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**The University of Chicago Graduate School of Business**

Dynamic Portfolio Optimization using Decomposition and Finite Element Methods John R. Birge Quantstar and The University of Chicago Graduate School of Business Quantstar QuantDay2007-New York

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**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) Quantstar QuantDay2007-New York

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**Outline General Model – Observations**

Dynamic Model Construction and Motivation Overview of approaches Decomposition Lagrangian and ADP methods Finite-Element Approach Conclusions Quantstar QuantDay2007-New York

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**Why Model Dynamically? Three potential reasons: Example 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? Quantstar QuantDay2007-New York

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**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 Quantstar QuantDay2007-New York

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**Efficient Frontier Some mix of risk-less and risky asset**

For 2-year returns: Quantstar QuantDay2007-New York

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**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 Quantstar QuantDay2007-New York

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**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 Stocks Up Stocks Down Quantstar QuantDay2007-New York

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**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 Target Shortfall Quantstar QuantDay2007-New York

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**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 Quantstar QuantDay2007-New York

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**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 policies Disadvantages: Dimensionality, approximation of V at some point needed, limited policy set may be needed, accuracy hard to judge Consistency questions: Policies optimal? Policies feasible? Consistent future value? Quantstar QuantDay2007-New York

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**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? Quantstar QuantDay2007-New York

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**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 ’, Sti for all t, i, ’, This says decision cannot depend on future. Quantstar QuantDay2007-New York

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**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 Quantstar QuantDay2007-New York

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**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) Quantstar QuantDay2007-New York

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**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 Quantstar QuantDay2007-New York

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**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 Quantstar QuantDay2007-New York

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**General 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 Quantstar QuantDay2007-New York

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**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) Quantstar QuantDay2007-New York

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**Simplified Finite Sample Model**

Assume p is fixed and random variables represented by sample xit for t=1,2,..,T, i=1,…,Nt with probabilities pit ,a(i) an ancestor of i, then model becomes (no chance constraints): minimize St=1T Si=1Nt pit ft(xa(i)t,xit+1, xit) s.t xit Î 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. Quantstar QuantDay2007-New York

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**Outline General Model – Observations**

Dynamic Model Construction and Motivation Overview of approaches Decomposition Lagrangian and ADP methods Finite Element Methods Conclusions . Quantstar QuantDay2007-New York

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**Solving 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) Quantstar QuantDay2007-New York

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**Standard 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 Quantstar QuantDay2007-New York

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**Outline General Model – Observations Overview Decomposition**

Lagrangian and ADP methods Finite Element Methods Conclusions Quantstar QuantDay2007-New York

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**Similar/Small Problem Structure: Dynamic Programming View**

Stages: t=1,...,T States: xt -> Btxt (or other transformation) Value function: Vt(xt) = E[Vt(xt,xt)] where xt is the random element and Vt(xt,xt) = min ft(xt,xt+1,xt) +Vt+1(xt+1) s.t. xt+1 Î Xt+1t(,xt) xt given Solve : iterate from T to 1 Quantstar QuantDay2007-New York

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**Linear Model Structure**

VN+1(xN) = 0, for all xN, -Vt,k(xt-1,a(k)) is a piecewise linear, convex function of xt-1,a(k) Quantstar QuantDay2007-New York 3

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**Decomposition Methods**

Benders idea Form an outer linearization of -Vt Add cuts on function : Feasible region (feasibility cuts) -Vt new cut (optimality cut) min at k : < -Vt LINEARIZATION AT ITERATION k Quantstar QuantDay2007-New York

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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 Quantstar QuantDay2007-New York 6

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**Sample Results Example performance LOG (CPUS) 4 Standard LP**

NESTED DECOMP. 3 2 1 LOG (NO. OF VARIABLES) PARALLEL: 60-80% EFFICIENCY IN SPEEDUP Other problems: similar results Quantstar QuantDay2007-New York

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**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) Quantstar QuantDay2007-New York

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**Abridged Nested Decomposition**

Donohue/JRB 2006 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 Quantstar QuantDay2007-New York 10

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**Outline General Model – Observations Overview of approaches**

Decomposition Lagrangian and ADP methods Finite Element Methods Conclusions Quantstar QuantDay2007-New York

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**Lagrangian-based Approaches**

General idea: Relax nonanticipativity (or perhaps other constraints) Place in objective Separable problems MIN E [ St=1T ft(xt,xt+1) ] xt Î Xt + E[w,x] + r/2||x-x||2 MIN E [ St=1T ft(xt,xt+1) ] s.t xt Î Xt xt nonanticipative 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) Quantstar QuantDay2007-New York

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**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¸ Quantstar QuantDay2007-New York

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**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 Quantstar QuantDay2007-New York

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**Outline General Model – Observations Overview of approaches**

Decomposition Lagrangian and ADP methods Finite element methods Conclusions Quantstar QuantDay2007-New York

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**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? Quantstar QuantDay2007-New York

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**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 Quantstar QuantDay2007-New York

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**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) Quantstar QuantDay2007-New York

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**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 Quantstar QuantDay2007-New York

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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 Quantstar QuantDay2007-New York

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Conclusions 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 Quantstar QuantDay2007-New York

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