Tools and components for optimisation and risk analysis

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Tools and components for optimisation and risk
analysis

• Professor Gautam Mitra
• Presented to Clarifi, New York.

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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

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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

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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

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Solvers FortMP/QP/QMIP

• FortMP processes quadratic programs (QP) and
  quadratic mixed integer programs (QMIP) using
• Branch and bound
• Branch and relax

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Solvers - FortSP

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

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Solvers FortMP MEX

• Matlab environment add-in
• Permits the use of FortMPs rich and robust
  optimising functionalities directly from Matlab
• Ideal for rapid application prototyping and for
  using in research environment

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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 ERx
• risk measure is cov(Ri,Rj)

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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

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Other restrictions

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

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Other restrictions

• Rebalancing model
• Threshold constraints
• where dj are binary decision variables

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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
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Models statistics
Model Asset universe Cardinality limit No of Factors No of G.O. constraints
Cfu508 508 150 22 556
Cfu525 525 50 26 579
Cfu1057 1,057 50 26 1,109
Cfu1533 1,533 50 26 1,614
Cfu9583 9,583 50 26 9,665
Elu250 1,528 250 21 1,569
Msci150 5,591 150 21 5,635
Msci75 855 75 21 1,057
MsciE100 543 100 24 591
US50 500 50 26 547
Ussa50 500 50 26 547
Msci75sc 855 75 21 1,057
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Models statistics
Model Rows Columns Non-zeros Q-rows Q-nzros
Cfu508 2,117 2,790 13,917 530 606
Cfu525 2,185 3,320 16,215 551 701
Cfu1057 4,311 6,502 31,438 1,083 1,233
Cfu1533 6,244 9,407 46,136 1,559 1,709
Cfu9583 38,445 48,127 242,941 9,609 9,759
Elu250 6,179 77,922 36,470 1,549 1,623
Msci150 22,434 28,236 135,182 5,612 5,702
Msci75 3,491 4,473 20,255 876 950
MsciE100 2,249 3,402 15,749 567 650
US50 2,078 2,620 13,588 526 620
Ussa50 2,078 2,692 13,732 526 620
Msci75sc 3,501 4,483 21,083 950 950
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Benchmarks

• Results obtained using FortMPs
• accelerated heuristic functionality

Model CPLEX CPLEX FortMP FortMP
Model Time Final objective Time Final objective
Cfu508 21.06 -.21349129640e-2 3.58 -.21357886E-02
Cfu525 600.03 -.18568727790e-2 21.06 -.18559756E-02
Cfu1057 7.87 -.45888022432e-2 15.39 -.45887767E-02
Cfu1533 152.64 -.26654366558e-2 121.72 -.26657032E-02
Cfu9583 1102.10 0.29656464844e-1 180.11 -.31311094E-02
Elu250 1.40 0.97411801870e-1 7.59 0.97411804E-01
Msci150 751.71 0.59109482779e-2 92.19 -.11110079E-02
Msci75 600.23 -.96858354537e-2 22.11 -.96916174E-02
MsciE100 600.01 -.20717965889e-2 21.63 -.20881854E-02
US50 0.06 0.29570565523 0.27 0.29570565
Ussa50 600.02 -.18411673090e-2 23.16 -.18477797E-02
Msci75sc 600.04 -.30444937767e-2 30.14 0.29694054
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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)

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AMPL Products Offer
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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

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

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AMPL-COM Object
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AMPL Studio-SPInE

• Seamlessly integrated into AMPL studio
  environment
• Extends AMPL language with constructs specific
  for modelling SP problems and interprets them

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Liability Determined Investment (LDI)Asset and
Liability Management (ALM)
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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
• Surpluswealthassets PV(liabilities)
  PV(goals)

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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

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LDI Tool Description

• Multi-Objective
• Minimise PV01 Deviations (Deterministic)
• Minimise Net PV Deviations (Stochastic)
• Maximise Surplus Wealth
• Minimise Initial Injected Cash
• Minimise Member Contributions

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Solving the Decision Model

• The decision problem can be formulated and
  processed as
• LP/IP
• SP with recourse
• Chance Constrained Programming
• Robust Optimisation

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Scenario tree structure
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Stochastic Features

• Scenario generation (User supplied)
• Asset prices
• Liabilities
• Ex-ante asset decisions
• Ex-post evaluation (Simulation)

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LDI Information Flow
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LDI Project Control
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LDI Information Flow
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LDI Optimising Engine
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LDI Information Flow
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Simulation and analysis