for international portfolio management q · random variables, dynamic adjustments of portfolio positions and currency hedging levels in response to changing information on prevailing

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European Journal of Operational Research xxx (2006) xxx–xxx

www.elsevier.com/locate/ejor

A dynamic stochastic programming modelfor international portfolio management q

Nikolas Topaloglou, Hercules Vladimirou *, Stavros A. Zenios

HERMES European Center of Excellence on Computational Finance and Economics, School of Economics and Management,

University of Cyprus, P.O. Box 20537, CY-1678 Nicosia, Cyprus

Received 1 February 2005; accepted 1 July 2005

Abstract

We develop a multi-stage stochastic programming model for international portfolio management in a dynamic setting.We model uncertainty in asset prices and exchange rates in terms of scenario trees that reflect the empirical distributionsimplied by market data. The model takes a holistic view of the problem. It considers portfolio rebalancing decisions overmultiple periods in accordance with the contingencies of the scenario tree. The solution jointly determines capital alloca-tions to international markets, the selection of assets within each market, and appropriate currency hedging levels. Weinvestigate the performance of alternative hedging strategies through extensive numerical tests with real market data.We show that appropriate selection of currency forward contracts materially reduces risk in international portfolios.We further find that multi-stage models consistently outperform single-stage models. Our results demonstrate that the sto-chastic programming framework provides a flexible and effective decision support tool for international portfoliomanagement.� 2006 Elsevier B.V. All rights reserved.

Keywords: Stochastic programming; Risk management; International portfolios; Scenarios

1. Introduction

Active management of international portfolios isof particular interest to multinational firms, finan-

0377-2217/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.ejor.2005.07.035

q Research partially supported by the HERMES Center ofExcellence on Computational Finance and Economics which isfunded by the European Commission (Grant ICA1-CT-2000-70015), and by a research grant from the University of Cyprus.

* Corresponding author. Fax: +357 22 892421.E-mail addresses: [email protected] (N. Topaloglou), hercu-

[email protected] (H. Vladimirou), [email protected] (S.A. Zenios).

Please cite this article in press as: N. Topaloglou et al., A dynRes. (2006), doi:10.1016/j.ejor.2005.07.035

cial intermediaries, institutional investors (e.g.,banks, insurance firms, mutual funds, pensionfunds) and high net-worth individuals. Investmentsin foreign securities are becoming accessible to awide range of investors. This is a consequence ofliberalization in capital flows, and advancementsin information technology that provide instanta-neously information from remote financial marketsand facilitate the execution of transactions. Inter-national investments provide certain benefits: (a)the prospect for higher profit in the event of favor-able performance of foreign markets, (b) widerscope for diversification, (c) reduced exposure to

.

amic stochastic programming model ..., Eur. J. Operat.

mailto:[email protected]




2 N. Topaloglou et al. / European Journal of Operational Research xxx (2006) xxx–xxx

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systematic risk due to generally low correlations ofinternational securities.

The potential benefits of international diversifica-tion were recognized early; see, for example, Grubel[11], Levy and Sarnat [18], and Solnik [22]. Due tocomparably low correlations of international securi-ties, improvements in reward-to-risk performancecan be achieved by holding internationally diversi-fied portfolios. Despite a general increase in the cor-relation of international markets in recent years as aresult of financial globalization and market integra-tion, international investments continue to providea wider scope for portfolio diversification than isavailable in a domestic market. Optimal manage-ment of international portfolios, especially in adynamic setting, continues to be a problem of greatinterest and practical significance.

Eun and Resnick [8] pointed out that interna-tional diversification decisions should take intoaccount the exposure to currency risk. Namely,the risk stemming from fluctuations of foreignexchange rates, as uncertain future exchange ratesdirectly affect the translation of returns from foreigninvestments into the investor’s base-currencyreturns. Given the volatility of exchange rates insome periods, currency risk can have a majorimpact on international portfolios.

Means to effectively hedge currency risk areactively studied. In a key early contribution, Black[3] suggested a universal hedge ratio for all inves-tors, implying that all foreign investments shouldbe hedged to the same proportion. Subsequent stud-ies examined more flexible currency hedging strate-gies. A number of empirical studies evaluated themerits of various approaches for hedging exchangerate risk, usually on the basis of forward currencyrates. We do not review this extensive literaturehere. Despite a general agreement that currencyhedging can mitigate portfolio risk, there is not aconsensus as to a universally optimal currency hedg-ing rule. The literature presents somewhat differentviews as to the optimal course of action for hedgingcurrency risk in international portfolios, dependingon the focus of each study with regard to variousfactors, such as: the investment opportunity set,the risk aversion preference and time horizon ofthe decision maker, the reference currency of theinvestor, the investment strategy (passive vs active),the distribution and stationarity of asset returnsand exchange rates in the timeframe of the study(i.e. the historical data used in calibrating the distri-butions), the realized market conditions (trends,

Please cite this article in press as: N. Topaloglou et al., A dyRes. (2006), doi:10.1016/j.ejor.2005.07.035

volatilities, correlations between markets) at thetime of simulations, the specific hedging strategiesthat were compared.

The overall conclusion is that the relative meritsof alternative hedging strategies remain mostly anempirical issue, they depend on the factors men-tioned above and are usually problem-dependent.These observations point to the need for flexibledecision approaches that can be adjusted in eachcase to the problem at hand. This study developsa general and flexible model to support interna-tional portfolio management decisions.

In practice, and more often than not in the liter-ature as well, the international portfolio manage-ment problem is addressed in an overlay manner.First, the capital allocation to various markets isdecided at the strategic level. The management ofthe capital allocations is then assigned to differenttrading desks with expertise in the respective mar-kets. Each trading desk has independent jurisdictionto tactically select specific securities within itsassigned market, and operationally manage itsown portfolio. Performance is typically comparedagainst some benchmark. Currency hedging is oftenviewed as a subordinate decision, and is taken lastto cover specific exposures of foreign asset positionsthat were decided previously. A global view of theproblem is not often taken, and possible cross-hedg-ing effects among portfolio positions are frequentlyignored. Changes in the portfolio structure are notalways coordinated with corresponding adjustmentsto the currency hedging positions. Important andinterrelated decisions are considered separatelyand sequentially.

Jorion [16] criticized this overlay approach andshowed that it is suboptimal to a holistic view thatconsiders all the interrelated decisions in a unifiedmanner. Here, we develop a comprehensive model-ling framework that jointly addresses the capitalallocation, the portfolio selection, and the currencyhedging decisions.

Filatov and Rappoport [9] were the first to sug-gest a selective hedging approach in which the hedgeratios may vary across currencies. They showed thatselective hedging can yield different optimal hedgeratios for each currency. Owing to this flexibility,the selective hedging approach dominates otherstrategies that tie in specific ways the hedge ratiosacross currencies (e.g., unitary and partial hedging,which selective hedging encompasses as specialcases). In their empirical tests, they also found thatthe optimal selective hedging policy can be different

namic stochastic programming model ..., Eur. J. Operat.

N. Topaloglou et al. / European Journal of Operational Research xxx (2006) xxx–xxx 3

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for investors with different reference currencies.Beltratti et al. [2] incorporated selective hedgingdecisions within a single-stage portfolio optimiza-tion model and empirically confirmed the superior-ity of selective hedging to unitary hedging. Theirmodel used a mean-absolute-deviation objective,and bootstrapping of historical data for scenariogeneration. Topaloglou et al. [25] extended thismodel for international asset allocation by using amore appropriate scenario generation procedure,and by adopting a risk measure that is suitable forthe asymmetric distributions exhibited by interna-tional asset returns and exchange rates.

Glen and Jorion [10] considered dynamic hedgingstrategies, and observed that the optimal hedgingpolicy may be temporally unstable. They concludedthat benefits can be obtained by following a statecontingent policy that varies the level and the struc-ture of hedging depending on the investment oppor-tunity set and on evolving information regarding thedistributions of asset returns and exchange rates.

Building on the findings of these studies, wedevelop a general and flexible modelling frameworkfor the active management of international portfo-lios in a dynamic setting. The primary features ofour model are: a multi-period decision frameworkthat considers information and decision dynamics,the ability to capture arbitrary stochastic evolutionsof the random factors by means of a discrete repre-sentation (scenario tree), state-contingent decisionsin conformity with the projected outcomes of therandom variables, dynamic adjustments of portfoliopositions and currency hedging levels in response tochanging information on prevailing economic con-ditions (distributions of the random variables asreflected in the scenario tree), operationalization ofcurrency hedging decisions by means of explicit for-ward exchange contracts for each currency that fullyencompass a selective hedging approach, a holisticview of the problem that accounts for decision inter-actions and considers the total risk exposure of theinternational portfolio, a risk measure that is suit-able for skewed and leptokurtic distributions ashave been observed for international indices andexchange rates, consideration of transaction costs.Alternative objectives to reflect the decision maker’srisk bearing preferences, as well as other operationalissues (e.g., managerial and regulatory conditions)can be easily incorporated with appropriate adjust-ments to the model.

The paper is organized as follows. In the next sec-tion we briefly describe the problem and introduce


the modelling choices for our solution approach.In Section 3 we describe the representation of uncer-tainty in the multi-stage portfolio optimizationmodel by means of a scenario tree. We present thekey statistics of the random variables based on his-torical data and discuss the scenario generation pro-cedure. In Section 4 we formulate the stochasticprogramming model for international portfoliomanagement and examine its features. In Section 5we describe the computational tests and we presentthe empirical results. In Section 6 we discuss thefindings of this study and directions for furtherresearch.

2. Problem description and modeling approach

We consider the problem of a decision makerwho is concerned with the active management of aset of financial assets (stock and bond indices in thisstudy) denominated in multiple currencies, so as togenerate profit while at the same time controllingthe downside risk exposure. The problem has adynamic structure that involves portfolio rebalanc-ing decisions at periodic intervals in response tonew information on market conditions (i.e., chang-ing perceptions regarding the distributions of ran-dom asset prices and exchange rates). Rebalancingdecisions are manifested in a sequence of successiverevisions of holdings through sales and purchases ofassets and currency exchange transactions in thespot market. Currency forward contracts can beemployed to hedge the currency risk of foreigninvestments; we consider forward contracts with aterm of a single period at each decision stage.

The decision maker starts with an initial portfo-lio and has full knowledge of the current asset pricesand exchange rates. Thus, individual asset holdings,as well as the entire portfolio, can be accurately val-ued. The decision maker must assess the potentialmovements of the asset prices and the exchangerates that affect the future value and risk exposureof his portfolio. His perceptions for such marketmovements are expressed in terms of a joint proba-bility distribution of the random variables; he mustproject the contingent evolution of the random vari-ables over the entire planning horizon. The portfo-lio rebalancing and currency hedging decisionsshould conform to these projections.

Clearly, the initial portfolio restructuring deci-sion has a direct effect in subsequent periods. Port-folio rebalancing decisions at later periods dependon the portfolio composition at that time, the pre-



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vailing market conditions at the time, and the per-ception for subsequent potential movements of therandom variables. Conversely, the flexibility forsubsequent portfolio rebalancing influences earlierdecisions. Thus, a multi-stage model is needed tocapture the information and decision dynamicsand their interaction.

Practical issues, such as transaction costs, mana-gerial and regulatory requirements should be incor-porated in the decision model. Transaction costs doplay a role in portfolio management and their effectmust be considered. Their omission can lead to highportfolio turnover with a consequent reduction ingains and a potential increase in risk exposure.The impact of transaction costs and the potentialbenefits of diversification are more properly cap-tured in a multi-period decision model.

The paradigm of multi-stage stochastic programswith recourse is particularly suitable for this prob-lem. Uncertainty in input parameters of stochasticprograms is represented by discrete scenarios thatdepict the joint co-variation of the random vari-ables. In multi-stage problems the progressive evo-lution of the random variables is expressed interms of a scenario tree. Scenarios can be generatedwith various approaches, and they are not restrictedto any particular distributional assumption. Asym-metric and fat-tailed distributions that are oftenobserved in practice can be captured by discreteapproximations. Indeed, this flexibility in the repre-sentation of uncertainty in input parameters is amajor advantage of stochastic programs.

Stochastic programs can accommodate differentobjective functions to capture the decision maker’srisk bearing preferences (e.g., utility functions, pen-alties on shortfalls and other risk measures, etc.)Moreover, they can incorporate managerial andregulatory requirements, especially when suchrequirements are expressed in terms of linear con-straints. Because of their flexibility, stochastic pro-grams have attracted the attention of researchersand practitioners alike and are being increasinglyapplied to diverse practical problems. Variousapplications of stochastic programs are docu-mented in the volume edited by Wallace and Zie-mba [30].

Financial modeling is a particularly fertile appli-cation domain for stochastic programming. Numer-ous important contributions have been made duringthe last 20 years. Financial applications of stochas-tic programming models have twice been among thefinalists for the Edelman Prize for best achievement


in Management Science in recent years (see, [5,19]).There have been numerous notable contributions ofstochastic programming applications to such diverseproblems as asset and liability management, portfo-lio management, insurance, pension funds, creditrisk management, etc. Some recent collections con-tain several representative contributions (e.g.,[31,30,28,32]).

We adopt the multistage stochastic programmingframework for the international portfolio manage-ment problem because of its flexible features thatwe discussed above. Multi-stage models help deci-sion makers gain useful insights and adopt moreeffective decisions. They shape decisions based onlonger-term potential benefits and avoid myopicreactions to short-term market movements thatmay prove risky. They determine appropriatedynamic contingency (recourse) decisions underchanging economic conditions that are representedby scenario trees. Our model encompasses all theimportant aspects of the problem and jointly deter-mines the allocation of capital to international mar-kets, the selection of specific asset holdings withineach market, and currency hedging decisions withforward contracts. Thus, all interrelated decisions,that are traditionally considered separately, are castin a unified and flexible framework.

As we indicate in Section 3.2, the returns of sev-eral international indices and the movements ofexchange rates exhibit asymmetric distributionswith fat tails. Consideration of these features isimportant in our study that aims to devise methodsfor controlling downside risk. We capture skewnessand excess kurtosis in the distributions of the ran-dom variables by applying a moment-matching sce-nario generation procedure. This proceduregenerates scenarios so that key statistics (specifi-cally, the first four marginal moments and the corre-lations) of the random variables match specifiedtarget values. We estimated the target values forthese statistics on the basis of historical data. Weverified through exhaustive tests that the scenariosets also satisfied the required no-arbitrageconditions.

We employed the conditional value-at-risk(CVaR) metric in the objective function in order tominimize the expected shortfall beyond the value-at-risk (VaR) of portfolio losses at the end of thehorizon. The choice of the objective function wasmade in light of the observed asymmetries andexcess kurtosis of key random variables. CVaR is acoherent risk measure, and is suitable for asymmet-



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ric distributions. This objective function enabled theeffective control of downside risk exposure. Alterna-tive means can be employed to control downsiderisk; for example, piecewise-linear convex penaltieson shortfall levels [5].

This study extends the work in Topaloglou et al.[25] in several important directions. First, it devel-ops a multi-stage stochastic programming modelthat allows the solution of dynamic interna-tional portfolio management models. Second, itoperationalizes the currency hedging decisions byincorporating explicit variables that transparentlydetermine the appropriate level of forward contractsfor each currency. Third, it adopts a flexible andeffective scenario generation procedure that pro-duces scenario sets that closely approximate theempirical joint distributions of asset returns andexchange rates.

3. Representation of uncertainty

The representation of uncertainty in inputparameters of the portfolio optimization model isa critical step in the modelling process. The keyrandom inputs in the international portfolio man-agement problem are the asset prices (or, equiva-lently, their returns) and the currency exchangerates at the trading dates within the planning hori-zon. Plausible evolutions of the random parametersduring the planning horizon are specified in terms of

NT

•••

•••

•••

•••

•••

•••

•••

p (n)

Rootnode

•••

•••

•••

•••

•••

•••

•••

•••

Sn

n

t=Tt=0 t=1 t=2 t=3Time period

Fig. 1. General form of a scenario tree.


discrete joint outcomes that are mapped to thenodes of a scenario tree, such as the one shown inFig. 1.

A major advantage of stochastic programs is thatthey are not restricted to any particular distribu-tional assumption for the random variables. Theycan accommodate arbitrary discrete distributionsthat are expressed by means of a scenario tree.Alternative scenario generation procedures for sto-chastic programs are reviewed in Dupacova et al.[6]. The key decisions in a scenario generationmethod are the choice of a statistical model andthe calibration of its parameters, usually on thebasis of market data or subjective expert opinions.The scenario set must conform to financial princi-ples; specifically, it must be free from arbitrageopportunities.

3.1. The scenario tree

The planning horizon is divided to periodst = 0,1, . . . ,T corresponding to the times at whichportfolio rebalancing decisions can be made. Thetree has a depth equal to the number of periods(decision stages); we use monthly trading periods.The root node (n = 0) corresponds to the initialstate at the present time (t = 0). All input data asso-ciated with the root node are known with certainty.The tree branches out from the root to depict pro-gressive outcomes in the values of the randomvariables at subsequent periods. The branches ema-nating from the root reflect the possible outcomesduring the first period (t = 1). Each postulated out-come is associated with an immediate successornode. Similarly, the branches emanating from anysubsequent node represent the discrete, condi-tional distribution of the random variables duringthe next time period. Each node reflects a possiblestate at the corresponding time period, and capturesa joint realization of the random variables at thattime.

Each scenario distinguishes a sequence of jointrealizations of the random variables during theplanning horizon. Thus, it has a one-to-one corre-spondence with a terminal node (leaf) of the tree.The constituent realizations of the random variablesat each period are identified by the nodes on theunique path from the root to the leaf node associ-ated with the scenario (such as the highlighted pathin Fig. 1). The scenario tree need not be symmetric,and it is rarely binomial as shown in Fig. 1 simplyfor illustration purposes.



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We use the following notation:N the set of nodes of the scenario tree;n 2 N a typical node of the scenario tree (n = 0

denotes the root node at t = 0);Nt � N the set of distinct nodes of the tree at time

period t = 0,1, . . . ,T;NT � N the set of leaf (terminal) nodes at the last

period T, that uniquely identify the scenar-ios;

p(n) 2 N the unique predecessor node of noden 2 Nn{0};

Sn � N the set of immediate successor nodes ofnode n 2 NnNT. This set of nodes repre-sents the discrete distribution of the ran-dom variables at the respective timeperiod, conditional on the state of node n;

pnt the conditional probability for the outcome

associated with the transition from the pre-decessor node p(n) to node n 2 N;

pn the probability of the state associated withnode n 2 N.

The probability, pn, of a certain node (state)n 2 N is determined by multiplying the conditionalprobabilities of the outcomes on the path from theroot to the specific node; i.e., by compounding theconditional probabilities of the constituent out-comes that lead to the specific state. The probabili-ties of all distinct nodes at any decision stage sum toone (i.e.,

Pn2N t

pn ¼ 1; t ¼ 0; 1; . . . ; T ). Also, theprobability of a node is equal to the sum of theprobabilities of its immediate successor nodes (i.e.,pn ¼

Pm2Sn

pm; 8n 2 N nNT ).The scenario tree represents the evolution of the

multi-variate random variables over the planninghorizon. We define distinct (recourse) variables,and associated constraints, to model the portfoliorebalancing decisions at each intermediate node ofthe scenario tree (i.e., "n 2 NnNT). At the leafnodes, NT, we only compute the corresponding ter-minal value of the portfolio under the respective sce-narios. The size of the resulting multi-stagestochastic program grows substantially with thenumber of tree nodes. In multi-stage problemsattention must be paid to limit the branching factorfrom each node—and, consequently, the total num-ber of scenarios—in order to keep the size of theoptimization program within computationally trac-table limits. Scenario reduction techniques havebeen proposed for this purpose (see, [7,12]).

Stochastic programs must conform to the logicalrequirement for non-anticipative decisions. That is,


scenarios that share common information history(outcomes) up to a particular time period—i.e., havecommon subpath of the scenario tree up to that per-iod—must yield identical decisions up to that per-iod. The non-anticipativity condition is explicitlyenforced in our model as decision variables aredefined for each node—instead of each path.

3.2. Scenario generation

We consider portfolios of stock and bond indicesdenominated in different currencies. Statistical anal-ysis of market data reveals that the random vari-ables (index returns and currency exchange rates)are correlated. Moreover, their historical values donot conform to normal distributions; they exhibitasymmetries and heavy tails. These observationsare consistent with the findings of other studies thatconsidered the returns of international financialassets; see, for example, Prakash et al. [20] and ref-erences therein. These features should be reflected inthe postulated scenario sets that should capture thestatistical characteristics of the random variables’empirical distribution. Effectively capturing theobserved skewness and excess kurtosis in the distri-bution of the random variables, as well as their cor-relations, becomes important. This becomes all themore necessary as we are concerned with controllingthe downside risk in the tail of the portfolio’s returndistribution.

In the empirical tests we consider investments infour markets: United States (US), Great Britain(UK), Germany (GR) and Japan (JP), comprisedof the following instruments in each market: a stockindex, denoted as Stk, and bond indices with threedifferent maturity bands: short—(1–3 years), inter-mediate—(3–7 years) and long-term (7–10 years),denoted Bnd1, Bnd3 and Bnd7, respectively. Thus,a total of 16 assets are considered in each portfolio.The problem is viewed from the perspective of a USinvestor. Repositioning of investments betweenmarkets entails spot currency exchange transac-tions. Moreover, forward currency exchange con-tracts—with a term equal to a decision period, i.e.,one month—are incorporated in the portfolio opti-mization model to (partly) hedge the currency riskof foreign investments. Hence, data for the spotand forward exchange rates of the foreign currenciesto USD are also needed.

Market values of the stock indices were obtainedfrom the Morgan Stanley Capital International, Inc.database (www.mscidata.com). The values for the


http://www.mscidata.com

Table 1Statistical characteristics of historical monthly data for domesticreturns of assets and proportional changes of spot currencyexchange rates over the period 01/1990–12/2001

Assetclass

Mean(%)

Std.Dev.(%)

Skewness Kurtosis Jarque–Berastatistic

Statistical characteristics of monthly domestic returns of assets

US.Stk 1.211 4.097 �0.413 3.649 6.11UK.Stk 1.025 4.254 �0.018 3.371 0.61GRStk 1.167 5.807 �0.561 4.301 13.92JP.Stk �0.170 6.881 0.017 3.836 2.45US.Bnd1 0.567 0.504 �0.006 2.765 0.74US.Bnd3 0.654 1.118 �0.112 2.626 1.03US.Bnd7 0.710 1.670 �0.090 2.912 0.30UK.Bnd1 0.680 0.714 0.995 7.461 150.82UK.Bnd3 0.741 1.349 0.510 4.631 29.45UK.Bnd7 0.793 1.880 0.134 3.390 1.53GR.Bnd1 0.493 0.480 0.288 4.451 31.24GR.Bnd3 0.552 0.930 �0.255 3.162 4.58GR.Bnd7 0.583 1.372 �0.663 3.887 22.87JP.Bnd1 0.283 0.481 0.663 4.708 12.98JP.Bnd3 0.416 1.121 �0.099 4.401 26.37JP.Bnd7 0.503 1.678 �0.519 5.434 47.62

Exchangerate

Statistical characteristics of monthly proportional spot exchange

rate changes

US to UK �0.116 2.894 �0.755 5.672 102.39US to GR �0.124 3.091 �0.215 3.399 5.69US to JP 0.030 3.615 0.942 6.213 105.27


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bond indices and the currency exchange rates werecollected from DataStream. All time series havea monthly time-step and cover the period fromApril 1988 through December 2001. From the dataof index prices we computed their correspondingmonthly returns (in domestic terms). Similarly, fromthe observed series of spot exchange rates we com-puted their corresponding monthly appreciationrates (proportional changes).

The statistics in Table 1 show that both thedomestic returns of the indices and the proportionalchanges of exchange rates exhibit skewed distribu-tions; they also exhibit considerable variance incomparison to their mean (especially the stock indi-ces). Jarque–Bera tests on these data indicate thatthe normality hypothesis cannot be accepted formost of them.1 The normal distribution cannotproperly capture the observed joint behavior ofthe financial time series. Extreme values are encoun-tered more often than predicted by the normaldistribution.

The correlations of the random variables over theperiod 01/1990–12/2001 are shown in Table 2.Observe that correlations of asset returns are muchlower across markets than within markets. Hence,international diversification can reduce the total riskof a portfolio.

Given the statistical characteristics of the randomvariables, we apply the moment-matching proce-dure of Høyland and Wallace [15] and Høylandet al. [14] to generate sets of scenarios so that keystatistics of the random variables match specifiedtarget values. Specifically, we match the followingstatistics: the first four marginal moments (mean,variance, skewness, and kurtosis), as well as the cor-relations of the monthly asset returns and currencyexchange appreciation rates. We estimate the targetvalues to be matched on the basis of historical data.

The user specifies a-priori the desired number ofscenarios, thus controlling the size of the result-ing portfolio optimization program. The model pre-sented in the next section is dynamic and involves anarbitrary number of decision stages. We generatethe scenarios incrementally, one stage at a time.

1 The Jarque–Bera statistic has a X2 distribution with twodegrees of freedom. Its critical values at the 5% and 1%confidence levels are 5.99 and 9.21, respectively. The normalityhypothesis is rejected when the Jarque–Bera statistic has a highervalue than the corresponding critical value at the respectiveconfidence level.


From observed market data we estimate the targetstatistics of the random variables for a monthly timeperiod—equal to a decision period of the model. Weapply the moment-matching procedure to generatethe joint realizations (a discrete multivariate distri-bution) of the random variables for the first period(t = 1)—i.e., for the branches emanating from theroot node. Similarly, for each subsequent node wegenerate a number of joint outcomes of the randomvariables for the next stage that match the same sta-tistical properties. We generate equiprobable out-comes from each node; this is not an absoluterequirement as the moment-matching method canalso generate outcomes with differing probabilities,at the expense of higher computational effort.

This scenario generation procedure does notaccount for possible intertemporal dependencies ofthe random variables (e.g., mean reversion, volatil-ity clustering, etc.). This should be the subject offurther research. Consideration of such effects isnot critical for problems with fairly short planninghorizons (<1 year) and monthly decision periods,


Tab

le2

Co

rrel

atio

ns

of

mo

nth

lyas

set

retu

rns

and

pro

po

rtio

nal

chan

ges

of

spo

tex

chan

gera

tes

com

pu

ted

fro

mh

isto

rica

lm

ark

etd

ata

ove

rth

ep

erio

d01

/199

0–12

/200

1

US

.Stk

UK

.Stk

GR

.Stk

JP.Stk

US

.Bnd1

US

.Bnd3

US

.Bnd7

UK

.Bnd1

UK

.Bnd3

UK

.

Bnd7

GR

.Bnd1

GR

.Bnd3

GR

.Bnd7

JP.Bnd1

JP.Bnd3

JP.

Bnd7

US

to UK

US

to GR

US

to JP

US

.Stk

1

UK

.Stk

0.69

1

GR

.Stk

0.58

0.60

1

JP.Stk

0.40

0.39

0.37

1

US

.Bnd1

0.18

0.07

�0.

08�

0.02

1

US

.Bnd3

0.21

0.09

�0.

050.

010.

961

US

.Bnd7

0.24

0.11

�0.

020.

010.

890.

971

UK

.Bnd1

0.13

0.40

0.03

0.00

0.38

0.37

0.35

1

UK

.Bnd3

0.21

0.48

0.11

0.03

0.41

0.44

0.44

0.94

1

UK

.Bnd7

0.26

0.50

0.16

0.05

0.42

0.47

0.49

0.84

0.97

1

GR

.Bnd1

0.05

0.20

0.03

�0.

030.

360.

390.

380.

590.

590.

561

GR

.Bnd3

0.13

0.25

0.12

0.01

0.42

0.48

0.50

0.54

0.63

0.64

0.92

1

GR

.Bnd7

0.17

0.27

0.20

0.02

0.45

0.53

0.56

0.48

0.61

0.67

0.77

0.94

1

JP.Bnd1

0.10

0.12

0.02

0.05

0.21

0.21

0.20

0.33

0.31

0.29

0.40

0.36

0.30

1

JP.Bnd3

0.11

0.09

0.01

0.05

0.22

0.25

0.26

0.24

0.27

0.27

0.37

0.38

0.33

0.91

1

JP.Bnd7

0.12

0.10

0.02

0.05

0.25

0.29

0.30

0.22

0.26

0.27

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ARTICLE IN PRESS


as we consider in this study. It becomes more impor-tant for problems with short decision stages (e.g.,daily) or very long planning horizons.

At the root node we know with certainty themarket asset prices, the spot exchange rates, andthe forward rates for the next period (i.e., onemonth). Using the projected asset returns and cur-rency appreciation rates for the first period, we com-pute the joint asset prices and spot currencyexchange rates for each node at the first stage(t = 1). Similarly, knowing the asset prices and spotexchange rates at a subsequent node of the scenariotree, as well as the projected asset returns and cur-rency appreciation rates for the branches emanatingfrom that node, we determine the asset prices andspot exchange rates at the end of the stage for eachimmediate successor node. For each node we alsospecify forward exchange rates for one-month for-ward currency transactions. These forward ratesare set equal to the expected value of the respectivespot exchange rates at the end of the period—i.e., bytaking the expectation of the spot exchange ratesover the immediate successor nodes. This is doneso as to ensure that the no-arbitrage conditionsare met for forward currency exchanges.

The multi-stage stochastic program can beextended to include longer-term forward currencyexchanges (i.e., spanning multiple decision stages).Additional variables would be needed to representthese forward currency contracts, coupling the cashbalance conditions in non-successive decision peri-ods. The longer-term forward exchange rates wouldhave to be carefully modelled consistently with theprojected spot exchange rates and the shorter-termforward exchange rates. We leave such an extensionfor a subsequent study.

In a critique of the moment-matching method,Klaasen [17] argued that it is possible to match themoments of random asset returns with a coarse dis-crete distribution that violates the no-arbitrage prin-ciple; he provided a small example with a triangulardistribution for a univariate random variable. Wetested exhaustively all the scenario sets used in ournumerical experiments and empirically verified thatthe no-arbitrage conditions were always satisfied.

The stochastic programming model in the nextsection is not restricted to the moment-matching sce-nario generation procedure. A user can employ analternative approach—such as the method of Hoch-reiter and Pflug [13]—that he finds preferable to gen-erate the scenarios of asset prices and exchange rates.Of course, the scenarios must effectively reflect the



ARTICLE IN PRESS

intended distributions of the random variables andmust satisfy fundamental financial principles (e.g.,must be arbitrage free). The moment-matching pro-cedure that we employed in the numerical validationtests meets these requirements.

4. International portfolio management model

The model determines a sequence of investmentdecisions at discrete points in time (monthly inter-vals). The portfolio manager starts with an initialportfolio and with a set of postulated outcomesregarding future states of the economy. This infor-mation is incorporated into a portfolio restructuringdecision. At the beginning of the next period themanager has at hand a seasoned portfolio. The com-position of the portfolio depends on the transactionsat the previous decision point; its value depends onthe outcome of asset returns and exchange rates inthe interim period. For every projected realizationof the random variables we end up at a different stateat the end of the period—associated with a descen-dant node. Another portfolio restructuring decisionis made at that node based on the portfolio at hand,and taking into account the subsequent possible out-comes of the random variables.

The problem is viewed from the perspective of aUS investor who may hold assets denominated inmultiple currencies. Without loss of generality, nodirect exchanges between foreign currencies are exe-cuted—either in the spot or in the forward market—in order to simplify the formulation of the modeland to reduce its data needs, as well as the numberof decision variables. All currency exchanges areexecuted with respect to the base currency. To repo-sition his investments from one market (currency) toanother, the investor must first convert to base cur-rency the proceeds of foreign asset sales in the mar-ket in which he reduces his presence and thenpurchase the foreign currency in which he wishesto increase his investments. The spot exchange ratesof foreign currencies to USD applicable at the deci-sion state (node) are used for the currency exchangetransactions. At the end of the holding period wecompute the state-dependent value of asset holdingsusing their projected prices at the respective treenode. The USD-equivalent value is determined byapplying the estimate of the appropriate spotexchange rate to USD at the same node.

The portfolio is exposed to market risk in thedomestic and foreign markets, as well as to currencyrisk for the foreign investments. To (partly) hedge


currency risk at each decision node, the investormay enter into forward currency exchange con-tracts. We allow forward currency exchanges witha term (maturity) of one period, i.e., one month.The key decision is the amount of the contract.The actual exchange takes place at the end of theperiod using the forward rate that is specified atthe time the decision is made (i.e., at the beginningof the period). The forward currency exchange con-tracts constitute hedging decisions to mitigate therespective exposure of foreign investments to cur-rency risk. The optimal selection of currency for-ward contracts is integrally incorporated in theportfolio management model.

The notion of ‘‘full hedging’’ that is oftenreferred to in the literature is not attainable exactlywith forward exchanges. This is because the value offoreign asset holdings at the end of any period is notknown with certainty at the beginning of the period,i.e., at the time that a forward contract is decided.This value depends on the realized asset returns dur-ing the interim period. Yet, the amount of the for-ward contract must be decided before the realizedasset returns are observed. Hence, forward currencyexchanges cannot hedge fully the currency riskexposure of foreign investments.

The stochastic programming model takes asinput the representation of uncertainty as capturedby the scenario tree. The model minimizes the con-ditional value-at-risk (CVaR) of portfolio losses atthe end of the planning horizon. That is, it mini-mizes the expected value in the tail (beyond a spe-cific percentile, a) of the portfolio losses at the endof the planning horizon. CVaR is the conditionalexpectation of excess shortfall beyond the value-at-risk of losses (VaR) at the respective percentilelevel. Unlike VaR, CVaR is a coherent risk measurein the sense of Artzner et al. [1], and is receivingincreasing attention in financial applications. Theincorporation of the CVaR function in the stochasticprogramming model is done along the lines ofRockafellar and Uryasev [21] (see, also, Topaloglouet al. [25]).

We define the following additional notation:Sets:

C0 set of markets (synonymously, countries,currencies);

‘ 2 C0 index of investor’s base (reference) currency(in our case USD);

C set of foreign markets; C = C0n{‘};Ic set of available investments (asset classes)

in market c 2 C0.


f Operational Research xxx (2006) xxx–xxx

ARTICLE IN PRESS

User-specified parameters:
a critical percentile for VaR and CVaR;l target (minimum) expected portfolio return
over the planning horizon.

Deterministic input data:h0

c initially available cash in currency c 2 C0

(deficit if negative);bic initial position (in number of units of face

value) in asset i 2 Ic of market c 2 C0;P 0

ic current market price of asset i 2 Ic,c 2 C0

(in units of domestic currency c);e0

c current spot exchange rate of currencyc 2 C0;

u0c currently quoted one-month forward ex-

change rate for foreign currency c 2 C;cic proportional transaction cost for sales or

purchases of asset i 2 Ic,c 2 C0;V0 total value of initial portfolio (i.e., initial

wealth, in units of reference currency):V 0 ¼

Pc2C0ðh0

c þP

i2IcbicP 0

icÞe0c .

Scenario-dependent data:pn probability of node n 2 N—we generate

symmetric trees with equiprobable scenar-ios, thus pn ¼ 1

jN t j ; 8n 2 N t; t ¼ 0; 1; . . . ; T ;hn

c exogenous cash inflow (liability if �ve) ofcurrency c 2 C0 at node n 2 N;

P nic price of asset i 2 Ic,c 2 C0 at node n 2 N (in

units of domestic currency c);en

c spot exchange rate for foreign currencyc 2 C at node n 2 N;

unc one-month forward exchange rate for for-

eign currency c 2 C at node n 2 N;Un

c upper bound on a forward contract in cur-rency c 2 C0 at node n 2 N (in units of thebase currency).

Decision variables:Portfolio rebalancing decisions are made at

nodes of the scenario tree except for the leaves; thusseparate variables are defined for each noden 2 NnNT to reflect the decisions made at the respec-tive node:xn

ic units of asset i 2 Ic,c 2 C0 purchased;vn

ic units of asset i 2 Ic,c 2 C0 sold;wn

ic units of asset i 2 Ic,c 2 C0 held in the port-folio after revision;

gnc expenditure of base currency, ‘, for pur-

chase of currency c 2 C in the spot market;qn

c revenue in base currency, ‘, from sale offoreign currency c 2 C in the spot market;

10 N. Topaloglou et al. / European Journal o


f nc amount of base currency, ‘, collected from

sale of currency c 2 C in the forward mar-ket (i.e., amount of forward contract, inunits of the base currency). A negative va-lue indicates a purchase of the foreign cur-rency in the forward market. Thesedecisions are made at node n 2 NnNT, butthe actual transaction is executed at theend of the respective period, i.e., at the suc-cessor nodes Sn.

Variables at the leaf nodes n 2 NT:Vn final value of the portfolio held at the end

of the planning horizon (in units of the basecurrency, ‘);

Rn return of the portfolio over the planninghorizon;

Ln portfolio loss over the planning horizon.

Auxiliary variables:yn portfolio shortfall in excess of VaR at leaf

node n 2 NT;z variable in definition of CVaR—equals to

VaR at the optimal solution.

All exchange rates are expressed as the equivalentamount of the base currency, ‘, for one unit of theforeign currency. Obviously, en

‘ ¼ un‘ ¼ 1; 8n 2 N .

Also gn‘ ¼ qn

‘ ¼ f n‘ ¼ 0; 8n 2 N ; these variables are

omitted in the actual implementation.We formulate the multi-stage stochastic pro-

gramming model for the dynamic internationalportfolio management problem as follows:

min f ¼ zþ 1

1� a

Xn2NT

pnyn; ð1aÞ

s:t: w0ic ¼ bic þ x0

ic � v0ic; 8c 2 C0; 8i 2 I c; ð1bÞ

wnic ¼ wpðnÞ

ic þ xnic � vn

ic; 8c 2 C0; 8i 2 I c;

8n 2 N n fNT [ 0g; ð1cÞ

h0‘ þ

Xi2I ‘

v0i‘P

0i‘ð1� cicÞ þ

Xc2C

q0c

¼Xi2I ‘

x0i‘P

0i‘ð1þ cicÞ þ

Xc2C

g0c ; ð1dÞ

h0c þ

Xi2Ic

v0icP

0icð1� cicÞ þ

g0c

e0c

¼Xi2Ic

x0icP

0icð1þ cicÞ þ

q0c

e0c

; 8c 2 C ; ð1eÞ



ARTICLE IN PRESS

hn‘ þ

Xi2I ‘

vni‘P

ni‘ð1� cicÞ þ

Xc2C

ðqnc þ f pðnÞ

c Þ

¼Xi2I ‘

xni‘P

ni‘ð1þ cicÞ þ

Xc2C

gnc ;

8n 2 N n fNT [ 0g; ð1fÞ

hnc þ

Xi2Ic

vnicP

nicð1� cicÞ þ

gnc

enc

¼Xi2Ic

xnicP

nicð1þ cicÞ þ

qnc

enc

þ f pðnÞc

upðnÞc

;

8c 2 C ; 8n 2 N n fNT [ 0g; ð1gÞ

Rn ¼V n

V 0

� 1; 8n 2 NT ; ð1hÞ

Ln ¼ �Rn; 8n 2 NT ; ð1iÞ

yn P Ln � z; 8n 2 NT ; ð1jÞ

yn P 0; 8n 2 NT ; ð1kÞ

xnic P 0; wn

ic P 0; 8c 2 C0 ; 8i 2 I c;

8n 2 N nNT ; ð1lÞ

0 6 v0ic 6 bic; 8c 2 C0 ; 8i 2 I c; ð1mÞ

0 6 vnic 6 wpðnÞ

ic ; 8c 2 C0; 8i 2 I c;

8n 2 N n fNT [ 0g; ð1nÞ

gnc P 0; qn

c P 0; 8c 2 C ; 8n 2 N nNT ;

ð1oÞ

f nc 6 Un

c ; 8c 2 C ; 8n 2 N nNT ; ð1pÞXn2NT

pnRn P l; ð1qÞ

where

V n ¼ hn‘ þXi2I‘

wpðnÞi‘ P n

i‘

þXc2C

f pðnÞc þ en

c hnc þXi2Ic

wpðnÞic P n

ic �f pðnÞ

c

upðnÞc

!( ); 8n 2NT :

ð2ÞThe model is a multi-stage stochastic linear programwith recourse. Decisions at any node explicitly de-pend on the corresponding postulated outcome forthe random variables and directly impact on deci-sions at descendant nodes. The model minimizesthe conditional value-at-risk of the portfolio lossesat the end of the horizon, while it parametrically sets


a minimum target, l, for the portfolio’s expected re-turn over the planning horizon. Expectations arecomputed over the set of terminal states (leaf nodes).The objective value, f, measures the CVaR ofportfolio losses at the end of the planning horizon,while the corresponding VaR of portfolio losses(at percentile a) is captured by the variable z; see [21].

Eqs. (1b) and (1c) are the balance conditions foreach asset, in each market, at the first and subse-quent decision stages, respectively. Eqs. (1d) and(1e) impose the cash balance conditions at the firststage; the former for the base currency ‘ and the lat-ter for the foreign currencies c 2 C. In each case,availability of funds stems from initially availablereserves, revenues from asset sales, and amountsreceived through incoming currency exchanges inthe spot market. Correspondingly, the uses of fundscover the expenditure for the purchase of assets andoutgoing currency exchanges in the spot market. Noholdings in cash are allowed after portfolio restruc-turing. Hence, we do not need to explicitly modelthe interest rates in each market. Similarly, Eqs.(1f) and (1g) impose the state-dependent cash bal-ance conditions in every currency at subsequentdecision stages. These equations additionallyaccount for forward currency exchange contractsthat are decided at the predecessor node.

Eqs. (1h) and (1i) define the portfolio return andthe portfolio loss at leaf node n 2 NT, respectively.The constraints (1j) and (1k) define the portfolio’sexcess shortfall, yn = max[0,Ln � z], over the plan-ning horizon for each scenario. The constraints in(1l), (1m) and (1n) disallow short positions in theassets and ensure that sales cannot exceed the quan-tities in the portfolio at hand. Constraints (1o)ensure that currency transactions in the spot marketare nonnegative. Constraint (1q) sets a minimumtarget l on the portfolio’s expected return over theplanning horizon. By parametrically changing thistarget level we generate solutions that trade offexpected return against total risk exposure (as cap-tured in the objective function).

The final value of the portfolio at leaf noden 2 NT is computed in (2). This equation takes intoconsideration exogenously available cash and reve-nues from the liquidation of asset holdings in eachmarket. The contribution of foreign investments tothe total value of the portfolio accounts for thesettlement of any outstanding forward contracts.The residual amount is valued in terms of thebase currency by using the projected spot exchangerates.



ARTICLE IN PRESS

The constraints in (1p) impose limits on the cur-rency forward contracts at every decision state. Weconsider alternative hedging policies by appropri-ately setting these bounds:

ðiÞ Unc ¼

Xi2Ic

encðwn

icPnicÞ; 8c 2 C ; 8n 2N nNT ; ð3Þ

ðiiÞ Unc ¼

Xm2Sn

pmemc ðXi2Ic

wnicP

micÞ; 8c 2 C ; 8n 2N nNT ;

ð4ÞðiiiÞ Un

c ¼1; 8c 2 C ; 8n 2N nNT ; ð5ÞðivÞ Un

c ¼ 0; 8c 2 C ; 8n 2N nNT : ð6Þ

The first case permits a forward contract to cover upto the current value of total asset holdings in therespective currency, with the foreign holdings val-ued at the time that the forward contract is decided.This alternative ignores the fact that the value of theforeign asset holdings will change due to the uncer-tain returns and exchange rates during the period.In the second case, the level of a forward contractis bounded by the expected value (in units of thebase currency) of the respective foreign asset hold-ings in the portfolio at the end of the decision peri-od—thus, the expectation is taken over theoutcomes at the successor nodes—so as to reflectthe expected exposure of the foreign asset positions.In the third case, no restriction is explicitly imposedon the level of forward contracts, which are thentreated simply as alternative investment opportuni-ties. In this case, forward positions are allowedregardless of the value of asset holdings in therespective currency. The last case sets the currencyforward contracts identically equal to zero (in fact,we eliminate from the model the variables f n

c in thiscase). This case corresponds to totally unhedgedinternational investments.

The proportional transaction costs cic for theassets create proportional bid-ask spreads2cicP

nic; 8c 2 C0; 8i 2 I c; 8n 2 N nNT . The effective

purchase price for an asset in a cash balance equa-tion for node n 2 NnNT is P n

icð1þ cicÞ, while theeffective sale price for the same asset is P n

icð1� cicÞ.As defined in the formulation, the model permits adifferent transaction cost for each asset. In the com-putational tests we used a constant proportionaltransaction cost cic = 0.05% for all assets (i.e.,"c 2 C0, "i 2 Ic). For currency exchanges (bothspot and forward) a transaction cost 0.01% wasincorporated in the respective exchange rates.

Starting with an initial portfolio, the multi-stageportfolio optimization model determines optimal


decisions for the contingencies of the scenario tree.The decisions at each node of the tree specify notonly the allocation of funds across markets but alsothe specific asset holdings in each market. More-over, currency forward contracts are determinedso as to (partly) hedge the currency risk exposureof the foreign investments during the holding period(i.e., until the next portfolio rebalancing decision).Thus, a number of decisions that are usually treatedseparately are cast here in a unified framework. Sim-ple variants of the model can reflect alternative pol-icies for hedging currency risks with the use ofcurrency forward contracts. The model provides atestbed so as to analyze empirically its effectivenessin managing international portfolios of stock andbond indices, and to compare the performance ofrisk hedging alternatives.

5. Empirical results

We implemented the multistage stochastic pro-gramming model in the General Algebraic Model-ing System (GAMS) [4]. We solved single-stageand two-stage instances of the model. The aims ofthe numerical experiments are: (i) to investigatethe efficacy of the stochastic programming modelas a practical decision support tool for managinginternational investment portfolios, (ii) to test alter-native risk hedging strategies so as to assess theirrelative performance in controlling currency risk,and (iii) to contrast the performance of the two-stage stochastic programming model with that ofits single-stage counterpart.

We examine the performance of various decisionstrategies in static as well as in dynamic tests toidentify the most promising tactics. In static testswe compare the risk-return profiles (efficient fron-tiers) generated with appropriate variants of themodel at a certain point in time. The optimizationmodels were run with alternative bounds on cur-rency forward contracts as specified in Eqs. (3)–(6). The static tests considered portfolio selectionproblems. The initial portfolio involved only a cashendowment in the base currency—which the modelsapportioned optimally to the available assets. Themodels were repeatedly run for various levels ofminimum expected return, l. The solutions tracethe corresponding efficient frontiers of expectedportfolio return vs. the CVaR risk metric of portfo-lio losses (at the a = 95% percentile) over the plan-ning horizon. The efficient frontiers are determined



ARTICLE IN PRESS

in-sample; that is, with respect to the postulatedscenarios.

The static tests yield useful insights regarding thepotential of the various decision strategies withrespect to the postulated distributions (scenarios)at a certain point in time. However, this potentialdoes not necessarily materialize in practice. Weadditionally ran dynamic tests to assess the perfor-mance of the models in backtesting simulations.The models were run, on a rolling horizon basis,at each successive month in the period 04/1998–11/2001 (i.e., for a total of 43 months). Starting withan initial cash endowment in the base currency inApril 1998, each model was executed to decide theinitial portfolio composition. The statistics of therandom variables, computed from their observedmarket values during the previous 10 years, werematched in generating the scenarios. Each modelwas solved and the first-stage decisions wererecorded. The clock then advanced one month.The realized return of the optimal portfolio wasdetermined on the basis of the revealed marketprices of the assets and the exchange rates. Any out-standing currency forward contracts were settledand the resulting cash positions in each currencywere updated accordingly. A new set of scenarioswas then generated by matching the statistics of

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

-2.0%0 0.0% 2.0% 4.0%

CVaR ( α=95

Exp

ecte

d R

etu

rn

Fig. 2. Risk-return efficient frontiers with two-stage mode


the random variables to their estimates from theirmarket values during the previous ten years. Withthe new scenarios as input, and using the portfoliocomposition and cash positions resulting from theprevious decisions as a starting point, the modelwas solved again. The process was repeated for eachsuccessive month and the ex post realized returnswere recorded. Thus, the backtesting simulationsdemonstrate the actual returns that would havebeen realized had the decisions of the models beenimplemented during the simulation period 04/1998–11/2001.

5.1. Assessment of hedging strategies

Fig. 2 presents the efficient frontiers for alterna-tive allowable levels of currency forward positions— controlled by the respective bounds in (3)–(6).This figure shows the tradeoffs between the expectedreturn and the CVaR measure of losses during atwo-month planning horizon for optimal portfoliosof the two-stage stochastic program. The two-stagemodel used 15,000 scenarios composed of 150 jointrealizations of the random variables in the firststage, each followed by a set of 100 further out-comes in the second stage. These tests consideredportfolio selection problems in August 2001. The

6.0% 10.0%8.0% 12.0%

%) of Portfolio Losses

Strategy 1

Strategy 2

Strategy 3

No Hedging

ls for alternative hedging strategies (August 2001).



ARTICLE IN PRESS

scenarios were generated so that the statistical prop-erties of the random variables in each stage matchedtheir empirical values for the 10 preceding years.The results obtained at this time are typical of themodel’s observed behavior at other periods.

The efficient frontier of the optimal unhedgedportfolios—i.e., without currency forward con-tracts—is clearly dominated by the efficient frontiersof the optimal, selectively hedged portfolios. Incor-porating decisions for currency forward positions inthe portfolio management model improves the risk-return profile of the resulting optimal portfolios asindicated by the shift of the efficient frontier to theleft. That is, for any value of target expected return,the optimal hedged portfolios exhibit a lower levelof risk. This potential benefit of risk reduction isincreasing for more aggressive targets of expectedportfolio return.

The three strategies that permit currency forwardcontracts reflect alternative views of the selectivehedging approach, as they allow the hedge ratiosto be different—and indeed they come out to be dif-ferent—across currencies. The third strategy exhib-its the dominating risk-return profile, as it allowsunrestricted use of currency forward positions.The other two strategies restrict the use of currencyforward contracts so as to (partly) cover the expo-sure in foreign currency investments. The first strat-egy (Eq. (3)) limits the levels of currency forwardcontracts to the respective current values of foreigninvestments in the revised portfolio, while the sec-ond strategy (Eq. (4)) bounds the levels of such con-tracts to the expected value of the respective foreignasset holdings at the end of the decision period.These two strategies exhibited almost indistinguish-able risk-return profiles.

We applied the same hedging tactics in backtest-ing experiments to investigate whether their poten-tial in the static tests actually materializes inpractice. Fig. 3 contrasts the ex post realized returnsof the different hedging policies over backtestingsimulations covering the period 04/1998–11/2001.The results were generated by successively applyingthe respective two-stage models in each month ofthe simulation period and implementing each timethe first-stage optimal decisions of the model. Eachinstance of the model again used 15,000 scenarios(150 outcomes for the first stage, each associatedwith 100 further outcomes for the second period)that were generated as described above.

The first graph in Fig. 3 presents the results forthe minimum risk case—i.e., when the models sim-


ply minimize the CVaR risk measure at the end ofthe planning horizon without imposing a target onexpected portfolio return. The results in the secondgraph were generated when a target l = 2% forexpected return over the two-month horizonwas imposed at each instance of the two-stage port-folio optimization model (i.e., for an aggressiveinvestor).

In the minimum risk case, all three selectivehedging strategies resulted in essentially the sameperformance. Only the most liberal (third) strategyfell a little behind in the fall of 1998, but traced clo-sely the performance of the other two strategies inall other periods. The performance of optimalunhedged portfolios was not very different eitherin this case. Optimal hedged portfolios exhibitedonly a slight advantage in comparison to the opti-mal unhedged portfolios, in the minimum risk case,as they demonstrated slightly more stable returnpaths. The optimal portfolios were positionedalmost exclusively in short-term government bondindices throughout these simulations. The modelsselected diversified portfolios of short-term interna-tional bond indices that weathered the storm of theSeptember 11, 2001 crisis unscathed, and actuallygenerated profits during that period. That crisisaffected primarily the stock markets for a shortperiod and had no material impact on thebond markets, especially the international bondmarkets.

The differences in the performance of the alter-native hedging strategies are more pronouncedwhen we use a more aggressive target for expectedreturn, as shown in the second graph of Fig. 3. Inthis case all three selective hedging strategies dem-onstrate material benefits from the reduction ofcurrency risk through the use of currency forwardpositions for hedging purposes. Their realizedreturn paths over the simulation period are discern-ibly more stable than the corresponding path of theoptimal unhedged portfolios. The internationalhedged portfolios were affected much less thanthe undhedged portfolios during market down-turns (e.g., 08/1998, 01–02/2000, 04/2001, 08–10/2001).

Again, the first two strategies demonstrated verysimilar ex post performance; with the secondstrategy being a very slight favorite. The third strat-egy—with unrestricted positions in currency for-wards—lagged a bit behind, particularly in periodsof down markets. In these simulations the modelsselected portfolios that varied more substantially


Minimum risk case

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

Jul-9

8

Oct-98

Jan-99

Apr-99

Jul-9

9

Oct-99

Jan-00

Apr-00

Jul-0

0

Oct-00

Jan-01

Apr-01

Jul-0

1

Oct-01

Time period

Cu

mm

ula

tive

Wea

lth

Cu

mm

ula

tive

Wea

lth

Strategy 1 Strategy 2 Strategy 3 No_Hedging

Time period

Strategy 1 Strategy 2 Strategy 3 No_Hedging

Target expected returns µ=2% over the planning horizon

0.95

1.00

1.05

1.10

1.15

1.20

1.25

Jul-9

8

Oct-98

Jan-99

Apr-99

Jul-9

9

Oct-99

Jan-00

Apr-00

Jul-0

0

Oct-00

Jan-01

Apr-01

Jul-0

1

Oct-01

Fig. 3. Ex-post realized performance of alternative hedging strategies in two-stage models. The first graph corresponds to the minimumrisk case, while for the second graph the target expected return at the horizon is l = 2%.


ARTICLE IN PRESS

over time, in comparison to the runs for the mini-mum risk case, in their attempt to meet the highexpected return target. The optimal portfolios alsoinvolved sizable positions in the US stock index


for most of the simulation period, and thus didnot avoid the effects of the crisis in September2001—even if more mildly than the unhedged port-folios. The strategies with controlled currency hedg-



ARTICLE IN PRESS

ing consistently attained more stable (less volatile)return paths compared to the returns of unhedgedportfolios and also had lower loses in times of severemarket downturns.

The results of the static and the dynamic testsshow that benefits can be gained, in terms of riskreduction, by internalizing decisions for currencyforward positions within the portfolio managementmodels. Regardless of minor details on how per-missible currency forward contracts may be con-trolled in the portfolio optimization models, weobserve that the controlled use of forward con-tracts has a positive impact on reducing risk. Thestochastic programming models prove to be usefuland practical decision support tools for interna-tional portfolio management. They provide aflexible framework for incorporating alternativerisk hedging strategies in a dynamic decisionsetting.

We adopted the second strategy for the tests thatare presented below as it exhibited more effectiveperformance than the other hedging alternatives.Recall that in this strategy the allowable positionsin one-month currency forwards are bounded bythe expected value of asset holdings in the respectivecurrency during the same term.

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

-2.0% 0.0% 2.0% 4.0% 6.0%

CVaR ( α=95%

Exp

ecte

d R

etu

rn

Fig. 4. Efficient frontiers for single- and


5.2. Comparison of single- and two-stage models

We now turn to a comparative assessment of sin-gle- and two-stage variants of the stochastic pro-gramming models. We first examine theperformance of the models in static tests on August2001. The 15,000 scenarios generated for the previ-ous static tests were used as input in these tests aswell. We set up and solved two-stage and single-stage instances of the portfolio selection model.The single-stage model had the same horizon (twomonths) and used the same scenarios as the two-stage model. Thus, the two models used exactlythe same information content in terms of the out-comes of the random variables (scenarios), and opti-mized the same risk measure (CVaR of portfoliolosses at the end of the two-month horizon), startingwith the same initial portfolio—a cash endowmentin the base currency only. The only fundamental dif-ference was that, unlike the single-stage model, thetwo-stage model allowed portfolio rebalancing deci-sions during the interim month.

The resulting risk-return efficient frontiers of thetwo models are shown in Fig. 4. The two-stagemodel exhibits a dominating risk-return profile overthe common two-month horizon of the two models;

Two-stage Model

Single-stage Model

10.0%8.0% 12.0% 14.0% 16.0%

) of Portfolio Losses

two-stage models (August 2001).



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for any expected return target, the two-stage modelyields optimal solutions with lower risk than thesolutions of the respective single-stage model. Theseresults confirm the intuition that when the two mod-els have the same information content and the samehorizon, then the two-stage model should producesuperior results owing to its additional flexibilityto incorporate rebalancing decisions at an interme-diate stage.

The intermediate decisions permit the reconfigu-ration of the portfolio during the planning horizonfor each descendant node of the root in the scenariotree, i.e., in response to each specific outcome of thefirst stage. The same principle applies to extensionsof the model with additional decision stages, as longas the postulated outcomes (scenarios) remain thesame over the planning horizon.

At the minimum-risk end of the frontier the ben-efit of the two-stage model over the single-stagemodel seems marginal. This is because at the mini-mum risk case both models exhibit a ‘‘flight tosafety’’, i.e., they select very similar conservativeportfolios composed of short-term internationalbond indices. However, the potential benefits forrisk reduction by adopting the two-stage model incomparison to its single-stage counterpart increasefor increasingly aggressive (higher) targets ofexpected return that dictate the selection of riskierportfolios. In these cases, the flexibility of an interimreadjustment of the portfolio during the planninghorizon in response to changing economic condi-tions carries a higher incremental value.

The results indicate that for a given representa-tion of uncertainty, and a specific horizon, it is pref-erable to allow portfolio rebalancing in as manystages as captured in the scenario tree rather thanto aggregate decision stages. Of course, this model-ling choice has significant implications on the sizeand computational complexity of the resulting sto-chastic programs.

Finally, we contrast the performance of the sin-gle- and two-stage models in dynamic, backtestingexperiments with real market data. Again, the mod-els were set up and executed repeatedly at successivetime periods according to the procedure we haveexplained earlier. From each instance of the modelswe kept and implemented only the optimal decisionsfor the first stage. The single-stage model has a hori-zon of one month, while the two-stage model has ahorizon of two months, partitioned into twomonthly decision stages. The scenarios were gener-ated on a rolling horizon basis; at each month the


scenarios were generated on the basis of the histor-ical data during the prior 10 years.

In order to assess the effect of adding information(i.e., additional outcomes in the scenario set) weexperimented with the following variants of themodels:

• a two-stage model that used 15,000 scenarios(150 · 100) generated as described before,

• a single-stage model that used the 150 first-stageoutcomes of the two-stage model as its scenarioset,

• a single-stage model with a finer representationof uncertainty for its monthly horizon, comprisedof 15,000 scenarios.

The first two models share the same informationregarding potential outcomes in the month ahead,but the two-stage model incorporates additionalinformation for potential outcomes in the followingmonth. The third model uses a much finer represen-tation for the distribution of the random variablesin the month ahead.

The ex post realized returns of these models dur-ing backtesting simulations are shown in Fig. 5. Thefirst graph corresponds to the experiments with min-imum risk models, i.e., models that minimize therisk measure without any constraint on targetexpected return. The second graph represents theuse of more aggressive return targets in the models(a target expected return l = 1% over the monthlyhorizon of the single-stage models, and a targetl = 2% over the two-month horizon of the two-stage model). In all cases, the two-stage modelachieved superior performance, while the single-stage model with the limited set of 150 scenariosproduced the worst performance. Clearly, the addi-tion of information in the representation of uncer-tainty—either with more scenarios for the single-stage model, or with the extension of the horizonto consider further outcomes in a second decisionperiod in a two-stage model—resulted in perfor-mance improvements. Hence, there is an addedvalue to finer specifications of distributions (scenar-ios) in these stochastic programming models.

We now compare the performance of single- andtwo-stage models that both use 15,000 scenarios. Inthe minimum risk case, their performance is quitesimilar. They achieve quite stable growth paths,with slight losses in only few instances during thesimulations. As we explained earlier, this is a conse-quence of the selection of very similar portfolios


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Two-Stage Single-stage (15000 scenarios) Single-stage (150 scenarios)

Two-Stage Single-stage (15000 scenarios) Single-stage (150 scenarios)

Fig. 5. Ex-post realized performance of single- and two-stage models. The first graph corresponds to the minimum risk case, while for thesecond graph the target expected returns during the planning horizon are l = 1% for the single-stage, and l = 2% for the two-stagemodels.


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(composed of the most secure assets) by both mod-els. But, as expected, the differences in realized per-formance are more evident when higher targetreturns are imposed, forcing the selection of riskierportfolios. In this case, the two-stage model yieldsa clearly superior performance.


Figs. 6 and 7 compare the compositions of theoptimal portfolios for the single- and two-stagemodels throughout the simulation period. Observethat in the minimum risk case (Fig. 6) both modelsselect very similar portfolios. These consist primar-ily of positions in the short-term US bond index


Fig. 6. Compositions of selectively-hedged international portfolios (minimum risk case) during backtesting simulations. The first graphshows the portfolios selected by the single-stage model, while the second graph shows the portfolios selected by the two-stage model.


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and hedged positions in the short-term governmentbond indices of the other three countries; theseinstruments exhibited the most stable performanceover the backtesting period. For the more aggressivetargets of expected return, Fig. 7 shows that themodels resort to more diversified portfolios that


include also holdings of stock indices (particularlyin the US stock index). The optimal portfolios ofthe two-stage model are more diversified and morestable over time compared to those of the single-stage model, thus indicating less active portfolioturnovers. This can be attributed to the look-ahead


Fig. 7. Compositions of selectively-hedged international portfolios (aggressive portfolios) during backtesting simulations. The first graphshows the portfolios selected by the single-stage model, while the second graph shows the portfolios selected by the two-stage model.


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feature of this model that has a longer horizon andconsiders longer-term effects in comparison to themyopic model when deciding the portfoliocomposition.

Finally, we compute some measures to comparethe overall performance of the models. Specifically,


we consider the following measures of the ex-postrealized monthly returns over the simulation period:geometric mean, standard deviation, Sharpe ratio,and the upside potential and downside risk (UP)ratio proposed by Sortino and van der Meeer [23].This ratio contrasts the upside potential against a


Table 3Statistical characteristics of realized monthly returns

Two-stagemodel

Single-stagemodel (15,000scenarios)

Single-stagemodel (150scenarios)

Performance measures of monthly realized returns (aggressive

models)

Geometricmean

0.460% 0.432% 0.430%

Standarddeviation

0.0124 0.0139 0.0143

Sharp ratio �0.0120 �0.0307 �0.0314UP_ratio 0.9692 0.9501 0.8317

Performance measures of monthly realized returns (minimum risk)

Geometricmean

0.581% 0.579% 0.484%

Standarddeviation

0.0025 0.0027 0.0042

Sharp ratio 0.4132 0.3796 0.0201UP_ratio 12.451 11.135 10.377


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specific target (benchmark) with the shortfall riskagainst the same target. We use the risk-free rateof one-month T-bills as the benchmark. The UP_ra-tio is computed as follows. Let rt be the realizedreturn of a portfolio in month t = 1, . . . ,k of thesimulation, where k = 43 is the number of monthsin the simulation period 04/1998–11/2001. Let qt

be the return of the benchmark (riskless asset) atthe same period. Then the UP_ratio is

UP ratio ¼1k

Pkt¼1 max½0; rt � qt�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1k

Pkt¼1ðmax½0; qt � rt�Þ2

q ð7Þ

The numerator is the average excess return com-pared to the benchmark, reflecting the upside poten-tial. The denominator is a measure of downside risk,as proposed in Sortino et al. [24], and can bethought of as the shortfall risk compared to thebenchmark.

The performance measures from the simulationresults are reported in Table 3. The aggressive mod-

Table 4Size and solution times of models

Model Number of constraints Number of va

Single-stage (150 scenarios) 325 360Single-stage (15,000

scenarios)30,026 30,060

Two-stage (15,000 scenarios) 36,782 39,969


els exhibited the worst measures. Each model has alower geometric mean and higher standard devia-tion in the aggressive case than its minimum riskinstance. The Sharpe ratios also have small negativevalues for the aggressive models, and the UP_ratiosare small, indicating a low upside potential relativeto downside risk. The expected return targets(�12.6% annually) that have been used for the mod-els in the aggressive case proved overly ambitiousand unattainable during the particular simulationperiod. We used them simply to distinguish the rel-ative behavior of the models at different levels ofrisk tolerance. The aggressive models attained aver-age annual returns of 5.7% (two-stage model) and5.3% (singe-stage model) during the simulationperiod. The models fared much better in the mini-mum risk case, yielding average annual returns of7.2% (two-stage) and 6.0% (single-stage), while alsoexhibiting significantly lower volatility of returnsduring this period as can be seen from the lowstandard deviation values (see also Fig. 5). Thereported returns of the models are net of transactioncosts.

These results show that the two-stage modeldominated its single-stage counterparts; it consis-tently achieved the best performance in terms ofboth higher growth and greater stability of returns.In all measures, the single-stage model with thefewer scenarios had the worst performance, whilethe performance of the single-stage model with thefiner scenario set was fairly close to that of thetwo-stage model. This observation implies thatthere is incremental benefit from increasing infor-mation content in stochastic programs, with ahigher branching factor in the first stage but alsowith the addition of a subsequent stage. The ques-tion of the sensitivity of the results to the relativebranching factors at different stages of a stochasticprogram, considering also the implications of thisdecision on the size and computational complexityof the resulting stochastic programs, remains aproblem-dependent empirical issue.

riables Number of nonzeros Approx. solution time(seconds)

3860 0.1375,111 55

444,499 88



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Table 4 presents the size and computationaleffort required to solve the stochastic linear pro-grams. The reported solution times reflect an aver-age for typical problem instances. The modelswere solved with IBM’s Optimization SubroutineLibrary (OSL) on an IBM RS/6000 44P worsksta-tion (Model 170 with a 400 MHz Power 3 Risk pro-cessor, 1Gb of RAM, running AIX 4.3). Observethe almost proportional increase in solution timewith the increasing number of scenarios for the sin-gle-stage model, and the larger size and requiredsolution time for the two-stage model in comparisonto a single-stage model with the same number of sce-narios. These solution times are by no means pro-hibitive for realistic instances of the models withtoday’s available computing technologies. Theproblems can be solved much more efficiently byemploying specialized algorithms that exploit thestructure of stochastic programs, and especially byresorting to parallel computing systems (e.g., see[29]). Issues of computational efficiency are not ofprimary concern in this study.

6. Conclusions and further research

We developed a stochastic optimizationapproach for managing international portfolios offinancial assets in a dynamic setting and demon-strated its practical viability through extensive com-putational experiments using real market data. Weformulated a multi-stage stochastic programmingmodel to address active, international portfoliomanagement problems. The model provides a usefuland flexible decision support framework as itencompasses many practical features. It considersdecisions over multiple periods to capture decisiondynamics, it accommodates a multi-stage represen-tation of the stochastic evolution of the randomvariables by means of a scenario tree, and itaccounts for transaction costs.

Our implementation of the model controls theportfolio’s total risk exposure by minimizing theconditional value-at-risk (CVaR) of losses at theend of the planning horizon, while imposing a targeton expected return over the planning horizon. TheCVaR objective minimizes excess shortfall beyondVaR. CVaR is a coherent risk measure and is appro-priate for asymmetric distributions. So, its choice isconsistent with the asymmetric and leptokurticreturn distributions exhibited by market data ofinternational stock and bond indices, as well as bycurrency exchange rates. However, the model can


be easily modified to accommodate alternativeobjective functions to reflect the decision maker’srisk-bearing preferences, or to incorporate addi-tional practical constraints (e.g., managerial or reg-ulatory requirements).

We employed a flexible and realistic scenario gen-eration method based on principles of moment-matching. This method generates scenarios so thatkey statistics of the random variables match theirobserved values so as to closely approximate theirempirical distributions. The scenarios of interna-tional stock and bond index returns and exchangerates reflect the correlations, as well as the skewnessand excess kurtosis that these financial variablesexhibit in historical data. Through exhaustive tests,we verified that the no-arbitrage conditions weresatisfied for all scenario sets that we used in numer-ical experiments. We have not observed a case inwhich the scenario generation method failed to meetthe no-arbitrage conditions. Other appropriate sce-nario generation methods can also be used in con-junction with the stochastic portfolio optimizationmodel.

A contribution of this study is the extension ofthe stochastic programming model for internationalportfolio management to a multi-stage, dynamicsetting. Another contribution concerns the internal-ization of decisions for currency forward positionsin the portfolio management model, as means formitigating currency risk. The inclusion of explicitdecisions for currency forward contracts enablesthe determination of optimal selective hedging deci-sions to control currency risk exposure of foreigninvestments.

The model determines jointly the allocation ofcapital to international markets, the transactionsto achieve an optimal portfolio composition (i.e.,the specific asset holdings in each market), as wellas the levels of appropriate currency forward con-tracts to minimize the portfolio’s total risk. Thus,the asset allocation, the portfolio selection, andthe currency risk hedging decisions, that are tradi-tionally considered separately, are here cast in a uni-fied decision framework.

The stochastic programming model provides aflexible framework to assess alternative risk hedgingtactics. We used variants of the model to investigatethe performance of alternative decision strategies inextensive empirical tests, both in static and dynamicexperiments. We demonstrated that controlled useof currency forward contracts materially contrib-utes to the reduction of risk of international portfo-



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lios. Selective hedging strategies proved effective incontrolling risk and generating stable return pathsin backtesting simulations with real market data.Our results also demonstrated the incremental ben-efits that can be gained from the adoption ofdynamic portfolio optimization models instead ofmyopic, single-period models. Two-stage variantsof the model exhibited superior performance incomparison to corresponding single-stage models,both in static as well as in dynamic tests. The two-stage models produced more diversified and stableportfolios, with lower volatility of returns andhigher resilience during market downturns, andwith lower turnover compared to the decisions ofmyopic models. Moreover, our results showed thatfiner specifications of distributions (scenarios) ofuncertain input parameters resulted in improvedsolutions of the stochastic programs.

The multi-stage stochastic programming modelprovides the foundation for the implementationand empirical investigation of additional decisionstrategies. Building on the fundamental structureof the stochastic programming model, we havedeveloped extensions that incorporate differenttypes of options in the international portfolio man-agement model. We use the stochastic programmingframework to empirically examine the effectivenessof option-based strategies to control exposure todifferent risk factors. In [26] we focus on the useof straight options, and quantos, on internationalstock indices as means of mitigating market risk,while in [27] we incorporate currency options inthe stochastic programming model and contrasttheir risk hedging performance with that of currencyforward contracts.

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for international portfolio management q · random variables, dynamic adjustments of portfolio positions and currency hedging levels in response to changing information on prevailing