Ri$kLab.CNTM,Slide1,Quantitative Risk Management - Aggregate Risk,Ming-Heng Zhang RiskLab.CN,Ri$kLab.CNTM,Slide2,Note That,The materials in this seminar are mainly picked from “Quantitative Risk Management Concept , Techniques and Tools” wroten by McNeil, A.J., Frey, R. and Paul Embrechts,Ri$kLab.CNTM,Slide3,Introduction,Definition of Aggregate Risk Considered as the risk of a portfolio, which could even be the entire position in risky assets of a financial enterprise (from a investment strategy view).Defined by total exposure of a bank to any single customer for both spot and forward contracts (from Reuters financial glossary).Defined as a summary judgment about the level of supervisory concern, incorporated with judgments about the quantity of risk and the quality of risk management or weighed of the relative importance of each and characterized as low, moderate, or high (from a supervisory perspective ).,Ri$kLab.CNTM,Slide4,Contents,Theoretical concepts in quantitative risk management that fall under the broad heading of aggregate risk.Defining aggregate riskMeasuring aggregate riskBounding an aggregate risk Allocating risk capital,Ri$kLab.CNTM,Slide5,Coherent Measures of Risk,Such coherent risk measure is that the properties that a good measure of risk should have with particular emphasis on aggregation properties.Purpose - list the properties that a good risk measure should have.Background retrospect to the axioms of coherence proposed for applications in financial risk management in the seminal paper by Artzner et al. (1999) who specified a number of axioms that any so-called coherent risk measure should satisfy by use of economic reasoning.,Ri$kLab.CNTM,Slide6,Axioms of Coherence for Risk,Definition - Formal Risk MeasurementGiven a probability space (,F, P) and a time horizon, the set of all random variables on (,F) denote by L0(,F,P), which they almost surely finite.A set M L0(,F, P) of rvs called as financial risks that has convex property, i.e., for any L1,L2 and M and R, then L1+L2M and LM, that can be interpreted as portfolio loss over some time horizon.Risk measures are real-value functions :MRDifferentia between loss severity and risk capital,Ri$kLab.CNTM,Slide7,(L) - Risk Capital,Notes Interpret (L) as the amount of capital that should be added to a position with loss given by L, so that the position becomes acceptable to an external or internal risk controller.Positions with (L) 0 are acceptable without injection of capital(无须重新注资).That positions with(L) 0, we have (L)= (L).Notes this property will be justified if we assume that the axiom 2 holds and the subadditivity implies that for any n in natural set, (nL)=(L+L+L) (L)+ (L)+(L)=n(L).The equation holds true on that fact no netting or diversification between the losses in this portfolio,Ri$kLab.CNTM,Slide11,Axiom 4 Monotonicity,Definition For L1,L2M such that L1L2, almost surely we have (L1) (L2).Notes The positions that lead to higher losses in every state of the world require more risk capital. (L1)0 for any L0 (0+0+0)= (0) n(0) for any n in natural set L1-L20 means that (L1-L2) 0 (L1 ) =(L1-L2+L2) (L1-L2)+(L2),Ri$kLab.CNTM,Slide12,Coherent Risk Measure,DefinitionA risk measure whose domain includes the convex cone M is called coherent (on M) if it satisfies the Axioms 1 - 4.Notes translation invariance (L+c)=(L)+c Subadditivity (L1+L2) (L1)+(L2) positive homogeneity (nL)=n(L) Monotonicity (L1)(L2) if L1L2,Ri$kLab.CNTM,Slide13,Remark on the Convex Risk Measures,Definition of convex risk measuresFor all L1,L2 M and 0,1, we have(L1+ (1-)L2) (L1)+(1-)(L2)Notes The economic justification of convex risk measures is again the idea that diversification reduce risk.Recently attracted a lot of attention See “Convex Risk Measures for Portfolio Optimization and Concept of Flexibility”,