Coherent Risk Measures Research Articles

Reinforcement learning with dynamic convex risk measures

AbstractWe develop an approach for solving time‐consistent risk‐sensitive stochastic optimization problems using model‐free reinforcement learning (RL). Specifically, we assume agents assess the risk of a sequence of random variables using dynamic convex risk measures. We employ a time‐consistent dynamic programming principle to determine the value of a particular policy, and develop policy gradient update rules that aid in obtaining optimal policies. We further develop an actor–critic style algorithm using neural networks to optimize over policies. Finally, we demonstrate the performance and flexibility of our approach by applying it to three optimization problems: statistical arbitrage trading strategies, financial hedging, and obstacle avoidance robot control.

Statistical inference for extreme extremile in heavy-tailed heteroscedastic regression model

As a least squares analogue of quantiles, extremiles define a coherent risk measure determined by weighted expectations instead of tail probabilities. Estimating extremiles of heavy-tailed variables in a regression framework is a challenging task, especially for dependent cases. This paper develops some methods for the estimation of extreme conditional extremiles in the framework of heteroscedastic regression model with heavy-tail noises, specifically, direct and indirect methods based on the conditional extremile estimators for the residuals. We also construct corresponding bias-reduced estimators and investigate their asymptotic properties compared to the original versions. Our mathematical assumptions are satisfied in the mean-variance regression model and heteroscedastic single-index model, which makes it possible to apply our result in a series of important examples. We demonstrate our results through a simulation study and real sets of insurance and financial data analyses.

Prediction, Risk Assessment and Comparison of Selected Emerging Markets' Stock Indices During COVID-19 Pandemic Using the Coherent Measure

In this study, we modeled the log-return of three emerging markets' stock indices, namely, Shanghai SSE, Russia MOEX, and Bombay Stock Exchange Sensex using the generalized hyperbolic family of distributions. We found the generalized hyperbolic family of distributions as the best fit for describing the probability density based on AIC and likelihood ratio test. The coherent risk measure, i.e., the expected shortfall, predicted using the best fit probability distribution, was used as a market risk quantification metric. During the COVID-19 period, the Indian stock market showed maximum market risk, followed by the Russian. The Chinese market showed the least market risk. Our experiment demonstrated a significant (p = 0.000) difference in the three markets concerning the coherent risk at different probability levels from 0.001 to 0.05 in the COVID-19 period using the Jonckheere-Terpstra test. The coherent market risk increased substantially in the Indian and Russian markets during the COVID-19 pandemic compared to the pre-COVID-19 period. However, in the Chinese market, we found that the coherent risk decreased during the COVID-19 period compared to the pre-COVID-19 period. We carried out the empirical study using the adjusted daily closing values of SSE, MOEX, and Sensex from July 2018 to July 2021 and dividing the data sets into pre-COVID-19 and COVID-19 periods based on the first emergence of the COVID-19 case.

Scenario decomposable subgradient projection method for two-stage stochastic programming with convex risk measures

Scenario decomposable subgradient projection method for two-stage stochastic programming with convex risk measures

On the General Deviation Measure and the Gini coefficient

AbstractThe General Deviation Measure introduces a progressive definition for financial risk measurement, which presents an alternative to the Coherent Risk Measure. This definition replaces the Translation Invariance and Monotonicity axioms with the Shift Invariance and Nonnegativity axioms, and it includes the Mean Absolute Deviation measure and other variations of the Value‐at‐Risk measurements. This research shows that Coherent Risk Measure holds an intrinsic contradiction regarding riskless assets, and it proves that the Gini coefficient is also a General Deviation Measure. These contributions improve the efficiency of risk measurement and asset pricing in the financial markets.

Risk-averse optimization of reward-based coherent risk measures

In real-world problems such as robotics, finance and healthcare, randomness is always present, thus, it is important to take risk into consideration in order to limit the chance of rare but dangerous events. The literature on risk-averse reinforcement learning has produced many different approaches to tackle the problem, but they either struggle to scale up to complex instances, or they exhibit irrational behaviors. Here we present two novel risk-averse objectives that are both coherent and easy to optimize: the reward-based mean-mean absolute deviation (Mean-RMAD) and the reward-based conditional value at risk (RCVaR). Instead of reducing the return risk, these measures minimize the per-step reward one. We prove that these risk measures bound the corresponding return-based risk measures, so that they can be also used as proxies for their return-based versions. We develop safe algorithms for these risk measures with guaranteed monotonic improvement, and their practical trust-region versions. Furthermore, we propose a decomposition for the RCVaR optimization problem into a sequence of risk-neutral problems. Finally, we conduct an empirical analysis on the introduced approaches, demonstrating their effectiveness in retrieving a variety of risk-averse behaviors on both toy problems and more challenging ones, such as a simulated trading environment and robotic locomotion tasks.

Capital allocation with multivariate convex risk measures

Capital allocation is of fundamental importance in risk management. Capital allocation problems with univariate risk measures have been extensively studied in the finance literature. In contrast, the studies about capital allocations with multivariate risk measures are quite insufficient in the literature. In this paper, we study capital allocations with multivariate coherent and convex risk measures. First, we propose the definitions of capital allocation problem and principle for general multivariate risk measures. Then based on the dual representations proved by [41], we provide the capital allocation principles with multivariate coherent and convex risk measures. Their properties are also investigated. For the case of multivariate convex risk measures, we also suggest a reasonable and effective capital allocation scheme. Finally, examples are given to illustrate the effectiveness of the proposed capital allocation scheme, where the involved convex risk measures are multivariate and univariate entropic risk measures, respectively.

The Prospect Theory and First Price Auctions: an Explanation of Overbidding

This paper attempted using the prospect theory to explain overbidding in first price auctions. The standard outlook in the literature on auctions is that bidders overbid, but the probability weighting functions are nonlinear as in the prospect theory, so they not only tend to underweight the probabilities of winning the auction but also overweight, so that there are overbidders and underbidders. This paper proves that to some extent, non-linear weighting functions do explain overbidding the risk-neutral Nash equilibrium valuation (RNNE). Furthermore, coherent risk measures, such as certainty equivalent and translation invariance, were used to show loss aversion among bidders, and in line with the prospect theory, convexity was also confirmed with sub-additivity, monotonicity and with positive homogeneity.

Polyhedral Coherent Risk Measure and Distributionally Robust Portfolio Optimization

Polyhedral Coherent Risk Measure and Distributionally Robust Portfolio Optimization

A theory for combinations of risk measures

We study combinations of risk measures under no restrictive assumption on the set of alternatives. We develop and discuss results regarding the preservation of properties and acceptance sets for the combinations of risk measures. One of the main results is the representation for resulting risk measures from the properties of both alternative functionals and combination functions. To that, we build on the development of a representation for arbitrary mixture of convex risk measures. In this case, we obtain a penalty that recalls the notion of inf-convolution under theoretical measure integration. As an application, we address the context of probability-based risk measurements for functionals on the set of distribution functions. We develop results related to this specific context. We also explore features of individual interest generated by our framework, such as the preservation of continuity properties, the representation of worst-case risk measures, stochastic dominance and elicitability. We also address model uncertainty measurement under our framework and propose a new class of measures for this task.

Dynamic portfolio selection with linear control policies for coherent risk minimization

This paper is concerned with a linear control policy for dynamic portfolio selection. We develop this policy by incorporating time-series behaviors of asset returns on the basis of coherent risk minimization. Analyzing the dual form of our optimization model, we demonstrate that the investment performance of linear control policies is directly connected to the intertemporal covariance of asset returns. To mitigate overfitting to training data (i.e., historical asset returns), we apply robust optimization. For this optimization, we prove that the worst-case coherent risk measure can be decomposed into the empirical risk measure and the penalty terms. Numerical results demonstrate that when the number of assets is small, linear control policies deliver good out-of-sample investment performance. When the number of assets is large, the penalty terms improve the out-of-sample investment performance.

TTRISK: Tensor train decomposition algorithm for risk averse optimization

AbstractThis article develops a new algorithm named TTRISK to solve high‐dimensional risk‐averse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the so‐called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush–Kuhn–Tucker (KKT) system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large‐scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID‐19. The results indicate that the risk‐averse framework is feasible with the tensor approximations under tens of random variables.

On Polyhedral Coherent Risk Measures and Portfolio Optimization Problems

Introduction. The problem of decision-making under risk and uncertainty lies in the use of adequate criteria for assessing their optimality, in particular, in an adequate risk assessment. Various functions are known that are used as risk measures. For technical systems, the probability of an accident (failure) is used, in insurance – the probability of bankruptcy, in finance – Value-at-Risk, etc. At present, the concept of a coherent risk measure (CRM), in which its basic properties are postulated, is widely recognized. The paper considers CRMs and their subset, the polyhedral CRMs (PCMRs), which have attractive properties and contain a number of important risk measures. Such risk measures are well defined on complete information about the stochastic distributions of random variables. However, applications usually contain only partial such information from observational data. This only allows one to describe the stochastic distribution by an ambiguity set (AS). For such a case, robust PCMR constructions intended for risk assessment at AS are considered in the paper. The computation of such PCRM constructions in the form of linear programming problems (LP) is described. To demonstrate the use of the PCRM apparatus, the problems of portfolio optimization on reward-risk ratio are considered, where reward and risk are estimated by the average return and some PCRM respectively for known stochastic distributions, and by their robust constructions under uncertainty with AS. It is described how in both these cases the portfolio optimization problems are reduced to appropriate LP problems. The purpose of the paper is to describe the PCRM apparatus for assessing risks under uncertainty with AS and demonstrating the effectiveness of its application to linear problems on the example of portfolio optimization problems. Results. The use of the PCRM apparatus for the case of uncertainty with AS in the form of appropriate robust constructions and their application to portfolio optimization problems on reward-risk ratio is described. The conditions under which these portfolio problems are reduced to the corresponding LP tasks are formulated. Conclusions. The PCRM apparatus can be effectively applied to linear optimization problems under uncertainty with AS, which is demonstrated by the example of portfolio optimization problems. The reduction of portfolio problems to LP problems allows one to effectively solve them using standard methods. Keywords: coherent risk measure, polyhedral coherent risk measure, CVaR, ambiguity set, portfolio optimization, linear programming problem.

Nonasymptotic Convergence Rates for the Plug-in Estimation of Risk Measures

Let ρ be a general law-invariant convex risk measure, for instance, the average value at risk, and let X be a financial loss, that is, a real random variable. In practice, either the true distribution μ of X is unknown, or the numerical computation of [Formula: see text] is not possible. In both cases, either relying on historical data or using a Monte Carlo approach, one can resort to an independent and identically distributed sample of μ to approximate [Formula: see text] by the finite sample estimator [Formula: see text] (μNdenotes the empirical measure of μ). In this article, we investigate convergence rates of [Formula: see text] to [Formula: see text]. We provide nonasymptotic convergence rates for both the deviation probability and the expectation of the estimation error. The sharpness of these convergence rates is analyzed. Our framework further allows for hedging, and the convergence rates we obtain depend on neither the dimension of the underlying assets nor the number of options available for trading.Funding: Daniel Bartl is grateful for financial support through the Vienna Science and Technology Fund [Grant MA16-021] and the Austrian Science Fund [Grants ESP-31 and P34743]. Ludovic Tangpi is supported by the National Science Foundation [Grant DMS-2005832] and CAREER award [Grant DMS-2143861].

Asset allocation for a DC pension plan with minimum guarantee constraint and hidden Markov regime-switching

This paper is devoted to the study of the asset allocation problem for a DC pension plan with minimum guarantee constraint in a hidden Markov regime-switching economy. Suppose that four types of assets are available in the financial market: a risk-free asset, a zero-coupon bond, an inflation-indexed bond and a stock. The expected return rate of the stock depends on unobservable economic states, and the change of states is described by a hidden Markov chain. In addition, the CIR process is used to describe the evolution of the nominal interest rate. The contribution rate is also assumed to be stochastic. The goal of investment management is to minimize the convex risk measure of the terminal wealth in excess of the minimum guarantee constraint. First, we transform the partially observable optimization problem into the one with complete information using the Wonham filtering technique and deal with the minimum guarantee constraint by constructing auxiliary processes. Furthermore, we derive the optimal investment strategy by the BSDE approach. Finally, some numerical results are presented to illustrate the impacts of some important parameters on investment behaviors.

Evaluating port efficiency dynamics: A risk-based approach

This study proposes a new methodology to quantify the efficiency dynamics of a port over time. While efficiency evaluation has gained full attention in port management, researchers conducting related studies are challenged by temporal variations observed in the collected data. Existing approaches have almost exclusively relied on multivariate normal distributional assumptions of the input and output data, but empirical evidence from real data shows that the port operations data demonstrate long-tail distributions and violate the distributional assumptions. In addition, many existing models are intractable (non-convex) or lack interpretability. Motivated by these challenges, we develop an optimization-based approach for efficiency measurement under uncertainty that is compatible with the conventional non-parametric method. In particular, inspired by the coherent risk measure, we create a risk-based index to measure the efficiency of any operating unit by comparing its observations against a benchmark that is guaranteed to be production possible under a certain risk level. To facilitate the computation of the index, we develop a risk-based port efficiency evaluation (RPE) model, which can be reformulated as an exponential cone program (ECP) and solved efficiently by off-the-shelf solvers. We test our model for a multipurpose port on a real dataset of 3,394 observations showing the proposed approach’s merits. We find that the port productivity peaks on Tuesday and Saturday and troughs on Friday. We also provide evidence for the Chinese New Year effect from a port management perspective and draw managerial insights from the study.

Risk-Averse Optimal Control in Continuous Time by Nesting Risk Measures

This paper extends dynamic control problems from a risk-neutral to a risk-averse setting. We establish a limit for consecutive risk-averse decision making by consistently and adequately nesting coherent risk measures. This approach provides a new perspective on multistage optimal control problems in continuous time. For the limiting case, we elaborate a new dynamic programming principle, which is risk averse, and give risk-averse Hamilton–Jacobi–Bellman equations by generalizing the infinitesimal generator. In doing so, we provide a constructive explanation of the driver g in g-expectation, a dynamic risk measure based on backward stochastic differential equations. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Project-ID 416228727 – SFB 1410].

On tail-risk measures for non-integrable heavy-tailed random variables

The assessment of risk for heavy-tailed distributions is a crucial question in various fields of application. An important family of risk measures is provided by the class of distortion risk (DR) measures which encompasses the Value-at-Risk and the Tail-Value-at-Risk measures. The Tail-Value-at-Risk is a coherent risk measure (which is not the case for the Value-at-Risk) but it is defined only for integrable quantile functions that is to say for heavy-tailed distributions with a tail index smaller than 1. Moreover, it is a matter of fact that the performance of the empirical estimator is strongly deteriorated when the tail index becomes close to 1. The main contribution is the introduction and the estimation of a new risk measure which is defined for all heavy-tailed distributions and which is tail-equivalent to a coherent DR measure when the tail of the underlying distribution is not too heavy. Its finite sample performance is discussed on a fire claims dataset.

Incorporating convex risk measures into multistage stochastic programming algorithms

Incorporating convex risk measures into multistage stochastic programming algorithms

Actuarial pricing with financial methods

The objective of this paper is twofold. On the one hand, the optimal combination of reinsurance and financial investment will be studied under a general framework. Indeed, there is no specific type of reinsurance contract, there is no specific dynamics of the involved financial instruments and the financial market does not have to be free of frictions. On the other hand, it will be pointed out how the optimal combination above may provide us with new premium principles making the insurer global risk vanish. The risk will be managed with a coherent risk measure, and the new premium principles will seem to reflect several properties, which are desirable from both the analytical and the economic perspectives. From the analytical viewpoint, the premium principles will be continuous, homogeneous and increasing. From the economic viewpoint, the premium principles will lead to cheaper prices with respect to both the insurance market and the financial one. In other words, the premium principles will make the insurer more competitive in prices under a null risk. General necessary and sufficient optimality conditions will be given, as well as closed forms for the solutions under appropriate assumptions. Several methods preventing unbounded optimization problems will warrant special attention, and one particular case will be more thoroughly studied, namely, the combination of the Black–Scholes–Merton pricing model with the conditional value at risk.