We extend the classical risk minimization model with scalar risk measures to the general case of set-valued risk measures. The problem we obtain is a set-valued optimization model and we propose a goal programming-based approach with satisfaction function to obtain a solution which represents the best compromise between goals and the achievement levels. Numerical examples are provided to illustrate how the method works in practical situations.
We present a general framework for portfolio risk management in discrete time, based on a replicating martingale. This martingale is learned from a finite sample in a supervised setting. The model learns the features necessary for an effective low-dimensional representation, overcoming the curse of dimensionality common to function approximation in high-dimensional spaces. We show results based on polynomial and neural network bases. Both offer superior results to naive Monte Carlo methods and other existing methods like least-squares Monte Carlo and replicating portfolios.
When applying Value at Risk (VaR) procedures to specific positions or portfolios, we often focus on developing procedures only for the specific assets in the portfolio. However, since this small portfolio risk analysis ignores information from assets outside the target portfolio, there may be significant information loss. In this paper, we develop a dynamic process to incorporate the ignored information. We also study how to overcome the curse of dimensionality and discuss where and when benefits occur from a large number of assets, which is called the blessing of dimensionality. We find empirical support for the proposed method.
Using particle system methodologies we study the propagation of financial distress in a network of firms facing credit risk. We investigate the phenomenon of a credit crisis and quantify the losses that a bank may suffer in a large credit portfolio. Applying a large deviation principle we compute the limiting distributions of the system and determine the time evolution of the credit quality indicators of the firms, deriving moreover the dynamics of a global financial health indicator. We finally describe a suitable version of the Central Limit Theorem useful to study large portfolio losses. Simulation results are provided as well as applications to portfolio loss distribution analysis.
We propose a generalization of the classical notion of the $V@R_{lambda}$ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The $V@R_{lambda}$ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on $mathcal{P}(% mathbb{R}).$
In this paper, we examine the effect of background risk on portfolio selection and optimal reinsurance design under the criterion of maximizing the probability of reaching a goal. Following the literature, we adopt dependence uncertainty to model the dependence ambiguity between financial risk (or insurable risk) and background risk. Because the goal-reaching objective function is non-concave, these two problems bring highly unconventional and challenging issues for which classical optimization techniques often fail. Using quantile formulation method, we derive the optimal solutions explicitly. The results show that the presence of background risk does not alter the shape of the solution but instead changes the parameter value of the solution. Finally, numerical examples are given to illustrate the results and verify the robustness of our solutions.
Davide La Torre
,Marco Maggis
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(2012)
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"A Goal Programming Model with Satisfaction Function for Risk Management and Optimal Portfolio Diversification"
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Marco Maggis Doctor
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