Dual representations of risk measures

Transcription

1 May 7, 2005

2 The variance and similar pure risk measures The standard deviation and similar pure risk measures The lower semi-variance and similar pure risk measures The lower semi-standard deviation and similar pure risk measures Minimal prediction error risk measures

3 Acceptability functionals - Basic properties Y a random variable defined on a probability space (Ω, F, P) (A1) Translation equivariance. for constant c. (A2) Strictness. A(Y + c) = A(Y ) + c A(Y ) E(Y ), where the equality sign holds iff Y is a constant. (A3) Concavity. A(λY 1 + (1 λ)y 2 ) λa(y 1 ) + (1 λ)a(y 2 ), for 0 λ 1.

4 Acceptability functionals - Additional properties (A4) Positive homogeneity. A(λY ) = λa(y ) for λ > 0. (A5) monotonicity w.r.t. first order stochastic dominance. Y 1 FSD Y 2 if E(U(Y 1 )) E(U(Y 2 )) for all monotonic integrable utility functions U. A is monotonic w.r.t the first order stochastic dominance, if Y 1 FSD Y 2 implies that A(Y 1 ) A(Y 2 ). (A6) monotonicity w.r.t. second order stochastic dominance. Y 1 SSD Y 2 if E(U(Y 1 )) E(U(Y 2 )) for all monotonic and concave integrable utility functions U. A is monotonic w.r.t the second order stochastic dominance, if Y 1 SSD Y 2 implies that A(Y 1 ) A(Y 2 ).

5 Coherent risk measures If ρ(y ) = A(Y ), where A is an acceptability functional, satisfying (A1) - (A5), then ρ is said to be coherent. If A is continuous w.r.t. convergence in probability and satisfies (A1) - (A5), then it has a representation A(Y ) = inf{e(yz) : Z Z} where Z is a set of probability densities containing the constant density 1 (Delbaen, 2000).

6 Superdifferential representations Let A be continuous w.r.t. convergence in probability. It has a superdifferential representation, if A(Y ) = inf{e(y Z) + α(z) : Z Z}, where Z is a set of (signed) densities. (Föllmer and Schied).. We consider only functionals, which depend on the distribution only, that is which satisfy A(Y 1 ) = A(Y 2 ), if Y 1 and Y 2 have the same distribution. A characterization of such functionals was given by Kusuoka (2001).

7 Suppose that A has a superdifferential representation (α, Z). Then (i) A depends only on the distribution, if the set Z is determined by distribution only, i.e. if Z and Z have the same distribution and Z Z then also Z Z and α(z) = α( Z). (ii) A is positively homogeneous if α = 0. (iii) A is monotonic w.r.t. first order stochastic dominance, if Z contains only nonnegative random variables. (iv) A positively homogeneous A is monotonic w.r.t. second order stochastic dominance, if Z contains only nonnegative densities and in addition is stable w.r.t. conditional expectations, i.e. Z Z implies that E(Z F) Z for any σ-algebra F.

8 Pure risk functionals (Deviation functionals) D is called pure risk functional or deviation functional, if it exhibits the following properties (D1) Translation invariance D(Y + c) = D(Y ) (D2) Strictness. D(Y ) 0, where the equality sign holds iff Y is a constant. (D3) Convexity. D(λY 1 + (1 λ)y 2 ) λd(y 1 ) + (1 λ)d(y 2 ), for 0 λ 1.

9 Besides, the risk functional may also exhibit the following properties (D4) Positive homogeneity. for λ 0. D(λY ) = λd(y ), (D5) monotonicity w.r.t. convex dominance. Y 1 is dominated w.r.t. convex dominance (in symbol: Y 1 CXD Y 2 ), if E(U(Y 1 )) E(U(Y 2 )) for all convex utility functions U. The pure risk measure D is called monotonic w.r.t the convex dominance, if Y 1 CXD Y 2 implies that D(Y 1 ) D(Y 2 ).

10 Subdifferential representations for deviation functionals Remark: D is a deviation functional, if and only if A(Y ) = E(Y ) D(Y ) is an acceptability functional. A fulfills (Ai) iff the pertaining D fulfills (Di), where i=1,2,3,4. A representation of D of the form D(Y ) = sup{e(y Z) β(z) : Z Z} is called a subdifferential representation of D.

11 continued (v) D is positively homogeneous if β = 0. (vi) D is monotonic w.r.t. convex dominance, if Z is stable w.r.t. conditional expectations and β is monotonic w.r.t. conditional expectation, i.e. β(z) β(e(z F)) for all σ-algebras F and all Z Z.

12 (Strassen 1965). Outline Coupling (i) The FSD-coupling: If Y 1 FSD Y 2, then one may construct a pair Ỹ1, Ỹ2 of random variables with the same marginal distributions as Y 1, Y 2, such that Ỹ 1 Ỹ2 a.s. (ii) The CXD-coupling: If Y 1 CXD Y 2, then one may construct a pair Ỹ1, Ỹ2 of random variables with the same marginal distributions as Y 1, Y 2, such that Ỹ 1 = E(Ỹ2 Ỹ1) a.s. (iii) The SSD-coupling. If Y 1 SSD Y 2, then one may construct a pair Ỹ1, Ỹ2 of random variables with the same marginal distributions as Y 1, Y 2, such that Ỹ 2 E(Ỹ 1 Ỹ 2 ) a.s.

13 The variance and similar pure risk measures The standard deviation and similar pure risk measures The lower semi-variance and similar pure risk measures The lower semi-standard deviation and similar pure risk measures Minimal prediction error risk measures The variance Var(Y ) = Y EY 2 2 and alike Let V p = E 1/p [ V p ]. (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. β(z) = p1 q q D q(z), thus D is convex and monotonic w.r.t. convex dominance (D5). (ii) A(Y ) := EY Y EY p p = inf{e(y Z) + p1 q q D q(z) : EZ = 1}. A is concave, but has none of the properties (A4) - (A6).

14 The variance and similar pure risk measures The standard deviation and similar pure risk measures The lower semi-variance and similar pure risk measures The lower semi-standard deviation and similar pure risk measures Minimal prediction error risk measures 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. Z = {Z : E(Z) = 0, D q (Z) 1} is stable w.r.t. conditional expectations and hence D is convex, homogeneous (D4) and monotonic w.r.t. convex dominance (D5). (ii) A(Y ) := EY Y EY p = inf{e(y Z) : E(Z) = 1, D q (Z 1) 1}. A is concave, positively homogeneous (A4), but is not monotonic in general. (iii) 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.

15 The variance and similar pure risk measures The standard deviation and similar pure risk measures The lower semi-variance and similar pure risk measures The lower semi-standard deviation and similar pure risk measures Minimal prediction error risk measures 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.

16 The variance and similar pure risk measures The standard deviation and similar pure risk measures The lower semi-variance and similar pure risk measures The lower semi-standard deviation and similar pure risk measures Minimal prediction error risk measures The lower semi-standard deviation and alike (i) D(Y ) = [Y EY ] p = sup{e(y Z) : E(Z) = 0, Z 1, Z 1 q 1}. (ii) A(Y ) = EY [Y EY ] p = inf{e(y Z) : E(Z) = 1, Z 0, Z q 1}. Thus A is positively homogeneous and monotonic w.r.t. SSD.

17 The variance and similar pure risk measures The standard deviation and similar pure risk measures The lower semi-variance and similar pure risk measures The lower semi-standard deviation and similar pure risk measures Minimal prediction error risk measures Minimal prediction error risk measures Let D(Y ) := inf{e[h(y a)] : a R}. Then D fulfills (D1) - (D3) and has the representation (i) D(Y ) := sup{e(yz) Eh (Z) : EZ = 0}, where h is the 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}.

18 The variance and similar pure risk measures The standard deviation and similar pure risk measures The lower semi-variance and similar pure risk measures The lower semi-standard deviation and similar pure risk measures Minimal prediction error risk measures Examples Example. 1 If we take h(u) = u α α u, then we get the D(Y ) = E(Y ) AV@R(Y ), resp. A(Y ) = AV@R α (Y ), where AV@R α is the average value-at-risk (conditional value-at-risk) AV@R α (Y ) = max{a 1 α E([Y a] ) : a R}. Since the dual of h is { h 0 α 1 (v) = α v 1 otherwise the dual representation of AV@R is AV@R α (Y ) = min{e(y Z) : E(Z) = 1, 0 Z 1/α}. Thus AV@R α is pos. homogeneous and monotonic w.r.t. SSD.

19 The variance and similar pure risk measures The standard deviation and similar pure risk measures The lower semi-variance and similar pure risk measures The lower semi-standard deviation and similar pure risk measures Minimal prediction error risk measures 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 h (v) = 1 4 v 2, we arrive at the well known formula Var(Y ) = sup{e(y Z) 1 Var(Z) : EZ = 0}. 4

20 The variance and similar pure risk measures The standard deviation and similar pure risk measures The lower semi-variance and similar pure risk measures The lower semi-standard deviation and similar pure risk measures Minimal prediction error risk measures Example 3. Let us take h(u) = u 2 1l u 0 + cu1l u>0. This asymmetric pure risk measure penalizes large negative deviation much higher than large positive deviations. We have that h (v) = if v > c 0 if 0 v c v 2 4 if v > 0 Thus the pertaining pure risk measure is D(Y ) = sup{e(yz) 1 4 E[(Z ) 2 ] : EZ = 0; Z c}.