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We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position $X$ to ensure that Expected Shortfall $ES_p(X)$ does not exceed a pre-specified threshold $g(p)$ for every probability level $pin[0,1]$. Through the choice of the benchmark risk profile $g$ one can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance.
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The 2008 mortgage crisis is an example of an extreme event. Extreme value theory tries to estimate such tail risks. Modern finance practitioners prefer Expected Shortfall based risk metrics (which capture tail risk) over traditional approaches like volatility or even Value-at-Risk. This paper provides a quantum annealing algorithm in QUBO form for a dynamic asset allocation problem using expected shortfall constraint. It was motivated by the need to refine the current quantum algorithms for Markowitz type problems which are academically interesting but not useful for practitioners. The algorithm is dynamic and the risk target emerges naturally from the market volatility. Moreover, it avoids complicated statistics like generalized pareto distribution. It translates the problem into qubit form suitable for implementation by a quantum annealer like D-Wave. Such QUBO algorithms are expected to be solved faster using quantum annealing systems than any classical algorithm using classical computer (but yet to be demonstrated at scale).
We shall provide in this paper good deal pricing bounds for contingent claims induced by the shortfall risk with some loss function. Assumptions we impose on loss functions and contingent claims are very mild. We prove that the upper and lower bounds of good deal pricing bounds are expressed by convex risk measures on Orlicz hearts. In addition, we obtain its representation with the minimal penalty function. Moreover, we give a representation, for two simple cases, of good deal bounds and calculate the optimal strategies when a claim is traded at the upper or lower bounds of its good deal pricing bound.
We present the Shortfall Deviation Risk (SDR), a risk measure that represents the expected loss that occurs with certain probability penalized by the dispersion of results that are worse than such an expectation. SDR combines Expected Shortfall (ES) and Shortfall Deviation (SD), which we also introduce, contemplating two fundamental pillars of the risk concept, the probability of adverse events and the variability of an expectation, and considers extreme results. We demonstrate that SD is a generalized deviation measure, whereas SDR is a coherent risk measure. We achieve the dual representation of SDR, and we discuss issues such as its representation by a weighted ES, acceptance sets, convexity, continuity and the relationship with stochastic dominance. Illustrations with real and simulated data allow us to conclude that SDR offers greater protection in risk measurement compared with VaR and ES, especially in times of significant turbulence in riskier scenarios.
We consider a multi-step algorithm for the computation of the historical expected shortfall such as defined by the Basel Minimum Capital Requirements for Market Risk. At each step of the algorithm, we use Monte Carlo simulations to reduce the number of historical scenarios that potentially belong to the set of worst scenarios. The number of simulations increases as the number of candidate scenarios is reduced and the distance between them diminishes. For the most naive scheme, we show that the L p-error of the estimator of the Expected Shortfall is bounded by a linear combination of the probabilities of inversion of favorable and unfavorable scenarios at each step, and of the last step Monte Carlo error associated to each scenario. By using concentration inequalities, we then show that, for sub-gamma pricing errors, the probabilities of inversion converge at an exponential rate in the number of simulated paths. We then propose an adaptative version in which the algorithm improves step by step its knowledge on the unknown parameters of interest: mean and variance of the Monte Carlo estimators of the different scenarios. Both schemes can be optimized by using dynamic programming algorithms that can be solved off-line. To our knowledge, these are the first non-asymptotic bounds for such estimators. Our hypotheses are weak enough to allow for the use of estimators for the different scenarios and steps based on the same random variables, which, in practice, reduces considerably the computational effort. First numerical tests are performed.
Quantiles and expected shortfalls are usually used to measure risks of stochastic systems, which are often estimated by Monte Carlo methods. This paper focuses on the use of quasi-Monte Carlo (QMC) method, whose convergence rate is asymptotically better than Monte Carlo in the numerical integration. We first prove the convergence of QMC-based quantile estimates under very mild conditions, and then establish a deterministic error bound of $O(N^{-1/d})$ for the quantile estimates, where $d$ is the dimension of the QMC point sets used in the simulation and $N$ is the sample size. Under certain conditions, we show that the mean squared error (MSE) of the randomized QMC estimate for expected shortfall is $o(N^{-1})$. Moreover, under stronger conditions the MSE can be improved to $O(N^{-1-1/(2d-1)+epsilon})$ for arbitrarily small $epsilon>0$.
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