Distributional Bounds for Portfolio Risk with Tail Dependence

Kunio So, Junichi Imai

Research output: Contribution to journalArticlepeer-review


The present paper proposes a new method for estimating portfolio risk by applying the concept of bounds to a dependence structure. We introduce four tail dependence measures as partial dependence information and derive bounds on the distribution of a non-decreasing function to obtain bounds on risk measures. We show that bounds on risk measures can be tightened significantly in the probability levels with which we are concerned, those for financial risk management. In the present paper, we provide theorems describing the distributional bounds of the proposed method and prove that these bounds are pointwise best-possible bounds. Furthermore, we calculate risk measures, i.e., value at risk and expected shortfall, from empirical return data and compare the effectiveness of the proposed model with that of typical parametric copula models.

Original languageEnglish
Pages (from-to)795-816
Number of pages22
JournalMethodology and Computing in Applied Probability
Issue number3
Publication statusPublished - 2014 Feb 14


  • Copulas
  • Fréchet bounds
  • Market risks
  • Risk management
  • Tail dependence

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability


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