Quantstar QuantDay2007-New York 1 Dynamic Portfolio Optimization using Decomposition and Finite Element Methods John R. Birge Quantstar and The University

Quantstar QuantDay2007-New York 1

Dynamic Portfolio Optimization using Decomposition and Finite

Element Methods

Dynamic Portfolio Optimization using Decomposition and Finite

Element Methods

John R. Birge

Quantstar and

The University of Chicago Graduate School of Business

www.ChicagoGSB.edu/fac/john.birge


Theme

• Models for Dynamic Portfolio Optimization are:– big (exponential growth in time and state)

– general (can model many situations)

– structured (useful properties somewhere)

• Some hope for solution by:– modeling the “right” way

– using structure wisely

– approximating (with some guarantees/bounds)


Outline• General Model – Observations• Dynamic Model Construction and Motivation• Overview of approaches• Decomposition• Lagrangian and ADP methods• Finite-Element Approach• Conclusions


Why Model Dynamically?

• Three potential reasons:– Market timing– Reduce transaction costs (taxes) over time– Maximize wealth-dependent objectives

• Example– Suppose major goal is $100MM to pay pension liability in 2 years– Start with $82MM; Invest in stock (annual vol=18.75%, annual

exp. Return=7.75%); bond (Treasury, annual vol=0; return=3%)– Can we meet liability (without corporate contribution)? – How likely is a surplus?


Alternatives

• Markowitz (mean-variance) – Fixed Mix– Pick a portfolio on the efficient frontier

– Maintain the ratio of stock to bonds to minimize expected shortfall

• Buy-and-hold (Minimize expected loss)– Invest in stock and bonds and hold for 2 years

• Dynamic (stochastic program)– Allow trading before 2 years that might change the mix

of stock and bonds


Efficient Frontier

• Some mix of risk-less and risky asset

• For 2-year returns:

00.050.10.150.20.250.30.350.4

0 0.1 0.2 0.3 0.4


Best Fixed Mix and Buy-and-Hold

• Fixed Mix: 27% in stock– Meet the liability 25%

of time (with binomial model)

• Buy-and-Hold: 25% in stock– Meet the liability 25%

of time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Stock Bond

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Stock Bond


Best Dynamic Strategy

• Start with 57% in stock

• If stocks go up in 1 year, shift to 0% in bond

• If stocks go down in 1 year, shift to 91% in stock

• Meet the liability 75% of time

0

0.1

0.2

0.3

0.4

0.5

0.6

Stock Bond

0

0.2

0.4

0.6

0.8

1

1.2

Stock Bond0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stock Bond

Stocks Up Stocks Down


Advantages of Dynamic Mix

• Able to lock in gains

• Take on more risk when necessary to meet targets

• Respond to individual utility that depends on level of wealth

TargetShortfall


Approaches for Dynamic Portfolios• Static extensions

– Can re-solve (but hard to maintain consistent objective)– Solutions can vary greatly– Transaction costs difficult to include

• Dynamic programming policies– Approximation– Restricted policies (optimal – feasible?) – Portfolio replication (duration match)

• General methods (stochastic programs)– Can include wide variety– Computational (and modeling) challenges


Dynamic Programming Approach• State: xt corresponding to positions in each asset (and possibly price,

economic, other factors)• Value function: Vt (xt)• Actions: ut • Possible events st, probability pst

• Find:

Vt (xt) = max –ct ut + Σst pstVt+1 (xt+1(xt,ut,st))Advantages: general, dynamic, can limit types of policiesDisadvantages: Dimensionality, approximation of V at some point

needed, limited policy set may be needed, accuracy hard to judgeConsistency questions: Policies optimal? Policies feasible? Consistent

future value?


Other Restricted Policy Approaches

• Kusy-Ziemba ALM model for Vancouver Credit Union

• Idea: assume an expected liability mix with variation around it; minimize penalty to meet the variation

• Formulation: min Σi ci xi + Σst pst(qst

+ yst+ + qst

- yst-)

s.t. Σi fits xi + yst+ - yst

- = lts all t and s; xi y >= 0, i = 1…n

Problems: Similar to liability matching. Consistency questions: Possible to purchase insurance at cost of penalties?

Best possible policy?


General Methods• Basic Framework: Stochastic Programming • Model Formulation:

Advantages:General model, can handle transaction costs, include tax lots, etc.

Disadvantages: Size of model, insight

Consistency questions: Price dynamics appropriate? objective appropriate? Solution method consistent?

max p(U(W( , T) )s.t. (for all ): k x(k,1, ) = W(o) (initial) k r(k,t-1, ) x(k,t-1, ) - k x(k,t, ) = 0 , all t >1; k r(k,T-1, ) x(k,T-1, ) - W( , T) = 0, (final); x(k,t, ) >= 0, all k,t;Nonanticipativity: x(k,t, ’) - x(k,t, ) = 0 if ’, St

i for all t, i, ’, This says decision cannot depend on future.


Model Consistency• Price dynamics may have inherent arbitrage

– Example: model includes option in formulation that is not the present value of future values in model (in risk-neutral prob.)

– Does not include all market securities available

• Policy inconsistency– May not have inherent arbitrage but inclusion of market

instrument may create arbitrage opportunity– Skews results to follow policy constraints

• Lack of extreme cases– Limited set of policies may avoid extreme cases that drive

solutions


Objective Consistency

• Examples with non-coherent objectives– Value-at-Risk – Probability of beating benchmark

• Coherent measures of risk – Can lead to piecewise linear utility function

forms– Expected shortfall, downside risk, or

conditional value-at-risk (Uryasiev and Rockafellar)


Model and Method Difficulties

• Model Difficulties– Arbitrage in tree– Loss of extreme cases– Inconsistent utilities

• Method Difficulties– Deterministic incapable on large problems– Stochastic methods have bias difficulties

• Particularly for decomposition methods• Discrete time approximations

– Stopping rules and time hard to judge


Resolving Inconsistencies

• Objective: Coherent measures (& good estimation)

• Model resolutions– Construction of no-arbitrage trees (e.g., Klaassen)

– Extreme cases (Generalized moment problems and fitting with existing price observations)

• Method resolutions– Use structure for consistent bound estimates

– Decompose for efficient solution


General Form in Discrete TimeGeneral Form in Discrete Time

• Find x=(x1,x2,…,xT) and p (allows for “robust formulation”) to

minimize Ep [ t=1Tft(xt,xt+1,p) ]

s.t. xt 2 Xt, xt nonanticipative, p2 P (distribution class) P[ ht (xt,xt+1,pt,) <= 0 ] >= a (chance constraint)

General Approaches:Simplify distribution (e.g., sample) and form a mathematical program:

• Solve step-by-step (dynamic program)• Solve as single large-scale optimization problem

Use iterative procedure of sampling and optimization steps


What about Continuous Time?

• Sometimes very useful to develop overall structure of value function

• May help to identify a policy that can be explored in discrete time (e.g., portfolio no-trade region)

• Analysis can become complex for multiple state variables

• Possible bounding results for discrete approximations (e.g., FEM approach)


Simplified Finite Sample Model

• Assume p is fixed and random variables represented by sample i

t for t=1,2,..,T, i=1,…,Nt

with probabilities pit ,a(i) an ancestor of i, then

model becomes (no chance constraints):minimize t=1

T i=1Nt pi

t ft(xa(i)

t,xit+1, i

t) s.t. xi

t Xit

Observations?• Problems for different i are similar – solving one may help to solve others• Problems may decompose across i and across t yielding

•smaller problems (that may scale linearly in size)•opportunities for parallel computation.


Outline

• General Model – Observations• Dynamic Model Construction and Motivation• Overview of approaches• Decomposition• Lagrangian and ADP methods• Finite Element Methods• Conclusions

.


Solving As Large-scale Mathematical ProgramSolving As Large-scale Mathematical Program

• Principles:– Discretization leads to mathematical program but large-scale– Use standard methods but exploit structure

• Direct methods– Take advantage of sparsity structure

• Some efficiencies

– Use similar subproblem structure• Greater efficiency

• Size– Unlimited (infinite numbers of variables)– Still solvable (caution on claims)


Standard ApproachesStandard Approaches• Sparsity structure advantage

– Partitioning– Basis factorization – Interior point factorization

• Similar/small problem advantage– DP approaches

• Decomposition:– Benders, l-shaped (Van Slyke – Wets)– Dantzig-Wolfe (primal version)– Regularized (Ruszczynski)

• Various sampling schemes (Higle/Sen stochastic decomposition, abridged nested decomposition)

• Approximate DP (Bertsekas, Tsitsiklis, Van Roy..)

– Lagrangian methods


Outline

• General Model – Observations

• Overview

• Decomposition

• Lagrangian and ADP methods

• Finite Element Methods

• Conclusions


Similar/Small Problem Structure: Dynamic Programming View

Similar/Small Problem Structure: Dynamic Programming View

• Stages: t=1,...,T

• States: xt -> Btxt (or other transformation)

• Value function:Vt(xt) = E[Vt(xt,t)] where

t is the random element and

Vt(xt,t) = min ft(xt,xt+1,t) +Vt+1(xt+1)

s.t. xt+1 Xt+1t(,t) xt given

• Solve : iterate from T to 1


Linear Model Structure 1 1 2 1

1 1 1

1

max

. .

0

c x V x

s t W x h

x

,

, , ,1, 1, ,t k t

t t k t k t kt a k t a kV x prob V x

, , 1 ,, ,1,

, , 1 , 1,

,

max,

. .

0

t t k t k t t kt k t kt a k

t t k t t k t t k t a k

t k

c x V xV x

s t W x h T x

x

Stage 1 Stage 2 Stage 3

x1 x32 3x2

• VN+1(xN) = 0, for all xN,

• -Vt,k(xt-1,a(k)) is a piecewise linear, convex function of xt-1,a(k)


Decomposition Methods• Benders idea

– Form an outer linearization of -Vt

– Add cuts on function :

-Vt

LINEARIZATION AT ITERATION kmin at k : < -Vt

new cut (optimality cut)

Feasible region

(feasibility cuts)


Nested Decomposition• In each subproblem, replace expected recourse function -Vt,k(xt-1,a(k))

with unrestricted variable t,k

– Forward Pass:• Starting at the root node and proceeding forward through the scenario tree, solve each node

subproblem

• Add feasibility cuts as infeasibilities arise

– Backward Pass• Starting in top node of Stage t = N-1, use optimal dual values in descendant Stage t+1 nodes to

construct new optimality cut. Repeat for all nodes in Stage t, resolve all Stage t nodes, then t t-1.

– Convergence achieved when

, , ,, , ,1,

, , 1 , 1,

, , , ,

, , ,

,

ˆ min~ Q ,

. .

0

t t k t k t kt k t k t kt a k

t t k t t k t t k t a k

t k t k t k t k

t k t k t k

t k

c xV x

s t W x h T x

E x e optimality cuts

D x d feasibility cuts

x

1 2 1V x


Sample Results Sample Results

Example performance

LOG (NO. OF VARIABLES)

LOG (CPUS)

3 4 5 6 71

2

3

4 Standard LP NESTED DECOMP.

PARALLEL: 60-80% EFFICIENCY IN SPEEDUP

Other problems: similar results


Decomposition Enhancements• Optimal basis repetition

– Take advantage of having solved one problem to solve others– Use bunching to solve multiple problems from root basis– Share bases across levels of the scenario tree– Use solution of single scenario as hot start

• Multicuts– Create cuts for each descendant scenario

• Regularization – Add quadratic term to keep close to previous solution

• Sampling– Stochastic decomposition (Higle/Sen)– Importance sampling (Infanger/Dantzig/Glynn)– Multistage (Pereira/Pinto, Abridged ND)


Abridged Nested Decomposition

• Incorporates sampling into the general framework of Nested Decomposition

• Assumes relatively complete recourse and serial independence

• Samples both the sub-problems to solve and the solutions to continue from in the forward pass through sample-path tree

Donohue/JRB 2006


Outline


• Overview of approaches

• Decomposition


• Finite Element Methods

• Conclusions


Lagrangian-based ApproachesLagrangian-based Approaches• General idea:

– Relax nonanticipativity (or perhaps other constraints)

– Place in objective

– Separable problems

MIN E [ t=1T ft(xt,xt+1) ]

s.t. xt Xt

xt nonanticipative

MIN E [ t=1T ft(xt,xt+1) ]

xt Xt

+ E[w,x] + r/2||x-x||2

Update: wt; Project: x into N - nonanticipative space as x

Convergence: Convex problems - Progressive Hedging Alg. (Rockafellar and Wets)Advantage: Maintain problem structure (e.g., network)


Approximate Dynamic Programming: Infinite Horizon

• Use LP solution of dynamic (Bellman) equation:

max (d,V) s.t. TV ¸ V for distribution d on x

• Approximate V with finite set of basis functions j, weights j

• LP for finite set becomes: Find to

max (d,) s.t. T¸


Solving ADP Form

• Bounds available (Van Roy, De Farias)• Discretizations:

– Discrete state space x– Use structure to reduce constraint set

• Use Duality: – Dual Form:

min max (d,) + (,T-)Can combine with outer approximation


Outline


• Overview of approaches

• Decomposition


• Finite element methods

• Conclusions


Continuous-Time Setup

• Suppose Vt(xt)=maxx2 X E[stT fu(xu|xt) du]

• Questions:– Can the form of Vt provide insight into the

effects of time discretization?

– When does Vt have useful structural properties?

– Can different methods of discretization provide better results than others?


Portfolio in Continuous Time

• Setup:– Value function, u

– PDE with no trade, ut + Gu – ru =0, u(T,x) given

– Define Mu(t,x)=supx’|x’2 Y(t,x) u(t,x’)

where Y(t,x) is the set of attainable portfolios from x at t


Variational Inequality Form

• General variational inequality form

ut+Gu-ru· 0, u-Mu¸ 0, (ut+Gu-ru)(u-Mu)=0

• Computational approach:– Apply high-order FEM methods for the

continuous regime– Use a sequential optimization process to

determine the free boundary (which then effectively determines the no-trade region)


FEM Approach• Local Discontinuous Galerkin Method

– Decomposes by time

– Parallel implementation

– Can achieve high order of accuracy

• Use Legendre polynomials as basis functions that then approximate the value function

• General result (Liu/JRB): ||u – uh||· C hk+1/2

for kth order polynomials


Extensions

• Increase the complexity of portfolio examples in higher dimensions

• Extend approach to other models governed by smooth dynamics plus non-smooth impulse-type controls

• Provide generalizations for other forms of stochastic programs


ConclusionsConclusions• Use of structure in solving large-scale dynamic

portfolio problems:– Repeated problems– Nonzero pattern for sparsity – Use of decomposition and sampling ideas– Potential for high-accuracy methods with FEM

• Computational results– Structure accelerates solution (and allows additional

complexity: asset types/transaction costs/etc)– Speedups possible in orders of magnitude over standard

software implementations

Documents

Quantstar QuantDay2007-New York 1 Dynamic Portfolio Optimization using Decomposition and Finite Element Methods John R. Birge Quantstar and The University