Transcript PORTFOLIO CONSTRUCTION BASED ON STOCHASTIC …

Tools and components for
optimisation and risk analysis
Professor Gautam Mitra
Presented to Clarifi, New York.
Outline
• Solvers (FortMP / FortSP)
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
–
–
–
–
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
Solvers – FortMP
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• 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
Solvers - FortMP
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• 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
Solvers – FortMP/QP/QMIP
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• FortMP processes quadratic programs
(QP) and quadratic mixed integer
programs (QMIP) using
– Branch and bound
– Branch and relax
Solvers - FortSP
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• Processes stochastic programming
problems with recourse using
– Benders decomposition (nested)
– Stochastic decomposition
• In contrast to deterministic equivalent,
these algorithms scale up
Solvers – FortMP MEX
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• 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
Portfolio Optimisation Model
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• 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)
Mean Variance model
Outline
Solvers
Porfolio Opt
• It can be expressed as a quadratic
program (QP):
max pT x  1 xT Qx
AMPL Suite
LDI/ALM
subject to
2
Ax  b
x0
• Can be refined adding more restriction
on the choice of the assets
Other restrictions
K
ri   i    ik f k  ei
• Factor model
K

Outline
k 1
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
subject to
y P2 .k   xi2 2i
i 1
y P.k 
N
x 
N
 x
i 1
k 1
N
i
i 1
i
i
f
ik
k
k  1,, K

• Index tracking model
N
2
2
(
x

b
)

 j j ej
j 1
n
y p sk   ( xs  bi ) ik fk
s 1
– where bj are normalized coefficients of
the chosen benchmark portfolio
Other restrictions
• Rebalancing model
Outline
Solvers

j
xj  x  x  x ,
b
j
s
j
j

 x ,
j  1,N.
j  1,, N ,
N
b
s
(
x

x
 j j )  t;
Porfolio Opt
AMPL Suite
LDI/ALM
x
j 1
• Threshold constraints

threshold
j
 j  x j  u j j
where δj are binary decision variables
Other restrictions
• Cardinality constraints
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
N
j C
– at most C assets are held 
j 1
• The last two constraints transforms the
QP problem in QMIP
• Non linear transaction cost
Segment 3
Segment 1
Segment 2
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.
Models statistics
Model
Asset
universe
Cardinality
limit
No of
Factors
Cfu508
508
150
22
556
Solvers
Cfu525
525
50
26
579
Porfolio Opt
Cfu1057
1,057
50
26
1,109
AMPL Suite
Cfu1533
1,533
50
26
1,614
LDI/ALM
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
No of G.O.
constraints
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
Porfolio Opt
Cfu1533
6,244
9,407
46,136
1,559
1,709
AMPL Suite
Cfu9583
38,445
48,127
242,941
9,609
9,759
LDI/ALM
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
Benchmarks
• Results obtained using FortMP’s
accelerated heuristic functionality
CPLEX
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
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
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
0.97411804E-01
AMPL Overview
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• 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)
AMPL Products Offer
Data
AMPL Studio
AMPL
Models
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
AMPL
Interactive
AMPL COM
Object
C# / C++ / VB /
VBA Application
SAMPL/SPInE
AMPL Studio
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• 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
AMPL Studio
Menu Bar
Editing Area
Outline
Solvers
Porfolio Opt
Workspace and
Model Explorer
AMPL Suite
LDI/ALM
Multifunctional Output
Console
AMPL-COM Object
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• 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
AMPL-COM Object
AMPL
Outline
Command
Solvers
Models
Solvers
Porfolio Opt
AMPL Suite
Model
LDI/ALM
Variables
Variable
Constraints
Constraint
Objectives
Objective
Options
Solution
SolTime
AMPL Studio-SPInE
• Seamlessly integrated into AMPL studio environment
• Extends AMPL language with constructs specific for
modelling SP problems and interprets them
Outline
Random
Parameters
Solvers
Porfolio Opt
AMPL Suite
Variables
LDI/ALM
Indices
Stages
Aggregations
Scenario Tree
Structure
Scenario
Probabilities
Constraints
Parameters
Objectives
Probabilistic
Constraints
Scenario
Index
Time Index
Liability Determined Investment (LDI)
Asset and Liability Management (ALM)
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
Scope and Purpose of LDI
Modelling System
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• 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)
LDI Tool Description
Outline
Solvers
• Cash flow matching over a long planning
horizon: up to 50 years or more
Porfolio Opt
AMPL Suite
LDI/ALM
• Portfolio mix: mainly fixed income,
derivatives (swaps) and if necessary
equities
LDI Tool Description
• Multi-Objective:
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
–
–
–
–
–
Minimise PV01 Deviations (Deterministic)
Minimise Net PV Deviations (Stochastic)
Maximise Surplus Wealth
Minimise Initial Injected Cash
Minimise Member Contributions
Solving the Decision Model
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
• The decision problem can be formulated
and processed as:
–
–
–
–
LP/IP
SP with recourse
Chance Constrained Programming
Robust Optimisation
Scenario tree structure
Expected Value
Wait and See
Two-Stage SP
Multi-Stage SP
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
-
Stochastic Features
• Scenario generation (User supplied):
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
– Asset prices
– Liabilities
• Ex-ante asset decisions
• Ex-post evaluation (Simulation)
LDI Information Flow
Information
Market Data
RISKWATCH +
Data Builder
RISKWATCH
+ Customer
Data Builder
Data
Sources
POPULATE DATA
Outline
Solvers
Porfolio Opt
Consolidator
+
Diagnostic
AMPL Suite
LDI/ALM
DATA MART
#.TAB
MODEL
DATA
Optimisers
- FortMP
- CPLEX
Simulation and
visualization to
RISKWATCH
AMPL Modelling
Environment
RESULTS
[AMPL Com Object + LDI Model File +
#.DAT File + #.RUN File]
LDI Project Control
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
LDI Information Flow
Information
Market Data
RISKWATCH +
Data Builder
RISKWATCH
+ Customer
Data Builder
Data
Sources
POPULATE DATA
Outline
Solvers
Porfolio Opt
AMPL Suite
Consolidator
+
Diagnostic
LDI/ALM
DATA MART
#.TAB
MODEL
DATA
Optimisers
- FortMP
- CPLEX
Simulation and
visualization to
RISKWATCH
AMPL Modelling
Environment
RESULTS
[AMPL Com Object + LDI Model File +
#.DAT File + #.RUN File]
LDI Optimising Engine
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM
LDI Information Flow
Information
Market Data
RISKWATCH +
Data Builder
RISKWATCH
+ Customer
Data Builder
Data
Sources
POPULATE DATA
Outline
Solvers
Porfolio Opt
AMPL Suite
Consolidator
+
Diagnostic
LDI/ALM
DATA MART
#.TAB
MODEL
DATA
Optimisers
- FortMP
- CPLEX
Simulation and
visualization to
RISKWATCH
AMPL Modelling
Environment
RESULTS
[AMPL Com Object + LDI Model File +
#.DAT File + #.RUN File]
Simulation and analysis
Outline
Solvers
Porfolio Opt
AMPL Suite
LDI/ALM