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Risk Measures Faculty of Science Department of Applied Mathematics, Computer Science and Statistics Utility based risk measures Jasmine Maes Promotor: Prof. Dr. D. Vyncke Supervisor: H. Gudmundsson Thesis submitted tot obtain the academic degree of Master of Science: Applied Mathematics Academic year 2015{2016 Acknowledgements First of all I would like to thank my supervisor Mr. Gundmundson for letting me come by his office whenever I felt like I needed it, for coming up with good ideas and for supporting me throughout the thesis. I also would like to thank my promotor prof. Vyncke for giving me advice when I asked for it, while still allowing me a lot of freedom. Last but not least I would like to thank my friends for listing to all my complaints when things didn't go as planned and when I got stuck, and my parents for their financial support during my education. The author gives permission to make this master thesis available for consultation and to copy parts of this master thesis for personal use. In the case of any other use, the limitations of the copyright have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation. Ghent, 1 June 2016. 2 Contents Preface 6 1 Mathematical representation of risk 8 1.1 Definitions and properties . .8 1.2 The acceptance set of a risk measure . 12 1.3 The penalty function . 16 1.4 Robust representation of convex risk measures . 20 2 An introduction to decision theory 24 2.1 The axioms of von Neumann-Morgenstern . 24 2.2 Risk and utility . 26 2.3 Certainty equivalents . 28 2.3.1 The ordinary certainty equivalent . 28 2.3.2 The optimised certainty equivalent . 29 2.3.3 The u-Mean certainty equivalent . 33 2.4 The exponential utility function . 33 2.5 Stochastic dominance . 35 3 Value at Risk and Expected shortfall 38 3.1 Value at Risk . 38 3.1.1 General properties . 39 3.1.2 Consistency with expected utility maximisation . 42 3.2 Expected shortfall . 45 3.2.1 General properties . 46 3.2.2 Consistency with expected utility maximisation . 49 4 Utility based risk measures 54 4.1 Utility based shortfall risk measures . 54 4.2 Divergence risk measures . 60 4.2.1 Construction and representation . 60 4.2.2 The coherence of divergence risk measures . 71 4.2.3 Examples . 75 4.3 The ordinary certainty equivalent as risk measure . 76 5 Utility functions 78 5.1 The power utility functions . 80 5.2 The exponential utility functions . 81 5.3 The polynomial utility functions . 84 4 5.4 The SAHARA utility functions . 88 5.5 The κ-utility functions . 96 Conclusion 102 A Dutch summary 104 B Additional computations 107 B.1 Computations regarding the SAHARA utility class . 107 B.1.1 Computation of the utility function . 107 B.1.2 Computation of the divergence function . 110 B.2 Computations regarding the κ-utility class . 112 B.2.1 Determining the asymptotic behaviour . 112 B.2.2 Computation of the divergence function . 114 5 Preface When choosing between different investment opportunities it is tempting to select the one which offers the highest expected return. However, this strategy would ignore the risk associated with that investment. Generally speaking we have that the larger the expected return of an investment, the larger the risk associated with it. Taking into account the risk of a particular investment is not only necessary to pick the best investment, but also to set up capital requirements. These capital requirements should create a buffer for potential losses of the investments. But how do we describe and measure this risk? We could of course try to describe the cumulative distribution or the density function of the investment. Although this would give us a lot of information about the risk involved, it could still be very difficult to compare different investment opportunities in terms of risk. But a more important problem is that it gives us too much information in some sense. Therefore it would be useful to summarize the distribution of the investment into a number, which represents the risk. These numbers can then be used to determine the necessary capital requirements. More formally if the stochastic variable X models the returns of an investment, then a risk measure is a mapping ρ : X 7! ρ(X) such that ρ(X) 2 R. Because a stochastic variable can be viewed as a function, a risk measure can be interpreted as a functional. We could therefore study risk measures by looking at them as purely mathematical objects. Using techniques and ideas from math- ematical analysis we could then analyse properties of these functionals. This is exactly what we will do in the first chapter of this thesis. Studying risk measures only from a purely mathematical point of view has the downside that it ignores the intuition behind it. The attitude towards risky alter- natives is a subjective matter determined by personal preferences. These personal preferences can be represented using so called utility functions. Utility functions are commonly used in economics to model how people make decisions under un- certainty. In the second chapter we will therefore introduce this decision theoretic framework and explain the necessary concepts of economics. Armed with both a strong mathematical and economic framework we will then apply these concepts to two commonly used risk measures in industry, Value at Risk and Expected Shortfall. This analysis will be the subject of the third chapter. The fourth chapter combines the axiomatic approach from the first chapter and the economic ideas from the second chapter and describes different ways in which utility functions can be used to construct risk measures. We will introduce utility based shortfall risk measures and divergences risk measures. Using ideas from mathematical optimisation we will link different utility based risk measures and discuss different representations of these risk measures. 6 After this discussion the question arises which utility function we should use to construct these utility based risk measures. Because utility functions represent personal preferences we do not believe that there is a straightforward answer to this question. However, the properties of the utility function used in the risk measures do affect this risk measure. The last chapter takes a closer look at different classes of utility functions which appear in literature and asses their properties in the context of utility based risk measures. 7 1 Mathematical representation of risk In this chapter we will look at risk measures from a solely mathematical point of view. We will define what a risk measure is, and what properties it should have. We will take a closer look at the concepts of the acceptance set and the penalty function. Finally we will introduce the robust representation of a risk measure. The contents of this chapter is largely based on the theorems found in [11]. 1.1 Definitions and properties Consider a probability space (Ω; F;P ). Where Ω represents the set of all possible scenario's, where F is a σ-algebra and where P is a probability measure. The future value of a scenario is uncertain and can be represented by a stochastic variable X. This is a function on the set of all possible scenario's to the real numbers, X :Ω ! R. Let X denote a given linear space of functions X :Ω ! R including the constants. A risk measure ρ is a mapping ρ : X! R. Our goal is to define ρ in such a way that it can quantify the risk of a market position X, such that it can serve as a measure to determine the capital requirement of X. That is the amount of capi- tal needed when invested in a risk-free manner will make the position acceptable. Using this interpretation of ρ(X), we would like to have a risk measure that has some likeable properties. First of all, if the value of the portfolio X is smaller then the value of the portfolio Y almost surely, then it would be logical that than you need more money to make the position of X acceptable, than to make the position of Y acceptable. This property is called monotonicity. Property 1. (Monotonicity) If X ≤ Y , then ρ(X) ≥ ρ(Y ). Secondly, it is logical to assume that to make the position X +m acceptable, where m is a risk-free amount, we need to have ρ(X) − m. This is precisely the amount 8 ρ(X) to make the position X acceptable reduced by the risk-free amount m we already had. This property is called translation invariance or cash invariance. Property 2. (Translation invariance) If m 2 R, then ρ(X + m) = ρ(X) − m. Definition 1.1. A mapping ρ : X! R is called a monetary risk measure if ρ satisfies both monotonicity and translation invariance. Here we would like to point out that some authors define a monetary risk such that ρ can also take the values of +1 and −∞. But then they use the additional property that ρ(0) is finite or even normalized ρ(0) = 0. In [11] we found the following lemma. Lemma 1.1. Any monetary risk measure ρ is Lipschitz continuous with respect to the supremum norm k · k, we have: jρ(X) − ρ(Y )j ≤ kX − Y k (1.1) Proof. We have that X − Y ≤ sup jX(!) − Y (!)j; !2Ω hence X ≤ Y +kX−Y k. Using monotonicity we find that ρ(X) ≥ ρ (Y + kX − Y k). Using translation invariance we get ρ(X) ≥ ρ(Y ) − kX − Y k.
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