is a multiobjective cubic function defined over the random variableω . As [200] notes on pp. 240–241, the expected rate of return is an ambiguousconcept. Unless otherwise stated, it will be used to refer to E[w] = μ . The maximum2 These limitations of the mean-variance approach were established at an early point by, e.g.,[12,75,129,182,183,279,330,391]. Rothschild and Stiglitz [347] showed from several perspectivesthat the usage of variance as a definition of risk is insufficient. More recent assessments of themean-variance approach include, e.g., [76, 77, 272, 293].
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value function for the firm is then determined for a given set of parameters (λ , κ ) > 0representing the degree for absolute risk aversion and absolute prudence. Typicalvalues of absolute risk aversion and prudence have been suggested to lie in therange of 1–5 [110, 111, 290].Equation 12.21 is independent of a firm’s wealth level. In particular, a change inwealth just causes a parallel shift for the model which will not affect the risk attitudeand the choice behavior of the firm. For symmetric distributions or distributions nothighly skewed, such that E[(v − E[v])2 ] > cE[(w − E[w])3 ], the model will be riskaverse. However, for highly positively skewed distributions, such that the skewnessof the distribution of derivative payoffs overwhelms the variance of the distributionof transaction exposure , i.e., E[(v − E[v])2 ] < cE[(w − E[w])3 ], the model will beruin-averse.
Chapter 13
Problem Statement and ComputationalComplexity
13.1 Problem StatementIn order to find possibly optimal combinations between spot, forward, and Europeanstraddle option contracts, it is proposed to embed the three-moment utility functionas formulated in Definition 12.9 in a single-period stochastic combinatorial optimization problem (SCOP) framework with linear constraints. Decisions are madesolely at t = 0 and cannot be revised in subsequent periods. The decision horizon isfinite and set to T = 1. Due to the current popularity of multiperiod financial models, a single-period description of the problem might be considered as disputable.In a dynamic modeling context, decisions are optimized in stages, because uncertaininformation is not revealed all at once. In our context this would require building anintegrated dynamic simulation model that additionally describes the future development of the forward rates as well as the option premiums over the next 12 months ina stochastic manner. Obviously, such an approach would be much more difficult tomodel than our single-factor approach, contain significantly higher model risk (threerisk factors plus interactions), and would require more computational resources. Itis therefore doubtful if such a model would indeed be advantageous, especially ifwe consider the underlying decision context. In practice, firms’ currency hedgingdecisions are hardly enforced by speculative behavior towards future price developments of hedging vehicles. Instead, they rather concentrate on reward/risk estimatesthat depend solely on the underlying risk factor, i.e., the exchange rate in our case.Any further assumptions or risks are usually avoided. Hence, although a multiperioddecomposition of reward and risk poses an interesting theoretical problem, we arguethat it is less relevant in practice. Therefore, only forward rates and option prices asknown in t = 0 are relevant. Future prices of hedging vehicles are irrelevant andthus, not considered. The single-period SCOP is stated as follows:1Problem 13.1. (BEST HEDGE). Given a probability space (Ω , Σ , P), where Ω isthe domain of random variables ω (typically a subset of Rk ), Σ is a family of events,1 For the general definition see [216].
C. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 13,c Springer-Verlag Berlin Heidelberg 2009
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that is subsets of Ω , and P is a probability distribution Σ with P(Ω ) = 1. Consideralso a finite set S of decision variables x. S is typically a subset of Rn . Given a finiteset of feasible solutions x, a real-valued utility function U(v(x, ω ), w(x, ω )) of thetwo variables (x, ω ) ∈ (S, Ω ) and constraint functions Hi , i = 1, 2, . . . , m mapping(x, ω ) ∈ (S, Ω ) to R findmax U(v(x, ω ), w(x, ω ))
(13.1)
x∈s
subject to
Hi (w, ω ) ≤ 0,
i = 1, . . . , m
where, U(v(x, ω ), w(x, ω )) is given byU(v, w) = E[w] − {λ Var[v] − κ Sk[w]}
(13.2)
= E[w] − {λ E[(v − E[v])2] − κ E[(w − E[w])3]}according to Definition 12.9, andT
v(x, w) = ∑ zt × (xts × ωt + xtF × f0,t + xtO × X0,t )t=1T
w(x, ω ) = ∑ zt (xtF × (ωt − f0,t )
(13.3)
t=1
+ xtO × (|ωt − X0,t | − p0,t ωt=0 ) − c × (xtF + xtO ))according to Definition 12.1.Single-period SCOP formulations have been originally proposed in the context ofmathematical programing applied to SCOPs, and this field is also called in the literature stochastic integer programing (SIP), a subset of the broader field of stochasticprograming [38]. Surveys on SIP include [181,225]. Single-period SCOPs are characterized by the fact that decisions, or equivalently, the identification of a possiblyoptimal solution is done before the actual realization of the random variables. Thisframework is applicable when a given solution may be applied with no modifications (or very small ones) once the actual realization of the random variables areknown.Global optimization is the task of finding the absolutely best set of decision variables to optimize its objective function. In general, there can be solutions that arelocally optimal but not globally optimal.Definition 13.1. A local maximizer x∗ of U(v(x, ω ), w(x, ω )) is a point such thatthere exists a neighborhood B of x∗ with:U(v(x∗ , ω ), w(x∗ , ω )) ≥ U(v(x, ω ), w(x, ω )), ∀x ∈ B
(13.4)
Many local maxima may exist with substantially different function values. For problems with multiple maxima we may be interested in finding the best maximum.
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Definition 13.2. The global maximization problem for a function U(v(x∗ , ω ),w(x∗ , ω )) ≥ U(v(x, ω ), w(x, ω )) is to find x∗ such that:U(v(x∗ , ω ), w(x∗ , ω )) ≥ U(v(x, ω ), w(x, ω )), ∀x ∈ S
(13.5)
All practical optimization methods, whether local or global, are iterative in natureand typically proceed from a starting point, which is an estimate of x∗ , via a sequence of points with increasing function value, until some termination condition issatisfied. Local optimization problems can be solved more easily than global onessince a local solution can be characterized by computable information at x∗ (positivedefiniteness of the Hessian and zero gradient), whilst for the case of a global optimum no such criteria exist, in general. The aim of global optimization is to find thepoints in S for which the function reaches its highest value, the global maximum.In order to investigate the computational complexity of Problem 13.1, we focuson the theory of NP-completeness [152, 320]. This theory is designed to be appliedonly to decision problems, i.e., problems whose solution is either “yes” or “no.”The standard formulation of a decision problem consists of two parts: the first partspecifies a generic instance of the problem in terms of various components such assets, graphs, functions, numbers, etc. The second part states a yes/no question askedin terms of the generic instance.Problem 13.2. (BEST HEDGE Decision Problem). Given a probability space(Ω , Σ , P) a finite set of feasible solutions x, a real-valued utility function U(w(x, ω ))of the two variables (x, ω ) ∈ (S, Ω ), and an integer bound B, is there a solution xsuch that(13.6)u(x) := EP (U(v(x, ω ), w(x, ω ))) ≤ B?(That is, is there a vector x such that a given utility target level can be realized?)
13.2 Computational Complexity Considerations13.2.1 Complexity of Deterministic Combinatorial OptimizationIf the exchange rate over the planning horizon was known today, then Problem 13.1would correspond to a deterministic combinatorial optimization problem (DCOP).All information would be available at the decision stage and could be used by anoptimization algorithm to find a possibly optimal solution. The concrete applicationof a solution found would lead exactly to the cost of the solution as computed bythe optimization algorithm. Therefore DCOPs are also considered static problemsbecause from the point of view of the decision maker, there is nothing else to bedecided after the optimization took place.Note that in Problem 13.1 the size of the choice set has already been reduced bypermitting only two kinds of hedging instruments. The size of the true choice set inProblem 13.1 is determined by the number of permitted hedging instruments n, the
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degree of discretization d, the speculation constraint, and the length of the planninghorizon T . In absence of speculation and possible budget constraints, a collection ofm = T × (n × (d + 1))! binary decisions is given, each regarding the purchase of acombination of instruments at price p. With budget I the firm faces a subset of I/pcombinations out of the complete set of binary decisions. The more budget is at thefirm’s disposal, the more “optionality” can be added to standard forward hedges,and thus the more combinations must be considered. However, absence of a budgetconstraint does not mean that optionality can be added infinitely. It is the speculationconstraint that restricts the firm in purchasing more forwards/options than are actually required to close the open positions. The speculation constraint therefore represents a bound on the number of combinations to be considered. At the maximum,the firm must answer m binary decision problems which takes time proportional to(n × d)! which is not a polynomial bound.Proposition 13.1. The deterministic version of Problem 13.1 is NP-hard as theMINIMUM COVER Problem is a special case of the deterministic decision versionof Problem 13.1.Proof. The result can be proven in a similar way as [156] prove their ConsumerProblem to be NP-complete. This is done by reducing the problem MINIMUMCOVER ([SP5]) as represented in [152], p. 222, to the deterministic version of Decision Problem 13.1.Problem 13.3. (MINIMUM COVER). Given a collection C of subsets of a finiteset S, and a positive integer K ≤ |C|, does C containa cover for S of size K or less,that is, a subset C ⊆ C with C ≤ |K| and such that c∈C c = S?Let there be given an instance of MINIMUM COVER: A finite set S ≡ {1, . . . , r}of length |S| = r, a collection of subsets C = {S1 , . . . , Sq } of length |C| = q, and apositive integer k ≤ q. Let (yi j )i≤q, j≤r denote the incidence matrix, namely yi j = 1if j ∈ Si and yi j = 0 if j ∈ Si .We now define the associated hedging problem. Let n = q. For i ≤ n, let pi = 1,and define I = t. Next, define u(x) by u(x1 , . . . , xn ) = Π j≤r ∑i≤n yi j xi . Finally, setū = 1. A bundle satisfies ∑i≤n pi xi ≤ 1 and u(x1 , . . . , xn ) ≥ ū if and only if ∑i≤n xi ≤ tand ∑i≤n yi j xi ≥ 1 for every j ≤ r. In other words, the firm has a feasible bundlex ≡ (x1 , . . . , xn ) obtaining the utility of 1 if and only if:1. No more than t products of 1, . . . , n are purchased at a positive quantity at x, and2. The subsets Si corresponding to the positive xi form a cover of S = {1, . . . , r}.The construction above can be performed in linear time. It is then left to showthat we have obtained a legitimate utility function u. Continuity holds becausethis is a well-defined function that is described by an algebraic formula. Sinceyi j ≥ 0, u is nondecreasing in the xi ’s. We turn to prove that it is quasi-concave.If there exists j ≤ r such that yi j = 0 for all i ≤ n, u(x1 , . . . , xn ) = 0, and u is quasiconcave. Let us therefore assume that this is not the case. Hence u is the productof r expressions, each of which is a simple summation of a nonempty subset of{x1 , . . . , xn }. On the domain {x|u(x) > 0}, define v = log(u). It is sufficient to show
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137
that v(x1 , . . . , xn ) = ∑ j≤r log(∑i≤n yi j xi ) is quasi-concave. Obviously, for every j ≤ r,log(∑i≤n yi j xi ) is a concave function, and the sum of concave functions is concave.This completes the proof of Proposition 13.1.Thus (assuming P = NP), it is impossible for a nondeterministic Turing machineto verify in polynomial time whether there exists a bundle of instruments such thata firm can obtain a given level of utility without violating its speculation constraint.The result suggests that it is hard to maximize a utility function over a large choiceset – even in the absence of uncertainty. As rational as investors can possibly be,it is unlikely that they can solve, in their minds, problems that prove intractablefor computer scientists equipped with the latest technology. Correspondingly, it isalways possible that a firm will fail to even consider some of the bundles available toit. In addition, utility maximization may not give an answer on how firms choose abundle in the feasible set. It follows, that one cannot simply teach firms to maximizetheir utility functions.
13.2.2 Complexity of Stochastic Combinatorial OptimizationRealistic utility maximization problems, such as Problem 13.1, are likely to be burdened with two sources of difficulty: first, the utility function may not be knownfor many bundles that have not been consumed. In this context, psychological literature suggests that people do not seem to be particularly successful in predictingtheir own well-being as a result of future consumption. That is, consumers do notexcel in affective forecasting (see [155, 214]). Firms may be uncertain about theirutility functions, and they may learn them through the experience of hedging. In thissense, a firm is faced with a familiar trade off between exploration and exploitation:trying new options in order to gain information and selecting among known optionsin an attempt to use this information for maximization of well-being. By contrast,Problem 13.1 ignores this difficulty of learning the utility function. We assume thatthe utility function is given, as an easily applicable formula, and that, given a particular bundle, there is uncertainty regarding the utility derived from it due to theexistence of uncertainty about the future exchange rate.One possibility with problem formulations involving uncertainty is to describeuncertain information by means of random variables of known probability distributions. Under this assumption, the optimization problem is stochastic, and the objective function strongly depends on the probabilistic structure of the model. Theadvantage of using SCOPs over DCOPs is that the solutions produced may be moreeasily and better adapted to practical situations where uncertainty cannot be neglected. However, this comes at a price: first, for a practical application of SCOPs,there is the need to assess probability distributions from real data or subjectively,which is a task that is far from trivial. Second, the objective function is typicallymuch more computationally demanding in SCOPs than in DCOPs. Although theproblem is formulated as a single-period SCOP, one can distinguish a time beforethe actual realization of the random variables, and a time after the random variables
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are revealed, because the associated random events happen. Hence, our problemformulation is consistent with the probabilistic combinatorial optimization problem (PCOP) framework as presented in [34, 35]. In PCOPs, one is confronted witha valid probabilistic data set in the first stage. In a second stage, the actual data setmaterializes. There are two possible strategies for solving this sort of problem: reoptimization and a priori optimization which have both been shown to be NP-completeproblems [34, 35].
13.2.2.1 Computational Complexity of ReoptimizationIn the above application, one finds that after solving a given instance of a combinatorial optimization problem, it becomes necessary to solve repeatedly many variationsof the instance solved originally. The most obvious approach in dealing with suchcases is to attempt to solve optimally (or near-optimally) every potential instance ofthe original problem. Such an approach is called a reoptimization strategy. Similarto [34], p. 91, who considered the probabilistic traveling salesman problem (PTSP),it can be shown that the proof of this problem is straightforward.Problem 13.4. (PTSP1). Given a set of instances D from a complete graph G =(V, E), |V | = n, a cost f : E → R, probability p(d) that the single instances materialize and a bound B, does there exist a tour τ such thatE[Lτ ] = ∑ p(d)Lτ (d) ≤ B?
(13.7)
d⊆D
If every possible subset of the node set V may or may not be present on any giveninstance of the optimization problem then there are 2n possible instances of theproblem, namely all the possible subsets of V . Hence, one has to compute O(2n )terms, each of which requires the evaluation of a classical TSP problem which isknown to be NP-complete (see Problem [ND22] in [152], pp. 211–212). Thus, evenin the simple case where we assume binary scenarios, an exponential number ofclassical TSPs has to be solved. Our underlying problem represents a generalizationof this idea including stochastic demands which are not only binary (demand of oneunit with a certain probability), but can be any random variable. Nevertheless, theproblem size will be exponential in n.
13.2.2.2 Computational Complexity of A Priori OptimizationThe reoptimization approach suffers from two disadvantages. Since the combinatorial optimization problem considered is NP-complete, one has to solve exponentiallymany instances of a hard problem. However, in financial applications it is usuallynecessary to find a solution to each new instance quickly, but one might not havethe required computing power or other resources for doing so. A priori optimization is a strategy that differs from reoptimization (see [35, 204, 243]). Given a set
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139
of first-stage probabilistically valid input data, one optimizes a priori over this datain order to be able to efficiently modify this solution as needed to fit the actuallyrealized second-stage data. Let D denote the first-stage probabilistically valid dataset and X(D) denote the a priori solution on D. Furthermore, let M denote a modification strategy which modifies the a priori solution so that it fits the second-stagerealized data. The value of the solution output by M for an a priori solution X(D)and a materialized valid data set d ⊆ D is then M(X(D), d). The a priori optimizationrule requires that we obtain an a priori solution X(D) over D such that the expectedoutcome of the modification strategy is minimizedE[M] = ∑ p(d)M(S(D), d)
(13.8)
d⊆D
It is important to note three points in this context. First, PCOP problems, such as thePTSP, differ with respect to the particular modification strategy M. Second, althoughthe design process of solving a PCOP problem is a two-stage procedure, the robustness criterion of an optimum expectation pertains only to the second stage outcome.Third, optimizing E[M] essentially requires probabilistic analysis of the modification strategy M over a distribution of input data. The decision problem of the a priorioptimization version of the PTSP can be expressed as follows.Problem 13.5. Given a set of instances D from a complete graph G = (V, E),|V | = n, a cost f : E → R, probability p(d) that the single instances materialize,an updating method M, and a bound B, does there exist a tour τ of length (“cost”) Lsuch thatE[LM(τ ) ] = ∑ p(d)LM(τ ) (d) ≤ B?(13.9)d⊆D
In other words, the goal is to minimize the “weighted average” over all probleminstances of the values LM(τ ) (d) obtained by applying the updating method M to thea priori solution t.Theorem 13.1. [34]. Problem PTSP2 is NP-complete.Noticeably, no general approximation technique has emerged under the PCOPframework. This is mainly due to the fact that PCOP concerns probabilistic analysis of a modification strategy over a distribution of input data, given an a priorisolution. Therefore, analysis is always heavily dependent on the characteristics ofthe modification strategy used, as much as on the structure of the underlying deterministic problem studied under the PCOP framework.
13.2.3 Objective Function CharacteristicsApart from structural reasons concerning the problem as a whole, the decision toincorporate skewness in evaluating hedging instruments makes the optimal selection procedure more complicated than a mean-variance based evaluation procedure.
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Whereas, the latter one implies evaluating a quadratic function, the mean-varianceskewness objective function is cubic, and practical problems arise when a nonconvex and nonsmooth objective function has to be optimized.2 Deterministic gradientsearch methods might be comfortable to solve the optimization problem, as longas the considered measures have sufficient differentiability properties and the function is monotonically in- or decreasing. However, the proof of such differentiabilityproperties can be rather difficult for random variables with a discrete probabilitydistribution. In addition, the mean-variance-skewness function may have multiplefeasible regions and multiple locally optimal points within each region. Thus, usingderivative or gradient information in order to determine the direction in which thefunction is increasing (or decreasing) may be time-consuming. On a one-processorcomputer, a gradient algorithm would have to be started many times with a different set of decision variables (“brute-force”), as one might be frequently stuck in alocal maxima and the situation at one possible solution gives very little informationabout where to look for a better solution. Many methods used to circumvent thisproblem have been restricted to goal programing or linear programing techniques.For example, [240] gave a goal programing procedure that performs portfolio selection based on competing and conflicting objectives by maximizing both expectedreturn and skewness while minimizing the risk associated with the return (i.e.,variance). Similarly, [253] provided a goal programing algorithm to solve a meanvariance-skewness model with the aid of the general Minkovski distance. Divergingfrom previous studies, [414] transformed the mean-variance-skewness model intoa parametric linear programing problem by maximizing the skewness under givenlevels of mean and variance. Likewise, [265] also transformed the mean-varianceskewness model with transaction costs into a linear programing problem and verifiedits efficiency via a numerical example. However, the main disadvantage of these algorithms is that they generally converge slowly, if at all [224]. It is impractical andinefficient to exhaustively enumerate all of the possible solutions and pick the bestone, even on a fast computer. Furthermore, most existing studies only present somenumerical examples with artificial data.
∈ S ⇒ α x + (1 − α )x ∈ S, 0 ≤ α ≤ 1. A function f (x) is convexon S if x , x ∈ S, 0 ≤ α ≤ 1, f (α x + (1 − α )x ) ≤ α f (x ) + (1 − α ) f (x ). In other words, the linejoining x and x is never below the function value at that point. The fact that the objective functionis nonconvex also implies that it is nonlinear, as it is a well known fact that a linear function isalways convex (and concave).
2 A set S ⊆ Rn is convex if: x , x
Chapter 14
Model Implementation
14.1 Simulation/OptimizationIn the absence of a tractable mathematical structure of Problem 13.1, we study thebehavior of heuristics, “quick and dirty” algorithms which return feasible solutionsthat are not necessarily optimal. In particular, we apply a methodology called Simulation/Optimization which is a general expression for solving problems where onehas to search for the settings of controllable decision variables that yield the maximum or minimum expected performance of a stochastic system as presented by asimulation model [146, 148]. For a compact picture of the field, see the reviews ofthe Winter Simulation Conference [147, 149].Given the analogies between the TSP and the PTSP, it is reasonable to expectthat, like in the TSP, a good heuristic for the problem may be obtained by theintegration of1. A solution construction algorithm generating candidate solutions2. A local search algorithm, which tries to improve as much as possible the candidate solutionThe sequence construction and improvement of a solution is repeated several timesuntil a good solution or some other termination criterion is not satisfied. With respect to Fig. 14.1, this involves running a simulation for an initial set of values,analyzing the results, changing one or more values, rerunning the simulation, andrepeating the process until a satisfactory (optimal) solution is found ([123], p. 13).This process can be very tedious and time-consuming since not all of these valuesimprove the objective function. The accuracy of the results therefore depends on thetime limit granted for searching, the number of trials per simulation, the number ofdecision variables, and complexity of the objective function. It has been shown thateven when there is no uncertainty, optimization can be very difficult if the numberof decision variables is large, and little is known about the structure of the objectivefunction. By definition, the main difference between the PTSP requiring simulationfor estimating the objective function and the TSP with an exactly computable objective function is that, in the first-mentioned case, it is not possible to decide withC. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 14,c Springer-Verlag Berlin Heidelberg 2009
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Optimization ModuleDetermine a newset of values fordecision variables
Simulation ModuleGenerate randomnumbers in orderto simulateexchange ratepaths
Calculateexchange ratedependentcashflows
Is objectivebetter thanpreviousones?
Performanceestimate
Yes
Memorize newbest result
Yes
Stop and prompt tocontinue
NoMax timeor numberofsimulationsreached?No
Fig. 14.1 Coordination of optimization and simulation
certainty whether a solution is better than another one. This can only be tested bystatistical sampling, obtaining a correct comparison result only with a certain probability. The PTSP therefore adds an additional complication to the TSP becausethe performance of a particular candidate solution cannot be evaluated exactly, butinstead must be estimated. It may be therefore not possible to conclusively determine if one candidate solution is better than another one which makes it difficultfor optimization algorithms to move in improving directions. Theoretically, one caneliminate this complication by generating so many replications, or such long runs,at each candidate solution that the performance estimate has essentially no variance.In practice, however, this could mean that very few alternative candidate solutionswill be explored due to the time required to simulate each one ([27], p. 488).
14.2 Simulation Model14.2.1 DatasetsGiven its launch on 1 January 1999, when the Euro became the currency for 11member states of the European Union, the Euro exchange rate has a rather short history. As a result, it is not straightforward to scrutinize the evolution of the Euro
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143
over this period against longer term trends. In order to overcome this obstacle,it has become common practice to use a proxy measure for the Euro for the period before its actual existence. Either the DEM [80], which was de facto the anchor currency among the European currencies participating in the exchange ratemechanism, or a “synthetic” Euro exchange rate [309], i.e., a weighted averageof the Euro legacy currencies has been commonly used. We decided for the firstalternative and construct the EUR/USD series for the period from January 1975to December 1998 by using the fixed EURP/DEM conversion rate agreed in 1998(1 EUR = 1.95583 DEM). From January 1999 until March 2007, we use the regularEUR/USD exchange rate. The data we use are historical monthly averaged nominalexchange rate data as obtained from Bloomberg which total to 387 values.
14.2.2 Component 1: EquilibriumAccording to economic theory, exchange rates are determined by relative price differentials in the long run. The relationship between price indices and the exchangerate is referred to as PPP which is one of the central doctrines in internationaleconomics (see Sect. 3.1). PPP is expressed by the price differential between twocountries, which is given by the differential between the consumer price index (CPI)in the United States and the CPI of the Eurozone. Monthly CPI data was obtainedfrom the OECD database for the same period of time as above. In order to adjust forthe level of the exchange rate, the price differential was multiplied by the respectiveaverage spot rates over the observation period1 TPt∗St(14.1)PPPt =∑T t=1Pt
14.2.2.1 Visual InspectionThe EUR/USD exchange rate series and the equilibrium rate as determined by (14.1)is shown in Fig. 14.2.Visual inspection seems to support that the EUR/USD real exchange rate oscillates around a mean in the long run. Between the January 1999 and October 2000,the EUR depreciated by around 20% in nominal effective terms. This reflected a significant weakening of the EUR against the currencies of major trading partners suchas the US dollar (−26%). After reaching a trough in October 2000, the EUR bottomed out. Following a period of protracted weakness, the EUR moved in 2002 ontoa steady recovery track to trade by March 2007 well above the levels observed at itslaunch, before reverting again somewhat in 2005 and also, well above equilibrium.The fact that rigorous statistical tests, such as the conventional unit root test, commonly fail to confirm such mean reverting properties [231, 359] should not preventus from drawing inferences from PPP analysis. For example, if a unit root would
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Fig. 14.2 EUR/USD equilibrium
indeed mimic the true data generating process of the real exchange rate, it wouldbehave like a