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Spectral Measures of Risk Coherence in theory and practice

Spectral Measures of Risk Coherence in theory and practice. Budapest – September 11, 2003. Subject of the talk: only finance (and a bit of statistic). Our investigation will be completely devoted to financial and statistical questions. The results will be however absolutely general.

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Spectral Measures of Risk Coherence in theory and practice

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  1. Spectral Measures of Risk Coherence in theory and practice Budapest – September 11, 2003

  2. Subject of the talk: only finance (and a bit of statistic) Our investigation will be completely devoted to financial and statistical questions. The results will be howeverabsolutely general

  3. Part 1: Defining a Risk Measure

  4. The qualitative concept of “risk” and “risk premium” Everybody has an innate feeling for financial risk .... ... more or less this ......

  5. Concept of Risk ? Risk Measure test requirements (axioms) on the risk measure fundamental shared principles How to define risk in a quantitative fashion ? ...

  6. Portfolio A + Portfolio B = Portfolio A + B The risk diversification principle The aggregation of portfolios has always the effect of reducing or at most leaving unchanged the overall risk. Risk of ( A + B ) is less or equal to Riskof (A) + Riskof (B)

  7. The diversification principle goes here (Monotonicity) ifthen (Positive Homogeneity) ifthen (Translational Invariance) (Subadditivity) Coherent Measures of Risk In the paper “Coherent measures of Risk” (Artzner et al. Mathematical Finance, July 1999)a set of axioms was proposed as the key properties to be satisfied by any “coherent measure of risk”.

  8. Strange as it may seem, this is the question most frequently asked to risk managers worldwide today Value at Risk (VaR): how it works • To compute VaR, we need to specify • A time horizon: for instanceone day. It represents the future period over which we measure the risks of a portfolio • A 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 in one day in the 5% worst cases” Or equivalently: “The VaR of a portfolio is the maximum loss that a portfolio can suffer in one day in the 95% best cases”

  9. Value at Risk (VaR): how it works

  10. The formidable advantages introduced by VaR Since 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 is Universal:VaR can be measured on portfolios of any type (greeks on the contrary are designed “ad hoc” for specific risks) 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) Probabilistic:VaR provides a loss and a probability occurrence (while greeks are “what if” measures, which tell us nothing on the probabilities of the “if”) 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 one No practitioner in 2003 would ever give up to these advantages anymore

  11. + = VaR = 2 VaR = 3 VaR = 10 The deadly sin of VaR Unfortunately however VaR Violates the subadditivity axiomand sois not coherent Or equivalently Violates the diversification principleand so for us it is not a risk measure at all In other words it may happen that …

  12. The source of all VaR’s troubles: neglecting the tail VaR doesn’t care what’s beyond the threshold. I do care !

  13. BANK business unit: Equities business unit: Forex business unit:Fixed Income Subadditivity and capital allocation Due to the lack of subadditivity, VaR appears to be unfit for determining the capital adequacy of 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. Capital reserves as if VaR = 10 ? VaR = 5 VaR = 3 VaR = 2

  14. We still wonder what concept of risk Value at Risk has in mind ! What is the concept of risk of VaR ? From an epistemologic point of view however, the main problem of VaR is not its lack of subadditivity but the very lack of any associated consistent set of axioms

  15. A natural question Is it possible to find coherent measures which are asversatile and flexible as VaR ? The answer is fortunately YES (… and they are also infinitely many …)

  16. Expected Shortfall as an improvement of VaR Definition of Expected Shortfall: “The ESof a portfolio is theaverage lossthat a portfolio can suffer in one day in the 5% worst cases” Remember that “The VaRof a portfolio is theminimum lossthat a portfolio can suffer in one day in the 5% worst cases” ES = the average of worst cases VaR = the best of worst cases

  17. Expected Shortfall: how it works ...does it make such a big difference ?

  18. 2001 : new definition of Expected Shortfall This measure is SUBADDITIVE and in fact COHERENTwith no hypotheses on the pdf 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.

  19. In this case ES is the average of the pdf defined by the darkened area only (5% worst) When TCE and ES differ while TCE is the average of the pdf defined by all the first two columns (>5% worst)

  20. Ordered statistics (= sorted data from worst to best) Estimating Expected Shortfall One can show that ES is indeed estimable in a consistent way as the “Average of 100% worst cases”.

  21. Example 1: a subadditivity violation of VaR Consider a Bond Aand suppose that, at maturity, there are three possible cases: 1) No default:it redeemsthe nominal (100 Euro)and the coupon (8 Euro) or 2) Soft default: it redeems only the nominal (100 Euro) but not the coupon or 3) Hard Default: it pays nothing

  22. A subadditivity violation of VaR Consider another Bond B perfectly identical to A, but issued by a different issuer Suppose now that the risks of the two bonds happen to be mutually exclusive, in the sense that if issuer A defaults, B does not, and vice-versa. Typical case: ANTICORRELATED RISKS = RISK REDUCTION IN CASE OF DIVERSIFICATION

  23. VaR dissuades from diversification ! ES advises diversification Risk Measurement

  24. Coherent risk measures display always convex risk surfaces with a unique global minimum and no local minima Risk surfaces ES VaR 100% A 50%-50% 100% B Non-coherent measures display in general risk surfaces affected by multiple (local) minima

  25. Example 2: a simple prototype portfolio Consider a portfolio made of n risky bonds all of which have a 3% default probability and suppose for simplicity that all the default probabilities are independent of one another. Portfolio = { 100 Euro invested in n independent identical distributed Bonds } Bond payoff = Nominal (or 0 with probability 3%) Question: let’s choose n in such a way to minimize the risk of the portfolio Let’s try to answer this question with a 5% VaR, ES and TCE (= ES (old)) with a time horizon equal to the maturity of the bond.

  26. “risk” versus number of bonds in the portfolio The surface of risk of ES has a single global minimum at n= and no fake local minima. ES just tell us: “buy more bonds you can” VaR and TCE suggest us NOT TO BUY the 13th, 28thor 47th bond because it would increase the risk of the portfolio .... (?) Are things better for large portfolios ???...

  27. ...but for large n the pdf should be normal and VaR coherent ... (!!!) Notice that the pdf really becomes normal-shaped for large N

  28. ... or not ? But convexity problems still remain !!!

  29. Part 2: Defining a space of risk measures

  30. A natural question: ... other coherent measures ? Is the Expected Shortfall an “isolated exception” or does it belong to a large class of coherent measures ? Is it possible to create new coherent measures starting from some given known ones? The answer is simple and allows to create a wide CLASS of coherent measures. Given n coherent measuresof risk 1, 2,... n any convex linear combination  = 1 1 + 2 2 + ...+ n n ( with k k = 1 and k>0 ) is another coherent measure of risk

  31. If any point represents a given coherent measure ... ... Then any other point in the generated “convex hull” is a new coherent measure of risk Geometrical interpretation Given n coherent measures, their most general convex combination is any of the points contained in the generated “convex hull”

  32. Set of all Expected Shortfalls with (0,1] Convex hull = New space of coherent measures Our strategy .... We already know infinitely many coherent measures of risk, namely all the possible -Expected Shortfalls for any value  between 0 e 1 In this way we can generate a new class of coherent measures. This class is defined “Spectral Measures of Risk”

  33. Definition: Spectral measure of risk with spectrum • is positive • is not increasing Theorem: the measure M(X) is coherent if and only if Spectral measures of risk: explicit characterization

  34. Worst cases Best cases The “Risk Aversion Function” (p) “(p) decreasing” explains the essence of coherence ...a measure is coherent only if it maps “bigger weights to worse cases” Any admissible (p) represents a possible legitimate rational attitude toward risk A rational investor may express her own subjective risk aversion through her own subjective(p) which in turns give her own spectral measure M (p): Risk Aversion Function It may thought of as a function which “weights” all cases from the worst to the best

  35. Expected Shortfall: Step function • positive • decreasing Value at Risk: Spike function (Dirac delta) • positive • not decreasing Risk Aversion Function (p) for ES and VaR

  36. Ordered statistics (= data sorted from worst to best) Discretized  function Estimating Spectral Measures of Risk It can be shown that any spectral measure has the following consistent estimator:

  37. DIFFERENT SPECTRAL MEASURES DIFFERENT PORTFOLIOS Tailoring Risks ! The Expected Shortfall is just one out of infinitely many possible Spectral Measures ES expresses just a specific risk aversion But is there a spectral measure which is optimal for all portfolios ? NO

  38. Coherent Measures Coherent but not Spectral Spectral Measures What are the distinguishing properties of Spectral measures ? A characterization of spectral measures among coherent measures via additional properties (axioms) would give us not only more information on Spectral Measures, but also useful information on NON-spectral measures.

  39. If “X is worse than Y in probability”, then its risk must be bigger The measure of risk depends ONLY from the probability distribution of X and it is therefore estimable from empirical data of X. If X and Y are “perfectly correlated”, then the risk of X+Y must be the sum of the risks of X and Y. (X+Y) = (X) + (Y) A fifth axiom ? A sixth one ? One can show that the Spectral Measures M are all the coherent measures which satisfy two additional conditions: (Kusuoka 2001, Tasche 2002) The first condition may be expressed alternatively as (Monotonicity w.r.t. “First Stochastic Dominance”) If Prob(X a)  Prob(Y a), aR then (Y)  (X) A coherent measure of risk which is NOT estimable is WCE (Artzner et al. 1997) (“Estimability from empirical data” or “law invariance”) It must be possible to estimate (X) from empirical data of X The second condition is: (“Comonotonic additivity”) If X and Y are comonotonic risks, then (X+Y) = (X) + (Y) A coherent measure of risk which is NOT Comonotonic Additive is semivariance (Fischer 2001)

  40. Part 3: A wider class: Convex Measures of Risk Handle with care

  41. Subadditivity fails Positive Homogeneity fails Liquidity Risk: when coherency violations make sense When an asset position in a portfolio has a size which is comparable with the capacity of the market (market depth) of absorbing a sudden sell off, we are in presence of liquidity risk. Selling large asset amounts moves market bids downwards. In this case clearly Risk (A+A) = Risk (2 A) > 2 Risk (A) = Risk(A) + Risk (A)

  42. (Monotonicity) ifthen (Translational Invariance) (Positive Homogeneity) ifthen (Subadditivity) (Convexity) Weaker Condition Convex Measures of Risk (or Weakly Coherent Measures of Risk) Heath, Follmer et al., Frittelli et al. define a larger class of measures which allow for possible coherency violations due to liquidity risk.

  43. Our point of view: some care is needed We believe however that in absence of liquidity risk, coherency violations are completely undesired for a measure of risk. The “small size limit” of a measure of risk should therefore be a (strongly) coherent measure of risk. • This observation • Rules out measures of risk which are intrinsically non coherent in their analytical dependence from pdf’s. • Forces a convex measure to carry possible coherency violations only through dimensional constants (typically the market depth di of each market’s asset Ai) • When each asset’s position is much smaller than its market depth we want the measure to be strongly coherent

  44. Convex measures: a step forward ? We are persuaded that convex measures of risk may represent a significant step forward in risk market practice provided that they respect the “small size coherent limit”. Otherwise, trying to take liquidity into account we may jeopardize the properties of coherency where it should hold in a strong sense. A convex measure “beyond coherency” is therefore typically NOT a smarter formula which allows coherency violations, because it should be sensitive to positions sizes. A convex measure “beyond coherency” is more probably a measure with a coherent analytical structure PLUS a database of each assets’ market depths to which the position sizes have to be compared in the search for illiquidities.

  45. A natural solution A natural way to define a convex measure satisfying the small size coherent condition is adding a coherent measure a liquidity charge • The liquidity charge C • Apply to illiquid assets only and contain their dimensional market depths. • Goes to zero in the liquid limit when all position becomes much smaller of its market depth. • We do not propose any specific modelling of the liquidity charge

  46. Part 4: Coherency and convexity: optimizing spectral measures

  47. Convexity of the “Risk Surface” Absence of local minima / Existence of a unique global minimum Coherency and Convexity in short Coherency of the Risk Measure

  48. PROBLEM ! A SORTING operation on data makes the dependence NOT EXPLICITLY ANALYTIC. Serious problems for any common optimizator. Minimizing the Expected Shortfall Let a portfolio of M assets be a function of their “weights” wj=1....M and let X=X(wi ) be its Profit & Loss. We want to find optimal weights by minimizing its Expected Shortfall In the case of a N scenarios estimator we have Notice: also in the case of non parametric VaR a SORTING operation is needed in the estimator and the same problem appears

  49. Notice: the SORTING operator on data has disappeared. The dependence on data is manifestly analytic. The Pflug-Uryasev-Rockafellar solution Pflug, Uryasev & Rockafellar (2000, 2001) introduce a function which is analytic,convex and piecewise linear in all its arguments. It depends on X(w) but also on an auxiliary variable  In the discrete case with N scenarios it becomes

  50. Properties of : the Pflug-Uryasev-Rockafellar theorem Minimizing  in its arguments (w,) amounts to minimizing ES in (w) only Moreover the parameter in the extremum takes the value of VaR(X(w)). (w) and ES(w) coincide but just in the minimum ! The auxiliary parameter in the minimum becomes the VaR

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