Technical Appendix Lecture 6: Risk and uncertainty

2010

Abstract: In the paper, we consider the application of the theory of probability metrics in several areas in the eld of nance. First, we argue that specially structured probability metrics can be used to quantify stochastic dominance relations. Second, the methods of the theory ...

International Journal of Theoretical and Applied Finance, 2008

This paper examines the properties that a risk measure should satisfy in order to characterize an investor's preferences. In particular, we propose some intuitive and realistic examples that describe several desirable features of an ideal risk measure. This analysis is the first step in understanding how to classify an investor's risk. Risk is an asymmetric, relative, heteroskedastic, multidimensional concept that has to take into account asymptotic behavior of returns, inter-temporal dependence, risk-time aggregation, and the impact of several economic phenomena that could influence an investor's preferences. In order to consider the financial impact of the several aspects of risk, we propose and analyze the relationship between distributional modeling and risk measures. Similar to the notion of ideal probability metric to a given approximation problem, we are in the search for an ideal risk measure or ideal performance ratio for a portfolio selection problem. We then emphasize the parallels between risk measures and probability metrics underlying the computational advantage and disadvantage of different approaches.

Finance and Stochastics, 2006

General deviation measures are introduced and studied systematically for their potential applications to risk management in areas like portfolio optimization and engineering. Such measures include standard deviation as a special case but need not be symmetric with respect to ups and downs. Their properties are explored with a mind to generating a large assortment of examples and assessing which may exhibit superior behavior. Connections are shown with coherent risk measures in the sense of Artzner, Delbaen, Eber and Heath, when those are applied to the difference between a random variable and its expectation, instead of to the random variable itself. However, the correspondence is only one-to-one when both classes are restricted by properties called lower range dominance, on the one hand, and strict expectation boundedness on the other. Dual characterizations in terms of sets called risk envelopes are fully provided.

2007

Extending the approach of Jouini et al. we define set-valued (convex) measures of risk and its acceptance sets. Using a new duality theory for set-valued convex functions we give dual representation theorems. A scalarization concept is introduced that has economical meaning in terms of prices of portfolios of reference instruments. Using primal and dual descriptions, we introduce new examples for set-valued measures of risk, e.g. set-valued expectations, Value at Risk, Average Value at Risk and entropic risk measure.

arXiv: Risk Management, 2020

Risk measures connect probability theory or statistics to optimization, particularly to convex optimization. They are nowadays standard in applications of finance and in insurance involving risk aversion. This paper investigates a wide class of risk measures on Orlicz spaces. The characterizing function describes the decision maker's risk assessment towards increasing losses. We link the risk measures to a crucial formula developed by Rockafellar for the Average Value-at-Risk based on convex duality, which is fundamental in corresponding optimization problems. We characterize the dual and provide complementary representations.

Report Eurandom, 2005

In this paper we compare two sets of risk measures with respect to the criteria of first and second order stochastic dominance. We observe that overall risk measures do not preserve consistent preference ordering between assets under the first order stochastic dominance rule, while the downside risk measures, with the exception of Expected Shortfall (es), do preserve a consistent preference ordering under first order stochastic dominance. Further, risk measures except es preserve consistent preference ordering between assets under the second order stochastic dominance rule, although for some of the downside risk measures such preference ordering is only partial.

Insurance Mathematics & Economics, 2006

The main purpose to study risk measures for portfolio vectors X = (X 1 , . . . , X d ) is to measure not only the risk of the marginals X i separately but to measure the joint risk of X caused by the variation of the components and their possible dependence. Thus an important property of risk measures for portfolio vectors is consistency with respect to various classes of convex and dependence orderings. From this perspective we introduce and study convex risk measures for portfolio vectors defined axiomatically and further introduce two natural and easy to interprete and calculate classes of examples of risk measures for portfolio vectors and investigate their consistency properties.

In this paper we compare overall as well as downside risk measures with respect to the criteria of first and second order stochastic dominance. While the downside risk measures, with the exception of tail conditional expectation, are consistent with first order stochastic dominance, overall risk measures are not, even if we restrict ourselves to two-parameter distributions. Most common risk measures preserve consistent preference orderings between prospects under the second order stochastic dominance rule, although for some of the downside risk measures such consistency holds deep enough in the tail only. In fact, the partial order induced by many risk measures is equivalent to sosd. Tail conditional expectation is not consistent with respect to second order stochastic dominance.

1998

Two methods are frequently used for modeling the choice among uncertain prospects: stochastic dominance relation and mean-risk approaches. The former is based on an axiomatic model of riskaverse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible trade-off analysis, but cannot model all risk-averse preferences. The seminal Markowitz model uses the variance as the risk measure in the mean-risk analysis which results in a formulation of a quadratic programming model. Following the pioneering work of Sharpe, many attempts have been made to linearize the mean-risk approach. There were introduced risk measures which lead to linear programming mean-risk models. This paper focuses on two such risk measures: the Gini's mean (absolute) difference and the mean absolute deviation. Consistency of the corresponding mean-risk models with the second degree stochastic dominance (SSD) is reexamined. Both the models are in some manner consistent with the SSD rules, provided that the trade-off coefficient is bounded by a certain constant. However, for the Gini's mean difference the consistency turns out to be much stronger than that for the mean absolute deviation. The analysis is graphically illustrated within the framework of the absolute Lorenz curves.

Journal of Mathematical Sciences, 1992

It is well known that the inequality #2 _> 2#~ holds for moments in the class of distributions with increasing intensity function, with equality being attained only for an exponential distribution. Thus we can talk of a characterization of the exponential distribution by the property of minimizing the functional #2-2#12 in the class of distributions with increasing intensity functions. Is it possible to characterize a more or less arbitrary distribution by the property of minimizing a suitable functional? The answer to this question is of course affirmative, since, for example, for each distribution G(x) we can consider the functional 5:a(F) = sup IF(x)-a(x)l X in the class of all distribution functions. It is clear that the minimum of 9:G(F) is attained for F = G and is zero. However the functional 9"a has a rather complicated structure. In what follows we shall construct functionals of simpler form (mean values of certain statistics) for which such a characterization is still possible. Functionals of this type turn out to be connected with certain metrics on classes of probability distributions. We now make some notational conventions. The letters X, Y,... will denote random variables (or vectors) that are as a rule independent. If X is a random variable, the symbol X t will always denote a random variable independent of X and having the same distribution. Theorem 1. Suppose the random variables X, X I, Y, yi are all independent of one another and that for some r e (0, 2) EJXI ~ < oc, Elyl r < c<).