Robust portfolio optimization with copulas

Conditional Value at Risk (CVaR) is widely used in portfolio optimization as a measure of risk. CVaR is clearly dependent on the underlying probability distribution of the portfolio. We show how copulas can be introduced to any problem that involves distributions and how they can provide solutions for the modeling of the portfolio. We use this to provide the copula formulation of the CVaR of a portfolio. Given the critical dependence of CVaR on the underlying distribution, we use a robust framework to extend our approach to Worst Case CVaR (WCVaR). WCVaR is achieved through the use of rival copulas. These rival copulas have the advantage of exploiting a variety of dependence structures, symmetric and not. We compare our model against two other models, Gaussian CVaR and Worst Case Markowitz. Our empirical analysis shows that WCVaR can asses the risk more adequately than the two competitive models during periods of crisis.     

Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin

We consider the classical risk model and carry out a sensitivity and robustness analysis of finite-time ruin probabilities. We provide algorithms to compute the related influence functions. We also prove the weak convergence of a sequence of empirical finite-time ruin probabilities starting from zero initial reserve toward a Gaussian random variable. We define the concepts of reliable finite-time ruin probability as a Value-at-Risk of the estimator of the finite-time ruin probability. To control this robust risk measure, an additional initial reserve is needed and called Estimation Risk Solvency Margin (ERSM). We apply our results to show how portfolio experience could be rewarded by cut-offs in solvency capital requirements. An application to catastrophe contamination and numerical examples are also developed.     

Robust asset allocation with conditional value at risk using the forward search

The well‐known Markowitz approach to portfolio allocation, based on expected returns and their covariance, seems to provide questionable results in financial management. One motivation for the pitfall is that financial returns have heavier than Gaussian tails, so the covariance of returns, used in the Markowitz model as a measure of portfolio risk, is likely to provide a loose quantification of the effective risk. Additionally, the Markowitz approach is very sensitive to small changes in either the expected returns or their correlation, often leading to irrelevant portfolio allocations. More recent allocation techniques are based on alternative risk measures, such as value at risk (VaR) and conditional VaR (CVaR), which are believed to be more accurate measures of risk for fat‐tailed distributions. Nevertheless, both VaR and CVaR estimates can be influenced by the presence of extreme returns. In this paper, we discuss sensitivity to the presence of extreme returns and outliers when optimizing the allocation, under the constraint of keeping CVaR to a minimum. A robust and efficient approach, based on the forward search, is suggested. A Monte Carlo simulation study shows the advantages of the proposed approach, which outperforms both robust and nonrobust alternatives under a variety of specifications. The performance of the method is also thoroughly evaluated with an application to a set of US stocks.     

On Coherent Risk Measures Induced by Convex Risk Measures

We study the close relationship between coherent risk measures and convex risk measures. Inspired by the obtained results, we propose a class of coherent risk measures induced by convex risk measures. The robust representation and minimization problem of the induced coherent risk measure are investigated. A new coherent risk measure, the Entropic Conditional Value-at-Risk (ECVaR), is proposed as a special case. We show how to apply the induced coherent risk measure to realistic portfolio selection problems. Finally, by comparing its out-of-sample performance with that of CVaR, entropic risk measure, as well as entropic value-at-risk, we carry out a series of empirical tests to demonstrate the practicality and superiority of the ECVaR measure in optimal portfolio selection.     

Risk-averse feasible policies for large-scale multistage stochastic linear programs

We develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints, assuming that only the first- and second-order moments as well as the support of the uncertain parameters are given. It is known that robust chance constraints can be conservatively approximated by Worst-Case Conditional Value-at-Risk (CVaR) constraints. We first prove that this approximation is exact for robust individual chance constraints with concave or (not necessarily concave) quadratic constraint functions, and we demonstrate that the Worst-Case CVaR can be computed efficiently for these classes of constraint functions. Next, we study the Worst-Case CVaR approximation for joint chance constraints. This approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations. The tightness depends on a set of scaling parameters, which can be tuned via a sequential convex optimization algorithm. We show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the Worst-Case CVaR can be evaluated efficiently if the scaling parameters are kept constant. We evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations.     

CVaR风险度量模型在投资组合中的运用

风险价值(VaR)是近年来金融机构广泛运用的风险度量指标，条件风险价值(CVaR)是VaR的修正模型，也称为平均超额损失或尾部VaR，它比VaR具有更好的性质。在本中，我们将运用风险度量指标VaR和CVaR，提出一个新的最优投资组合模型。介绍了模型的算法，而且利用我国的股票市场进行了实证分析，验证了新模型的有效性，为制定合理的投资组合提供了一种新思路。     

Newsvendor solutions via conditional value-at-risk minimization

In this paper, we consider the minimization of the conditional value-at-risk (CVaR), a most preferable risk measure in financial risk management, in the context of the well-known single-period newsvendor problem, which is originally formulated as the maximization of the expected profit or the minimization of the expected cost. We show that downside risk measures including the CVaR are tractable in the problem due to their convexity, and consequently, under mild assumptions on the probability distribution of products’ demand, we provide analytical solutions or linear programming (LP) formulation of the minimization of the CVaR measures defined with two different loss functions. Numerical examples are also exhibited, clarifying the difference among the models analyzed in this paper, and demonstrating the efficiency of the LP solutions.     

Robust Bayesian analysis in partially ordered plausibility calculi

In Robust Bayesian analysis one attempts to avoid the ‘Dogma of Precision’ in Bayesian analysis by entertaining a set of probability distributions instead of exactly one. The algebraic approach to plausibility calculi is inspired by Cox's and Jaynes' analyses of plausibility assessment as a logic of uncertainty. In the algebraic approach one is not so much interested in different ways to prove that precise Bayesian probability is inevitable but rather in how different sets of assumptions are reflected in the resulting plausibility calculus. It has repeatedly been pointed out that a partially ordered plausibility domain is more appropriate than a totally ordered one, but it has not yet been completely resolved exactly what such domains can look like. One such domain is the natural robust Bayesian representation, an indexed family of probabilities.We show that every plausibility calculus embeddable in a partially ordered ring is equivalent to a subring of a product of ordered fields, i.e., the robust Bayesian representation is universal under our assumptions, if extended rather than standard probability is used.We also show that this representation has at least the same expressiveness as coherent sets of desirable gambles with real valued payoffs, for a finite universe.     

Diversification-consistent data envelopment analysis with general deviation measures

We propose new efficiency tests which are based on traditional DEA models and take into account portfolio diversification. The goal is to identify the investment opportunities that perform well without specifying our attitude to risk. We use general deviation measures as the inputs and return measures as the outputs. We discuss the choice of the set of investment opportunities including portfolios with limited number of assets. We compare the optimal values (efficiency scores) of all proposed tests leading to the relations between the sets of efficient opportunities. Strength of the tests is then discussed. We test the efficiency of 25 world financial indices using new DEA models with CVaR deviation measures.     

Optimal reinsurance under dynamic VaR constraint

This paper deals with the optimal reinsurance strategy from an insurer’s point of view. Our objective is to find the optimal policy that maximises the insurer’s survival probability. To meet the requirement of regulators and provide a tool to risk management, we introduce the dynamic version of Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR) and worst-case CVaR (wcCVaR) constraints in diffusion model and the risk measure limit is proportional to company’s surplus in hand. In the dynamic setting, a CVaR/wcCVaR constraint is equivalent to a VaR constraint under a higher confidence level. Applying dynamic programming technique, we obtain closed form expressions of the optimal reinsurance strategies and corresponding survival probabilities under both proportional and excess-of-loss reinsurance. Several numerical examples are provided to illustrate the impact caused by dynamic VaR/CVaR/wcCVaR limit in both types of reinsurance policy.     

Maximising entropy on the nonparametric predictive inference model for multinomial data

The combination of mathematical models and uncertainty measures can be applied in the area of data mining for diverse objectives with as final aim to support decision making. The maximum entropy function is an excellent measure of uncertainty when the information is represented by a mathematical model based on imprecise probabilities. In this paper, we present algorithms to obtain the maximum entropy value when the information available is represented by a new model based on imprecise probabilities: the nonparametric predictive inference model for multinomial data (NPI-M), which represents a type of entropy-linear program. To reduce the complexity of the model, we prove that the NPI-M lower and upper probabilities for any general event can be expressed as a combination of the lower and upper probabilities for the singleton events, and that this model can not be associated with a closed polyhedral set of probabilities. An algorithm to obtain the maximum entropy probability distribution on the set associated with NPI-M is presented. We also consider a model which uses the closed and convex set of probability distributions generated by the NPI-M singleton probabilities, a closed polyhedral set. We call this model A-NPI-M. A-NPI-M can be seen as an approximation of NPI-M, this approximation being simpler to use because it is not necessary to consider the set of constraints associated with the exact model.     

证券投资组合优化问题的强健性的锥优化方法(英)

本文研究了具有强健性的证券投资组合优化问题.模型以最差条件在值风险为风险度量方法,并且考虑了交易费用对收益的影响.当投资组合的收益率概率分布不能准确确定但是在有界的区间内,尤其是在箱型区间结构和椭球区域结构内时,我们可以把具有强健性的证券投资组合优化问题的模型分别转化成线性规划和二阶锥规划形式.最后,我们用一个真实市场数据的算例来验证此方法.     

证券投资组合优化问题的强健性的锥优化方法(英文)

本文研究了具有强健性的证券投资组合优化问题.模型以最差条件在值风险为风险度量方法,并且考虑了交易费用对收益的影响.当投资组合的收益率概率分布不能准确确定但是在有界的区间内,尤其是在箱型区间结构和椭球区域结构内时,我们可以把具有强健性的证券投资组合优化问题的模型分别转化成线性规划和二阶锥规划形式.最后,我们用一个真实市场数据的算例来验证此方法.     

Mean-CVaR模型下的资产组合和破产风险控制

为了克服尾部风险测度CVaR模型本身的不足，并且给“如何实现资产组合的破产风险与期望利润的最优配置”问题提供一个更加符合现实的答案，本文在CVaR模型基础上，通过把风险资本的来源内生于资本禀赋以及把风险资本的机会成本引入利润函数的方式提出了线性Mean—CVaR模型。同时，本文通过对“上证50”成分股进行选择的实证分析给出了由线形Mean—CVaR模型得到的更加合理的资产组合与资本储备。     

Composite multilinearity,epistemic uncertainty and risk achievement worth

Risk achievement worth is one of the most widely utilized importance measures. RAW is defined as the ratio of the risk metric value attained when a component has failed over the base case value of the risk metric. Traditionally, both the numerator and denominator are point estimates. Relevant literature has shown that inclusion of epistemic uncertainty (i) induces notable variability in the point estimate ranking and (ii) causes the expected value of the risk metric to differ from its nominal value. We investigate the conditions under which the equality of the nominal and expected values of a reliability risk metric holds. We then study how the presence of epistemic uncertainty affects RAW and the associated ranking. We propose an extension of RAW (called ERAW) which allows one to obtain a ranking robust to epistemic uncertainty. We discuss the properties of ERAW and the conditions under which it coincides with RAW. We apply our findings to a probabilistic risk assessment model developed for the safety analysis of NASA lunar space missions.     

A general importance sampling algorithm for estimating portfolio loss probabilities in linear factor models

This paper develops a novel importance sampling algorithm for estimating the probability of large portfolio losses in the conditional independence framework. We apply exponential tilts to (i) the distribution of the natural sufficient statistics of the systematic risk factors and (ii) conditional default probabilities, given the simulated values of the systematic risk factors, and select parameter values by minimizing the Kullback–Leibler divergence of the resulting parametric family from the ideal (zero-variance) importance density. Optimal parameter values are shown to satisfy intuitive moment-matching conditions, and the asymptotic behaviour of large portfolios is used to approximate the requisite moments. In a sense we generalize the algorithm of Glasserman and Li (2005) so that it can be applied in a wider variety of models. We show how to implement our algorithm in the $t$ copula model and compare its performance there to the algorithm developed by Chan and Kroese (2010). We find that our algorithm requires substantially less computational time (especially for large portfolios) but is slightly less accurate. Our algorithm can also be used to estimate more general risk measures, such as conditional tail expectations, whereas Chan and Kroese (2010) is specifically designed to estimate loss probabilities.     

Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity

We consider the problem of optimal portfolio choice using the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures for a market consisting of n risky assets and a riskless asset and where short positions are allowed. When the distribution of returns of risky assets is unknown but the mean return vector and variance/covariance matrix of the risky assets are fixed, we derive the distributionally robust portfolio rules. Then, we address uncertainty (ambiguity) in the mean return vector in addition to distribution ambiguity, and derive the optimal portfolio rules when the uncertainty in the return vector is modeled via an ellipsoidal uncertainty set. In the presence of a riskless asset, the robust CVaR and VaR measures, coupled with a minimum mean return constraint, yield simple, mean-variance efficient optimal portfolio rules. In a market without the riskless asset, we obtain a closed-form portfolio rule that generalizes earlier results, without a minimum mean return restriction.     

An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution

This paper proposes a unified framework to solve distributionally robust mean-risk optimization problem that simultaneously uses variance, value-at-risk (VaR) and conditional value-at-risk (CVaR) as a triple-risk measure. It provides investors with more flexibility to find portfolios in the sense that it allows investors to optimize a return-risk profile in the presence of estimation error. We derive a closed-form expression for the optimal portfolio strategy to the robust mean-multiple risk portfolio selection model under distribution and mean return ambiguity (RMP). Specially, the robust mean-variance, robust maximum return, robust minimum VaR and robust minimum CVaR efficient portfolios are all special instances of RMP portfolios. We analytically and numerically show that the resulting portfolio weight converges to the minimum variance portfolio when the level of ambiguity aversion is in a high value. Using numerical experiment with simulated data, we demonstrate that our robust portfolios under ambiguity are more stable over time than the non-robust portfolios.     

On a robust risk measurement approach for capital determination errors minimization

We propose a robust risk measurement approach that minimizes the expectation of overestimation plus underestimation costs. We consider uncertainty by taking the supremum over a collection of probability measures, relating our approach to dual sets in the representation of coherent risk measures. We provide results that guarantee the existence of a solution and explore the properties of minimizer and minimum as risk and deviation measures, respectively. An empirical illustration is carried out to demonstrate the use of our approach in capital determination.     

Variable-sized uncertainty and inverse problems in robust optimization

In robust optimization, the general aim is to find a solution that performs well over a set of possible parameter outcomes, the so-called uncertainty set. In this paper, we assume that the uncertainty size is not fixed, and instead aim at finding a set of robust solutions that covers all possible uncertainty set outcomes. We refer to these problems as robust optimization with variable-sized uncertainty. We discuss how to construct smallest possible sets of min–max robust solutions and give bounds on their size.A special case of this perspective is to analyze for which uncertainty sets a nominal solution ceases to be a robust solution, which amounts to an inverse robust optimization problem. We consider this problem with a min–max regret objective and present mixed-integer linear programming formulations that can be applied to construct suitable uncertainty sets.Results on both variable-sized uncertainty and inverse problems are further supported with experimental data.