Next Article in Journal
Modified Munich Chain-Ladder Method
Previous Article in Journal
Information-Based Trade in German Real Estate and Equity Markets
Article Menu

Export Article

Open AccessArticle
Risks 2015, 3(4), 599-623;

Dependence Uncertainty Bounds for the Expectile of a Portfolio

RiskLab, Department of Mathematics, ETH Zurich, 8092 Zürich, Switzerland
Faculty of Economics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Bruxelles, Belgium
Author to whom correspondence should be addressed.
Academic Editor: Andrea Consiglio
Received: 4 August 2015 / Accepted: 4 December 2015 / Published: 10 December 2015
Full-Text   |   PDF [372 KB, uploaded 10 December 2015]   |  


We study upper and lower bounds on the expectile risk measure of risky portfolios when the joint distribution of the risky components is not fully specified. First, we summarize methods for obtaining bounds when only the marginal distributions of the components are known, but not their interdependence (unconstrained bounds). In particular, we provide the best-possible upper bound and the best-possible lower bound (under some conditions), as well as numerical procedures to compute them. We also derive simple analytic bounds that appear adequate in various situations of interest. Second, we study bounds when some information on interdependence is available (constrained bounds). When the variance of the portfolio is known, a simple-to-compute upper bound is provided, and we illustrate that it may significantly improve the unconstrained upper bound. We also show that the unconstrained lower bound cannot be readily improved using variance information. Next, we derive improved bounds when the bivariate distributions of each of the risky components and a risk factor are known. When the factor induces a positive dependence among the components, it is typically possible to improve the unconstrained lower bound. Finally, the unconstrained dependence uncertainty spreads of expected shortfall, value-at-risk and the expectile are compared. View Full-Text
Keywords: expectiles; convex order; elicitability; coherence; dependence expectiles; convex order; elicitability; coherence; dependence

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

Share & Cite This Article

MDPI and ACS Style

Jakobsons, E.; Vanduffel, S. Dependence Uncertainty Bounds for the Expectile of a Portfolio. Risks 2015, 3, 599-623.

Show more citation formats Show less citations formats

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Risks EISSN 2227-9091 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top