Multivariate stress scenarios and solvency

TY - JOUR

T1 - Multivariate stress scenarios and solvency

AU - McNeil, Alexander J.

AU - Smith, Andrew D.

PY - 2012/5

Y1 - 2012/5

N2 - We show how the probabilistic concepts of half-space trimming and depth may be used to define convex scenario sets Qα for stress testing the risk factors that affect the solvency of an insurance company over a prescribed time period. By choosing the scenario in Qα which minimizes net asset value at the end of the time period, we propose the idea of the least solvent likely event (LSLE) as a solution to the forward stress testing problem. By considering the support function of the convex scenario set Qα, we establish theoretical properties of the LSLE when financial risk factors can be assumed to have a linear effect on the net assets of an insurer. In particular, we show that the LSLE may be interpreted as a scenario causing a loss equivalent to the Value-at-Risk (VaR) at confidence level α, provided the α-quantile is a subadditive risk measure on linear combinations of the risk factors. In this case, we also show that the LSLE has an interpretation as a per-unit allocation of capital to the underlying risk factors when the overall capital is determined according to the VaR. These insights allow us to define alternative scenario sets that relate in similar ways to coherent measures, such as expected shortfall. We also introduce the most likely ruin event (MLRE) as a solution to the problem of reverse stress testing.

AB - We show how the probabilistic concepts of half-space trimming and depth may be used to define convex scenario sets Qα for stress testing the risk factors that affect the solvency of an insurance company over a prescribed time period. By choosing the scenario in Qα which minimizes net asset value at the end of the time period, we propose the idea of the least solvent likely event (LSLE) as a solution to the forward stress testing problem. By considering the support function of the convex scenario set Qα, we establish theoretical properties of the LSLE when financial risk factors can be assumed to have a linear effect on the net assets of an insurer. In particular, we show that the LSLE may be interpreted as a scenario causing a loss equivalent to the Value-at-Risk (VaR) at confidence level α, provided the α-quantile is a subadditive risk measure on linear combinations of the risk factors. In this case, we also show that the LSLE has an interpretation as a per-unit allocation of capital to the underlying risk factors when the overall capital is determined according to the VaR. These insights allow us to define alternative scenario sets that relate in similar ways to coherent measures, such as expected shortfall. We also introduce the most likely ruin event (MLRE) as a solution to the problem of reverse stress testing.

KW - Convex analysis

KW - Risk measures

KW - Scenario sets

KW - Solvency II

KW - Stress testing

UR - http://www.scopus.com/inward/record.url?scp=84862796619&partnerID=8YFLogxK

U2 - 10.1016/j.insmatheco.2011.12.005

DO - 10.1016/j.insmatheco.2011.12.005

M3 - Article

AN - SCOPUS:84862796619

SN - 0167-6687

VL - 50

SP - 299

EP - 308

JO - Insurance: Mathematics and Economics

JF - Insurance: Mathematics and Economics

IS - 3

ER -