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This paper investigates the coherent risk measure of normal mixture distributions. The main result shows that the mean-risk portfolio optimization problem of some widely-used normal mixture distributions can be reduced to a quadratic programming problem which has closed form of solution by fixing the location parameter and skewness parameter. In addition, the worst-case value at risk in the robust portfolio optimization can also be calculated directly. Finally the conditional value at risk is considered as an example of coherent risk measure; and we obtain the marginal contribution to risk for a portfolio based on normal mixture model.

This paper provides an efficient and accurate approximation for the distorted expectation of a risk factor when its density function or characteristic function is given in an analytical form. The fast Fourier transform (FFT) algorithm is used to set up an approximation for the distorted density function with a truncated sum of its Fourier-cosine series expansion on a finite interval. The resulting truncation approximation leads to an analytical approximation for the distorted expectation. The proposed approach can be used in different branches of applications of distorted expectations, such as in risk management, portfolio optimization and conic finance.

In this paper, we study the extent to which any risk measure can lead to superadditive risk assessments, implying the potential for penalizing portfolio diversification. For this purpose we introduce the notion of extreme-aggregation risk measures. The extreme-aggregation measure characterizes the most superadditive behavior of a risk measure, by yielding the worst-possible diversification ratio across dependence structures. One of the main contributions is demonstrating that, for a wide range of risk measures, the extreme-aggregation measure corresponds to the smallest dominating coherent risk measure. In our main result, it is shown that the extreme- aggregation measure induced by a distortion risk measure is a coherent distortion risk measure. In the case of convex risk measures, a general robust representation of coherent extreme-aggregation measures is provided. In particular, the extreme-aggregation measure induced by a convex short- fall risk measure is a coherent expectile. These results show that, in the presence of dependence uncertainty, quantification of a coherent risk measure is often necessary, an observation that lends further support to the use of coherent risk measures in portfolio risk management.

In this paper we study the aggregate risk of inhomogeneous risks with dependence uncertainty, evaluated by a generic risk measure. We say that a pair of risk measures are asymptotically equivalent if the ratio of the worst-case values of the two risk measures is almost one for the sum of a large number of risks with unknown dependence structure. The study of asymptotic equivalence is particularly important for a pair of a non-coherent risk measure and a coherent risk measure, since the worst-case value of a non-coherent risk measure under dependence uncertainty is typically very difficult to obtain. The main contribution of this paper is that we establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.

Elicitability has recently been discussed as a desirable property for risk measures. Kou and Peng (2014) showed that an elicitable distortion risk measure is either a Value-at-Risk or the mean. We give a concise alternative proof of this result, and discuss the conflict between comonotonic additivity and elicitability.

A basic risk hypothesis for system throughput capacity in the presence of risk is proposed. It is expressed as a basic risk equation , derived in the paper, and governs all nongrowth, nonevolving, agent-directed systems. The basic risk equation shows how expected throughput capacity increases linearly with positive risk of loss of throughput capacity. The conventional standard deviation risk measure, from financial systems, may be used. A proposed new measure, the mean-expected loss risk measure with respect to the hazard-free case, is shown to be more appropriate for systems in general. The concept of an efficient system environment is also proposed. The well-known financial risk equation, hitherto deduced empirically, may be derived from the basic risk equation. When there is both risk exposure and resource sharing, the basic risk equation may be combined with a resource-sharing equation that governs how throughput capacity changes with the resource-sharing level. The basic risk equation also allows for risk elimination and reduction. All quantities in the equation are precisely defined, and their units are specified. The risk equation reduces to a useful numerical expression in practice.

We relate value at risk (VaR) to mean-variance analysis and examine the economic implications of using a mean-VaR model for portfolio selection. When comparing two mean-variance efficient portfolios, the higher variance portfolio might have less VaR. Consequently, an efficient portfolio that globally minimizes VaR may not exist. Surprisingly, we show that it is plausible for certain risk-averse agents to end up selecting portfolios with larger standard deviations if they switch from using variance to VaR as a measure of risk. Therefore, regulators should be aware that VaR is not an unqualified improvement over variance as a measure of risk.

We study a space of coherent risk measures M/ obtained as certain expansions of coherent elementary basis measures. In this space, the concept of ‘‘risk aversion function’’ / naturally arises as the spectral representation of each risk measure in a space of functions of confidence level probabilities. We give necessary and sufficient conditions on / for M/ to be a coherent measure. We find in this way a simple interpretation of the concept of coherence and a way to map any rational investor’s subjective risk aversion onto a coherent measure and vice-versa. We also provide for these measures their discrete versions M ðNÞ / acting on finite sets of N independent realizations of a r.v. which are not only shown to be coherent measures for any fixed N, but also consistent estimators of M/ for large N. 2002 Elsevier Science B.V. All rights reserved.

In this paper, we propose two constructions which allow dependence between the risks of an insurance portfolio in the individual risk model. In the first construction, each risk’s experience is influenced by an individual and a collective risk factor, as well as a class factor if the portfolio is divided into different classes. The second construction uses copulas. The impact on the cumulative distribution function of the aggregate claim amount and on the stop-loss premium is presented via numerical examples.

Financial risk management takes an important part of continuing financial globalization. From the point of financial risk management, financial risk should be controlled at the right level. Considering the characteristics of financial time series, we construct the PGARCH-EVT-Copula model that includes different aspects of statistical features in measuring the risk. With this model, we measure Value at Risk and Expected Shortfall of the futures portfolio and compare them in the risk measurement and testify the reliability with the help of Monte-Carlo simulation method. Finally, we draw a conclusion that at 95% confidence level, Expected Shortfall can better estimate the risk of assets price extreme changing. This paper provides a risk management method for stabilizing the financial market.

Coherent distortion risk measure is needed in the actuarial and financial fields in order to provide incentive for active risk management. The purpose of this study is to propose extended versions of Wang transform using skew normal distribution functions. The main results show that the extended version of skew normal distortion risk measure is coherent and its transform satisfies the classic capital asset pricing model. Properties of the stock price model under log-skewnormal and its transform are also studied. A simulation based on the skew normal transforms is given for a insurance payoff function.

The particular subject of this paper, is to construct a general framework that can consider and analyse in the same time the upside and downside risks. This paper offers a comparative analysis of concept risk measures, we focus on quantile based risk measure (ES and VaR), spectral risk measure and distortion risk measure. After introducing each measure, we investigate their interest and limit. Knowing that quantile based risk measure cannot capture correctly the risk aversion of risk manager and spectral risk measure can be inconsistent to risk aversion, we propose and develop a new distortion risk measure extending the work of Wang (J Risk Insurance 67, 2000) and Sereda et al. (Handbook of Portfolio Construction 2012). Finally we provide a comprehensive analysis of the feasibility of this approach using the S&P500 data set from 01/01/1999 to 31/12/2011.

There is no consistency in VaR measure that may lead to the failure of risk management, this shortcoming can be overcome with Expected shortfall (ES) measure. Since the presence of heteroscedasticity in the capital gains sequence results that variable distribution changes over time, so it is necessary to build GARCH-ES for measure of short-term international capital flows (SCF) risk in China. The empirical results show: Measure under EGARCH-ES models appeared failure cases occur in the short-term international capital flows under extreme circumstances; In general or extreme cases, the accuracy of GARCH (1, 1)-ES and GARCH-M-ES model is better than the former measure. The accuracy of the test results showed that: in the negative range, ES measure appeared more failed.

GlueVaR risk measures defined by Belles-Sampera et al. (2014) generalize the traditional quantile-based approach to risk measurement, while a subfamily of these risk measures has been shown to satisfy the tail-subadditivity property. In this paper we show how GlueVaR risk measures can be implemented to solve problems of proportional capital allocation. In addition, the classical capital allocation framework suggested by Dhaene et al. (2012) is generalized to allow the application of the Value-at-Risk (VaR) measure in combination with a stand-alone proportional allocation criterion (i.e., to accommodate the Haircut allocation principle). Two new proportional capital allocation principles based on GlueVaR risk measures are defined. An example based on insurance claims data is presented, in which allocation solutions with tail-subadditive risk measures are discussed.

This note remedies a risk measure, which was proposed by the work of Jan and Wang (2012). They used property of martingale to measure idiosyncratic risk, and illustrated that it is better than the measurements of variance and semivariance. However, their risk measure can¡¯t distinguish between the assets whose return rising firstly and then declining, and the assets whose return declining firstly and then rising. In this note, I propose a remedied method, which puts more weight to the recent return¡¯s variation, and demonstrate that the new weighting risk measure is more close to the investor risk conception.

This paper studies a new risk measure derived from the expected area in red introduced in Loisel (2005). Specifically, we derive various properties of a risk measure defined as the smallest initial capital needed to ensure that the expected time-integrated negative part of the risk process on a fixed time interval [ 0 , T ] ( T can be infinite) is less than a given predetermined risk limit. We also investigate the optimal risk limit allocation: given a risk limit set at a company level for the sum of the expected areas in red of all lines, we determine the way(s) to allocate this risk limit to the subsequent business lines in order to minimize the overall capital needs.

This paper proposes a risk measure for a portfolio of European-style derivative securities over a fixed time horizon under the Black–Scholes economy. The proposed risk measure is scenario-based along the same line as [3]. The risk measure is constructed by using the risk-neutral probability ([Formula: see text]-measure), the physical probability ([Formula: see text]-measure) and a family of subjective probability measures. The subjective probabilities are introduced by using Girsanov's theorem. In this way, we provide risk managers or regulators with the flexibility of adjusting the risk measure according to their risk preferences and subjective beliefs. The advantages of the proposed measure are that it is easy to implement and that it satisfies the four desirable properties introduced in [3], which make it a coherent risk measure. Finally, we incorporate the presence of transaction costs into our framework.

This paper discusses a class of risk measures developed from a risk measure recently proposed for insurance pricing. This paper reviews the distortion function approach developed in the actuarial literature for insurance risk. The proportional hazards transform is a particular case. The relationship between this approach to risk and other approaches including the dual theory of choice under risk is discussed. A new class of risk measures with suitable properties for asset allocation based on the distortion function approach to insurance risk is developed. This measure treats upside and downside risk differently. Properties of special cases of the risk measure and links to conventional portfolio selection risk measures are discussed. © 2000 Elsevier Science B.V. All rights reserved.

The paper reports the results of an experiment on individual investors’ risk perception in a financial decision making context under two different modes of information presentation (framings). One way to reduce the complexity of a risky decision situation is to focus on risk measures, e.g. the statistical moments of a risky alternative’s distribution. There is a huge number of propositions about which risk measure is to be used from a theoretical point of view. Many of these models are based on the variance as risk measure. But since the symmetrical nature of variance does not capture the common notion of risk as something undesired there has been much discontent with this approach. More recently, lower partial moments (LPMs) have been rediscovered as a more suitable risk measure. They reflect the popular negative meaning of risk since they only take negative deviations from a reference point to measure risk. The purpose of this paper is to examine experimentally people’s risk perception in a financial context. The focus is on the correspondence of risk perceptions with specific LPMs. The main findings can be summarized as follows. First, symmetrical risk measures like variance can be clearly dismissed in favor of shortfall measures like LPMs. Second, the reference point (target) of individuals for defining losses is not a distribution’s mean but the initial price in a time series of stock prices. Third, the LPM which explains risk perception best is the LPM 0 , i.e. the probability of loss. Fourth, the framing of price distributions (histograms versus charts) exerts a significance influence on average risk ratings, the latter being higher for the histogram framing. Fifth, positive deviations from an individual reference point tend to decrease perceived risk.

The paper studies connections between risk aversion and martingale measures in a discrete-time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b= 1; that is, in the case of the log-utility, the optimal dynamic portfolio defines the numeraire portfolio.

We discuss the concept of the risk measure as an expectation using a probability distortion, and classify the standard risk measures according to their associated distortion functions. Using two examples, we explore the features of the different measures.

In this paper we study methods for measuring risk. First, we introduce a conditional risk measure and point out that it is a coherent risk measure. Using the Bayesian statistical idea a subjective risk measure is defined. In some special cases, closed form expressions for the risk measures can be obtained. The credibility theory can be used to relax the strong assumptions on the model and prior distributions, and to obtain approximated risk measure formulas. Applications in both finance and insurance are discussed.

This paper presents a portfolio selection model based on the idea of approximation. The model describes a portfolio by its decumulative distribution curve and a preference structure by a family of convex indifference curves. It prescribes the optimal portfolio as the one whose decumulative curve has the highest tangent indifference curve. The model extends the mean–variance model in the sense that it does not restrict the return distributions of assets to be normal. While under the assumption of normality, the model simplifies to the mean–variance model. The model has a measure of risk attitudes that resembles the Arrow–Pratt measure while combining both wealth and probability attitudes. Using this measure, we show that the smaller the curvature of a value function and the larger the curvature of a weighting function, the more risk averse an agent.

this paper gives a general exposition of the subject of Value at Risk (VaR), which is now considered as a standard measure of market risks. It is defined as the maximal loss of the portfolio for a given probability over a given period. This measure is sensitive to the tails of the distribution of returns; extreme value theory is used here to quantify this phenomenon.

The Value at risk (VaR) measure the risk of loss associated to financial assets. For a given time period (normally ranging from 1 to 10 years) and a with a given probability confidence (generally equal to 95 or 99%); this measure represents the maximum loss the investor can suffer when holding financial assets. The time horizon used to calculate the VaR depends on the investment duration; the value at risk is used to compute the minimum capital requirements necessary to compensate losses resulting from market risk, according to the BIS banking regulation.

By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).

We consider a vector-valued multivariate risk measure that depends on the user's profile given by the user's utility. It is constructed on the basis of weighted-mean trimmed regions and represents the solution of an optimization problem. The key feature of this measure is convexity. We apply the measure to the portfolio selection problem, employing different measures of performance as objective functions in a common geometrical framework.

We propose a class of distortion measures based on contagion from an external “scenario” variable. The dependence between the scenario and the variable whose risk is measured is modeled with a copula function with horizontal concave sections. Special cases are the perfect dependence copula, which generates expected shortfall, the Marshall–Olkin family and the Placket family. As an application, we evaluate distortion measures bank liabilities with respect to a country risk scenario in the current European debt crisis.

In this paper, we take an axiomatic approach to characterize insurance prices in a competitive market setting. We present four axioms to describe the behavior of market insurance prices. From these axioms it follows that the price of an insurance risk has a Choquet integral representation with respect to a distorted probability (Yaari, 1987). We propose an additional axiom for reducing compound risks. This axiom determines that the distortion function is a power function.

Nowadays one of the most studied issues in economic or finance field is to get the best possible return with the minimum risk. Therefore, the objective of the paper is to select the optimal investment portfolio from SP500 stock market and CBOE Interest Rate 10-Year Bond to obtain the minimum risk in the financial market. For this purpose, the paper consists of: 1) the marginal density distribution of the two financial assets is described with kernel density estimation to get the "high-picky and fat-tail" shape; 2) the relation structure of assets is studied with copula function to describe the correlation of financial assets in a nonlinear condition; 3) value at risk (VaR) is computed through the combination of Copula method and Monte Carlo simulation to measure the possible maximum loss better. Therefore, through the above three steps methodology, the risk of the portifolio is described more accuratIy than the conventional method, which always underestimates the risk in the finicial market. So it is necessary to pay attention to the happening of extreme cases like "Black Friday 2008" and appropriate investment allocation is a wise strategy to make diversification and spread risks in financial market.

This paper studies an optimization problem involving pay-offs of (perhaps dynamic) investment strategies. The pay-off is the decision variable, the expected pay-off is maximized and its risk is minimized. The pricing rule may incorporate transaction costs and the risk measure is continuous, coherent and expectation bounded.We will prove the necessity of dealing with pricing rules such that there exists an essentially bounded stochastic discount factor that must also be bounded from below by a strictly positive value. Otherwise, good deals will be available to traders, i.e. depending on the selected risk measure, investors can choose pay-offs whose (risk, return) will be as close as desired to (−1,1) or (−1,1). This pathological property still holds for vector risk measures (i.e. if we minimize a vector-valued function whose components are risk measures). It is worth pointing out that, essentially, bounded stochastic discount factors are not usual in the financial literature. In particular, the most famous frictionless, complete and arbitrage-free pricing models imply the existence of good deals for every continuous, coherent and expectation bounded (scalar or vector) measure of risk, and the incorporation of transaction costs will not guarantee the solution of this caveat This research was partially supported by RD_Sistemas SA, Welzia Management SGIIC SA, and Ministerio de Economía, Spain (grants ECO2009-14457-C04 and ECO2012-39031-C02-01) Publicado

This paper examines a class of premium functionals which are (i) comonotonic additive and (ii) stochastic dominance preservative. The representation for this class is a transformation of the decumulative distribution function. It has close connections with the recent developments in economic decision theory and non-additive measure theory. Among a few elementary members of this class, the proportional hazard transform seems to stand out as being most plausible for actuaries.

In this paper we propose a standard measure of risk that is based on the converted expected utility of normalized lotteries with zero-expected values. This measure of risk has many desirable properties that characterize the notion of risk. It is very general and includes many previously proposed measures of risk as special cases. Moreover, our standard measure of risk provides a preference-based and unified method for risk studies. Since the standard measure of risk is compatible with the measure of expected utility, it can be used explicitly or implicitly in an expected utility model. Under a condition called risk Independence, a decision could be made by explicitly trading off between risk and value, which offers an alternative representation of the expected utility model, named the standard risk-value model. Finally, we discuss some other applications of the standard measure of risk and extensions of our risk-value tradeoff framework for descriptive decision making.

As the two important form of financial market, the risk of financial securities, such as stocks and bonds, has been a hot topic in the financial field; at the same time, under the influence of many factors of financial assets, the correlation between portfolio returns causes more research. This paper presents Copula-SV-t model that it uses SV-t model to measure the edge distribution, and uses the Copula-t method to obtain the high-dimensional joint distribution. It not only solves the actual deviation with using the ARCH family model to calculate the portfolio risk, but also solves the problem to overestimate the risk with using extreme value theory to study financial risk. Through the empirical research, the conclusion shows that the model describes better assets and tail characteristics of assets, and is more in line with the reality of the market. Furthermore, Empirical evidence also shows that if the portfolio is relatively large degree of correlation, the ability to disperse portfolio risk is relatively weakness.

Evaluating risk measures, premiums, and capital allocation based on dependent multi-losses is a notoriously difficult task. In this paper, we demonstrate how this can be successfully accomplished when losses follow the multivariate Pareto distribution of the second kind, which is an attractive model for multi-losses whose dependence and tail heaviness are influenced by a heavy-tailed background risk. A particular attention is given to the distortion and weighted risk measures and allocations, as well as their special cases such as the conditional layer expectation, tail value at risk, and the truncated tail value at risk. We derive formulas which are either of closed form or follow well-defined recursive procedures. In either case, their computational use is straightforward.

The dichotomy between risk analytics and pricing is well known amongst financial practitioners and researchers. For risk analysis, such as computing value at risk and credit exposures, expectations of future values must be computed under the real world measure. For pricing, expectations are computed under a risk neutral measure. This means that for calculations such as value at risk (VaR) and credit exposures (EEs, EPEs, etc) on derivative portfolios, risk factors are evolved to the horizon date under the real world measure, at which point the portfolio is repriced under a risk neutral measure.Simulation under the real world measure followed by repricing under a risk neutral measure is computationally intensive, especially when the repricing requires Monte Carlo. Because of the computational effort involved, shortcuts are often taken. One common shortcut is to assume the real world measure is the risk neutral measure, so that calculations can be done under one measure. This particular shortcut is especially problematic as it leads to results varying wildly depending on the numeraire chosen.Here we detail methods of avoiding this problem by combining the real world measure with the risk neutral measure. We present an application to risk analytics which speeds up the calculations by orders of magnitude, changing O(n^2) calculations to O(n) with a far smaller scaling constant. This allows the computation of exposures under the real world measure, obviating the need for the dangerous practice of computing exposures under a risk neutral measure.

This paper proposes a new premium principle, where risk loadings are imposed by a proportional decrease in the hazard rates. This premium principle is scale invariant and additive for layers. It is shown that this principle will generate stop-loss contracts as optimal reinsurance arrangements in a competitive market when the reinsurer is less risk-averse than the direct insurer. Finally, increased limits factors are calculated based on this principle.

Despite the fact that the value-at-risk (VaR) measure has been adopted as a standard risk measure in the financial industry, it has a number of deficiencies recognized by financial professionals. It is not a coherent risk measure. This is because it does not satisfy the subadditivity property requirement of a coherent risk measure. That is, there are cases in which the portfolio VaR is larger than the sum of the VaRs of the portfolio constituents, supporting the view that VaR cannot be used as a true risk measure. Unlike VaR, the average value-at-risk measure (AVaR)—also referred to as conditional value-at-risk and expected shortfall—is a coherent risk measure and has other advantages that result in its greater acceptance in risk modeling. Keywords: average value-at-risk; coherent risk measures; average value-at-risk; conditional value-at-risk; expected shortfall; The Student's; distribution

A risk measure that has been widely accepted since the 1990s is the value-at-risk (VaR). In the late 1980s, it was integrated by JP Morgan on a firmwide level into its risk-management system. In the mid-1990s, the VaR measure was approved by regulators as a valid approach to calculating capital reserves needed to cover market risk. The Basel Committee on Banking Supervision released a package of amendments to the requirements for banking institutions, allowing them to use their own internal systems for risk estimation. In this way, capital reserves, which financial institutions are required to keep, could be based on the VaR numbers computed internally by an in-house risk management system. Generally, regulators demand that the capital reserve equal the VaR number multiplied by a factor between 3 and 4. Thus, regulators link the capital reserves for market risk directly to the risk measure. In practice, there are several approaches for estimating VaR. Keywords: value-at-risk; tail probability; RiskMetrics Group; historical method; hybrid method; Monte Carlo method; selection of a statistical model; estimation of the statistical model parameters; generation of scenarios from the fitted model; calculation of portfolio risk; coherent risk measure; monotonicity; positive homogeneity property

Portfolio selection problem is one of the most important issues within financial risk management and decision making. It concerns both, financial institutions and their regulator/supervisor bodies. A very challenging question in this context is whether there is some impact of alternative dependency/concordance measures on the efficiency of optimal portfolios. Therefore, the alternative ways of portfolio comparisons were developed, among them a stochastic dominance approach is one of the most popular one. In particular, the definition of the second-order stochastic dominance (SSD) relation uses comparisons of either twice cumulative distribution functions or expected utilities. Alternatively, one can define SSD relation using cumulative quantile functions or conditional value at risk. The task of this paper is therefore to examine and analyze the SSD efficiency of min-var portfolios that are selected on the basis of alternative concordance matrices set up on the basis of either Spearman rho or Kendall tau. It is empirically documented that only Pearson measure in Markowitz model identified a portfolio that can be of interest for at least one risk averse investor. Moreover, a portfolio based on Kendall measure is very poor (at least in terms of SSD efficiency).

Distorted expectations can be expressed as mixtures of quantiles. In this note, we show that this statement is essentially true, but that one has to be careful with the correct formulation of it. Furthermore, the proofs of the additivity property for distorted expectations of a comonotonic sum that appear in the literature often do not cover the case of a general distortion function. We present a straightforward proof for the general case, making use of the appropriate expressions for distorted expectations in terms of mixtures of quantiles. ispartof: FEB Research Report AFI_1273 pages:1-11 status: published

An important problem in quantitative risk measurement in finance and insurance is to determine (sharp) bounds for the distribution function of the joint portfolio \(\sum _{i=1}^{n}X_{i}\) of a risk vector \(X\,=\,(X_{1},\ldots ,X_{n})\) where the marginal distribution functions \(F_{i}\,\sim \,X_{i}\) are known but the dependence between the components is unspecified.

Linear optimization problems are investigated whose parameters are uncertain. We apply coherent distortion risk measures to capture the violation of restrictions. Such a model turns out to be appropriate for many applications and, principally, for the mean-risk portfolio selection problem. Each risk constraint induces an uncertainty set of coefficients, which comes out to be a weighted-mean trimmed region. We consider a problem with a single constraint. Given an external sample of the coefficients, the uncertainty set is a convex polytope that can be exactly calculated. If the sample is i.i.d. from a general probability distribution, the solution of the stochastic linear program (SLP) is a consistent estimator of the SLP solution with respect to the underlying probability. An efficient geometrical algorithm is proposed to solve the SLP.

This chapter presents a multi-criteria portfolio model with the expected return as a performance measure and the expected worst-case return as a risk measure. The problems are formulated as a single-objective linear program, as a bi-objective linear program, and as a triple-objective mixed integer program. The problem objective is to allocate the wealth on different securities to optimize the portfolio return. The portfolio approach has allowed the two popular financial engineering percentile measures of risk, value-at-risk (VaR) and conditional value-at-risk (CVaR) to be applied. The decision-maker can assess the value of portfolio return, the risk level, and the number of assets, and can decide how to invest in a real-life situation comparing with ideal (optimal) portfolio solutions. The concave efficient frontiers illustrate the trade-off between the conditional value-at-risk and the expected return of the portfolio. Numerical examples based on historical daily input data from the Warsaw Stock Exchange are presented and selected computational results are provided. The computational experiments prove that both proposed linear and mixed integer programming approaches provide the decision-maker with a simple tool for evaluating the relationship between the expected and the worst-case portfolio return.

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel's classical notion of qualitative robustness is not suitable for risk measurement and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz space. This concept of robustness captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results that are of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for {\psi}-weak convergence.

Statistics & Risk Modeling, 29 (2), ISSN:2193-1402, ISSN:2196-7040

Pursuing our previous work in which the classical notion of increasing convex stochastic dominance relation with respect to a probability has been extended to the case of a normalised monotone (but not necessarily additive) set function also called a capacity, the present paper gives a generalization to the case of a capacity of the classical notion of increasing stochastic dominance relation. This relation is characterized by using the notions of distribution function and quantile function with respect to the given capacity. Characterizations, involving Choquet integrals with respect to a distorted capacity, are established for the classes of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to a given generalized stochastic dominance relation. Moreover, under suitable assumptions, a "Kusuoka-type" characterization is proved for the class of monetary risk measures having the properties of comonotonic additivity and consistency with respect to the generalized increasing convex stochastic dominance relation. Generalizations to the case of a capacity of some well-known risk measures (such as the Value at Risk or the Tail Value at Risk) are provided as examples. It is also established that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity.