The original motivation of this work comes from a classic problem in finance and insurance: that of computing the value-at-risk (VaR) of a portfolio of dependent risky positions, i.e. the quantile at a certain level of confidence of the loss distribution. In fact, it is difficult to overestimate the importance of the concept of VaR in modern finance and insurance: it has been recommended, although with several warnings, as a measure of risk and the basis for capital requirement determination both by the guidelines of international committees (such as Basel 2 and 3, Solvency 2 etc.) and the internal models adopted by major banks and insurance companies. However the actual computation of the VaR of a portfolio constituted by several dependent risky assets is often a hard practical and theoretical task. To this purpose here we prove the convergence of a geometric algorithm (alternative to Monte Carlo and quasi Monte Carlo methods) for computing the value-at-risk of a portfolio of any dimension, i.e. the distribution of the sum of its components, which can exhibit any dependence structure. Moreover, although the original motivation is financial, our result has a relevant measure-theoretical meaning. What we prove, in fact, is that the H-measure of a d-dimensional simplex (for any d≥2 and any absolutely continuous with respect to Lebesgue measure H) can be approximated by convergent algebraic sums of H-measures of hypercubes (obtained through a self-similar construction).

`http://hdl.handle.net/2158/986206`

Titolo: | Computing the distribution of the sum of dependent random variables via overlapping hypercubes |

Autori interni: | GALEOTTI, MARCELLO |

Data di pubblicazione: | 2015 |

Rivista: | DECISIONS IN ECONOMICS AND FINANCE |

Abstract: | The original motivation of this work comes from a classic problem in finance and insurance: that of computing the value-at-risk (VaR) of a portfolio of dependent risky positions, i.e. the quantile at a certain level of confidence of the loss distribution. In fact, it is difficult to overestimate the importance of the concept of VaR in modern finance and insurance: it has been recommended, although with several warnings, as a measure of risk and the basis for capital requirement determination both by the guidelines of international committees (such as Basel 2 and 3, Solvency 2 etc.) and the internal models adopted by major banks and insurance companies. However the actual computation of the VaR of a portfolio constituted by several dependent risky assets is often a hard practical and theoretical task. To this purpose here we prove the convergence of a geometric algorithm (alternative to Monte Carlo and quasi Monte Carlo methods) for computing the value-at-risk of a portfolio of any dimension, i.e. the distribution of the sum of its components, which can exhibit any dependence structure. Moreover, although the original motivation is financial, our result has a relevant measure-theoretical meaning. What we prove, in fact, is that the H-measure of a d-dimensional simplex (for any d≥2 and any absolutely continuous with respect to Lebesgue measure H) can be approximated by convergent algebraic sums of H-measures of hypercubes (obtained through a self-similar construction). |

Handle: | http://hdl.handle.net/2158/986206 |

Appare nelle tipologie: | 1a - Articolo su rivista |

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