individuals end up paying more than their actuarial fair premiums to make up for the excess losses generated by high-risk individuals. In a competitive insurance market, the most risky of the low-risk consumers are forced to enter the residual market as their adjusted premium becomes higher than the premium set by the regulator. As a result the size and excess losses of the residual market increase until a new, higher premium is set by the regulator. The model can be applied to similar price-regulated markets in order to make inferences about population movements between risk groups, expected increases in low-risk premiums and the time or level of the next action to be taken by the regulator. Keywords: Price Regulation, Assigned Risk Pool, Insurance, Moral Hazard, Loss Models.
33
The Solvency II Actuary
Kathryn Morgan Financial Services Authority,
25 The North Colonnade, Canary Wharf,
London, E14 5HS [email protected]
Annette Olesen
PricewaterhouseCoopers LLP 6 Hay's Lane,
London, SE1 2HB
Abstract: Solvency II is a fundamental change for the insurance industry. As a profession we have a key part to play to fulfill the requirement for an Actuarial Function as set out in Article 47 of the framework Directive, and the challenge to get involved more widely in the risk management of the organisation to potentially assist and play a key role the assessment of regulatory capital requirements. These parts of the solvency framework have not been assigned solely to the Actuarial Function but are areas where we face the challenge of demonstrating that we can add value and that we should be involved.
We need to drive the development of methodologies and deal with a number of fairly complex technical challenges not all of which we do today. This paper presents some of those challenges. Over and above this, we must be able to clearly communicate complex issues and results to our colleagues and senior management.
We believe that Solvency II will fundamentally change and set standards for our work for years to come.
34
The one-year non-life insurance risk
Ohlsson, Esbjorn∗ & Lauzeningks, Jan†
Abstract
With few exceptions, the literature on non-life insurance reserve
risk has been devoted to the ultimo risk, the risk in the full run-
off of the liabilities. This is in contrast to the short time horizon
in models for the total risk of the insurance company, and in
particular the one-year risk perspective taken in the Solvency II
project, and in the computation of risk margins with the Cost-of-
Capital method. This paper aims at clarifying the methodology
for the one-year risk; in particular we describe a simulation ap-
proach to the one-year reserve risk. We also discuss the one-year
premium risk and the premium reserve. Finally, we initiate a
discussion on the role of risk margins and discounting for the
reserve and premium risk.1
Keywords
Reserve risk, premium risk, Solvency II, IFRS 4 phase II, risk
margin, Dynamic Financial Analysis, stochastic reserving.
∗Lansforsakringar Alliance, SE-106 50 Stockholm, Sweden, E-mail:
[email protected]†Gothaer Allgemeine Versicherung AG, Germany1Any opinions expressed in this paper are the authors’ own, and do not constitute
policy of our employers.
1
35
Model risk in claims reserving within Tweedie’s
compound Poisson models
Gareth W. Peters1,2 Pavel V. Shevchenko2,∗ Mario V. Wuthrich3
1 UNSW Mathematics and Statistics Department, Sydney, 2052, Australia2 CSIRO Mathematical and Information Sciences, Sydney, Locked Bag 17,
North Ryde, NSW, 1670, Australia3 ETH Zurich, Department of Mathematics, CH-8092 Zurich, Switzerland∗ presenting/corresponding author, e-mail: [email protected]
Abstract
In this paper we examine the claims reserving problem using Tweedie’s compound
Poisson model. We develop maximum likelihood and Bayesian Markov chain Monte
Carlo simulation approaches to fit the model and then compare estimated models
under different scenarios. The key point we demonstrate relates to comparison
of reserving quantities with and without model uncertainty incorporated into the
prediction. We consider both the model selection problem and the model averaging
solutions for the predicted reserves. As a part of this process we also consider the
sub problem of variable selection to obtain a parsimonious representation of the
model being fitted.
Keywords: Claims reserving, model risk, Tweedie’s compound Poisson model,
Bayesian analysis, model selection, model averaging, Markov chain Monte Carlo.
36
Expected shortfall of claims amounts: some practical aspects†
Frédéric Planchet∗ Pierre-E. Thérondα
ISFA – Laboratoire SAF β Université de Lyon - Université Claude Bernard Lyon 1
WINTER & Associés γ
Abstract
In this paper, we focus on the practical aspects of the methodologies which enable the calculation of the expected shortfall of claims amounts. If many theoretical results exist on the subject, the operational setting for applications is seldom discussed. One shows that the choice of the method to adopt requires some particular investigations. Moreover the results obtained are overall not very robust when the only extreme value theory is used (without making an assumption on the structure of the tail distribution). KEYWORDS: Reserving, extreme values.
Espérance du coût des sinistres au-delà d’un seuil : quelques aspects pratiques
Résumé
Le présent article s’intéresse à la robustesse des méthodes pratiques de calcul de l’espérance du coût des sinistres au-delà d’un seuil. Si de nombreuses réflexions théoriques existent sur le sujet, la mise en œuvre opérationnelle est rarement discutée. On montre que le choix de la méthode à retenir nécessite des investigations particulières et que les résultats obtenus sont globalement peu robustes lorsque l’on s’en tient à l’utilisation de la théorie des valeurs extrêmes. MOTS-CLEFS : Provisionnement, valeurs extrêmes.
† This paper will be presented in English. ∗ Corresponding author. Contact : [email protected] α Contact : [email protected] β Institut de Science Financière et d’Assurances (ISFA) - 50 avenue Tony Garnier - 69366 Lyon Cedex 07 – France. γ WINTER & Associés – 43-47 avenue de la Grande Armée - 75116 Paris et 18 avenue Félix Faure - 69007 Lyon – France.
37
A Dynamic Model of a Non-Life Insurance Portfolio
Colin M. Ramsay
University of Nebraska-Lincoln
Finance Department — 210 CBA
Lincoln NE 68588-0426, USA
E-mail: [email protected]
April 17, 2008
Abstract
There are two major problems with most actuarial models used in risk theory: (i)
they view an insurance company’s operation as static, i.e., policyholders remains con-
stant, and (ii) they assume the demand for insurance is perfectly inelastic, i.e., potential
policyholders are insensitive to the size of the premium charged. These assumptions
are clearly false in practice. As a step toward more realistic mathematical models, this
paper introduces a simple dynamic model of an insurance portfolio where number of
insured policyholders is a stochastic process and the rate of sale of new policies de-
pends on the premium charged. We assume the insurer is a pure monopolist facing
various types of costs (to obtain and maintain policies, and to process claims) and a
linear demand function for its insurance policies. The insurer’s profit maximizing pre-
mium is determined and is compared to the traditional actuarial premium based on the
equivalence principle. It turns out that the optimal premium is the average of the equiv-
alence principle premium and the maximum premium a policyholder would be willing
to pay, and that the equivalence principle premium priced portfolio is expected to be
twice as large as the profit maximized portfolio. In addition, expressions are provided
for the portfolio’s value, value-at-risk, and tail value-at-risk. Finally, we consider the case
where there are several insurers with different costs selling identical policies and operat-
ing within a Cournot oligopoly framework and provide an expression for each insurer’s
profit maximizing premium.
Key words and phrases: demand function, new sales, profit maximization, value-at-risk,
tail value-at-risk, monololy, Cournot oligopoly
1
38
MODELLING AGGREGATE NON-LIFE UNDERWRITING RISK: STANDARD FORMULA VS INTERNAL MODEL
SAVELLI Nino CLEMENTE Gian Paolo Catholic University of Milan Catholic University of Milan
Largo Gemelli, 1 – 20123 Milano Largo Gemelli, 1 – 20123 Milano Email: [email protected] Email: [email protected]
Abstract The main target of this paper is to analyse the risk profile of a multi-line non-life insurer. A risk theoretical simulation model is then applied with the aim to predict the risk capital regarding only premium risk. A systematic comparison has been performed between Risk Based Capital obtained by the application of an Internal Model and the corresponding Solvency Capital Requirement as provided by the Solvency II-QIS3 standard formula for different insurers according to dimension and risk distribution. Finally the paper discusses the dependence problem: losses from different line of business are linked by different approaches. At this regard the dependence effect on RBC is examined comparing the QIS3 aggregation formula (using correlation matrix) with Internal Model results. Furthermore different results are obtained applying either elliptical copula functions and approximation formula based on linear correlation
Keywords: Aggregation and dependency in non-life insurance, Premium Risk, Internal Model, Solvency II-QIS3 Standard Formula.
39
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40
Bornhuetter–Ferguson as aGeneral Principle of Loss Reserving
By Klaus D. Schmidt
Lehrstuhl fur VersicherungsmathematikTechnische Universitat Dresden
D–01062 Dresden
e–mail: [email protected]
Abstract
It is the purpose of this paper to demonstrate that a straightforward extensionof the Bornhuetter–Ferguson method provides a general principle comprisingvarious methods of loss reserving which are based on run–off triangles. Themost prominent methods underlying the Bornhuetter–Ferguson principle arethe chain–ladder method and the more general loss–development method, aswell as the additive method and the more general Cape–Cod method.Besides its force of unifying these and other methods of loss reserving, theBornhuetter–Ferguson principle can be used also to generate new methodsof loss reserving, to evaluate the range of predictors depending on differentsources of information, and to compare the portfolio under consideration withthe market portfolio.The present paper summarizes results of Schmidt [2006] and of Schmidt andZocher [2008]; see also Radtke and Schmidt [2004].
Keywords: Additive method, Bornhuetter–Ferguson principle, Bornhuetter–Ferguson method, Cape–Cod method, chain–ladder method, developmentpattern, loss–development method, multiplicative model.
41
1
Carbon, complexity and insurance
Nick Silver
Abstract
The purpose of this paper is to look at the insurance needs of the carbon market. The
carbon market is an artificial market created under the Kyoto Protocol to trade
emissions reductions.
The aim of the carbon market is to reduce anthropogenic carbon emissions to mitigate
against dangerous climate change. The first part of the paper gives a description of the
market. Drawing on complexity economics, I then go on to argue that the markets are
likely to be of limited success in achieving their aims.
The paper then describes how carefully designed insurance products, such as a
delivery wrap or a carbon cat bond, could improve the effectiveness of the market by
reducing investment risk and hence enhancing capital flows into carbon abatement
projects.
Keywords
Climate change, carbon trading, Clean Development Mechanism, insurance,
complexity economics, emissions targets, basis risk
Nick Silver, FIA, Parhelion Underwriting Limited
E-mail: [email protected]
42
XL Property rating : a reinsurance pricing tool
combining experience and exposure rating for
property excess of loss treaties
M. Snoussi∗, S. Desmedt∗‡, X. Chenut∗ and J.F. Walhin¶§
∗ Secura, 9-31 Avenue des Nerviens, B-1040 Brussels, Belgium‡ Corresponding author, Tel: +32 2 5048228, e-mail: [email protected]¶ Fortis, Montagne du Parc 3 (1MA3F), B-1000 Brussels, Belgium§ Institut des Sciences Actuarielles, Universite Catholique de Louvain, 6 rue des Wallons,
B-1348 Louvain-la-Neuve, Belgium
Abstract
Experience rating and exposure rating are traditionally considered to be inde-pendent but complementary methods for pricing property per risk excess of lossreinsurance. Both techniques have their strengths and limitations. The purpose ofexperience rating is to forecast the losses borne by the reinsurer based on histori-cal claims information, possibly corrected for the current economic environment.Most of the traditional experience rating methods require that the relative portfo-lio composition remains constant over time, both regarding the risk types and theinsured values of the risks. Mathematical models are often used to make extrap-olations to price unused or rarely used capacity. These methods do not take intoaccount the composition of the portfolio which generates exposure in the upperregion of the programme. Exposure rating methods take the profile of the port-folio as a starting point. In theory, this method should allow to perform pricing,even if no loss experience is available. In practice, exposure pricing also has itslimitations.
In a paper presented at the 10th IME conference in Leuven, Desmedt and Walhinshowed that these two methods may be efficiently combined in order to improvethe accuracy of property excess of loss pricing. They developed an experiencerating method, based on historical profile information and exposure curves to de-rive reliable measures for the evolution of the claim frequency and severity abovedifferent thresholds. For pricing unused or rarely used capacity, exposure ratingis used, based on a calibration on the experience of a working layer.
Based on these principles, a reinsurance pricing tool has been developed using theSASr development environment. Starting from historical portfolio profiles andclaims information, this tool allows to analyse the imported information, tune thepricing parameters, and finally price the working layers with modified experiencerating, taking historical profile information into account, and the upper layerswith calibrated exposure rating. This tool is also characterized by a high degreeof flexibility, in the sense that it allows to easily test various assumptions with aminimum of manipulations. This often proves to be useful in a context where the
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quality of information is not always optimal, and consequently assumptions have
to be made.
In the presentation, we will first recall the basics of the methodology. After-
wards, we will illustrate the various functionalities of the tool by going through a
real-life quotation of a property per risk excess of loss reinsurance treaty.
Keywords
Experience Rating, Exposure Rating, Property per Risk Excess of Loss Reinsur-
ance, Exposure Curves, Reinsurance Pricing Tool
2
44
Claim Counts Modeling and Stable Distributions.
0
Claim Counts Modeling and Stable Distributions.
José L. Vilar-Zanón ([email protected])* Antonio Heras-Martínez
José A. Gil-Fana
Departamento de Economía Financiera y Actuarial. Universidad Complutense de Madrid
Spain
Abstract: In this communication we show how the generalized Poisson Pascal (G.P.P.) distribution may be a very good choice when trying to model claim counts. This is a very critical step in the modeling process of an experience rating scheme such as Bonus-Malus Systems (BMS), for any conclusion will depend on this early choice. Neither parametric models like the negative binomial or Poisson mixed by Inverse Gaussian, nor nonparametric ones like the good risk-bad risk model are enough to model real world. The application of the G.P.P. may be of a great help when trying to model claim counts with a high skewness, as was indicated in Panjer, Willmot (1992) and Klugman, Panjer, Willmot (1998). The difficulty to overcome when using it is the numerical handling of stable distributions. Keywords: Claim counts, counting distributions, stable distributions, compound Poisson distributions, mixed Poisson distributions, portfolio structure, Bonus-malus systems, generalized Poisson Pascal distribution.
This research was granted by Ministerio de Educación y Ciencia under project SEJ2005-06744.
* Departamento de Economía Financiera y Actuarial. Facultad de CC. Económicas y Empresariales. Universidad Complutense de Madrid. Campus de Somosaguas. 28223, Pozuelo de Alarcón, Madrid. Spain.,
45
Securitisation and Pricing of Flood Insurance:
A market consistent approach
Luiz Vitiello and Ser-Huang Poon∗
June 3, 2008
Abstract
The idea in this paper was conceived in 2003 when severe floods in Kent, England,
triggered many insurance companies declared large residential areas in the region as unin-
surable zone. For the individual household actors in the economy, this is a classic example
of market breakdown. Here, we show one way to price flood insurance that is consistent
with pricing theory that is fundamental in Finance. In this framework, market is complete
in which a pricing kernel exists. With the assumption that floods can be modelled using
rainfall data, prices of flood insurance cover can be priced as options on a transformed
gamma distribution. The resulting pricing formulae is closed form and preference free.
To make our pricing framework market consistent, an asset specific pricing kernel can be
inferred from prices of insurance contracts of household in other part of the country. Using
rainfall precipitation data for England and Wales from 1766 to 2007, collected from UK
Met Office, we demonstrate the flexibility of the transformed gamma option pricing model
in pricing different types of flood driven payoff functions.
Keywords: Catastrophe insurance, transformed gamma distribution, pricing kernel.
JEL classification: G12, G13, G22.
∗Corresponding author. Please send all correspondence to Manchester Business School, Craw-ford House, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Tel: +44 161 2750431, Fax: +44 161 275 4023, Email: [email protected]. We are grateful to MichaelBrennan, Richard Stapleton and Konstantinos Vonatsos for valuable insights. We would like tothank the seminar participants at Singapore Management University for helpful comments.
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46
Modelling dependence of interest rates, inflation rates and stock market returns
Hans Waszink AAG MBA MSc Waszink Actuarial Advisory Ltd.
Sunnyside, Lowden Hill, Chippenham, UK [email protected]
Abstract: In the first part of this article, an approach to model the value of an outstanding, discounted liability under the impact of uncertain interest and inflation rates is discussed. Interest and inflation rates are modeled separately as time series to take into account autocorrelation. Subsequently, the dependence between interest and inflation is modeled using copulas. The goodness of fit of some copulas can be evaluated on the basis of historic data using a quantile plot. This is done for the Gumbel, Clayton and Independent copulas. The Gumbel copula, which gives the best fit, is then compared with the Normal copula to show that the two copulas are very similar with the parameters chosen. The distribution of the required reserve is shown under four different copula assumptions: comonotonicity, which represent the best case, countermonotonicity which represents the worst case, and the Gumbel and Normal copulas which represent more realistic scenarios. The choice of copula has considerable impact on the higher percentiles of the required reserve, and the adopted approach is effective in selecting a suitable copula for the modelling of two underlying variables. In the second part of the article, the application of copulas to the modelling of three dependent variables (interest, inflation and stock market return) is investigated. As the Clayton and Gumbel copulas only have a single parameter, they prove to be less suitable for the modelling of more than two dependent variables. The normal copula appears more suitable in this case as the dependence between each modelled variable can be set separately by a different parameter in the correlation matrix. Keywords: copulas, dependence, autocorrelation, time series, stochastic modelling, goodness of fit, interest, inflation, stock market returns, discounting.
47
A study of solvency capital requirements forgeneral insurers in China
Zhigang Xie
48
Clustering in Ratemaking: with Application in Territories
Clustering
Ji Yao
Zurich Financial Services
126 Hagley Road
Birmingham, B16 9PF, UK
Tel: 0044-(0)121 697 4416
E-mail: [email protected]
Abstract
Clustering methods are briefly reviewed and their applications in insurance rate-making are discussed in this paper. First, the reason for clustering and the consideration in choosing clustering methods in insurance ratemaking are discussed. Then clustering methods, including partitioning, hierarchical, density-based and grid-based methods, are reviewed and particularly the problem of applying these methods directly in insurance ratemaking is discussed. An exposure-adjusted hybrid (EAH) clustering method is proposed, which may alleviate some of these problems. Results from EAH approach are presented step by step using the UK motor data. The limitations and other considerations of clustering are followed in the end.
Keywords: Clustering, ratemaking, generalized linear modeling, territory analysis, data mining.
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49
Bayesian Approach for Prediction Error in Chain-Ladder
Claims Reserving
Ji Yao
Zurich Financial Services
126 Hagley Road
Birmingham, B16 9PF, UK
Tel: 0044-(0)121 697 4416
E-mail: [email protected]
Abstract
Bayesian approach is applied to calculate the prediction error in chain ladder claim
reserving. The philosophy of Bayesian approach to prediction error is first introduced
and compared with Frequentist approach. All parameters in the model are then
estimated by Bayesian approach, where three types of prior distributions are used.
Posterior distribution could be standard student t-distribution and inverse Gamma
distribution under non-informative and conjugate prior distribution. Finally, theory is
illustrated by numerical examples. It concludes that prediction error by Bayesian
approach is much higher than that by Frequentist approach.
Keywords. Bayesian approach; Prediction error; Chain ladder; Reserving; Student t
distribution; Inverse Gamma distribution.
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50
A generalized linear discrete time model for managing the
solvency interaction and singularities arising from potential
regulatory constraints imposed within a portfolio of different
insurance products
Alexandros A. Zimbidis+, Athanasios A. Pantelous+*
and Grigoris I. Kalogeropoulos *
+Department of Statistics, Athens University of Economics and Business
*Department of Mathematics, University of Athens
Emails: [email protected], [email protected], [email protected]
Abstract
We consider a typical portfolio of different insurance products and investigate the pricing pro-
cess using the framework of a generalized linear discrete time model. Moreover, we assume
that, due to regulatory constraints, the resulting system is singular and calculate the solution
using the tools of matrix pencil theory. Finally, we present a numerical application for two
different portfolios.
Key-words: Pooling of risks, Solvency Interaction, Singularities, Matrix pencil theory, De-
scriptor Control Systems; Non Causality
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