Risk measure

The present paper focuses on a particular methodology for measuring the occupational health and safety risks in tourism companies by numerical risk coefficients. The methodology is based on the achievements of several US and international research and innovation development programs, such as: "SSP", "ISSP", "F/A-18" and "AFMC". Under this methodology, the occupational health and safety risks are regarded as a function of three variable factors: the likelihood of occurrence of a negative, risk event; the consequences from the realization of this very same risk event and the immediacy of occurrence of the risk event in respect of time. The values of each of these three variables are being measured by the help of score card estimation tables and are being presented as per a zero-referent scale of 1 to 10 or of 1 to 100. Thus a final estimation of the value of a certain occupational health and safety risk can be achieved. The advantages and the opportunities for improving of this methodology on an enterprise level are reviewed as well. The main advantage which is pointed out is that the methodology could by used even by inexperienced tourism company which has small or small or no past record of health and safety risk measurement. In the addition to the risk measurement and quantification methodology, a set of criteria for acceptance of the occupational health and safety risks are also presented. This includes: the usage of risk matrixes, the "f-N" curves and the "ALARP" principle.

The present paper focuses on a particular methodology for measuring the occupational health and safety risks in tourism companies by numerical risk coefficients. The methodology is based on the achievements of several US and international research and innovation development programs, such as: "SSP", "ISSP", "F/A-18" and "AFMC". Under this methodology, the occupational health and safety risks are regarded as a function of three variable factors: the likelihood of occurrence of a negative, risk event; the consequences from the realization of this very same risk event and the immediacy of occurrence of the risk event in respect of time. The values of each of these three variables are being measured by the help of score card estimation tables and are being presented as per a zero-referent scale of 1 to 10 or of 1 to 100. Thus a final estimation of the value of a certain occupational health and safety risk can be achieved. The advantages and the opportunities for improving of this methodology on an enterprise level are reviewed as well. The main advantage which is pointed out is that the methodology could by used even by inexperienced tourism company which has small or small or no past record of health and safety risk measurement. In the addition to the risk measurement and quantification methodology, a set of criteria for acceptance of the occupational health and safety risks are also presented. This includes: the usage of risk matrixes, the "f-N" curves and the "ALARP" principle.

In this paper, we attempt to invent a new way to understand risk, measure it, and weigh its consequences. We attempt to design a rational process for risk-taking; a process that gives the system dynamicist the ability to define what may happen in the future and then to choose among alternatives.

Procyclicality of historical risk measure estimation means that one tends to over-estimate future risk when present realized volatility is high and vice versa under-estimate future risk when the realized volatility is low. Out of it different questions arise, relevant for applications and theory: What are the factors which affect the degree of procyclicality? More specifically, how does the choice of risk measure affect this? How does this behaviour vary with the choice of realized volatility estimator? How do different underlying model assumptions influence the pro-cyclical effect? In this paper we consider three different well-known risk measures (Value-at-Risk, Expected Shortfall, Expectile), the r-th absolute centred sample moment, for any integer $r>0$, as realized volatility estimator (this includes the sample variance and the sample mean absolute deviation around the sample mean) and two models (either an iid model or an augmented GARCH($p$,$q$) model). We show that the st...

Basel II and Solvency 2 both use the Value-at-Risk (VaR) as the risk measure to compute the Capital Requirements. In practice, to calibrate the VaR, a normal approximation is often chosen for the unknown distribution of the yearly log returns of financial assets. This is usually justified by the use of the Central Limit Theorem (CLT), when assuming aggregation of independent and identically distributed (iid) observations in the portfolio model. Such a choice of modeling, in particular using light tail distributions, has proven during the crisis of 2008/2009 to be an inadequate approximation when dealing with the presence of extreme returns; as a consequence, it leads to a gross underestimation of the risks. The main objective of our study is to obtain the most accurate evaluations of the aggregated risks distribution and risk measures when working on financial or insurance data under the presence of heavy tail and to provide practical solutions for accurately estimating high quantiles of aggregated risks. We explore a new method, called Normex, to handle this problem numerically as well as theoretically, based on properties of upper order statistics. Normex provides accurate results, only weakly dependent upon the sample size and the tail index. We compare it with existing methods.

Expected shortfall (ES) has been widely accepted as a risk measure that is conceptually superior to value-at-risk (VaR). At the same time, however, it has been criticized for issues relating to backtesting. In particular, ES has been found not to be elicitable, which means that backtesting for ES is less straightforward than, for example, backtesting for VaR. Expectiles have been suggested as potentially better alternatives to both ES and VaR. In this paper, we revisit the commonly accepted desirable properties of risk measures such as coherence, comonotonic additivity, robustness and elicitability. We check VaR, ES and expectiles with regard to whether or not they enjoy these properties, with particular emphasis on expectiles. We also consider their impact on capital allocation, an important issue in risk management. We find that, despite the caveats that apply to the estimation and backtesting of ES, it can be considered a good risk measure. As a consequence, there is no sufficient evidence to justify an all-inclusive replacement of ES by expectiles in applications. For backtesting ES, we propose an empirical approach that consists of replacing ES by a set of four quantiles, which should allow us to make use of backtesting methods for VaR.

Extreme losses of portfolios with heavy-tailed components are studied in the framework of multivariate regular variation. Asymptotic distributions of extreme portfolio losses are characterized by a functional γ ξ = γ ξ (Ψ, α) of the tail index α, the spectral measure Ψ, and the vector ξ of portfolio weights. Existence, uniqueness, and location of the optimal portfolio are analysed and applied to the minimization of risk measures. It is shown that diversification effects are positive for α > 1 and negative for α < 1. Strong consistency and asymptotic normality are established for a semiparametric estimator of the mapping ξ → γ ξ . Strong consistency is also established for the estimated optimal portfolio.

The optimal risk allocation problem, equivalently the optimal risk sharing problem, in a market with n traders endowed with risk measure 1 , . . . , n is a classical problem in insurance and mathematical finance. This problem however makes only sense under a condition motivated from game theory which is called Pareto equilibrium. There are many situations of practical interest, where this condition does not hold. This is the case if the risk measures are based on essential different views towards risk. In this paper we introduce and analyze a meaningful extension of the optimal risk allocation (risk sharing) problem without assuming the equilibrium condition. The main point of this is to introduce a suitable and well motivated restriction on the class of admissible allocations which prevents effects of artificial 'risk arbitrage'. As a result we obtain a new coherent risk measure which describes the inherent risk which remains after using admissible risk exchange in an optimal way.

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.

Much of the recent literature on risk measures is concerned with essentially bounded risks in L ∞ . In this paper we investigate in detail continuity and representation properties of convex risk measures on L p spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on L p we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for L p risks.

Distributions for returns are used to compute the capital charge for portfolios in investment banks. The mainstream definition of returns is based on closing prices and neglects the important effects of intraday trading activity on the losses . In this paper we introduce "minimal returns", a definition of returns that accounts for intraday trading and gives a worst-case approach on losses. We suggest an appropriate distribution for minimal returns that can be used to compute Value at Risk and coherent risk measures, as suggested by .

The study analysis properties of types of proportional reinsurance cover (such as quota share reinsurance, variable quota share reinsurance, surplus reinsurance and quota share with surplus reinsurance). Also, assessed the impact of proportional reinsurance on the Egyptian insurance Market. The study used risk measures, the Value-at-Risk (VaR) and the Conditional-TailExpectation (CTE) of aggregate claims distributions, these risk measures have been used to the determination of optimal reinsurance. This study Estimated optimal reinsurance by maximize the Return on Risk Adjusted Capital of the retained risk (RORAC).

We use the theory of cooperative games for the design of fair insurance contracts. An insurance contract needs to specify the premium to be paid and a possible participation in the benefit (or surplus) of the company. It results from the analysis that when a contract is exposed to the default risk of the insurance company, ex-ante equilibrium considerations require a certain participation in the benefit of the company to be specified in the contracts. The fair benefit participation of agents appears as an outcome of a game involving the residual risks induced by the default possibility and using fuzzy coalitions.

In this paper, we attempt to invent a new way to understand risk, measure it, and weigh its consequences. We attempt to design a rational process for risk-taking; a process that gives the system dynamicist the ability to define what may happen in the future and then to choose among alternatives.

Individuals differ significantly in their willingness to take risks, partly due to genetic differences. We explore how risk taking behavior correlates with different versions of the dopamine receptor D4 gene (DRD4). We focus on risk taking in the card game contract bridge, and economic risk taking as proxied by a financial gamble. We also explore selfreported general risk taking, and self-reported behavior in risk-related activities. Our participants are serious tournament bridge players, which gives them substantial experience in risk taking. We find some evidence that men with a 7-repeat allele (7R+) of DRD4 take more overall risk in bridge than individuals without this allele (7R-), and strong evidence that 7R+ men take more economic risk in an investment game. Interestingly, these relationships are not found in the women in our study. Although the number of 7R+ women in our sample is low, our results may reflect a gender difference in how the 7R+ genotype affects behavior. Bridge masterpoints measure past success, thus reflecting playing skill and experience. We show that masterpoint level modulates the effect of the DRD4 gene in men in a highly important manner. We find that higher ranked 7R+ men take significantly more good risks and significantly fewer bad risks than other men, whereas the opposite is found for less-expert 7R+ men. This is the first study to distinguish between advantageous and disadvantageous risk taking. We identify a strong interaction among desirable risk taking behavior, measured success, and genetic variation. Considering other risk measures, we find no difference between 7R+ and 7R-individuals in general risk taking or in any of a number of other riskrelated activities. Our results indicate that the dopamine system plays an important role in explaining individual differences in risk taking in bridge and economic risk taking among men. Little relationship is found in other activities involving risk or among women.

A phase transition in high-dimensional random geometry is analyzed as it arises in a variety of problems. A prominent example is the feasibility of a minimax problem that represents the extremal case of a class of financial risk measures, among them the current regulatory market risk measure Expected Shortfall. Others include portfolio optimization with a ban on short-selling, the storage capacity of the perceptron, the solvability of a set of linear equations with random coefficients, and competition for resources in an ecological system. These examples shed light on various aspects of the underlying geometric phase transition, create links between problems belonging to seemingly distant fields, and offer the possibility for further ramifications.

Portfolio selection has a central role in finance theory and practical applications. The classical approach uses the standard deviation (variance) as risk measure, while several other alternatives have also been introduced in the literature. Due to its computational advantages, portfolio optimization based on absolute deviation looks particularly interesting and it is widely used in practice. For the practical implementation of any variant, however, one needs to estimate the parameters from finite return series, which inevitably introduces measurement noise that in turn affects the portfolio selection. Although much research has concentrated on investigating the noise in the classical model, little attention has been devoted to the case of absolute deviation. In this paper, we study the effect of estimation noise in the case of absolute deviation based portfolio optimization. We show that the key parameter determining the effect of noise is the ratio of the time series length and portfolio size and that the level of noise is higher than in the classical variance-based model. This points out the importance of finding out when theoretically ,,better" portfolio selection models can truly outperform the basic ones.

It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio cannot be optimized under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability

Investors who optimize their portfolios under any of the coherent risk measures are naturally led to regularized portfolio optimization when they take into account the impact their trades make on the market. We show here that the impact function determines which regularizer is used. We also show that any regularizer based on the norm L p with p > 1 makes the sensitivity of coherent risk measures to estimation error disappear, while regularizers with p < 1 do not. The L 1 norm represents a border case: its "soft" implementation does not remove the instability, but rather shifts its locus, whereas its "hard" implementation (equivalent to a ban on short selling) eliminates it. We demonstrate these effects on the important special case of Expected Shortfall (ES) that is on its way to becoming the next global regulatory market risk measure.

We address the problem of portfolio optimization under the simplest coherent risk measure, i.e. the expected shortfall. As it is well known, one can map this problem into a linear programming setting. For some values of the external parameters, when the available time series is too short, the portfolio optimization is ill posed because it leads to unbounded positions, infinitely short on some assets and infinitely long on some others. As first observed by Kondor and coworkers, this phenomenon is actually a phase transition. We investigate the nature of this transition by means of a replica approach.

We analyze the performance of RiskMetrics, a widely used methodology for measuring market risk. Based on the assumption of normally distributed returns, the RiskMetrics model completely ignores the presence of fat tails in the distribution function, which is an important feature of financial data. Nevertheless, it was commonly found that RiskMetrics performs satisfactorily well, and therefore the technique has become widely used in the financial industry. We find, however, that the success of RiskMetrics is the artifact of the choice of the risk measure. First, the outstanding performance of volatility estimates is basically due to the choice of a very short (one-period ahead) forecasting horizon. Second, the satisfactory performance in obtaining Value-at-Risk by simply multiplying volatility with a constant factor is mainly due to the choice of the particular significance level.

The optimization of large portfolios displays an inherent instability to estimation error. This poses a fundamental problem, because solutions that are not stable under sample fluctuations may look optimal for a given sample, but are, in effect, very far from optimal with respect to the average risk. In this paper, we approach the problem from the point of view of statistical learning theory. The occurrence of the instability is intimately related to over-fitting which can be avoided using known regularization methods. We show how regularized portfolio optimization with the expected shortfall as a risk measure is related to support vector regression. The budget constraint dictates a modification. We present the resulting optimization problem and discuss the solution. The L2 norm of the weight vector is used as a regularizer, which corresponds to a diversification "pressure". This means that diversification, besides counteracting downward fluctuations in some assets by upward fluctuations in others, is also crucial because it improves the stability of the solution. The approach we provide here allows for the simultaneous treatment of optimization and diversification in one framework that enables the investor to trade-off between the two, depending on the size of the available data set.

We study the sensitivity to estimation error of portfolios optimized under various risk measures, including variance, absolute deviation, expected shortfall and maximal loss. We introduce a measure of portfolio sensitivity and test the various risk measures by considering simulated portfolios of varying sizes N and for different lengths T of the time series. We find that the effect of noise is very strong in all the investigated cases, asymptotically it only depends on the ratio N/T , and diverges at a critical value of N/T , that depends on the risk measure in question. This divergence is the manifestation of a phase transition, analogous to the algorithmic phase transitions recently discovered in a number of hard computational problems. The transition is accompanied by a number of critical phenomena, including the divergent sample to sample fluctuations of portfolio weights. While the optimization under variance and mean absolute deviation is always feasible below the critical value of N/T , expected shortfall and maximal loss display a probabilistic feasibility problem, in that they can become unbounded from below already for small values of the ratio N/T , and then no solution exists to the optimization problem under these risk measures. Although powerful filtering techniques exist for the mitigation of the above instability in the case of variance, our findings point to the necessity of developing similar filtering procedures adapted to the other risk measures where they are much less developed or nonexistent. Another important message of this study is that the requirement of robustness (noise-tolerance) should be given special attention when considering the theoretical and practical criteria to be imposed on a risk measure.

It is shown that the axioms for coherent risk measures imply that whenever there is a pair of portfolios such that one of them dominates the other in a given sample (which happens with finite probability even for large samples), then there is no optimal portfolio under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability was first discovered in the special example of Expected Shortfall which is used here both as an illustration and as a springboard for generalization.

We show that including a term which accounts for finite liquidity in portfolio optimization naturally mitigates the instabilities that arise in the estimation of coherent risk measures on finite samples. This is because taking into account the impact of trading in the market is mathematically equivalent to introducing a regularization on the risk measure. We show here that the impact function determines which regularizer is to be used. We also show that any regularizer based on the norm [Formula: see text] with [Formula: see text] makes the sensitivity of coherent risk measures to estimation error disappear, while regularizers with [Formula: see text] do not. The [Formula: see text] norm represents a border case: its “soft” implementation does not remove the instability, but rather shifts its locus, whereas its “hard” implementation (including hard limits or a ban on short selling) eliminates it. We demonstrate these effects on the important special case of expected shortfall (ES) w...

This study investigates whether voluntary management disclosure of earnings forecasts influences investors' long-term assessment of firm risk and firm value. We control for possible endogeneity between various firm-specific characteristics and the voluntary issuing of different types of management earnings forecasts by utilizing a two-stage Heckman treatment analysis. We find a significant negative relationship between the issuance of management earnings forecasts and a variety of risk measures including idiosyncratic risk, stock return volatility, beta, and bid-ask spreads suggesting the issuance of management earings forecasts reduces information asymmetry and thus reduces risk. We find a significant positive relationship between management earnings forecasts and firm value as captured by Tobin's Q when the issuance of earnings is more precise (i.e., point estimates versus ranges or qualitative guidance), and a stronger positive relationship when the precise forecasts are also credible, i.e., when actual earnings meet or exceed management forecasts.

We present a new approach to handle dependencies within the general framework of case-control designs, illustrating our approach by a particular application from the field of genetic epidemiology. The method is derived for parent-offspring trios, which will later be relaxed to more general family structures. For applications in genetic epidemiology we consider tests on equality of allele frequencies among cases and controls utilizing well-known risk measures to test for independence of phenotype and genotype at the observed locus. These test statistics are derived as functions of the entries in the associated contingency table containing the numbers of the alleles under consideration in the case and the control group. We find the joint asymptotic distribution of these entries, which enables us to derive critical values for any test constructed on this basis. A simulation study reveals the finite sample behavior of our test statistics.

We address the statistical estimation of composite functionals which may be nonlinear in the probability measure. Our study is motivated by the need to estimate coherent measures of risk, which become increasingly popular in finance, insurance, and other areas associated with optimization under uncertainty and risk. We establish central limit formulae for composite risk functionals. Furthermore, we discuss the asymptotic behavior of optimization problems whose objectives are composite risk functionals and we establish a central limit formula of their optimal values when an estimator of the risk functional is used. While the mathematical structures accommodate commonly used coherent measures of risk, they have more general character, which may be of independent interest.

We derive representations of higher order dual measures of risk in L p spaces as suprema of integrals of Average Values at Risk with respect to probability measures on (0, 1] (Kusuoka representations). The suprema are taken over convex sets of probability measures. The sets are described by constraints on the dual norms of certain transformations of distribution functions. For p = 2, we obtain a special description of the set and we relate the measures of risk to the Fano factor in statistics.

Tail risk measures such as the Value-at-Risk (VaR) are being advocated as conceptually appropriate statistical and economical alternatives to dispersion measures of risk such as the standard deviation. VaR and dispersion risk measures are applied to assess the revenue risk reduction potential of an index rainfall insurance. VaR and dispersion measures indicate that a Rainfall Index Insurance Contract reduces revenue risk.

The question of pricing and hedging a given contingent claim has a unique solution in a complete market framework. When some incompleteness is introduced, the problem becomes however more dicult. Several approaches have been adopted in the literature to provide a satisfactory answer to this problem, for a particular choice criterion. Among them, Hodges and Neuberger proposed in 1989 a method based on utility maximization. The price of the contingent claim is then obtained as the smallest (resp. largest) amount leading the agent indierent between selling (resp. buying) the claim and doing nothing. The price obtained is the indierence seller's (resp. buyer's) price. Since then, many authors have used this approach, the exponential utility function being most often used (see for instance, El Karoui and Rouge [51], Becherer [11], Delbaen et al. [39] , Musiela and Zariphopoulou [93] or Mania and Schweizer [89]...). In this chapter, we also adopt this exponential utility point of view to start with in order to nd the optimal hedge and price of a contingent claim based on a non-tradable risk. But soon, we notice that the right framework to work with is not that of the exponential utility itself but that of the certainty equivalent which is a convex functional satisfying some nice properties among which that of cash translation invariance. Hence, the results obtained in this particular framework can be immediately extended to functionals satisfying the same properties, in other words to convex risk measures as introduced by Föllmer and Schied [53] and or by Frittelli and Gianin [57]. Starting with a utility maximization problem, we end up with an equivalent risk measure minimization in order to price and hedge this contingent claim. Moreover, this hedging problem can be seen as a particular case of a more general situation of risk transfer between dierent agents, one of them consisting of the nancial market. Therefore, we consider in this 1 The authors would like to thank their respective husband, Laurent Veilex and Faycal El Karoui, for their constant support and incredible patience.

We develop a methodology to optimally design a financial issue to hedge non-tradable risk on financial markets. Economic agents assess their risk using monetary risk measure. The inf-convolution of convex risk measures is the key transformation in solving this optimization problem. When agents' risk measures only differ from a risk aversion coefficient, the optimal risk transfer is amazingly equal to a proportion of the initial risk. For dynamic risk measures defined through their local specifications using BSDE's, their inf-convolution is equivalent to that of their associated drivers. In this case, it is also possible to characterize the optimal risk transfer.

We show that stochastic recovery always leads to counter-intuitive behaviors in the risk measures of a CDO tranche -namely, continuity on default and positive credit spread risk cannot be ensured simultaneously. We then propose a simple recovery variance regularization method to control the magnitude of negative credit spread risk while preserving the continuity on default. 1 The 15-30% CDX tranche is often referred to as the super senior tranche, but it is not "super senior" in our definition.

The class of all law invariant, convex risk measures for portfolio vectors is characterized. The building blocks of this class are shown to be formed by the maximal correlation risk measures. We further introduce some classes of multivariate distortion risk measures and relate them to multivariate quantile functionals and to an extension of the average value at risk measure.

The optimal risk allocation problem or equivalently the problem of risk sharing is the problem to allocate a risk in an optimal way to n traders endowed with risk measures 1 , . . . , n . This problem has a long history in mathematical economics and insurance. In the first part of the paper we review some mathematical tools and discuss their applications to various problems on risk measures related to the allocation problem like to monotonicity properties of optimal allocations, to optimal investment problems or to an appropriate definition of the conditional value at risk. We then consider the risk allocation problem for convex risk measures i . In general the optimal risk allocation problem is well defined only under an equilibrium condition. This condition can be characterized by the existence of a common scenario measure. We formulate a meaningful modification of the optimal risk allocation problem also for markets without assuming the equilibrium condition and characterize optimal solutions. The basic idea is to restrict the class of admissible allocations in a proper way.

We establish various extensions of the comonotone improvement result of . Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion. Annals of Operations Research 52, 97-106] which are of interest for the risk sharing problem. As a consequence we obtain general results of the comonotonicity of Pareto optimal risk allocations using risk measures consistent with the stochastic convex order.

The paper gives an overview of mathematical models and methods used in financial risk management; the main area of application is credit risk. A brief introduction explains the mathematical issues arising in the risk management of a portfolio of loans. The paper continues with a formal overview of credit risk management models and discusses axiomatic approaches to risk measurement. We close with a section on dynamic credit risk models used in the pricing of credit derivatives. Mathematical techniques used stem from probability theory, statistics, convex analysis and stochastic process theory.

We extend the classical risk minimization model with scalar risk measures to the general case of set-valued risk measures. The problem we obtain is a set-valued optimization model and we propose a goal programming-based approach with satisfaction function to obtain a solution which represents the best compromise between goals and the achievement levels. Numerical examples are provided to illustrate how the method works in practical situations.

In this thesis, we study the problems of risk measurement, valuation and hedging of financial positions in incomplete markets when an insufficient number of assets are available for investment (real options). We work closely with three measures of risk: Worst-Case Scenario (WCS) (the supremum of expected values over a set of given probability measures), Value-at-Risk (VaR) and Average Value-at-Risk (AVaR), and analyse the problem of hedging derivative securities depending on a non-traded asset,

Using regular variation to define heavy tailed distributions, we show that prominent downside risk measures produce similar and consistent ranking of heavy tailed risk. Thus regardless of the particular risk measure being used, assets will be ranked in a similar and consistent manner for heavy tailed assets.

We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper [11]. These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures on the optional σ-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for different notions of time consistency. In particular we show how "bubbles" may appear in the dynamic penalization, and how they cause a breakdown of asymptotic safety of the risk assessment procedure.

Different safety measures adopted by governments across the globe require the estimates of willingness to pay of the people to swap wealth for a reduction in the probability of death and injury. The approximation of these trade-offs are employed in assessing the cost-benefit analysis of environmental issues, public safety measures on highways and roads, medical treatments, and many other areas. Economists term a trade-off between money and fatality risks as the Value of a Statistical Life (VSL). The Value of Statistical Life and Limb is generally predicted using one of the three main approaches. The first is by the compensating wage differentials that workers must be paid to take riskier jobs [Viscusi and Aldy (2003)]. The second approach examines other behaviours where people weigh costs against risks [Blomquist (2004)] and the third is through contingent valuation surveys where respondents report their willingness to pay (WTP) to obtain a specified reduction in mortality risks. Th...

We formulate a new theory of expected utility in which risk and uncertainty is modelled by the usage of a so called event space which is a natural generalisation of a state space. The basic idea is that the decision maker for each group of related decisions creates a "small world" (a local state space) from the events in the "grand world" (the event space). We introduce a set of preference axioms similar in spirit to the Savage axioms, and show that they lead to a a more general expected utility theory, where in each "small world" risk is described by an additive probability measure. All local risk measures appear as restrictions of a common integrated additive expectation functional defined on the "grand world". A benefit of the theory is that it allows for an intuitive distinction between risk and uncertainty. In particular, there is a numerical measure of the degree of uncertainty aversion associated with a given preference relation, which can be calculated. We illustrate the use of the theory for the Ellsberg paradox and for a no-trade portfolio decision problem, none of which can be captured using standard expected utility theory. JEL classification: D8 and G12.

We formulate a new theory of expected utility in which risk and uncertainty is modelled by the usage of a so called event space which is a natural generalisation of a state space. The basic idea is that the decision maker for each group of related decisions creates a &quot;small world&quot; (a local state space) from the events in the &quot;grand world&quot; (the event space). We introduce a set of preference axioms similar in spirit to the Savage axioms, and show that they lead to a a more general expected utility theory, where in each &quot;small world&quot; risk is described by an additive probability measure. All local risk measures appear as restrictions of a common integrated additive expectation functional defined on the &quot;grand world&quot;. A benefit of the theory is that it allows for an intuitive distinction between risk and uncertainty. In particular, there is a numerical measure of the degree of uncertainty aversion associated with a given preference relation, which ...

Actuaries and other managers of risk identify factors in modeling insurance risks because (i) they feel that these factors may cause the outcome of a risk or (ii) that the factors can be managed, thus allowing analysts a degree of control over the system risk. The purpose of this paper is to propose a framework for quantifying the relative importance of a source of risk. The intent is that, with a quantitative measure of relative importance, risk managers will be able to sharpen their intuition about the relative importance of risks and be better custodians of financial security systems. This paper proposes a measure of relative importance that has its roots in both the statistics and economics literatures. The measure is intuitively appealing when assessing the effectiveness of basic risk management techniques including risk exchange, pooling and financial risk management. The risk measure is also shown to be useful in multivariate situations where several factors affect a risk simultaneously. The paper illustrates this usefulness by considering a pool of policies that is subject to mortality, the risk of a disaster that is common to all policies and to a common investment environment.

Recently equal risk pricing, a framework for fair derivative pricing, was extended to consider dynamic risk measures. However, all current implementations either employ a static risk measure that violates time consistency, or are based on traditional dynamic programming solution schemes that are impracticable in problems with a large number of underlying assets (due to the curse of dimensionality) or with incomplete asset dynamics information. In this paper, we extend for the first time a famous off-policy deterministic actor-critic deep reinforcement learning (ACRL) algorithm to the problem of solving a risk averse Markov decision process that models risk using a time consistent recursive expectile risk measure. This new ACRL algorithm allows us to identify high quality time consistent hedging policies (and equal risk prices) for options, such as basket options, that cannot be handled using traditional methods, or in context where only historical trajectories of the underlying assets are available. Our numerical experiments, which involve both a simple vanilla option and a more exotic basket option, confirm that the new ACRL algorithm can produce 1) in simple environments, nearly optimal hedging policies, and highly accurate prices, simultaneously for a range of maturities 2) in complex environments, good quality policies and prices using reasonable amount of computing resources; and 3) overall, hedging strategies that actually outperform the strategies produced using static risk measures when the risk is evaluated at later points of time.

In this paper, we consider the problem of equal risk pricing and hedging in which the fair price of an option is the price that exposes both sides of the contract to the same level of risk. Focusing for the first time on the context where risk is measured according to convex risk measures, we establish that the problem reduces to solving independently the writer and the buyer's hedging problem with zero initial capital. By further imposing that the risk measures decompose in a way that satisfies a Markovian property, we provide dynamic programming equations that can be used to solve the hedging problems for both the case of European and American options. All of our results are general enough to accommodate situations where the risk is measured according to a worst-case risk measure as is typically done in robust optimization. Our numerical study illustrates the advantages of equal risk pricing over schemes that only account for a single party, pricing based on quadratic hedging (i.e. -arbitrage pricing), or pricing based on a fixed equivalent martingale measure (i.e. Black-Scholes pricing). In particular, the numerical results confirm that when employing an equal risk price both the writer and the buyer end up being exposed to risks that are more similar and on average smaller than what they would experience with the other approaches.

Recently equal risk pricing, a framework for fair derivative pricing, was extended to consider dynamic risk measures. However, all current implementations either employ a static risk measure that violates time consistency, or are based on traditional dynamic programming solution schemes that are impracticable in problems with a large number of underlying assets (due to the curse of dimensionality) or with incomplete asset dynamics information. In this paper, we extend for the first time a famous off-policy deterministic actor-critic deep reinforcement learning (ACRL) algorithm to the problem of solving a risk averse Markov decision process that models risk using a time consistent recursive expectile risk measure. This new ACRL algorithm allows us to identify high quality time consistent hedging policies (and equal risk prices) for options, such as basket options, that cannot be handled using traditional methods, or in context where only historical trajectories of the underlying assets are available. Our numerical experiments, which involve both a simple vanilla option and a more exotic basket option, confirm that the new ACRL algorithm can produce 1) in simple environments, nearly optimal hedging policies, and highly accurate prices, simultaneously for a range of maturities 2) in complex environments, good quality policies and prices using reasonable amount of computing resources; and 3) overall, hedging strategies that actually outperform the strategies produced using static risk measures when the risk is evaluated at later points of time.