What Is a Good Risk Measure: Bridging the Gaps between Robustness, Subadditivity, Prospect Theory, and Insurance Risk Measures

Transcription

1 What Is a Good Risk Measure: Bridging the Gaps between Robustness, Subadditivity, Prospect Theory, and Insurance Risk Measures C.C. Heyde 1 S.G. Kou 2 X.H. Peng 2 1 Department of Statistics Columbia University 2 Department of IEOR Columbia University Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 1 / 37

2 Outline 1 Introduction Review of Risk Measures 2 Motivation and Examples What is an Axiom Some Concepts of the Law Examples and Tail Conditional Median 3 Reasons to Relax Subadditivity Review of Existing Reasons Two New Reasons: Prospect Theory and Robustness 4 Natural Risk Statistic Axioms and Characterization of Natural Risk Statistic Comparing the Natural Risk Statistic, Coherent Risk Measure and Insurance Risk Measure Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 2 / 37

3 Outline 1 Introduction Review of Risk Measures 2 Motivation and Examples What is an Axiom Some Concepts of the Law Examples and Tail Conditional Median 3 Reasons to Relax Subadditivity Review of Existing Reasons Two New Reasons: Prospect Theory and Robustness 4 Natural Risk Statistic Axioms and Characterization of Natural Risk Statistic Comparing the Natural Risk Statistic, Coherent Risk Measure and Insurance Risk Measure Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 3 / 37

4 Measure of Risk Maps a R.V. to a Real Number Risk: future loss of a position random variable X Measure of risk: mapping from a set of risks to the real line X ρ(x) Examples: margin requirement for financial trading capital requirement against market risk insurance risk premium Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 4 / 37

5 Coherent Measure of Risk The First Measure: Coherent Measure of Risk (Artzner et al. 1999, Huber 1981) Translation invariance: ρ(x + a) = ρ(x) + a Positive homogeneity: ρ(λx) = λρ(x),λ 0 Monotonicity: ρ(x) ρ(y), if X Y Subadditivity: ρ(x + Y) ρ(x) + ρ(y) Characterization of coherent measure of risk Maximum expected loss Acceptance sets ρ(x) = sup E P [X] P P ρ(x) = min{m : X m acceptance set} Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 5 / 37

6 Coherent Measure of Risk The First Measure: Coherent Measure of Risk (Artzner et al. 1999, Huber 1981) Translation invariance: ρ(x + a) = ρ(x) + a Positive homogeneity: ρ(λx) = λρ(x),λ 0 Monotonicity: ρ(x) ρ(y), if X Y Subadditivity: ρ(x + Y) ρ(x) + ρ(y) Characterization of coherent measure of risk Maximum expected loss Acceptance sets ρ(x) = sup E P [X] P P ρ(x) = min{m : X m acceptance set} Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 5 / 37

7 Insurance Measure of Risk X and Y are comonotonic if and only if ω 1,ω 2 Ω,(X(ω 1 ) X(ω 2 ))(Y(ω 1 ) Y(ω 2 )) 0 The Second Measure: Insurance Measure of Risk (Wang et al. 1997) Law invariance: ρ(x) = ρ(y), if X and Y have the same distribution Monotonicity: ρ(x) ρ(y), if X Y Comonotonic additivity: ρ(x + Y) = ρ(x) + ρ(y), if X and Y are comonotonic Continuity: lim ρ(max(x d, 0)) = d 0 ρ(x+ ), lim ρ(min(x, d)) = ρ(x) d lim ρ(max(x, d)) = ρ(x) d Scale normalization: ρ(1) = 1 Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 6 / 37

8 Insurance Measure of Risk X and Y are comonotonic if and only if ω 1,ω 2 Ω,(X(ω 1 ) X(ω 2 ))(Y(ω 1 ) Y(ω 2 )) 0 The Second Measure: Insurance Measure of Risk (Wang et al. 1997) Law invariance: ρ(x) = ρ(y), if X and Y have the same distribution Monotonicity: ρ(x) ρ(y), if X Y Comonotonic additivity: ρ(x + Y) = ρ(x) + ρ(y), if X and Y are comonotonic Continuity: lim ρ(max(x d, 0)) = d 0 ρ(x+ ), lim ρ(min(x, d)) = ρ(x) d lim ρ(max(x, d)) = ρ(x) d Scale normalization: ρ(1) = 1 Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 6 / 37

9 Insurance Measure of Risk (Continued) It does not require subadditivity ρ( ) is an insurance measure of risk if and only if ρ( ) has a Choquet integral representation: ρ(x) = Xd(g P) = 0 (g(p(x > t)) 1)dt+ where g( ) is increasing, g(0) = 0, g(1) = 1 0 g(p(x > t))dt It does not generate scenarios, unlike the coherent risk measure Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 7 / 37

10 Comonotonic Additivity Does Not Hold in Two Scenarios Probability space: (Ω = {ω 1,ω 2,ω 3 }, F, P), P(ω i ) = 1 3 Two comonotonic random variables Two distortion function (Z(ω 1 ), Z(ω 2 ), Z(ω 3 )) = (3, 2, 4), Y = Z 2 g 1 ( 1 3 ) = 0.5, g 1( 2 3 ) = 1.0; g 2( 1 3 ) = 0.72, g 2( 2 3 ) = 0.8 Risk measure incorporating two scenarios: { } ρ(x) = max X d(g 1 P), X d(g 2 P) Only strict comonotonic subadditivity holds! ρ(z + Y) = 9.28 < ρ(z) + ρ(y) = = 9.3 Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 8 / 37

11 Outline 1 Introduction Review of Risk Measures 2 Motivation and Examples What is an Axiom Some Concepts of the Law Examples and Tail Conditional Median 3 Reasons to Relax Subadditivity Review of Existing Reasons Two New Reasons: Prospect Theory and Robustness 4 Natural Risk Statistic Axioms and Characterization of Natural Risk Statistic Comparing the Natural Risk Statistic, Coherent Risk Measure and Insurance Risk Measure Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 9 / 37

12 What Is an Axiom Two Definitions in Cambridge English Dictionary: (1) FORMAL: a statement or principle which is generally accepted to be true, but is not necessarily so. (2) SPECIALIZED: a formal statement or principle in mathematics, science, etc., from which other statements can be obtained. Axioms are not theorems. By definition, they cannot be proved right or wrong. It is useful to provide alternative axioms so that people can have a choice. Axioms should be as general as possible; otherwise, there is almost no difference between axioms and assumptions. Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 10 / 37

13 Concepts of the Law 101 Large literature on the subject. See, e.g., the textbook by H. Hart (1994) The Concept of Law. (1) Legal Realism. A law is only a guideline for judges and enforcement officers, i.e. a law is only intended to be the average of what the judges and officers will decide. This requests the robustness of the law. (2) Legal Positivism. A law should reflect the society norm. This requests consistency to the social behavior of the society. (3) The Separability Thesis. Law and internal standards (e.g. morality) are conceptually distinct. We should expect laws to be less restrictive than internal standards. Here we talk about the risk measures for law and regulation, not internal risk measures within firms. For the external risk measures, it has to be robust and consistent with what the society practices. Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 11 / 37

14 An Example of Traffic Limit Not by studying physics of road conditions 85 rule Recognizing that people constantly drive 5 to 10 miles beyond the traffic limit. Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 12 / 37

15 Example: Value-at-risk (VaR) Value-at-risk (VaR) at level α: VaR α (X) = α-quantile of X, α [95%, 99.9%] Value-at-risk does not satisfy subadditivity VaR α (X + Y) VaR α (X) + VaR α (Y), in general Inconsistency: 99% VaR is the primary measure of risk imposed by Basel Accord, but it is excluded by coherent measure of risk Value-at-risk satisfies axioms for the insurance risk measure Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 13 / 37

16 Example: Tail Conditional Expectation (TCE) Tail conditional expectation (TCE) at level α (Artzner et.al, 1999): TCE α (X) = E[X X VaR α (X)] Tail conditional expectation is also called conditional value-at-risk (Rockafellar & Uryasev, 2002) and expected shortfall (Acerbi, 2002) Tail conditional expectation is a coherent measure of risk, at least for continuous random variables Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 14 / 37

17 Tail conditional median (TCM) Tail conditional median (TCM) at level α: TCM α (X) = median[x X VaR α (X)] = VaR 1+α(X) 2 Tail conditional median is equal to VaR at a higher level Tail conditional median does NOT satisfy subadditivity Tail conditional median is more robust than tail conditional expectation Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 15 / 37

18 Tail conditional median (TCM) Tail conditional median (TCM) at level α: TCM α (X) = median[x X VaR α (X)] = VaR 1+α(X) 2 Tail conditional median is equal to VaR at a higher level Tail conditional median does NOT satisfy subadditivity Tail conditional median is more robust than tail conditional expectation Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 15 / 37

19 Tail conditional median (TCM) Tail conditional median (TCM) at level α: TCM α (X) = median[x X VaR α (X)] = VaR 1+α(X) 2 Tail conditional median is equal to VaR at a higher level Tail conditional median does NOT satisfy subadditivity Tail conditional median is more robust than tail conditional expectation Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 15 / 37

20 TCM and TCE for Auto Insurance Claim α TCE α TCM α TCE α TCM α TCE α TCM α TCE α 99.0% % 98.5% % 98.0% % 97.5% % 97.0% % 96.5% % 96.0% % 95.5% % 95.0% % Which one is better, Tail Conditional Median or Tail Conditional Expectation? Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 16 / 37

21 Outline 1 Introduction Review of Risk Measures 2 Motivation and Examples What is an Axiom Some Concepts of the Law Examples and Tail Conditional Median 3 Reasons to Relax Subadditivity Review of Existing Reasons Two New Reasons: Prospect Theory and Robustness 4 Natural Risk Statistic Axioms and Characterization of Natural Risk Statistic Comparing the Natural Risk Statistic, Coherent Risk Measure and Insurance Risk Measure Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 17 / 37

22 Reason 1: Tail Subadditivity of VaR If X and Y have elliptical distributions, then (Emberchts, et.al, 1998) VaR α (X + Y) VaR α (X) + VaR α (Y) VaR has subadditivity in the tail region (Daníelsson et al. 2005) If X and Y have jointly regularly varying tails with tail index bigger than 1, then there exists α 0 (0, 1), such that VaR α (X + Y) VaR α (X) + VaR α (Y), α (α 0, 1) VaR has subadditivity in simulations when α [95%, 99%]. (Daníelsson et al. 2005) The couterexamples for subadditivity may be too narrow and may be pathological. Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 18 / 37

23 Reason 2: A Merger May Create Extra Risk Merger of two firms removes the fire walls between them and hence increase the risk of complete breakdown of the two firms as a whole. (Dhaene et al. 2005) By splitting a risky trading business into separate sub-firms, the parent firm can reduce the risk of bankruptcy. Example: collapse of Britain s Barings Bank is caused by a single trader, Nick Leeson, in Singapore. Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 19 / 37

24 Reason 3: Psychological Theory of Uncertainty and Risk Rational people systematically violate the axioms of expected utility theory, e.g., the Ellsberg paradoxes. Prospect theory (Kahneman & Tversky, 1979, 1992, 1993) is able to explain the major violations of expected utility theory Four patterns of risk attitudes: People are risk seeking for losses of moderate or high probability It is better to postulate preference axioms on comonotonic random variables rather than on arbitrary random variables Non-comonotonic random loss can violate subadditivity Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 20 / 37

25 An Example Violating Subadditivity A fair coin is flipped Three positions: H: losing $10,000 if it comes up head" T: losing $10,000 if it comes up tail" S: losing $5,000 for sure ρ(s) = 5, 000, ρ(h) = ρ(t) ρ(h) < ρ(s) because people are risk seeking in face of loss ρ(h + T) = ρ(sure loss of 10,000) = 10, 000 Subadditivity does not hold: ρ(h) + ρ(t) < 10, 000 = ρ(h + T)! Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 21 / 37

26 An Example Violating Subadditivity A fair coin is flipped Three positions: H: losing $10,000 if it comes up head" T: losing $10,000 if it comes up tail" S: losing $5,000 for sure ρ(s) = 5, 000, ρ(h) = ρ(t) ρ(h) < ρ(s) because people are risk seeking in face of loss ρ(h + T) = ρ(sure loss of 10,000) = 10, 000 Subadditivity does not hold: ρ(h) + ρ(t) < 10, 000 = ρ(h + T)! Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 21 / 37

27 Reason 4: from Robustness Viewpoint Enforcement of risk measure: A regulator imposes a risk measure Each firm calculates the level of risk and reports to the regulator Each firm has the freedom to choose its own internal models For example, Basel Accord imposes 99%-VaR as the risk measure, banks can use their own internal models in the calculation The risk measure should be robust with respect to underlying models, otherwise the firms can choose a model which produces a low level of risk Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 22 / 37

28 Reason 4: from Robustness Viewpoint (continued) Another issue: How to specify a correct model? Difficulty: heaviness of tail distribution is hard to determine With 5,000 observations one can not distinguish between the Laplace distribution and the T-distributions. (Heyde & Kou, 2004) Since we can not specify the correct model to some extent, the risk measure should have robustness with respect to models. The risk measure should also be robust with respect to data Tail Conditional Expectation (TCE) is sensitive to the model and outliers in the data Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 23 / 37

29 Reason 4: from Robustness Viewpoint (continued) Another issue: How to specify a correct model? Difficulty: heaviness of tail distribution is hard to determine With 5,000 observations one can not distinguish between the Laplace distribution and the T-distributions. (Heyde & Kou, 2004) Since we can not specify the correct model to some extent, the risk measure should have robustness with respect to models. The risk measure should also be robust with respect to data Tail Conditional Expectation (TCE) is sensitive to the model and outliers in the data Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 23 / 37

30 TCE is Sensitive to the Heaviness of Tail Distribution TCM and TCE for Laplace and T-distributions, α [99%, 99.9%] 9 8 TCE Laplace T 3 T 5 T TCM Laplace T 3 T 5 T TCE α 6 TCM α log(1 α) log(1 α) Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 24 / 37

31 TCE is Sensitive to the Heaviness of Tail Distribution (continued) TCM and TCE for Laplace and T-distributions, α [95%, 99%] TCE TCM Laplace T 3 T 5 T Laplace T 3 T 5 T 12 TCE α 3 TCM α log(1 α) log(1 α) Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 25 / 37

32 TCM is More Robust Than TCE Influence function of an estimator T(F): T((1 ε)f + εδ x ) T(F) IF(x, T,F) = lim. ε 0 ε TCM has bounded influence function: sup IF(u, TCM α, X) < u TCE has unbounded influence function: sup IF(u, TCE α, X) = u Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 26 / 37

33 Outline 1 Introduction Review of Risk Measures 2 Motivation and Examples What is an Axiom Some Concepts of the Law Examples and Tail Conditional Median 3 Reasons to Relax Subadditivity Review of Existing Reasons Two New Reasons: Prospect Theory and Robustness 4 Natural Risk Statistic Axioms and Characterization of Natural Risk Statistic Comparing the Natural Risk Statistic, Coherent Risk Measure and Insurance Risk Measure Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 27 / 37

34 Axioms for Natural Risk Statistic Data x = (x 1,..., x n ) R n Risk statistic: a mapping from the data set to the real line: Axioms for natural risk statistic: ˆρ : R n R x ˆρ( x) Positive homogeneity and translation invariance: ˆρ(a x + b) = aˆρ( x) + b, x R n, a 0, b R Monotonicity: ˆρ( x) ˆρ(ỹ), if x ỹ, i.e., x i y i, i = 1,...,n. Comonotonic subadditivity: ˆρ( x + ỹ) ˆρ( x) + ˆρ(ỹ), if x and ỹ are comonotonic, i.e., (x i x j )(y i y j ) 0 for any i j. Empirical law invariance: ˆρ((x 1,..., x n )) = ˆρ((x i1,..., x in )), for any permutation (i 1,..., i n ). Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 28 / 37

35 Axioms for Natural Risk Statistic Data x = (x 1,..., x n ) R n Risk statistic: a mapping from the data set to the real line: Axioms for natural risk statistic: ˆρ : R n R x ˆρ( x) Positive homogeneity and translation invariance: ˆρ(a x + b) = aˆρ( x) + b, x R n, a 0, b R Monotonicity: ˆρ( x) ˆρ(ỹ), if x ỹ, i.e., x i y i, i = 1,...,n. Comonotonic subadditivity: ˆρ( x + ỹ) ˆρ( x) + ˆρ(ỹ), if x and ỹ are comonotonic, i.e., (x i x j )(y i y j ) 0 for any i j. Empirical law invariance: ˆρ((x 1,..., x n )) = ˆρ((x i1,..., x in )), for any permutation (i 1,..., i n ). Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 28 / 37

36 Axioms for Natural Risk Statistic Data x = (x 1,..., x n ) R n Risk statistic: a mapping from the data set to the real line: Axioms for natural risk statistic: ˆρ : R n R x ˆρ( x) Positive homogeneity and translation invariance: ˆρ(a x + b) = aˆρ( x) + b, x R n, a 0, b R Monotonicity: ˆρ( x) ˆρ(ỹ), if x ỹ, i.e., x i y i, i = 1,...,n. Comonotonic subadditivity: ˆρ( x + ỹ) ˆρ( x) + ˆρ(ỹ), if x and ỹ are comonotonic, i.e., (x i x j )(y i y j ) 0 for any i j. Empirical law invariance: ˆρ((x 1,..., x n )) = ˆρ((x i1,..., x in )), for any permutation (i 1,..., i n ). Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 28 / 37

37 Axioms for Natural Risk Statistic Data x = (x 1,..., x n ) R n Risk statistic: a mapping from the data set to the real line: Axioms for natural risk statistic: ˆρ : R n R x ˆρ( x) Positive homogeneity and translation invariance: ˆρ(a x + b) = aˆρ( x) + b, x R n, a 0, b R Monotonicity: ˆρ( x) ˆρ(ỹ), if x ỹ, i.e., x i y i, i = 1,...,n. Comonotonic subadditivity: ˆρ( x + ỹ) ˆρ( x) + ˆρ(ỹ), if x and ỹ are comonotonic, i.e., (x i x j )(y i y j ) 0 for any i j. Empirical law invariance: ˆρ((x 1,..., x n )) = ˆρ((x i1,..., x in )), for any permutation (i 1,..., i n ). Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 28 / 37

38 Representation Theorem for Natural Risk Statistic Theorem Let x (1),..., x (n) be the order statistics of the data x = (x 1,..., x n ) with x (n) being the largest. Then ˆρ is a natural risk statistic if and only if there exists a set of weights W = { w = (w 1,..., w n )} R n with each w W satisfying n i=1 w i = 1 and w i 0, 1 i n, such that ˆρ( x) = sup { w W n w i x (i) }, x R n. i=1 Natural risk statistic includes VaR and TCM Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 29 / 37

39 Representation Theorem for Natural Risk Statistic Theorem Let x (1),..., x (n) be the order statistics of the data x = (x 1,..., x n ) with x (n) being the largest. Then ˆρ is a natural risk statistic if and only if there exists a set of weights W = { w = (w 1,..., w n )} R n with each w W satisfying n i=1 w i = 1 and w i 0, 1 i n, such that ˆρ( x) = sup { w W n w i x (i) }, x R n. i=1 Natural risk statistic includes VaR and TCM Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 29 / 37

40 Another Representation via Acceptance Sets Representation of natural risk statistic via acceptance sets: ˆρ( x) = inf{m x m A}, x R n. A statistical acceptance set is a subset of R n : The acceptance set A contains R n where R n = { x Rn x i 0, i = 1,...,n}. The acceptance set A does not intersect the set R n ++ where R n ++ = { x R n x i > 0, i = 1,...,n}. If x and ỹ are comonotonic and x A, ỹ A, then λ x + (1 λ)ỹ A, for λ [0, 1]. The acceptance set A is positively homogeneous, i.e., if x A, then λ x A for all λ 0. If x ỹ and ỹ A, then x A. If x A, then (x i1,...,x in ) A for any permutation (i 1,..., i n ). Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 30 / 37

41 Another Representation via Acceptance Sets Representation of natural risk statistic via acceptance sets: ˆρ( x) = inf{m x m A}, x R n. A statistical acceptance set is a subset of R n : The acceptance set A contains R n where R n = { x Rn x i 0, i = 1,...,n}. The acceptance set A does not intersect the set R n ++ where R n ++ = { x R n x i > 0, i = 1,...,n}. If x and ỹ are comonotonic and x A, ỹ A, then λ x + (1 λ)ỹ A, for λ [0, 1]. The acceptance set A is positively homogeneous, i.e., if x A, then λ x A for all λ 0. If x ỹ and ỹ A, then x A. If x A, then (x i1,...,x in ) A for any permutation (i 1,..., i n ). Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 30 / 37

42 Coherent Risk Statistic and Representation Axioms for coherent risk statistic Positive homogeneity and translation invariance: ˆρ(a x + b) = aˆρ( x) + b, x R n, a 0, b R Monotonicity: ˆρ( x) ˆρ(ỹ), if x ỹ, i.e., x i y i, i = 1,...,n. Subadditivity: ˆρ( x + ỹ) ˆρ( x) + ˆρ(ỹ), for x and ỹ. Proposition: ˆρ is a coherent risk statistic if and only if there exists a set of weights W = { w = (w 1,...,w n )} R n, such that ˆρ( x) = sup { w W n w i x i }, x R n. i=1 Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 31 / 37

43 Coherent Risk Statistic and Representation Axioms for coherent risk statistic Positive homogeneity and translation invariance: ˆρ(a x + b) = aˆρ( x) + b, x R n, a 0, b R Monotonicity: ˆρ( x) ˆρ(ỹ), if x ỹ, i.e., x i y i, i = 1,...,n. Subadditivity: ˆρ( x + ỹ) ˆρ( x) + ˆρ(ỹ), for x and ỹ. Proposition: ˆρ is a coherent risk statistic if and only if there exists a set of weights W = { w = (w 1,...,w n )} R n, such that ˆρ( x) = sup { w W n w i x i }, x R n. i=1 Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 31 / 37

44 Law Invariant Coherent Risk Statistic Axioms for law invariant coherent risk statistic Positive homogeneity and translation invariance: ˆρ(a x + b) = aˆρ( x) + b, x R n, a 0, b R Monotonicity: ˆρ( x) ˆρ(ỹ), if x ỹ, i.e., x i y i, i = 1,...,n. Subadditivity: ˆρ( x + ỹ) ˆρ( x) + ˆρ(ỹ), for x and ỹ. Empirical law invariance: ˆρ((x 1,..., x n )) = ˆρ((x i1,..., x in )), for any permutation (i 1,..., i n ). Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 32 / 37

45 Representation Theorem for Law Invariant Coherent Risk Statistic Theorem ˆρ is a law invariant coherent risk statistic if and only if there exists a set of weights W = { w = (w 1,..., w n )} R n with each w W satisfying n i=1 w i = 1 and w i 0, 1 i n and w 1 w 2... w n, such that ˆρ( x) = sup { w W n w i x (i) }, x R n. i=1 Assigning larger weights to larger observations leads to less robust risk statistics Natural risk statistic is more robust Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 33 / 37

46 Representation Theorem for Law Invariant Coherent Risk Statistic Theorem ˆρ is a law invariant coherent risk statistic if and only if there exists a set of weights W = { w = (w 1,..., w n )} R n with each w W satisfying n i=1 w i = 1 and w i 0, 1 i n and w 1 w 2... w n, such that ˆρ( x) = sup { w W n w i x (i) }, x R n. i=1 Assigning larger weights to larger observations leads to less robust risk statistics Natural risk statistic is more robust Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 33 / 37

47 Insurance Risk Statistic Axioms for insurance risk statistic Monotonicity: ˆρ( x) ˆρ(ỹ), if x ỹ, i.e., x i y i, i = 1,...,n. Comonotonic additivity: ˆρ( x + ỹ) = ˆρ( x) + ˆρ(ỹ), if x and ỹ are comonotonic. Scale normalization: ˆρ(1) = 1. Empirical law invariance: ˆρ((x 1,..., x n )) = ˆρ((x i1,..., x in )), for any permutation (i 1,..., i n ). Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 34 / 37

48 Representation Theorem for Insurance Risk Statistic Theorem ˆρ is an insurance risk statistic if and only if there exists a single weight w = (w 1,...,w n ) with w i 0 for i = 1,...,n and n i=1 w i = 1, such that n ˆρ( x) = w i x (i), x R n. i=1 Insurance risk statistic is just one L-statistic. Insurance risk statistic cannot incorporate different scenarios. Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 35 / 37

49 Summary We propose a new risk measure, the natural risk statistic It requires comonotonic subadditivity instead of subadditivity It provides an axiomatic justification for VaR It includes tail conditional median (TCM) which is more robust than TCE. It is able to incorporate scenario analysis. Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 36 / 37

50 Thank you! Heyde, Kou and Peng (Columbia Univ.) Natural Risk Statistic 37 / 37