Read the article of Armstrong & Brigo (2019) attached to this assignment.

https://api.semanticscholar.org/CorpusID:159194532

1. Analyze the article in a critical way

2. Explain the concept of value at risk and expected shortfall

3. What are the requirement of Basel 3 and how they differ from Basel 2.

4. How did the authors derive the relationship between VaR and ES on one hand and the utility function from the other hand

Journal of Banking and Finance 101 (2019) 122-135 Contents lists available at ScienceDirect Journal of Banking and Finance journal homepage: www.elsevier.com/locate/jbf ELSEVIER Chacko Risk managing tail-risk seekers: VaR and expected shortfall vs S-shaped utility John Armstrong", Damiano Brigob.* -Department of Mathematics King's College London United Kingdom Department of Mathematics Imperial College London 180 Queen's Gate London SW7 ZAZ Lhited Kingdom ARTICLE INFO Artide history: Received 16 April 2018 Accepted 19 January 2019 Available online 29 January 2019 JEI dassgication: DRI GII G13 ABSTRACT We consider marker players with tail-risk-seeking behaviour modelled by S-shaped utility, as introduced by Kahneman and Tversky. We argue that risk measures such as value at risk (VaR) and expected shordall (ES) are ineffective in constraining such players, as such measures cannot reduce the traders expected S- shaped utilities. Indeed, when designing payoffs aiming to maximize utility under a VaR or ES risk limit, the players will attain the same supremum of expected utility with or without VaR or ES limits. By con- trast, we show that risk management constraints based on a second more conventional concave utility function can reduce the maximum 5-shaped utility that can be achieved by the investor. Indeed, product designs leading to progressively larger 5-shaped utilities will lead to progressively lower expected con- straining conventional utilities, violating the related risk limit. These results hold in a variety of market models, including the Black Scholes options model, and are particularly relevant for risk managers given the historical role of VaR and the endorsement of ES by the Basel committee in 2012-2013. 2019 Elsevier B.V. All rights reserved. Keywords: Optimal product design under risk constraints Value at risk constraints Expected shortfall constraints Concave utility constraints S-Shaped utility maximization Limited liability investors Tail-risk-seeking investors Effective risk constraints Concave utility risk constraints 1. Introduction We consider the effects of imposing risk constraints on a trader in the case when that trader exhibits risk-seeking behaviour over losses or has limited liability, When making investment decisions, a trader is constrained in how much funds they can use in setting up the investment (giving rise to a budget or price constraint). In addition a trader is typ- ically limited on how much risk of future losses the investment may entail, Risk limits are common in banking and represent a tool for risk control. A natural risk limit to adopt is a risk limit based on classic risk measures such as Value at Risk (VaR) and Expected Shortfall (ES). The trader is then constrained to choose an invest- ment whose VaR or ES is below a given limit, set by the bank risk manager when approving the trade, The question we seek to answer in this paper is whether such limits are effective when the trader exhibits tail-risk-seeking be- haviour or has some form of limited liability. We consider a risk limit to be "effective" if it has an economic impact upon the trader, in the sense that imposing the limit decreases the expected util- ities that the trader can achieve. We model the tail-risk-seeking behaviour and/or limited liability by assuming that the utility of the trader is given by an 5-shaped function of the payoff of their investments. We assume that the trader acts to maximize their ex- pected utility, subject to the budget constraint and any risk con- straints. We find that in market models including the benchmark Black-Scholes model, neither VaR nor Es constraints are effective in curbing such risk-seeking behaviour. This result is particularly important in the light of the fact that ES has been officially en- dorsed and suggested as a risk measure by the Basel committee in 2012-2013 (Basel Committee on Banking Supervision, 2013: 2016). partly for its coherent risk measure" properties (Artzner et al., 1999; Acerbi and Tasche, 2002). We are grateful to Doctor Dirk Tasche for helpful suggestions and discussion on the first version. We thank Professor Xunyu Zhou who, during our seminar at Columbia University in NY on November 29, 2017, provided us with a number of references we had missed in our literature review and that have been included in the introduction of this updated version. Corresponding author. E-mail addresses: john. Larmstrongklacuk (J. Armstrong). damiano.brigolt imperial.ac.uk (D. Brigo) https://doi.org/10.1016/jbankfin 2019.01.010 0378-4266 2019 Elsevier B.V. All rights reserved Journal of Banking and Finance 101 (2019) 122-135 Contents lists available at ScienceDirect Journal of Banking and Finance journal homepage: www.elsevier.com/locate/jbf ELSEVIER Chacko Risk managing tail-risk seekers: VaR and expected shortfall vs S-shaped utility John Armstrong", Damiano Brigob.* -Department of Mathematics King's College London United Kingdom Department of Mathematics Imperial College London 180 Queen's Gate London SW7 ZAZ Lhited Kingdom ARTICLE INFO Artide history: Received 16 April 2018 Accepted 19 January 2019 Available online 29 January 2019 JEI dassgication: DRI GII G13 ABSTRACT We consider marker players with tail-risk-seeking behaviour modelled by S-shaped utility, as introduced by Kahneman and Tversky. We argue that risk measures such as value at risk (VaR) and expected shordall (ES) are ineffective in constraining such players, as such measures cannot reduce the traders expected S- shaped utilities. Indeed, when designing payoffs aiming to maximize utility under a VaR or ES risk limit, the players will attain the same supremum of expected utility with or without VaR or ES limits. By con- trast, we show that risk management constraints based on a second more conventional concave utility function can reduce the maximum 5-shaped utility that can be achieved by the investor. Indeed, product designs leading to progressively larger 5-shaped utilities will lead to progressively lower expected con- straining conventional utilities, violating the related risk limit. These results hold in a variety of market models, including the Black Scholes options model, and are particularly relevant for risk managers given the historical role of VaR and the endorsement of ES by the Basel committee in 2012-2013. 2019 Elsevier B.V. All rights reserved. Keywords: Optimal product design under risk constraints Value at risk constraints Expected shortfall constraints Concave utility constraints S-Shaped utility maximization Limited liability investors Tail-risk-seeking investors Effective risk constraints Concave utility risk constraints 1. Introduction We consider the effects of imposing risk constraints on a trader in the case when that trader exhibits risk-seeking behaviour over losses or has limited liability, When making investment decisions, a trader is constrained in how much funds they can use in setting up the investment (giving rise to a budget or price constraint). In addition a trader is typ- ically limited on how much risk of future losses the investment may entail, Risk limits are common in banking and represent a tool for risk control. A natural risk limit to adopt is a risk limit based on classic risk measures such as Value at Risk (VaR) and Expected Shortfall (ES). The trader is then constrained to choose an invest- ment whose VaR or ES is below a given limit, set by the bank risk manager when approving the trade, The question we seek to answer in this paper is whether such limits are effective when the trader exhibits tail-risk-seeking be- haviour or has some form of limited liability. We consider a risk limit to be "effective" if it has an economic impact upon the trader, in the sense that imposing the limit decreases the expected util- ities that the trader can achieve. We model the tail-risk-seeking behaviour and/or limited liability by assuming that the utility of the trader is given by an 5-shaped function of the payoff of their investments. We assume that the trader acts to maximize their ex- pected utility, subject to the budget constraint and any risk con- straints. We find that in market models including the benchmark Black-Scholes model, neither VaR nor Es constraints are effective in curbing such risk-seeking behaviour. This result is particularly important in the light of the fact that ES has been officially en- dorsed and suggested as a risk measure by the Basel committee in 2012-2013 (Basel Committee on Banking Supervision, 2013: 2016). partly for its coherent risk measure" properties (Artzner et al., 1999; Acerbi and Tasche, 2002). We are grateful to Doctor Dirk Tasche for helpful suggestions and discussion on the first version. We thank Professor Xunyu Zhou who, during our seminar at Columbia University in NY on November 29, 2017, provided us with a number of references we had missed in our literature review and that have been included in the introduction of this updated version. Corresponding author. E-mail addresses: john. Larmstrongklacuk (J. Armstrong). damiano.brigolt imperial.ac.uk (D. Brigo) https://doi.org/10.1016/jbankfin 2019.01.010 0378-4266 2019 Elsevier B.V. All rights reserved