Acceptability functionals, risk capital functionals and risk deviation functionals: Primal and dual properties for one-and multiperiod models

Transcription

1 Acceptability functionals, risk capital functionals and risk deviation functionals: Primal and dual properties for one-and multiperiod models

2 Preliminary remark: risk measures risk functionals translation invariance, equivariance, antivariance law invariant version independent utility, safety functional acceptability functional

3 Acceptability functionals - Basic properties An acceptability functional is a real valued functional defined on a space of random variables Y on (Ω, F 1, P) (TE) Translation equivariance: A(Y + c) = A(Y ) + c (DE) Dominated by expectation: A(Y ) E(Y ), equality iff Y is a constant. (CC) Concavity: A(λY 1 + (1 λ)y 2 ) λa(y 1 ) + (1 λ)a(y 2 ), for 0 λ 1. (PH) Positive homogeneity: A(λY ) = λa(y ) for λ > 0. (MON)Monotonicity: Y 1 Y 2 implies that A(Y 1 ) A(Y 2 ).

4 Acceptability functionals - Additional properties If A is version independent, the we define in addition (MFSD) Monotonicity w.r.t. first order stochastic dominance. Y 1 FSD Y 2 implies that A(Y 1 ) A(Y 2 ). (MSSD) monotonicity w.r.t. second order stochastic dominance. Y 1 SSD Y 2 implies that A(Y 1 ) A(Y 2 ). (CC) compound convexity. E[A(E(Y F))] A(Y ).

5 Risk capital functionals. ρ(y ) = A(Y ), where A is an acceptability functional (Typical property: translation antivariance) Risk deviation functionals. D(Y ) = E(Y ) A(Y ), where A is an acceptability functional (Typical property: translation invariance)

6 A = E D translation equivariant dominated by expectation concave positively homogeneous monotonic w.r.t FSD monotonic w.r.t. SSD compound convexity D = E A translation invariant nonnegative convex positively homogeneous monotonic w.r.t. CXD (convex dominance) compound concavity

7 Superdifferential representations Concave functions are infima of linear functions. Upper semicontinuous concave functions are characterized by the fact that they coincide with their bidual. Superdifferential representation. A(Y ) = inf{e(y Z) + α(z) : Z Z}, (Delbaen, Föllmer, Schied and many others). Example. { E[Y ], E[([Y ] A(Y ) := ) 2 ] < +,, otherwise. A is proper, concave, monotone and translation-equivariant on L 1. However, A is not u.s.c. on Z and A ++ (Y ) = E[Y ] for every Y L 1.

8 (Z, α)-representations We may assume that Z lie in a linear function space and that α(z) = for Z / Z and therefore avoid to write down the set Z of admissible supergradients explicitly, if we wish. α is not unique, if α 1 and α 2 have the same l.s.c. convex minorant, then they generate the same functional. Standard representation: α l.s.c. proper convex, Z = {Z : α(z) < } is then convex, closed. α(z) = A + = inf{e(y Z) + A( Y ) : Y Y}..

9 for A Assume that A is version independent and has a superdifferential standard representation (Z, α). A(Y ) = inf{e(y Z) + α(z) : Z Z}. Suppose further that the probability space is not atomic. Then (i) A is positively homogeneous iff α = 0 on Z. (ii) A is monotonic w.r.t. first order stochastic dominance, iff Z contains only nonnegative random variables. (iii) A is monotonic w.r.t. second order stochastic dominance, iff Z contains only nonnegative r.v s, is stable w.r.t. conditional expectations (i.e. Z Z implies that E(Z F) Z) and α is monotonic w.r.t. conditional expectations (i.e. α(e(z F)) α(z) ) for all Z and any σ-algebra F.

10 for D Suppose that D has a superdifferential representation (Z, β) with β convex l.s.c. and Z Then D(Y ) = sup{e(y Z) β(z) : Z Z}. (iv) D is positively homogeneous iff β = 0. (v) D is monotonic w.r.t. convex dominance, iff Z is stable w.r.t. conditional expectations and β is monotonic w.r.t. conditional expectation.

11 The variance Var(Y ) = Y EY 2 2 and alike (i) D(Y ) := Y EY p p = sup{e(y Z) p1 q q D q(z) : EZ = 0}, where D q (Z) = inf{ Z a q q : a R} and 1/p + 1/q = 1. (ii) A(Y ) := EY Y EY p p = inf{e(y Z) + p1 q q D q(z) : EZ = 1}. D is convex and monotonic w.r.t. convex dominance and A is concave, but has none of the properties (PH), (MFSD), (MSSD).

12 The standard deviation Y EY 2 and alike (i) D(Y ) := Y EY p = sup{e(y Z) : E(Z) = 0, D q (Z) 1}, where 1/p + 1/q = 1. (ii) A(Y ) := EY Y EY p = inf{e(y Z) : E(Z) = 1, D q (Z 1) 1}. D is positively homogeneous and monotonic w.r.t. convex dominance. A is positively homogeneous, but is not monotonic in general. The functional EY 1 2 Y EY 1 is monotonic w.r.t. SSD, because it has the representation inf{e(y Z) : EZ = 1; a : a 1 2, 1 Z a 1 2 a.s. }, which implies that all feasible Z are nonnegative.

13 The lower semi-variance and alike The lower semi-variance is Var (Y ) = [Y EY ] 2 2 (i) D(Y ) := [Y EY ] p p = sup{e(y Z) p1 q q E[( essup Z Z)q ] : E(Z) = 0}. (ii) A(Y ) := EY [Y EY ] p p = inf{e(y Z) + p1 q q E[(Z essinf Z)q ] : E(Z) = 1}. A is concave, but neither positively homogeneous nor monotonic in general.

14 The lower semi-standard deviation and alike (i) D(Y ) = [Y EY ] p = sup{e(y Z) : E(Z) = 0, Z 1, essup Z Z q 1}. (ii) A(Y ) = EY [Y EY ] p = inf{e(y Z) : E(Z) = 1, Z 0, Z essinf Z q 1}. A is positively homogeneous and monotonic w.r.t. SSD.

15 Minimal prediction error risk measures Let D(Y ) := inf{e[h(y a)] : a R}. Then D is translation invariant, convex and nonnegative and has the representation (i) D(Y ) := sup{e(y Z) Eh (Z) : EZ = 0}, where h is the Fenchel dual of h. h (v) = sup{uv h(u) : u R}. (ii) A(Y ) := EY D(Y ) = inf{e(y Z) E[h (1 Z)] : E(Z) = 1}.

16 Examples for minimal prediction error risk functionals Example. 1 If we take h(u) = u γ γ u, then we get that D(Y ) = AV@RD γ (Y ) = E(Y ) AV@R γ (Y ), resp. A(Y ) = AV@R γ (Y ), where AV@R γ is the average value-at-risk AV@R γ (Y ) = max{a 1 γ E([Y a] ) : a R}. Since the dual of h is { 0 h γ 1 (v) = γ v 1 otherwise the dual representation of AV@R is AV@R γ (Y ) = inf{e(y Z) : E(Z) = 1, 0 Z 1/γ}. Thus AV@R γ is positively homogeneous and monotonic w.r.t. SSD.

17 The newsboy story Assume that a newsboy has to decide about how many copies of a newspaper he should buy. The buying price is b < 1, the selling price is 1. Unsold copies may be returned for a price of r < b. The acceptability functional is his expected profit: A(Y ) = max{e[min(y, x) + r[y x] bx] : x R}. Acceptability requires an optimal action! A(Y ) = max{e[x [Y x] + r[y x] bx] : x R} = (1 b) max{e[x 1 r 1 b [Y x] ] : x R} = (1 b)av@r γ (Y ) where γ = 1 b 1 r. D(Y ) := E(Y ) Georg A(Y Ch. ) Pflug = be(y Acceptability ) + (1and risk b)av@r γ (Y ).

18 Example 2. If we take h(u) = u 2, then we get D(Y ) = Var(Y ). Since the dual of h(u) = u 2 is we arrive at the representation h (v) = 1 4 v 2, Var(Y ) = sup{e(y Z) 1 Var(Z) : EZ = 0}. 4

19 Distortion risk functionals Let A(Y ) = E[Y k(1 F Y (Y ))], where k is nonnegative, strictly monotone and continuous function on [0, 1] and F Y (u) = P{Y u} is the distribution function of Y. The dual representation is A(Y ) = inf{e(y Z) : Z Z}, where Z = {Z : k(z) is uniformly distributed in [0,1]}. Since Z consists only of nonnegative variables, A is monotonic w.r.t. FSD and positively homogeneous. Z is however not convex. Assume that k(z ) has uniform [0,1] distribution and let Z = {Z : Z CXD Z }. Then Z is the closed convex hull of Z. Since Z is stable w.r.t. conditional expectations, A is monotonic w.r.t. SSD (De Giorgi 2002)).

20 Assume that A is version independent. Let F be some sub σ-algebra and introduce A(Y, F). Extend the list of properties for acceptability functionals with two more: (IM) Information monotonicity. If F F, then A(Y, F) A(Y, F) (FI) Full information property. If Y is F measurable, then A(Y ) = E(Y ).

21 Example: newsboy with partial information A(Y, F) = max{e[min(y, X )+r[y X ] bx ] : X is F measurable}. has properties (IM),(FI). A(Y, F) = be(y ) + (1 b)e[av@r γ (Y F)].

22 Multiperiod acceptability and risk functionals Y 1,..., Y T discounted cash-flow adapted to (F t ). An acceptability functional A(Y 1,..., Y T, F 1,..., F T ) is a real valued concave functional defined on a space of adapted processes (Y t ) on (Ω, F 1, P) (MTE) Multiperiod Translation equivariance. A(Y 1,..., Y t + c t,..., Y T, F 1,..., F T ) = E(c t ) +A(Y 1,..., Y T, F 1,..., F T ) if c t is F t 1 mb. (MMON) Monotonicity. If Y ( t 1) Y (2) t, t = 1,..., T, then A(Y (1) 1,..., Y (1) T, F 1,..., F T ) A(Y (2) 1,..., Y (2) T, F 1,..., F T ).

23 A proper concave, monotonic u.s.c. multiperiod acceptability functional A has the representation A(Y ) = inf Z Z { T t=1 } E[Y t Z t ]+α(z) : Z t 0, E[Z t F t 1 ] = 1, t = 1,..., T Conversely, if A can be represented in the above form for some function α : Z R, then A is an upper semicontinuous and concave multiperiod acceptability functional. (with Werner Römisch)

24 A has the monotonicity property (MMON), if all supergradients Z t are nonnegative. It has the full information property, if α(z) 0 and α(1,..., 1) = 0.

25 The multiperiod newsboy problem Decision: commitment in period t 1 for consumption of x t in period t. Consumption gives NPV 1. Shortfall costs are u t > 1, surplus is carries over with loss of (1 l t ). Final surplus gives NPV of l T < 1. Let K t is the random surplus carried from period t to period t + 1. We have K 0 = 0 and K t = [l t 1 K t 1 + Y t x t ] +, t = 1,..., T. The shortfall M t at period t is given by M t = [l t 1 K t 1 + Y t x t ]. The net present value of the whole operation is T (x t u t M t ) + l T K T t=1

26 The decision maker s optimization problem defines the acceptability functionals [ T ] A(Y 1,..., Y T, F 0,..., F T 1 ) = sup{e (x t u t M t ) + l T K T : t=1 x t is F t 1 -mb for t = 1,..., T }. A clairvoyant may choose the decision x t as F t measurable and gets [ T ] A(Y 1,..., Y T, F 1,..., F T ) = sup{e (x t u t M t ) + l T K T : t=1 x t is F T -mb for t = 1,..., T } = Thus D = T t=1 E(Y t) A are the costs for not being clairvoyant. T E(Y t ) t=1

27 Supergradient representation T A(Y 1,..., Y T, F 0,..., F T ) = inf{ E(Y t Z t ) : Z t is F t mb. for t = 1,.. t=1 E(Z t F t 1 ) = 1, t = 1,..., T a.s. l t Z t u t, t = 1,..., T a.s.} Comparing this to the supergradient representation of AV@R AV@R γ (Y ) = inf{e(y Z) : E(Z) = 1, 0 Z 1/γ}. one gets that T T A(Y 1,..., Y T, F 0,..., F T ) = E(Y t )+ (1 l t )E[AV@R γt (Y t F t 1 )] t=1 t=1

28 where Likewise, defining Outline D(Y 1,..., Y T, F 0,..., F T ) = γ t = (1 l t )/(u t l t ), t = 1,..., T. T E(Y t ) A(Y 1,..., Y T, F 0,..., F T ) t=1 we get the subgradient representation T D(Y 1,..., Y T, F 0,..., F T ) = sup{ E(Y t Z t ) : Z t is F t mb. for t = 1,. t=1 E(Z t F t 1 ) = 0, t = 1,..., T a.s. 1 u t Z t 1 l t, t = 1,..., T a.s.}

29 Comparing this to the subgradient representation of γ (Y ) = inf{e(y Z) : E(Z) = 0, γ 1 γ Z 1} one sees that T D(Y 1,..., Y T, F 0,..., F T ) = (1 l t )E[AV@RD γt (Y t F t )] t=1

30 Kusuoka representation F t = σ(i t ), where I t is a tree process, i.e. σ(i t ) σ(i t+1 ). (I t ) is given modulo bijective transformations. A does only depend on the distribution of I t (modulo bijective transformations) and the conditional distributions of Y t given I t. Then = A(Y 1,..., Y T, F 1,..., F T ) T { 1 } E [AV@R γ (Y t I t )dm(γ I t ) + α t (I t )] dγ t=1 0