VaR vs. Expected Shortfall

Transcription

1 VaR vs. Expected Shortfall Risk Measures under Solvency II Dietmar Pfeifer (2004)

2 Risk measures and premium principles a comparison VaR vs. Expected Shortfall Dependence and its implications for risk measures A (frightening?) example from theory A (frightening?) example from the real world Conclusions 2

3 1. Risk measures and premium principles a comparison A premium principle H resp. a risk measure R is a non-negative mapping on Z, the set of non-negative risks, with the property X Y P = P H( X) = H( Y) for all XY, Z, i.e. the premium resp. the risk measure depends only on the distribution of the risk. Example: H( X) = R( X) = E( X) [Expected risk; net risk premium; average risk] 3

4 1. Risk measures and premium principles a comparison A premium principle H is called positively loaded (pl), if: positively homogeneous (ph), if: additive (ad), if: H( X) E( X) for all X Z ; HcX ( ) = chx ( ) for all c 0 and X Z ; H( X + Y) = H( X) + H( Y) for all XY, Z, with X, Y being stochastically independent; 4

5 total loss bounded (tb), if: where Dietmar Pfeifer 1. Risk measures and premium principles a comparison { } H( X) ϖ : = sup x F ( x) < 1 for all X Z, X F X denotes the cumulative distribution function of X; X stochastically increasing (si), if: H( X) H( Y) for all XY, Z being stochastically ordered, i.e. F ( x) F ( x) for all x. X Remark: Each additive premium principle H also is translation invariant (ti), i.e. H( X + c) = H( X) + c for all X Z and c. Y 5

6 1. Risk measures and premium principles a comparison Let δ 0. Then the premium principle H with H( X) = (1 + δ) E( X), X Z is called expectation principle (ExP) with safety loading factor δ. Let δ 0. If the variance Var( X ) exists, then the premium principle H with H( X) = E( X) + δvar( X), X Z is called variance principle (VaP) with safety loading factor δ. Let δ 0. If the variance Var( X ) exists, then the premium principle H with H( X) = E( X) + δ Var( X), X Z is called standard deviation principle (StP) with safety loading factor δ. 6

7 1. Risk measures and premium principles a comparison Let g : H with be strictly increasing and convex. Then the premium principle ( ( )) H( X) = g E g( X), X Z is called mean value principle (MvP) w.r.t. g. Let g : be strictly increasing. Then the premium principle H with + + is called Esscher principle (EsP) w.r.t g. [ ( )] E[ g( X) ] EX gx H( X) =, X Z 7

8 1. Risk measures and premium principles a comparison Let α ( 0,1 ). Then the premium principle H with 1 ( α) { X α} + H( X) = F 1 = inf x F ( x) 1, X Z X is called percentile principle (PcP) at risk level α. This premium principle is also known as Value at Risk at risk level α resp. as VaR α ( PML). 8

9 1. Risk measures and premium principles a comparison property principle pl ph ad tb si ExP yes yes yes no yes VaP yes no yes no no StP yes yes no no no MvP yes no no yes yes EsP yes no no yes no PcP (VaR) no yes no yes yes Each premium principle can in principle also be considered as a risk measure. However, more specific attributes of risk measures have been developed in the context of the solvency capital for banking and insurance companies ( Basel II, Solvency II). 9

10 1. Risk measures and premium principles a comparison A risk measure R is called coherent according to ARTZNER, DELBAEN, EBER und HEATH, if it possesses the following properties: R is positively homogeneous (ph), i.e. R is translation invariant (ti), i.e. R ist sub-additive (sa), i.e. R is increasing (in), i.e. R( cx ) = c R( X ) for all c 0 and X Z ; R( X + c) = R( X) + c for all X Z and c ; R( X + Y) R( X) + R( Y) for all XY, Z ; R( X) R( Y) for all X, Y Z with X Y. 10

11 2. VaR vs. Expected Shortfall The international discussion of risk measures as a basis for the determination of the target capital for Solvency II (IAA, DAV, SST) concerning the total risk S = X (from assets and liabilities) clearly focuses on Value at Risk: VaRα = F 1 ( 1 α) = inf { x + F ( x) 1 α} S S n i= 1 i ( Life Insurance) and ( Nonlife Insurance). Expected Shortfall: ESα= E( S S> VaRα) 11

12 2. VaR vs. Expected Shortfall 12

13 2. VaR vs. Expected Shortfall Common Pro s in favour of Expected Shortfall against Value at Risk: ES α is a coherent risk measure, VaR α is not ( sub-additivity) ES α provides a quantification of the potential (high) loss, VaR α does not ES α enables a risk-adjusted additive capital allocation through ( i ) EXi: α = E X S> VaR α Theoretically o.k., but does it work in practice? 13

14 2. VaR vs. Expected Shortfall 14

15 3. Dependence and its implications for risk measures A function C of n variables on the unit n-cube [0,1] n is called a copula if it is a multivariate distribution function that has continuous uniform margins. Fréchet-Hoeffding bounds: max( u + + u n + 1, 0) Cu (,, u ) min( u,, u ) 1 n 1 n 1 n 15

16 3. Dependence and its implications for risk measures Theorem (Sklar). Let H denote a n-dimensional distribution function with margins F,,. 1 F n Then there exists a copula C such that for all real ( x1,, x n ), ( F( x, F ( )) H( x,, x ) = C ), x. 1 n 1 1 n n If all the margins are continuous, then the copula is unique, and is determined uniquely on the ranges of the marginal distribution functions otherwise. Moreover, if we denote 1 1 by F1,, Fn the generalized inverses of the marginal distribution functions, then for every ( u1,, u n ) in the unit n-cube, 1 1 ( F1 ( ) Cu (,, u) = H u ), F ( u ). 1 n 1 n n 16

17 3. Dependence and its implications for risk measures Familiar examples of copulas: Gauß: 1 1 Φ ( u1 ) Φ ( u n ) 1 1 tr 1 = 1 n v μ Σ v μ 1 n (2 π) det( ) 2 Σ C ( u,, u ) exp ( ) ( ) dv dv, Φ Σ positive-definite n Student s t: ν + n 1 1 ν+ n t ( u1 ) t ( u ) Γ ν ν n tr 1 C ( u,, u ) = 1 + ( ) Σ ( ) dv dv, t 1 n 1 n ν v μ v μ n ν Γ ( πν) det( Σ) 2 Σ positive-definite, ν 17

18 3. Dependence and its implications for risk measures Familiar examples of copulas (cont.): Clayton: n 1/ θ θ CCl ( u1,, un) = ui n+ 1, θ > 0 i= 1 Gumbel: n 1/ θ θ CGu ( u1,, un ) = exp ( ln( ui )), θ 1 i= 1 Frank: 1 n θu i θ e 1 CFr ( u1,, un) = ln 1 ( e 1 ), θ 0 θ θ + > i= 1 e 1 18

19 3. Dependence and its implications for risk measures General message: dependence resp. copula has essential influence on risk measures: Example: heavy-tailed risk distributions, Pareto-type with shape parameter 1 density f( x) =, x x 1 λ = : 2 cumulative distribution function 1 F( x) = 1, x 0 1+ x 19

20 3. Dependence and its implications for risk measures Case 1: two independent risks X,Y of the same type: f X z + Y( z) = 2 (2 + z) 1+ z 1 1+ z 3 Case 2: two maximally positively dependent risks X,Y of the same type: f X + Y 1 ( z) = z / z 3 Case 3: two maximally negatively dependent risks X,Y of the same type: f X+ Y ( z) = 4+ z 2 3+ z 1 6+ z 3+ z 4z z 1+ z ( )

21 3. Dependence and its implications for risk measures negative dependence positive dependence independence plot of survival functions for cases 1,2,3 21

22 3. Dependence and its implications for risk measures negative dependence ES α = α positive dependence VaR α independence plot of survival functions for cases 1,2,3 22

23 3. Dependence and its implications for risk measures Exact calculation of Value at Risk: Case 1: two independent risks X,Y of the same type: 4 2 VaR α = 2 α 1+ 1 α 2 2 Case 2: two maximally positively dependent risks X,Y of the same type: 2 VaR α = 2 2 α Case 3: two maximally negatively dependent risks X,Y of the same type: 4 4 VaR α = 2 α (2 α)

24 3. Dependence and its implications for risk measures Consequence: 1. Solvency capital for one portfolio consisting of two independent risks of the same type is strictly larger than the sum of the solvency capitals for two portfolios, each consisting of a single risk! no diversification effect! 24

25 3. Dependence and its implications for risk measures 2. Solvency capital for one portfolio consisting of two independent risks of the same type is asymptotically equivalent (for large return periods) to the solvency capital for one portfolio consisting of two negatively dependent risks of the same type! independence close to worst case! 25

26 3. Dependence and its implications for risk measures Example (cont.): heavy-tailed risk distributions, Pareto-type with shape parameter λ = 2: 2 density f( x) =, x 0 3 (1 + x) 1 cumulative distribution function F( x) = 1, x 0 2 (1 + x) here: EX ( + Y) = 2 26

27 3. Dependence and its implications for risk measures Case 1: two independent risks X,Y of the same type: f X+ Y 2 ( + z+ z ) 48ln(1 + z) 4z ( z) = + (2 + z) (2 + z) ( 1 + z) (1 + z) Case 2: two maximally positively dependent risks X,Y of the same type: f X + Y ( z) 8 z = 3 ( 2+ ) (1 + 8 z) 3 Case 3: two maximally negatively dependent risks X,Y of the same type: no closed form available 27

28 3. Dependence and its implications for risk measures Calculation of Value at Risk, special case α = 0,99 (i.e. 100 year return period): Case 1: two independent risks X,Y of the same type: VaR α = 14,14 ES = 28, 72 (numerical evaluation / simulation) α Case 2: two maximally positively dependent risks X,Y of the same type: VaR α = 18 ES = 38 (exact calculation) Case 3: two maximally negatively dependent risks X,Y of the same type: VaR α α = 13,15 ES = 27, 24 (estimated by simulation) α rule of thumb: for λ 2: ES VaR 0,99 0,99 2! 28

29 4. A (frightening?) example from theory We consider a portfolio with two risks X and Y and their distributions given by x PX ( = x) 0,90 0,09 0,01 y 1 5 (amounts in Mio. ) PY ( = y) 0,20 0,80 Risk level: α = 0,01 corresponding to a return period T of 100 years Risk X is "dangerous", risk Y is "harmless" 29

30 4. A (frightening?) example from theory Distribution of total risk S = X + Y under independence: s PS ( = s) 0,180 0,018 0,720 0,072 0,002 0,008 PS ( s) 0,180 0,198 0,918 0,990 0,992 1 This implies: VaR α = 8 and 101 0, , 008 ES α = = 104,2 0,01 x y 1 5 P( X = x) 0,90 0,09 0,01 PY ( = y) 0,20 0,80 30

31 4. A (frightening?) example from theory Risk-based capital allocation with ES α : s PS ( = s) 0,180 0,018 0,720 0,072 0,002 0,008 PS ( s) 0,180 0,198 0,918 0,990 0,992 1 ( ) ( ) EX = E X S > 8 = 100 EY = E Y S > 8 = 4,2 α α x y 1 5 PX ( = x S> 8) PY ( = y S> 8) = PY ( = y) 0,20 0,80 consequence: insufficient capital in 80% of all cases! 31

32 4. A (frightening?) example from theory What is the return period T corresponding to a risk (loss) of ES α = 104,2? s PS ( = s) 0,180 0,018 0,720 0,072 0,002 0,008 PS ( s) 0,180 0,198 0,918 0,990 0, T = = = ,992 0, 008 Consequence: a risk-based capital allocation with ES α = 104,2 increases the former return period of 100 years by only 25% to 125 years, while the capital requirement is 13-times as much as with a capital allocation based on VaR α = 8!! 32

33 4. A (frightening?) example from theory Proportional risk-based capital allocation with VaR α : s PS ( = s) 0,180 0,018 0,720 0,072 0,002 0,008 PS ( s) 0,180 0,198 0,918 0,990 0,992 1 EX α E( X S 8) 1,182 E( Y S 8) 4, 2 = 8 = 8 = 1, 757 EY = 8 = 8 = 6, 243 α ES ( S 8) 5, 382 ES ( S 8) 5, 382 x y 1 5 P( X = x S 8) 0,909 0,091 0 PY ( = y S 8) = PY ( = y) 0,20 0,80 consequence: insufficient capital in 9,1% of all cases! 33

34 4. A (frightening?) example from theory Optimal risk-based capital allocation: s PS ( = s) 0,180 0,018 0,720 0,072 0,002 0,008 PS ( s) 0,180 0,198 0,918 0,990 0,992 1 EX α = 3 EY = 5 α x y 1 5 P( X = x S 8) 0,909 0,091 0 PY ( = y S 8) = PY ( = y) 0,20 0,80 consequence: 8 Mio. capital cover both risks at α = 0, 01 optimally! 34

35 4. A (frightening?) example from theory Influence of dependencies ( copulas) on ES α and VaR α : PX ( = xy, = y) x = 1 x = 3 x = 100 y = 1 a a + b + 0,19 0,01 b 0,2 y = 5 0,9 a a b 0,1 b 0,8 0,90 0,09 0,01 with side conditions 0< b < 0,01 0,1+ b< a< 0,19 + b 35

36 4. A (frightening?) example from theory Distribution of total risk S = X + Y under dependence: s PS ( = s) a a + b + 0,19 0,9 a a b 0,1 0,01 b b PS ( s) a b + 0,19 1,09 a + b 0,99 1 b 1 implying VaR α = 8 and ES α 101 (0, 01 b) b = = b 0,01 Consequence: VaR α remains unchanged, ES α varies between 101 and 105! 36

37 4. A (frightening?) example from theory Possible reduction of ES α by reinsurance with priority VaR α : (net) reinsurance premium: + ( ) (( ) ) 0,93 4 ( 0,93 0,97) RV = α ES VaR = E S VaR = + b α a α α [reinsurance premium: add safety loading] Consequence: target capital reduces to roughly 10 Mio only! 37

38 Example company portfolio: Dietmar Pfeifer 5. A (frightening?) example from the real world location 1 location 2 34 years of data 18 years of data windstorm windstorm hailstorm climatic dependence (?) flooding spatial dependence 38

39 Marginal analysis location 1: > Indexing > Detrending Dietmar Pfeifer 5. A (frightening?) example from the real world 39

40 Distribution fitting location 1, windstorm: Dietmar Pfeifer 5. A (frightening?) example from the real world Fréchet distribution, max. deviation: 0,36637 Loglogistic distribution, max. deviation: 0,64749 Pearson type V distribution, max. deviation: 0,37747 Lognormal distribution, max. deviation: 0,

41 Anderson-Darling test: 5. A (frightening?) example from the real world Distribution type: Fréchet; test statistic: 0,18444 α 0,25 0,1 0,05 0,025 0,01 critical value 0,458 0,616 0,732 0,848 1,004 Distribution type: Pearson type V; test statistic: 0,17911 α 0,25 0,1 0,05 0,025 0,01 critical value 0,485 0,655 0,783 0,913 1,078 Distribution type: Loglogistic; test statistic: 0,34706 α 0,25 0,1 0,05 0,025 0,01 critical value 0,423 0,559 0,655 0,763 0,899 Distribution type: Lognormal; test statistic: 0,52042 α 0,25 0,1 0,05 0,025 0,01 critical value 0,459 0,616 0,734 0,853 1,011 41

42 2 χ test: Distribution type: Fréchet; test statistic: 0,58824 Dietmar Pfeifer 5. A (frightening?) example from the real world d.f. α 0,25 0,15 0,1 0,05 0,01 3 critical value 4,108 5,317 6,251 7,815 11,345 5 critical value 6,626 8,115 9,236 11,070 15,086 Distribution type: Pearson Type V; test statistic: 0,94118 d.f. α 0,25 0,15 0,1 0,05 0,01 3 critical value 4,108 5,317 6,251 7,815 11,345 5 critical value 6,626 8,115 9,236 11,070 15,086 Distribution type: Loglogistic; test statistic: 3,41176 d.f. α 0,25 0,15 0,1 0,05 0,01 3 critical value 4,108 5,317 6,251 7,815 11,345 5 critical value 6,626 8,115 9,236 11,070 15,086 Distribution type: Lognormal; test statistic: 3,41176 d.f. α 0,25 0,15 0,1 0,05 0,01 3 critical value 4,108 5,317 6,251 7,815 11,345 5 critical value 6,626 8,115 9,236 11,070 15,086 42

43 5. A (frightening?) example from the real world Summary of marginal statistical analysis: location 1 34 years of data location 2 18 years of data windstorm: Fréchet windstorm: Fréchet hailstorm: Lognormal climatic dependence (?) flooding: Lognormal spatial dependence 43

44 5. A (frightening?) example from the real world Dependence analysis: Spatial dependence location 1 / location 2 (windstorm): location 2 location 2 44

45 5. A (frightening?) example from the real world Estimation methods for bivariate Gumbel copula C ( u, v) λ : λ λ {( ) ( ln ) 1/ λ } Cλ ( u, v) = exp ( ln u + v), 0 < uv, 1, with a structural parameter λ 1. The corresponding density is given by ( ln u) ( ln v) c uv C uv C uv kuv kuv u v uv 2 λ 1 λ 1 1/ λ 2 1/ λ λ(, ) = λ(, ) = λ(, ) (, ) λ 1 + (, ) λ λ with kuv (, ) = ( ln u) + ( ln v),0 < uv, 1. For λ = 1, the independence copula is obtained. 45

46 5. A (frightening?) example from the real world Method I: Use a functional relationship between the correlation of suitably transformed data and λ. This procedure is documented in REISS AND THOMAS (2001), p. 240f. and relies on the fact that, when the original distributions are of negative exponential type, i.e. the marginal c.d.f. s of X and Y are x e, x 0 FX( x) = FY( x) = 1, x > 0 for x, then the correlation ρλ ( ) between X and Y is given by 2 1 Γ + λ ρλ ( ) = Γ+ λ for λ 1. 46

47 5. A (frightening?) example from the real world ρλ ( ) = 0, 2746 λ =1,

48 5. A (frightening?) example from the real world Method II: Use the method of maximum-likelihood. For this purpose, consider the function n ((, ),, (, ); λ) : ln c ( u, v) 1 1 n n λ i i L u v u v n = = i= 1 1/ λ 1 (, ) ( λ 1) { ln( ln ) ln( ln )} 2 ln ( (, i i i i i i) ) n n n = k u v + u + v + k u v + λ i= 1 i= 1 k= 1 + ln λ 1 + k u, v ln u + ln v 1/ λ ( ) i i ( i i) i= 1 i= 1 n as a function of λ and find its argmax, i.e. the value of λ that maximizes L ( u, v ),,( u, v ); λ given the data ( u, v ),,( u, v ). ( ) 1 1 n n 1 1 n n 48

49 5. A (frightening?) example from the real world log-likelihood-function and its derivative for storm data λ = 1,

50 5. A (frightening?) example from the real world copula densities cuv (, ;1,3392) und cuv (, ;1,33347) [truncated above] 50

51 5. A (frightening?) example from the real world Dependence analysis: Other dependencies windstorm / hailstorm / flooding: empirical copula (scatterplot) empirical copula (scatterplot) empirical copula (scatterplot) 1,0 1,0 1,0 0,8 0,8 0,8 hailstorm 0,6 0,4 flooding 0,6 0,4 hailstorm 0,6 0,4 0,2 0,2 0,2 0,0 0,0 0,0 0,0 0,2 0,4 0,6 0,8 1,0 windstorm 0,0 0,2 0,4 0,6 0,8 1,0 windstorm 0,0 0,2 0,4 0,6 0,8 1,0 flooding 1 0,2226 0,3782 Gauß copula: Σ= 0, ,3341 0,3782 0,

52 5. A (frightening?) example from the real world 1,6E9 PML (VaR) as a function of return period T 1,5E9 1,4E9 1,3E9 1,2E9 PML (VaR) in 1,1E9 1E9 9E8 8E8 7E8 6E8 5E8 4E return period T PML VaR ES (!) 100 = 0,01 = 823 Mio 0,01 = 1880 Mio 52

53 6. Conclusions Some Con s against Expected Shortfall: ES α is based on the average of losses above VaR α and can thus be rigorously motivated only by the Law of Large Numbers. However, this is not very meaningful from an economic point of view since defaults are just single events. ES α may thus lead to economically meaningless risk-based capital allocations, which in particular do not provide the correct allocations of risks in the nor- 1 α 100% of the years). mal situation (i.e. in ( ) Compared with VaR α, ES α does not increase the default return period significantly, although the capital requirement might be significantly higher. 53

54 6. Conclusions ES α enforces insurance companies to buy reinsurance to a significantly higher extend than today. ES α is definitely not appropriate for portfolios with rare, but potentially very large losses (e.g. natural perils: windstorm, flooding, earthquake, ) ES α is extremely sensitive to the statistical estimation of marginal distributions. ES α is sensitive to dependence structures in the risks. 54

55 References P. ARTZNER, F. DELBAEN, J.M. EBER, D. HEATH (1999): Coherent measures of risk. Math. Finance 9, U. CHERUBINI, E. LUCIANO, W. VECCHIATO (2004): Copula Methods in Finance. Wiley, N.Y. M.A.H. DEMPSTER (ED.) (2002): Risk Management: Value at Risk and Beyond. Cambridge Univ. Press, Cambridge. S.A. KLUGMAN, H.H. PANJER, G.E. WILLMOT (2004): Loss Models. From Data to Decisions. Wiley, N.Y. R.-D. REISS, M. THOMAS (2001): Statistical Analysis of Extreme Values. With Applications to Insurance, Finance, Hydrology and Other Fields. Birkhäuser, Basel. H. ROOTZEN, C. KLÜPPELBERG (1999): A single number can t hedge against economic catrastrophes. Ambio. Special Issue on Risk Assessment An International Perspective. Vol. 26, No. 6,