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Dynamic Risk Measurement, with an Application to a Pension Fund Setting

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Dynamic Risk Measurement, with an Application to a Pension Fund Setting Dynamic Risk Measurement, with an Application to a Pension Fund Setting by Servaas van Bilsen 24.26.44 A thesis submitted in partial fulfillment of the requirements for the degree of Master in Quantitative Finance and Actuarial Sciences Faculty of Economics and Business Administration Tilburg University July, 2010 Supervised by: dr. R.J.A. Laeven prof. dr. J.M. Schumacher Abstract The financial crisis has highlighted that economic agents have not been successful in associating capital requirements with the risks un- dertaken by financial institutions. The connection between capital requirements and risk-taking has to be improved. This thesis focuses on the improvement of risk measurement. First, we provide a litera- ture review on static and dynamic convex measures of risk. An explicit convex measure of risk is given by the entropic risk measure. We study this risk measure and its connection with stochastic differential equa- tions. As an extension to the entropic risk measure, we also investigate a risk measure that allows for some degree of ambiguity in choosing a probabilistic model of some random variable. The literature has often focused on risk measurement for random variables. To take into ac- count the dynamic fluctuation of intermediate cash flows, we develop an entropic risk measure for random processes. Finally, we apply some measures of risk to a pension fund setting. Keywords: Static Measures of Risk, Dynamic Measures of Risk, Time-consistency, Entropic Risk Measure, Ambiguity, Random Pro- cesses, Pension Funds, Backward Stochastic Differential Equations ii Contents 1 Introduction 1 2 Static Risk Measures 6 2.1 Monetary, Convex and Coherent Risk Measures . .6 2.2 Robust Representation Theorems . .9 3 Dynamic Risk Measures in Discrete Time 13 3.1 The Setup and Notation . 14 3.2 Risk Measures in a Conditional Framework . 15 3.3 Time-Consistency Properties of Risk Measures . 18 3.4 Conditional Risk Measures on L1 ............... 21 4 Some Dynamic Measures of Risk 22 4.1 The Entropic Risk Measure: Definition and Properties . 23 4.1.1 Modeling Ambiguity . 29 4.2 The Entropic Risk Measure for Random Processes . 31 5 A Numerical Exercise: A Pension Fund Setting 35 5.1 One Source of Uncertainty . 37 5.2 A More Realistic Model . 43 6 Final Remarks 49 7 Appendix 50 7.1 Proofs . 50 7.2 Solving BSDEs: A Numerical Approach . 58 7.2.1 One-dimensional Case . 59 7.2.2 Multi-dimensional Case . 60 7.2.3 A Transformation Method . 61 7.3 Figures . 62 iii List of Figures 1 Solvency Buffer: Current Situation vs. Proposals . 37 2 Expected Entropic Risk Measure Process - Funding level 100% ................................... 62 3 Expected Entropic Risk Measure Process - Funding level 70% 62 4 Scenario Entropic Risk Measure Process . 63 5 Expected Risk Measure Process based on Quantile Function . 63 6 Expected Entropy Coherent Risk Measure Process (with Am- biguity) . 64 7 Expected Risk Measure Process - Entropic Risk Measure vs. Entropic Risk Measure for Random Processes . 64 8 Scenario - Entropic Risk Measure vs. Entropic Risk Measure for Random Processes . 65 9 Expected Entropic Risk Measure Process - Three Different Pension Funds . 65 10 Correlation between Interest Rate Risk and Equity Risk . 66 11 Scenario - Entropic Risk Measure vs. Entropic Risk Measure for Random Processes . 66 iv 1 Introduction The variability of some financial variable is traditionally regarded as an in- dicator of risk. Any measure of dispersion can then be viewed as a quantifi- cation of risk. Markowitz [54] was one of the first who adopted the standard deviation as a measure of risk. Although the standard deviation does not fully capture downside risk, it is still a widely used measure of risk in finan- cial economics. The downside risk is, however, taken into account by other measures of risk such as Expected Shortfall or Value at Risk (in short, VaR). In situations where agents care more about downside losses, the downside risk measures are seemingly a major improvement over traditional risk mea- sures. Some of those downside risk measures suffer, however, from being stable and do neither encourage diversification1 nor account for the size of extremely large losses. Furthermore, VaR is an important measure of risk which is part of the capital adequacy rules laid out in the Basel Accord. Especially inspired by the Basel Accords on Banking Supervision, Artzner et al. [4] [5] started an axiomatic analysis of measures of risk. They defined a measure of risk in the context of an acceptance set C. This set includes all the future net worths (i.e. monetary payoffs) that are `acceptable'. The measure of risk can then be interpreted as the smallest amount of money that would have to be added to the future net worth X 2 X to make it acceptable. This naturally leads to an axiom of cash invariance (i.e. after adding some fixed amount of money to X, the amount of money needed to make X acceptable, is reduced by the same amount of money). The riskiness of X clearly depends on the choice of the acceptance set C. A risk measure of this type (i.e. a risk measure that satisfies cash invariance) often emerges in the context of a capital requirement to regulate a financial position. By way of illustration, consider the Financial Assessment Framework for Dutch pension funds. This regulatory framework specifies the acceptance set in such a way that the probability of underfunding (i.e. funding level2 below 100%) in one year from now should at all times be less than 2.5%. Dutch pension funds should keep extra capital in reserve such that the one-year ahead funding level is contained in the acceptance set. Given the definition of a risk measure in terms of an acceptance set C, Artzner et al. [4] [5] introduced the axiomatic class of coherent risk measures 1E.g. VaR is not compatible with subadditivity (diversification) when losses are not elliptically distributed. 2The relative value of the pension fund assets and nominal liabilities, expressed as a percentage figure. 1 on a finite probability space (i.e. the sample space of the future net worth is supposed to be finite). A coherent risk measure satisfies four basic axioms. Furthermore, the authors argued that the riskiness of a future net worth X does not depend on a fixed probability P, but on X itself. In other words, model risk - the risk that the choice of the distribution of the future values of X is not correct - is taken into account. They concluded that any coherent measure of risk appears as given by a `worst case method' in a framework of probability measures. Their ideas, and especially the robust representation theorems for coherent risk measures, trace back to Huber [45]. Delbaen [27] extended the axiomatic class of coherent risk measures to arbitrary probability spaces. F¨ollmer& Schied [37] and Frittelli & Rosazza Gianin [40] criticized the axiom of positive homogeneity as investigated in Artzner et al. [4] [5]. They removed this axiom and introduced the general axiomatic class of convex risk measures. Under some regularity conditions, convex duality can be used to deduce a robust representation theorem in terms of the convex conjugate. The convex conjugate corresponds with the Fenchel-Legendre transform on the dual space X 0 of X . In the context of risk measurement, the convex conjugate is called a penalty function. This function gives different probability measures varying impact in the robust representation theorem (i.e. some probability measures might be more plausible than others). In the aforementioned references, the central object of study is the single-period future net worth X. Static risk measurement arising from a stream of payoffs is analyzed in for instance Cheridito et al. [19] [20], Cvitanic & Karatzas [26] and Scandolo [64]. Section 2 provides a literature review on static measures of risk. In case of static risk measurement, the riskiness of X is only monitored at the beginning of the evaluation period. So, the risk assessment can not be adjusted during the evaluation period. A natural extension would then be to evaluate the riskiness of X at a finite collection of points in time. In fact, the information flow - which is modeled by a filtration - plays a crucial role. The riskiness of X is updated at each point in time when new information is gathered. These considerations give rise to the notion of a conditional risk measure, i.e. a risk measure that only depends on that part of the future that is not ruled out by new information. Detlefsen & Scandolo [30] studied the class of convex risk measures in a conditional framework. They introduced the notion of a dynamic convex risk measure, which is basically a family of successive conditional convex risk measures. A dynamic risk measure is very relevant for the financial industry. For instance, a dynamic risk measure can be used to determine the premium 2 for an insurance contract or to assess the riskiness of a dynamic financial policy (in the absence of complete markets). Risk measurement in a multi- period framework is also studied - among many others - by Artzner et al. [6], Cheridito et al. [21], Cheridito & Kupper [22], F¨ollmer& Penner [39], Riedel [62], Roorda et al. [63] and Wang [68]. A dynamic risk measure should be consistent over time (i.e. what is preferred at one point in time should be consistent with what is preferred at another point in time). The notion of time-consistency can be understood in many ways. A risk measure is generally said to be strongly time-consistent if it can be either computed directly or in two steps backwards in time (i.e. recursively). However, many other (weaker) notions of time-consistency have been studied in the literature.
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