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Tools and components for optimisation and risk analysis

Tools and components for optimisation and risk analysis. Professor Gautam Mitra Presented to Clarifi, New York. Outline. Solvers (FortMP / FortSP) Linear / Mixed Integer (LP / IP) Quadratic / Mixed (QP / QMIP) Stochastic optimization (SP) FortMP-MEX Matlab add-on

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Tools and components for optimisation and risk analysis

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  1. Tools and components for optimisation and risk analysis Professor Gautam Mitra Presented to Clarifi, New York.

  2. Outline • Solvers (FortMP / FortSP) • Linear / Mixed Integer (LP / IP) • Quadratic / Mixed (QP / QMIP) • Stochastic optimization (SP) • FortMP-MEX Matlab add-on • Portfolio Optimisation model and engine • Modelling Systems (AMPL Suite) • AMPL Studio • AMPL COM • AMPL SPInE • Liability Determined Investment (LDI) / Asset and Liability Management ALM

  3. Solvers – FortMP • FortMP is a large scale optimiser • Rich functionality • Robust solution algorithm [Math Programming Article] • Solves medium to large models • Not suitable for very large hyper sparse model • Available in stand-alone and library versions

  4. Solvers - FortMP • Barrier and sparse simplex algorithms • Solves variable separable programming including special ordered sets of Type 1 and Type 2 (SOS1 and SOS2) problems • Extends LP to process MIP problems • Branch and bound • Cutting planes • Pre processing techniques

  5. Solvers – FortMP/QP/QMIP • FortMP processes quadratic programs (QP) and quadratic mixed integer programs (QMIP) using • Branch and bound • Branch and relax

  6. Solvers - FortSP • Processes stochastic programming problems with recourse using • Benders decomposition (nested) • Stochastic decomposition • In contrast to deterministic equivalent, these algorithms scale up

  7. Solvers – FortMP MEX • Matlab environment add-in • Permits the use of FortMP’s rich and robust optimising functionalities directly from Matlab • Ideal for rapid application prototyping and for using in research environment

  8. Portfolio Optimisation Model • An optimum asset allocation strategy explores a return and risk (pareto) efficient frontier and in this respect is a two objective (linear return and quadratic risk) constrained optimisation problem. • The Mean-Variance model is the basic portfolio optimisation model which • linear part is E[Rx] • risk measure is cov(Ri,Rj)

  9. Mean Variance model • It can be expressed as a quadratic program (QP): max • Can be refined adding more restriction on the choice of the assets

  10. Other restrictions • Factor model subject to • Index tracking model • where bj are normalized coefficients of the chosen benchmark portfolio

  11. Other restrictions • Rebalancing model • Threshold constraints where δj are binary decision variables

  12. Other restrictions • Cardinality constraints • at most C assets are held • The last two constraints transforms the QP problem in QMIP • Non linear transaction cost Segment 3 Segment 1 Segment 1: Steep initial cost or set up cost. Segment 2: Nearly linear incremental cost over a range. Segment 3: Steep increase in cost of the asset for large volumes of transaction. Segment 2

  13. Models statistics

  14. Models statistics

  15. Benchmarks • Results obtained using FortMP’s • accelerated heuristic functionality

  16. AMPL Overview • AMPL: comprehensive and powerful algebraic modelling language for linear and non-linear optimisation problems • Optimal for rapid prototyping and model development • Extended to express stochastic optimisation models (SAMPL)

  17. Data AMPL Models AMPL Interactive AMPL COM Object C# / C++ / VB / VBA Application AMPL Studio SAMPL/SPInE AMPL Products Offer

  18. AMPL Studio • Integrated modelling system based on AMPL language. • Benefits: • Rich and user-friendly graphical interface • Compact and easy database connection • Workspace management • Model (set / variables) explorer • Seamless integration through memory interaction with various solvers

  19. AMPL Studio Menu Bar Editing Area Workspace and Model Explorer Multifunctional Output Console

  20. AMPL-COM Object • Object Oriented Component Library, based on Microsoft COM software technologies • Rationale: • Utilise the features of a programming language and AMPL individually as well as in combination • Benefits: • Enable to build powerful DSS applications • Hide Models from End Users • Accessible the full AMPL features within any major development environments

  21. Command AMPL Models Solvers Model Solution SolTime Variables Constraints Objectives Options Objective Variable Constraint AMPL-COM Object

  22. AMPL Studio-SPInE Time Index Scenario Tree Structure Scenario Probabilities Scenario Index Probabilistic Constraints Stages Aggregations Random Parameters • Seamlessly integrated into AMPL studio environment • Extends AMPL language with constructs specific for modelling SP problems and interprets them Parameters Indices Variables Constraints Objectives

  23. Liability Determined Investment (LDI)Asset and Liability Management (ALM)

  24. Scope and Purpose of LDI Modelling System • Balance cash in-flow streams of asset returns and asset sales with cash out-flow streams of liability obligation as well as asset purchases • Objective: maximise surplus wealth or terminal wealth at the end of the planning period: Surpluswealth=assets – PV(liabilities) – PV(goals)

  25. LDI Tool Description • Cash flow matching over a long planning horizon: up to 50 years or more • Portfolio mix: mainly fixed income, derivatives (swaps) and if necessary equities

  26. LDI Tool Description • Multi-Objective: • Minimise PV01 Deviations (Deterministic) • Minimise Net PV Deviations (Stochastic) • Maximise Surplus Wealth • Minimise Initial Injected Cash • Minimise Member Contributions

  27. Solving the Decision Model • The decision problem can be formulated and processed as: • LP/IP • SP with recourse • Chance Constrained Programming • Robust Optimisation

  28. Scenario tree structure

  29. Stochastic Features • Scenario generation (User supplied): • Asset prices • Liabilities • Ex-ante asset decisions • Ex-post evaluation (Simulation)

  30. Information • RISKWATCH + Customer Data Builder RISKWATCH + Data Builder DataSources Market Data POPULATE DATA DATA MART#.TAB MODEL DATA Consolidator+Diagnostic Optimisers - FortMP - CPLEX AMPL Modelling Environment[AMPL Com Object + LDI Model File + #.DAT File + #.RUN File] Simulation and visualization to RISKWATCH RESULTS LDI Information Flow

  31. LDI Project Control

  32. Information • RISKWATCH + Customer Data Builder RISKWATCH + Data Builder DataSources Market Data POPULATE DATA DATA MART#.TAB MODEL DATA Consolidator+Diagnostic Optimisers - FortMP - CPLEX AMPL Modelling Environment[AMPL Com Object + LDI Model File + #.DAT File + #.RUN File] Simulation and visualization to RISKWATCH RESULTS LDI Information Flow

  33. LDI Optimising Engine

  34. Information • RISKWATCH + Customer Data Builder RISKWATCH + Data Builder DataSources Market Data POPULATE DATA DATA MART#.TAB MODEL DATA Consolidator+Diagnostic Optimisers - FortMP - CPLEX AMPL Modelling Environment[AMPL Com Object + LDI Model File + #.DAT File + #.RUN File] Simulation and visualization to RISKWATCH RESULTS LDI Information Flow

  35. Simulation and analysis

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