swans and other catastrophes

Transcription

1 2. Risk management in L 0, black swans and other catastrophes José Garrido Department of Mathematics and Statistics Concordia University, Montreal, Canada III Congreso Internacional de Actuaría Universidad de los Andes November 21 22, 2011 (joint work with Prof. A. Balbás, U. Carlos III of Madrid, Spain) Research funded by the Natural Sciences and Engineering Research Council of Canada (NSERC)

2 2. Risk management in L 0 [Source: The (CAS) Actuarial Review, May 2011]

3 2. Risk management in L 0 Mathematics and the actuary

4 2. Risk management in L 0 [Source: The (CAS) Actuarial Review, August 2011]

5 2. Risk management in L 0 The perfect answer and the actuary

6 2. Risk management in L 0 Abstract Risk measures are commonly used now in actuarial and financial risk management alike, for problems such as pricing, reinsurance, capital allocations, portfolio management or credit risk.

7 2. Risk management in L 0 Abstract Risk measures are commonly used now in actuarial and financial risk management alike, for problems such as pricing, reinsurance, capital allocations, portfolio management or credit risk. With the notable exception of Value at Risk (VaR), most well accepted measures apply only to risks with finite moments. Mathematically this restricts the set of risks random variables to L p, for some p 1, which excludes heavy tailed risks in L 0.

8 2. Risk management in L 0 Abstract Risk measures are commonly used now in actuarial and financial risk management alike, for problems such as pricing, reinsurance, capital allocations, portfolio management or credit risk. With the notable exception of Value at Risk (VaR), most well accepted measures apply only to risks with finite moments. Mathematically this restricts the set of risks random variables to L p, for some p 1, which excludes heavy tailed risks in L 0. Without getting into the subjective choice of what properties are reasonable for a risk measure, we revisit the risk management problem for heavy tailed risks through a personal survey of ideas in functional analysis and convex optimization.

9 2. Risk management in L 0 Overview: 1.1 Introduction 1.2 Random variables in L p 1.3 Risk measures in L p 2. Risk management in L Heavy tail risks 2.2 Random variables in L Risk measures in L 0

10 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p 1.1 Introduction Does randomness really exist in nature or is it just a convenient mathematical model?

11 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p 1.1 Introduction Does randomness really exist in nature or is it just a convenient mathematical model? For instance, the weather,

12 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p 1.1 Introduction Does randomness really exist in nature or is it just a convenient mathematical model? For instance, the weather, genetic mutations,

13 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p 1.1 Introduction Does randomness really exist in nature or is it just a convenient mathematical model? For instance, the weather, genetic mutations, birth/death?

14 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p 1.1 Introduction Does randomness really exist in nature or is it just a convenient mathematical model? For instance, the weather, genetic mutations, birth/death? Oxford English Dictionary 1 randomness: Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice; haphazard. 1

15 What is a risk? 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p

16 What is a risk? 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p (Source: a life threatening hazard?

17 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p What is a heavy tail risk?

18 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p What is a heavy tail risk? OPINION SHAUN WANG The Coming Storms On a sunny day in the fall of 2008, the natural surroundings of my Atlanta home were so tranquil and beautiful that it made me forget for a moment all the financial market turmoil that was going on. Fast forward to early The U.S. economy now appears to be on a gradual recovery, especially if you look at the impressive run up of the stock markets. Suddenly on March 11, 2011, a devastating earthquake and tsunami hit Japan. The horrific scenes of natural disaster and human tragedy made my stomach wrench. Now, looking ahead to the coming years, I sense an impending storm gathering. It is likely a perfect storm, combining natural disasters on a larger scale than the Japan earthquake and a second-dip market meltdown worse than the fall 2008 financial crisis. The clouds for the coming storm are visible: 1) Four of the five costliest earthquakes and tsunamis of the last 30 years have occurred in the last 13 months. There is a geophysical linkage (via crustal plates) of Chile, New Zealand, Japan, and the Northwest U.S. as the Pacific Ring of Fire. Given the recent earthquakes in Haiti (January 12, 2010), Chile (February 27, 2010), New Zealand (September 4, 2010, and February 21, 2011), and Japan (March 11, 2011), the conditional probability of an earthquake within the next year in the northwest U.S. has increased significantly due to the changing pressures on the crustal plates. 1,2 2) The world is on a brink of severe shortage of food and water, due to growing demand that supplies cannot keep up with. The world population has reached a new peak of nearly seven billion. The population growth rate is faster than exponential growth, given that the one billion mark was first reached around the year This population growth coincides with industrial consumptions for agricultural and water resources. Meanwhile the earth is facing the prospect of severe drought in the arable land areas (despite the floods in Australia). 3 Severe drought and water scarcity will affect the food availability and cause spikes in food prices. 3) There are other potential threats of natural and manmade disasters. NASA warns that solar flares from a huge space storm might cause devastation. Solar flares would be like a bolt of lightning and may cause disruptions to the communication and navigation systems. AR readers may have seen the profile of Nolan Asch s nonactuarial pursuits in the February issue ( Saving the World from Asteroids ). These may be less likely than the ring of fire earthquake scenarios, but nevertheless represent possible 1 earth-core.html. 2 3 The Actuarial Review [Source: The (CAS) Actuarial Review, May 2011]

19 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p

20 Risk 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Wikipedia 1 : Risk is the potential that a chosen action or activity (including the choice of inaction) will lead to a loss (an undesirable outcome). The notion implies that a choice having an influence on the outcome exists (or existed). Potential losses themselves may also be called risks. Almost any human endeavour carries some risk, but some are much more risky than others. 1

21 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Mathematical formalism A financial risk is defined as a loss random variable. A random variable X : Ω R is a real valued function, defined on a sample space Ω, that is F-measurable (a σ-algebra of events). A probability measure P, on F, completes the definition of a probability space (Ω, F, P) and induces a probability measure P on the Borel sets σ(r): P{X x} = P{ω Ω; X (ω) x}, x R and a probability distribution F X of X on R as follows: F X (x) = P{X x}, x R.

22 Examples: 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p A financial loss is, for instance, the difference X t = A P t between the acquisition cost of a portfolio, A, and its current market value P t.

23 Examples: 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p A financial loss is, for instance, the difference X t = A P t between the acquisition cost of a portfolio, A, and its current market value P t. A pension fund manager needs to assess the risk of his/her portfolio. What losses are tolerable, which ones are not?

24 Examples: 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p A financial loss is, for instance, the difference X t = A P t between the acquisition cost of a portfolio, A, and its current market value P t. A pension fund manager needs to assess the risk of his/her portfolio. What losses are tolerable, which ones are not? How can he/she measure the loss risk?

25 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p 1.2 Random variables in L p Take p [1, ) and q (1, ) to be conjugate numbers with q = if p = 1. 1 p + 1 q = 1

26 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p 1.2 Random variables in L p Take p [1, ) and q (1, ) to be conjugate numbers with q = if p = 1. 1 p + 1 q = 1 L p (L q ) the usual Banach space of random variables such that E( X p ) < (E( X q ) <, or if p = 1, then X is essentially bounded, as q = ).

27 The dual 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Note that these form a decreasing sequence of spaces 1 p 1 p 2 = L p 1 L p 2.

28 The dual 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Note that these form a decreasing sequence of spaces 1 p 1 p 2 = L p 1 L p 2. By Riesz representation theorem: L q is the dual space of L p (Rudin, 1973, McG-H).

29 1.3 Risk measures in L p 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Risk measures are commonly characterized through axioms: homogeneity, translation invariance, sub additivity, etc. These being properties of the risk function.

30 1.3 Risk measures in L p 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Risk measures are commonly characterized through axioms: homogeneity, translation invariance, sub additivity, etc. These being properties of the risk function. In the mathematical finance literature, risk measures are also represented through classical optimization arguments used in functional analysis. The focus being on the space of random variables (risks), rather than on the properties of a particular risk measure.

31 1.3 Risk measures in L p 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Risk measures are commonly characterized through axioms: homogeneity, translation invariance, sub additivity, etc. These being properties of the risk function. In the mathematical finance literature, risk measures are also represented through classical optimization arguments used in functional analysis. The focus being on the space of random variables (risks), rather than on the properties of a particular risk measure. The two approaches are equivalent!

32 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Families of risk measures A risk measure is a function ρ : L p R used to control the risk level of loss X L p, say at the end of a period [0, T ].

33 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Families of risk measures A risk measure is a function ρ : L p R used to control the risk level of loss X L p, say at the end of a period [0, T ]. What real valued function ρ is a good risk measure?

34 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Families of risk measures A risk measure is a function ρ : L p R used to control the risk level of loss X L p, say at the end of a period [0, T ]. What real valued function ρ is a good risk measure? To extend the discussion to insurance losses consider general risk functions used by traders/insurers to control the risk of a final wealth X L p at the end of a period [0, T ].

35 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Families of risk measures A risk measure is a function ρ : L p R used to control the risk level of loss X L p, say at the end of a period [0, T ]. What real valued function ρ is a good risk measure? To extend the discussion to insurance losses consider general risk functions used by traders/insurers to control the risk of a final wealth X L p at the end of a period [0, T ]. The set is convex. ρ = {Y L q ; E(XY ) ρ(x ), X L p },

36 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Families of risk measures A risk measure is a function ρ : L p R used to control the risk level of loss X L p, say at the end of a period [0, T ]. What real valued function ρ is a good risk measure? To extend the discussion to insurance losses consider general risk functions used by traders/insurers to control the risk of a final wealth X L p at the end of a period [0, T ]. The set is convex. ρ = {Y L q ; E(XY ) ρ(x ), X L p }, With the representation theorem of risk measures of Rockafellar et al. (2006, Fin. & Stoch.) the following theorem can be proved.

37 Theorem Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p The 2 following assertions are equivalent: 1. ρ is convex and σ(l q, L p )-compact, ρ(x ) = max{ E(XY ) : Y ρ } holds for every X L p, there exists Ẽ 0 in R such that ρ {Y L q ; E(Y ) = Ẽ}, and the constant (0-variance) random variable Y = Ẽ a.s. belongs to ρ.

38 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Theorem 1 (... continued) 2. ρ is continuous and satisfies: (a) (translation invariance) ρ(x + k) = ρ(x ) Ẽk, for every X L p and k R, (b) (sub aditivity) ρ(x 1 + X 2 ) ρ(x 1 ) + ρ(x 2 ), for every X 1, X 2 L p, (c) (homogeneity) ρ(αx ) = αρ(x ), for every X L p and α > 0, (d) (mean dominated) ρ(x ) Ẽ E(X ), for every X L p.

39 Coherent risk measures 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p It is easily shown that if ρ satisfies (a) (d) above with Ẽ = 1 then it is coherent in the sense of Artzner et al. (1999, Math. Fin.). That is ρ is decreasing if and only if ρ L q + = {Y L q ; P(Y 0) = 1}.

40 Examples 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Interesting examples of coherent continuous risk measures satisfying (a) (d) above with Ẽ = 1 are (among many others): Conditional Value at Risk (CVaR): defined in L 1 CVaR α (X ) = E [ X X < VaR α (X ) ], where VaR α (X ) = inf{x ; P(X x) > 1 α} can be extended even on L 0, but is not coherent,

41 More examples 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Weighted Value at Risk (WVaR): defined on L 1 WVaR g (X ) = 1 0 VaR t(x )dg(t), for an appropriate weighting function such that g(1) g(0) = 1,

42 More examples 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Weighted Value at Risk (WVaR): defined on L 1 WVaR g (X ) = 1 0 VaR t(x )dg(t), for an appropriate weighting function such that g(1) g(0) = 1, for example the Dual Power Transform (DPT a (X )) is based on WVaR for g(t) = 1 (1 t) a, a > 1,

43 More examples 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Weighted Value at Risk (WVaR): defined on L 1 WVaR g (X ) = 1 0 VaR t(x )dg(t), for an appropriate weighting function such that g(1) g(0) = 1, for example the Dual Power Transform (DPT a (X )) is based on WVaR for g(t) = 1 (1 t) a, a > 1, or the measure of Wang (2000, J.Risk & Ins.) for g(t) = Φ [ a + Φ 1 (t) ], a < 0, is defined on L 2.

44 More examples 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Weighted Value at Risk (WVaR): defined on L 1 WVaR g (X ) = 1 0 VaR t(x )dg(t), for an appropriate weighting function such that g(1) g(0) = 1, for example the Dual Power Transform (DPT a (X )) is based on WVaR for g(t) = 1 (1 t) a, a > 1, or the measure of Wang (2000, J.Risk & Ins.) for g(t) = Φ [ a + Φ 1 (t) ], a < 0, is defined on L 2. Weighted Conditional Value at Risk (WCVaR): WCVaR a (X ) = 1 CVaR 0 t(x )dg(t), of Rockafeller et al. (2006, Fin. & Stoch.) is also defined on L 1.

45 Other representations 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p As properties (a) (d) do not draw consensus, other families of risk measures have been proposed, again motivated by natural properties these must satisfy, and then justified through representation theorems like Theorem 1.

46 Other representations 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p As properties (a) (d) do not draw consensus, other families of risk measures have been proposed, again motivated by natural properties these must satisfy, and then justified through representation theorems like Theorem 1. Particular examples are the expectation bounded and the deviation risk measures of Rockafeller et al. (2006, Fin. & Stoch.), which are continuous risk measures satisfying (a) (d) above with Ẽ = 0, such as the p deviation, given by ρ(x ) = [ E ( E(X ) X p ) ] 1/p,

47 Deviation measures 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p or the downside p semi deviation: ρ(x ) = again, among many others. [ E ( max{e(x ) X, 0} p ) ] 1/p,

48 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Other applications of this representation Such representations of risk measures based on convex optimization have been successfully used in: reinsurance: Balbás et al. (2009, IME and 2011, EJOR) study the optimal (minimal risk) reinsurance design for a broad class of risk measures,

49 1.1 Introduction 2. Risk management in L Random variables in L p 1.3 Risk measures in L p Other applications of this representation Such representations of risk measures based on convex optimization have been successfully used in: reinsurance: Balbás et al. (2009, IME and 2011, EJOR) study the optimal (minimal risk) reinsurance design for a broad class of risk measures, credit risk: Okhrati et al. (2011, pre-print) define a ρ arbitrage for bond markets under general (finite variation Lévy) contingent claims. Among other things, it can be used for credit rating of corporate bonds.

50 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 2. Risk management in L 0 All the above risk measures require at least a finite mean of the final wealth; X L 1.

51 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 2. Risk management in L 0 All the above risk measures require at least a finite mean of the final wealth; X L 1. Actuarial risk can be heavy tailed (reinsurance costs in Japan). Figure: Pareto distributions (Source: distribution)

52 2.1 Heavy tail risks 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 For X Pareto(k, θ): θ E(X ) = k 1, k > 1, θ V(X ) = E(X ) (k 2), k > 2, VaR α (X ) = θ [ (1 α) 1/k 1 ], CVaR α (X ) = VaR α (X ) + θ(1 α) 1/k, k > 1. k 1

53 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Financial heavy tail risks Financial risks can exhibit heavier tails than the exponential decaying tails of normal distributions (credit risk crisis, black swans).

54 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Financial heavy tail risks Financial risks can exhibit heavier tails than the exponential decaying tails of normal distributions (credit risk crisis, black swans). Figure: Symmetric α-stable distributions (Source: distributions)

55 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 What is a heavy tail random variable? In the actuarial literature a heavy tail distribution usually means a subexponential distribution: 1 F 2 (x) lim x 1 F (x) = 2.

56 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 What is a heavy tail random variable? In the actuarial literature a heavy tail distribution usually means a subexponential distribution: 1 F 2 (x) lim x 1 F (x) = 2. Subexponential distributions do not admit a moment generating function, which is used by some as a definition of heavy tail.

57 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 What is a heavy tail random variable? In the actuarial literature a heavy tail distribution usually means a subexponential distribution: 1 F 2 (x) lim x 1 F (x) = 2. Subexponential distributions do not admit a moment generating function, which is used by some as a definition of heavy tail. Subexponential distributions form a wide class that includes slowly varying tail distributions: 1 F (x) x γ C(x), x.

58 Other definitions 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Some authors call light tailed the random variables that admit a moment generating function only on a limited domain: E(e tx ) <, t ( γ 1, γ 2 ), such as the exponential/gamma or the inverse Gaussian on R +.

59 Other definitions 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Some authors call light tailed the random variables that admit a moment generating function only on a limited domain: E(e tx ) <, t ( γ 1, γ 2 ), such as the exponential/gamma or the inverse Gaussian on R +. Others called them medium tailed.

60 Other definitions 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Some authors call light tailed the random variables that admit a moment generating function only on a limited domain: E(e tx ) <, t ( γ 1, γ 2 ), such as the exponential/gamma or the inverse Gaussian on R +. Others called them medium tailed. In any case, the consensus is that distributions that admit a moment generating over all of R are light tail and have finite moments E( X p ) < of all orders (Poisson, normal).

61 Other definitions 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Some authors call light tailed the random variables that admit a moment generating function only on a limited domain: E(e tx ) <, t ( γ 1, γ 2 ), such as the exponential/gamma or the inverse Gaussian on R +. Others called them medium tailed. In any case, the consensus is that distributions that admit a moment generating over all of R are light tail and have finite moments E( X p ) < of all orders (Poisson, normal). Then in statistics and in applied probability they sometimes call heavy tailed any random variable that is not light tailed.

62 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L Random variables in L 0 Is it realistic to use random variables with E( X ) =?

63 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L Random variables in L 0 Is it realistic to use random variables with E( X ) =? Or is it just a convenient model for heavy tailed losses with random variables in L 0, the set of all random variables on (Ω, F, P)?

64 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 (a) Risk theory

65 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 (a) Risk theory (b) L 0 risks

66 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 (a) Risk theory (b) L 0 risks

67 Dual of L Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Can Theorem 1 be extended beyond L 1 to the larger L 0?

68 Dual of L Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Can Theorem 1 be extended beyond L 1 to the larger L 0? In other words, can we define coherent risk measures or deviation measures for risks with infinite expectation?

69 Dual of L Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Can Theorem 1 be extended beyond L 1 to the larger L 0? In other words, can we define coherent risk measures or deviation measures for risks with infinite expectation? The answer to the first question is no: for 0 r < 1, L r is a metric but not Banach space (see Rudin, 1973).

70 Dual of L Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Can Theorem 1 be extended beyond L 1 to the larger L 0? In other words, can we define coherent risk measures or deviation measures for risks with infinite expectation? The answer to the first question is no: for 0 r < 1, L r is a metric but not Banach space (see Rudin, 1973). Its dual reduces to zero if P is atomless, so ρ {0} and Theorem 1 leads to trivial risk measures.

71 Theorem Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Let P be atomless and 0 r < 1. If ρ : L r R is continuous and satisfies (b) and (c) of Theorem 1, then ρ = 0.

72 Theorem Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Let P be atomless and 0 r < 1. If ρ : L r R is continuous and satisfies (b) and (c) of Theorem 1, then ρ = 0. The proof follows from Delbaen (2000). He proposes approximations to substitute for the lack of characterization.

73 Theorem Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Let P be atomless and 0 r < 1. If ρ : L r R is continuous and satisfies (b) and (c) of Theorem 1, then ρ = 0. The proof follows from Delbaen (2000). He proposes approximations to substitute for the lack of characterization. The only risk measures that can be used on L 0 are those based on percentiles, like VaR, as there are no continuous functions on L r, hence no dual elements.

74 Naive solution 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Statisticians routinely use variance reduction transforms, such as ln X or X. Actuaries use similar tail reduction transforms (Y = 1/X ).

75 Naive solution 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Statisticians routinely use variance reduction transforms, such as ln X or X. Actuaries use similar tail reduction transforms (Y = 1/X ). These may lead to simple approximations when X / L 1 : E(X ) = E( 1 Y ) 1 [ σ(y )] 1 +, E(Y ) 3 where σ(y) = V(Y )/E(Y ) 2 is the coefficient of variation of Y.

76 2.3 Risk measures in L Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Nešlehová, Chavez Demoulin and Embrechts (2006, JOR): Define empirical risk measures for loss random variables that include the Pareto(α, θ) case with bad exponent α < 1.

77 2.3 Risk measures in L Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Nešlehová, Chavez Demoulin and Embrechts (2006, JOR): Define empirical risk measures for loss random variables that include the Pareto(α, θ) case with bad exponent α < 1. These random variables show up in operational risk studies.

78 2.3 Risk measures in L Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Nešlehová, Chavez Demoulin and Embrechts (2006, JOR): Define empirical risk measures for loss random variables that include the Pareto(α, θ) case with bad exponent α < 1. These random variables show up in operational risk studies. If, for risk capital allocation, there is a need for concave (even coherent) risk measures then there is no immediate solution for this problem.

79 Other penguins Delbaen (2009, MF): 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 We show that when a real valued risk measure is defined on a solid, rearrangement invariant space of random variables, then necessarily it satisfies a weak compactness, also called continuity from below, property, and the space necessarily consists of integrable random variables...

80 Other penguins Delbaen (2009, MF): 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 We show that when a real valued risk measure is defined on a solid, rearrangement invariant space of random variables, then necessarily it satisfies a weak compactness, also called continuity from below, property, and the space necessarily consists of integrable random variables As a result we see that a risk measure defined for, say, Cauchy distributed random variable, must take infinite values for some of the random variables.

81 Other penguins Delbaen (2009, MF): 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 We show that when a real valued risk measure is defined on a solid, rearrangement invariant space of random variables, then necessarily it satisfies a weak compactness, also called continuity from below, property, and the space necessarily consists of integrable random variables As a result we see that a risk measure defined for, say, Cauchy distributed random variable, must take infinite values for some of the random variables. Uses conjugate Young functions to transform X L 0 into an element of an Orlicz space. The risk measure is E(X ) (linear, positive) when restricted to the Orlicz space.

82 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Other penguins (... continued) In Balbás & G (2011) we look for a representation, like that in Theorem 1, but in L r, for 0 r < 1, that can overcome Theorem 2.

83 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Other penguins (... continued) In Balbás & G (2011) we look for a representation, like that in Theorem 1, but in L r, for 0 r < 1, that can overcome Theorem 2. We propose an extension ρ : L p ( k j=1 U j) R, as described in the following theorem.

84 Theorem Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Consider 0 r < 1 p < and a family of linear spaces U 1,... U k, such that U j L r, for j = 1, 2,..., k, and {U 1,..., U k, L p } Fréchet (locally convex, complete and metric) satisfying: u j U j, for j = 1,..., k k j=1 u j = 0 for every (u 1,..., u k ) L p. } u 1 = = u k = 0

85 Theorem Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Consider 0 r < 1 p < and a family of linear spaces U 1,... U k, such that U j L r, for j = 1, 2,..., k, and {U 1,..., U k, L p } Fréchet (locally convex, complete and metric) satisfying: u j U j, for j = 1,..., k k j=1 u j = 0 } u 1 = = u k = 0 for every (u 1,..., u k ) L p. Now suppose that ρ j : L p R are continuous and satisfy the properties of (sub aditivity) ρ j (v 1 + v 2 ) ρ j (v 1 ) + ρ j (v 2 ), v 1, v 2 L p, (1) and (homogeneity) ρ j (αv) = αρ j (v), v L p. (2)

86 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Theorem 3 (... continued) Then there exists ρ : L p ( k j=1 U j) R such that: a) ρ extends every ρ j,

87 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Theorem 3 (... continued) Then there exists ρ : L p ( k j=1 U j) R such that: a) ρ extends every ρ j, b) ρ is continuous and also satisfies (1) and (2),

88 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Theorem 3 (... continued) Then there exists ρ : L p ( k j=1 U j) R such that: a) ρ extends every ρ j, b) ρ is continuous and also satisfies (1) and (2), c) if θ : L p ( k j=1 U j) R satisfies a) and b), above, and θ ρ, then θ = ρ (that is ρ is minimal).

89 Conclusion 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 With the construction in Theorem 3, extensions of coherent/expectation bounded risk measures are feasible beyond L 1.

90 Conclusion 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 With the construction in Theorem 3, extensions of coherent/expectation bounded risk measures are feasible beyond L 1. In particular, an extension of CVaR is possible which amounts to the CVaR for risks in L 1 and VaR for heavy tailed risks, through a special parameter.

91 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Thank you for your attention!

92 Bibliography 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Artzner, P., Delbaen, F., Eber, J-M. and D. Heath (1999) Coherent measures of risk, Mathematical Finance, 9, 3, Balbás, A., Balbás, B. and A. Heras (2009) Optimal reinsurance with general risk measures involving risk measures, European Journal of Operational Research, 1 29, Balbás, A., Balbás, B. and A. Heras (2011) Stable solutions for optimal reinsurance problems, Insurance: Mathematics and Economics, 44, Balbás, A. and J. Garrido (2011) Heavy tails and risk measures, preprint. Delbaen, F. (2000) Coherent risk measures, preprint.

93 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Bibliography (... continued) Delbaen, F. (2009) Risk measures for non integrable random variables, Mathematical Finance, 19, Nešlehová, J., Chavez Demoulin, V. and Embrechts, P. (2006) Infinite-mean models and the LDA for operational risk, Journal of Operational Risk, 1, 1, Okhrati, R., Balbás, A. and J. Garrido (2011) Defaultable claims under finite variation Lévy processes, preprint, submitted to Stochatic Processes and their Applications. Rockafellar, R.T., Uryasev, S. and M. Zabarankin (2006) Generalized deviations in risk analysis, Finance and Stochastics, 10, Rudin, W. (1973) Functional Analysis, McGraw Hill.

94 Bibliography (... end) 2.1 Heavy tail risks 2. Risk management in L Random variables in L Risk measures in L 0 Wang, S.S. (2000) A class of distortion operators for financial and insurance risks, Journal of Risk Insurance, 6, 7,