Person:
Balbás Aparicio, Raquel

First Name

Raquel

Last Name

Balbás Aparicio

URI

https://hdl.handle.net/20.500.14352/74932

Affiliation

Universidad Complutense de Madrid

Faculty / Institute

Ciencias Económicas y Empresariales

Department

Economía Financiera, Actuarial y Estadística

Area

Economía Financiera y Contabilidad

Identifiers

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Now showing 1 - 9 of 9

VaR as the CVaR sensitivity: Applications in risk optimization
(Journal of Computational and Applied Mathematics, 2016) Balbás, Alejandro; Balbás, Beatriz; Balbás Aparicio, Raquel
VaR minimization is a complex problem playing a critical role in many actuarial and financial applications of mathematical programming. The usual methods of convex programming do not apply due to the lack of sub-additivity. The usual methods of differentiable programming do not apply either, due to the lack of continuity. Taking into account that the CVaR may be given as an integral of VaR, one has that VaR becomes a first order mathematical derivative of CVaR. This property will enable us to give accurate approximations in VaR optimization, since the optimization VaR and CVaR will become quite closely related topics. Applications in both finance and insurance will be given.
Bidual representation of expectiles
(Risks, 2023) Balbás De La Corte, Alejandro; Balbás, Beatriz; Balbás Aparicio, Raquel; Charron, Jean-Philippe
Downside risk measures play a very interesting role in risk management problems. In particular, the value at risk (VaR) and the conditional value at risk (CVaR) have become very important instruments to address problems such as risk optimization, capital requirements, portfolio selection, pricing and hedging issues, risk transference, risk sharing, etc. In contrast, expectile risk measures are not as widely used, even though they are both coherent and elicitable. This paper addresses the bidual representation of expectiles in order to prove further important properties of these risk measures. Indeed, the bidual representation of expectiles enables us to estimate and optimize them by linear programming methods, deal with optimization problems involving expectile-linked constraints, relate expectiles with VaR and CVaR by means of both equalities and inequalities, give VaR and CVaR hyperbolic upper bounds beyond the level of confidence, and analyze whether co-monotonic additivity holds for expectiles. Illustrative applications are presented.
Differential equations connecting VaR and CVaR
(Journal of Computational and Applied Mathematics, 2017) Balbás, Alejandro; Balbás, Beatriz; Balbás Aparicio, Raquel
The Value at Risk (VaR) is a very important risk measure for practitioners, supervisors and researchers. Many practitioners draw on VaR as a critical instrument in Risk Management and other Actuarial/Financial problems, while supervisors and regulators must deal with VaR due to the Basel Accords and Solvency II, among other reasons. From a theoretical point of view VaR presents some drawbacks overcome by other risk measures such as the Conditional Value at Risk (CVaR). VaR is neither di¤erentiable nor sub-additive because it is neither continuous nor convex. On the contrary, CVaR satisfies all of these properties, and this simplifies many analytical studies if VaR is replaced by CVaR. In this paper several di¤erential equations connecting both VaR and CVaR will be presented. They will allow us to address several important issues involving VaR with the help of the CVaR properties. This new methodology seems to be very e¢ cient. In particular, a new VaR Representation Theorem may be found, and optimization problems involving VaR or probabilistic constraints always have an equivalent di¤erentiable optimization problem. Applications in VaR, marginal VaR, CVaR and marginal CVaR estimates will be addressed as well. An illustrative actuarial numerical example will be given.
Good deals and benchmarks in robust portfolio selection
(European Journal of Operational Research, 2016) Balbás, Alejandro; Balbás, Beatriz; Balbás Aparicio, Raquel
This paper deals with portfolio selection problems under risk and ambiguity. The investor may be ambiguouswith respect to the set of states of nature and their probabilities. Both static and discrete or continuous timedynamic pricing models are included in the analysis. Risk and ambiguity are measured in general settings. Theconsidered risk measures contain, as particular cases, the usual deviations and the coherent and expectationbounded measures of risk.Four contributions seem to be reached. Firstly, necessary and sufficient optimality conditions are given. Sec-ondly, the portfolio selection problem may be frequently solved by linear programming linked methods, de-spite the fact that risk and ambiguity cannot be given by linear expressions. Thirdly, if there is a market priceof risk then there exists a benchmark that creates a robust capital market line when combined with the risk-less asset. The global risk of every portfolio may be divided into systemic and specific. Moreover, if there is noambiguity with respect to the states of nature (only their probabilities are uncertain), then classical CAPM-formulae may be found. Fourthly, some recent pathological findings for ambiguity-free analyses also apply inambiguous frameworks. In particular, there may exist arbitrage free markets such that the ambiguous agentcan guarantee every expected return with a maximum risk bounded from above by zero,i.e., the capitalmarket line (risk, return) becomes vertical. For instance, in the (non-ambiguous) Black and Scholes modelthis property holds for important risk measures such as the absolute deviation or the CVaR. Nevertheless, inambiguous settings, adequate increments of the ambiguity level will allow us to recover capital market linesconsistent with the empirical evidence. The introduction of ambiguity may overcome several caveats of manyimportant pricing models.
Optimal reinsurance under risk and uncertainty
(Insurance: Mathematics and Economics, 2014) Balbás, Alejandro; Balbás, Beatriz; Balbás Aparicio, Raquel; Heras Martínez, Antonio José
This paper deals with the optimal reinsurance problem if both insurer and reinsurer are facing risk and uncertainty, though the classical uncertainty free case is also included. The insurer and reinsurer degrees of uncertainty do not have to be identical. The decision variable is not the retained (or ceded) risk, but its sensitivity with respect to the total claims. Thus, if one imposes strictly positive lower bounds for this variable, the reinsurer moral hazard is totally eliminated. Three main contributions seem to be reached. Firstly, necessary and sufficient optimality conditions are given in a very general setting. Secondly, the optimal contract is often a bang–bang solution, i.e., the sensitivity between the retained risk and the total claims saturates the imposed constraints. Thirdly, the optimal reinsurance problem is equivalent to other linear programming problem, despite the fact that risk, uncertainty, and many premium principles are not linear. This may be important because linear problems may be easily solved in practice, since there are very efficient algorithms.
Project number: 89
Las matemáticas empresariales en el marco de cualificaciones del Espacio Europeo de Educación Superior (QF-EHEA)
(2016) García Pineda, María Pilar; Heras Martínez, Antonio; Blanco García, Susana; Balbás Aparicio, Raquel; Rebollo Castillo, Francisco Javier
Haremos comparaciones entre las metodologías docentes de las asignaturas de Matemáticas Empresariales impartidas en las principales universidades europeas, proponiendo medidas para la mejora de la calidad de estas asignaturas en nuestra universidad.
Bidual representation of expectiles
(Risks, 2023) Balbás De La Corte, Alejandro; Balbás, Beatriz; Balbás Aparicio, Raquel; Charron, Jean-Philippe
Downside risk measures play a very interesting role in risk management problems. In particular, the value at risk (VaR) and the conditional value at risk (CVaR) have become very important instruments to address problems such as risk optimization, capital requirements, portfolio selection, pricing and hedging issues, risk transference, risk sharing, etc. In contrast, expectile risk measures are not as widely used, even though they are both coherent and elicitable. This paper addresses the bidual representation of expectiles in order to prove further important properties of these risk measures. Indeed, the bidual representation of expectiles enables us to estimate and optimize them by linear programming methods, deal with optimization problems involving expectile-linked constraints, relate expectiles with VaR and CVaR by means of both equalities and inequalities, give VaR and CVaR hyperbolic upper bounds beyond the level of confidence, and analyze whether co-monotonic additivity holds for expectiles. Illustrative applications are presented.
Optimal Reinsurance: A Risk Sharing Approach
(Risks, 2013) Balbás, Alejandro; Balbás, Beatriz; Balbás Aparicio, Raquel
This paper proposes risk sharing strategies, which allow insurers to cooperate and diversify non-systemic risk. We deal with both deviation measures and coherent risk measures and provide general mathematical methods applying to optimize them all. Numerical examples are given in order to illustrate how efficiently the non-systemic risk can be diversified and how effective the presented mathematical tools may be. It is also illustrated how the existence of huge disasters may lead to wrong solutions of our optimal risk sharing problem, in the sense that the involved risk measure could ignore the existence of a non-null probability of “global ruin” after the design of the optimal risk sharing strategy. To overcome this caveat, one can use more conservative risk measures. The stability in the large of the optimal sharing plan guarantees that “the global ruin caveat” may be also addressed and solved with the presented methods.
Project number: 104
Las Matemáticas Empresariales en el marco Erasmus Mundus
(2017) García Pineda, María Pilar; Heras Martínez, Antonio José; Blanco García, Susana; Balbás Aparicio, Raquel; García Villalba, Luis Javier; Riomoros Callejo, María Isabel; Portela García-Miguel, Javier; Sandoval Orozco, Ana Lucila; Rebollo Castillo, Francisco Javier
La creciente importancia de los métodos cuantitativos en las ciencias económicas y empresariales nos motiva a proponer una revisión detallada de los syllabus de las materias de matemáticas que se imparten en el Grado de Administración y Dirección de Empresas, con el objetivo de Investigar las correspondencias entre nuestros syllabus y los de las mas importantes universidades a nivel internacional (en el marco Erasmus Mundus). La investigación que proponemos llevará a cabo comparaciones exhaustivas de los temarios de esta categoría de asignaturas y sus metodologías docentes, y detectará las posibles discrepancias existentes en este tipo de estudios dependiendo de la universidad que los imparte. En una segunda fase, estudiaremos las causas de las posibles diferencias detectadas y, finalmente, produciremos un sistema capaz de sugerir medidas concretas que solventen los posibles problemas detectados.