St.Petersburg Spring School in Risk Management, Insurance and Finance

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General Information About the School

The main objective of the School is to disseminate knowledge and results of latest research in the area of Risk Management, Insurance, Finance, and related fields. The target audience is academics, students and practitioners from Russia and other countries of the former Soviet Union, although participants from the rest of the world are very welcome to join the School as well.

This School is included in the CPD system of the Guild of Actuaries (Russia).

The School is held at the European University at St.Petersburg (EUSP) in March or April each year.

The working language of the School is English.

Contact Information:

Ms Lina Korotkova

Department of Economics
European University at St. Petersburg
3 Gagarinskaya str.
St. Petersburg, 191187


Phone/fax: +7 (812) 386-7632

Please refer to year tabs for further relevant information on past and future schools.


The Fourth School to be held at the European University at St.Petersburg (EUSP) will differ from the schools in previous years in that it will be centred around a single topic of life assurance and pensions. The topic of the School 2015 is OPTIONS AND GUARANTEES IN LIFE INSURANCE AND PENSION PRODUCTS.

The School dates are March 30 - April 1, 2015. Participants requiring a visa to Russia must register before January 15, 2015. Other participants must register before March 16, 2015.

To register, please send your name, affiliation and contact information to the e-mail address given below in the contact details.

As in the previous years the School is free to partake in, i.e. there is no participation fee in 2015.

The Organising Committee

The Programme

Many modern insurance and pension products are designed as packages, whose items may be either included or not in the product actually purchased by the client. For example: the endowment insurance which can include various rider benefits and options, the Universal Life insurance, the Variable Annuities, the presence of possible LTC benefits in pension products. Appropriate modelling tools are then needed for valuating these products, and appropriate actions are required for managing the insurance portfolios.
The School first aims at presenting the main features of various insurance and pension products (Insurance Day). Special attention is then placed on the assessment and the management of risks inherent in a life annuity portfolio or pension plan (Risk Management Day). Finally, valuation of life insurance contracts in a contingent-claims framework is discussed, with a special focus on variable annuity products (Finance day).

Click on a particular day below to see more details.

March 30th, 2015: Insurance day (6 hours)


Building up the Post-Rretirement Income


Professor Ermanno Pitacco, University of Trieste, Italy

Summary of the Course

The benefits provided by many modern life insurance and pension products imply a wide range of “guarantees” and hence risks borne by the insurance company (or the pension fund). Guarantees and inherent risks clearly emerge in recent scenarios, in particular because of volatility in the financial markets and trends in mortality / longevity. Appropriate modeling tools are then needed for pricing and reserving. Hence, a progressive shift from expected present values, and their prominent role in life insurance (and pension) calculations, to more modern and complex approaches, in particular the Enterprise Risk Management (ERM) based approach, is currently updating the actuarial toolkit.

However the implementation of complex mathematical methods often constitutes, on the one hand, an obstacle on the way towards sound pricing and reserving principles. On the other hand, facing the risks by charging very high premiums trivially reduces the insurer’s market share. Then, alternative solutions can be suggested by an appropriate product design which aims at sharing risks between the insurer and the policyholders. Interesting examples are provided by the design of life annuities as regards the longevity risk, and by profit participation mechanisms as regards the financial market risks.

In this course, we first focus on various arrangements which aim at building the post-retirement income, and involve either the accumulation phase, or the payout phase, or both. Various products are available on financial and insurance markets, each product with a specific guarantee structure (Conventional Life Annuities either immediate or deferred, Variable annuities, withdrawal plans, etc.). We then shift to a range of specific annuity products, stressing the relevant features: Advanced Life Delayed Annuity (ALDA), Ruin Contingent Life Annuity (RCLA), Variable Annuities (VA). Finally, we focus on some arrangements for the payout phase: the life annuity with a guarantee period, the value-protected life annuity (that is, providing “capital protection”), progressive annuitization schemes, life annuities combined with Long Term Care (LTC) benefits, longevity-linked life annuities. We conclude with a short introduction to the special-rate annuities (e.g. the impaired-life annuities).


1.1    Introduction
1.2    Packaging guarantees and options in insurance products
1.3    Product design: sharing the market risk in participating endowment insurance
1.4    Standard life annuities
1.5    Propensity to annuitize
1.6    Life annuity and death benefits
1.7    Last survivor annuities
1.8    Annuitization strategies
1.9    Guarantees in life annuities: basic structures
1.10    Non-standard life annuities (ALDA, RCLA, long-term care benefits, variable annuities, longevity-linked life annuities)
1.11    Underwritten (special-rate) life annuities

March 31st, 2015: Risk Management day (6 hours)


Risk Management in a Longevity Risk Scenario


Professor Ermanno Pitacco, University of Trieste, Italy

Summary of the Course

The calculation procedures proposed by the traditional actuarial mathematics rely, in a modern perspective, on deterministic models, because only expected values are usually addressed. The need for a sound assessment of the insurer's risk profile (as, in particular, emerges from new solvency standards) conversely suggests a comprehensive approach to the formal representation of the life insurance and annuity business, more general than that provided by traditional actuarial mathematics.

Enterprise Risk Management (ERM) suggests a comprehensive approach and provides a sound framework for dealing with life insurance and annuity business, and in particular with the related technical issues. In this framework, risk identification, risk assessment and impact assessment constitute basic steps towards the choice of appropriate tools for managing the risk themselves. Besides the traditional tools given by reinsurance and capital allocation, a risk management perspective suggests as effective tools, for example, a careful product design, an appropriate hedging involving opposite exposures to mortality/longevity risks, etc.

In this course we mainly focus on the application of ERM principles to life annuity and pension business. Starting from biometric issues (the longevity dynamics and the related uncertainty), we then describe a set of risk management actions for a life annuity portfolio or a pension fund. Several numerical examples illustrate the impact of longevity dynamics and the effect of capital allocation. We conclude stressing the need for monitoring mortality experience, and presenting a model for Bayesian inference on future mortality trends.


2.1    Introduction
2.2    Biometric assumptions for life annuities: projected life tables
2.2    The longevity dynamics: individual vs aggregate longevity risk
2.3    Modelling issues: from expected present values to ERM
2.4    Risk management actions
2.5    Risk assessment and impact assessment for pensions and life annuities
2.6    Monitoring: inference from longevity experience

April 1st, 2015: Finance day (6 hours)


Valuation of Life Insurance Contracts in a Contingent-Claims Framework


Professor Anna Rita Bacinello, University of Trieste, Italy

Summary of the Course

The object of life and pension insurance is to provide an adequate level of financial protection against uncertain events affecting the life of the insured. In particular, this protection can be addressed to the family of the insured and cover the case of her (early) death, or it can be addressed to the insured herself in order to get an adequate income after retirement. Of course both these objectives can also be pursued together. Especially in the second case, the protection takes the form of a long-term contract, where the financial component is predominant. In order to face the more and more aggressive competition with financial markets, the insurance industry offers specific and very flexible products such as, e.g., variable annuities, that allow the policyholders to benefit from high yields in financial markets without taking (all) the risk connected with them. Hence the problem of adequately assessing the value of the guarantees, that usually take the form of embedded derivatives, is very crucial, along with the necessity of singling out suitable hedging strategies for them.

In this course we start by recalling some generalities on life insurance contracts. Then we pass to the analysis of a unit-linked policy with minimum guarantees that was first studied by Boyle, Brennan and Schwartz in the second half of the seventies of the twentieth century. In particular, these authors were the first who recognised that a guaranteed benefit can be decomposed in terms of European options, once the mortality risk is diversified away, and merged together the traditional actuarial techniques with the then recent results on option pricing theory initiated by Black, Merton and Scholes in the first half of the same decade. After that we pass to the analysis of some extensions of the previous model that allow for different contracts, or benefit structures, or types of guarantees, as well as other embedded options such as, e.g., the surrender option and the guaranteed annuity option.

We focus, in particular, on the surrender option, a typical American-style contingent-claim that entitles the policyholder to early terminate a life insurance contract receiving a cash amount, called surrender value. As for any kind of American derivative, this option does not admit a closed-form valuation formula, so that a numerical approach is called for. We provide a very general valuation framework for a life insurance contract embedding such an option, based on the Least Squares Monte Carlo technique.

Then we pass to the analysis of variable annuities, that package different types of options and guarantees. Basically, these products are unit-linked investment policies providing a post-retirement income. The guarantees, commonly referred to as GMxBs (namely, Guaranteed Minimum Benefits of type `x'), include minimum benefits both in case of death and survival. After providing a detailed description of the most common guarantees offered by variable annuities, we introduce the alternative assumptions underlying the policyholder behavior (static, mixed and dynamic approach) and propose a unifying framework for treatment and valuation of variable annuities under quite general model assumptions. Then we concentrate on the valuation of Guaranteed Minimum Withdrawal Benefits with the dynamic approach, and formulate the valuation model as a discrete stochastic control problem which is solved with a dynamic programming algorithm.


3.1    Introduction
3.2    Unit-linked products with minimum guarantees
3.3    The Boyle-Brennan-Schwartz model and related extensions
3.4    The surrender option
3.5    The structure of variable annuities: guarantees and options
3.6    Valuation of variable annuities under alternative policyholders’ behaviors

Photo Report of the School 2015

References for the Course

The three courses are mainly based on the following material:

A.R. Bacinello, F. Ortu (1993a): “Pricing equity-linked life insurance with endogenous minimum guarantees”, Insurance: Mathematics and Economics, vol. 12, n. 3, p. 245-258

A.R. Bacinello (2005): “Endogenous model of surrender conditions in equity-linked life insurance”, Insurance: Mathematics and Economics, vol. 37, n.2, p. 270-296

A.R. Bacinello (2008): “A full Monte Carlo approach to the valuation of the surrender option embedded in life insurance contracts”, Mathematical and Statistical Methods for Insurance and Finance, C. Perna and M. Sibillo (eds.), Springer, p. 19-26

A.R. Bacinello, E. Biffis, P. Millossovich (2009): “Pricing life insurance contracts with early exercise features”, Journal of Computational and Applied Mathematics, vol. 233, n.1, p. 27-35

A.R. Bacinello, E. Biffis, P. Millossovich (2010): “Regression-based algorithms for life insurance contracts with surrender guarantees”, Quantitative Finance, vol. 10, n. 9, p. 1077-1090

A.R. Bacinello, P. Millossovich, A. Olivieri, E. Pitacco (2011): “Variable annuities: a unifying valuation approach”, Insurance: Mathematics & Economics, vol. 49, p. 285-297

A.R. Bacinello, P. Millossovich, A. Olivieri, E. Pitacco (2012): “Variable Annuities as Life Insurance Packages: A Unifying Approach to the Valuation of Guarantees”, Actuarial and Financial Mathematics Conference. Interplay between Finance and Insurance, M. Vanmaele et al (eds.) , Koninklijke Vlaamse Academie van Belgie voor Wetenschappen en Kunsten, Brussel, p. 3-15

A.R. Bacinello, P. Millossovich, A. Montealegre (2014): “The Valuation of GMWB Variable Annuities under Alternative Fund Distributions and Policyholder Behaviours”, Scandinavian Actuarial Journal, Available at:

A. Olivieri, E. Pitacco (2008): “Assessing the cost of capital for longevity risk”, Insurance: Mathematics & Economics, vol. 42, n. 3, p. 1013–1021

A. Olivieri, E. Pitacco (2009): “Solvency requirements for life annuities allowing for mortality risks: internal models versus standard formulas”, in: M. Cruz (Ed.), The Solvency II Handbook. Developing ERM frameworks in insurance and reinsurance companies, Risk Books: p. 371-397

A. Olivieri, E. Pitacco (2009): “Stochastic mortality: the impact on target capital”, ASTIN Bulletin, vol. 39, n. 2, p. 541-563. doi: 10.2143 /AST.39.2.2044647

A. Olivieri, E. Pitacco (2011): “Life tables in actuarial models: from the deterministic setting to a Bayesian approach”, AStA Advances in Statistical Analysis, vol. 96, p. 127–153

E. Pitacco (2007): “Mortality and longevity: a risk management perspective”, Invited lecture at the 1st IAA Life Section Colloquium, Stockholm (available at:

E. Pitacco, M. Denuit, S. Haberman, A. Olivieri (2009): Modelling longevity dynamics for pensions and annuity business, Oxford University Press

E. Pitacco (2012): “From benefits to guarantees: looking at life insurance products in a new framework”, CEPAR Working Paper 2012 / 26, University of New South Wales, Sydney. Available at:

E. Pitacco (2013): “Biometric risk transfers in life annuities and pension products: A survey”, CEPAR Working Paper 2013 / 25, University of New South Wales, Sydney, Available at:


The Third School is to be held at the European University at St.Petersburg (EUSP) and is organised with the generous support from Barclays Bank and JTI.

The School dates are March 20-22, 2014. Participants requiring a visa to Russia must register before January 15, 2014. Other participants must register before March 1, 2014.

To register, please send your name, affiliation and contact information to the e-mail address given below in the contact details.

As in two previous years the School is free to partake in, i.e. there is no participation fee in 2014.

The Organising Committee

The Programme

Click on a particular day below to see more details.

20 March, 2014: Risk Management day (6 hours)


From Risk Measures to Solvency Accounting


Professor Karl-Theodor Eisele, University of Strasbourg, France

Summary of the Course

Our starting point are convex risk measures (or their negative values: concave risk assessments) and their robust representations in the one- and multi-period case. They may be regarded as possible answers to the problem of Knightian uncertainty or model risk. Several examples will be given; among them the representation of asymptotically stable assessments. Of particular interest are time-consistent risk assessments, the representations of which use in the coherent case m-stable sets of probabilities.

Our next step is to regard risk assessments of non-tradeable assets or liabilities in the presence of a financial market. Here, a central point to be discussed is the so-called supervisory arbitrage. If supervisory arbitrage is excluded, we are led to the problem of the existence of optimal replicating portfolios (ORP). Under ORPs, risk assessments are market-consistent and thus define their market-consistent envelope.

It turns out that this concept will become the essential tool of prudent supervision of banks and insurances. Based on a business plan e.g. of an insurance company, we introduce the concepts of supervisory provision and – margin. Next, we divide these items into their parts assigned to external capital and to own funds, respectively. This will lead us to the notions of risk margin and supervisory capital requirement. The relation between them is the cost-of-capital ratio. With the help of a bid-price operator, we give examples how the cost-of-capital ratio is calculated in the case of a finite risk horizon or in the stationary infinite case.

Finally we present the internal model of the European Solvency II project in its bottom-up version, and show how it coincides with the top-down version where the concepts worked-out so far will be used.

(Participants should have a good knowledge about measure and probability theory, including Lp spaces.)

21 March, 2014: Finance day (6 hours)

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Analysis of the Monte Carlo CVA and Basel III CVA During and After the Financial Crisis of 2008-2009


Dr Olga Pavlova, Managing Partner at Grizzly Bear Capital, London, UK

Summary of the Course

The course will focus on the main driving forces behind counterparty risk for a derivatives portfolio.  First, we shall run through main simulation techniques for CVA. Then we shall compare the simulated CVA with Basel III CVA using historical data from 2008 to present. We shall comment on strengths and weaknesses of the Basel approximation in different market conditions using the simulated CVA as a benchmark.

A more detailed summary will appear here in due course.

About the lecturer:

Olga Pavlova holds a PhD in Mathematics and a Master in Finance degrees. Her work experience in the City started at Deutsche Bank (London) on the Market Risk Modelling team. She then joined the Quantitative Risk Analytics team at Barclays Capital (London). There, she was a Global Head of Credit Risk Analytics, covering risk modelling for all derivative classes. Among other projects, she represented the bank in regulatory approvals of Basel II and III capital requirements.  She subsequently joined UBS (London) as a Head of Quantitative Business Analysis. In this most recent role, Olga looked at various capital optimisation techniques for portfolios of derivatives. She currently works as an independent consultant giving lectures on various topics in financial modelling with the main emphasis being on counterparty risk. She also supervises masters degree theses at the Department of Financial Engineering, Imperial College (London).

22 March, 2014: Insurance day (6 hours)


Claims Reserving in Non-Life Insurance and its Prediction Uncertainty


Professor Mario V. Wüthrich, ETH Zurich, Switzerland

Summary of the Course

Lecture 1: The claims reserving problem and the chain-ladder method
We introduce the claims reserving problem and the claims reserving terminology. Claims reserving is one of the most important tasks in a non-life insurance company. It aims at predicting cash flows of the outstanding loss liabilities. These predictions are needed for pricing of insurance products, for accouting of insurance business and for risk management purposes. We discuss these problems and we give first techniques that help to predict these cash flows.

Lecture 2: Stochastic chain-ladder model and prediction uncertainty
We present the stochastic chain-ladder model for claims reserving. The stochastic chain-ladder model is the most popular stochastic model that is used for claims reserving. On the one hand it is simple and on the other hand it provides quite reasonable predictions. We discuss this model and analyze the quantification of prediction uncertainty.

Lecture 3: The claims development result for solvency considerations
The quantification of prediction uncertainty is usually done over the entire run-off of the outstanding loss liabilities (long-term behavior). For modern solvency considerations we should also investigate the short-term behavior, i.e. by which amount the claims reserves may change over the next accounting year. This leads to the consideration of the claims development result. We discuss this claims development result and analyze the quantification of prediction uncertainty in this short-term view.

Lecture 4: Dependence modeling on claims reserving triangles
All the previous considerations are done under rather restrictive model assumptions about dependence, namely one makes strong assumptions about independence. In this lecture we relax these independence assumptions and we analyze the sensitivities of the results by allowing for dependence. We will observe that this increases the confidence bounds substantially.

Photo Report of the School 2014


The Second School is to be held by the European University at St.Petersburg (EUSP) and is organised with the generous support from Barclays Bank and JTI.

The School dates are April 1-3, 2013. Participants must register before March 15, 2013.

To register, please send your name, affiliation and contact information to the e-mail address given below in the contact details.

As in 2012 the School is free to partake in, i.e. there is no participation fee in 2013.

The Organising Committee

The Programme

Click on a particular day below to see more details.

April 1st, 2013: Risk Management day (6 hours)


Comparison and Contrast of Risk Measures to Decision Principles, Solvency Capital and Insurance Premium Principles


Professor Marc Goovaerts, Katholieke Universiteit Leuven, Belgium

Summary of the Course

The talk is divided into four parts:

   Part 1. Risk Measures. In actuarial research, distortion, mean value and Haezendonck risk measures are concepts that are usually treated separately. In this part we indicate and characterize the relation between the different risk measures, as well as their relation to convex risk measures. While it is known that the mean value principle can be used to generate premium calculation principles, we will show how they also allow to generate solvency calculation principles.  Moreover, we explain the role provided for the distortion risk measures as an extension of the Tail Value-at-Risk (TVaR) and Conditional Tail Expectation (CTE). 

   Part 2. Decision Principles. We argue that a distinction exists between risk measures and decision principles. Though both are functions assigning a real number to a random variable, we think there is a hierarchy between the two concepts. Risk measures operate on the first “level”, quantifying the risk in the situation under consideration, while decision principles operate on the second “level”, often being derived from the risk measure. We illustrate this distinction with several canonical examples of economic situations encountered in insurance and finance. Special attention is paid to the role of axiomatic haracterizations in determining risk measures and decision principles. Some new axiomatic characterizations of families of risk measures and decision principles are also presented.

    Part 3. Solvency capital. We examine properties of risk measures that can be considered to be in line with some “best practice” rules in insurance, based on solvency margins. We give ample motivation that all economic aspects related to an insurance portfolio should be considered in the definition of a risk measure. As a consequence, conditions arise for comparison as well as for addition of risk measures. We demonstrate that imposing properties that are generally valid for risk measures, in all possible dependency structures, based on the difference of the risk and the solvency margin, though providing opportunities to derive nice mathematical results, violates best practice rules. We show that so-called coherent risk measures lead to problems. In particular we consider an exponential risk measure related to a discrete ruin model, depending on the initial surplus, the desired ruin probability

    Part 4. Insurance principles. The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(S) and w (S,p) and another exogenous parameter a. Minimizing a general Markov bound leads to a unifying equation: E(w(S,p))=aE(v(S)). For any random variable, the risk measure p is the solution to the unifying equation. By varying the functions w and v, we derive the mean value principle, the zero-utility premium principle, the Swiss premium principle, Tail VaR, Yaari’s dual theory of risk, mixture of Esscher principles and more.

April 2nd, 2013: Insurance day (6 hours)


Health Insurance: Actuarial Models


Professor Ermanno Pitacco, University of Trieste, Italy

Summary of the Course

The talk presents various technical aspects of health insurance products (including sickness benefits, disability annuities, and so on), focusing on the basic actuarial structures. Special attention is paid to the following topics:

1. The need for health-related insurance covers

2. Products in the area of health insurance

3. Between Life and Non-Life insurance: the actuarial structure of health insurance products

4. Actuarial models for sickness benefits

5. Actuarial models for disability benefits (Income Protection and Long Term Care)

6. Some problems in current scenarios

April 3rd, 2013: Finance day (6 hours)


Applications of Backward Stochastic Differential Equations with Jumps to Pricing and Hedging of Contingent Claims in Incomplete Markets


Professor Łukasz Delong, Warsaw School of Economics, Poland

Summary of the Course

In the first part of the course we recall basic techniques of stochastic calculus for Brownian motions and random measures. We next introduce Backward Stochastic Differential Equations with jumps (BSDEs with jumps). We investigate theoretical properties of BSDEs which are crucial for financial modelling. We comment on analytical and numerical methods for solving BSDEs. In particular, we discuss methods based on Itô formula and partial integro-differential equations, Malliavin derivatives and Least Squares Monte Carlo.

In the second part of the course we show how to use Brownian motions and random measures for modelling equity, default and insurance risks. We focus on the problem of pricing and hedging of contingent claims in incomplete financial markets. Since in an incomplete market perfect hedging is not possible, we investigate different pricing and hedging objectives, including superhedging, quadratic hedging under an equivalent martingale measure, quadratic hedging under a real-world measure, hedging under local mean-variance Markowitz risk measure, robust hedging under model risk, no-good-deal pricing. We show how to solve these optimization problems and define optimal prices and hedging strategies by applying BSDEs. Advantages of applying BSDEs are discussed.

Photo Report of the School 2013


The School was held at the European University at St.Petersburg on April 2-4, 2012.

Participants had to register before March 15, 2012. Participation was free for all participants.

The Organising Committee

The Programme

Click on a particular day below to see more details.

April 2nd, 2012: Finance day (6 hours)


Superhedging Strategies for Index Options and Measuring Systemic Risk in Stock Markets


Professor Jan Dhaene, Katholieke Universiteit Leuven, Belgium


Summary of the Course

We investigate static super-replicating strategies for European-type call options written on a stock index, that is a weighted sum of stock prices. Both the infinite market case (where prices of the plain vanilla options are available for all strikes) and the finite market case (where only a finite number of plain vanilla option prices are observed) are considered. We show how to construct a portfolio consisting of the plain vanilla options on the different stocks, whose pay-off super-replicates the pay-off of the index option. As a consequence, the price of the super-replicating portfolio is an upper bound for the price of the index option. The super-hedging strategy is model-free in the sense that it is expressed in terms of the observed option prices on the individual stocks. In a second part of the course, we introduce an easy to calculate measure for systemic risk in stock markets. This measure is baptized the Herd Behavior Index (HIX). It is model-independent and forward looking, based on observed option data. In order to determine the degree of systemic risk or herd behavior in a stock market one should compare the observed market situation with the extreme (theoretical) situation under which the whole system is driven by a single factor. The Herd Behavior Index (HIX) is defined as the ratio of an option-based estimate of the risk-neutral variance of the market index and an option-based estimate of the corresponding variance of this extreme market situation. The HIX can be determined for any market index provided an appropriate series of vanilla options is traded on this index as well as on its components. As an illustration, we determine historical values of the 30-days implied Herd Behavior Index for the Dow Jones Industrial Average.

April 3rd, 2012: Risk Management day (6 hours)



Principles and Methods of Capital Allocation for Enterprise Risk Management


Professor Emiliano Valdez, University of Connecticut, USA

Summary of the Course

There is a growing need and interest among financial institutions not only to determine the total company capital requirement, but also to allocate this total capital across various business units or product lines. The term "capital allocation" has been used to refer to a fair and equitable sub-division of this total capital requirement; it is indeed similar in concept to a fair division of capital in a diversified portfolio of investments. In this talk, we will examine the principles behind capital allocation for enterprise risk management: what is considered fair and equitable, what are the regulatory requirements, and what methods can be used. Along these principles, we examine a unifying framework for allocating the aggregate capital of a financial firm to its various business units or product lines. This unifying ap-proach relies on an optimization argument, requiring that the weighted sum of measures for the deviations of the business unit's losses from their respective allocated capital be minimized. In essence, this leads us to some degree of fairness and equity because this requires capital to be close to the risk that necessitates holding it. Additionally, this approach is very flexible in the sense that different forms of the objective function can reflect alternative definitions of company risk tolerance. Owing to this flexibility, the general framework reproduces several capital allocation methods that appear in the literature and allows for alternative interpretations and possible extensions. Several examples will be discussed to illustrate and to understand the implications of the results arising from these various methods.

April 4th, 2012: Insurance day (6 hours)


The chain-ladder method: influence analysis, robustification and diagnostic tool


Tim Verdonck, Katholieke Universiteit Leuven, Belgium

Summary of the Course

The chain-ladder method is a widely used technique to forecast the reserves that have to be kept regarding claims that are known to exist, but for which the actual size is unknown at the time the reserves have to be set. Such claims are often represented in a run-off triangle and hence the goal of claims reserving is to obtain predictions for the lower part of the triangle. Several traditional actuarial methods to complete a run-off triangle, such as the chain-ladder method, can be described by one Generalized Linear Model (GLM). In practice it can be easily seen that even one outlier can lead to a huge over- or un-derestimation of the overall reserve when using the chain-ladder method. This indicates that individual claims can be very influential when determining the chain-ladder estimates. The effect of contamination can be mathematically analyzed by calculating influence functions in the GLM framework corresponding to the chain-ladder method. It is proven that the influence functions are unbounded, confirming the sensitivity of the chain-ladder method to outliers. Robust alternatives are introduced to estimate the future claims reserves in a more outlier resistant way. Based on the influence functions and the robust estimators, a diagnostic tool is presented highlighting the influence of every individual claim on the classical chain-ladder estimates. With this tool it is possible to detect immediately which claims have an abnormally positive or negative influence on the reserve estimates. The robust methodology and the diagnostic tool will be illustrated on some artificial and real data sets.

Photo Report of the School 2012

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