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"Value at Risk"
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Partially ordered data sets and a new efficient method for calculating multivariate conditional value-at-risk
Recent studies in Lee and Prékopa (Oper Res Lett 45:19–24, 2017) and Lee (Oper Res Lett 45:1204–1220, 2017) showed that a union of partially ordered orthants in Rn can be decomposed only into the largest and the second largest chains. This allows us to calculate the probability of the union of such events in a recursive manner. If the vertices of such orthants designate p-level efficient points, i.e., the multivariate quantile or the multivariate value-at-risk (MVaR) in Rn, then the number of them, say N, is typically very large, which makes it almost impossible to calculate the multivariate conditional value-at-risk (MCVaR) introduced by Prékopa (Ann Oper Res 193(1):49–69, 2012). This is because it takes O(2N) in case of N MVaRs in Rn to find the exact value of MCVaR. In this paper, upon the basis of ideas in Lee and Prékopa (Oper Res Lett 45:19–24, 2017) and Lee (Oper Res Lett 45:1204–1220, 2017), together with proper adjustments, we study efficient methods for the calculation of the MCVaR without resorting to an approximation. In fact, the proposed methods not only have polynomial time complexity but also computes the exact value of MCVaR. We also discuss additional benefits MCVaR has to offer over its univariate counter part, the conditional value-at-risk, by providing numerical results. Numerical examples are presented with computing time in both cases of given population and sample data sets.
Journal Article
Entropic Value-at-Risk: A New Coherent Risk Measure
This paper introduces the concept of
entropic value-at-risk
(EVaR), a new coherent risk measure that corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the
value-at-risk
(VaR) as well as the
conditional value-at-risk
(CVaR). We show that a broad class of stochastic optimization problems that are computationally intractable with the CVaR is efficiently solvable when the EVaR is incorporated. We also prove that if two distributions have the same EVaR at all confidence levels, then they are identical at all points. The dual representation of the EVaR is closely related to the Kullback-Leibler divergence, also known as the relative entropy. Inspired by this dual representation, we define a large class of coherent risk measures, called
g-entropic
risk measures. The new class includes both the CVaR and the EVaR.
Journal Article
Distortion Risk Measures of Increasing Rearrangement
Increasing rearrangement is a rewarding instrument in financial risk management. In practice, risks must be managed from different perspectives. A common example is the portfolio risk, which often can be seen from at least two perspectives: market value and book value. Different perspectives with different distributions can be coupled by increasing rearrangement. One distribution is regarded as underlying, and the other distribution can be expressed as an increasing rearrangement of the underlying distribution. Then, the risk measure for the latter can be expressed in terms of the underlying distribution. Our first objective is to introduce increasing rearrangement for application in financial risk management and to apply increasing rearrangement to the class of distortion risk measures. We derive formulae to compute risk measures in terms of the underlying distribution. Afterwards, we apply our results to a series of special distortion risk measures, namely the value at risk, expected shortfall, range value at risk, conditional value at risk, and Wang’s risk measure. Finally, we present the connection of increasing rearrangement with inverse transform sampling, Monte Carlo simulation, and cost-efficient strategies. Butterfly options serve as an illustrative example of the method.
Journal Article
Risk-Based Loan Pricing: Portfolio Optimization Approach with Marginal Risk Contribution
We consider a lender (bank) that determines the optimal loan price (interest rate) to offer to prospective borrowers under uncertain borrower response and default risk. A borrower may or may not accept the loan at the price offered, and both the principal loaned and the interest income become uncertain because of the risk of default. We present a risk-based loan pricing optimization framework that explicitly takes into account the marginal risk contribution, the portfolio risk, and a borrower’s acceptance probability. Marginal risk assesses the incremental risk contribution of a prospective loan to the bank’s overall portfolio risk by capturing the dependencies between the prospective loan and the existing portfolio and is evaluated with respect to the value-at-risk and conditional value-at-risk measures. We examine the properties and computational challenges of the formulations. We design a reformulation method based on the concavifiability concept to transform the nonlinear objective functions and to derive equivalent mixed-integer nonlinear reformulations with convex continuous relaxations. We also extend the approach to multiloan pricing problems, which feature explicit loan selection decisions in addition to pricing decisions. We derive formulations with multiple loans that take the form of mixed-integer nonlinear problems with nonconvex continuous relaxations and develop a computationally efficient algorithmic method. We provide numerical evidence demonstrating the value of the proposed framework, test the computational tractability, and discuss managerial implications.
This paper was accepted by Chung Piaw Teo, optimization.
Journal Article
Beyond cash-additive risk measures: when changing the numéraire fails
We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified
eligible asset
to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numéraire. However, discounting does not work in all financially relevant situations, especially when the eligible asset is a defaultable bond. In this paper, we fill this gap by allowing general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on value-at-risk and tail value-at-risk on
L
p
spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules.
Journal Article
Can One Reinforce Investments in Renewable Energy Stock Indices with the ESG Index?
Studies on the environmental, social, and governance (ESG) index have become increasingly important since the ESG index offers attractive characteristics, such as environmental friendliness. Scholars and institutional investors are evaluating if investment in the ESG index can positively change current portfolios. It is crucial that institutional investors seek related assets to diversify their investments when such investors create funds in the renewable energy sector, which is highly related to environmental issues. The ESG index has proven to be a good investment choice, but we are not aware of its performance when combined with renewable energy securities. To uncover this nature, we investigate the dependence structure of the ESG index and four renewable energy indices with constant and time-varying copula models and evaluate the potential performance of using different ratios of the ESG index in the portfolio. Criteria such as risk-adjusted return, standard deviation, and conditional value-at-risk (CVaR) show that the ESG index can provide satisfactory results in lowering the potential CVaR and maintaining a high return. A goodness-of-fit test is then used to ensure the results obtained from the copula models.
Journal Article
Cyber risk quantification: Investigating the role of cyber value at risk
The aim of this paper is to deepen the application of value at risk in the cyber domain, with particular attention to its potential role in security investment valuation. Cyber risk is a fundamental component of the overall risk faced by any organization. In order to plan the size of security investments and to estimate the consequent risk reduction, managers strongly need to quantify it. Accordingly, they can decide about the possibility of sharing residual risk with a third party, such as an insurance company. Recently, cyber risk management techniques are including some risk quantile-based measures that are widely employed in the financial domain. They refer to value at risk that, in the cyber context, takes the name of cyber value at risk (Cy-VaR). In this paper, the main features and challenging issues of Cy-VaR are examined. The possible use of this risk measure in supporting investment decisions in cyber context is discussed, and new risk-based security metrics are proposed. Some simple examples are given to show their potential
Journal Article
Optimal risk sharing in insurance networks
We discuss the impact of risk sharing and asset–liability management on capital requirements. Our analysis contributes to the evaluation of the merits and deficiencies of different risk measures. In particular, we highlight that the class of V@R-based risk measures allows for a substantial reduction of the total capital requirement in corporate networks that share risks between entities. We provide case studies that complement previous theoretical results and demonstrate their practical relevance. For large networks, optimal asset–liability management is often contrary to those strategies that are desirable from a regulatory point of view.
Journal Article
Procyclical Leverage and Value-at-Risk
The availability of credit varies over the business cycle through shifts in the leverage of financial intermediaries. Empirically, we find that intermediary leverage is negatively aligned with the banks' Value-at-Risk (VaR). Motivated by the evidence, we explore a contracting model that captures the observed features. Under general conditions on the outcome distribution given by extreme value theory (EVT), intermediaries maintain a constant probability of default to shifts in the outcome distribution, implying substantial deleveraging during downturns. For some parameter values, we can solve the model explicitly, thereby endogenizing the VaR threshold probability from the contracting problem.
Journal Article