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

Tools and components for optimisation and risk analysis

Professor Gautam Mitra

Presented to Clarifi, New York.

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

• 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

Outline

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Solvers – FortMP/QP/QMIP

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Solvers - FortSP

• Processes stochastic programming problems with recourse using – Benders decomposition (nested)– Stochastic decomposition

• In contrast to deterministic equivalent, these algorithms scale up

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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)

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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

0

subject to

x

bAx

Qxxxp TT

2

1

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Other restrictions

• Factor model

subject to

• Index tracking model

– where bj are normalized coefficients of the chosen benchmark portfolio

K

kikikii efr

1

K

k

N

iikP ixy

1 1

222.

N

ifikikP Kkxyk

1. ,,1

N

iiix

1

N

jejj j

bx1

22)(

n

sfkikisskp

bxy1

)(

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Other restrictions

• Rebalancing model

• Threshold constraints

where δj are binary decision variables

.,1, Njxx j

,,,1, Njxxxx jsj

bjj

N

j

sj

bj txx

1

;)(

jjjjthresholdj ux

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Other restrictions

• Cardinality constraints– at most C assets are held

• The last two constraints transforms the QP problem in QMIP

• Non linear transaction cost

N

jj C

1

Segment 3

Segment 2

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.

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Models statistics

ModelAsset

universeCardinality

limitNo 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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Benchmarks

ModelCPLEX FortMP

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

• Results obtained using FortMP’s

accelerated heuristic functionality

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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)

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

AMPL Products Offer

DataAMPL Models

AMPL Interactive

AMPL Studio

AMPL COM Object

C# / C++ / VB / VBA Application

SAMPL/SPInE

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

AMPL StudioMenu Bar

Editing Area

Multifunctional Output Console

Workspace and Model Explorer

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

AMPL-COM Object AMPL

Models Solvers Command

Model

Variables Constraints Objectives Options SolTime Solution

Variable Constraint Objective

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

AMPL Studio-SPInE• Seamlessly integrated into AMPL studio environment

• Extends AMPL language with constructs specific for modelling SP problems and interprets them

Time Index

Scenario Tree

Structure Scenario Probabiliti

esScenario

Index

Probabilistic

Constraints

Stages Aggregatio

ns

Random Paramete

rs

Parameters

Indices

Variables

Constraints

Objectives

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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)

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

LDI Tool Description

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Solving the Decision Model

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

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Scenario tree structure

-

Expected Value Wait and See

Multi-Stage SPTwo-Stage SP

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Stochastic Features

• Scenario generation (User supplied):– Asset prices– Liabilities

• Ex-ante asset decisions

• Ex-post evaluation (Simulation)

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

LDI Information Flow

RESULTSSimulation and visualization to RISKWATCH

Market Data RISKWATCH + Data Builder

Information

DataSources

RISKWATCH + Customer Data

Builder

POPULATE DATA

Consolidator+

Diagnostic

DATA MART

#.TAB

AMPL Modelling Environment

[AMPL Com Object + LDI Model File + #.DAT File + #.RUN File]

Optimisers

- FortMP- CPLEX

MODEL DATA

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

LDI Project Control

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt



Information

DataSources


Builder

POPULATE DATA

Consolidator+

Diagnostic

DATA MART

#.TAB




Optimisers

- FortMP- CPLEX

MODEL DATA

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

LDI Optimising Engine

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt



Information

DataSources


Builder

POPULATE DATA

Consolidator+

Diagnostic

DATA MART

#.TAB



Optimisers

- FortMP- CPLEX


MODEL DATA

Outline

Solvers

AMPL Suite

LDI/ALM

Porfolio Opt

Simulation and analysis

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

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Transcript of Tools and components for optimisation and risk analysis Professor Gautam Mitra Presented to Clarifi,...