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Mathematics and Financial Economics
Published in: Mathematics and Financial Economics 4/2021

30-04-2021

Risk management with expected shortfall

Author: Pengyu Wei

Published in: Mathematics and Financial Economics | Issue 4/2021

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Abstract

This article studies optimal, dynamic portfolio and wealth/consumption policies of expected utility-maximizing investors who must also manage market-risk exposure which is measured by expected shortfall (ES). We find that ES managers can incur larger losses when losses occur, compared to benchmark managers. A general-equilibrium analysis reveals that the presence of ES managers increases the market volatility during periods of significant financial market stress. We propose weighted shortfall, a coherent and moreover spectral risk measure, that can rectify the shortcomings of ES.
previous article Diffusion bank networks and capital flows
next article Supermartingale deflators in the absence of a numéraire

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Appendix
Available only for authorised users
Footnotes
1
WS is a spectral risk measure (see [1]) with the “risk spectrum” \({F_{\xi } ^{-1}(1-z)}/{\int _0^{\alpha } F_{\xi } ^{-1}(1-s)ds}.\) WS posses several desirable properties, such as monotonicity, translation-invariance, positive homogeneity, sub-additivity, law-invariance, and comonotonic additivity. In particular, it is a coherent risk measure.
 
2
With a slight abuse of notation, we use \(i=1\) to denote the first agent who is a benchmark agent in all three economies and \(i=2\) to denote the second agent who is a benchmark agent in the benchmark economy, an ES agent in the ES economy, and a WS agent in the WS economy.
 
3
From now on we denote by \(F_{\xi }^{-1}\) the quantile function of \(\xi (T-)\).
 
4
If there are multiple stocks, the equity market value will be defined as the sum of values of all stocks.
 
5
For comparison purpose, we additionally consider the VaR economy in [7] but with an infinite time horizon.
 
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Metadata
Title
Risk management with expected shortfall
Author
Pengyu Wei
Publication date
30-04-2021
Publisher
Springer Berlin Heidelberg
Published in
Mathematics and Financial Economics / Issue 4/2021
Print ISSN: 1862-9679
Electronic ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-021-00298-x

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