Consistent risk measures for portfolio vectors

2017

The use of risk measures and its applications in portfolio optimisation Resham Sivnarain Magister Scientiae In the Department of Mathematics and Applied Mathematics University of Pretoria 2017 Supervisor: Prof. E. Maré In this dissertation, we study the application of risk measures to portfolio optimisation. A risk measure is a functional over the set of random portfolio returns mappings . We present the various risk measures in this dissertation within an axiomatic framework. Although Value-at-Risk (VaR) has been widely used, the Conditional-Value-at-Risk (CVaR) has become the more popular risk measure since it is a coherent and convex risk measure. We solve a CVaR based optimisation model that is used for portfolio optimisation and hedging a target portfolio. Additionally, we solve a CVaR based optimisation model with cost considerations included in the objective function. Further, we include alternative risk measures such as distortion, spectral, drawdown and coherent-distortion ...

Mathematical Finance, 2007

The optimized certainty equivalent (OCE) is a decision theoretic criterion based on a utility function, that was first introduced by the authors in 1986. This paper re-examines this fundamental concept, studies and extends its main properties, and put it in perspective to recent concepts of risk measures. We show that the negative of the OCE naturally provides a wide family of risk measures that fits the axiomatic formalism of convex risk measures. Duality theory is used to reveal the link between the OCE and the ϕ-divergence functional (a generalization of relative entropy), and allows for deriving various variational formulas for risk measures. Within this interpretation of the OCE, we prove that several risk measures recently analyzed and proposed in the literature (e.g., conditional value of risk, bounded shortfall risk) can be derived as special cases of the OCE by using particular utility functions. We further study the relations between the OCE and other certainty equivalents, providing general conditions under which these can be viewed as coherent/convex risk measures. Throughout the paper several examples illustrate the flexibility and adequacy of the OCE for building risk measures.

Insurance: Mathematics and Economics, 2004

Many types of insurance premium principles and/or risk measures can be characterized by means of a set of axioms which in many cases are arbitrarily chosen and not always in accordance with economic reality. In the present paper we generalize Yaari's risk measure based on some less stringent axioms. In addition we derive translation invariant minimal Orlicz risk measures, which we call Haezendonck risk measures, and obtain sufficient conditions on the risk measure of Bernoulli risks to fulfill additivity and superadditivity properties for Orlicz premium principles. Keywords: Consistent risk measures, Haezendonck risk measure, Yaari's dual theory of choice under risks 1 Introduction Recently, in Goovaerts et al. (2003 a ) it was argued that risk measures should be selected in an appropriate way in order to reflect the basic economic underlying reality. Indeed several examples can be given, which are relevant to real life insurance problems where evidently the properties that the risk measures should have are determined by the realities of the actuarial applications.

2012

Equivalent characterizations of multi-portfolio time consistency are deduced for closed convex and coherent set-valued risk measures on L_d^p(\Omega,\mathcal F_T,\mathbb P) with image space in the power set of L_d^p(\Omega,\mathcal F_t,\mathbb P). In the convex case, multi-portfolio time consistency is equivalent to a cocycle condition on the sum of minimal penalty functions. In the coherent case, multi-portfolio time consistency is equivalent to a generalized version of stability of the dual variables. As examples, the set-valued entropic risk measure with constant risk aversion coefficient is shown to satisfy the cocycle condition for its minimal penalty functions, the set of superhedging portfolios in markets with proportional transaction costs is shown to have the stability property and in markets with convex transaction costs is shown to satisfy the composed cocycle condition, and a multi-portfolio time consistent version of the set-valued average value at risk, the composed AV@R, is given and its dual representation deduced.

The European Journal of Finance, 2011

The problem of risk portfolio optimization with translation-invariant and positive-homogeneous risk measures, which includes value-at-risk (VaR) and tail conditional expectation (TCE), leads to the problem of minimizing a combination of a linear functional and a square root of a quadratic functional for the case of elliptical multivariate underlying distributions. In this paper we provide an explicit closed-form solution of this minimization problem, and the condition under which this solution exists. The results are illustrated using data of 10 stocks from NASDAQ/Computers. The distance between the VaR and TCE optimal portfolios is investigated.

The minimization of general risk or dispersion measures is becoming more and more important in Portfolio Choice Theory. There are two major reasons. Firstly, the lack of symmetry in the returns of many assets provokes that the classical optimization of the standard deviation may lead to dominated strategies, from the point of view of the second order stochastic dominance. Secondly, but not less important, many institutional investors must respect legal capital requirements, which may be more easily studied if one deals with a risk measure related to capital losses. This paper proposes a new method to simultaneously minimize several risk or dispersion measures. The representation theorems of risk measures are applied to transform the general risk minimization problem in a minimax problem, and later in a linear programming problem between infinite-dimensional Banach spaces. Then, new necessary and sufficient optimality conditions are stated and a simplex-like algorithm is developed. T...

Insurance: Mathematics and Economics, 2012

Tail condition expectation Minimization of root of quadratic functional Elliptical family a b s t r a c t Risk portfolio optimization, with translation-invariant and positive-homogeneous risk measures, leads to the problem of minimizing a combination of a linear functional and a square root of a quadratic functional for the case of elliptical multivariate underlying distributions.

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.

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.

2008

Several polyhedral risk measures have been recently introduced leading to Linear Programming (LP) models for portfolio optimization. In this paper we study LP solvable portfolio optimization models based on the tail Gini's mean difference risk measurement. We use combinations of the Conditional Value at Risk (CVaR) measures to get some approximations to the tail Gini's mean difference with the advantage of being computationally much simpler than the Gini's measure itself. We introduce the weighted CVaR (WCVaR) measures defined as simple combinations of a very few CVaR measures with specific type of weights settings which relates the WCVaR measure to the tail Gini's mean difference. This allows us to use a few tolerance levels as only parameters specifying the entire WCVaR measures while the corresponding weights are automatically predefined by the requirements of the corresponding tail Gini's measure. All the studied models are SSD consistent and LP computable. We analyze both the theoretical properties of the models and their performances on the real-life data.