Coherent Risk Measures Research Articles

Dynamic Risked Equilibrium

We study a competitive partial equilibrium in markets where risk-averse agents solve multistage stochastic optimization problems formulated in scenario trees. The agents trade a commodity that is produced from an uncertain supply of resources. Both resources and the commodity can be stored for later consumption. Several examples of a multistage risked equilibrium are outlined, including aspects of battery and hydroelectric storage in electricity markets, distributed ownership of competing technologies relying on shared resources, and aspects of water control and pricing. The agents are assumed to have nested coherent risk measures based on one-step risk measures with polyhedral risk sets that have a nonempty intersection over agents. Agents can trade risk in a complete market of Arrow-Debreu securities. In this setting, we define a risk-trading competitive market equilibrium and establish two welfare theorems. Competitive equilibrium will yield a social optimum (with a suitably defined social risk measure) when agents have strictly monotone one-step risk measures. Conversely, a social optimum with an appropriately chosen risk measure will yield a risk-trading competitive market equilibrium when all agents have strictly monotone risk measures. The paper also demonstrates versions of these theorems when risk measures are not strictly monotone.

Coherent risk measures alone are ineffective in constraining portfolio losses

We show that coherent risk measures alone are ineffective in curbing the behaviour of investors with limited liability or excessive tail-risk seeking behaviour if the market admits statistical arbitrage opportunities which we term ρ-arbitrage for a risk measure ρ. We show how to determine analytically whether such ρ-arbitrage portfolios exist in complete markets and in the Markowitz model. We also consider realistic numerical examples of incomplete markets and determine whether Expected-Shortfall arbitrage exists in these markets. We find that the answer depends heavily upon the probability model selected by the risk manager but that it is certainly possible for expected shortfall constraints to be ineffective in realistic markets. Since value at risk constraints are weaker than expected shortfall constraints, our results can be applied to value at risk.

A new coherent multivariate average-value-at-risk

A new operator for handling the joint risk of different sources has been presented and its various properties are investigated. The problem of risk evaluation of multivariate risk sources has been studied, and a multivariate risk measure, so-called multivariate average-value-at-risk, , is proposed to quantify the total risk. It is shown that the proposed operator satisfies the four axioms of a coherent risk measure while reducing to one variable average-value-at-risk, , in case N = 1. In that respect, it is shown that is the natural extension of to N-dimensional case maintaining its axiomatic properties. We further show is flexible by giving the investor the option to choose the risk level of each random loss i differently. This flexibility is novel and can not be achieved applying univariate with corresponding risk level α to the sum of the risk marginals. The framework is applicable for Gaussian mixture models with dependent risk factors that are naturally used in financial and actuarial modelling. A multivariate tail variance and its connection with is also presented via Chebyshev inequality for tail events. Examples with numerical simulations are also illustrated throughout.

COHERENT RISK MEASURE ON L0: NA CONDITION, PRICING AND DUAL REPRESENTATION

The NA condition is one of the pillars supporting the classical theory of financial mathematics. We revisit this condition for financial market models where a dynamic risk-measure defined on [Formula: see text] is fixed to characterize the family of acceptable wealths that play the role of nonnegative financial positions. We provide in this setting a new version of the fundamental theorem of asset pricing and we deduce a dual characterization of the super-hedging prices (called risk-hedging prices) of a European option. Moreover, we show that the set of all risk-hedging prices is closed under NA. At last, we provide a dual representation of the risk-measure on [Formula: see text] under some conditions.

Mean‐ portfolio selection and ‐arbitrage for coherent risk measures

AbstractWe revisit mean‐risk portfolio selection in a one‐period financial market where risk is quantified by a positively homogeneous risk measure . We first show that under mild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact. However, unlike in classical mean‐variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation ‐arbitrage, and prove that it cannot be excluded—unless is as conservative as the worst‐case risk measure. After providing a primal characterization of ‐arbitrage, we focus our attention on coherent risk measures that admit a dual representation and give a necessary and sufficient dual characterization of ‐arbitrage. We show that the absence of ‐arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual representation of . A special case of our result shows that the market does not admit ‐arbitrage for Expected Shortfall at level if and only if there exists an EMM such that .

TEST FOR CHANGES IN THE MODELED SOLVENCY CAPITAL REQUIREMENT OF AN INTERNAL RISK MODEL

AbstractIn the context of the Solvency II directive, the operation of an internal risk model is a possible way for risk assessment and for the determination of the solvency capital requirement of an insurance company in the European Union. A Monte Carlo procedure is customary to generate a model output. To be compliant with the directive, validation of the internal risk model is conducted on the basis of the model output. For this purpose, we suggest a new test for checking whether there is a significant change in the modeled solvency capital requirement. Asymptotic properties of the test statistic are investigated and a bootstrap approximation is justified. A simulation study investigates the performance of the test in the finite sample case and confirms the theoretical results. The internal risk model and the application of the test is illustrated in a simplified example. The method has more general usage for inference of a broad class of law-invariant and coherent risk measures on the basis of a paired sample.

Dependent conditional value-at-risk for aggregate risk models

Risk measure forecast and model have been developed in order to not only provide better forecast but also preserve its (empirical) property especially coherent property. Whilst the widely used risk measure of Value-at-Risk (VaR) has shown its performance and benefit in many applications, it is in fact not a coherent risk measure. Conditional VaR (CoVaR), defined as mean of losses beyond VaR, is one of alternative risk measures that satisfies coherent property. There have been several extensions of CoVaR such as Modified CoVaR (MCoVaR) and Copula CoVaR (CCoVaR). In this paper, we propose another risk measure, called Dependent CoVaR (DCoVaR), for a target loss that depends on another random loss, including model parameter treated as random loss. It is found that our DCoVaR provides better forecast than both MCoVaR and CCoVaR. Numerical simulation is carried out to illustrate the proposed DCoVaR. In addition, we do an empirical study of financial returns data to compute the DCoVaR forecast for heteroscedastic process of GARCH(1,1). The empirical results show that the Gumbel Copula describes the dependence structure of the returns quite nicely and the forecast of DCoVaR using Gumbel Copula is more accurate than that of using Clayton Copula. The DCoVaR is superior than MCoVaR, CCoVaR and CoVaR to comprehend the connection between bivariate losses and to help us exceedingly about how optimum to position our investments and elevate our financial risk protection. In other words, putting on the suggested risk measure will enable us to avoid non-essential extra capital allocation while not neglecting other risks associated with the target risk. Moreover, in actuarial context, DCoVaR can be applied to determine insurance premiums while reducing the risk of insurance company.

Aggregation of opinions and risk measures

A ranking ≿ over a set of alternatives is an aggregation of experts' opinions (AEO) if it depends on the experts' assessments only. We study both those rankings that result from pooling Bayesian experts and those that result from pooling possibly non-Bayesian experts. In the non-Bayesian case, we allow for the simultaneous presence of experts that may display very different attitudes toward uncertainty. We show that a unique axiom along with a mild regularity condition fully characterize those AEO rankings which are “generalized averages” of experts' opinions, in the sense that the average is obtained by using a capacity rather than a probability measure. We call these rankings non-linear pools. We consider a number of special cases such as linear pools (Stone, 1961), concave/convex pools (Crès et al., 2011), quantiles and pools of equally reliable experts. We then apply our results to the theory of risk measures. Our application can be viewed as a generalization of the robust approach (Glasserman and Xu, 2014) to risk measurement in that it allows both for a more general notion of “model” and more general aggregation rules. We show that a wide class of risk measures can be regarded as non-linear pools. Not only does this class include all coherent risk measures (Artzner et al., 1999), but also measures like the Value-at-Risk, which fail subadditivity. We also briefly discuss the possibility of extending our findings to include the convex risk measures of Föllmer and Schied (2002) as well as their non-subadditive extensions.

Functional estimation of extreme conditional expectiles

Quantiles and expectiles can be interpreted as solutions of convex minimization problems. Unlike quantiles, expectiles are determined by tail expectations rather than tail probabilities, and define a coherent risk measure. For these reasons, among others, they have recently been the subject of renewed attention in actuarial and financial risk management. The challenging problem of estimating extreme expectiles, whose order converges to one as the sample size increases, is considered given a functional covariate. A functional kernel estimator of extreme conditional expectiles is constructed by writing expectiles as quantiles of a different distribution. The asymptotic properties of the estimators are studied in the context of conditional heavy-tailed distributions. Different ways of estimating the functional tail index, as a way to extrapolate the estimates to the very far conditional tails, are examined. A numerical illustration of the finite-sample performance of the estimators is provided on simulated and real datasets.

Impact of capacity mechanisms and demand elasticity on generation adequacy with risk-averse generation companies

While on-going and future developments will likely increase the short-term elasticity of the electricity demand, capacity markets are often put forward as a possible remedy to ensure generation adequacy. In this paper, we provide a model formulation of a stochastic non-cooperative capacity planning problem reflecting short-term elastic energy demand, risk-averse investors, and a capacity market. The Conditional-Value-At-Risk is used as a coherent risk-measure. For solving the model, we deploy an Alternating Direction Method of Multipliers. We analyze the impact of short-term demand elasticity on the social welfare, agents’ surpluses, and Loss-of-Load-Expectation of an energy-only market and a market complemented by a capacity market. Moreover, we consider the impact of set capacity targets. For the energy-only market, we show that short-term demand elasticity even amplifies the negative effect of risk aversion on social welfare. A capacity market can be used to hedge against the negative effects of risk aversion, even if the capacity target is not set ideally. Furthermore, we show that a capacity market can be used to mitigate welfare transfer to investors and lower the loss-of-load-expectation induced by scarcity times.

Dynamic Risk Measures for Processes via Backward Stochastic Differential Equations Associated with Lévy Processes.

In this paper, we study the dynamic risk measures for processes induced by backward stochastic differential equations driven by Teugel’s martingales associated with Lévy processes (BSDELs). The representation theorem for generators of BSDELs is provided. Furthermore, the time consistency of the coherent and convex dynamic risk measures for processes is characterized by means of the generators of BSDELs. Moreover, the coherency and convexity of dynamic risk measures for processes are characterized by the generators of BSDELs. Finally, we provide two numerical examples to illustrate the proposed dynamic risk measures.

COHERENT RISK MEASURES AND NORMAL MIXTURE DISTRIBUTIONS WITH APPLICATIONS IN PORTFOLIO OPTIMIZATION

This paper investigates the coherent risk measure of a class of normal mixture distributions which are widely-used in finance. The main result shows that the mean-risk portfolio optimization problem with these 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, we show that the efficient frontier of the portfolio optimization problem can be extended to three dimensions in this case. The worst-case value-at-risk in the robust portfolio optimization can also be calculated directly. Finally, the conditional value-at-risk (CVaR) is considered as an example of coherent risk measure. We obtain the marginal contribution to risk for a portfolio based on the normal mixture model.

Constrained Risk-Averse Markov Decision Processes

We consider the problem of designing policies for Markov decision processes (MDPs) with dynamic coherent risk objectives and constraints. We begin by formulating the problem in a Lagrangian framework. Under the assumption that the risk objectives and constraints can be represented by a Markov risk transition mapping, we propose an optimization-based method to synthesize Markovian policies that lower-bound the constrained risk-averse problem. We demonstrate that the formulated optimization problems are in the form of difference convex programs (DCPs) and can be solved by the disciplined convex-concave programming (DCCP) framework. We show that these results generalize linear programs for constrained MDPs with total discounted expected costs and constraints. Finally, we illustrate the effectiveness of the proposed method with numerical experiments on a rover navigation problem involving conditional-value-at-risk (CVaR) and entropic-value-at-risk (EVaR) coherent risk measures.

Estimation of the Portfolio Risk from Conditional Value at Risk Using Monte Carlo Simulation

Portfolio risk shows the large deviations in portfolio returns from expected portfolio returns. Value at Risk (VaR) is one method for determining the maximum risk of loss of a portfolio or an asset based on a certain probability and time. There are three methods to estimate VaR, namely variance-covariance, historical, and Monte Carlo simulations. One disadvantage of VaR is that it is incoherent because it does not have sub-additive properties. Conditional Value at Risk (CVaR) is a coherent or related risk measure and has a sub-additive nature which indicates that the loss on the portfolio is smaller or equal to the amount of loss of each asset. CVaR can provide loss information above the maximum loss. Estimating portfolio risk from the CVaR value using Monte Carlo simulation and its application to PT. Bank Negara Indonesia (Persero) Tbk (BBNI.JK) and PT. Bank Tabungan Negara (Persero) Tbk (BBTN.JK) will be discussed in this study. The daily closing price of each BBNI and BBTN share from 6 January 2019 to 30 December 2019 is used to measure the CVaR of the two banks' stock portfolios with this Monte Carlo simulation. The steps taken are determining the return value of assets, testing the normality of return of assets, looking for risk measures of returning assets that form a normally distributed portfolio, simulate the return of assets with monte carlo, calculate portfolio weights, looking for returns portfolio, calculate the quartile of portfolio return as a VaR value, and calculate the average loss above the VaR value as a CVaR value. The results of portfolio risk estimation of the value of CVaR using Monte Carlo simulation on PT. Bank Negara Indonesia (Persero) Tbk and PT. Bank Tabungan Negara (Persero) Tbk at a confidence level of 90%, 95%, and 99% is 5.82%, 6.39%, and 7.1% with a standard error of 0.58%, 0.59%, and 0.59%. If the initial funds that will be invested in this portfolio are illustrated at Rp 100,000,000, it can be interpreted that the maximum possible risk that investors will receive in the future will not exceed Rp 5,820,000, Rp 6,390,000 and Rp 7,100,000 at the significant level 90%, 95%, and 99%

Real-Valued Systemic Risk Measures

We describe the axiomatic approach to real-valued Systemic Risk Measures, which is a natural counterpart to the nowadays classical univariate theory initiated by Artzner et al. in the seminal paper “Coherent measures of risk”, Math. Finance, (1999). In particular, we direct our attention towards Systemic Risk Measures of shortfall type with random allocations, which consider as eligible, for securing the system, those positions whose aggregated expected utility is above a given threshold. We present duality results, which allow us to motivate why this particular risk measurement regime is fair for both the single agents and the whole system at the same time. We relate Systemic Risk Measures of shortfall type to an equilibrium concept, namely a Systemic Optimal Risk Transfer Equilibrium, which conjugates Bühlmann’s Risk Exchange Equilibrium with a capital allocation problem at an initial time. We conclude by presenting extensions to the conditional, dynamic framework. The latter is the suitable setup when additional information is available at an initial time.

Contracts in electricity markets under EU ETS: A stochastic programming approach

The European Union Emission Trading Scheme (EU ETS) is a cornerstone of the EU's strategy to fight climate change and an important device for plummeting greenhouse gas (GHG) emissions in an economically efficient manner. The power industry has switched to an auction-based allocation system at the onset of Phase III of the EU ETS to bring economic efficiency by negating windfall profits that have been resulted from grandfathered allocation of allowances in the previous phases. In this work, we analyze and simulate the interaction of oligopolistic generators in an electricity market with a game-theoretical framework where the electricity and the emissions markets interact in a two-stage electricity market. For analytical simplicity, we assume a single futures market where the electricity is committed at the futures price, and the emissions allowance is contracted in advance, prior to a spot market where the energy and allowances delivery takes place. Moreover, a coherent risk measure is applied (Conditional Value at Risk) to model both risk averse and risk neutral generators and a two-stage stochastic optimization setting is introduced to deal with the uncertainty of renewable capacity, demand, generation, and emission costs. The performance of the proposed equilibrium model and its main properties are examined through realistic numerical simulations. Our results show that renewable generators are surging and substituting conventional generators without compromising social welfare. Hence, both renewable deployment and emission allowance auctioning are effectively reducing GHG emissions and promoting low-carbon economic path.

Bayes risk, elicitability, and the Expected Shortfall

AbstractMotivated by recent advances on elicitability of risk measures and practical considerations of risk optimization, we introduce the notions of Bayes pairs and Bayes risk measures. Bayes risk measures are the counterpart of elicitable risk measures, extensively studied in the recent literature. The Expected Shortfall (ES) is the most important coherent risk measure in both industry practice and academic research in finance, insurance, risk management, and engineering. One of our central results is that under a continuity condition, ES is the only class of coherent Bayes risk measures. We further show that entropic risk measures are the only risk measures which are both elicitable and Bayes. Several other theoretical properties and open questions on Bayes risk measures are discussed.

On continuity in risk-averse bilevel stochastic linear programming with random lower level objective function

We study bilevel stochastic linear programs with fixed lower level feasible set and random follower's goal function and show that the parametrized random variable arising as the upper level outcome depends continuously on the leader's decision w.r.t. any Lp-norm with p∈[1,∞). This entails continuity of the objective function for a class of models involving convex risk measures defined on appropriate Lp-spaces and allows to formulate verifiable sufficient conditions for the existence of optimal solutions.

Coherent portfolio performance ratios

In Quantitative Finance 2016, Chen, Hu and Lin (CHL) claimed the following: ‘ … there is yet no coherent risk measure related to investment performance.’ (p. 682). Our paper suggests and analyzes four coherence axioms that portfolio performance ratios should satisfy. Our Portfolio Riskless Translation Invariance axiom must be satisfied to assure separation of the objective decision to optimize a portfolio’s risky composition from the subjective decision to optimize the weight of the portfolio's level of risk-free asset. Performance ratios with fixed thresholds other than the risk-free rate do not satisfy this axiom, allowing portfolio managers to affect an ex-ante performance ratio merely by changing the proportion of the risk-free asset in the portfolio rather than by improving the composition of the portfolio’s risky components. The magnitude of this potential drawback is examined using S&P-500 stock index data. Replacing the fixed threshold, T, with a threshold that equals γ times the portfolio’s risk premium plus (1-γ) times the risk-free rate, eliminates the above shortcoming for any selected γ. In addition, using performance ratios with threshold rather than fixed T, assures consistency of performance ratios of effective stochastic dominance and risk-free asset rules.

Robust spectral risk optimization when the subjective risk aversion is ambiguous: a moment-type approach

Choice of a risk measure for quantifying risk of an investment portfolio depends on the decision maker’s risk preference. In this paper, we consider the case when such a preference can be described by a law invariant coherent risk measure but the choice of a specific risk measure is ambiguous. We propose a robust spectral risk approach to address such ambiguity. Differing from Wang and Xu (SIAM J Optim 30(4):3198–3229, 2020), the new robust model allows one to elicit the decision maker’s risk preference through pairwise comparisons and use the elicited preference information to construct an ambiguity set of risk spectra. The robust spectral risk measure (RSRM) is based on the worst case risk spectrum from the set. To calculate RSRM and solve the associated optimal decision making problem, we use a technique from Acerbi and Simonetti (Portfolio optimization with spectral measures of risk. Working paper, 2002) to develop a new computational approach which is independent of order statistics and reformulate the robust spectral risk optimization problem as a single deterministic convex programming problem when the risk spectra in the ambiguity set are step-like. Moreover, we propose an approximation scheme when the risk spectra are not step-like and derive a bound for the model approximation error and its propagation to the optimal decision making problems. Some preliminary numerical test results are reported about the performance of the robust model and the computational scheme.