Risk Measures and Optimal Portfolio Selection (with applications to elliptical distributions)

Transcription

1 ! " Risk Measures and Optimal Portfolio Selection (with applications to elliptical distributions) Jan Dhaene Emiliano A. Valdez Tom Hoedemakers Katholieke Universiteit Leuven, Belgium University of New South Wales, Sydney, Australia Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 1/278

2 Table of Contents: 1 Solvency Capital, Risk Measures and Comonotonicity... pp Comonotonicity and Optimal Portfolio Selection... pp Elliptical Distributions - An Introduction... pp Tail Conditional Expectations for Elliptical Distributions... pp Bounds for Sums of Non-Independent Log-Elliptical Random Variables... pp Capital Allocation and Elliptical Distributions... pp Convex Bounds for Scalar Products of Random Variables (With Applications to Loss Reserving and Life Annuities)... pp Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 2/278

3 * %) /%).* -, ) & Lecture No. 1 Solvency Capital, Risk Measures and Comonotonicity Jan Dhaene %$# +* # * '() Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 3/278

4 7 26 < 26 ;7 : Risk measures Risk: random future loss. Risk Measure: mapping from the set of quantifiable risks to the real line: X ρ(x). Actuarial examples: premium principles, technical provisions (liabilities), solvency capital requirements. In sequel: ρ(x) measures the upper tails of the d.f Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 4/278

5 D?C I?C HD G F Insurance company risk taxonomy Financial risks: asset risks (credit risks, market risks), liability risks (non-cathastrophic risks, catastrophic risks). Operational risks: business risks, event risks.?>= ED = D ABC Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 5/278

6 Q LP VLP UQ T S P M Required vs. available capital Required capital: required assets ρ(x) minus liabilities L(X), to ensure that obligations can be met: Different kinds of capital: K(X) = ρ(x) L(X). regulatory capital: you must have, rating agency capital: you are expected to have, economic capital: you should have, available capital: you actually have. LKJ RQ J Q NOP Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 6/278

7 ^ Y] cy] b^ a ` ] Z Required vs. available capital Parameters: default probability, time horizon, run-off vs. wind-up vs. going concern, valuation of liabilities: mark-to-model, valuation of assets: mark-to-market. Total balance sheet capital approach: ρ(x) = L(X) + K(X). YXW _^ W ^ [\] Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 7/278

8 k fj pfj ok n m j g The quantile risk measure Quantiles: Q p (X) = inf {x R F X (x) p}, p (0, 1). 1 F (x) X p Q (X) p x fed lk d k hij Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 8/278

9 x sw } sw x { z w t The quantile risk measure Determining the required capital by we have K(X) = Q 0.99 (X) L(X), K(X) = inf {K Pr [X > L(X) + K] 0.01}. Q p (X) = F 1 X (p) = V ar p(x). Meaningful when only concerned about frequency of default and not severity of default. Does not answer the question how bad is bad? srq yx q x uvw Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 9/278

10 Š ˆ Tail Value-at-Risk and Conditional Tail Expectation Tail Value-at-Risk: T V ar p (X) = 1 1 p 1 p Q q (X) dq, p (0, 1). Determining the required capital by K(X) = T V ar 0.99 (X) L(X), we define bad times if X in cushion [Q 0.99 (X), T V ar 0.99 (X)]. Conditional Tail Expectation: CT E p (X) = E [X X > Q p (X)], p (0, 1). CT E p = expectation of the top (1 p)% losses. ~ ~ ƒ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 10/278

11 Ž Relations between risk measures Expected Shortfall: ESF p (X) = E [ (X Q p (X)) + ], p (0, 1). ESF p (X) = expectation of shortfall in case required capital K(X) = Q p (X) L(X). Relations: T V ar p (X) = Q p (X) p ESF p(x), CT E p (X) = Q p (X) + CT E p (X) = T V ar FX (Q p (X))(X). 1 1 F X (Q p (X)) ESF p(x), Œ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 11/278

12 Ÿ šž šž Ÿ ž Relations between risk measures When F X is continuous: CT E p (X) = T V ar p (X). 1 p ESF (X) p F (x) X Q (X) p TVaR (X) p x š Ÿ Ÿ œ ž Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 12/278

13 «± ««Normal random variables Let X N ( µ, σ 2). Quantiles: Q p (X) = µ + σ Φ 1 (p). where Φ denotes the standard normal distribution function. Expected Shortfall: ESF p (X) = σ Φ ( Φ 1 (p) ) σ Φ 1 (p) (1 p). Conditional Tail Expectation: CT E p (X) = µ + σ Φ ( Φ 1 (p) ) 1 p. ª«Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 13/278

14 ¹ ¾ ½¹ ¼» µ Lognormal random variables Let ln X N ( µ, σ 2). Quantiles: Q p (X) = e µ+σ Φ 1 (p). Expected Shortfall: ESF p (X) = e µ+σ2 /2 Φ ( σ Φ 1 (p) ) e µ+σ Φ 1 (p) (1 p). Conditional Tail Expectation: CT E p (X) = e µ+σ2 /2 Φ ( σ Φ 1 (p) ) 1 p. ³² º¹ ² ¹ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 14/278

15 Æ ÁÅ ËÁÅ ÊÆ É È Å Â Risk measures and ordering of risks Ordering of risks: Stochastic dominance: X st Y F X (x) F Y (x) for all x. Stop-loss order: X sl Y E[(X d) + ] E[(Y d) + ] for all d. Convex order: X cx Y X sl Y and E[X] = E[Y ]. ÁÀ ÇÆ Æ ÃÄÅ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 15/278

16 Ó ÎÒ ØÎÒ Ó Ö Õ Ò Ï Risk measures and ordering of risks Stochastic dominance vs. ordered quantiles: X st Y Q p (X) Q p (Y ) for all p (0, 1). Stop-loss order vs. ordered TVaR s: X sl Y T V ar p (X) T V ar p (Y ) for all p (0, 1). ÎÍÌ ÔÓ Ì Ó ÐÑÒ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 16/278

17 à Ûß åûß äà ã â ß Ü Comonotonicity A set S R n is comonotonic for all x and y in S either x y or x y holds. ÛÚÙ áà Ù à ÝÞß Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 17/278

18 í èì òèì ñí ð ï ì é Comonotonicity A set S R n is comonotonic for all x and y in S either x y or x y holds. èçæ îí æ í êëì Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 17/278

19 ú õù ÿõù þú ý ü ù ö Comonotonicity A set S R n is comonotonic for all x and y in S either x y or x y holds. õôó ûú ó ú øù Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 17/278

20 Comonotonicity A set S R n is comonotonic for all x and y in S either x y or x y holds. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 17/278

21 Comonotonicity A set S R n is comonotonic for all x and y in S either x y or x y holds. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 17/278

22 ! & %! $ # "!! Comonotonicity A set S R n is comonotonic for all x and y in S either x y or x y holds. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 17/278

23 ). ) /. '. +,- *) Comonotonicity A set S R n is comonotonic for all x and y in S either x y or x y holds. A comonotonic set is a thin set. (' Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 17/278

24 6 ; 6 :?; > = : < ; 4 ; 89: 76 Comonotonicity A random vector (X 1,..., X n ) is comonotonic (X 1,..., X n ) has a comonotonic support. 54 Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 18/278

25 C H C G M G LH K J G I H A H EFG DC Comonotonicity A random vector (X 1,..., X n ) is comonotonic (X 1,..., X n ) has a comonotonic support. BA Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 18/278

26 P U P T Z T YU X W T V U N U RST QP Comonotonicity A random vector (X 1,..., X n ) is comonotonic (X 1,..., X n ) has a comonotonic support. ON Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 18/278

27 ] b ] a g a fb e d a c b [ b _`a ^] Comonotonicity A random vector (X 1,..., X n ) is comonotonic (X 1,..., X n ) has a comonotonic support. Comonotonicity is very strong positive dependency structure. Comonotonic r.v. s are not able to compensate each other. (Y c 1,..., Y c n ) is the comonotonic counterpart of (Y 1,..., Y n ). \[ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 18/278

28 j o j n t n so r q n p o h o lmn kj Characterizations of comonotonicity Notations: U : uniformly distributed on the (0, 1). X = (X 1,..., X n ). Comonotonicity of a random vector: X is comonotonic X = d ( F 1 X 1 (U),..., F 1 X n (U) ) There exists a r.v. Z, and non-decreasing functions f 1,..., f n such that X d = (f 1 (Z),, f n (Z)), Pr [X x] = min {F X1 (x 1 ), F X2 (x 2 ),..., F Xn (x n )}. The Fréchet bound: Pr [Y x] min {F Y1 (x 1 ), F Y2 (x 2 ),..., F Yn (x n )}. The upper bound is reachable in the class of random vectors with given marginals. ih Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 19/278

29 w w { { ~ { } u yz{ xw Comonotonicity and correlation Corr[X, Y ] = 1 (X, Y ) is comonotonic. The class of all random couples with given marginals always contains comonotonic couples, does not always contain perfectly correlated couples. Risk sharing schemes: { Z, Z d X = d, Z > d, Y = { 0, Z d Z d, Z > d. X and Y are comonotonic, but not perfectly correlated. vu Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 20/278

30 ˆ Ž ˆ Œ ˆ Š ˆ Comonotonic bounds for sums of dependent r.v. s Theorem: For any (X 1, X 2,..., X n ) and any Λ, we have n E [X i Λ] cx i=1 n X i cx i=1 n i=1 F 1 X i (U). Notations: S = n i=1 X i. S l = n i=1 E [X i Λ] = lower bound. S c = n i=1 F 1 X i (U) = comonotonic upper bound. If all E [X i Λ] are functions of Λ, then S l is a comonotonic sum. ƒ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 21/278

31 š Risk measures and comonotonicity Additivity of risk measures of comonotonic sums: n n Q p ( Xi c ) = Q p (X i ). T V ar p ( i=1 n Xi c ) = i=1 i=1 n T V ar p (X i ). i=1 Sub-additivity of risk measures: Any risk measure that preserves stop-loss order is additive for comonotonic risks is sub-additive: ρ(x + Y ) ρ(x) + ρ(y ). Examples: TailVaR p is sub-additive. CTE p, Q p and ESF p are NOT sub-additive. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 22/278

32 ž ž œ Ÿž Distortion risk measures Expectation of a r.v.: E[X] = 0 with F X (x) = Pr[X > x]. [1 F X (x)] dx + 0 F X (x) dx, œ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 23/278

33 ««µ ³ ² ± «Distortion risk measures 1 II F (x) X E[X] = I II I 0 x ª Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 24/278

34 ½ ¼  ¼ Á½ À ¼ ¾ ½ ½ º»¼ ¹ Distortion risk measures Expectation of a r.v.: E[X] = 0 with F X (x) = Pr[X > x]. [1 F X (x)] dx + Distortion function: g : [0, 1] [0, 1] is a distortion function g is, g(0) = 0 and g(1) = 1. 0 F X (x) dx, Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 25/278

35 Å Ê Å É Ï É ÎÊ Í Ì É Ë Ê Ã Ê ÇÈÉ ÆÅ Distortion risk measures: g(x) concave g(x) x 1 g(x) 0 1 x ÄÃ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 26/278

36 Ò Ò Ö Ü Ö Û Ú Ù Ö Ø Ð ÔÕÖ ÓÒ Distortion risk measures Expectation of a r.v.: E[X] = 0 with F X (x) = Pr[X > x]. [1 F X (x)] dx + Distortion function: g : [0, 1] [0, 1] is a distortion function g is, g(0) = 0 and g(1) = 1. Distortion risk measure: 0 F X (x) dx, ρ g [X] = 0 [ 1 g ( FX (x) )] dx + 0 g ( FX (x) ) dx. ρ g [X] = distorted expectation of X. ÑÐ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 27/278

37 ß ä ß ã é ã èä ç æ ã å ä Ý ä áâã àß Distortion risk measures: g(x) x II 1 F (x) X II' g(f (x)) X E[X] = I (II+II') ρ [X] = (I+I') II E[X] g I I' 0 x ÞÝ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 28/278

38 ì ñ ì ð ö ð õñ ô ó ð ò ñ ê ñ îïð íì Examples of distortion risk measures Expectation: X E[X]. g(x) = x, 0 x 1. 1 g(x) 0 1 x ëê Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 29/278

39 ù þ ù ý ý þ ý ÿ þ þ ûüý úù Examples of distortion risk measures The quantile risk measure: X Q p (X). g(x) = I (x > 1 p), 0 x 1. 1 g(x) 0 1 p 1 x ø Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 30/278

40 Examples of distortion risk measures Tail Value-at-Risk: X T V ar p (X). g(x) = min ( ) x 1 p, 1, 0 x 1. 1 g(x) 0 1 p 1 x Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 31/278

41 Examples of distortion risk measures Conditional Tail Expectation: X CT E p (X). is NOT a distortion risk measure. Expected Shortfall: X ESF p (X). is NOT a distortion risk measure. Stoch. dominance vs. ordered distortion risk measures: X st Y ρ g [X] ρ g [Y ] for all distortion functions g. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 32/278

42 % $ * $ )% ( ' $ & % % "#$! The Wang transform risk measure Problems with TVaR p : no incentive for taking actions that increase the distribution function for outcomes smaller than Q p, accounts for the ESF does not adjust for extreme low-frequency, high severity losses. The Wang transform risk measure : with X ρ gp (X), 0 < p < 1, g p (x) = Φ [ Φ 1 (x) + Φ 1 (p) ], 0 x 1. offers a possible solution. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 33/278

43 /01.- The Wang transform risk measure Examples: if X is normal: ρ gp (X) = Q p (X). if X is lognormal: ρ gp (X) = Q Φ [Φ 1 (p)+ σ 2 ] (X).,+ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 34/278

44 :? : > D > C? B A 8? <=> ;: Properties of distortion risk measures Additivity for comonotonic risks: ρ g [X c 1 + X c X c n] = n ρ g (X i ). i=1 Positive homogeneity: for any a > 0, ρ g [ax] = aρ g [X]. Translation invariance: ρ g [X + b] = ρ g [X] + b. Monotonicity: X Y ρ g [X] ρ g [Y ]. 98 Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 35/278

45 G L G K Q K PL O N K M L E L IJK HG Concave distortion risk measures Concave distortion risk measures: ρ g ( ) is a concave distortion risk measure if g is concave. T V ar p ( ) is concave, Q p ( ) not. SL-order vs. ordered concave distortion risk measures: X sl Y ρ g [X] ρ g [Y ] for all concave g. FE Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 36/278

46 T Y T X ^ X ]Y \ [ X Z Y R Y VWX UT The Beta distortion risk measure Problem with TVaR p : For any concave g, ρ g strongly preserves stop-loss order g is strictly concave. T V ar p does not strongly preserve stop-loss order. The Beta distribution: (a > 0, b > 0) F β (x) = 1 β (a, b) x 0 t a 1 (1 t) b 1 dt, 0 x 1. The Beta distortion risk measure: X ρ Fβ (X). ρ Fβ strictly preserves stop-loss order provided 0 < a 1, b 1 and a and b are not both equal to 1. A PH-transform risk measure: Wang (1995). a = 0.1 and b = 1. SR Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 37/278

47 a f a e k e jf i h e g f _ f cde ba Sub-additivity of risk measures Merging decreases the insolvency risk : (X + Y ρ [X] ρ [Y ]) + (X ρ [X]) + + (Y ρ [Y ]) + Sub-additivity is allowed to some extent. Concave distortion risk measures are sub-additive: ρ g [X + Y ] ρ g [X] + ρ g [Y ]. Q p is not sub-additive, T V ar p is sub-additive. Optimality of T V ar p : T V ar p (X) = min {ρ g (X) g is concave and ρ g Q p }. `_ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 38/278

48 n s n r x r ws v u r t s l s pqr on Axiomatic characterization of risk measures A risk measure is "Artzner-coherent if it is sub-additive, monotone, positive homogeneous and translation invariant. Q p is not coherent. Concave distortion risk measures are coherent. The Dutch risk measure: ρ(x) = E [X] + E [ (X E [X]) + ]. ρ(x) is coherent, but not comonotonic-additive ρ(x) is NOT a distortion risk measure. Coherent or not? Markowitz (1959): We might decide that in one context one basic set of principles is appropriate, while in another context a different set of principles should be used. ml Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 39/278

49 { { ƒ y }~ { Distortion risk measures for sums of dependent r.v. s Approximations for sums of dependent r.v. s: S = n i=1 X i with given marginals, but unknown copula. S l = n E [X i Λ] cx S cx i=1 n i=1 F 1 X i (U) = S c Approximations for ρ g [S]: (if all E [X i Λ] are in Λ) ρ g [S c ] = ρ g [S l] = n ρ g [X i ], i=1 n ρ g [E (X i Λ)]. i=1 If g is concave: ρ g [ S l ] ρ g [S] ρ g [S c ]. zy Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 40/278

50 ˆ ˆ Œ Œ Œ Ž Š Œ ˆ Application: provisions for future payment obligations Problem description Consider a payment obligation of 1 per year, due at times 1, 2,..., 20, Let e Y (i) be the discount factor over [0, i]: e Y (i) e (Y 1+Y Y i ). Assume the yearly returns Y j are i.i.d. and normal distributed with parameters µ = 0.07 and σ = 0.1. The stochastic provision is defined by S = 20 i=1 e (Y 1+Y Y i ). Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 41/278

51 š Ÿ žš œ š š Provisions for future payment obligations Convex bounds for S = 20 Let Λ = 20 i=1 Y i Then where i=1 e Y (i) 20 j=i e jµ and r i = corr [Λ, Y (i)] > 0. S l cx S cx S c S l = S c = n e E[Y (i)] r i σ Y (i) Φ 1 (U)+ 1 2 (1 r2 i )σ2 Y (i), i=1 n e E[Y (i)]+ σ Y (i) Φ 1 (U). i=1 Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 42/278

52 «ª Provisions for future payment obligations Provision (or total capital requirement) The provision for this series of future obligations is set equal to ρ g [S] Approximate ρ g [S] by ρ g [S c ] = ρ g [S l] = n ρ g [X i ], i=1 n ρ g [E (X i Λ)]. i=1 Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 43/278

53 ³ ¹ ³ ³ µ ±²³ Provisions for future payment obligations The Quantile-provision principle: ρ g [S] = Q p [S] Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 44/278

54 ¼ Á ¼ À Æ À ÅÁ Ä Ã À  Á º Á ¾ À ½¼ Provisions for future payment obligations The CTE-provision principle: ρ g [S] =TVaR p [S] p TVAR p [S l ] TVAR p [S] TVAR p [S c ] »º Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 45/278

55 É Î É Í Ó Í ÒÎ Ñ Ð Í Ï Î Ç Î ËÌÍ ÊÉ Theories of choice under risk Expected utility theory: von Neumann & Morgenstern (1947). Prefer loss X over loss Y if E [u(w X)] E [u(w Y )], u(x) = utility of wealth-level x, function of x. Risk aversion: u is concave. Yaari s dual theory of choice under risk: Yaari (1987). Prefer loss X over loss Y if f(q) = distortion function. Risk aversion: f is convex. ρ f [w X] ρ f [w Y ], ÈÇ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 46/278

56 Ö Û Ö Ú à Ú ßÛ Þ Ý Ú Ü Û Ô Û ØÙÚ Ö Compare theories of choice under risk Transformed expected wealth levels: E[w X] = E[u(w X)] = ρ f [w X] = Ordering of risks: Q 1 q (w X) dq, u [Q 1 q (w X)] dq, Q 1 q (w X) df(q). In both theories, stochastic dominance reflects common preferences of all decision makers. In both theories, stop-loss order reflects common preferences of all risk-averse decision makers. ÕÔ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 47/278

57 ã è ã ç í ç ìè ë ê ç é è á è åæç äã References (see Dhaene, Denuit, Goovaerts, Kaas, Vyncke (2002a). The concept of comonotonicity in actuarial science and finance: Theory, Insurance: Mathematics & Economics, vol. 31(1), Dhaene, Denuit, Goovaerts, Kaas, Vyncke (2002b). The concept of comonotonicity in actuarial science and finance: Applications, Insurance: Mathematics & Economics, vol. 31(2), Dhaene, Vanduffel, Tang, Goovaerts, Kaas, Vyncke (2003). Capital requirements, risk measures and comonotonicity âá Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 48/278

58 ð õ ð ô ú ô ùõ ø ô ö õ î õ òóô ñð Lecture No. 2 Comonotonicity and Optimal Portfolio Selection Jan Dhaene ïî Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 49/278

59 ý ý û ÿ þý Introduction Strategic portfolio selection: For a given savings and/or consumption pattern over a given time horizon, identify the best allocation of wealth among a basket of securities. The Terminal Wealth problem: Saving for retirement. A loan with an amortization fund with random return. The Reserving problem: The after retirement problem. Technical provisions. Capital requirements. üû Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 50/278

60 Introduction The Buy and Hold strategy: Keep the initial quantities constant. A static strategy. The Constant Mix strategy: Keep the initial proportions constant. A dynamic strategy. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 51/278

61 ! Comonotonicity Notations: U: uniformly distributed on (0, 1). X = (X 1,..., X n ). F 1 X (p) = Q p [X] = VaR p [X]= inf {x R F X (x) p}. Comonotonicity of a random vector: X is comonotonic there exist non-decreasing functions f 1,..., f n and a r.v. Z such that X d = [f 1 (Z),..., f n (Z)]. Comonotonicity: very strong positive dependency structure. Comonotonic r.v. s cannot be pooled. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 52/278

62 $ ) $ (. ( -), + ( * ) " ) &'( %$ Comonotonic bounds for sums of dependent r.v. s Theorem: For any X and any Λ, we have n E [X i Λ] cx i=1 n X i cx i=1 n i=1 F 1 X i (U). Notations: S = n i=1 X i. S l = n i=1 E [X i Λ] = lower bound. S c = n i=1 F 1 X i (U) = comonotonic upper bound. If all E [X i Λ] are increasing functions of Λ, then S l is a comonotonic sum. #" Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 53/278

63 ; 5 : / Performance of the comonotonic approximations Local comonotonicity: Let B(τ) be a standard Wiener process. The accumulated returns exp [µτ + σ B(τ)], will be almost comonotonic. The continuous perpetuity: exp [µ (τ + τ) + σ B (τ + τ)] S = 0 exp [ µτ σ B(τ)] dτ has a reciprocal Gamma distribution. 0/ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 54/278

64 > C > B H B GC F E B D C < Numerical illustration: µ = 0.07 and σ = Circles: Plot of (Q p [S], Q p [S l ]) =< Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 55/278

65 K P K O U O TP S R O Q P I P MNO LK Numerical illustration p Q p [S l ] Q p [S] Q p [S c ] JI Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 56/278

66 X ] X \ b \ a] ` _ \ ^ ] V ] Z[\ YX The Black-Scholes setting 1 risk-free and m risky assets: dp 0 (t) P 0 (t) = r dt dp i (t) P i (t) = µ i dt + d j=1 σ ij dw j (t) with ( W 1 (τ),..., W d (τ) ) : independent standard Brownian motions. WV Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 57/278

67 e j e i o i nj m l i k j c j ghi fe The Black-Scholes setting Equivalent formalism: dp 0 (t) P 0 (t) dp i (t) P i (t) = r dt = µ i dt + σ i db i (t) with ( B 1 (τ),..., B m (τ) ) correlated standard Brownian motions. dc Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 58/278

68 r w r v v {w z y v x w p w tuv sr The Black-Scholes setting Return of asset i in year k: P i (k) = P i (k 1) e Y i k Y i k normal distributed with E [ Yk i ] = µi 1 2 σ2 i and Var [ Yk i ] = σ 2 i Independence over the different years: k l Y i k and Y j l Dependence within each year: Cov are independent. [ ] Yk i, Y j k Assumptions: µ r1 and Σ is positive definite. = (Σ) ij qp Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 59/278

69 ƒ ƒ ˆ ƒ } ƒ Investment strategies Constant mix strategies: π (t) = (π 1, π 2,..., π m ) with π i = fraction invested in risky asset i, 1 m π i = fraction invested in riskfree asset. i=1 Fractions time-independent. Dynamic trading strategies. Requires continuously rebalancing. ~} Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 60/278

70 Œ Œ Š Ž Œ Investment strategies The portfolio return process: Merton (1971). P (t) = price of one unit of (π 1, π 2,..., π m ). dp (t) P (t) = µ (π) t + σ (π) db(t) with B(τ) a standard Brownian motion and µ (π) = r + π T ( µ r 1 ), σ 2 (π) = π T Σ π Yearly portfolio returns: P (k) = P (k 1) e Y k(π) The Y k (π) are i.i.d. normal with E [Y k (π)] = µ (π) 1 2 σ2 (π), Var [Y k (π)] = σ 2 (π) Š Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 61/278

71 ž ž Ÿ ž ž œ š Markowitz mean-variance analysis The mean-variance efficient frontier: max π is obtained for the portfolio π σ = σ µ (π) subject to σ (π) = σ Σ 1 (µ r1 ) (µ r1 ) T Σ 1 (µ r1 ) with (µ µ (π σ ) T ) = r + σ r1 Σ 1 (µ r1 ) Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 62/278

72 «ª ª «ª ««ª Markowitz mean-variance analysis: r < µ ( π (m)) µ(π) r π (m) σ(π) Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 63/278

73 ³ ³ ½ ¼» º ¹ ± µ ³ Markowitz mean-variance analysis: r < µ ( π (m)) µ(π) r π (m) σ(π) ²± Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 63/278

74 À Å À Ä Ê Ä ÉÅ È Ç Ä Æ Å ¾ Å ÂÃÄ ÁÀ Markowitz mean-variance analysis: r < µ ( π (m)) µ(π) r π (m) σ(π) ¾ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 63/278

75 Í Ò Í Ñ Ñ ÖÒ Õ Ô Ñ Ó Ò Ë Ò ÏÐÑ ÎÍ Markowitz mean-variance analysis: r < µ ( π (m)) µ(π) π (t) r π (m) σ(π) ÌË Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 63/278

76 Ú ß Ú Þ ä Þ ãß â á Þ à ß Ø ß ÜÝÞ ÛÚ Markowitz mean-variance analysis The Capital Market Line and the Sharpe ratio: µ (π σ ) = r + ( µ ( π (t) ) r σ ( π (t)) ) σ. Two Fund Separation Theorem: π σ = ( µ (π σ ) r µ ( π (t)) r ) π (t). ÙØ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 64/278

77 ç ì ç ë ñ ë ðì ï î ë í ì å ì éêë èç Saving and terminal wealth Problem description: α 0, α 1,..., α n : positive savings at times 0, 1, 2,..., n. Investment strategy: π(t) = (π 1, π 2,..., π m ). Wealth at time j: W j (π) = W j 1 (π) e Y j(π) + α j with W 0 (π) = α 0. What is the optimal investment strategy π? Depends on target capital and probability level. æå Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 65/278

78 ô ù ô ø þ ø ýù ü û ø ú ù ò ù ö ø õô Approximating Terminal Wealth Terminal wealth W n (π): W n (π) = n α i e Y i+1(π)+y 2 (π)+ +Y n (π) = i=0 n i=0 X i The comonotonic upper bound for W n (π): W c n (π) = n i=0 F 1 X i (U) A comonotonic lower bound for W n (π): n n Wn l (π) = E X i Y j (π) i=0 j=1 j 1 k=0 α k e k µ(π) Convex ordering: W l n(π) cx W n (π) cx W c n(π) óò Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 66/278

79 ÿ Optimal investment strategies Terminal wealth W n (π): W n (π) = n α i e Y i+1(π)+y i+2 (π)+ +Y n (π) i=0 Utility Theory: Von Neumann & Morgenstern (1947). max π E [u (W n (π))] Yaari s dual theory of choice under risk: Yaari (1987). max π Ef [W n (π)] where E f is determined with f (Pr (W n (π) > x)), convexity of f corresponds with risk aversion. ÿ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 67/278

80 Optimal investment strategies Reduced optimization problem: For σ (π 1 ) = σ (π 2 ) and µ (π 1 ) < µ (π 2 ), we have that Hence, max π W n (π 1 ) st W n (π 2 ). E [u (W n (π))] = max σ E [u (W n (π σ ))] and max π Ef [W n (π)] = max σ E f [W n (π σ )]. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 68/278

81 % $ # "! The Target Capital Distorted expectations: for f(x) = { 0 : x p 1 : x > p, the distorted expection E f [W n (π)] reduces to Q 1 p [W n (π)] = sup {x Pr [W n ( π) > x] p}. Problem: d.f. of W n (π) too cumbersome to work with curse of dimensionality dependencies Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 69/278

82 ( - (, 2, 1-0 /,. - & - *+, )( Maximizing the Target Capital, for a given p Optimal investment strategy: π follows from max π Q 1 p [W n (π)] Approximation:the approximation π l for π follows from max σ ] Q 1 p [Wn l (π σ ) with ] Q 1 p [Wn(π l σ ) = n α i e (n i) [µ(π σ ) 1 2 r2 i (πσ )σ 2 ] n i r i (π σ )σφ 1 (p) i=0 '& Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 70/278

83 5 : 5 9? 9 >: = < 9 ; : 3 : Numerical illustration Available assets: 1 riskfree asset with r = risky assets with and The tangency portfolio: µ 1 = 0.06, σ 1 = 0.10 µ 2 = 0.10, σ 2 = 0.20 Corr [ Yk 1, Y k 2 ] = 0.5 π (t) = ( 5 9, 4 ) (, µ π (t)) = 7 ( 9 90, σ π (t)) = Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 71/278

84 B G B F L F KG J I F H G DEF CB Numerical illustration Yearly savings: α 0 =... = α 39 = 1 Terminal wealth: W 40 (π) = 39 i=0 e Y i+1(π)+y 2 (π)+ +Y 40 (π) Optimal investment strategy: max π Q 0.05 [W 40 (π)] A@ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 72/278

85 [ Z O T O S Y S XT W V S U T M T QRS PO Numerical illustration quantile target capital Q 0.05 [W n (π σ )] as a function of the proportion invested in π (t) dots: Q 0.05 [W s n (π σ )], solid: Q 0.05 W l n (π σ ), dashed: Q 0.05 [W c n (π σ )] NM Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 73/278

86 ^ c ^ b h b gc f e b d c \ c `ab _^ Numerical illustration Minimizing the savings effort per unit of Target Capital: The optimal investment strategy π is defined as the one that minimizes α (π) in Q 1 p [α (π) 39 i=0 e Y i+1(π)+y 2 (π)+ +Y 40 (π) ] = 1. ]\ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 74/278

87 k p k o u o tp s r o q p i p mno lk Numerical illustration minimal savings amount optimal risky proportion p Solid line (left scale): minimal yearly savings amount as a function of p. Dashed line (right scale): optimal proportion invested in the tangency portfolio. ji Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 75/278

88 x } x } ~ } v } z{ yx Other optimization criteria Maximizing the Target Capital for a given probability level p: with max π CLTE 1 p [W n (π)] CLTE 1 p [X] = E [X X < Q 1 p [X]] Maximizing p for a given Target Capital K: max π Pr [W n (π) > K] wv Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 76/278

89 Š ŽŠ Œ Š ƒ Š ˆ Provisions for future liabilities Problem description: α 1,..., α n : positive payments, due at times 1,..., n. R 0 = initial provision established at time 0. Investment strategy: π (t) = (π 1, π 2,..., π m ). Provision at time j: R j (R 0, π) = R j 1 (R 0, π) e Y j(π) α j with R 0 (R 0, π) = R 0. What is the optimal investment strategy π? Answer depends on initial provision R 0 and probability level p. ƒ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 77/278

90 œ š The stochastic provision Definition: S (π) = n α i e (Y 1(π)+Y 2 (π)+ +Y (π)) i. i=1 Relation: R n (R 0, π) = (R 0 S (π)) e (Y 1(π)+ +Y n (π)). An investment strategy π is only acceptable if Pr [R n (R 0, π) 0] is large enough. Relation: Pr [R n (R 0, π) 0] = Pr [S (π) R 0 ]. PROBLEM: d.f. of S (π) too cumbersome to work with. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 78/278

91 Ÿ Ÿ Ÿ Comonotonic approximations for S (π) The comonotonic upper bound for S (π): S (π) cx S c (π). A comonotonic lower bound for S (π): [ S l (π) = E S (π) S l cx S (π). S l (π) is a comonotonic sum. n j=1 Y j (π) n k=j α k e k[µ(π) σ2 (π)] ]. ž Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 79/278

92 ± µ± ³ ² ± ª ± Optimal investment strategies The Initial Provision: Definition: R 0 (π) = E g [S (π)] where S (π) is the Stochastic Provision. E g [ ] is a distortion risk measure. If g is concave, then E g [ ] is a coherent risk measure. The optimal investment strategy: (π, R0 ) follows from R 0 = min π E g [S (π)] «ª Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 80/278

93 ¹ ¾ ¹ ½ à ½ ¾ Á À ½ ¾ ¾»¼½ º¹ Reduced optimization problem For σ (π 1 ) = σ (π 2 ) and µ (π 1 ) < µ (π 2 ), we have that S (π 2 ) st S (π 1 ). Hence, min π E g [S (π)] = min σ E g [S (π σ )]. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 81/278

94 Æ Ë Æ Ê Ð Ê ÏË Î Í Ê Ì Ë Ä Ë ÈÉÊ ÇÆ Minimizing the Initial Provision, for a given p The p - quantile provision principle: If investment strategy = π, then R 0 (π) = Q p [S (π)] = inf {x Pr [R n (x, π) 0] p}. Optimal strategy: (π, R0 ) follows from R 0 = min π Q p [S (π)]. Approximation: (π l, R0 l ) follows from [ ] R0 l = min Q p S l (π σ ). σ ÅÄ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 82/278

95 Ó Ø Ó Ý ÜØ Û Ú Ù Ø Ñ Ø ÕÖ ÔÓ Numerical illustration Available assets: 1 riskfree asset with r = risky assets with and The tangency portfolio: µ 1 = 0.06, σ 1 = 0.10 µ 2 = 0.10, σ 2 = 0.20 Corr [ Yk 1, Y k 2 ] = 0.5 π (t) = ( 5 9, 4 ) (, µ π (t)) = 7 ( 9 90, σ π (t)) = ÒÑ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 83/278

96 à å à ä ê ä éå è ç ä æ å Þ å âãä áà Numerical illustration Yearly consumptions: α 1 =... = α 40 = 1. Stochastic provision: S (π) = 40 i=1 e (Y 1(π)+Y 2 (π)+ +Y i (π)). Optimal investment strategy: Approximation: R 0 = min π Q p [S (π)]. R l 0 = min σ Q p [S (π σ )]. ßÞ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 84/278

97 í ò í ñ ñ öò õ ô ñ ó ò ë ò ïðñ îí Numerical illustration minimal reserve optimal risky proportion p Solid line (left scale): minimal initial provision R0 l as a function of p. Dashed line (right scale): optimal proportion invested in the tangency portfolio. ìë Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 85/278

98 ú ÿ ú þ þ ÿ ÿ ø ÿ üýþ þ ûú Other optimization criteria Minimizing the Initial Provision, given p: with R 0 = min π CTE p [S (π)] CTE p [X] = E [X X > Q p [X]]. Maximizing p for a given Initial Provision R 0 : p = max π Pr [R n (R 0, π) > 0]. ùø Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 86/278

99 Generalizations Investment restrictions: are taken into account by redefining the set of efficient portfolios. Yaari s dual theory: The final wealth problem can be solved for general distorted expectations. Distortion risk measures: The initial provision can be defined in terms of general distortion risk measures. Stochastic sums: How to avoid outliving your money? Positive and negative payments: The savings - retirement problem. Other distributions: Lévy-type or Elliptical-type distributions Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 87/278

100 Some references ( [1] Dhaene, Denuit, Goovaerts, Kaas, Vyncke (2002a). The concept of comonotonicity in actuarial science and finance: Theory. Insurance: Mathematics & Economics, vol. 31(1), [2] Dhaene, Denuit, Goovaerts, Kaas, Vyncke (2002b). The concept of comonotonicity in actuarial science and finance: Applications. Insurance: Mathematics & Economics, vol. 31(2), [3] Dhaene, Vanduffel, Goovaerts, Kaas, Vyncke (2004). Comonotonic approximations for optimal portfolio selection problems. (forthcoming) [4] Dhaene, Vanduffel, Tang, Goovaerts, Kaas, Vyncke (2003). Risk measures and comonotonicity. (submitted) Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 88/278

101 & ' # ( * )( 23, ),1 1 # ' (. "4 " Lecture No. 3 Elliptical Distributions - An Introduction Emiliano A. Valdez & ' % " $ 0$ # " "! " " & / +*, - - * 1 Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 89/278

102 5 < = 9 B? 6 BG G 9 = 6> D 8J Elliptical Distributions This family coincides with the family of symmetric distributions in the univariate case (e.g. normal, Student-t) and can be characterized using either: characteristic generator density generator References: Landsman and Valdez (2003) Tail Conditional Expectations for Elliptical Distributions, North American Actuarial Journal. Valdez and Dhaene (2004) Bounds for Sums of Non-Independent Log-Elliptical Random Variables, work in progress. Valdez and Chernih (2003) Wang s Capital Allocation Formula for Elliptically-Contoured Distributions, Insurance: Mathematics & Economics. 5 6 < = 6 8 :; F: < E A@B C 8 6 HI 6 G Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 90/278

103 K R S O T V UT X U L X] ] O S LT Z N` Why Elliptical Distributions? Provides a rich class of multivariate distributions that share several tractable properties of the multivariate normal. Student t, Laplace, Logistic, etc. Linear combinations of components of multivariate elliptical is again elliptical (Important for modelling yearly returns, and for constructing the conditioning variable.) Allows more flexibility to model multivariate extremes and other forms of non-normal dependency structures. Fat extremes, tail dependence. Some studies show that light tailness of normal show its inadequacies to model extreme credit default events. K L R S L N PQ \P O N N M N N R [ WVX Y N L ^_ Y V L ] Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 91/278

104 a h i e j l kj n k b ns s e i bj p dv Some Notation Consider an n-dimensional random vector X = (X 1, X 2,..., X n ) T. Distribution function: F X (x 1, x 2,..., x n ) = P (X 1 x 1,..., X n x n ) Density function: f X (x 1, x 2,..., x n ) = n F X (x 1, x 2,..., x n ) x 1 x n Characteristic function: ϕ X (t) = E [ exp ( ix T t )] = E [exp (i n k=1 X kt k )] Moment generating function: M X (t) = E [ exp ( X T t )] = ϕ X ( it) Covariance matrix: Cov (X) = (Cov (X i, X j )) for i, j = 1,..., n a b h i b d fg rf e d d c d d h q mln o d b tu o l b s Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 92/278

105 w ~ { x { x zœ Multivariate Normal Family It is well-known that the joint density of a multivariate normal X is given by f X (x) = c [ n exp 1 ] Σ 2 (x µ)t Σ 1 (x µ). The normalizing constant is given by c n = (2π) n/2. Its characteristic function is ϕ X (t) = exp ( it T µ 1 2 tt Σt ) = exp ( it T µ ) exp ( 1 2 tt Σt ) And its covariance is Cov (X) = Σ. w x ~ x z } ˆ { z z y z z ~ ƒ z x Š x Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 93/278

106 Ž š Ž šÿ Ÿ Ž œ Multivariate Normal - continued Define the characteristic generator as ψ (t) = e t and density generator as g n (u) = e u The density can then be written as f X (x) = c n g n [ 1 ] Σ 2 (x µ)t Σ 1 (x µ) and its characteristic function as ϕ X (t) = exp ( it T µ ) ψ ( 1 2 tt Σt ). Ž Ž ž š Ž Ÿ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 94/278

107 ª «µ µ «² Class of Elliptical Distributions X has multivariate elliptical distribution, X E n (µ, Σ,ψ), if char. function can be expressed as ϕ X (t) = exp(it T µ)ψ ( 1 2 tt Σt ) for some column-vector µ, n n positive-definite matrix Σ. If density exists, it has the form f X (x) = c [ ] n 1 g n Σ 2 (x µ)t Σ 1 (x µ), for some function g n ( ) called the density generator. ª «ª ³ ± ± µ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 95/278

108 ¹ À Á ½ Â Ä ÃÂ Æ Ã º ÆË Ë ½ Á ºÂ È ¼Î Elliptical Distributions - continued The normalizing constant c n can be explicitly determined by transforming into polar coordinates and we have c n = Γ (n/2) (2π) n/2 [ 0 x n/2 1 g n (x)dx] 1. Thus, we see the condition 0 x n/2 1 g n (x)dx < guarantees g n as density generator. Note that for a given characteristic generator ψ, the density generator g and/or the normalizing constant c may depend on the dimension of the random vector X. ¹ º À Á º ¼ ¾ ʾ ½ ¼ ¼» ¼ ¼ À É ÅÄÆ Ç ¼ º ÌÍ Ç Ä º Ë Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 96/278

109 Ï Ö Ó Ø Ú ÙØ Ü Ù Ð Üá á Ó ÐØ Þ Òä Some Properties If mean exists, it will be If covariance exists, it will be E (X) = µ. Cov (X) = ψ (0) Σ. Let A be some m n matrix of rank m n and b some m-dimensional column-vector. Then AX + b E m ( Aµ + b,aσa T, g m ). Define the sum S = X 1 + X X n = e T X, where e is a column vector of ones with dimension n. Then S E n ( e T µ, e T Σe, g 1 ). Ï Ð Ö Ð Ò ÔÕ àô Ó Ò Ò Ñ Ò Ò Ö ß ÛÚÜ Ý Ò Ð âã Ý Ú Ð á Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 97/278

110 å ì í é î ð ïî ò ï æ ò é í æî ô èú Multivariate Student-t Family Density generator: g n (u) = p > n/2 and k p is some constant. ( 1 + u k p ) p where parameter Density: f X (x) = ] p c n [1 + (x µ)t Σ 1 (x µ) Σ 2k p Normalizing constant: c n = Γ(p) Γ(p n/2) (2πk p) n/2 If p = (n + m) /2 where n, m are integers, and k p = m, we get the traditional form of the multivariate Student t with density: f X (x) = Γ ( n+m 2 (πm) n/2 Γ ( m 2 ) [ ) Σ 1 + (x µ)t Σ 1 (x µ) m ] ( n+m 2 ) å æ ì í æ è êë öê é è è ç è è ì õ ñðò ó è æ øù ó ð æ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 98/278

111 û ÿ ü ÿ ü þ Generalized Student-t Distribution 1 Density: f X (x) = σ 2k p B(1/2,p 1/2) B (, ) is the beta function. [1 + (x µ)2 2k p σ 2 ] p, where For p > 3/2, usually k p = (2p 3)/2 becaue it leads to the important property that V ar (X) = σ 2. For 1/2 < p 3/2, variance does not exist and k p = 1/2. Note for example in the case where p = 1, we have standard Cauchy distribution: f X (x) = 1 σπ [ 1 + (x µ)2 σ 2 ] 1. It is well-known that mean and variance for this distribution does not exist. û ü ÿ þ ý þ þ ü þ þ þ ü ü Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 99/278

112 # # & Density Functions of GST - Figure p = p = 5 normal f(x) 0.3 p = p = x "! $% # Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 100/278

113 '. / ( / (0 6 *< Multivariate Logistic Family Density generator: g (u) = e u (1+e u ) 2 Density: f X (x) = c n Σ [ ] exp 1 2 (x µ)t Σ 1 (x µ) { [ ]} exp 1 2 (x µ)t Σ 1 (x µ) Normalizing constant: c n = (2π) n/2 ( 1) j 1 j 1 n/2 j=1 1 ' (. / ( *,- 8, + * * ) * * * ( :; 5 2 ( 9 Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 101/278

114 = D E A F H GF J G > JO O A E >F Multivariate Exponential Power Family Density generator: g (u) = e rus, for r, s > 0 Density: f X (x) = c { n exp r Σ 2 [ ] s } (x µ) T Σ 1 (x µ) Normalizing constant: c n = sγ (n/2) (2π) n/2 Γ (n/2s) rn/2s When r = s = 1, this reduces to multivariate normal. When s = 1/2 and r = 2, we have Double Exponential or Laplace distributions. = > D E BC D M IHJ > PQ K H > O Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 102/278

115 W [ T\ b S Z [ W \ ^ ]\ ` ] T `e e Vh Normal Bivariate Densities - Figure 2 Normal Student t Logistic Laplace S T Z [ T V XY dx W V V U V V Z c _^` a V T fg a ^ T e Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 103/278

116 i p q m r t sr v s j v{ { m q jr x l~ Lecture No. 4 Tail Conditional Expectations for Elliptical Distributions Emiliano A. Valdez i j p q j l no zn m l l k l l p y utv w l j } w t j { Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 104/278

117 ƒ ˆ Š ˆ Œ Œ ƒ ˆ Ž Introduction Developing a standard framework for risk measurement is becoming increasingly important. This paper is about a risk measure called tail conditional expectations and their explicit forms for the family of elliptical distributions. This family coincides with the family of symmetric distributions in the univariate case (e.g. normal, Student-t) and can be characterized using either: characteristic generator density generator ƒ ŠŒ Š Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 105/278

118 œ ž Ÿž Ÿ ž ª Introduction - continued We introduce the notion of a cumulative generator which plays a key role in computing tail conditional expectations. We extended the ideas into the multivariate framework allowing us to decompose the total of the tail conditional expectations into its various constituents. decomposing the total into an allocation formula Landsman and Valdez (2003) œ š š œ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 106/278

119 «² ³ µ µ ½ ½ ³ º À Risk Measure A risk measure ϑ is a mapping from the space of random variables L to the set of real numbers: ϑ : X L R. Some useful properties of a risk measure: 1. Monotonicity: X 1 X 2 with probability 1 = ϑ (X 1 ) ϑ (X 2 ). 2. Homogeneity: ϑ (λx) = λϑ (X) for any non-negative λ. 3. Subadditivity: ϑ (X 1 + X 2 ) ϑ (X 1 ) + ϑ (X 2 ). 4. Translation Invariance: ϑ (X + α) = ϑ (X) + α for any constant α. Some consequences: ϑ (0) = 0; a X b = a ϑ (X) b; ϑ (X ϑ (X)) = 0. «² ³ ± ¼ ²» ¹ ¾ ¹ ½ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 107/278

120 Á È É Å Ê Ì ËÊ Î Ë Â ÎÓ Ó Å É ÂÊ Ð ÄÖ The Tail Conditional Expectation Notation: X : loss random variable; F X (x) : distribution function; F X (x) = 1 F X (x): tail function; x q : q-th quantile with F X (x q ) = 1 q The tail conditional expectation (TCE) is T CE X (x q ) = E (X X > x q ). Other names used: tail-var, conditional VAR Value-at-risk: x q = Q q (X) Expected Shortfall: E [ (X x q ) + ] = ESFq (X) Relationships: T CE X (x q ) = x q +E (X x q X > x q ) = x q q E [ (X x q ) + ] Á Â È É Â Ä ÆÇ ÒÆ Å Ä Ä Ã Ä Ä È Ñ ÍÌÎ Ï Ä Â ÔÕ Ï Ì Â Ó Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 108/278

121 Þ ß Û à â áà ä á Ø äé é Û ß Øà æ Úì TCE for Univariate Elliptical Let X E 1 ( µ, σ 2, g ) so that density f X (x) = c σ g [ 1 2 where c is the normalizing constant. ( x µ ) ] 2 σ Since X is elliptical distribution, the standardized random variable Z = (X µ) /σ will have a standard elliptical distribution function F Z (z) = c z g ( 1 2 u2) du, with mean 0 and variance σz 2 = 2c 0 u 2 g ( 1 2 u2) du = ψ (0), if they exist. Define the cumulative density generator: G (x) = c x 0 g (u) du and denote G (x) = G ( ) G (x). Ø Þ ß Ø Ú ÜÝ èü Û Ú Ú Ù Ú Ú Þ ç ãâä å Ú Ø êë å â Ø é Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 109/278

122 í ô õ ñ ö ø ö ú î úÿ ÿ ñ õ îö ü ð - continued The tail conditional expectation of X is T CE X (x q ) = µ + λ σ 2 where λ is λ = 1 σ G ( 1 2 z2 q) F X (x q ) = 1 σ G ( 1 2 z2 q) F Z (z q ) and z q = (x q µ) /σ. Moreover, if the variance of X exists, then 1 σ G ( 1 Z 2 2 z2) has the sense of a density of another spherical random variable Z and λ has the form λ = 1 σ f Z (z q ) F Z (z q ) σ2 Z. í î ô õ î ð òó þò ñ ð ð ï ð ð ô ý ùøú û î ð û ø î ÿ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 110/278

123 Some Examples Normal Distribution: λ = 1 σ ϕ (z q) 1 Φ (z q ) where ϕ ( ) and Φ ( ) denote respectively the density and distribution functions of a standard normal distribution. Notice that Z is simply the standard normal variable Z. Student-t: λ = 2p 5 2p 3 f Z ( ) 2p 5 2p 3 z q; p 1 F Z (z q ; p) only for the case where p > 5/2. Here, Z scaled GST with parameter p 1. is simply a Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers p. 111/278