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Models and Tools for Portfolio Planning

APMOD 2006. Models and Tools for Portfolio Planning. Plenary Talk APMOD 2006 June 19 Gautam Mitra also included research results of: John Beasley, Ken Darby-Dowman, Frank Ellison Cormac Lucas, Diana Roman Acknowledgement: Industry sponsors, UBS Investment Research, INSIGHT ( HBOS ).

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Models and Tools for Portfolio Planning

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  1. APMOD 2006 Models and Tools for Portfolio Planning Plenary Talk APMOD 2006 June 19 Gautam Mitra also included research results of: John Beasley, Ken Darby-Dowman, Frank Ellison Cormac Lucas, Diana Roman Acknowledgement: Industry sponsors, UBS Investment Research, INSIGHT ( HBOS )

  2. Outline Outline 2. Modelling and solving: key issues 2.1 Modelling issues 2.2 Methods and algorithms • Background: current status 3. Computational results: a selection 4. New developments 4.1 Shaping the distribution I 4.2 Shaping the distribution II 5. Summary 6. References

  3. Background: current status A historical perspective • Markowitz ..mean-variance 1952,1959 • Hanoch and Levy 1969, valid efficiency criteria • individual’s utility function • Kallberg and Ziemba’s study.. alternative utility functions • Sharpe ..single index market model 1963 • Arrow- Pratt.. absolute risk aversion

  4. Mean-Variance Model - Markowitz (1952,1959) - alternative formulation • QP (Arrow-Pratt absolute risk aversionparameter)

  5. Background: current status A historical perspective..cont • Sharpe 64, Lintner 65, Mossin 66…CAPM model • Rosenberg 1974 multifactor model • Ross.. Arbitrage Pricing Theory(APT) multifactor equilibrium model • Text Books: Elton & Gruber also Grinold & Kahn • LP formulation 1980s.. computational tractability • Konno MAD model.. also weighted goal program • Perold 1984 survey…

  6. Modelling Issues Portfolio Planning • Mean variance Current practice and R&D focus: • Factor model • Rebalancing with turnover limits • Index Tracking(+enhanced indexation) [Style input and goal oriented model] • Cardinality of stock held: threshold constraints • Cardinality of trades: threshold constraints

  7. Modelling Issues Portfolio Planning • Tracking error as a constraint…[discuss ] Emerging industry focus and research challenges: • Nonlinear transaction cost /market impact[discuss ] • Trade scheduling =algorithmic trading.. [discuss ] • Resampled efficient frontier • Risk attribution and risk budgeting

  8. Modelling and solving: key issues Portfolio Models • Mean-VarianceMAX S.T • Factor ModelMAX S.T

  9. Modelling and solving: key issues • Index TrackerMax • Turnover Limit Rebalancing model • G.O. Side Constraints

  10. Modelling and solving: key issues • Integer Decision Variables & Cardinality Constraints □ □ □ Specifies cardinality limit to no. of assets x

  11. Modelling and solving: key issues Groups of discrete logical variables B = whether to buy S = whether to sell (only if x*>0 ) K = whether to keep part or all the initial hold (only if x* > 0)Threshold constraints for the -variables are: minB xBB minS xSS minK x K  where min is the minimum weight that can be traded or held. Cardinality constraints are: B + S <= Tmax (maximum trades) (B|x*=0) + K <= Hmax (maximum stocks) And extra constraints that may be needed B + S <= 1, (only if x* > 0)

  12. Modelling and solving: key issues Discrete Constraint Efficient Frontier (DCEF) (3) Example: • 4 stock universe, EF • introduce cardinality constraint, i.e. build a portfolios containing 2 stocks only Effect: Discontinuities in the Efficient Frontier

  13. Modelling and solving: key issues Discrete Constraint Efficient Frontier (DCEF) (2) Why discontinuities? • take investment opportunity set • delete all inefficient portfolios (dominated points)

  14. Modelling and solving: key issues MV-global mean-variance portfolio E – a portfolio on (global) efficient frontier ( = risk of benchmark) P – on Tracking Error frontier but higher risk!

  15. Trade scheduling Statement of the problem: Current holding is defined as pre trade portfolio rebalance the portfolio ( buy and sell amounts ) => trade orders to dealers/brokers this leads to post trade portfolio Trading risk is the volatility of a long/short Portfolio = portfolio of trade orders

  16. Trade scheduling = algorithmic trading We want to minimise the loss of value in getting our portfolio from what it is to what we want it to be..! • Trading N shares of one stock • Trading Ni shares of i=1,..m stocks • Trading Ni shares of i=1,..m stocks over t = 1,…T periods

  17. Trade scheduling Trade cost analysis: Agency cost Bid/ask spread Market impact ( my trade moves the price) Trend cost (other people’s trade moves the price…may be in my favour ) Influences: Trade in Ford impacts Trade in GM Consider the two sets of orders -Buy 5 million Ford// buy 5 million GM -Buy 5 million Ford// sell 5 million GM

  18. Modelling and solving: key issues Non-linear transaction cost Cost Segment 1 Segment 3 Segment 2 Transaction amount

  19. Modelling and solving: key issues Actual T-costs Buy cost Sell cost Buy rate Sell rate Buy Amount Sell Amount Cost Slope r4 Slope r3 v4 Slope r2 Slope r1 v3 v2 v1 a1 a2 a3 a4 Separable programming = special ordered sets

  20. Computational models WE MOVE BACK TO THE MODELS WITH WHICH WE ARE CURRENTLY PREOCCUPIED..!

  21. Methods and algorithms QP for Portfolio Planning • Problem Statement: KKT optimality conditions • Solution of continuous convex QP • LP extended to QP: Wolfe’s Algorithm • Beale, Dantzig, Cottle and Van-de-Panne’s pivotal method • Hohenbalken’s Method: Successive LPs • Goldfarb’s dual approach • Interior point method (IPM)

  22. Methods and algorithms most general statement of the quadratic program : is a statement of the general, QP problem. For a given variable index j, when absent specifies default value , when absent specifies a default value , that is, no finite upper bound.

  23. Methods and algorithms QP for Portfolio Planning • Convex Quadratic Programming Maximize: Subject to:

  24. Methods and algorithms The Lagrangean function Karush Kuhn-Tucker ( KKT ) conditions

  25. Methods and algorithms The Algorithm • Sparse-Simplex And Symmetry/skew symmetry in the generalised version Optimum via complementary bases Complementary bases: Better C.B. via neighbouring bases (1 pair basic, 1 pair non-basic)

  26. Methods and algorithms Algorithm sequence • Primal Version • b values are feasible • c infeasible • Dual version • c values feasible • b infeasible • 1. If the tableau in Standard form choose incoming as the dual complement of the infeasible primal. • 2. If the tableau is nearly complementary nonStandard form • take as the incoming variable the primal • 3. Follow normal LP procedure to update column • 4. Select outgoing variable by ratio test and include the complement in the test. • Terminates**** Either =>Primal and Dual feasibility [optimal ] • or =>unbounded ray [dual]… primal infeasible

  27. Methods and algorithms Family of models • Efficient Frontier: Succession of λ-values in monotonic sequence Succession of models --- each very similar to its predecessor • Models laid out on B&B Tree Continuous ( 0 ≤  ≤ 1 ) Discrete ( =0 or 1) by succession of FIXES□ Succession of fixes Succession of models – each very similar to its predecessor

  28. Methods and algorithms Family of models…in sequence A sequence of models P0, P1,… Pk Let optimum bases for these problems be denoted Let number of iterations be denoted Let number of basis changes from neighbour be denoted Typically is of the order of 1 percent or less

  29. Methods and algorithms Return Risk Family of risk problems in the efficient frontier

  30. Methods and algorithms RQP DECRESASING Z BOUNDED BRANCH AND BOUND INTEGER SOLUTION

  31. Methods and algorithms Warm Start • Basis Save (compact form) • Restart by factorising the basis • Basis Factors Save • Restart and continue (in situ) • Solution Save • Restart with “CRASH” and “PUSH” • External Save • Supply “SAVE” material to start run

  32. Methods and algorithms Branch fix & Relax In the search as takes the value zero, also becomes zero, the values of and all become fixed, and constraints (1) and (2) and (3) now become redundant; these may now be relaxed.

  33. Methods and algorithms [Optimum]Basis Prediction • Take Primal non-zeros into the basis • Add dual complements in the opposite status • When updating the model • Fix a variable • Primal variable leaves basis • Dual complement enters basis • Free a constraint • Primal constraint (logical) enters basis • Dual complement leaves basis

  34. Computational result: a selection The Results:The ModelsSolver PerformanceEfficient FrontierBranch, Fix and Relax

  35. Computational result: a selection The Models

  36. Computational result: a selection Relaxed QP Performance

  37. Computational result: a selection Integer Stage Performance

  38. Computational result: a selection Continuous Efficient Frontier (Cfu9583)

  39. Computational result: a selection Discrete EF (Cfu508)

  40. Computational result: a selection Branch, Fix and Relax: Cfu9583 illustration

  41. Computational result: a selection Summary

  42. Computational result: a selection Comparisons with CPLEX

  43. Shaping the distribution I Distribution properties of a portfolio …shaping the distribution

  44. Shaping the distribution I Three principles of choice(under uncertainty) • Expected utility maximisation • Mean-risk model • Stochastic dominance The first approach: Portfolio construction based on stochastic dominance and target return distributions

  45. Shaping the distribution I First and Second Order Stochastic Dominance Definition: Rx is preferred to Ry with respect to FSD (Notation: Rx>FSD Ry) if and only if Fx(r)Fy(r), for every r, with at least one strict inequality. Intuitively, this describes the preference for larger outcomes. Definition: Rx is preferred to Ry with respect to SSD (Notation: Rx>SSD Ry) if and only if F(2)x(r)F(2)y(r), for every r, with at least one strict inequality, Where: .

  46. Shaping the distribution I First and Second Order Stochastic Dominance- discrete random variables Consider now the special case where the two random variables Rx and Ry are discrete with T equally probable outcomes. In order to make the comparison, we rank the outcomes of Rx and Ry in ascending order to obtain two T-dimensional vectors: (1,…, T) and (1,…, T) such that 1…T and 1…T. Rx>FSD Ry if and only if ii for every i{1,…,T} with at least one strict inequality. Rx>SSD Ry if and only if for every i{1,…,T} with at least one strict inequality.

  47. Shaping the distribution I Motivation and contribution The attempts to combine the practicality of mean-risk models with the theoretical solid basis of stochastic dominance  mean-risk models consistent with SD. • The proposed model that finds a portfolio which is: • non-dominated with respect to SSD ( optimal for every rational and risk averse investor). • has a return distribution close to a reference (target, goal, aspiration) distribution which is introduced as input data. The entire distribution of portfolio return is taken into account, not just summary statistics. This is achieved using a computationally tractable procedure which uses linear inequalities and LP [Goal] models.

  48. Shaping the distribution I Motivation and contribution Close to a reference distribution means “close or better”: 3 cases: 1. The reference distribution is SSD efficient  portfolio whose return distribution is the reference distribution. 2. The reference distribution is not SSD efficient  portfolio whose return distribution is SSD efficient - better than reference distribution (outperforms the goal!) 3. The reference distribution is not attainable (too high outcomes) portfolio whose return distribution is SSD efficient and comes close to the reference distribution.

  49. SSD efficiency and multiple criteria optimisation The portfolio selection problem: case of T equally probable scenarios. portfolio x  return (f1(x),…,fT(x)): random variable seen as a T-dimensional vector, where (the portfolio return in scenario i=1,…,T)

  50. SSD efficiency and multiple criteria optimisation How do we express that z1=(f1(x1),…fT(x1)) dominates z2=(f1(x2),…fT(x2)) with respect to SSD relation? The sum of the worst k outcomes of z1 is greater than the the sum of the worst k outcomes of z2, k=1,…,T. Equivalently: the CVaR of z1 at confidence level k/T is smaller than the the CVaR of z2, at confidence level k/T, k=1,…,T. Example: Order the outcomes (1,4,3,2) and (3,5,0,2) (1,2,3,4) and (0,2,3,5) Cumulate the outcomes Pareto comparison (1,3,6,10) and (0,2,5,10) (1,4,3,2) dominates (3,5,0,2) with respect to SSD

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