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This paper presents an approach to shortest path minimization for graphs with random weights of arcs. To deal with uncertainty we use the following risk measures: Probability of Exceedance (POE), Buffered Probability of Exceedance (bPOE), Value-at-Risk (VaR), and Conditional Value-at-Risk (CVaR). Minimization problems with POE and VaR objectives result in mixed integer linear problems (MILP) with two types of binary variables. The first type models path, and the second type calculates POE and VaR functions. Formulations with bPOE and CVaR objectives have only the first type binary variables. The bPOE and CVaR minimization problems have a smaller number of binary variables and therefore can be solved faster than problems with POE or VaR objectives. The paper suggested a heuristic algorithm for minimizing bPOE by solving several CVaR minimization problems. Case study (posted at web) numerically compares optimization times with considered risk functions.
Jeremy D. Jordan; Stan Uryasev. Shortest path network problems with stochastic arc weights. Optimization Letters 2021, 1 -20.
AMA StyleJeremy D. Jordan, Stan Uryasev. Shortest path network problems with stochastic arc weights. Optimization Letters. 2021; ():1-20.
Chicago/Turabian StyleJeremy D. Jordan; Stan Uryasev. 2021. "Shortest path network problems with stochastic arc weights." Optimization Letters , no. : 1-20.
Systemic risk is the risk that the distress of one or more institutions trigger a collapse of the entire financial system. We extend CoVaR (value-at-risk conditioned on an institution) and CoCVaR (conditional value-at-risk conditioned on an institution) systemic risk contribution measures and propose a new CoCDaR (conditional drawdown-at-risk conditioned on an institution) measure based on drawdowns. This new measure accounts for consecutive negative returns of a security, while CoVaR and CoCVaR combine together negative returns from different time periods. For instance, ten 2% consecutive losses resulting in 20% drawdown will be noticed by CoCDaR, while CoVaR and CoCVaR are not sensitive to relatively small one period losses. The proposed measure provides insights for systemic risks under extreme stresses related to drawdowns. CoCDaR and its multivariate version, mCoCDaR, estimate an impact on big cumulative losses of the entire financial system caused by an individual firm’s distress. It can be used for ranking individual systemic risk contributions of financial institutions (banks). CoCDaR and mCoCDaR are computed with CVaR regression of drawdowns. Moreover, mCoCDaR can be used to estimate drawdowns of a security as a function of some other factors. For instance, we show how to perform fund drawdown style classification depending on drawdowns of indices. Case study results, data, and codes are posted on the web.
Rui Ding; Stan Uryasev. CoCDaR and mCoCDaR: New Approach for Measurement of Systemic Risk Contributions. Journal of Risk and Financial Management 2020, 13, 270 .
AMA StyleRui Ding, Stan Uryasev. CoCDaR and mCoCDaR: New Approach for Measurement of Systemic Risk Contributions. Journal of Risk and Financial Management. 2020; 13 (11):270.
Chicago/Turabian StyleRui Ding; Stan Uryasev. 2020. "CoCDaR and mCoCDaR: New Approach for Measurement of Systemic Risk Contributions." Journal of Risk and Financial Management 13, no. 11: 270.
We propose an approach to solving the problem of optimizing the reliability of complex systems using Buffered Probability of Exceedance (bPOE). As a research subject, we consider the model of optimal control of oscillations of a hinged beam with random defects. This example shows that minimizing bPOE in reliability optimization problems is more preferable than minimizing the classical probability of exceedance.
G. M. Zrazhevsky; A. N. Golodnikov; S. P. Uryasev; A. G. Zrazhevsky. Application of Buffered Probability of Exceedance in Reliability Optimization Problems*. Cybernetics and Systems Analysis 2020, 56, 476 -484.
AMA StyleG. M. Zrazhevsky, A. N. Golodnikov, S. P. Uryasev, A. G. Zrazhevsky. Application of Buffered Probability of Exceedance in Reliability Optimization Problems*. Cybernetics and Systems Analysis. 2020; 56 (3):476-484.
Chicago/Turabian StyleG. M. Zrazhevsky; A. N. Golodnikov; S. P. Uryasev; A. G. Zrazhevsky. 2020. "Application of Buffered Probability of Exceedance in Reliability Optimization Problems*." Cybernetics and Systems Analysis 56, no. 3: 476-484.
We consider the problem of finding checkerboard copulas for modeling multivariate distributions. A checkerboard copula is a distribution with a corresponding density defined almost everywhere by a step function on an m-uniform subdivision of the unit hyper-cube. We develop optimization procedures for finding copulas defined by multiply-stochastic matrices matching available information. Two types of information are used for building copulas: 1) Spearman Rho rank correlation coefficients; 2) Empirical distributions of sums of random variables combined with empirical marginal probability distributions. To construct checkerboard copulas we solved optimization problems. The first problem maximizes entropy with constraints on Spearman Rho coefficients. The second problem minimizes some error function to match available data. We conducted a case study illustrating the application of the developed methodology using property and casualty insurance data. The optimization problems were numerically solved with the AORDA Portfolio Safeguard (PSG) package, which has precoded entropy and error functions. Case study data, codes, and results are posted at the web.
Viktor Kuzmenko; Romel Salam; Stan Uryasev. Checkerboard copula defined by sums of random variables. Dependence Modeling 2020, 8, 70 -92.
AMA StyleViktor Kuzmenko, Romel Salam, Stan Uryasev. Checkerboard copula defined by sums of random variables. Dependence Modeling. 2020; 8 (1):70-92.
Chicago/Turabian StyleViktor Kuzmenko; Romel Salam; Stan Uryasev. 2020. "Checkerboard copula defined by sums of random variables." Dependence Modeling 8, no. 1: 70-92.
Standard methods of fitting finite mixture models take into account the majority of observations in the center of the distribution. This paper considers the case where the decision maker wants to make sure that the tail of the fitted distribution is at least as heavy as the tail of the empirical distribution. For instance, in nuclear engineering, where probability of exceedance (POE) needs to be estimated, it is important to fit correctly tails of the distributions. The goal of this paper is to supplement the standard methodology and to assure an appropriate heaviness of the fitted tails. We consider a new Conditional Value-at-Risk (CVaR) distance between distributions, that is a convex function with respect to weights of the mixture. We have conducted a case study demonstrating e˚ciency of the approach. Weights of mixture are found by minimizing CVaR distance between the mixture and the empirical distribution. We have suggested convex constraints on weights, assuring that the tail of the mixture is as heavy as the tail of empirical distribution.
Giorgi Pertaia; Stan Uryasev. Fitting heavy-tailed mixture models with CVaR constraints. Dependence Modeling 2019, 7, 365 -374.
AMA StyleGiorgi Pertaia, Stan Uryasev. Fitting heavy-tailed mixture models with CVaR constraints. Dependence Modeling. 2019; 7 (1):365-374.
Chicago/Turabian StyleGiorgi Pertaia; Stan Uryasev. 2019. "Fitting heavy-tailed mixture models with CVaR constraints." Dependence Modeling 7, no. 1: 365-374.
The paper considers optimization algorithms for location planning, which specifies positions of facilities providing demanded services. Examples of facilities include hospitals, restaurants, ambulances, retail and grocery stores, schools, and fire stations. We reduced the initial problem to approximation of a discrete distribution with a large number of atoms by some other discrete distribution with a smaller number of atoms. The approximation is done by minimizing the Kantorovich–Rubinstein distance between distributions. Positions and probabilities of atoms of the approximating distribution are optimized. The algorithm solves a sequence of optimization problems reducing the distance between distributions. We conducted a case study using Portfolio Safeguard (PSG) optimization package in MATLAB environment.
Viktor Kuzmenko; Stan Uryasev. Kantorovich–Rubinstein Distance Minimization: Application to Location Problems. Dynamics of Disasters 2019, 59 -68.
AMA StyleViktor Kuzmenko, Stan Uryasev. Kantorovich–Rubinstein Distance Minimization: Application to Location Problems. Dynamics of Disasters. 2019; ():59-68.
Chicago/Turabian StyleViktor Kuzmenko; Stan Uryasev. 2019. "Kantorovich–Rubinstein Distance Minimization: Application to Location Problems." Dynamics of Disasters , no. : 59-68.
A popular risk measure, conditional value-at-risk (CVaR), is called expected shortfall (ES) in financial applications. The research presented involved developing algorithms for the implementation of linear regression for estimating CVaR as a function of some factors. Such regression is called CVaR (superquantile) regression. The main statement of this paper is: CVaR linear regression can be reduced to minimizing the Rockafellar error function with linear programming. The theoretical basis for the analysis is established with the quadrangle theory of risk functions. We derived relationships between elements of CVaR quadrangle and mixed-quantile quadrangle for discrete distributions with equally probable atoms. The deviation in the CVaR quadrangle is an integral. We present two equivalent variants of discretization of this integral, which resulted in two sets of parameters for the mixed-quantile quadrangle. For the first set of parameters, the minimization of error from the CVaR quadrangle is equivalent to the minimization of the Rockafellar error from the mixed-quantile quadrangle. Alternatively, a two-stage procedure based on the decomposition theorem can be used for CVaR linear regression with both sets of parameters. This procedure is valid because the deviation in the mixed-quantile quadrangle (called mixed CVaR deviation) coincides with the deviation in the CVaR quadrangle for both sets of parameters. We illustrated theoretical results with a case study demonstrating the numerical efficiency of the suggested approach. The case study codes, data, and results are posted on the website. The case study was done with the Portfolio Safeguard (PSG) optimization package, which has precoded risk, deviation, and error functions for the considered quadrangles.
Alex Golodnikov; Viktor Kuzmenko; Stan Uryasev. CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles. Journal of Risk and Financial Management 2019, 12, 107 .
AMA StyleAlex Golodnikov, Viktor Kuzmenko, Stan Uryasev. CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles. Journal of Risk and Financial Management. 2019; 12 (3):107.
Chicago/Turabian StyleAlex Golodnikov; Viktor Kuzmenko; Stan Uryasev. 2019. "CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles." Journal of Risk and Financial Management 12, no. 3: 107.