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CMSDOI 10.1007/s10287-006-0035-7

O R I G I NA L PA P E R

On multistage Stochastic Integer Programmingfor incorporating logical constraints in assetand liability management under uncertainty

Laureano F. Escudero · Araceli Garín ·María Merino · Gloria Pérez

© Springer-Verlag 2006

Abstract We present a model for optimizing a mean-risk function of the ter-minal wealth for a fixed income asset portfolio restructuring with uncertainty inthe interest rate path and the liabilities along a given time horizon. Some logicalconstraints are considered to be satisfied by the assets portfolio. Uncertaintyis represented by a scenario tree and is dealt with by a multistage stochasticmixed 0-1 model with complete recourse. The problem is modelled as a splittingvariable representation of the Deterministic Equivalent Model for the stochas-tic model, where the 0-1 variables and the continuous variables appear at anystage. A Branch-and-Fix Coordination approach for the multistage 0–1 programsolving is proposed. Some computational experience is reported.

Keywords Multistage scenario tree · Assets and liabilities · Stochastic IntegerProgramming · Branch-and-Fix Coordination · Mean-risk function

L. F. Escudero (B)Centro de Investigación Operativa, Universidad Miguel Hernández, Elche (Alicante), Spaine-mail: [email protected]

A. GarínDpto. de Economía Aplicada III,Universidad del País Vasco, Bilbao (Vizcaya), Spaine-mail: [email protected]

M. Merino · G. PérezDpto. de Matemática Aplicada, Estadística e I.O, Universidad del País Vasco,Leioa (Vizcaya), Spaine-mail: [email protected]

G. Péreze-mail: [email protected]

L.F. Escudero et al.

AMS Subject Classification 90C15 · 90C11 · 90C06

1 Introduction: statement of the problem

Given a time horizon, a set of fixed income assets for trading, cash availabilityand a set of liabilities to satisfy, the asset and liability management (ALM) prob-lem, see e.g. Sodhi (2005), is concerned with restructuring a financial portfolio(including long and short positions in cash) over the time horizon to maximizean objective function, e.g. the terminal wealth. The deterministic version of theproblem can be represented by a mixed 0-1 model. Given the state of the artin optimization solving this problem should not present major difficulties formoderate size instances, at least. The difficulty lies in the uncertainty of themain parameters, namely the interest rate path and the liabilities over a giventime horizon. Uncertainty can be represented by a scenario tree and it can bedealt with by using stochastic programming approaches.

More specifically the problem has the following form: given a set of lia-bilities to satisfy over the time horizon and a scenario tree for the uncertainparameters, the aim is to determine the trading (i.e. buying and selling) volumeof a set of fixed-income financial assets in each scenario group at each stage,such that the trading and portfolio structuring constraints are satisfied andthe maximization of an objective function is achieved. Most authors considerthe problem with only continuous variables, see e.g., Kouwenberg and Zenios(2006), Ziemba (2003), Ziemba and Mulvey (1998) and the references herein,among others.

Moreover, very frequently the modelization of the logical constraints in theportfolio is required, see e.g., Konno and Yamamoto (2005) in a deterministiccontext. See also Drijver et al. (2003) in the stochastic context. These constraintsimpose upper bounds on the number of classes of assets to be considered in theportfolio and the number of assets to be traded in any time period. They alsoforce conditional lower bounds on the volume to be traded in periods. Then,it results that the stochastic continuous programming problem is convertedto a Stochastic Integer Programming (SIP) problem. See Klein Haneveld andvan der Vlerk (1999), Römisch and Schultz (2001) and Schultz (2003) for goodstate-of-the art surveys in the field. These approaches use Benders (1962) andLagrangian Decomposition schemes.

Although most approaches deal with the optimization of the objective func-tion expected value, there are some that additionally deal with mean-risk func-tions. They consider semi-deviations, see Ogryczak and Ruszczynski (1999),excess probabilities, see Dert (1998) and Schultz and Tiedemann (2004), andvalue-at risk (VaR) and conditional VaR, see Rockafellar and Uryasev (2000)and Schultz and Tiedemann (2006), as risk measure-based functions to be opti-mized. For a weighted minimization of (square) legal violations in fixed portfoliomodels, see Fleten et al. (2002).

The problem dealt with in this paper can be viewed as an extension of theALM model considered in Ahn et al. (1995) to integrate logical constraints. In

On multistage Stochastic Integer Programming for incorporating logical constraints

this sense we present the splitting variable representation of the DeterministicEquivalent Model (DEM) for the mixed 0–1 stochastic model, where the 0–1variables and the continuous variables appear at any stage. The objective func-tion to be maximized is the mean–risk function given by the composite functionof the expected value of terminal wealth and the probability of reaching a giventerminal wealth target.

On the other hand, an extension of the branch-and-fix Coordination (BFC)scheme that we present in Alonso-Ayuso et al. (2003a), Alonso-Ayuso et al.(2003b), is considered for multistage mixed 0–1 problem solving. The algorithmuses the twin node family (TNF) concept. It is especially designed to coordinateand reinforce the branching variable and the branching node selections at eachBF tree. Note: a BF tree is a branch-and-bound tree for a given scenario cluster,being a cluster a set of scenarios with given characteristics. The trees resultfrom the relaxation of the nonanticipativity constraints in the splitting variablerepresentation of the DEM. Additionally, the proposed approach considers thecompact representation of the DEM at each TNF integer set, i.e., each set ofso-called integer Branch-and-Fix (BF) nodes where the nonanticipativity con-straints of the 0–1 variables are satisfied. By fixing those variables to the nodes’values, the DEM has only continuous variables. The structure of the remain-ing model is used to satisfy the nonanticipativity constraints of the continuousvariables associated with each scenario group in the scenario tree and, so, toobtain the LP optimal solution for the given TNF integer set. Moreover, veryfew approaches deal with multistage mixed 0-1 integer ALM, where both typesof variables appear at any stage of the time horizon, see Drijver et al. (2003),Lulli and Sen (2004), Ntaimo and Sen (2005) among others. An important fea-ture of our approach with respect to some other stochastic integer approachesis that it addresses large-scale multistage environments where both continuousand 0–1 variables appear at any stage of the time horizon. Some computationalexperience is reported to assess the quality of the stochastic solution obtainedby our approach in comparison to the solution obtained by a state-of-the artoptimization engine.

The remainder of the paper is organized as follows. Section 2 introduces thescenario tree structure dealt with in the paper. Section 3 presents the mixed 0-1model for any scenario of the ALM problem. Section 4 introduces the objec-tive function to be optimized, the scenario cluster and the TNF environments.Section 5 presents the LP models to be used in the TNF integer sets. Section 6presents the BFC implementation that we propose. Section 7 reports on thecomputational experience. Section 8 concludes.

2 Scenario tree representation

The uncertainty in the stochastic parameters is to be dealt with via a scenariotree approach. To illustrate this concept, consider Fig. 1: each node in the figurerepresents a point in time where a decision can be made. Once a decision is


Fig. 1 Scenario tree

made, some contingencies can arise (e.g., in this example there are three contin-gencies for time period t = 2), and information related to these contingenciesis available at the beginning of the stage (here, time period). This informationstructure is visualized as a tree, where each root-to-leaf path represents onespecific scenario and corresponds to one realization of the whole set of theuncertain parameters. Two scenarios belong to the same group in a given stageif they have the same realizations of the uncertain parameters up to that stage.According to the nonanticipativity principle, see Rockafellar and Wets (1991),both scenarios should have the same value for the related variables with thetime index up to the given stage. Each scenario group is associated with a nodein the scenario tree.

The following notation is used in the model:

T , number of stages (in this case, time periods) over the time horizon.�, set of scenarios that represent the stochasticity of the uncertain parame-ters. Note: the scenarios are consecutively numbered.G, set of scenario groups. Note: the groups are consecutively numbered.Gt, set of scenario groups in stage t, for t = 1, . . . , T . Note: |G1| = 1 and1 ∈ G1.�g, set of scenarios that belong to group g, for g ∈ G.t(g), stage of scenario group g, such that g ∈ Gt(g).π(g), immediate ancestor node of node g, such that π(g) ∈ Gt(g)−1, forg ∈ G − G1.Sg, set of successor nodes of node g.N g, set of ancestor nodes of node g, including itself.

3 A mixed 0-1 model for ALM under a given scenario

The following notation is used for the sets, deterministic and stochastic param-eters, and continuous and 0–1 variables for the ALM model for a given scenarioω ∈ �.

Sets:


I, set of assets.J , set of asset classes (e.g. government funds, domestic and internationalassets, industry sectors).Ij, set of assets that belong to class j, such that Ij ⊆ I, j ∈ J .

Deterministic parameters:

Fi, face value of asset i, for i ∈ I.Ti, maturity period for asset i, for i ∈ I, such that ∃i ∈ I : Ti ≥ T .T j, maturity period for asset class j, such that T j = maxi∈Ij{Ti}, for j ∈ J .Ki, issuing period for asset i, where 0 ≤ Ki < Ti, for i ∈ I. By convention,Ki = 0, provided that the asset has been issued before time period t = 1.Kj, issuing period for asset class j, such that Kj = mini∈Ij{Ki}, for j ∈ J .Yiτ , yield of asset i at (the end of) time period τ for Ki < τ ≤ Ti, i ∈ I.P+

it , P−it , fixed transaction costs for buying and selling any volume of asset i

at (the end of) time period t, respectively, for Ki ≤ t < Ti, i ∈ I.aj, unit transaction cost for asset class j, for j ∈ J .fj, weighting factor on the interest rate for computing the value of the assetsin set Ij, for j ∈ J . This is used to weight the discount rate for obtaining thevalue of a given asset. It can be interpreted as the implied risk penalty for aparticular asset class, see Ben-Dov et al. (1992), among others.Zi0, volume of asset i in the portfolio at the beginning of the time horizon,for i ∈ I.X+

it , X−it , conditional minimum volumes of asset i that are allowed to be

bought and sold at time period t, respectively, for max{1, Ki} ≤ t < min{T , Ti},i ∈ I.X

+it , X

−it , maximum volumes of asset i that are allowed to be bought and sold

at time period t, respectively, for max{1, Ki} ≤ t < min{T , Ti}, i ∈ I.Fjt, Fjt, conditional minimum and maximum face values that are allowedfor asset class j to be kept in the portfolio at (the end of) time period t,respectively, for max{1, Ki} ≤ t ≤ min{T , T j − 1}, j ∈ J .C+

t , C+t , minimum and maximum allowed net cash at (the end of) time

period t, for 1 ≤ t ≤ T .C

−t , maximum allowed cash borrowing at time period t, for 1 ≤ t < T .

N, maximum number of asset classes that can be in the portfolio at (the endof) any period over the time horizon.N̂t, maximum number of assets for trading at time period t, for 1 ≤ t ≤ T .wg, (strictly positive) weight associated with scenario group g, for g ∈ G,such that wg = ∑

ω∈�g wω, where wω is the weight assigned to scenario ω.

Stochastic parameters for scenario group g, for g ∈ G:

rg, interest rate. The scenarios for the interest rate path over the time horizoncan be generated from the binomial lattice approach given in Black et al.(1990) as in Zenios (1993). See other schemes in Kouwenberg and Zenios(2006) and Ziemba (2003) and references herein.Eg

jτ , value at (the end of) time period t(g) of one unit of the yield of asset

class j at (the end of) time period τ , being Egjτ = 1 for g such that t(g) = τ


and, otherwise,

Egjτ =

∑

i∈Sg:t(i)=τ

wi

wg

∏

k ∈ N it(g) < t(k) ≤ τ

(1 + fjrk)−1 Kj ≤ t(g) < τ ≤ T j, j ∈ J .

Pgi , estimated unit value of asset i at (the end of) time period t, Ki ≤ t ≤

Ti, t = t(g), i ∈ I, such that it can be expressed as

Pgi =

∑

t≤τ≤Ti

Yiτ Egjτ i ∈ Ij, j ∈ J .

Lg, exogenous net cash inflow/outflow (liability).P

+gi = P

gi + aj, estimated unit cost of asset i, for Ki ≤ t < Ti, i ∈ Ij, j ∈ J .

P−gi = P

gi −aj, estimated unit revenue of asset i, for Ki < t < Ti, i ∈ Ij, j ∈ J .

Variables under scenario ω, for ω ∈ �:

X+ωit , volume of asset i to be bought at time period t, for max{1, Ki} ≤ t <

min{T , Ti}, i ∈ I.X−ω

it , volume of asset i to be sold at time period t, for Ki < t < min{T , Ti},i ∈ I.Zω

it , volume of asset i to be kept in the portfolio after trading at time periodt, for max{1, Ki} ≤ t ≤ min{T , Ti − 1}, i ∈ I.C+ω

t , net cash to be kept after asset trading and return cashing at time periodt, for 1 ≤ t ≤ T . (C+ω

0 = 0).C−ω

t , cash shortfall (to be borrowed) at time period t, for 1 ≤ t < T .(C−ω

0 = C−ωT = 0).

γ +ωit , 0–1 variable, which takes the value 1 if asset i is selected to be bought

at time period t and 0 otherwise, for max{1, Ki} ≤ t < min{T , Ti}, i ∈ I.γ −ω

it , 0–1 variable, which takes the value 1 if asset i is selected to be sold attime period t and 0 otherwise, for Ki < t < min{T , Ti}, i ∈ I.δω

jt , 0–1 variable, which takes the value 1 if asset class j is kept in the port-

folio at (the end of) time period t and 0 otherwise, for max{1, Kj} ≤ t ≤min{T , T j − 1}, j ∈ J .

Under a given scenario ω, the goal is to determine the trading volume(X+ω, X−ω) and then the assets’ volume Zω in the portfolio, net cash C+ω

and cash shortfall C−ω (if any), over the time horizon to maximize the terminalwealth, for ω ∈ �. Notice that the terminal wealth is computed by accumulatingthe final cash and the estimated value of any asset that remains in the portfolio.

If g is taken to be the scenario group in Gt, such that ω ∈ �g for given scenarioω and time period t, the mixed 0-1 program for scenario w ∈ � is as follows,

max C+ωT +

∑

i∈I:Ti>TP

−gi Zω

iT , (1)


where g ∈ GT : ω ∈ �g, s.t.

Zωi,t−1 + X+ω

it − X−ωit = Zω

it ∀ max{1, Ki} ≤ t ≤ min{T , Ti − 1}, i ∈ I (2)

C+ωt−1(1 + rg) − C−ω

t−1(1 + rg) +∑

i∈I:Ki<t≤Ti

YitZωi,t−1

+∑

i∈I:Ki<t<Ti

(P

−gi X−ω

it − P−it γ

−ωit

) −∑

i∈I:Ki≤t<Ti

(P

+gi X+ω

it + P+it γ

+ωit

)

−C+ωt + C−ω

t = Lg ∀1 ≤ t ≤ T (3)

Fjtδωjt ≤

∑

i∈Ij:Ki≤t<Ti

FiZωit ≤ Fjtδ

ωjt ∀1 ≤ t ≤ min{T , T j − 1}, j ∈ J (4)

∑

j∈J :Kj≤t<T j

δωjt ≤ N ∀1 ≤ t ≤ min{T , T j − 1} (5)

X+it γ

+ωit ≤ X+ω

it ≤ X+it γ

+ωit ∀ max{1, Ki} ≤ t < min{T , Ti}, i ∈ I (6)

X−it γ

−ωit ≤ X−ω

it ≤ X−it γ

−ωit ∀Ki < t < min{T , Ti}, i ∈ I (7)

∑

i∈I:Ki≤t<Ti

γ +ωit +

∑

i∈I:Ki<t<Ti

γ −ωit ≤ N̂t ∀1 ≤ t ≤ T (8)

γ +ωit + γ −ω

it ≤ 1 ∀Ki < t < min{T , Ti}, i ∈ I (9)

γ +ωit ≤ δω

jt ∀ max{1, Ki} ≤ t < min{T , Ti}, i ∈ I, where j ∈ J : i ∈ Ij

(10)

Zωit ≥ 0 ∀ max{1, Ki} ≤ t ≤ min{T , Ti − 1}, i ∈ I (11)

C+t ≤ C+ω

t ≤ C+t ∀1 ≤ t ≤ T (12)

0 ≤ C−ωt ≤ C

−t ∀1 ≤ t < T (13)

γ +ωit ∈ {0, 1} ∀ max{1, Kj} ≤ t < min{T , Ti}, i ∈ I (14)

γ −ωit ∈ {0, 1} ∀Ki < t < min{T , Ti}, i ∈ I (15)

δωjt ∈ {0, 1} ∀ max{1, Kj} ≤ t ≤ min{T , T j − 1}, j ∈ J (16)

Constraints (2) give the balance equations for the asset volume at the end of thetime periods. Constraints (3) give the balance equations for the cashflow at anytime period, comprising by the cash returns, the cashflow borrowing, the assetreturns, the transaction cost requirements and the liabilities. Notice that thebalance equation for the cashflow at the last period of the time horizon is givenonly by the cash returns, the cashflow borrowing, the asset returns and the lia-bilities. Constraints (4) force a conditional minimum and an allowed maximumon the total face value of the asset classes to be kept in the portfolio at any timeperiod. Constraints (5) ensure that the number of asset classes in the portfoliowill not exceed the allowed maximum at any time period. Constraints (6) and(7) bound the trading volume of each asset, if any. Constraints (8) ensure thatthe number of assets to be traded will not exceed the allowed maximum atany time period. The logical constraints (9) are 0–1 redundant but tighten the


LP relaxation of the original model. The bounds on the variables are given in(10)–(16).

4 Mean-risk function, scenario clusters, twin node families

Given the stochasticity of the main parameters over the time horizon, an ap-proach is needed to model decision making for restructuring the asset portfoliounder uncertainty. To that end, assume the following splitting variable represen-tation via scenario of the DEM for the mixed 0–1 multi-stage stochastic modelwith complete recourse. Note: The nonanticipativity principle is satisfied for thevariables whose related scenarios belong to the same scenario group.

The objective is to maximize the mean-risk function given by a compositefunction comprising by the expected terminal wealth and the weighted proba-bility of reaching a given terminal wealth target, subject to the scenario relatedconstraints (2)–(16) plus the nonanticipativity constraints (17) ∀ω, ω′ ∈ �g :ω �= ω′, g ∈ Gt for given time periods t.

X+ωit − X+ω′

it = 0 ∀ max{1, Ki} ≤ t < min{T , Ti}, i ∈ IX−ω

it − X−ω′it = 0 ∀Ki < t < min{T , Ti}, i ∈ I

Zωit − Zω′

it = 0 ∀ max{1, Ki} ≤ t ≤ min{T , Ti − 1}, i ∈ IC+ω

t − C+ω′t = 0 ∀1 ≤ t ≤ T

C−ωt − C−ω′

t = 0 ∀1 ≤ t < Tγ +ω

it − γ +ω′it = 0 ∀ max{1, Ki} ≤ t < min{T , Ti}, i ∈ I

γ −ωit − γ −ω′

it = 0 ∀Ki < t < min{T , Ti}, i ∈ Iδω

jt − δω′jt = 0 ∀ max{1, Kj} ≤ t ≤ min{T , T j − 1}, j ∈ J .

(17)

The mean-risk function based model can be expressed as

ZIP = max QE + ρQP s.t. constraints (2)–(17), (18)

where ρ is a positive weighting parameter, QE is the expected terminal wealthand QP is the Reaching Probability, such that

QE =∑

ω∈�

wω

⎛

⎝C+ωT +

∑

i∈I:Ti>TP

−gi Zω

iT

⎞

⎠ (19)

and

QP = P

⎛

⎝ω ∈ � : C+ωT +

∑

i∈I:Ti>TP

−gi Zω

iT ≥ φ

⎞

⎠ , (20)

where g : ω ∈ �g for g ∈ GT , and φ is the terminal wealth target.


Based on an approach for modelling the Reaching Probability inspired inSchultz and Tiedemann (2004), a more amenable expression of (18) for com-putational purposes, at least, can be as follows:

ZIP = max QE + ρ∑

ω∈�

wωνω

s.t. C+ωT +

∑

i∈I:Ti>TP

−gi Zω

iT ≥ φνω ∀ω ∈ �

constraints (2)–(17),νω ∈ {0, 1} ∀ω ∈ �, (21)

where νω is a 0–1 variable, such that

νω={

1, if the terminal wealth under scenario ω is not smaller than target φ

0, otherwise ∀ω ∈ �.

Notice that a scenario with negative terminal wealth is not allowed in (21)Relaxing (or dualizing) constraints (17) in model (21), results in |�| indepen-

dent scenario related mixed 0–1 models. The splitting variable representationis very amenable to Lagrangian Decomposition approaches, see e.g., Schultz(2003), Ziemba (2003) and references herein. It is also very appropriate for theBFC approach, to be presented below.

Let us compact the model to solve for a general linear objective function. Forthis purpose let xω

t and yωt give the n - and m-vectors of the 0–1 and continuous

variables, respectively, related to stage t under scenario ω, for t ∈ T and ω ∈ �.The DEM can be expressed as follows:

ZIP= max∑

ω∈�

∑

t∈Twω

(aω

t xωt + cω

t yωt)

s.t. A′ωt xω

t−1 + Aωt xω

t + B′ωt yω

t−1 + Bωt yω

t = bωt ∀ω ∈ �, t ∈ T

xωt − xω′

t = 0 ∀ω, ω′ ∈ �g, g ∈ Gt, t ∈ Tyω

t − yω′t = 0 ∀ω, ω′ ∈ �g, g ∈ Gt, t ∈ T

xωt ∈ {0, 1}n, yt ∈ R

+m ∀ω ∈ �, t ∈ T ,(22)

where aωt and cω

t are the row vectors of the objective function coefficients,Aω

t , A′ωt , B′ω

t and Bωt are the constraint matrices and bω

t is the rhs vector forω ∈ �, t ∈ T . Note 1: The expression for the Reaching Probability in model(21) is not shown in model (22) for the sake of simplicity of exposition. Note 2:A

′ωt = A

′ω′t and Aω

t = Aω′t , and the same for the B-matrices and the a-, b- and

c- vectors for ω, ω′ ∈ �g, g ∈ Gt, t ∈ T .We can see that the relaxation of the nonanticipativity constraints

xωt − xω′

t = 0yω

t − yω′t = 0

(23)


∀ω, ω′ ∈ �g, g ∈ Gt, t ∈ T in model (22) results in a set of |�| independent mixed0–1 models, where (24) is the model for scenario ω ∈ �.

max∑

t∈Twω

(aω

t xωt + cω

t yωt)

s.t. A′ωt xω

t−1 + Aωt xω

t + B′ωt yω

t−1 + Bωt yω

t = bωt ∀t ∈ T

xωt ∈ {0, 1}n, yt ∈ R

+m ∀t ∈ T . (24)

It is clear that the relaxation of the nonanticipativity constraints (23) is notrequired for all pairs of scenarios in order to obtain computational efficiency.Let us refer to a scenario cluster as a set of scenarios whose nonanticipativi-ty constraints are explicitly included in the model to be solved. The numberof scenarios to be considered in a given model thus basically depends on thedimensions of the scenario related model. Let q denote the number of scenarioclusters to be considered. The criterion for scenario clustering in the sets, say,�1, . . . , �q, such that �p ∩ �p′ = ∅, p, p′ = 1, . . . , q : p �= p′ and � = ∪q

p=1�p

could be alternatively based on the smallest internal deviation of the uncertainparameters, the greatest deviation, etc. Moreover we favor the allocation in thesame cluster of the scenarios with the most scenario groups in common. Asadditional notation, let Gp ⊆ G denote the set of scenario groups for cluster p,such that �g ⋂

�p �= ∅ means that g ∈ Gp.The model to be considered for scenario cluster p = 1, . . . , q can be expressed

by the following compact representation,

ZIPp = max

∑

g∈Gp

wg(agxg + cgyg)

s.t. A′gxπ(g) + Agxg + B′gyπ(g) + Bgyg = bg ∀g ∈ Gp

xg ∈ {0, 1}n, yg ∈ R+m ∀g ∈ Gp, (25)

where Ag and Bg are the constraint matrices for the x- and y-variables relatedto scenario group g, respectively, A′g and B′g are the constraint matrices forthe x- and y-variables related to the immediate ancestor of scenario group g,respectively, bg is the rhs vector, and xg and yg are the vectors of the variablesrelated to scenario group g, for g ∈ Gp. Notice that model (25) for q = 1 is DEM(22) and it is model (24) for q = �.

The q problems (25) are linked by the nonanticipativity constraints:

xg − xg′ = 0 (26)

yg − yg′ = 0 (27)

∀g ∈ Gp, g′ ∈ Gp′, such that p �= p′ and g = g′. Let xg and xg′be named vectors

of common variables.As an illustrative example, take the case whose scenario tree is shown in

Fig. 2, and assume that q = 3 such that the scenario clusters are structured


Fig. 2 Scenario tree. Compact representation

Fig. 3 Scenario cluster tree

as shown in Fig. 3. �1 = {8, 9}, �2 = {10} y �3 = {11} and the sets of sce-nario groups are G1 = {1, 2, 4, 5, 8, 9}, G2 = {1, 3, 6, 10} and G3 = {1, 3, 7, 11}.Notice that the nonanticipativity constraints related to the variables x2 and y2

are explicitly satisfied in the model scheme, leaving it to the algorithmic schemeto satisfy the constraints for the variables x1, y1, x3 and y3.

We can execute, say, a Branch-and-Bound procedure for ensuring the inte-grality condition in the models (25), for p = 1, . . . , q. However, instead ofobtaining the optimal solution independently, we propose in Alonso-Ayusoet al. (2003a), Alonso-Ayuso et al. (2003b) an approach so-called BFC. Ourextension is designed to coordinate the selection of the branching node andbranching variable for each scenario cluster–related BF tree, such that therelaxed constraints (26) are satisfied when fixing the appropriate variables toeither one or zero. The approach proposed also coordinates and reinforces the


Fig. 4 Twin node families

scenario cluster-related BF node pruning, the variable fixing and the objectivefunction value bounding of the subproblems attached to the nodes.

Let Rp denote the BF tree associated with scenario cluster p and Ap theset of nodes in Rp for p = 1, . . . , q. Any two nodes, say, a ∈ Ap and a′ ∈ Ap′are said to be twin nodes with respect to a scenario group if the paths fromtheir root nodes to each of them in their own BF trees Rp and Rp′, respec-tively, either have not yet branched on/fixed their common variables xg and xg′,∀g ∈ Gp, g′ ∈ Gp′ or have the same 0–1 values for their branched on/fixed com-mon variables. Notice that in order to satisfy the nonanticipativity constraints(26), the branching on/fixing common variables must be on the same 0–1 valuefor the twin nodes. A TNF, say, Hf is a set of nodes, such that any one is a twinnode to all the other members of the family, for f ∈ F , where F is the set ofTNFs; see Fig. 4.

A TNF integer set is a set of TNFs, one per BF tree, such that the x-variableshave 0–1 values and the constraints (26) are satisfied. An integer TNF integerset is a TNF integer set where all 0–1 variables have been branched/fixed. Acontinuous TNF integer set is a TNF integer set where there is a 0–1 variablethat has not been branched on/fixed.

5 Auxiliary models in TNF integer sets

5.1 On the LP model attached to a TNF integer set

The LP model after fixing in model (22) the x-variables to the 0–1 values, say,xg from the optimal solution of the models attached to a TNF integer set is asfollows:

ZTNF =∑

g∈Gwgagxg + max

yg

∑

g∈Gwgcgyg

s.t. B′gyπ(g) + Bgyg = bg − A′gxπ(g) − Agxg ∀g ∈ Gyg ∈ R

+m ∀g ∈ G. (28)


Notice that the solution of model (28) satisfies the nonanticipativity con-straints (27). The model is very amenable to a nested Benders decompositionscheme, see e.g. Birge (1985), Gassmann (1990), Birge and Louveaux (1997),Laporte and Louveaux (2002) and others. Moreover, we use the plain Simplexmethod in the computational experimentation reported below.

5.2 On the LP model attached to a continuous TNF integer set

Note the following additional notation. J now gives the set of indices of thevariables in any vector xg, and J g ⊆ J gives the set of indices of the variablesin vector xg that have already been branched on/fixed to the 0–1 value for sce-nario group g, for g ∈ G. By slightly abusing the notation, let xg

j denote the jthx-variable for scenario group g, for j ∈ J , g ∈ G.

The model can be expressed as follows:

ZCON= max∑

g∈Gwg(agxg + cgyg)

s.t. A′gxπ(g) + Agxg + B′gyπ(g) + Bgyg = bg ∀g ∈ Gyg ∈ R

+m ∀g ∈ G0 ≤ xg

j ≤ 1 ∀j ∈ J − J g, g ∈ G

xgj = xg

j ∀j ∈ J g, g ∈ G. (29)

The solution of model (29) also satisfies the nonanticipativity constraints(26)–(27).

6 BFC implementation

Different types of implementations can be considered within the BFC algo-rithmic framework presented in the previous section. Here we introduce theversion implemented to perform the computational experimentation reportedin the next section.

We have chosen a random branching ordering for the variables (in partic-ular, the order of inputting them into the model). The ν variables have thelowest branching priority. We have chosen the depth first strategy for the TNFbranching selection, with “branching on the zeros” first. Once a “branch onthe zeros” has been pruned, a “branch on the ones” for the same variable isperformed. If the related TNF is also pruned, a bactracking on the x-variablesand the scenario groups is performed until another “branch on the zeros” isdetected. At that TNF a “branching on the ones” is performed for the chosenx-variable to satisfy the nonanticipativity constraints (26).

Notice that a TNF may be pruned due to any of the following reasons: (a)the mixed 0–1 scenario cluster model (25) attached to a given node member isinfeasible, (b) the solution of the models (25) for the set of scenario clusterssatisfies the nonanticipativity constraints (26) and (27), (c) there is a guarantee


that a better solution than the incumbent one cannot be obtained from the bestdescendant TNF integer set (in our current implementation, it is based on theobjective function value, also called the solution value), (d) The LP model (28)attached to the TNF integer set is infeasible or its solution value is not betterthan the incumbent solution value, if is an integer TNF integer set, and (e) seebelow for the case for a continuous TNF integer set.

The solution to be obtained by solving the LP model (28) attached to aTNF integer set could be the incumbent solution. However, as said above, thatdoes not necessarily mean that it should be pruned. A better solution still maybe obtained by branching on the non-yet fixed x-variables in the given con-tinuous TNF integer set. Notice that the solution space defined by model (28)is included in the solution space defined by model (29). So, the TNF can bepruned if ZTNF = ZCON, or ZCON is not better (in our case, greater) than theincumbent solution value, since there is a guarantee that there is not a bettersolution than ZCON from its descendant nodes.

To gain computational efficiency, the optimization of model (29) should notbe performed for a small number of the x-variables already branched on/fixedin the given TNF. Let λ denote the minimum fraction of the branched/fixedx-variables that is required for optimizing model (29), where 0 ≤ λ ≤ 1.

To present the BFC procedure, let ZIP

denote the upper bound of the solu-tion value of the original model (22) to be obtained from the best descendant

TNF integer set for a given family. It can be computed as ZIP = ∑

p=1,...,q ZIPp .

By convention, ZIPp = −∞, ZTNF = −∞ and ZCON = −∞ for the related

infeasible problems.The procedure is as follows:

Step 0 Initialize ZIP := −∞.Step 1 Solve the LP relaxation of the original model (22), and compute ZLP.

Initialize ZIP

:= ZLP.Step 2 If there is any x-variable that takes a continuous value, then goto Step

3. Otherwise, the optimal solution to the original problem (22) has

been found and, so, ZIP := ZIP

and stop.Step 3 Initialize g := 1 and goto Step 5.Step 4 If g = |G|, then goto Step 10. Reset g := g + 1.Step 5 Set i := 1 and goto Step 7.Step 6 If i = n, then goto Step 4. Set i := i + 1.Step 7 Branch xg

i := 0, ∀p = 1, . . . , q : g ∈ Gp.Step 8 Solve the mixed 0–1 problems (25), ∀p = 1, . . . , q : g ∈ Gp, and com-

pute ZIP

. If ZIP ≤ ZIP, then goto Step 10. If there is any x-variable

that takes different values for the same scenario group in some of the qscenario clusters, then goto Step 6. If all the variables yg take the samevalue for all scenario clusters p = 1, . . . , q : g ∈ Gp, then update ZIP :=Z

IPand goto Step 10.

Step 9 Locally fix the x-variables at the values obtained in Step 8 and solvethe LP model (28) to satisfy the nonanticipativity constraints (27) for


the current TNF integer set. Notice that the solution value is denotedby ZTNF. Update ZIP := max{ZTNF, ZIP}. If

∑g∈G |J g| < λn|G|, then

goto Step 6. Solve the LP model (29) for the current continuous TNFinteger set, if any. Notice that the solution value is denoted by ZCON. IfZTNF �= ZCON and ZCON > ZIP, then goto Step 6.

Step 10 Prune the current TNF.Step 11 If xg

i has been branched on 0, ∀p = 1, . . . , q : g ∈ Gp, then goto Step 14.Reset i := i − 1.

Step 12 If i > 0, then goto Step 11.Step 13 Reset g := g − 1. If g = 0 then ZIP is the optimal solution, so, stop.

Reset i := n and goto Step 11.Step 14 Branch xg

i := 1, ∀p = 1, . . . , q : g ∈ Gp. Goto Step 8.

Figure 5 shows the BFC procedure graphically.

7 Computational results

In this section we present the results of the computational experiment obtainedwhile optimizing the model ALM by using the BFC approach presented in theprevious section.

The scenario generation is given via a discrete probability distribution, whereall scenarios and exact probabilities are known. For each scenario group wehave defined the associated weight, wg, ∀g ∈ G, as well as the following twoparameters:

1. The interest rate rg for the scenario groups ∀g ∈ G, which are generated byusing the binomial lattice approach given in Black et al. (1990). Differentinterest rates are generated for different scenario groups with the bino-mial lattice fitted to the corresponding term structures. The jth type can beexpressed as rω

tj = rtj(kt)ω, where rtj, and (kt)

ω represent the base type andthe volatility in time period t and scenario ω, respectively. So, the differenttrajectories are sampled through the binomial lattice to obtain the set ofscenarios and the related weights.

2. The liabilities Lg for the scenario groups ∀g ∈ G, which are randomlygenerated.

These parameters are generated automatically and systematically under eachscenario group based on a realistic pilot case that is randomly perturbed. Weshould mention that the algorithmic framework presented in this paper hasbeen developed to test out the SIP approach proposed in the paper for ALMwith logical constraints. It is not intended to serve as a basis for drawing eco-nomic conclusions for decision making in managing portfolios of given assetsand liabilities.

Table 1 gives the dimensions of the pilot cases. The headings are as follows:|I|, number of assets; |J |, number of asset classes; S, number of stages; |T |, num-ber of periods; |�|, number of scenarios; |G|, number of scenario groups; |�p|and |Gp|, number of scenarios and number of scenario groups in each cluster,


Fig. 5 Branch-and-Fix Coordination procedure. (where NA means the satisfaction of the nonan-ticipativity constraints)

respectively, where |�|/|�p| is the number q of clusters to be considered. Allscenario trees are symmetric ones. We can observe that the time periods aregrouped in stages. The heading tree gives the structure of the scenario treestarting from stage 2, so that the number of outlooks in each stage from eachancestor node in the tree is shown. As an example 3 · 22 means that stage 1 hasone outlook (in all cases, by construction), stage 2 has three outlooks and stages3 and 4 have two outlooks each from their ancestor nodes. The parameters usedare ρ = 4.0 × 106 and φ = 3.0 × 105 for case P1, ρ = 20, 000 and φ = 1, 000 forthe other cases, and λ = 1.


Table 1 Test bed dimensions

Case |I| |J | T tree |�| |G| |�p| |Gp|

P1 9 3 5 3 · 22 12 34 12 34P2 9 3 6 22 · 3 12 43 6 22P3 9 3 8 25 32 127 16 64P4 9 3 9 25 32 159 16 80P5 9 3 10 25 32 191 16 96P6 20 2 8 25 32 127 8 33P7 20 2 9 25 32 159 8 41P8 20 2 10 25 32 191 8 49P9 50 10 6 33 27 94 9 32P10 6 3 10 28 256 767 16 51P11 50 5 9 25 32 159 4 22P12 10 5 12 27 128 767 4 28P13 50 5 10 25 32 191 4 26P14 30 5 7 33 · 4 108 364 36 122P15 60 15 6 33 · 4 108 256 36 86P16 50 5 11 25 32 223 4 30P17 60 30 6 33 · 4 108 256 36 86P18 50 10 7 34 81 283 9 33P19 40 10 7 33 · 4 108 364 36 122P20 60 10 7 34 81 283 9 33

The cases are split into four categories. The first includes the cases with amaximum of 4,000 0-1 variables, the second and third categories include caseswith a number of 0-1 variables between 4,000 and 10,000, and 10,000 and 20,000,respectively, and the fourth category includes cases with more than 20,000 0-1variables.

Our algorithmic approach is implemented in a Visual C++.NET experimen-tal code. It uses the optimization engine IBM OSL v3.0 to solve the LP modelsin Steps 1 and 9 and the mixed 0–1 models in Step 8. The computational exper-iments were conducted on a Pentium III, 733 Mhz, under the Windows XPoperating system.

Table 2 gives the dimensions of the DEM (22) and the scenario cluster relatedmodel (25), both compact representations, where the headings are: m, numberof constraints; n01, number of 0-1 variables; nc, number of continuous variables;nel, number of nonzero elements in the constraint matrix; and dens, constraintmatrix density (in %). Note that the dimensions of the cases are very high.

Table 3 shows the main results of our computational experiment for givenvalues of the number of scenario clusters. The headings are as follows: ZLP,solution value of the LP relaxation of the original problem; ZIP, solution valueof the original problem; GAP, optimality gap defined as ZLP−ZIP

ZLP (in %); TLP,T, and TOSL, the elapsed time (secs) to obtain the LP solution to the originalproblem, the time to obtain the optimal solution by using the BFC scheme andthe time to obtain the optimal solution by the plain use of the optimizationengine for solving the DEM, respectively.


Table 2 Model dimensions. Compact representation

Case |G| Deterministic equivalent model Scenario cluster model

m n01 nc nel dens m n01 nc nel dens

P1 34 1689 465 749 5618 0.27 1689 465 749 5618 0.27P2 43 2337 654 1010 7760 0.20 1191 333 515 3944 0.39P3 127 7165 2018 3066 23812 0.07 3605 1015 1543 11970 0.13P4 159 9469 2690 3994 31428 0.05 4757 1351 2007 15778 0.10P5 191 11773 3362 4922 39044 0.04 5909 1687 2471 19586 0.08P6 127 14609 4002 6542 50928 0.03 3791 1038 1698 13188 0.13P7 159 19313 5346 8526 67248 0.03 4967 1374 2194 17268 0.10P8 191 24017 6690 10510 83568 0.02 6143 1710 2690 21348 0.08P9 94 26245 7347 11511 90961 0.02 8897 2489 3905 30775 0.05P10 767 28091 7915 12006 90643 0.02 1895 535 806 6095 0.24P11 127 47615 13317 20886 167121 0.01 6592 1844 2890 23060 0.07P12 767 54543 16093 21846 177529 0.01 2010 594 802 6514 0.23P13 191 59231 16677 25750 207729 0.01 8044 2264 3498 28136 0.06P14 364 60454 16718 26870 210002 0.01 20240 5596 8998 70272 0.02P15 256 73560 20028 33464 254990 0.01 24702 6726 11273 85548 0.02P16 223 70847 20037 30614 248337 0.01 9496 2684 4106 33212 0.05P17 256 78000 22248 33464 261650 0.01 26202 7476 11236 87798 0.02P18 283 79408 22251 34785 275371 0.01 9270 2599 4057 32059 0.05P19 364 81984 23108 35620 282042 0.00 27450 7736 11928 94382 0.02P20 283 94328 26281 41645 328991 0.01 11010 3069 4857 38299 0.04

The first conclusion that can be drawn from the results shown in Table 3is that our approach also obtains the optimal solution in all cases we haveexperimented with. Generally speaking, the optimization engine requires lesscomputational effort than the approach proposed when the cases have smalldimensions. In other words, it seems that the greater the dimensions of thecases, the better the performance of the proposed approach is, especially, con-sidering that our testing has been made with an experimental code. Notice thatthe elapsed time of the approach proposed is smaller by one order of magnitudethan the time required by OSL for some of the cases in the third and fourthcategories. (Cases P10 and P14 could not be solved by OSL within the timelimit, 6 h).

Another interesting observation from Table 3 is that the GAP is near to zeroin most of the test cases. This result is entirely different from the result that canbe obtained when the LP relaxation of the original problem is also includedby the relaxation of the nonanticipativity constraints (i.e., the solution value ofthe LP models LPp, p = 1, ..., q). We have not reported the related GAP thatis obtained by using this other approach but its value is very frequently greaterthan 100%.

Tables 4 and 5 show the performance of the BFC approach for differentvalues of the q-parameter and then different dimensions of model (25) forthe cases P9 and P16. Observe how sensitive the elapsed time for the solu-tion to the problem is relative to the number q of scenario clusters. Thisresult is also observed in the two stage approach reported in Escudero et al.(2006).


Table 3 Stochastic solution

Caso q ZLP ZIP GAP TLP T TOSL

P1 1 317836.98 317836.98 0.0000 0 8 8P2 2 321208.66 321208.66 0.0000 1 7 5P3 2 50199.24 50199.12 0.0002 3 23 15P4 2 51794.86 51794.74 0.0002 1 33 27P5 2 53468.86 53468.74 0.0002 7 40 44P6 4 88388.52 88388.26 0.0003 7 64 79P7 4 92047.61 92047.35 0.0003 14 93 111P8 4 95085.37 95085.10 0.0003 23 128 148P9 3 87552.00 87551.94 0.0001 18 273 322P10 16 53036.01 53035.89 0.0002 65 16735 –P11 8 200087.03 200086.36 0.0003 15 545 610P12 32 79618.04 79617.83 0.0003 214 2393 6099P13 8 209684.64 209683.95 0.0003 122 641 879P14 3 68956.28 68956.23 0.0001 208 3206 –P15 3 109143.83 109143.75 0.0001 44 359 7172P16 8 219746.61 219745.90 0.0003 172 885 1104P17 3 107620.11 107620.03 0.0000 54 279 2494P18 9 97088.86 97080.79 0.0083 150 922 1186P19 3 84866.63 84866.63 0.0000 58 269 7143P20 9 112572.16 112572.08 0.0001 76 242 325

Table 4 Performance of the BFC approach. Case P9

q |�p| |Gp| m n01 nc nel dens T

27 1 6 1767 501 761 6023 0.27 6809 3 12 3363 943 1471 11569 0.14 3523 9 32 8897 2489 3905 30775 0.05 2731 27 94 26245 7347 11511 90961 0.02 687

Table 5 Performance of the BFC approach. Case P16

q |�p| |Gp| m n01 nc nel dens T

32 1 11 3532 1001 1521 12293 0.14 134316 2 17 5399 1527 2332 18843 0.09 9678 4 30 9496 2684 4106 33212 0.05 8854 8 57 18053 5103 7806 63219 0.03 9282 16 112 35530 10046 15358 124502 0.01 9941 32 223 70847 20037 30614 248337 0.01 1668

8 Conclusions and future work

In this paper we present a model for ALM under uncertainty with logicalconstraints. It is formulated as a stochastic multistage mixed 0–1 model withcomplete recourse. The scenario tree based approach that we propose hasproved to be a useful mechanism for representing uncertainty. The goal is tomaximize a mean-risk function that consists of the expected terminal wealth


and the weighted probability of reaching a given terminal wealth. A new BFCscheme is presented for cases where the 0–1 and continuous variables appear atany stage. The computational experiment shows that the new approach obtainsthe optimal solution in a remarkably short elapsed time, which is time muchshorter than the time required by the plain utilization of the optimizer OSL forlarge-scale cases. As future work we are planning to perform a computationalcomparison of the quality of the solution by varying the weighting factor forthe reaching probability and the terminal wealth target. The implementation ofa nested Benders Decomposition approach to solving the LP models in Steps 1and 9 of the approach proposed is also planned to replace the current plain useof the Simplex method.

Acknowledgements This research has been partially supported by grants from Grupo consolid-ado 9/UPV 00038.321-13631/2001 from UPV, 1/BBVA 00038.16421/2004 from Fundación BBVA,MEC 05/168 from Ministerio de Educación y Ciencia and GRUPOS79/04 from the GeneralitatValenciana, Spain.

References

Ahn S, Escudero LF, Guignard-Spielberg M (1995) On modeling financial trading under interestrate uncertainty. In: Optimization in industry 3. Wiley, New York, pp 127–144

Alonso-Ayuso A, Escudero LF, Ortuño MT (2003a) BFC, a Branch-and-Fix Coordination algo-rithmic framework for solving some types of stochastic pure and mixed 0-1 programs. Eur JOper Res 151:503–519

Alonso-Ayuso A, Escudero LF, Garín A, Ortuño MT, Pérez G (2003b) An approach for strategicsupply chain planning based on stochastic 0-1 programming. J Glob Optim 26:97–124

Ben-Dov Y, Mayre L, Pica V (1992) Mortgage valuation models at Prudential Securities. Interfaces2:55–71

Benders JF (1962) Partitioning procedures for solving mixed variables programming problems.Numer Math 4:238–252

Birge JR (1985) Decomposition and partitioning methods for multi-stage stochastic linear pro-grams. Oper Res 33:989–1007

Birge JR, Louveaux FV (1997) Introduction to stochastic programming. Springer, Berlin Heidel-berg New York

Black F, Derman E, Toy W (1990) A one factor model of interest rates and its application to treasurybond options. Financ Anal J Jan/Feb 33–39

Dert CK (1998) A dynamic model for asset and liability management for defined benefit pensionfunds. In: Ziemba WT, Mulvey JM (eds)Worldwide asset and liability Modeling CambridgeUniversity Press, London, pp 501–536

Drijver SJ, Klein Haneveld WK, van der Vlerk MH (2003) Asset liability management modellingusing multistage Mixed-Integer Stochastic Programming. In: Scherer B (ed) Asset and liabilitymanagement tools, risk books. pp 309–324

Escudero LF, Garín A, Merino M, Pérez G (2006) A two stage stochastic integer programmingapproach as a mixture of Branch-and-Fix Coordination and Benders Decomposition schemes.Ann Oper Res (in press)

Fleten SE, Hoyland K, Wallace SW (2002) The performance of stochastic dynamic and fixed mixporfolio models. Eur J Oper Res 140:37–49

Gassmann HI (1990) MSLiP: a computer code for the multistage stochastic linear programmingproblem. Math Program 47:407–423

Klein Haneveld WK, van der Vlerk MH (1999) Stochastic integer programming: General modelsand algorithms. Ann Oper Res 85:39–57

Konno H, Yamamoto R (2005) Integer programming aproaches in mean-risk models. ComputManage Sci 2:339–351


Kouwenberg R, Zenios SA (2006) Stochastic programming models for asset liability management.In: Zenios SA, Ziemba WT (eds) Handbook of asset and liability management, Vol. 1. North-Holland, Amsterdam, pp 253–303

Laporte G, Louveaux FV (2002) An integer L-shaped algorithm for the capacitated vehicle routingproblem with stochastic demands. Oper Res 50:415–423

Lulli G, Sen S (2004) A Branch-and-Price algorithm for multi-stage stochastic integer programmingwith application to stochastic batch-sizing problems. Manage Sci 50:786–796

Ntaimo L, Sen S (2005) The million variable “march” for stochastic combinatorial optimization. JGlob Optim 32:385–400

Ogryczak W, Ruszczynski A (1999) From stochastic dominace to mean-risk models: semi-deviationsand risk measures. Eur J Oper Res 116:33–50

Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21–41Rockafellar RT, Wets RJ-B (1991) Scenario and policy aggregation in optimisation under uncer-

tainty. Math Oper Res 16:119–147Römisch W, Schultz R (2001) Multi-stage stochastic integer programs: An introduction. In:

Grötschel M, Krumke SO, Rambau J (eds) Online optimization of large scale systems. Springer,Berlin Heidelberg New York pp 581–600

Schultz R(2003) Stochastic programming with integer variables’. Math Program Ser B 97:285–309Schultz R, Tiedemann S (2004) Risk aversion via excess probabilities in stochastic programs with

mixed-integer recourse. SIAM J Optim 14:115–138Schultz R, Tiedemann S (2006) Conditional Value-at-Risk in stochastic programs with mixed integer

recourse. Math Program Ser B 105:365–386Sodhi MS (2005) LP modeling for asset-liability management: a survey of choices and simplifica-

tions. Oper Res 53:181–196Zenios SA (1993) A model for portfolio management with mortgage-backed securities. Ann Oper

Res 43:337–356Ziemba WT (2003) The stochastic programming approach to asset, liability and wealth manage-

ment. AIMR, CharlottevilleZiemba WT, Mulvey JM (eds) (1998) Worldwide asset and liability modeling, Cambridge University

Press, London

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