Distributional Bounds for Portfolio Risk with Tail Dependence

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.