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1、Spectral Measures of RiskCoherence in theory and practiceBudapest September 11, 2003Subject of the talk: only finance (and a bit of statistic)Risk Management QuestionsFinancialStatisticalProbabilisticComputationalWhat do Imeasure ?How do Iestimate it ?Whathypotheses should Imake ?How can I carry out
2、 thecomputation (in time) ?Our investigation will be completely devoted to financial and statistical questions. The results will be however absolutely generalPart 1:Defining a Risk MeasureThe qualitative concept of “risk” and “risk premium”Everybody has an innate feeling for financial risk . more or
3、 less this How to define risk in a quantitative fashion ? .fundamental shared principlesrequirements (axioms) on the risk measureConcept of RiskRisk Measure?test The risk diversification principleThe aggregation of portfolios has always the effect of reducing or at most leaving unchanged the overall
4、 risk.+=Portfolio APortfolio BPortfolio A + BRisk of ( A + B ) is less or equal to Risk of (A) + Risk of (B)Coherent Measures of Risk(Monotonicity) if then(Positive Homogeneity) if then (Translational Invariance) (Subadditivity)In the paper “Coherent measures of Risk” (Artzner et al. Mathematical Fi
5、nance, July 1999) a set of axioms was proposed as the key properties to be satisfied by any “coherent measure of risk”.The diversification principle goes here Value at Risk (VaR): how it worksTo compute VaR, we need to specifylA time horizon: for instance one day. It represents the future period ove
6、r which we measure the risks of a portfoliolA confidence level: for instance a 5% probability. It represents the fraction of future worst case scenarios of the portfolio that we want to single out.The definition of VaR is then:“The VaR of a portfolio is the minimum loss that a portfolio can suffer i
7、n one day in the 5% worst cases”Strange as it may seem, this is the question most frequently asked to risk managersworldwide todayOr equivalently:“The VaR of a portfolio is the maximum loss that a portfolio can suffer in one day in the 95% best cases”Value at Risk (VaR): how it worksThe formidable a
8、dvantages introduced by VaRSince its appearance, VaR turned out to be a more flexible instrument w.r.t. more traditional measures of risk such as the “greeks” or “sensitivities”, because VaR is1.Universal: VaR can be measured on portfolios of any type (greeks on the contrary are designed “ad hoc” fo
9、r specific risks)2.Global: VaR summarize in a single number all the risks of a portfolio (IR, FX, Equity, Credit, ) (while we need many greeks to detect them all)3.Probabilistic: VaR provides a loss and a probability occurrence (while greeks are “what if” measures, which tell us nothing on the proba
10、bilities of the “if”)4.Expressed in Lost Money: VaR is expressed in the best of possible units of measures: LOST MONEY. Greeks have peculiar and less transparent u.o.m.A VaR-based portfolio risk report is exceedingly clearer than a greeks-based oneNo practitioner in 2003 would ever give up to these
11、advantages anymore The deadly sin of VaRUnfortunately however VaR1.Violates the subadditivity axiom and so is not coherent2.Or equivalently2.Violates the diversification principle and so for us it is not a risk measure at all3.In other words it may happen that +=VaR = 2VaR = 3VaR = 10The source of a
12、ll VaRs troubles: neglecting the tailVaR doesnt care whats beyond the threshold. I do care !Subadditivity and capital allocationBANKbusiness unit: Fixed Incomebusiness unit: Equitiesbusiness unit: ForexDue to the lack of subadditivity, VaR appears to be unfit for determining the capital adequacy of
13、a bank.In a financial institution made of several branches, it is common (or it might be unavoidable for practical reasons) to perform the risk measurements in each branch separately, reporting the results to a central Risk Management dept. VaR = 5VaR = 3VaR = 2Capital reserves as if VaR = 10 ?What
14、is the concept of risk of VaR ?From an epistemologic point of view however, the main problem of VaR is not its lack of subadditivitybut the very lack of any associated consistent set of axiomsWe still wonder what concept of risk Value at Risk has in mind !A natural questionIs it possible to find coh
15、erent measures which are as versatile and flexible as VaR ?The answer is fortunately YES( and they are also infinitely many )Expected Shortfall as an improvement of VaRDefinition of Expected Shortfall:“The ES of a portfolio is the average loss that a portfolio can suffer in one day in the 5% worst c
16、ases”Remember that“The VaR of a portfolio is the minimum loss that a portfolio can suffer in one day in the 5% worst cases”ES = the average of worst casesVaR = the best of worst casesExpected Shortfall: how it works .does it makesuch a bigdifference ?Is the Expected Shortfall coherent ?The original definition of Expected Shortfall (also known as Tail Conditional Expectation TCE) is This measure is also NON - SUBADDITIVE in general and so NON - COHERENT.2001 : new defin
