The Impact of Correlation on (Range) Value-at-Risk

42 Pages Posted: 14 Dec 2021

See all articles by Carole Bernard

Carole Bernard

Grenoble Ecole de Management; Vrije Universiteit Brussel (VUB)

Corrado De Vecchi

Vrije Universiteit Brussel (VUB)

Steven Vanduffel

Vrije Universiteit Brussel (VUB)

Date Written: October 17, 2021

Abstract

The assessment of portfolio risk is often explicitly (e.g., the square root formula under Basel III) or implicitly (e.g., credit risk portfolio models) driven by the marginal distributions of the risky components and the correlations amongst them. We assess the extent by which such practice is prone to model error. In the case of a sum of n = 2 risks, we investigate under which conditions the unconstrained Value-at-Risk (VaR) bounds (which are the maximum and minimum VaR for S = X_1+X+2+...+X_n when only the marginal distributions of the X_i are known) coincide with the (constrained) VaR bounds when in addition one has information on some measure of dependence (e.g., Pearson correlation or Spearman's rho). We find that both bounds coincide if the measure of dependence takes value in an interval that we explicitly determine. For probability levels used in risk management, we show that using correlation information has typically no effect on the highest possible VaR whereas it can affect the lowest possible VaR. In the case of a general sum of two or more risks, we derive Range Value-at-Risk (RVaR) bounds under an average correlation constraint (in addition to the knowledge of the marginal distributions). While these bounds are not best-possible in general, we show that they are in the case of a sum of three or more standard uniformly distributed risks. As far as we know, this result is the first that provides an explicit best-possible bound on RVaR for a general sum of three or more risks (uniformly distributed) under a correlation constraint.

Keywords: Risk bounds, Value-at-Risk, Pearson correlation, Spearman's rho, Kendall's tau

Suggested Citation

Bernard, Carole and De Vecchi, Corrado and Vanduffel, Steven, The Impact of Correlation on (Range) Value-at-Risk (October 17, 2021). Available at SSRN: https://ssrn.com/abstract=3944648 or http://dx.doi.org/10.2139/ssrn.3944648

Carole Bernard

Grenoble Ecole de Management ( email )

12, rue Pierre Sémard
Grenoble Cedex, 38003
France

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
http://www.vub.ac.be/
Brussels, 1050
Belgium

Corrado De Vecchi (Contact Author)

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
http://www.vub.ac.be/
Brussels, 1050
Belgium

Steven Vanduffel

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, Brabant 1050
Belgium

HOME PAGE: http://www.stevenvanduffel.com

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
183
Abstract Views
1,000
Rank
329,392
PlumX Metrics