Estimation of the Marginal Expected Shortfall

The Mean when a Related Variable is Extreme

J. Cai, J.H.J. Einmahl, L.F.M. de Haan, C. Zhou

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Abstract

Abstract: Denote the loss return on the equity of a financial institution as X and that of the entire market as Y . For a given very small value of p > 0, the marginal expected shortfall (MES) is defined as E(X | Y > QY (1−p)), where QY (1−p) is the (1−p)-th quantile of the distribution of Y . The MES is an important factor when measuring the systemic risk of financial institutions. For a wide nonparametric class of bivariate distributions, we construct an estimator of the MES and establish the asymptotic normality of the estimator when p ↓ 0, as the sample size n → ∞. Since we are in particular interested in the case p = O(1=n), we use extreme value techniques for deriving the estimator and its asymptotic behavior. The finite sample performance of the estimator and the adequacy of the limit theorem are shown in a detailed simulation study. We also apply our method to estimate the MES of three large U.S. investment banks.
Original languageEnglish
Place of PublicationTilburg
PublisherEconometrics
Number of pages28
Volume2012-080
Publication statusPublished - 2012

Publication series

NameCentER Discussion Paper
Volume2012-080

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Expected Shortfall
Extremes
Estimator
Bivariate Distribution
Equity
Extreme Values
Quantile
Limit Theorems
Asymptotic Normality
Sample Size
Asymptotic Behavior
Simulation Study
Entire
Denote
Estimate

Keywords

  • Asymptotic normality
  • extreme values
  • tail dependence

Cite this

Cai, J., Einmahl, J. H. J., de Haan, L. F. M., & Zhou, C. (2012). Estimation of the Marginal Expected Shortfall: The Mean when a Related Variable is Extreme. (CentER Discussion Paper; Vol. 2012-080). Tilburg: Econometrics.
Cai, J. ; Einmahl, J.H.J. ; de Haan, L.F.M. ; Zhou, C. / Estimation of the Marginal Expected Shortfall : The Mean when a Related Variable is Extreme. Tilburg : Econometrics, 2012. (CentER Discussion Paper).
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abstract = "Abstract: Denote the loss return on the equity of a financial institution as X and that of the entire market as Y . For a given very small value of p > 0, the marginal expected shortfall (MES) is defined as E(X | Y > QY (1−p)), where QY (1−p) is the (1−p)-th quantile of the distribution of Y . The MES is an important factor when measuring the systemic risk of financial institutions. For a wide nonparametric class of bivariate distributions, we construct an estimator of the MES and establish the asymptotic normality of the estimator when p ↓ 0, as the sample size n → ∞. Since we are in particular interested in the case p = O(1=n), we use extreme value techniques for deriving the estimator and its asymptotic behavior. The finite sample performance of the estimator and the adequacy of the limit theorem are shown in a detailed simulation study. We also apply our method to estimate the MES of three large U.S. investment banks.",
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Cai, J, Einmahl, JHJ, de Haan, LFM & Zhou, C 2012 'Estimation of the Marginal Expected Shortfall: The Mean when a Related Variable is Extreme' CentER Discussion Paper, vol. 2012-080, Econometrics, Tilburg.

Estimation of the Marginal Expected Shortfall : The Mean when a Related Variable is Extreme. / Cai, J.; Einmahl, J.H.J.; de Haan, L.F.M.; Zhou, C.

Tilburg : Econometrics, 2012. (CentER Discussion Paper; Vol. 2012-080).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Estimation of the Marginal Expected Shortfall

T2 - The Mean when a Related Variable is Extreme

AU - Cai, J.

AU - Einmahl, J.H.J.

AU - de Haan, L.F.M.

AU - Zhou, C.

N1 - Pagination: 28

PY - 2012

Y1 - 2012

N2 - Abstract: Denote the loss return on the equity of a financial institution as X and that of the entire market as Y . For a given very small value of p > 0, the marginal expected shortfall (MES) is defined as E(X | Y > QY (1−p)), where QY (1−p) is the (1−p)-th quantile of the distribution of Y . The MES is an important factor when measuring the systemic risk of financial institutions. For a wide nonparametric class of bivariate distributions, we construct an estimator of the MES and establish the asymptotic normality of the estimator when p ↓ 0, as the sample size n → ∞. Since we are in particular interested in the case p = O(1=n), we use extreme value techniques for deriving the estimator and its asymptotic behavior. The finite sample performance of the estimator and the adequacy of the limit theorem are shown in a detailed simulation study. We also apply our method to estimate the MES of three large U.S. investment banks.

AB - Abstract: Denote the loss return on the equity of a financial institution as X and that of the entire market as Y . For a given very small value of p > 0, the marginal expected shortfall (MES) is defined as E(X | Y > QY (1−p)), where QY (1−p) is the (1−p)-th quantile of the distribution of Y . The MES is an important factor when measuring the systemic risk of financial institutions. For a wide nonparametric class of bivariate distributions, we construct an estimator of the MES and establish the asymptotic normality of the estimator when p ↓ 0, as the sample size n → ∞. Since we are in particular interested in the case p = O(1=n), we use extreme value techniques for deriving the estimator and its asymptotic behavior. The finite sample performance of the estimator and the adequacy of the limit theorem are shown in a detailed simulation study. We also apply our method to estimate the MES of three large U.S. investment banks.

KW - Asymptotic normality

KW - extreme values

KW - tail dependence

M3 - Discussion paper

VL - 2012-080

T3 - CentER Discussion Paper

BT - Estimation of the Marginal Expected Shortfall

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CY - Tilburg

ER -

Cai J, Einmahl JHJ, de Haan LFM, Zhou C. Estimation of the Marginal Expected Shortfall: The Mean when a Related Variable is Extreme. Tilburg: Econometrics. 2012. (CentER Discussion Paper).