Dynamic Risk Management with Markov Decision Processes

Dynamic risk management with Markov decision processes Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN der FakultÄatfÄurMathematik, UniversitÄatKarlsruhe (TH) genhemigte DISSERTATION von Dipl.-Wirtschaftsmath. Andr¶ePhilipp Mundt aus Pinneberg Tag der mÄundlichen PrÄufung:21. November 2007 Referentin: Prof. Dr. Nicole BÄauerle Korreferent: Prof. Dr. Ulrich Rieder Karlsruhe 2007 Preface During almost four years of work at the Kompetenzzentrum Versicherungswissen- schaften GmbH, Leibniz UniversitÄatHannover, and the Institut fÄurStochastik, UniversitÄatKarlsruhe (TH), many people supported me. First of all, I would like to thank my supervisor Nicole BÄauerle.She draw me to the subject of dynamic risk management and made many fruitful suggestions for the research treated in this work. Furthermore, she always took time for me and encouraged me when I was stuck and came to her with questions and problems. Secondly, let me thank Ulrich Rieder for being the second advisor and for useful discussions. From all the other people, I ¯rstly and most importantly have to mention Anja Blatter. She read almost the whole manuscript and made many valuable comments that led to improvements in formulations and the layout. Furthermore, I have to thank Gunther Amt and Bruno Ebner for helping me improve parts of the thesis and Mirko KÄotterand Lars Michael Ho®mann for useful discussions. Altogether, my colleagues and former colleagues from the Kompetenzzentrum Versicherungswissenschaften GmbH, the Institut fÄurMathematische Stochastik in Hanover and the Institut fÄurStochastik in Karlsruhe always provided a friendly and enjoyable atmosphere at work and during working breaks. When I moved to Hanover and Karlsruhe they made it very easy for me to settle in there. Finally, I have to mention my family and in particular my parents for supporting my studies and their con¯dence in my ability to complete my diploma and this thesis. Karlsruhe, October 2007 Andr¶eMundt vii Contents Introduction xi 1 Static and conditional risk measures 1 1.1 Model and de¯nition .......................... 1 1.2 Examples ................................ 4 1.3 Further properties and representations ................ 8 1.4 Remarks on conditional risk measures and measurability . 11 2 Portfolio optimization with constraints in a binomial model 15 2.1 The model ................................ 15 2.2 Risk minimization ............................ 18 2.2.1 The unconstrained case ..................... 19 2.2.2 A constraint on the ¯nal value . 21 2.3 Utility maximization .......................... 34 2.3.1 Comparison with the risk minimization . 34 2.3.2 The unconstrained case ..................... 36 2.3.3 Intermediate constraints .................... 37 3 Dynamic risk measures 51 3.1 An overview on the literature ..................... 51 3.2 De¯nitions, axioms and properties ................... 55 3.3 Stable sets of probability measures . 62 4 A risk measure by Pﬂug and Ruszczy¶nski 67 4.1 De¯nition of the dynamic risk measure . 67 4.2 Properties of the dynamic risk measure . 70 4.3 Solution via Markov decision processes . 73 4.4 A stable representation result ..................... 84 4.5 Martingales & co. ............................ 90 ix 5 A Bayesian control approach 93 5.1 The model ................................ 93 5.2 A Bayesian control approach to dynamic risk measures . 94 5.3 Explicit solutions ............................100 5.3.1 The case when the parameter # is known . 101 5.3.2 Beta distributions as initial distribution . 102 5.4 Comparison of value functions . 109 5.4.1 A comparison result for general distributions . 109 5.4.2 Examples ............................117 A Absolutely continuous probability measures 125 B The Beta distribution 129 Bibliography 131 Introduction Dealing with risk is particularly important for ¯nancial corporations such as banks and insurance companies, since these face uncertain events in various lines of their business. There exist several characterizations of risk which do not always have to describe distinct issues. This could be market risk, credit risk or operational risk, but also model risk or liquidity risk. Insurance companies additionally have to deal with underwriting risk, for example. The aim of integrated risk management is to manage some or all kinds of potential risk simultaneously. According to McNeil et al. (2005), the more general notion of risk management describes a \discipline for living with the possibility that future events may cause adverse e®ects", while banks or insurance companies are even able to \manage risks by repacking them and transferring them to markets in customized ways". Consequently, risk management can be regarded as a \core competence" of such companies. Historically, methods of dealing with risk go back to mean{variance optimization problems. These were ¯rst considered by de Finetti (1940) (see Barone (2006) for a translated version of the original article in Italian), whereas the issue only became famous with the work of Nobel laureate Markowitz (1952). Before, a main focus was on the mean of the underlying quantities, whereas these works brought the risk component in terms of the variance into play. Currently, some discussion about the contribution of the two authors can be observed in the literature. First, Markowitz (2006) himself claims that de Finetti did not solve the problem for the general case of correlated risk. However, this argument is contrasted in Pressacco and Sera¯ni (2007), where the authors reason that de Finetti solved the problem, but under a regularity assumption which is not always ful¯lled. An important tool in risk management is the implementation of risk measures, in particular ones which go beyond the variance. Since the risk is modelled by random quantities, i. e. random variables or, in a dynamic framework, stochastic processes, it is often possible to estimate or approximate the distribution of such positions. However, the actual realization of the risk remains uncertain. One way to quantify the risk beyond its distribution is to use risk measures. Risk measures assign a value to a risk at a future date that can be interpreted as a present monetary value or present and future monetary values if dynamic risk measures are applied. Let us now review some regulatory aspects and describe how risk measures are applied in risk management so far. This aspect started with the ¯rst Basel accord and its amendment which introduced the risk measure Value{at{Risk for calculat- xi ing the regulatory capital in internal models. Furthermore, special emphasis is put on credit and operational risk within the context of Basel II. While these accords aim to stabilize the international banking business, similar regulatory frameworks are introduced for the insurance sector via Solvency 2, having a more local or na- tional character. In particular, the aim is to regulate by law what kind of risk measures should be used to calculate the companies' target capital in order to ensure a certain level of solvency. Special emphasis is put on the coherent risk measure Average Value{at{Risk, sometimes denoted as Expected Shortfall, which is used in Solvency 2 instead of Value{at{Risk in Basel II. It has the advantage of being subadditive for general risks, therefore awarding diversi¯cation (see Chap- ter 1 for a more thorough investigation). Value{at{Risk ful¯lls this property only in an elliptical world. Consequently, using Value{at{Risk is not always the wrong thing to do but dangerous if applied inappropriately. Although many problems are similar for the banking and insurance sector respec- tively, there are some distinctions between these two kinds of ¯rms. Banks mainly deal with bounded risks, e. g. when facing credit risk. On the other hand, insurance companies often have to consider unbounded risks, e. g. when heavy{tailed distributed ¯nancial positions are present. To address both situations, we always treat integrable but not necessarily bounded risks in this work. Furthermore, a main issue will be to develop risk management tools for dynamic models. These naturally occur when considering portfolio optimization problems or in the context of developing reasonable risk measures for ¯nal payments or even stochastic processes. We consider only models in discrete time and denote these approaches with dynamic risk management. In dynamic economic models, one often faces a Markovian structure of the underlying stochastic processes. Hence, a method we will frequently use is the theory of Markov decision processes (MDP), sometimes also addressed as dynamic pro- gramming (DP) introduced in Bellman (1952). In this work, one ¯eld where the method is applied is dynamic portfolio optimization. This is a standard approach. However, the theory can even be used to solve optimization problems that occur in economically motivated de¯nitions of dynamic risk measures. One can even go further to de¯ne a reasonable class of dynamic risk measures for a model with incomplete information. Thorough investigations on Markov decision processes can be found e. g. in Hinderer (1970), Puterman (1994) or Hern¶andez-Lermaand Lasserre (1996). Further applications beyond economics are possible in biology or telecommunications, for example. An MDP is de¯ned as follows. First, one needs a dynamical system of interest which can be inﬂuenced over time by a risk manager through the choice of certain decision variables. Then, one has to de¯ne the state space of the process of interest, the action

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