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This article surveys the appropriateness of Value at Risk as a tool for managing trading portfolios. We introduce various calculation methods and give a synopsis of their pros and cons with respect to accuracy, implementational aspects as well as suitability for resource allocation and optimization. We contribute to existing approaches for capital allocation and provide an optimization model. Finally, we demonstrate shortcomings of Value at Risk as a measure of risk. The focus is on compatibility with decision theory and its relation to coherent measures of risk. Thereby, we discuss alternatives to Value at Risk such as Lower Partial Moment One or tail conditional expectation. We give some reasons to prefer the latter as an alternative to Value at Risk and suggest that under certain conditions the tail conditional expectation is compatible with second order stochastic dominance.

Risk control has become one of the major concern of financial institutions. The need for adequate statistical tools to measure and anticipate the amplitude of the potential moves of financial markets is clearly expressed, in particular for derivative markets. Classical theories, however, are based on simplified assumptions -- such as Gaussian statistics -- and lead to a systematic (and sometimes dramatic) underestimation of real risks. We summarize a few basic notions in probability theory which are useful in a financial context: Statistics of Extremes, Sums of Random Variables, Central Limit Theorems and Deviations, Correlations and Dependence, and a brief introduction to Random Matrix Theory. This self-contained text corresponds to the first chapter of our book, `Theory of Financial Risk'., 50 pages, LaTeX + 10 EPS figures

Risk is a concept which matters to many issues in economics and finance. The range of risk measures proposed goes from classics like variance to modern approaches like Value-at-Risk (VaR). In this paper, after a short characterization of manager’s intuitive notion of risk, an overview of those risk measures is given which try to measure risk in a standardized way independent of individually varying perception. Then, it is shown that all these measures including Value-at-Risk, basically, are special cases of a certain well-known family of risk measures. From this point of view, the most critical features of each measure, particularly of VaR, become immediately evident.

Value-at-Risk (VaR) has emerged as the standard tool for measuring and reporting financial market risk. Currently, more than eighty commercial vendors offer enterprise or trading risk management systems which report VaR-like measures. Risk managers are therefore often left with the daunting task of having to choose from this plethora of risk measures. Accordingly, this paper develops framework for answering the following questions about VaRs: 1) How can a risk manager test that the VaR measure at hand is properly specified, given the history of asset returns? 2) Given two different VaR measures, how can the risk manager compare the two and pick the best in a statistically meaningful way? Finally, 3) how can the risk manager combine two or more different VaR measures in order to obtain a single statistically superior measure? The usefulness of the methodology is illustrated in an application to daily returns on the S&P500. In the application, competing VaR measures are calculated from either historical or option-price based volatility measures, and the VaRs are then tested and compared.

We provide evidence for risk management as a value creation activity. To test this proposition, we introduce risk control in portfolio decision making where in order to assess risk we developed a VaR model that is able to take in multidimensional risks. Then we ran out a simulation according to Italian banking operational procedures: our evidence shows a better performance both as total return and as Sharpie ratio. We argue that Basle standard requirements are less heavier than ones granted by internal models, modified by prudential coefficients.

This study focuses on the concept of perceived risk and its relations to behavioural intentions and behaviour. Most research on health related behaviour employs a measure of unconditional risk. Given the inconsistent findings concerning the relationship between perceived risk and behavioural measures in the health literature, we investigated the usefulness of a conditional risk estimate as an alternative measure. In total, 247 heterosexual visitors of a STD-clinic participated in this study. Hundred participants had private partners only, 147 engaged in prostitution contacts. The relationship between the unconditional measure of perceived risk and behavioural intentions was found to be dependent upon sample-differences. However, it appeared that when a conditional risk estimate was used, a significant positive relation with intention was observed, regardless of the risk-status of the sample. Given the results of the present study, and the closer resemblance of the conditional risk measure to the ...

Complex systems are subject to failure with increased use and degradation. The risk process is the stochastic dynamic process of system failures and their severities. This paper considers aggregate risk measures for the risk process of complex systems in the context of stochastic ordering. The aggregation follows from the accumulation of losses from a series of failure events. The emphasis is on second-order risk measures which account for risk aversion as defined by concave utilities. A second-order measure termed the adjusted risk priority number (ARPN) is presented. The measure is constructed from well-known statistics: rate of failures, average severity of failures, and the Gini Index for severity of failures. The ARPN is contrasted with the traditional risk priority number (RPN) defined by the rate and average severity. The computation and use of the measures is illustrated with a spectrum of failure data from commercial aircraft in the USA.

The paper uses a discrete random variable and two sequences of nonnegative random variables formulating a stochastic model which is the maximum of a random number of stochastic multiplicative models. Sufficient conditions for the distribution function of the model are established. The paper also provides applications of this model assessing certain risk control operations applied to a random number of systems which have been affected by a major risk. Moreover, the paper makes clear that a concept of extreme value theory can contribute to the improvement of the applicability of risk management in a class of systems.

Value-at-Risk (VaR) is a popular risk-metric for reporting financial exposure, for evaluating fund/manager performance and for regulatory disclosures. Yet, VaR is not a coherent risk measure because it is not sub-additive. This paper applies the methodology of risk budgeting to determine if VaR qualifies as a coherent risk measure. We show that the tools of risk budgeting allow VaR to be treated as a coherent risk measure, even though it does not restore sub-additivity. The main finding is that the additional analysis provided by risk budgeting means that VaR is a useful tool even if it is not sub-additive.

Beta—the systematic risk—is generally accepted as a measure for the risk involved in holding a portfolio of risky securities. It will be shown in this paper that, because beta is measured by regressing one multiplicative variable (the rate of return of a security or a portfolio) on another multiplicative variable (the market), in the long run, the systematic risk will approach either zero or infinity. It will also be shown that in the long run the unsystematic risk will dominate the systematic risk, and that, regardless of the value of this latter risk. This implies that for investors with long planning horizon the information conveyed in the systematic risk of an investment is rather limited.

This paper develops a matrix-measure of multivariate risk aversion which is related to a notion of risk premium and states the restrictions that must be imposed upon the matrix-measures of two utility functions in order that one require a higher risk premium than another for every small multivariate risk. A necessary and sufficient condition for comparability of global attitude towards risk is that the local restrictions hold over the entire domain. The usefulness of the measures of risk aversion is discussed within the context of a multivariate risk-sharing problem. UNIVARIATE MEASURES OF ABSOLUTE and relative risk aversion were introduced in the seminal works of Pratt [8] and Arrow [1], and have since become indispensable tools for the analysis of risk bearing in situations involving unidimensional risks. Recent years witnessed numerous attempts to generalize various aspects of the Pratt-Arrow notions to the case of multivariate risk (Kihlstrom and Mirman [6], Keeney [5], Duncan [2], Paroush [7], and Stiglitz [9]). The univariate case is qualitatively different from the multivariate case in that the ordinal preferences of all decision makers are identical, whereas in the multivariate case the preference orderings may differ among decision makers. This fact has two important consequences. First, while the analysis of risk bearing in situations involving univariate risks turns out to depend solely upon the cardinal properties of the function representing the ordinal preferences of the decision makers involved, in situations involving multivariate risks, the ordinal preferences themselves play an important role in addition to that of the function representing them. Second, in the univariate case the local measure of absolute risk aversion permits a complete ordering of individuals according to the relation "at least as risk averse as," whereas in the multivariate case such ordering is not in general independent of the specific risk under consideration. Pratt [8] has shown that the univariate measure of absolute risk aversion satisfies a formal relation to a number which has the interpretation of risk premium. In this study I introduce a matrix measure of absolute multivariate risk aversion which satisfies a similar formal criterion. It should be noted at the outset, however, that in the multivariate case the risk premium may be regarded as a vector of commodities (see, for example, Duncan [2, 3]), and interpersonal comparison of risk aversion is not directional independent. I.e., the ranking of risk

(Originally published in Journal of Economic Theory (1983), 29, 347-352) - This paper is concerned with generalised scalar measures of risk aversion. A measure R which may meaningfully be applied to both unidimensional risks (risk in income or wealth) and multidimensional risks has been constructed. In case of iden- tical preferences, we have also constructed an alternative measure of risk aversion R* which is shown to be related to the Khilstrom-Mirman measure. This relationship explains the nature of the Khilstrom-Mirman measure. Journal of Economic Literature Classification Numbers: 021, 022, 026.

This paper seeks to get behind specific contextual referents of risky situations to consider characteristics of risk that apply to many situations. It is guided by previous theoretical and empirical research in perceived risk, and focuses on the joint effects on risk of loss probability and the distribution of losses. The approach taken follows modern axiomatic theory by proposing conditions on a relation “is at least as risky as” between pairs of probability distributions over an outcome variable. Several sets of axioms for risk that characterize different forms for risk measurement are presented.

Sharpe's, Treynor's and Jensen's measures have been extensively used for the performance evaluation of mutual funds or portfolios. These three widely used performance measures have been found to be highly correlated with their corresponding risk measures by a number of empirical studies. This paper focuses the investigation on the possible sources of the bias associated with the empirical relationship between the estimated Sharpe's measure and its estimated risk measure. In general, the sample size, the investment horizon and the market conditions are three important factors in determining the strong relationship between the ex-post Sharpe's measure and its estimated risk surrogate. The interesting findings of this study are as follows: (1) the estimated Sharpe's measure is uncorrelated with the estimated risk measure either when the risk free rate of interest equals the expected return on the market portfolio over the sample period or when the sample size is infinite, (2) the estimated Sharpe's measure is positively (or negatively) correlated with the estimated risk measure if the risk-free rate of interest is greater than (or less than) the expected return on the market portfolio, (3) an observation horizon shorter than the true investment horizon can reduce the dependence of the estimated Sharpe's measure on its estimated risk measure, and (4) an observation horizon longer than the true investment horizon will magnify the dependence. The results have indicated that, in conducting empirical research, a shorter observation horizon and a large sample size should be used to reduce the bias associated with the estimated Sharpe's measure.

This paper considers the risk premium for one risk when other background risk is present. In a mean-variance setting, we examine the conditions under which the risk premium will be negative. These conditions consider the variance of the considered risk in relation to the covariance of the considered risk with the background risk. We consider the cases of both multiplicative and additive risks. A breakdown of the total risk premium into several components is established by removing each source of risk consecutively. The case of multiplicative risks is shown to be quite different from the additive-risk case in that the individual's total willingness-to-pay for the removal of all risk on a sequential basis does not necessarily equal the willingness-to-pay for removal of all risk simultaneously.

This paper explores a problem with the use of the entropy measure of investment performance. The commonly used entropy measure ignores the dispersion of security frequency classes used in the calculation of entropy. Therefore, state-value weighting of the entropy is proposed and tested using a portfolio selection heuristic algorithm. The results indicate that weighting the entropy value will increase the investment performance of the entropy risk measure.

A measure for the relative degree of risk aversion in higher dimensions is proposed. It is shown that the measure possesses all the properties considered by Kihlstrom and Mirman. Finally a generalisation of the concepts of increasing, decreasing constant relative degrees of risk aversion to a class of non-homothetic utility functions is made.

(1987). A Note on the Measurement of Risk in a Portfolio. Financial Analysts Journal: Vol. 43, No. 2, pp. 61-69.

THE PURPOSE OF THIS PAPER is to determine the degree of nonstationarity in the risk levels and performance measures of mutual fund portfolios and reconcile several observed inconsistencies in the literature concerning a bias in risk-adjusted performance measures. A necessary condition for the estimation of the Jensen [17, 18] risk-adjusted performance measure and any statistical inference conerning that measure is that the mutual fund's systematic risk remain constant over time.' This assumption is unsatisfactory because mutual funds, generally being actively managed, should have a changing level of systematic risk as a result of the buying and selling decisions of their managers. For example, if a manager can make better than average forecasts of future realizations on market factors, he is expected to adjust portfolio risk in anticipation of market movements. The level of portfolio systematic risk can be substantially changed in either direction by altering the proportions of high and low risk securities in the portfolio. The important point is that the risk level of a managed portfolio is a decision variable. We should expect to observe nonstationary risk levels2 of managed portfolios as a result of timing

The risk premium (conveniently normalized) is defined as the measure of risk aversion. This measure does not require any relevant assumption in the theory of choice under uncertainty except the existence of a certainty equivalent. In particular, the independence axiom is not required. The measure of risk aversion of an action is provided not only for the case with one commodity and two consequences but also for the case with many commodities and consequences. The measure of mean risk aversion of all actions with given consequences is introduced and the local measure of risk aversion is obtained by making all these consequences approach the consequence under consideration. This measure is demonstrated to be zero when the von Neumann-Morgenstern utility function exists. In this case a measure of risk aversion of the second order is introduced, which turns out to be equal to the Arrow-Pratt absolute index when there is only one commodity and similar to the generalized measures proposed by several authors when there are many commodities and two consequences.

This paper introduces conditional risk analysis as a new approach to extend the standard risk analysis method of Hertz. An exercise in profit planning is used to illustrate the special features of conditional risk analysis: total risk measurement, risk decomposition, factor outcome analysis and variable significance analysis. Mathematical expressions are presented for performing the various analyses.

A special structure optimization model is presented which includes many of the single variable risk problems that are encountered in operational problems. A risk function is assumed which is a piece-wise linear function of some random variable whose distribution is known; one seeks the value of the decision variable which minimizes expected risk. In this paper are presented the necessary and sufficient conditions for this optimization for random variables which are either continuously or discretely distributed. The important special case of a continuous risk function is discussed; multiple risk problems with a joint constraint are analyzed; and the change in policy for a small change in the distribution of the random variable is investigated. Examples illustrate the application of the model.

THIS PAPER DEFINES two related three-parameter risk measures and shows that common risk measures of variance, semi-variance, mean absolute deviation, and probability of an outcome worse than some disaster level are all special cases of one of these three-parameter risk measures and that standard deviation is a special case of the other. It is established that the selection of any of these risk measures has implicitly involved decisions about: (1) a reference level of wealth about which deviations will be measured; (2) the relative importance of large versus small deviations; (3) the outcomes that should be included in the risk measures. Properties of these risk measures are derived and issues in the specification of risk measures are studied.

The classical conditional expectation with respect to σ-algebras, on probability measure spaces, has been extended for infinite measure spaces. In this paper we consider conditional expectations with respect to δ-rings, on arbitrary measure spaces. An additional condition has to be imposed in order to insure the uniqueness of the conditional expectation. The existence of the conditional expectation is proved for functions in Lp with 1 ≤ p < ∞, and, for localizable measures, also in L∞.The properties of the classical conditional expectation remain true in the general case, sometimes with some modifications.

A crucial property for dynamic risk measures is the time consistency. In this paper, a characterization of time consistency in terms of a “cocycle condition” for the minimal penalty function is proved for general dynamic risk measures continuous from above. Then the question of the regularity of paths is addressed. It is shown that, for a time consistent dynamic risk measure normalized and non-degenerate, the process associated with any bounded random variable has a cadlag modification, under a mild condition always satisfied in the case of continuity from below. When normalization is not assumed, a right continuity condition on the penalty has to be added. Applying these results and using right continuous BMO martingales, families of not necessarily normalized dynamic risk measures leading to cadlag paths, and allowing for jumps, are exhibited.

Current robust risk measures or portfolio selection models are usually derived under the worst-case analysis, which makes the investment decision too conservative and could not reflect the change of uncertainty sets with respect to different market environ- ments. We use the regime switching technique to describe the time-varying uncertainty set of the first and second order moments, and propose two kinds of robust risk measures: worst regime risk measure and mixed worst-case risk measure. These new risk measures have good properties and the robust portfolio selection models derived from them can be efficiently solved in polynomial time. Empirical results show the reasonability and efficiency of our new models.

The present paper is intended to selected aspects of approximation, respectively quantification of market risk exposure of financial instruments. Featured approximation methods or quantifying the level of risk are first described in theory and then applied to real data. Specifically, the shares of Oracle, Coca Cola and Pfizer term in January 2012 to February 2015. The price development is on a daily basis. An integral part of the paper is a graphical representation of the results and their comparison. The distribution function of the random variable representing the risk is not known and therefore it must be estimated. In the present paper we will assume that the theoretical probability distribution of losses is not known to us and we have to estimate, i.e. apply called nonparametric estimates.

This paper proposes two mixed models to study optimal saving in the presence of two types of risk: income risk and background risk. In the first model the income risk is a fuzzy number and the background risk is a random variable. In the second model the income risk is a random variable and the background risk is a fuzzy number. For these models three notions of precautionary saving are defined as indicators of the changes induced by the income risk and the background risk on the choice of optimal saving.

The study of risk aversion of an agent confronted by a risk situations with several parameters is an important topic of risk theory. It is tackled traditionally with probabilistic methods. When these do not offer an appropriate shaping we can use Zadeh's possibility theory. In this paper a possibilistic model of risk aversion with several parameters is proposed. The notion of possibilistic risk premium vector is introduced as a measure of an agent's risk aversion to a situation with several risk parameters. The main result of the paper is an approximate calculation formula of this indicator. The way we can apply this model in risk aversion evaluation in grid computing is sketched out.

A decision-making problem of investment into a profitable object in a catastrophic risk area is considered. By a catastrophic risk is meant the probability of severe yet unlikely losses. As a risk hedging mechanism, an insurance fund is considered that is replenished by a part of profit and is used for object renewal. It is shown that methods of insurance mathematics can be used to assess the risk to lose the object. For the plant loss probability as a function of the insurance reserve, integral equations are derived. They can be solved by successive approximations.

We use survey data on the well-being of individuals to measure attitudes toward risk. Risk neutrality cannot be rejected by the data.

This study is concerned with establishing the determinants of banks' exposure to risk and with predicting risk in banking. Using the COMPUSTAT data base, prediction rules have been developed for two aspects of risk: systematic risk (risk that is related to covariance with the market portfolio) and residual risk (the aggregate of specific risk and extra-market covariance). For each type of risk, several models have bean estimated: one model employs only measures of the asset and liability characteristics of the bank; a second employs these characteristics and other data taken from annual reports; a third model adds the history of the behavior of the price at the bank's common stock. The central conclusion of the study is that systematic and residual risk in banks can be predicted from predetermined data. Prediction rules estimated in this way can serve a useful function in monitoring bank risk. Further, the predictive significance of each variable serves as a measure of the appropriateness of that variable as an indicator of risk, and hence as a target for regulation.

A dimensional model of perceived risk and a model of risk acceptance based on risk rates are proposed in this paper. In line with the proposed model of perceived risk, risk is a linear combination of the three basic dimensions of a risky situation: the amount and probability of loss and the amount of gain. It is assumed that psychological transformations are made on these dimensions. According to the model of acceptance, acceptance is judged by making a trade-off between perceived risk and the amount of gain. These models have been investigated in two experiments, in which risk judgments and acceptance rates for a set of descriptions of risky investments were collected from managers in Poland. The proposed dimensional model of perceived risk was compared to the distributional models of risk and to the risk models based on the expectation principle. The best fit was obtained for the proposed model. It was also found that perceived risk was useful in predicting acceptance rates. Better fits were obtained for the models of acceptance, based on perceived risk, than for the expected/weighted utility models, including bilinear models.

The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(S) and φ(S, π) and another exogenous parameter α ≤ 1. Minimizing a general Markov bound leads to the following unifying equation: E [φ (S, π)] = αE [v (S)].For any random variable, the risk measure π is the solution to the unifying equation. By varying the functions φ and v, the paper derives the mean value principle, the zero-utility premium principle, the Swiss premium principle, Tail VaR, Yaari's dual theory of risk, mixture of Esscher principles and more. The paper also discusses combining two risks with super-additive properties and sub-additive properties. In addition, we recall some of the important characterization theorems of these risk measures.

In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied. We propose a characterization of a particular class of coherent risk measures defined in [Art99]. The considered premium principles are obtained by expansion of TVaR measures, consequently they look very interesting in insurance pricing where TVaR measures are frequently used to value tail risks.

related some of the common risk measures used in investment analysis. The measures of variance, semi-variance, mean absolute deviation, and probability of an outcome worse than some specified level were all shown to be specific cases of his general three-parameter risk measure. It is the purpose of this note to show that the functions used in the other major theoretical risk analysis method, stochastic dominance, can be derived as another special case of Stone's three-parameter risk measure. His measure was defined as

A normative method for risk management decision-making is developed from risk preference (utility) theory. Risk is measured by a new device called the "Risk Profile Curve." It represents a whole spectrum of value measurements, one for each risk aversion level. Best deductible and aggregate retention levels are examined. An equivalence between dominance in risk profile curves and the traditional stochastic dominance criteria is established for simply related loss distributions. Risk profile curves are shown to give more useful and practical decisionmaking information than that given by the usual stochastic dominance or expected utility results. Risk Management, a rapidly growing specialized field of management, is developing an awareness of its specialized structure and features. Yet its growth will be hampered without a more precise and operational definition of risk. Risk involves money and therefore, is partly a financial concept. Because in addition it involves uncertainty and therefore probability, it also is partly a statistical concept. However, it does not seem to be adequately represented by an expected value concept. Variance of the probability distribution of loss size may be closer to the true nature of risk. Yet, because of the extreme asymmetry of typical loss distributions, variance seems to be of limited use as a measure of risk.

Financial economists often test security pricing models for the destabilizing effects of informational uncertainty. In particular, Bar-Yosef and Brown (1977) examined the market model and concluded that systematic risk is concave during stock-split event periods. Unsystematic risk, however, was assumed constant. Further steps are taken in this article. A generalization of the random coefficients model of Hildreth and Houck (1968) is conducted, and case-by-case testing for event-period concavity of both systematic and unsystematic risk is conducted. In contrast to past findings, the results support concavity for unsystematic risk, not systematic risk.