and the development of data mining and optimization techniques for selecting the appropriate kernel. In light of this research,it would also be interesting to see if the dominance of hyperbolic SVMs can beconfirmed in further empirical investigations on financial market return prediction.Moreover, if we consider the evidence for second moment nonlinear dependenciesfrom Sect. 8.2.4, another interesting avenue of future research could be to use SVMsin order to predict financial market volatilities.
Chapter 11
Introduction
International investing and trade has one unintended consequence: the creation ofcurrency risk which may cause the local currency value of a firm’s foreign receivables, liabilities or investments to fluctuate dramatically because of pure currencyspot movements. In their corporate risk management survey, [364] stated that industrial corporations rank foreign exchange risks as the most costly risk with 93%of firms reporting some kind of foreign exchange exposure, and, on average, firmsjudging between a quarter and a third of their revenues, costs and cash flows as being exposed to movements in exchange rates.Protection against foreign exchange risks may be achieved through internal or external hedging. Internal hedging refers to the exploitation of possibilities of changing the currency of cash outflows to better align them with inflows. Instruments include invoicing imports and exports in the home currency, contracting currency andforeign exchange clauses, speeding up or slowing down payments (leading or lagging), matching or netting claims, currency reserves, changing debt/claim structures,adjusting credit conditions and prices, etc. [109, 396]. This can be done by changing vendors, by relocating production facilities abroad or by foreign-denominateddebt, that is, by funding itself in the foreign currency. Such hedges can act as substitutes to currency derivatives [154, 307, 331]. External hedging instruments, besidesderivatives, include export factoring and forfaiting, international leasing, and foreign exchange insurance [109]. However, because hedging normally refers to theuse of off-balance-sheet instruments, that is forward-based and option-based contracts in the case of exchange rate hedging, only such currency derivatives shall beconsidered in the following.Academic literature on currency and commodity hedging has developed severaltheories to explain an individual’s incentive to hedge. The characteristics of the oldest concept go back to the works of [190,215,226] and are as follows. First, hedgingmade the hedger’s position certain. He sold his product forward and made deliveryunder the contract. The sole purpose for an individual to hedge was certainty, or inother words, reduction of income variability. Keynes [226] is explicit that hedgingeliminates risk:If this [futures] price shows a profit on his costs of production, then he can go full steamahead, selling his product forward and running no risk. [. . . ]. If supply and demand areC. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 11,c Springer-Verlag Berlin Heidelberg 2009
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balanced, the spot price must exceed the forward price by the amount which the producer isready to sacrifice in order to hedge himself, i.e., to avoid the risk of price fluctuations duringhis production period.
Hicks [190], although taking over Keynes’s theory speaks of the risk of a pricechange being reduced rather than avoided:The ordinary businessman only enters into a forward contract if by doing so he can hedge –that is to say the forward transaction lessens the riskiness of his position.
Kaldor [215] describes hedgers as[. . . ] those who have certain commitments independent of any transactions in the forwardmarket, [. . . ] and who enter the forward market in order to reduce the risks arising out ofthese commitments.
He therefore speaks of hedging as reducing rather than eliminating risks. The second characteristic of the early concept of hedging is that a hedger does not act onhis expectations, i.e., he always sold/bought forward irrespective of any expectations about the future price. Keynes, Hicks, and Kaldor do not say explicitly thathedgers have no expectations, but they do not give the expected price any role in thehedger’s actions. Regarding currency risk, [322] advocated a fully hedged currencyposition on the basis of foreign currency risk not offering a commensurate return.In what they deem a “free lunch,” they argue that as a result of its zero long-termexpected return, currency risk can be removed without the portfolio suffering anyreduction in long-term return. Therefore, many analysts incorrectly came to the conclusion that the availability of extreme liquidity (and hence low bid-ask spreads), along term zero return and an apparent lack of predictive power of academic currencymodels meant that investors and corporations should naively remove currency riskby implementing passive hedges back into the base currency as one could reducevolatility without paying for it.Froot et al. [144] took exactly the opposite approach and suggested that investorsshould do nothing and leave investments unhedged and unmanaged. He argued thatover long investment horizons, real exchange rates revert back to their means according to the theory of PPP and investors should maintain an unhedged foreigncurrency position. He also concludes that even over shorter horizons, the small transaction costs and counterparty risks associated with maintaining a currency hedgeadd up over time and cause the optimal hedge ratio to decline as the investmenttimeframe increases. Still, [144] does acknowledge that real exchange rates may deviate from their theoretical fair value over shorter horizons and currency hedgingin this context is beneficial in dampening volatility. One decade earlier, [105] challenged the Miller–Modigliani hypothesis – which states that the issue of hedgingand corporate risk management, in general, is closely linked to the irrelevance ofa firm’s financial policy in complete and frictionless capital markets [298] – in itspure form and demonstrated that hedging can add value if markets are imperfect andsomewhat disconnected from neoclassical parity conditions. In the 1990s, a moredetailed theoretical discussion evolved about these market imperfections and their
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effect on corporate practices for hedging foreign exchange risk and other financialrisks [144, 145, 307, 370]. Market imperfections that justify static risk managementpolicies are:• Information asymmetry [93, 94]: Management knows about the firm’s exposureposition much better than investors. Thus, the management of the firm, not itsinvestors, should manage exchange exposure.• Managerial interests [370]: Large portions of managers’ wealth are related tothe well-being of the firm in the form of income and, possibly, share ownership.Since managers are not fully diversified, they have an incentive to hedge the risksinherent in their position.• Differential transaction costs: The firm is in a position to acquire low-cost hedgeswhereas transaction costs for individual investors can be substantial. In addition,the firm has access to hedging tools that are not available to private investors.• Default costs [370]: If default costs are significant, corporate hedging would bejustifiable because it will reduce the probability of default. Perception of a reduced default risk, in turn, can lead to a better credit rating and lower financingcosts.• Progressive corporate taxes [284, 370]: Under progressive corporate tax rates,stable before-tax earnings lead to lower corporate taxes than volatile earningswith the same average value. This happens, because under progressive tax rates,the firm pays more taxes in high-earning periods than it saves in low-earningperiods.• Financing investments [144]: Firms should hedge to ensure they always havesufficient cashflow to fund their planned investment program.Apart from these reasons, recent empirical studies find that aspirations of risk management programs also instill a culture within an organization which improves riskbased decision making among employees [364]. However, risk management alsoincorporates several difficulties. First, it creates transaction costs associated with thepurchase of derivatives and insurance contracts. Second, potential upside in hedgedrisks may be eliminated. Third, establishing a risk management function creates direct organizational costs such as salaries for staff and costs of IT systems. Fourth,risk management also creates additional risks, for example, the possibility of tradinglosses. Finally, risk management is inherently difficult itself and therefore hard tocommunicate to investors and board members.The hedging policy to time currency and commodity markets with forward contracts has been referred to as selective hedging [376]. This technique supports theearly ideas of [422] who argued that hedging is essentially not a risk reduction technique only, but speculation in the basis which is given by the difference betweenthe spot and forward price. He makes clear that hedging in his sense contains aspeculative element when he says ([422], p. 320):Such discretionary hedging involving a firm in the practice of both hedging and speculation,[. . . ] seems to be especially prevalent among dealers and processors who handle commodities such as wool and coffee.
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For his theoretical work on hedging, Working is considered as a pioneer for manyresearchers arguing that individuals hedge to maximize profits, taking into accountexpected changes in prices. The association in the literature of hedging with the motive of an uncertain profit is a reasonable consequence of the fact that hedging doesnot result in certainty. If hedging leaves risk, then expectations become relevant;and if a loss from hedging is expected it would be unwise to hedge. On the otherhand, if a profit is expected it would appear advantageous to hedge. It is interestingto notice, that although financial markets are, per definition, not the core businessof nonfinancial companies, they seem to be confident enough to take views on thefuture development of the exchange rate that will turn out to increase the firms’ cashflows. Empirical evidence suggests that European firms are more inclined than USfirms to accept open foreign exchange positions based on exchange rate forecasts[47]. In fact, selective hedging is also the approach that most German companiesadopt [159]. In its corporate risk management survey, [364] have lately supportedthis result stating thatalthough risk management of FX rates [. . . ] is common, risks are rarely totally eliminated.
It must be concluded that firms implicitly reject the efficient market hypothesis in itssemi-strong version, by forming different expectations than those conveyed by forward prices.1 Under the efficient market view of forward prices, hedgers are unlikelyto profit consistently from forward pricing strategies, so risk aversion becomes theprimary motive for using forward markets. However, if hedgers have different expectations from those portrayed by forward prices, then their use of forward marketscan be for the purpose of increasing profits as a result of those differing expectations,rather than managing risk. A large body of research has derived optimal hedgingrules under this assumption [213, 242, 305, 321, 352]. However, one can also arguethat financial markets in reality display a high degree of information efficiency because so many private and professional market participants are continuously strivingto gain access to new and better information and to analyze the available information very carefully. This argument may be seen as a warning for selective hedgers.In order to achieve the goal of increasing cashflows, firms willingly accept the riskof currency losses due to the open positions. The speculative nature of the selectivehedging strategy has been pointed out very sharply by [252], pp. 198–199:In fact, to the extent that it includes a speculative element by factoring possible gains intothe hedging decision, [selective hedging] differs little from staking the assistant treasurerwith a sum of money to be used to speculate on stock options, pork bellies or gold.
Stulz [376] stretches that selective hedging will increase shareholder value ifmanagers have an informational advantage relative to other market participants.However, if managers believe they have informational advantages when they do not,selective hedging will merely result in an increase in the variability of cash flowsthat could potentially reduce shareholder value. In fact, [65] reports that economicgains to selective hedging are very small and less than for an alternative technical1 A strict interpretation of the efficiency hypothesis is not very plausible because in this casenobody would have an incentive to invest in the production or analysis of new information.
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trading strategy. He also finds no evidence that selective hedging leads to superioroperating or financial performance, nor is it associated with proxies for superiormarket information. These results emphasize that the main advantage of the selective hedging strategy, namely the granted flexibility for the decision-maker toactively react to market events, may inhibit a lot of risk. Finding an acceptable orpossibly optimal balance between risk and reward is therefore relevant for any firmpursuing a selective strategy to hedging its currency risk.In this context, the popular concept of maximizing income subject to a givenlevel of risk [279], or alternatively, maximizing the level of expected utility derived from the activities being considered [411] becomes relevant.2 The two approaches of hedging in order to reduce/eliminate risk [190, 215, 226] and hedgingin order to maximize profit [422] become special cases of Markowitz’s portfoliotheoretic approach when either risk or return is considered to be extremely important. While the mean, as the central moment of a probability distribution, iscommonly accepted as a measure of (average) return, the adequacy of differentkinds of risk measures has been and is still being discussed to an increasing extent[3,4,17,177,212,279,294,315,334,339]. Mean-variance analysis is consistent withthe expected utility theorem only when returns are normally distributed and thedecision-maker has a quadratic utility function [279]. Mean-variance results mayalso be acceptable if higher moments than the second are small or relatively unimportant to the decision-maker. However, mean-variance or mean-quantile-basedperformance measures, such as mean-value-at-Risk, may be misleading if decisionmakers want to incorporate products with nonlinear payoff profiles, such as options,in their hedging strategies. In this case, symmetrical measures of risk, like the variance or the standard deviation are not suitable, as option positions typically followan asymmetrical risk-return-profile [52, 53]. In addition, quantile-based risk measures are not able to account for asymmetry. Furthermore, it has become a stylizedfact in experimental economics that individuals perceive risk in a nonlinear fashion.Evidence suggests that most individuals perceive a low probability of a large lossto be far more risky than a high probability of a small loss [229]. This empiricalfinding expresses a preference towards positively skewed probability distributions.While above studies have enhanced our understanding of what motivates individuals or corporations to manage risk, less attention has been directed towardsunderstanding precisely how a firm should hedge. One problem is that currency riskis a rather broad and subjective concept. For example, the risk of a trader/speculatoris different to a hedger’s perception of risk. Furthermore, there may even existdifferent priorities about managing risk among hedgers. Consider the case of amultinational nonfinancial firm. First, since there naturally exist different kindsof foreign currency exposure, such as transaction exposure, translation exposureand economic exposure, all of which may be conflicting, firms may have different perceptions on what kind of exposure matters most to them. Second, hedging2 Expected utility theory states that the decision maker chooses between risky or uncertainprospects by comparing their expected utility values. This can be achieved by specifying a utility function G : R → R and then optimizing its expected value.
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decisions, are often based on currency managers’ fundamental beliefs as to howcurrency markets operate. Naturally, these beliefs will vary from individual to individual. As in all decision-making processes involving human beings, these beliefscan be engraved in cultural norms, traditions, or political factors. This means thatpresented with exactly the same analysis, two different people might come out withtotally different interpretations of what it means to them, and what it would requirein terms of action. Due to this paradigm it is practically impossible to come up witha universal answer to the question of how a firm should hedge. A large body ofresearch, rooted in economics, international finance, portfolio theory, statistics, andoperations research has explored the optimal hedging of foreign exchange risks. Dueto the very specific nature of foreign exchange risk, this literature is often tailoredto a very specific industry, such as agriculture, electricity, insurance, banking, or industrial firms exporting or importing goods to or from a foreign market. Dependingon the nature of the context, hedging models may therefore contain very specificviews, assumptions, and modeling techniques. For instance, hedging models maynot only differ by sources of uncertainty and instruments permitted, but also by theirformal descriptions. Since a comprehensive discussion would go beyond the scopeof the dissertation, we will only summarize the kind of literature that has motivatedour approach. In corporate contexts, it is often essential to use risk management toengineer cash flows. Hence, an important part of this thesis is the comparison andcombination of alternative instruments and tactics for managing currency risk. Wedistinguish between linear contracts where the payoff of the contract is a straightline (forwards), and nonlinear contracts, where the payoff is not (options). There isno consensus in the literature regarding a universally preferable strategy to hedgecurrency risk, although the majority of results indicate that currency forwards generally yield better performance than single protective put options [7, 121, 282].Important early insights into the simultaneous choice of linear and nonlinearinstruments were provided by agricultural risk-management literature. Lapan et al.[242] proposed a one-period model wherein utility-maximizing managers face pricerisk (but not quantity risk) and choose among both forwards and options when making their hedging decisions. Assuming normally distributed prices (which allowfor negative prices), they showed that the optimal hedging position will consistonly of forward contracts as options become redundant. Under the assumptionof nonincreasing risk-aversion and a specific strike price, they conclude that aspeculative forward position is generally accompanied by a long straddle optionposition. Lence et al. [251] extended this model into a multiperiod framework andfound that in the presence of unbiased forward and options prices, options play animportant role. Another extension of the [242] model can be found in [352] whonote that the optimal hedging position usually will include options, in addition toforward contracts. Reference [323] has analyzed the optimal position in forwardsand straddles by comparative-static analyses plus numeric examples for variousrisk-utility functions. His main result is that the optimal mix of linear and nonlinear instruments depends on the specific stochastic exchange rate process that isassumed by a firm, as well as on its degree of risk aversion. Volosov et al. [408]formulate a two-stage stochastic programing decision model for foreign exchange
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exposure management that is based on a vector error correction model in order topredict the random behavior of the forward, as well as spot rates connecting USDand GBP. The model computes currency hedging strategies which provide rollingdecisions of how much forward contracts should be bought and how much shouldbe liquidated. Backtesting results show that the computed strategies improve thepassive “spot only” strategy considerably. However their analysis does not includeoption instruments which are frequently used in practice. Topaloglou at al. [395] extended their own multistage dynamic stochastic programing model for internationalportfolio, as introduced in [394], by introducing positions in currency options tothe decision set at each stage in addition to forward contracts. They find empiricalevidence that portfolios with optimally selected forward contracts outperform thosethat involve a single protective put option per currency. However, it is also foundthat trading strategies involving suitable combinations of currency options have thepotential to produce better performance. In summary, while many of these modelsare able to explain certain empirical features of hedging practices (such as partialhedging, nonlinear or dynamic strategies, etc.), none appear to explicitly accountfor all observed features. In particular, most models appear to come up short in twoareas. First, most models do not predict substantial time-series variation in hedgeratios. Second, the models do not explicitly model the impact of managers’ marketviews. In our opinion, these two phenomena are likely closely related and it is forthis reason that we subsequently explore the potential impact of managerial viewson hedging behavior.We demonstrate how industrial corporations and investors can develop an approach to managing currency transaction risk, thereby adding real economic valuefrom currency fluctuations. Our objective is to find a possibly optimal combinationof linear and nonlinear financial instruments to hedge currency risk over a planningperiod such that the expected utility at the planning horizon is maximized. Werequire the goal function to address the conflicting empirical finding that firms dolike to try to anticipate events, but that they also cannot base risk management onsecond-guessing the market. Our analysis therefore argues that a way to understandcorporate hedging behavior is in the context of speculative motives that could arisefrom either overconfidence or informational asymmetries. For this purpose, we assume the firm to have different future expectations than those implied by derivativeprices. If investors believe that the currency is going to move in an unfavorable direction, what should they use as a tool to hedge? If they expect the currency to movefavorably, but are not entirely sure, should they use a different risk managementstrategy? When addressing these questions, one must recognize that traditionalmean-variance or mean-quantile-based performance measures may be misleadingif products with nonlinear payoff profiles, such as options, are used [52, 53]. Inaddition, it has become a stylized fact that individuals perceive risk in a nonlinearfashion. Preferences for positive skewness have been shown to hold theoreticallyby [10, 362], and empirically by, e.g., [229, 258, 372, 373]. For these reasons, wepropose to embed a mean-variance-skewness utility maximization framework withlinear constraints in a single-period Stochastic Combinatorial Optimization Problem (SCOP) formulation in order to find optimal combinations for forward and
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European straddle option contracts given preferences among objectives. We showthat the proposed SCOP is computationally hard and too difficult to be solved analytically in a reasonable amount of time. More specifically, the problem is NP-hardand cannot be solved in polynomial time which directly suggests that solutions canonly be approximated [152, 320]. Second, computation of the objective function isnonconvex and nonsmooth. The resulting optimization problem therefore becomesfairly complex as it exhibits multiple local extrema and discontinuities.In order to attack the given problem and derive near-optimal decisions withina reasonable amount of time, we propose a simulation/optimization procedure.Simulation/optimization has become a large area of current research in informatics,and is a general expression for solving problems where one has to search for thesettings of controllable decision variables that yield the maximum or minimumexpected performance of a stochastic system as presented by a simulation model[148]. For modeling the EUR/USD exchange rate a smooth transition nonlinearPPP reversion model is presented. The simulation model is very attractive in thepresent context and unique to our knowledge. It addresses both, the first and thesecond PPP puzzle, and provides a theoretically valid and visually intuitive viewon the corridor of future EUR/USD spot development. The key feature is a smoothtransition function [387, 403] which allows for smooth transition between exchangerate regimes, symmetric adjustment of the exchange rate for deviations above andbelow equilibrium, and the potential inclusion of a neutral corridor where the exchange rate does not mean revert but moves sideways. Another advantage is that itrequires estimating only two parameters, namely the speed of mean reversion andexchange rate volatility.For the task of optimization, we propose the use of a metaheuristic combinatorial search algorithm. Metaheuristics are modern heuristics which have beenfirst introduced by [162]. For a good and recent overview of metaheuristics incombinatorial optimization, we refer to [46]. In short, a metaheuristic refers toan intelligent master strategy that guides and modifies other heuristic methods toproduce solutions beyond those that are normally generated in a quest for localoptimality. The specific metaheuristic we use is scatter search, a generalized formof path relinking [166, 167] which has proven to be highly successful for a varietyof known problems, such as vehicle routing [18, 337, 338], tree problems [68, 423],mixed integer programing [165] or financial product design [81]. The seminal ideasof scatter search originated in the late 1960s. A first description appeared in [161],the modern version of the method is described in [239].In order to show our simulation/optimization model’s applicability in a practicalcontext, a situation is presented, where a manufacturing company, located in theEU, sells its goods via a US-based subsidiary to the end-customer in the US. Sinceit is not clear what the EUR/USD spot exchange rate will be on future transactiondates, the subsidiary is exposed to foreign exchange transaction risk under theassumption that exposures are deterministic. We take the view that it is important to establish whether optimal risk management procedures offer a significantimprovement over more ad hoc procedures. For the purpose of model validation,historical data backtesting was carried out and it was assessed whether the optimized
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mean-variance-skewness approach is able to outperform passive strategies such asunitary spot, forward, and straddle, as well as a mixed strategy over time. Thepassive strategies are nonpredictive since they do not allow the hedge to time-vary,but are fixed-weight strategies and therefore, do not use the history of market information. We compare the alternative strategies in dynamic backtesting simulationsusing market data on a rolling horizon basis. The strategies were evaluated bothin terms of their ex ante objective function values – as well as in terms of ex postdevelopment of net income. We find that the optimized mean-variance-skewnessstrategy provides superior risk-return results in comparison to the passive strategiesif earnings risk is perceived asymmetrically in terms of downside risk. The pureforward strategy is found to have the lowest return per unit of earnings risk, whereasthe straddle strategy and the 1/3 strategy reveal similar risk-return characteristics.Our results demonstrate that first, currency hedging is a hard problem from acomputational complexity perspective and optimal solutions can be approximated atbest. We demonstrate through extensive numerical tests the viability of a simulation/optimization model as a decision support tool for foreign exchange management.We find in our experiments that scatter search is a search method that is both aggressive and robust. It is aggressive because it finds high-quality solutions early inthe search. It is robust because it continues to improve upon the best solution whenallowed to search longer. Our approach to hedging foreign exchange transactionrisk selectively is based on exchange rate expectations, considers real market dataand incorporates flexible weights. It is found that the approach adds value in termsof reducing risk and enhancing income (which contrasts literature). Interestingly,it also contrasts the finding that currency forward contracts generally yield betterresults in comparison to options. The main reason for this is the forward’s abilityto transfer income across states of nature, concentrating the payments on the firm’sdownside. Thus, a smaller hedge ratio is required, which makes risk managementless costly.The chapter is organized as follows: in Chap. 12 we introduce the principles usedto express preferences over probability distribution functions, including the normaldistribution and the concept of expected utility maximization. In particular, Chap. 12introduces how preferences over probability distributions can be realized using financial instruments and defines the composition of the probability distributionfunction to be optimized. Chapter 13 gives the exact formal representation of thecurrency hedging problem and describes its computational complexity. Chapter 14describes the idea of simulation/optimization, including a formal description ofthe exchange rate simulation model, as well as a description of the metaheuristicoptimization routine proposed. Finally, Chap. 15 describes the real-world contextin which the model was applied, as well as the backtesting approach, evaluationprocedure, computational tests and the empirical results.
Chapter 12
Preferences over Probability Distributions
12.1 Currency Hedging Instruments12.1.1 ForwardA forward contract is an agreement between two parties to buy/sell an asset at acertain future time for a certain price. One of the parties to a forward contract assumes a long position and agrees to buy the underlying asset on a certain specifiedfuture date for a certain specified price. The other party assumes a short positionand agrees to sell the asset on the same date for the same price. A forward contracttherefore has a symmetric distribution of rights and obligations. The payoff from along position in a forward contract on one unit of an asset isFT = ωT − f0,T
(12.1)
where f0,T is the price in t = 0 for a forward contract with maturity T , and ωT isthe spot price of the asset at maturity of the contract. Long positions enable hedgersto protect themselves against price increases in the currency. Conversely, the payofffrom a short position in a forward contract on one unit of an asset isFT = f0,T − ωT
(12.2)
Short positions protect hedgers against price decreases. The fact that the payoffsresulting from long and short positions in forward contracts can be symmetricallypositive or negative is illustrated in Fig. 12.1 [200]. The gain, when the value of theunderlying asset moves in one direction, is equal to the loss, when the value of theasset moves by the same amount in the opposite direction. Since it costs nothingto enter into a forward contract, the payoff from the contract is also the firm’s totalgain or total loss from the contract.A forward contract fixes the nominal exchange rate and the price until settlement.Thereby, hedging prevents transaction exposure. Since a forward contract does nothedge changes in real foreign exchange rates, economic exposure still remains.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 12,c Springer-Verlag Berlin Heidelberg 2009
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Fig. 12.1 Payoff functions from a forward contract
There can still be some foreign exchange gain or loss. When real exchange ratesremain stable, long-term fixing of the nominal exchange rates can even create aneconomic exposure that would not have existed otherwise [332]. Since forwardscan be privately arranged, that is traded over-the-counter, the terms of such a forward, for example, the maturity date and the characteristics of the underlying asset,can be customized for the users. However, liquidity and transaction costs are greaterbecause of this customization [154].According to the principle of risk neutral valuation, the market value of a forwardcontract at t < T is a linear functionFt = e−r f (T −t) Et [ωT − f0,t ]
(12.3)
In t = 0 the forward price f0,T is determined such that its market value equals zero,F0 = e−r f ×T Et [ωT − f0,T ] = 0
(12.4)
It follows that the forward price f0,T equals the risk-neutral expected value of theexchange rate from a capital market view.f0,T = E [ωT ]
(12.5)
Forward contracts are widely used instruments in currency hedging since they areable to inexpensively transfer risk, and reduce cashflow volatility [40, 299]. In theory, a full forward hedge is optimal, if the firm does not have any concrete expectations about the future course of the exchange rate and believes in market consensus[190, 215, 226, 322]. On the other hand, if a firm dares to deviate from risk-neutralmarket expectations, this would imply a speculative engagement which must result in a forward position deviating from the full hedge [422]. Typically, a selectivehedger would enter into a full hedge if he expected an adverse future exchange rate
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development. The brighter his expectations on the future exchange rate become, themore he would reduce the hedge amount. If expectations were totally optimistic, theexposure would be left open.