Combinatorial Data Mining Method for Multi-Portfolio Stochastic Asset Allocation

Transcription

1 Combinatorial for Stochastic Asset Allocation Ran Ji, M.A. Lejeune Department of Decision Sciences July 8, 2013

2 Content Class of Models with Downside Risk Measure Class of Models with

3 of multiple portfolios simultaneously Impose joint effects on the sub-portfolios Consider the benefits (return/risk) of global portfolio Financial Application Generalized manager conducting multiple portfolios in different asset classes simultaneously 3 Class of Models with

4 Class of Models with Model 1: Maximize the expected return of global portfolio max µ j x ij (1) s.t. P { i I j J ξ j x ij d i, i I j S i } p (2) j S i x ij = a i i I (3) x ij 0 i I, j J (4) J = set of all assets in market; I = set of all sub-portfolios; S i = set of available assets in sub-portfolio i; x ij = proportion of capital invested in asset j of sub-portfolio i; µ j = expected return of asset j; ξ j = random loss (negative return) of asset j; d i = accepted loss level for sub-portfolio i; a i = proportion of capital invested in sub-portfolio i; p = probability that sub-portfolios losses below the accepted level; 4 Class of Models with

5 Class of Models with Model 2: Minimize the variance of global portfolio min x T Σx s.t. (2) (4) Model 3: Multiobjective model: maximizing the expected return of the global portfolio while minimizing the sum of loss levels of the sub-portfolios max µ j x ij α d i i I j J i I s.t. (2) (4) Model 4: Maximize the joint probability p that loss of each sub-portfolio to be below the threshold level d i 5 Class of Models with max p s.t. (2) (4)

6 Joint Probabilistic Constraint P ξ j x ij d i, i I p j S i A multi-row chance constraint with random technology matrix with finite support. A joint discrete distribution for asset returns. Each row corresponds to a specific sub-portfolio whose random loss is required to be below a threshold level. The problem is usually non-convex and NP-hard. 6 Class of Models with

7 Example Consider a multi-portfolio optimization problem with 3 assets and 2 sub-portfolios. The first sub-portfolio contains assets 1 and 2, the second sub-portfolio contains assets 1 and 3. max 0.005(x 11 + x 12 ) x x 32 { } ξ1 x 11 + ξ 2 x 0.04 s.t. P 0.7 ξ 1 x 12 + ξ 3 x x 11 + x = 0.40 x 12 + x 32 = 0.60 x 11, x, x 12, x Class of Models with

8 Example The 3-dimensional random vector ξ accepts 10 equally likely realizations l k. l k j represents the loss of asset j in realization k. k l1 k l2 k l3 k F 1 (l1 k) F 2(l2 k) F 3(l3 k) F (l k ) Class of Models with The feasible set of probabilistic constraint is the union of several polyhedrons, and it is usually non-convex.

9 Logical Analysis of Data (LAD): Combination of optimization, combinatorics and Boolean functions. Class of Models with 9 Four Steps of Boolean Method recombinations Binarization of probability distribution Derivation of minorant MIP reformulations

10 Definition A realization of loss l k is called a p-sufficient loss realization if and only if F (l k ) p and is called a p-insufficient loss realization if F (l k ) < p. If a realization l k is p-sufficient, we have P(ξ l k ) p, then lj k x ij d i P ξ j x ij d i p j S i j S i To obtain the optimal solution to the problem, we must not only focus on the existing realizations, but should also consider all points that can possibly be p-sufficient. Let: C j = {l k j : l k j F 1 j (p), k Ω} j J The set C provides all points that can possibly be p-sufficient: Class of Models with 10 C = C 1 C j C J which we name recombinations.

11 Binarization Process The binarization process is the mapping R J {0, 1} n of a numerical vector l k into an n-binary vector β k = [β k 11,..., β k 1n 1,..., β k jm,..., β k jn j,...] such that the value of each component β jm is defined with respect to the cut point c jm as follows: Class of Models with 11 β k jm = { 1 if l k j c jm 0 otherwise where c jm denote the m th cut point associated with component ξ j, and we arrange them in ascending order such that m < m c jm < c jm, m = 1,..., n j, j J

12 Definition of sufficient-equivalent consistent cut points C e = J C j, where C j = {lj k : k Ω + } j=1 C e ensures no pair of p-sufficient and p-insufficient recombinations will share the same binary image. The binarization process allows the representation of combination of (F, p), the probability distribution F and the prescribed probability level p as a partially defined Boolean function (pdbf). Partially Defined Boolean Function(pdBf) Class of Models with 12 The function g( Ω + B Ω B ) is a partially defined Boolean function (pdbf) if there exists two disjoint subsets Ω + B and Ω B such that ( Ω + B Ω B ) = Ω B {0, 1} n, with the sets of true points Ω + B and false points Ω B.

13 Set of and Boolean Vectors Class of Models with 13

14 Concept of Minorant Objective of Minorant To solve the problem more efficiently, we are aiming at identifying tighter, or possibly minimal sufficient conditions for the probabilistic constraint to hold. To derive mathematical formulations with linear constraints to define a separating structure that includes at least one of the p-sufficient recombinations without any p-insufficient ones. Definition of Minorant A Boolean function f is a minorant of a pdbf g( Ω + B, Ω B ) if Ω B F(f ). A Boolean function f is a tight minorant of a pdbf g( Ω + B, Ω B ) if f is a minorant of g( Ω + B, Ω B ) and T(f ) Ω + B. Class of Models with 14

15 Derivation of Tight Minorant Formulations of Threshold Tight Minorant A threshold tight minorant f of g( Ω + B, Ω B ) contains J non-zero weights. Every feasible solution of the following system of inequalities n j j J m=1 w jm β k jm J 1 k Ω B (5) n j m=1 w jm = 1 j J (6) w jm {0, 1} j J, m = 1,, n j (7) Class of Models with 15 defines a tight minorant f with integral separating structure (w, J ) {0, 1} n.

16 Quadratic Programming Reformulation Recall that if a realization l k is p-sufficient, then lj k x ij d i P ξ j x ij d i p j S i j S i The p-sufficient recombinations lj k can be expressed by nj m=1 c jmw jm. The probabilistic constraint is transformed into QP Reformulation max s.t. c jm w jm x ij d i j S i m=1 n j n j j S i m=1 j S i i I µ i x ij c jm w jm x ij d i (3); (4); (5); (6); (7) i I i I Class of Models with 16

17 Mixed-Integer Programming Reformulations max s.t. µ i x ij i I j J j S i y j d i i I y j c jm x ij (1 w jm )c jnj y j 0 n j w jm βjm k J 1 j J m=1 n j m=1 w jm = 1 w jm {0, 1} x ij = a i j S i x ij 0 j J, m = 1,..., n j j J k Ω B j J j J, m = 1,..., n j i I i I, j J Class of Models with 17

18 Data Laboratory Collect returns of 15 equity indices and 5 bond indices ranging from 1991 to 2010 To construct 4 sub-portfolios US S&P500, Dow Jones Industrial, NASDAQ, NYSE, S&P100 EU CAC40, DAX, FTSE100, OMSX30, SMI AS Hengseng Index, NEKKEI 225, SSE, Straits Times, IPC Bond Global Aggregate, US Aggregate Bond, Global Treasury Bond, US Treasury, US Government Class of Models with 18

19 Randomly generated instances with different values of a i and d i. Lower bounds of loss level d i of sub-portfolios Table : Computational Results for Models J p CPU Time (s) d 1 d 2 d 3 d Class of Models with 19

20 Propose a class of probabilistic downside risk models for multi-portfolio optimization Propose a Boolean modeling method to reformulate the stochastic problem as a mixed-integer programming problem Computational testing for Boolean method to solve multi-portfolio problems Future work: 1. Include other risk measures to sub-portfolio constraints 2. Include the impact of transaction costs 3. More computational testing on multi-portfolio optimization models Class of Models with 20

21 Appendix: Transformation from QP into MIP n j n j With c jm w jm x ij d i, let y j c jm w jm x ij, j S i j J m=1 m=1 n j Since there is exactly one w jm could be =1 ( w jm = 1 j J), y j c jm w jm x ij, m=1 j J, m = 1,..., n j Consider an m, such that w im = 1 and w im = 0, m m. { cjm x y j c jm w jm x ij = ij if m = m 0 otherwise Include a big value of M to express the formulations as follows: y j c jm x ij (1 w jm )M jm y j 0 j J, m = 1,..., n j j J M jm is supposed to be the upper bound of c jm x ij, which is c jnj. Therefore, the quadratic constraint can be expressed as follows: y j c jm x ij (1 w jm )c jnj j J, m = 1,..., n j Class of Models with