Risk managing tail-risk seekers: VaR and expected shortfall vs S-shaped utility

TY - JOUR

T1 - Risk managing tail-risk seekers: VaR and expected shortfall vs S-shaped utility

AU - Armstrong, John

AU - Brigo, Damiano

PY - 2019/4/1

Y1 - 2019/4/1

N2 - We consider market 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 shortfall (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 contrast, we show that risk management constraints based on a second more conventional concave utility function can reduce the maximum S-shaped utility that can be achieved by the investor. Indeed, product designs leading to progressively larger S-shaped utilities will lead to progressively lower expected constraining 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.

AB - We consider market 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 shortfall (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 contrast, we show that risk management constraints based on a second more conventional concave utility function can reduce the maximum S-shaped utility that can be achieved by the investor. Indeed, product designs leading to progressively larger S-shaped utilities will lead to progressively lower expected constraining 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.

KW - Concave utility constraints

KW - Concave utility risk constraints

KW - Effective risk constraints

KW - Expected shortfall constraints

KW - Limited liability investors

KW - Optimal product design under risk constraints

KW - S-Shaped utility maximization

KW - Tail-risk-seeking investors

KW - Value at risk constraints

UR - http://www.scopus.com/inward/record.url?scp=85061594575&partnerID=8YFLogxK

U2 - 10.1016/j.jbankfin.2019.01.010

DO - 10.1016/j.jbankfin.2019.01.010

M3 - Article

SN - 0378-4266

VL - 101

SP - 122

EP - 135

JO - Journal of Banking and Finance

JF - Journal of Banking and Finance

ER -