The Second School in Risk Management, Insurance, and Finance took place on April 1–3, 2013 with generous support from Barclays Bank and JTI. The European University welcomed on its premises Marc Goovaerts (Katholieke Universiteit Leuven, Belgium), Ermanno Pitacco (University of Trieste, Italy), and Łukasz Delong (Warsaw School of Economics, Poland). The topics covered by the lecturers were «Comparison and Contrast of Risk Measures to Decision Principles, Solvency Capital and Insurance Premium Principles», «Health Insurance: Actuarial Models», and «Applications of Backward Stochastic Differential Equations with Jumps to Pricing and Hedging of Contingent Claims in Incomplete Markets».
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
byProfessor 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.
The Organising Committee
Partners and Supporters
The School greatly benefited from generous donations from
Photo Report of the School 2013