Optimization of nonstandard risk functionals

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2 Risk functionals Properties Dual properties A collection of risk functionals Algorithms

3 Properties Dual properties A collection of risk functionals 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 Properties Dual properties A collection of risk functionals 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 ). A is called version independent, if Y 1 L = Y2 implies that A(Y 1 ) = A(Y 2 ).

5 Properties Dual properties A collection of risk functionals Risk functionals 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 (TI) Translation invariance: D(Y + c) = D(Y ) (MCXD) Monotonicity w.r.t. convex dominance. Y 1 CXD Y 2 implies that D(Y 1 ) D(Y 2 ).

6 Properties Dual properties A collection of risk functionals 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. Example. A(Y ) = inf{e(y Z) + α(z) : Z Z}. { 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.

7 Properties Dual properties A collection of risk functionals Standard 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}.

8 Properties Dual properties A collection of risk functionals Dual properties 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.

9 Properties Dual properties A collection of risk functionals Dual properties 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.

10 Properties Dual properties A collection of risk functionals Dual representations Deviation functional D Dual representation Properties Y EY p p sup{e(y Z) p1 q D q q(z) : EZ = 0} MCXD Y EY p sup{e(y Z) : E(Z) = 0, D q(z) 1} MCXD [Y EY ] p p sup{e(y Z) p1 q E[( essup Z Z) q q ] : E(Z) = 0} MCXD [Y EY ] p sup{e(y Z) : E(Z) = 0, Z 1, essup Z Z q 1} MCXD Acceptability functional A Dual representation Properties EY Y EY p p inf{e(y Z) + p1 q D q q (Z) : EZ = 1} EY Y EY p inf{e(y Z) : E(Z) = 1, D q (Z 1) 1} EY [Y EY ] p p inf{e(y Z) + p1 q E[(Z essinf Z) q q ] : E(Z) = 1} EY [Y EY ] p inf{e(y Z) : E(Z) = 1, Z 0, Z essinf Z q 1} MSSD Here D q (Z) = inf{ Z a q q : a R} and 1/p + 1/q = 1.

11 Properties Dual properties A collection of risk functionals Minimal loss (ML) deviation risk functionals Let k be convex, nonnegative, k(x) = 0 iff x = 0. D k (Y ) := inf{e[k(y a)] : a R}. History: Laplace, Gauss (1820), A. Wald (1948), Huber (1970 ies). D k is translation invariant, convex and nonnegative and has the representation D k (Y ) := sup{e(y Z) E[k (Z)] : EZ = 0}, where k is the Fenchel dual of k, k (v) = sup{uv k(u) : u R}. A k (Y ) := EY D k (Y ) = inf{e(y Z) E[k (1 Z)] : E(Z) = 1}.

12 Properties Dual properties A collection of risk functionals Examples for minimal loss functionals k(u) = u γ γ u, D k (Y ) = AV@RD γ (Y ) = E(Y ) AV@R γ (Y ), A k (Y ) = AV@R γ (Y ) = max{a 1 γ E([Y a] ) : a R} α G 1 = 1 α Y (u) du 0 = inf{e(y Z) : E(Z) = 1, 0 Z 1/γ}. AV@R γ is positively homogeneous and monotonic w.r.t. SSD. k(u) = u 2, D k (Y ) = Var(Y ). Var(Y ) = sup{e(y Z) 1 4 Var(Z) : EZ = 0}.

13 Properties Dual properties A collection of risk functionals Distortion acceptability functionals A H (Y ) = 1 0 G 1 (p) dh(p) (version independent) where H is a distribution function of a nonnegative measure on [0,1]. History: Deneberg(1965), Yaari (1987), Wang (2000). 1 H distorted df. original df. G G(0)

14 Properties Dual properties A collection of risk functionals Examples for distortion functionals E(Y ) = 1 0 G 1 (p) dp H(p) = p V@R α (Y ) = G 1 (α) H(p) = 1l [α,1] (p) AV@R α (Y ) = 1 α α 0 G 1 (p) dp H(p) = min(p/α, 1) 1 0 G 1 (p)h(p) dp : Yaari s dual functional H(p) = p 0 h(q) dq Power distortion H(p) = p r 0 < r < 1 Wang distortion H(p) = Φ(Φ 1 (p + λ)) λ > 0

15 Properties Dual properties A collection of risk functionals Proposition. If H is concave with H(p) = p 0 h(q) dq, then A H (Y ) = inf{e(y Z) : Z = h(u), where U is uniformly [0,1] distributed}. which is the same as A H (Y ) = inf{e(y Z) : Z CXD Z, where Z = h(u), Here we use the following result: Lemma. with U uniformly [0, 1] distributed }. conv {Z : Z L = V } = {Z : Z CXD V }.

16 Algorithms Y x = x ξ, where x are the portfolio weights. Model A-C - acceptability as constraint: Maximize (in x) : x T E(ξ) subject to A(Y x ) q x T 1l = 1 x 0 Model D-C - deviation as constraint: Maximize (in x) : x T E(ξ) subject to D(Y x ) v x T 1l = 1 x 0

17 Algorithms Primal procedures for Minimal Loss functionals The problem Maximize (in x) : x T E(ξ) subject to inf{e[k(x T ξ a)] : a R} v x T 1l = 1 x 0 is formulated as a problem with one additional variable and a convex constraint Maximize (in x and a): x T E(ξ) subject to E[k(x T ξ a)] v x T 1l = 1 x 0

18 Algorithms Dual procedures for concave acceptability functionals Let A(Y ) = inf{e(y Z) + α(z) : Z Z}. The problem Maximize (in x) : x T E(ξ) subject to A(Y x ) q x T 1l = 1 x 0 is reformulated as a semi-infinite linear program Maximize (in x) : x T E(ξ) subject to E(Y x Z) + α(z) q for all Z Z x T 1l = 1 x 0

19 Algorithms An iterative algorithm 1. Set Z =. 2. Solve Maximize (in x) : x E(ξ) subject to E(Y x Z) + α(z) q for all Z Z x T 1l = 1 x 0 3. Solve inf{e(y x Z) + α(z) : Z Z}, add the minimizer function to Z and goto 2. or stop.

20 Algorithms A closer look to this algorithm for distortion functionals Consider the inner problem inf{e(y x Z) : Z = h(u), where U is uniformly [0,1] distributed}. for a decreasing h. Set η s = s/s (s 1)/S h(p) dp. By Hoeffding s Lemma, E(Y x Z) is minimized for all Z which have a given distribution, if Y x and Z are antimonotone, i.e. have the Fréchet lower bound copula C(p, q) = max(p + q 1, 0). If Y x takes the values y s, s = 1,... S with probability 1/S each, then the joint distribution of Y x and the minimizer Z takes the value (y s, η k ) if y s is the k-th smallest among (y 1,..., y S ). Thus the inner problem does not need optimization, it just needs sorting.

21 Algorithms as constraint is a distortion functional, but for a nonconcave H. There is no dual representation. For to solve we follow two paths: Smoothing Censoring Maximize (in x) : x E(ξ) subject to V@R α (x ξ) q x 1l = 1 x 0 (1)

22 Algorithms Smoothing A. Gaivoronski, G. Pflug: Value-at-risk in portfolio optimization: properties and computational approach, Journal of Risk, 7 (2) Winter 2004/05, 1-31 (2005) VaR SVaR, slight smoothing SVaR, more smoothing

23 Algorithms The censoring idea Let [Y ] c be the left censored variable [Y ] c = max(y, c). Proposition. AV@R α (Y ) = V@R α (Y ) 1 α G(G 1 (α)) AV@R α ([Y ] c ) = max(v@r α (Y ), c) 1 α [G(G 1 (α)) G(c)] + where G(u) = u G(v) dv The AV@R α ([Y ] c ) line (dashed) and the identity c c (solid)

24 Algorithms The censoring algorithm Let f (x) be a nonconcave function and let f (x) be a concave minorant of f. We want to solve max{r x : f (x) q : x S}. We start with solving x 0 = argmax {r x : f (x) q : x S}. For every z S (the simplex) with f (z) > q we construct a closed convex trust region C z,m and a concave function f z,m such that z C z,m f z,m (x) f (x) for x C z,m Here m is a parameter governing the size of the trust region. Iterative algorithm: x n+1 = argmax {r x : f xn,m(x) q : x C xn,m, x S}. Here f (x) = V@R α (Y x ), f (x) = AV@R α (Y x ), f z,m is an AV@R function for partially left censored data.

25 Algorithms A typical example 500 weekly data from NYSE, 6 assets. Constraint: Loss of more than 7% only in at most 5% of cases Maximal expected weekly return Improvement AV@R relaxation: 0.57% Censoring algorithm: 0.8% 40 % Constraint: Loss of more than 8% only in at most 5% of cases Maximal expected weekly return Improvement AV@R relaxation: 0.68% Censoring algorithm: 0.9% 35 %

26 Algorithms

27 Algorithms

28 Algorithms First, the efficient frontier was plotted (lower concave curve). Then, the return was improved by relaxing the to a V@R-constraint. One sees that the return may be considerably improved.