Solving real-life portfolio problem using stochastic programming and Monte-Carlo techniques Martin Branda Charles University in Prague Faculty of Mathematics and Physics International Conference on Computational Management Science July 28-30, 2010, Vienna Martin Branda (Charles University) 2010 1 / 36
Contents 1 General formulation and approximation schema 2 Investment problem formulations 3 Sample approximation using Monte-Carlo techniques 4 Numerical comparison Martin Branda (Charles University) 2010 2 / 36
General formulation and approximation schema Contents 1 General formulation and approximation schema 2 Investment problem formulations 3 Sample approximation using Monte-Carlo techniques 4 Numerical comparison Martin Branda (Charles University) 2010 3 / 36
General formulation and approximation schema Formulation and approximation schema using stochastic programming 1. 2. 3. Stochastic Sample Solution programming approximation validation formulation (SA) Program with a random − → Chance constrained − → SA CCP − → Reliability factor problem (CCP) ց ↓ Penalty function − → SA PFP − → Reliability problem (PFP) Martin Branda (Charles University) 2010 4 / 36
General formulation and approximation schema Goal We will compare the ability of both sample approximated problems to generate a feasible solution of the original problem chance constrained problem. Similar study was performed by J. Dupaˇ cov´ a, et al. (1991) and E. ˇ Zampachov´ a (2009). Martin Branda (Charles University) 2010 5 / 36
Investment problem formulations Contents 1 General formulation and approximation schema 2 Investment problem formulations 3 Sample approximation using Monte-Carlo techniques 4 Numerical comparison Martin Branda (Charles University) 2010 6 / 36
Investment problem formulations Investment problem We consider 13 most liquid assets which are traded on the main market ( SPAD ) on Prague Stock Exchange. Weekly returns from the period 6th February 2009 to 10th February 2010 are used to estimate the parameters of distributions. Suppose that the small investor trades assets on the ”mini-SPAD” market. This market enables to trade ”mini-lots” (standardized number of assets) with favoured transaction costs. Martin Branda (Charles University) 2010 7 / 36
Investment problem formulations Loss random variable The random loss function n n � � Z ( x , y , R ) = − ( R i − c i ) Q i x i + f i y i , i =1 i =1 Q i the quotation of the ”mini-lot” of security i , f i the fixed transaction costs , c i the proportional transaction costs , R i the random return of the security i , truncated normal distribution and multivariate skewed t-distribution (A. Azzalini et al (2003)) were used to model the random returns, x i the number of ”mini-lots”, y i binary variables which indicate, whether the security i is bought or not . Martin Branda (Charles University) 2010 8 / 36
Investment problem formulations Set of feasible solutions The set of feasible solutions contains a budget constraint and the restrictions on the minimal and the maximal number of ”mini-lots” which can be bought, i.e. { ( x , y ) ∈ N n × { 0 , 1 } n X = B l ≤ � n i =1 (1 + c i ) Q i x i + � n i =1 f i y i ≤ B u , l i y i ≤ x i ≤ u i y i , i = 1 , . . . , n } , where B l and B u are the lower and the upper bound on the capital available for the portfolio investment, l i > 0 and u i > 0 are the lower and the upper number of units for each security i . Martin Branda (Charles University) 2010 9 / 36
Investment problem formulations Portfolio problem with a random factor n n � � ” min ( x , y ) ∈ X ” − ( R i − c i ) Q i x i + f i y i i =1 i =1 or equivalently ” ( r , x , y ) ∈ R × X ” r min n n � � − ( R i − c i ) Q i x i + f i y i ≤ r i =1 i =1 But the random returns R i are random ... Martin Branda (Charles University) 2010 10 / 36
Investment problem formulations Formulation and approximation schema using stochastic programming 1. 2. 3. Stochastic Sample Solution programming approximation validation formulation (SA) Program with a random − → Chance constrained − → SA CCP − → Reliability factor problem (CCP) ց ↓ Penalty function − → SA PFP − → Reliability problem (PFP) Martin Branda (Charles University) 2010 11 / 36
Investment problem formulations Chance constrained problem Value at Risk problem The chance constrained portfolio problem can be formulated as follows ( r , x , y ) ∈ R × X r min n n � � � � P − ( R i − c i ) Q i x i + f i y i ≤ r ≥ 1 − ε, i =1 i =1 for some ε ∈ (0 , 1). Martin Branda (Charles University) 2010 12 / 36
Investment problem formulations Solving chance constrained problems In general, the feasible region is not convex even if the functions are convex, it is even not easy to check feasibility because it leads to computations of multivariate integrals. Hence, we will try to reformulate the chance constrained problem using a penalty function and incorporate it into the objective function... Martin Branda (Charles University) 2010 13 / 36
Investment problem formulations Penalty function portfolio problem Corresponding penalty function problem using the simple penalty function [ · ] + is n n � + � � � ( r , x , y ) ∈ R × X r + N · E min − ( R i − c i ) Q i x i + f i y i − r . i =1 i =1 Setting N = 1 / (1 − ε ) we obtain the CVaR problem, c.f. R.T. Rockafellar, S. Uryasev (2002). Martin Branda (Charles University) 2010 14 / 36
Investment problem formulations Asymptotic equivalence of chance constrained and penalty function problems Under the following assumptions, the asymptotic equivalence of the problems can be shown: Continuity of constraints and probabilistic functions. Compactness of the fixed set of feasible solutions. Existence of a permanently feasible solution . See Y.M. Ermoliev, et al (2000) - one joint chance constraint and particular penalty function (sum of positive parts), M.B., J. Dupaˇ cov´ a (2008) - one joint chance constraint and arbitrary penalty function, M.B. (2010A) - several joint chance constraints and arbitrary penalty functions. Martin Branda (Charles University) 2010 15 / 36
Investment problem formulations Deterministic vs. stochastic penalty method Deterministic - penalizes the infeasibility with respect to the decision vector, cf. M.S. Bazara, et al. (2006). Stochastic - penalizes violations of the constraints jointly with respect to the decision vector and to the random parameter... Martin Branda (Charles University) 2010 16 / 36
Sample approximation using Monte-Carlo techniques Contents 1 General formulation and approximation schema 2 Investment problem formulations 3 Sample approximation using Monte-Carlo techniques 4 Numerical comparison Martin Branda (Charles University) 2010 17 / 36
Sample approximation using Monte-Carlo techniques Formulation and approximation schema using stochastic programming 1. 2. 3. Stochastic Sample Solution programming approximation validation formulation (SA) Program with a random − → Chance constrained − → SA CCP − → Reliability factor problem (CCP) ց ↓ Penalty function − → SA PFP − → Reliability problem (PFP) Martin Branda (Charles University) 2010 18 / 36
Sample approximation using Monte-Carlo techniques Sample approximated chance constrained problem (SA CCP) The problem can be formulated as a large mixed-integer linear program using additional binary variables z s , s = 1 , . . . , S ϕ CPP , S ˆ = min ( r , x , y , z ) ∈ R × X ×{ 0 , 1 } S r γ n n � � ( R s − i − c i ) Q i x i + f i y i − M (1 − z s ) ≤ r , i =1 i =1 S 1 � z s ≥ 1 − γ, S s =1 for some level γ ∈ (0 , 1) ( γ � = ε in general). Martin Branda (Charles University) 2010 19 / 36
Sample approximation using Monte-Carlo techniques Sample approximated penalty function problem (SA PFP) The variables which are necessary to replace the positive parts are nonnegative continuous and the resulting problem is S r + N ϕ S � ˆ = min S · v s N ( r , x , y , v ) ∈ R × X × R S + s =1 n n � � ( R s v s ≥ − i − c i ) Q i x i + f i y i − r . i =1 i =1 for some penalty parameter N > 0. Martin Branda (Charles University) 2010 20 / 36
Sample approximation using Monte-Carlo techniques Estimated sample sizes Estimates for the sample sizes which are necessary to generate a lower bound for the optimal value of the original chance constrained problem, cf. S. Ahmed, A. Shapiro (2008): ( γ − ε ) 2 ln 1 2 ε S ≥ δ , a feasible solution of the original chance constrained problem, slight modification of J. Luedtke, S. Ahmed (2008): � � � 2 ln 1 � 2 � � 2 D S ≥ δ + 13 ln 116 + 13 ln 2 + ln + ln , ( ε − γ ) 2 ( ε − γ ) τ which is based on the decomposition of the set of feasible solutions into the integer and real bounded part, where | X | ≤ 116 13 · 2 13 , and we set τ = 10 − 6 , D = 2 · 10 6 (difference between the worst loss and the best profit for the distribution with bounded support). Martin Branda (Charles University) 2010 21 / 36
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