Ann Oper Res (2018) 266:35–69 https://doi.org/10.1007/s10479-017-2645-6 S.I.: ANALYTICAL MODELS FOR FINANCIAL MODELING AND RISK MANAGEMENT 1 1 Jörgen Blomvall · Jonas Ekblom Published online: 1 November 2017 © The Author(s) 2017. This article is an open access publication Abstract We develop a stochastic programming framework for hedging currency and inter- est rate risk, with market traded currency forward contracts and interest rate swaps, in an environment with uncertain cash ﬂows. The framework captures the skewness and kurtosis in exchange rates, transaction costs, the systematic risks in interest rates, and most importantly, the term premia which determine the expected cost of different hedging instruments. Given three commonly used objective functions: variance, expected shortfall, and mean log proﬁts, we study properties of the optimal hedge. We ﬁnd that the choice of objective function can have a substantial effect on the resulting hedge in terms of the portfolio composition, the resulting risk and the hedging cost. Further, we ﬁnd that unless the objective is indiffer- ent to hedging costs, term premia in the different markets, along with transaction costs, are fundamental determinants of the optimal hedge. Our results also show that to reduce risk properly and to keep hedging costs low, a rich-enough universe of hedging instruments is critical. Through out-of-sample testing we validate the ﬁndings of the in-sample analysis, and importantly, we show that the model is robust enough to be used on real market data. The proposed framework offers great ﬂexibility regarding the distributional assumptions of the underlying risk factors and the types of hedging instruments which can be included in the optimization model. Keywords Stochastic programming · Currency hedging · Term premia · Uncertain cash ﬂows · Risk management B Jonas Ekblom jonas.ekblom@liu.se Jörgen Blomvall jorgen.blomvall@liu.se Linköping University, Linköping, Sweden 123 36 Ann Oper Res (2018) 266:35–69 1 Introduction The use of ﬁnancial derivatives for hedging purposes is widespread among non-ﬁnancial ﬁrms and is considered as an integral part of the business and investment decisions of many companies. Speciﬁcally, the hedging of foreign exchange and interest rate risks are of high relevance for many ﬁrms. Considering the practical importance of hedging, one might guess that ﬁrms can rely on a well developed academic literature for advice on how to hedge. Cor- porate hedging has actually attracted considerable attention in academic literature; however, the focus has mainly been on understanding why ﬁrms should hedge and the characteristics of ﬁrms that hedge versus those who do not. Considering the theoretical justiﬁcations for hedging and the empirical evidence on the value of hedging, surprisingly little is known about how ﬁrms should hedge in practice. Examples of questions that we have only partial answers to are: the impact of empirically documented premia in e.g. the interest rate market on optimal hedging decisions; the value inherent in an expanded asset universe; whether risks should be partially or fully hedged; and not least, how a model supporting the hedging decision faced by market practitioners can be built considering e.g. transactions costs, trading at market prices, and ﬂexibility in modeling choices. In this study, we take a ﬁrst step towards addressing these questions by building a stochastic programming framework for determining optimal hedging decisions, given stochastic cash ﬂows in multiple currencies and exposure to foreign exchange and interest rate risks. The ﬁrst step in determining optimal hedging decisions is to deﬁne the problem, which requires us to understand ﬁrst of all why ﬁrms should hedge. The famous result of Modigliani and Miller (1958) implies that, given the assumption of perfect capital markets, ﬁnancial decisions such as hedging do not create ﬁrm value. The important lesson from this result is not that ﬁnancial decisions are irrelevant, but that the assumption of perfect capital mar- kets must fail to hold for ﬁnancial decisions to create ﬁrm value. A long strand of literature has suggested different market frictions as rationales for corporate hedging. Such rationales include: costs of ﬁnancial distress (Smith and Stulz 1985), convex tax functions (Smith and Stulz 1985), costly external ﬁnancing (Froot et al. 1993), informational asymmetry between ﬁrm and shareholders (DeMarzo and Dufﬁe 1991), and more generally, non-linearities in the function describing ﬁrm value (Mackay and Moeller 2007). The main lesson from this strand of literature is that costly states, in which ﬁrm value is destroyed due to cap- ital market imperfections, do exist, which implies that ﬁrm value is a non-linear function of proﬁts. Within the ﬁrm value maximization paradigm, the objective for the managers of the ﬁrm is to maximize the expected ﬁrm value which is itself a non-linear function of proﬁts. Unless variation in proﬁts is inﬁnitely costly, the optimal hedging decision is not to minimize variation in proﬁts, but to determine the optimal trade-off between risk and the costs of hedging. This fundamental insight is central in the proposed framework for corporate hedging where in addition to transaction costs, we carefully consider the The International Swaps and Derivatives Association (ISDA) survey on derivatives usage in 2009 shows that over 94% of the world’s 500 largest companies (Fortune Global 500) use derivatives for risk management purposes, and that foreign exchange and interest rate derivatives are the two most widely traded instruments (88% and 83% of the ﬁrms respectively). According to the ISDA insight survey in April 2015, almost 90% of the ﬁrms consider the use of derivatives to be important for their business and investment decisions. Based on a study that included not only large ﬁrms (7319 ﬁrms in 50 countries). Bartram et al. (2009) show that over 60% of the ﬁrms use derivatives. Given a concave function of proﬁts and/or a convex cost function, hedging is simply justiﬁed by Jensen’s inequality. We use a wider interpretation of ﬁrm proﬁts than is typical in accounting literature. 123 Ann Oper Res (2018) 266:35–69 37 empirical research on documented premia in the interest rate and foreign exchange mar- kets. There is an extensive empirical literature that investigates expected interest and foreign exchange rates. The existence of time-varying term premia on the interest rate market is well documented. Two of the most inﬂuential contributions are those of Fama and Bliss (1987)and Cochrane and Piazzesi (2005). A common ﬁnding in these and other papers on term premia is that some measure of the slope of the term structure predicts excess bond returns. The literature on foreign exchange rate expectations is substantial. From the uncovered interest rate parity (UIP), it follows that the expected future spot rate is equal to the forward rate. However, it has been documented that forward-spot differen- tials and subsequent changes in the spot rates are uncorrelated, or even negatively correlated (see e.g. Fama 1984; Engel 1996), which strongly reject the UIP. This failure of the UIP is often referred to as the forward premium puzzle and is what makes currency carry trades, i.e. the investment in high-interest-rate currencies funded by low-interest-rate cur- rencies, proﬁtable on average. A growing literature on carry trades shows that their excess returns are partly a compensation for exposure to crash or skewness risk (see e.g. Brun- nermeier et al. 2008), but that carry trades hedged against crash risk remain proﬁtable (see e.g. Jurek 2014). The impact of carry trades on foreign exchange dynamics is signiﬁcant. However, to predict foreign exchange rates, the simple random walk model remains pre- eminant, as concluded in the survey on foreign exchange rate expectations by Jongen et al. (2008). An important implication of the empirically observed properties of interest and for- eign exchange rates is that the expected return of different hedging instruments varies with respect to both maturity and currency. Hence, the inclusion of hedging instruments spanning over a large set of maturities in all relevant currencies is important not only for the purpose of reducing risk, but is also necessary to enable ﬁnding the optimal hedge, considering risk as well as costs. We include in total 66 different hedging instruments in the optimization model. These are yearly spaced currency forward contracts and inter- est rate swaps in the relevant markets. The hedging decision is complicated not least by the combinatorial property of the problem, where (possibly inﬁnitely) many combinations of ﬁnancial contracts can be used to control the same risk, but importantly, at different costs. Theoretical models of interest and foreign exchange rate dynamics that are consistent with empirical ﬁndings are key inputs in the proposed framework for optimal hedging. A complication in ﬁnancial modeling is the fact that theoretical models in general do not provide a perfect ﬁt to market prices, which causes model arbitrage if we allow trading at market prices. However, stochastic programming allows us to determine the optimal allocation of market traded contracts and still consider the important results from theoretical models. The use of market prices offers the opportunity to utilize relative price advantages in the market, and enables the optimal positions to be entered to these prices. Within the stochastic programming framework, we model term structure dynamics using the essentially afﬁne model deﬁned in Duffee (2002) which is ﬂexible enough to capture empirically observed properties of term premia. Consistent with the ﬁndings of non-normality for foreign exchange returns (see e.g. Westerﬁeld 1977) and possible conditional skewness documented in the carry-trade literature (see e.g. Brunnermeier et al. 2008), we model foreign exchange rates by a Poisson jump- Sarno et al. (2012) deﬁne an afﬁne multi-currency term structure model and documents in-sample predictive ability favorable to the random walk. However, as argued in the article, their focus is not to provide forecasting models and out-of-sample performance has therefore not been investigated. 123 38 Ann Oper Res (2018) 266:35–69 diffusion model with stochastic volatility. Motivated by the ﬁndings in Jongen et al. (2008), we set the drift to zero. The true objective relating ﬁrm value and proﬁts is unknown, and so we must rely on approximations. Popular objective functions used in academic literature as well as in industry include measures of dispersion, tail-risk measures, and concave functions inspired by utility theory. We study speciﬁcations of the ﬁrm objective of all three types; namely variance, expected shortfall, and mean log proﬁts. We view the main contributions of this study to be (i) the development of an optimization framework that is robust enough to be used for hedging on real market data, (ii) the inves- tigation of the impact from different objectives on the properties of the optimal hedge, and (iii) the ﬁndings on the importance of the asset universe made available for the risk manager, and the impact of term premia on the optimal hedge. 2 Literature The literature on foreign exchange risk management predominantly addresses the situation of a known (deterministic) cash ﬂow. The optimal hedge of a single known cash ﬂow, expressed as the ratio between futures contracts and foreign currency exposure, has been shown to be close, but not necessarily equal, to one, as documented in e.g. Ederington (1979). The case of an uncertain future cash ﬂow is addressed in Kerkvliet and Moffett (1991), where the hedge ratio that minimizes the variance of a cash and futures portfolio is derived. However, the result is restricted to the special case of a single cash ﬂow and is based on simplifying assumptions such as no interest rate risk. Another aspect of uncertain cash ﬂows, namely the arrival of new information, is investigated in Eaker and Grant (1985). In their paper, the optimal hedge is determined as the solution to a stochastic dynamic programming problem using forward contracts. Compared to Kerkvliet and Moffett (1991), as well as Eaker and Grant (1985), we consider a more general setting with multi-currency cash ﬂows, interest rate risk, market prices, transaction costs, a larger set of hedging instruments and term premia. However, we do not address the arrival of new information, which would require a multi-stage stochastic programming model, and we obtain no closed-form optimal hedge ratio. The paper that we ﬁnd to be closest to ours is Volosov et al. (2005), in which a stochastic programming model for hedging of foreign exchange exposure in a deterministic single- currency cash ﬂow stream is formulated. Uncertainty is restricted to random ﬂuctuations in future spot and forward rates. As in previous studies, interest rate risk is disregarded and only currency forward contracts are included. Using a multi-objective function, minimizing trans- action costs and deviations from treasury targets while maximizing expected terminal wealth, to solve the problem, they ﬁnd, through ex-post simulations, a considerable improvement in the risk-return proﬁle compared to the no-hedging strategy. Topaloglou et al. (2002) build a stochastic programming model for international asset allocation. By integrating the hedging decision with the portfolio selection problem, they show an improved risk-return proﬁle from international diversiﬁcation. In Topaloglou et al. (2008), the international asset allocation problem is addressed in a dynamic setting using a multi-stage stochastic programming model. Through ex-post simulation, they demonstrate the stochastic programming framework to be a ﬂexible and effective decision tool by showing that the risk of international portfolios is considerably reduced with this framework. 123 Ann Oper Res (2018) 266:35–69 39 3 The hedging problem In this section we give a general formulation of the corporate hedging problem and present a suitable method for solving this class of decision problems. Within the ﬁrm value maxi- mization paradigm the objective for the managers is to maximize the expected value of the ﬁrm. However, due to frictions in the markets, ﬁrm value itself is a nonlinear function of ﬁrm proﬁts. Determining the exact shape of this function is a nontrivial task even in the case of a single ﬁrm, and so ﬁnding a general functional form for a larger family of ﬁrms is likely to be an impracticality. In literature as well as in industry, different approximations for the objective of a ﬁrm are chosen. These show varying concordance with the properties of costs caused by ﬁnancial frictions. Three popular choices are tail-risk measures (e.g. expected shortfall), measures of dispersion (e.g. variance, absolute semi-deviation), and utility functions (e.g. power utility). These alternative speciﬁcations capture different properties of costs resulting from ﬁnancial frictions. If the cost of ﬁnancial distress is the only friction, then tail-risk measures are likely to be suitable as they focus on controlling low-proﬁt states. Minimization of variance (standard deviation) is optimal if variation in proﬁts is inﬁnitely costly. This is seldom the case in practice, but measures of variation may provide good decisions in environments where the impact of expected returns and third and higher moments are negligible. If ﬁnancial frictions destroy value in bad (low-proﬁt) as well as good (high-proﬁt) states of nature, then a concave function, potentially inspired by utility theory, may provide a good approximation. In our numerical analysis, we work with objective function speciﬁcations of all three types: expected shortfall; variance; and the natural logarithm. Although the nature of the hedging problem is sequential, we work in a single-period framework. The motivation for this is two-fold. First, for dynamic portfolio choice prob- lems with transaction costs, Brown and Smith (2011) show that single-period stochastic programming models can give decisions that are close to optimal. Secondly, there is a trade- off between the number of stages and the number of outcomes per stage, and we know from e.g. Kaut et al. (2007) that even in a single-period model the required number of scenarios for a stable solution is likely to be large. Although this is interesting for future research, multiple stages introduce additional difﬁculties, and to focus on the challenges already inherent in the corporate hedging problem, in this study we work in a single-period framework. The optimal hedging decision, formulated as a single-period optimization problem within the ﬁrm value maximization paradigm, can now be formulated. For this purpose, let y denote the decision vector, i.e. the number (and type) of ﬁnancial derivatives to trade, and ξ a random vector describing the uncertainty in the decision problem. We denote the (stochastic) next-period ﬁrm proﬁt by Π(y,ξ), so that the corporate hedging problem can be written as max E[ f (Π (y,ξ))] (1) s.t. y ∈ Y for some concave function f , and with Y denoting the set of constraints on the decision variable y.The set Y typically contains cash balance constraints and potential restrictions on trading that are dictated by the ﬁnancial policy of the ﬁrm, such as limitations on short- selling. Kaut et al. (2007) studies the stability of a model minimizing expected shortfall in a ﬁnancial application and shows that at least 5,000 scenarios are needed for stability of the model. 123 40 Ann Oper Res (2018) 266:35–69 Stochastic Scenario Optimization Historical processes generation problem data Fig. 1 Steps involved in formulating a stochastic programming model For the vast majority of real-world applications the hedging problem in (1) is analytically intractable, and so has to be solved with numerical techniques. Stochastic programming provides a suitable framework for this purpose, and solves the problem in (1) by discretizing the distribution of the random vector ξ, e.g. through Monte Carlo simulation. Given the discretization, the population mean in the objective of (1) is approximated by the sample counterpart which is known as the sample average approximation. The resulting optimization problem is a (large scale) deterministic non-linear program, commonly referred to as the deterministic equivalent of (1). The main steps in formulating and solving the stochastic programming model are illustrated in Fig. 1. When it is available, historical data is usually an important source of guidance on the modeling of uncertainty in stochastic programming applications. For the ﬁnancial case that we study, observed empirical properties of interest rates and foreign exchange rates are crucial when choosing suitable stochastic processes to model the risk faced by a ﬁrm. Scenarios for the uncertain parameters are generated by means of Monte Carlo simulation, which is a popular scenario generation method in stochastic programming. Given a ﬁnite set of possible realizations for the random parameters, the optimal hedge can be determined as the solution to the deﬁned optimization problem. 3.1 Problem instance The speciﬁc instance that we study is inspired by a real case of a Swedish company (henceforth referred to as the case company) that serves as a supplier to a US company (referred to as the customer). The supplier contract stipulates that over the following ten years the customer has the right, but not the obligation, to buy (a limited number of) products at a ﬁxed price in USD. This corresponds to the customer holding a set of American call options on the products, with strike prices equal to the agreed product prices and time to maturities of up to 10 years. The exposure, from what corresponds to written call options, creates large risks for the case company if it is left unhedged. The manufacturing costs for the case company are paid in SEK, GBP and EUR, while the cash inﬂows from sales are in USD. Cash in- and outﬂows are spread over the 10 years covered by the agreement. We model the cash ﬂows in all four currencies as annually spaced at 1, 2,..., 10 years from the time of the ﬁrst hedging decision, and we assume a 1-week hedging horizon. The customer is obliged to notify the case company on a weekly basis about the expected order sizes, but may in the case of new information give updates more frequently. The information about expected future cash ﬂows is used by the case company to monitor the value of the order book. We let T denote the set of times for the project and hedge portfolio cash ﬂows, and E ={EUR/SEK, GBP/SEK, USD/SEK} the set of exchange rates with SEK as the term currency. Further, we let c (e) determine the base currency of exchange rate e ∈ E (e.g. c (EUR/SEK) = EUR). The net value of the order book and the hedge portfolio at time t can then be expressed as Alternative scenario generation methods include moment matching (Høyland and Wallace 2001; Høyland et al. 2003), quasi Monte Carlo methods (Niederreiter 1992), scenario reduction (Dupaco ˇ vá et al. 2003; Heitsch and Römisch 2003; Bertocchi et al. 2006), and optimal discretization (Pﬂug 2001). 123 Ann Oper Res (2018) 266:35–69 41 ⎛ ⎞ ⎝ ⎠ Π = d D + C t SEK,t,τ SEK,t,τ SEK,t,τ, j τ ∈T j =1 ⎛ ⎞ ⎝ ⎠ + f d D + C , (2) e,t c (e),t,τ c (e),t,τ c (e),t,τ, j b b b e∈E τ ∈T j =1 where f is the spot exchange rate e ∈ E, at time t; d is the discount factor in currency e,t c b(e),t,τ c (e), for time τ , observed at time t; C is the cash ﬂow forecast in currency c (e),at b c (e),t,τ, j b time τ , for project j = 1,..., M, given the information obtained up to time t;and D c (e),t,τ is the cash ﬂow in currency c (e), at time τ , which results from the hedge portfolio held at time t. The risk faced by the case company derives from the distribution of changes in the net proﬁts, i.e. ΔΠ = Π − Π . Hence, we see from (2) that risk arises from the combined t +1 t +1 t effect of changes in spot exchange rates, movements in the term structures of interest rates, and uncertainty in future cash ﬂows. We include 2, 3,..., 10 year interest rate swaps in all four markets and 1, 2,..., 10 year currency forward contracts in the three exchange rates with SEK as the term currency. The hedging decision concerns choosing the portfolio of interest rate swaps and currency forward contracts, generating D , so as to control the b(e),t,τ risk from the uncertain cash ﬂows and thereby create additional ﬁrm value. 4 The optimization framework In this section, we move on from the general description of stochastic programming models in Fig. 1 and propose a framework for the speciﬁc problem of hedging currency and interest rate risk in an environment with uncertain cash ﬂows. The ﬁrst step in setting up the framework is to deﬁne the optimization problem, and given this we can determine the necessary input in terms of scenarios. Next, we make distributional assumptions for the risk factors and choose a suitable scenario generation method. This is followed by parameter estimation for the stochastic processes, and generation of scenarios. Finally, we solve the optimization problem, given the scenarios of the company cash ﬂows and the hedging instruments. An illustration of the framework from the perspective of the ﬂow of data, starting with the collection of historical data and the steps involved in generating scenarios and ending with the determination of the optimal hedge, is presented in Fig. 2. For the purpose of estimating the dynamics of interest and foreign exchange rates, we use synchronized historical weekly data on forward rate agreements, interest rate swaps, and foreign exchange rates. Parameters for the dynamic models of foreign exchange rates can be estimated directly from the historical data available, while the interest rate models ﬁrst require complete term structures on each of the four markets to be determined. The reason for this is simply that the interest rate models require historical observations of ﬁxed maturity yields that are not directly observable in the market. Estimation of the complete term structures, if made carefully, also has the important advantage of reducing the impact of noise present in the prices of individual contracts. The choice of the term structure estimation method, and speciﬁcally its ability to reduce noise in the raw data, has a big impact on the parameter estimation and thus on the quality of the scenarios generated. More generally, in a stochastic programming framework, the properties of the scenarios is of great importance for the quality of the optimal decision. The parameter estimation is performed separately for each interest rate market and foreign exchange rate, while the scenarios are generated collectively using a joint copula. By separating the estimation of the univariate distributions 42 Ann Oper Res (2018) 266:35–69 Data Company FRA & IRS FX rates collection cash ﬂows Term structure EUR GBP SEK USD estimation EUR/SEK Parameter EUR GBP SEK USD GBP/SEK estimation USD/SEK Term structure, Scenario FX,and cash- generation ﬂow scenarios Valuation of IRSs, Scenario FX forwards, pricing and cash ﬂows Solving for Optimization the optimal hedge Fig. 2 Optimization framework for corporate hedging and the modeling of dependencies, we are able to estimate the multivariate distribution of all risk factors. The dynamic model for the company cash ﬂows is based on distributional assumptions made by the case company. Although some data exists, we do not have enough historical data available to statistically estimate the distribution of the uncertain cash ﬂows. Hence, the best estimate we can obtain is likely to be based on the expert opinions from managers within the company. Given the scenarios, we can determine the value of all hedging instruments as well as the present value of the cash ﬂows in each state of the future. In addition to the future prices and values, we also need the current market prices and present values, which we determine from term structure estimates and foreign exchange rate observations on the day of the hedging decision. One of the advantages of the proposed framework is its ﬂexibility, as it allows for replace- ment of different components, e.g. the assumed stochastic processes, the scenario generation technique, the set of included ﬁnancial contracts, or the optimization model. Speciﬁcally, the framework allows us to study the impact of changing different components on the optimal hedge. One such experiment is to let one or several risk factors be deterministic, something which can help us to understand the importance of modeling uncertainty in the different risk factors. Next, we deﬁne the optimization problem, which is followed by a description of the parameter estimation and the scenario generation for interest rates, foreign exchange rates, and cash ﬂows respectively. The technical details of the pricing of currency forwards and IRSs can be found in “Appendix A”, and the details of the scenario generation is presented in “Appendix B”. 123 Ann Oper Res (2018) 266:35–69 43 4.1 The stochastic programming model We deﬁne the hedging problem as a two-stage (single-period) stochastic programming model without recourse decisions. Hence, decisions are made only at the initial stage, and the objec- tive is deﬁned over the possible realizations in the second stage. The objective is to maximize the expected ﬁrm value given exposure to (stochastic) cash ﬂows, and we study alternative speciﬁcations for the function relating ﬁrm value and proﬁts. Speciﬁcally, we consider the problem of forming a hedge portfolio from an asset universe of currency forward contracts and interest rate swaps. The asset set contains in total 66 different hedging instruments, made up of 30 forward contracts (3 exchange rates, 1,..., 10 years) and 36 interest rate swaps (4 markets, 2,..., 10 years). In addition to the currency forwards and interest rate swaps, the ﬁrm can also choose to hold cash that grows with the risk-free rate over the 1-week hedging horizon. We use the following sets and indices for the purpose of deﬁning the optimization problem, Sets: C ={EUR, GBP, SEK, USD} currencies; E ={EUR/SEK, GBP/SEK, USD/SEK} FX rates. Indices: c ∈ C currencies; e ∈ E FX rates; τ ∈ {1,..., 10} years forward in time; { } i ∈ 1,..., N scenarios. The decision variables of the model are the number of long and short contracts in currency forwards and interest rate swaps. Firm proﬁt is an auxiliary variable which we deﬁne as the sum of the hedge portfolio, the present value of uncertain cash ﬂows, and the holdings in cash. Auxiliary variables are also used for the purpose of deﬁning expected shortfall. The notation chosen for the decision and auxiliary variables is, Decision variables: F,L x number of long forwards in FX rate e with maturity in τ years; e,τ F,S x number of short forwards in FX rate e with maturity in τ years; e,τ I,L x number of receiver IRSs in currency c with maturity in τ years; c,τ I,S x number of payer IRSs in currency c with maturity in τ years. c,τ Auxiliary variables: z ﬁrm proﬁt in scenario i; y excess loss over VaR (at optimum) in scenario i; ζ help variable for calculation of expected shortfall (=VaR at optimum); ν post-decision holdings in cash. Trading in currency forwards is an agreement for a future transaction which is deﬁned to have zero value at the initiation of the contract, and so only requires contract values to be calculated in the second stage. The values of interest rate swaps are modeled as the prices of corresponding coupon bonds, and so have non-zero values also at the initial stage. All For valuation of an IRS at a date when the ﬂoating rate is ﬁxed (ﬁxing date) it is sufﬁcient to consider only the ﬁxed leg. 123 44 Ann Oper Res (2018) 266:35–69 prices are expressed in SEK. For the purpose of determining expected shortfall, we calculate the expected proﬁts at the initial stage from the pre-decision holdings in cash, the value of previously traded contracts, and the expected future cash ﬂows. To summarize we have, Deterministic parameters: R risk-free growth rate; z initial (expected) proﬁt; I,L P initial price of receiver IRSs in currency c, with maturity in τ years; c,τ,0 I,S P initial price of payer IRSs in currency c, with maturity in τ years; c,τ,0 α conﬁdence level in calculation of expected shortfall; h pre-decision holdings in cash. The scenario data, that together with the probabilities specify the multivariate distribution of the random variables at the end of the model horizon, are the values of long and short positions in the currency forwards and interest rate swaps, and the present value of the cash ﬂows. We collect the present value of the cash ﬂows and (if they exist) the values of previously traded contracts in a single random variable. Market spreads on swap rates and currency forward prices are included in the model, and create the difference in value between long and short positions. All asset values are expressed in SEK. “Appendix A” presents the exact pricing formulas. Stochastic parameters: p probability of scenario i; F,L P value of long forward in FX rate e, with mat. in τ years, in scen. i; e,τ,i F,S P value of short forward in FX rate e, with mat. in τ years, in scen. i; e,τ,i I,L P value of receiver IRSs in currency c, with mat. in τ years, in scen. i; c,τ,i I,S P value of payer IRSs in currency c, with mat. in τ years, in scen. i; c,τ,i b present value of company cash ﬂows in scenario i. Motivated by the lack of a single theoretically and empirically well-founded ﬁrm objective, we study alternative functional speciﬁcations which relate ﬁrm proﬁts and value. We deﬁne the problem for a general objective, f (z , y ), and present the instances that we study below. i i With the given notation, the general optimization problem can be formulated as: max p f (z , y ) (3a) i i i i =1 I,L I,S I,L I,S s.t. ν = h − P x − P x (3b) c,τ c,τ c,τ,0 c,τ,0 c∈C τ =2 I,L I,L I,S I,S z = P x − P x c,τ,i c,τ c,τ,i c,τ c∈C τ =2 F,L F,S F,L F,S + P x − P x + Rν + b i = 1,..., N (3c) e,τ e,τ e,τ,i e,τ,i e∈E τ =1 I,L I,S x , x ≥ 0 c ∈ C,τ = 2,..., 10 (3d) c,τ c,τ F,L F,S x , x ≥ 0 e ∈ E,τ = 1,..., 10 (3e) e,τ e,τ y ∈ Y (3f) 123 Ann Oper Res (2018) 266:35–69 45 We determine the cash holding after transactions and the proﬁts in the different scenarios by constraints (3b)and (3c) respectively. To model the transaction costs, we use separate variables for long and short positions, which from their construction are required to be non- negative by (3d)and (3e). Finally, we use the set Y in (3f) to model expected shortfall. We study three instances of the optimization model in (3), where we use different functional speciﬁcations of the objective. First, the minimum variance hedge is obtained by setting f (z ) =− (z −¯z) , (4) i i where z ¯ is the sample mean of z ,..., z . Second, we study minimization of expected 1 N shortfall. As shown in Rockafellar and Uryasev (2000), this problem can be formulated as a linear program, which with our notation yields f (z , y ) =− ζ + y , (5) i i α i 1 − α and Y = {y |y = max (z R − z − ζ , 0) , i = 1,..., N } , (6) i i 0 i α where we have deﬁned the loss relative to the initial wealth, z , capitalized by the risk-free rate. In addition to these two popular measures of risk, we study the maximization of expected logarithmic proﬁts. Being strictly concave on its whole domain, this objective function may potentially capture the aggregate impact from all ﬁnancial frictions affecting the ﬁrm. Finally, we study the maximization of expected logarithmic proﬁts given constraints on expected shortfall. Through a Lagrangian relaxation, this problem can be equivalently formulated as a trade-off between expected logarithmic proﬁts and expected shortfall by deﬁning f (z , y ) = λ ln(z ) − (1 − λ) ζ + y , (7) i i i α i 1 − α and with Y deﬁned as in (6). Note that λ = 0 gives minimization of expected shortfall and λ = 1 maximization of expected logarithmic proﬁts. The cases with 0 <λ< 1 correspond to the maximization of expected logarithmic proﬁts given constraints on expected shortfall. 4.2 Scenario generation Scenario generation produces a set of future states for the stochastic parameters in the stochas- tic programming model, and is used to represent the uncertainty inherent in the model. The stochastic parameters in the stochastic programming model are determined by the interest rate term structures in the four markets, the three exchange rates with SEK as the term cur- rency, and the project cash ﬂows. With three risk factors on each interest rate market, we have in total 16 random variables made up of 12 state variables, 3 exchange rates, and the order size representing the uncertainty in the project cash ﬂows. Scenarios are generated by Monte Carlo simulation with a weekly discretization of the assumed continuous stochas- tic processes, where a Gaussian copula is used to preserve the historical correlation of the uncertain parameters. The modeling choices and the parameter estimation for the interest, foreign exchange, and cash ﬂow models are discussed below, and a detailed mathematical description of the scenario generation is given in “Appendix B”. The application of robust optimization to risk and portfolio choice problems is being studied in an increasing stream of literature, see eg. Kapsos et al. (2017). An alternative is to formulate the model as a multiple-objective optimization problem. For a literature review on this topic, see Masmoudi and Abdelaziz (2017). In Kopa et al. (2016) a multiple-objective formulation is studied under stochastic dominance constraints. 123 46 Ann Oper Res (2018) 266:35–69 4.2.1 Interest rate modeling We need an interest rate model that allows us to represent the uncertainty in the term structure while capturing the important term premia. In the choice of model, a ﬁrst natural requirement is arbitrage-free modeling. Afﬁne term structure models allow this while remaining analyti- cally tractable. An important contribution in the class of afﬁne term structure models is the essentially afﬁne model proposed by Duffee (2002), which is a model carefully speciﬁed to be consistent with empirical properties of term premia. Compared to the standard class of afﬁne term structure models, the essentially afﬁne model of Duffee (2002) improves the forecasting potential of term premia by a more ﬂexible speciﬁcation of the market prices of risk. Within this framework, we are able to model the term structure with term premia included, while still allowing modeling which uses market prices. Model instance In the class of afﬁne term structure models, a large number of model spec- iﬁcations are possible. As in Duffee (2002), we choose to model three state variables. This choice is motivated by the ﬁndings in Litterman and Scheinkman (1991)who show that approximately 95–99% of term structure movements can be explained by only three factors. Among the class of essentially afﬁne models with three state variables, the Gaussian model, in which no state variables affect the instantaneous variance of the state variable vector, offers the greatest ﬂexibility for the estimation of time varying term premia. The Gaussian model is referred to as the m = 0 model in Duffee (2002), where results from a statistical comparison of models estimated with three state variables verify that this model is the best at ﬁtting time varying term premia. By imposing normalization on the Gaussian model as in Duffee (2002), which follow the setup of the canonical completely afﬁne models in Dai and Singleton (2000), the dynamics of the model instance used in this study can be written as A(τ )−B(τ ) X P(X ,τ) = e , (8) r = δ + δ X , (9) t 0 t dX =−KX dt + dW , (10) t t t where P(X ,τ) is the zero coupon bond price with time to maturity τ , r the short rate, and t t X ≡ X , X , X the state-price vector at time t.In (8), A(τ ) and B(τ ) are scalar and t t,1 t,2 t,3 vector valued functions respectively. The short rate process is afﬁne in the state-price vector X , with δ denoting a scalar and δ ∈ R . The mean-reversion of the state-price dynamics t 0 3×3 in (10) is governed by the lower triangular matrix K ∈ R , while uncertainty is generated by a three dimensional Brownian motion, W ≡ W , W , W , under the real-world t t,1 t,2 t,3 probability measure. Fully integrated multi-currency term structure models following the pioneering work of Backus et al. (2001) allow for arbitrage-free term structure modeling of multiple currencies, and are hence very attractive from a theoretical point of view. Simultaneous estimation of parameters describing the dynamics on multiple markets does however create additional challenges. Note that we observe multiple local optima, where some are unrealistic, already for a single market. See Duffee (2002)and Dai and Singleton (2000) for a more detailed description of this model. 123 Ann Oper Res (2018) 266:35–69 47 Table 1 Interest rate data collected from Thomson Reuters Eikon EUR GBP SEK USD FRA 3 × 6, 6 × 9, 9 × 12 3 × 6, 6 × 9, 9 × 12 FRA1-6 3 × 6, 6 × 9, 9 × 12 IRS (years) 2–10 2–10 2–10 2–10 We use weekly mid-quotes for forward rate agreements (FRA) and interest rate swaps (IRSs). The data range is from September 19th, 1997 to March 27th, 2017 Interest rate data Interest rate curves are estimated for Euro (EUR), Pound Sterling (GBP), Swedish Krona (SEK) and US Dollar (USD). Table 1 presents the contracts used to estimate yield curves consistent with market prices. To estimate stable interest rate curves we use Blomvall (2017), min h( f (t )) + z E z e e 2 e f (t ),z (11) s.t. g ( f (t )) + z = ρ e e f (t ) ≥ 0 where a smooth forward rate curve f (t ) is estimated, and where deviations, z , from market yields, ρ, are penalized with the diagonal weighting matrix E . The roughness is measured by h,and g is a function which transforms the forward rate curve f (t ) to a yield for instrument e, j j. The forward rate curve is discretized with one forward rate per day, Δ = 1/365, and roughness is measured as a numerical approximation of the integral of the squared second order derivative of f (t ), n −2 1 2 f − f f − f Δ + Δ τ +1 τ τ τ −1 τ −1 τ h( f ) = ϕ − , (12) 2 Δ + Δ Δ Δ 2 τ −1 τ τ τ −1 τ =1 where n is the number of (discretized) forward rates. The main difference to traditional term structure estimation methods is that here, the inﬁnite-dimensional optimization problem is discretized, instead of constraining the forward rate curve to speciﬁc parameterizations such as the quartic spline. With a daily discretization, all instruments can be dealt with exactly, and the behavior of the forward rate term structure can be handled by choosing ϕ . The problem is a large-scale non-linear optimization problem which can be solved efﬁciently by algorithms such as Manzano and Blomvall (2004). Even though many starting solutions have been tried, the algorithm converges to the same optimum, which indicates that the global optimum is identiﬁed. To estimate forward rate curves of high quality, a careful choice of the parameter values is generally required. The forward rate curve is more variable in the short end, and to capture this, a time varying ϕ is usually preferred. However, for this application, the simple constant choice ϕ = 1 is sufﬁcient, since only yearly spot rates are required and the large number of contracts in the short end makes the forward rate curve more variable in that region. The weighting of the pricing errors should depend on the amount of noise in the market prices. For the forward rate agreement and interest rate swap market, the noise is of similar level for all instruments, which makes equal weighting appropriate. If too much emphasis is on repricing the instruments, unrealistic forward rate curves will be created. If too little emphasis is on repricing the instruments there will be systematic pricing errors in the instruments, as the forward rate curve becomes too rigid. In extensive tests the choice E = 100I,where I is the identity matrix, has proven to be a good choice that creates realistic forward rate curves that do not contain systematic pricing errors. 123 48 Ann Oper Res (2018) 266:35–69 Parameter estimation Following Duffee (2002), we estimate parameters using quasi- maximum likelihood (QML). The estimation is implemented by assuming that yields with maturities 3 months, 5 years and 10 years are measured without error, while yields with matu- rities 6 months, 3 years and 7 years are assumed to be measured with error. The choice of maturities is motivated by the desire to achieve uniform distribution over the term structure. The QML estimation technique produces a non-convex optimization problem with a large number of local maxima. We implement the optimization problem in AMPL, using IPOPT to determine optimal parameters. Following Duffee (2002), starting solutions are randomly generated from a multivariate normal distribution with diagonal covariance matrix, and with mean and variances set to ’plausible’ values. Using this technique, 10,000 starting solutions are generated for each market. The 20 starting solutions with the highest QML values are then used for optimization in AMPL. The solution on each market with the highest QML value, after optimization in AMPL, is used as model parameters. Table 2 reports estimated parameters for the Gaussian essentially afﬁne term structure models along with the QML values for each market. We identify four main sources of difﬁculties in estimating the parameters. First, the QML optimization problem is non-convex, giving rise to a large number of local maxima, some of which produce unrealistic model properties. Different starting solutions are likely to produce a diverse set of parameter estimates, but with similar QML values, which give signiﬁcantly different model properties. We handle this by excluding parameter estimates that produce unrealistic model properties in terms of expected future interest rates, as discussed below. Second, the estimation method used, following Duffee (2002), where three yields (n = 3) are assumed to be measured without error, while the other yields in the data set are assumed to be measured with error, is not economically well justiﬁed. Different choices of the set of trusted yields produce different parameter estimates, and there is no justiﬁcation for a certain choice of the set of trusted yields. However, the estimation method provides a way of taking the information contained in more than n yields into consideration. Due to the risk of an ill- conditioned measurement error covariance matrix, the number of yields with measurement error included is, as in Duffee (2002), restricted to three. Third, the length and the quality of the raw data is critical for the parameter estimation. We use weekly observations of interest rate data on the four markets over a period of approximately 16 years, compared to the monthly observations of U.S. data over a period of almost 47 years used by Duffee (2002) (which is based on McCulloch and Kwon 1993; Bliss 1996). Fourth and ﬁnally, the structure of the mean reverting Gaussian essentially afﬁne term structure model implies a long run expected short rate. This varies signiﬁcantly in different local maxima and it is independent of the current state, making the expected short rate estimation critically dependent on the sample period used, as discussed below. The expected short rate at time T conditional on r follows from (9)as E [r |r ]= δ + δ E [X |X ], (13) T t 0 T t where the conditional expectation of X given X , derived in Duffee (2002), is given by T t E [X |X ]= exp (−K (T − t ))X , (14) T t t and where exp (−K (T − t )) is the matrix exponential. Stationarity requirements on the feedback matrix K , i.e., requirements of positive eigenvalues for K , imply that E [X |X ]→ T t 0as T →∞.From (13)and (14) we then obtain that E [r |r ]→ δ as T →∞. To guarantee T t 0 realistic model properties, the long run expected short rate is used as a ﬁlter in the selection of starting solutions. A comparison of expected future short rates and forward rates for EUR, 123 Ann Oper Res (2018) 266:35–69 49 Table 2 Parameters from quasi maximum likelihood estimation of the normalized Gaussian essentially afﬁne term structure model in (8)–(10) QML value EUR GBP SEK USD 27,577 27,630 27,388 26,691 δ 0.0307 0.0265 0.0260 0.0373 δ 0.0098 −0.0184 0.0090 0.0150 −0.0004 0.0082 −0.0027 −0.0058 −0.0072 0.0046 −0.0046 0.0044 K 3.5992 0 0 2.1912 0 0 2.2290 0 0 1.7345 0 0 1· K 1.9482 0.2569 0 0.1671 0.3750 0 0.0362 0.4954 0 0.2635 0.1531 0 2· K 0.1578 −0.2766 0.7810 0.0051 −1.7227 1.7165 −0.0890 −1.1309 1.2601 −1.8862 −1.1329 0.2878 3· λ −0.0437 −0.5277 −0.0029 −0.1857 0.3103 1.7161 −0.2220 0.2122 1.9281 0.0751 −0.0331 0.9570 λ −3.6749 0.1251 −0.1899 −2.2719 0.1629 −0.0941 −2.4305 0.2905 −0.0164 −1.6776 0.1104 0.0488 1· λ −1.7203 −0.0498 0.1559 −0.3179 −0.3505 0.0966 −0.0928 −0.4788 0.1290 −0.1027 −0.4093 0.0056 2· λ 0.1703 0.0577 −0.6329 0.2143 0.4123 0.2340 −1.0128 0.2745 0.1463 −0.5077 −0.0738 0.3858 3· L*1.1159 0 0 2.5846 0 0 1.9560 0 0 2.2055 0 0 1· L*0.8517 0.8620 0 1.0094 0.5291 0 0.9427 0.4272 0 1.7288 0.5226 0 2· L * −0.4512 −0.7336 0.3992 −0.1465 −0.2886 0.2247 −0.0405 −0.3680 0.4007 −0.6914 −0.3279 0.2970 3· Parameters are estimated using 828 weekly yields from September 19th, 1997 to July 26th, 2013. The lower triangular matrix L is the Cholesky decomposition of the measurement- error covariance matrix C = LL (see “Appendix B”). *L is scaled by a factor of 10 , thereby increasing the number of signiﬁcant digits presented 50 Ann Oper Res (2018) 266:35–69 EUR GBP 3.5 4.5 Forward rates Forward rates Expected short rates Expected short rates 3.5 2.5 2.5 1.5 1.5 0.5 0.5 0 0 02468 10 02468 10 Maturity (years) Maturity (years) (a) (b) USD SEK 3.5 Forward rates Forward rates Expected short rates Expected short rates 2.5 1.5 0.5 02468 10 02468 10 Maturity (years) Maturity (years) (d) (c) Fig. 3 The forward rate curves (%) in a–d are calculated from observed market prices of FRAs and IRSs on July 26th, 2013 using the model in (11), with E = 100I and ξ = 1/365. Expected short rates are given by e t (13) using the parameter estimates in Table 2 GBP, SEK and USD on July 26th, 2013, as implied by the parameter estimates in Table 2,is presented in Fig. 3. The main point of interest in Fig. 3 is whether the expected short rates, or the term premia, represented by the difference between forward rates and expected short rates, are reasonable representations of market expectations. There is no universal answer to this question, but there is empirical evidence to help us understand the properties of term premia. The documentation of term premia, which is closely connected to tests of the expectations hypothesis, is a well investigated area of research. Early references documenting the existence of term premia and rejecting the expectations hypothesis for the U.S. market include Fama and Bliss (1987) and Campbell and Shiller (1991). However, for the short end of the yield curve, there is evidence for term premia being close to zero (see e.g. Longstaff 2000). In a cross-country study by Wright (2011), term premia are documented for ten industrialized countries, with the four markets in this study included. Wright (2011) ﬁnds that term premia have declined globally over the period 1990–2009, and even turned negative in some countries. The term premia decline is to a large extent explained by monetary policy changes during the sample The Euro zone is represented by Germany. 123 Ann Oper Res (2018) 266:35–69 51 period, which have reduced the uncertainty about future inﬂation. The supporting evidence for a global movement from high to low interest rate economies with declining term premia, suggests weaknesses in the model deﬁned in Duffee (2002) when using a sample period covering different monetary regimes. The long-run expected short rate, δ ,issampleperiod sensitive, as it acts as a mean reverting level for the chosen estimation period. This motivates using the level of δ as a ﬁlter to reduce the effect of the identiﬁed model weakness. To summarize, empirical evidence suggests that the forward rate and expected short rate curves in Fig. 3 should almost coincide for short term maturities. However, for long term maturities, non-zero term premia have empirical support. The results produced by the essentially afﬁne term structure model imply that only USD shows a close match between forward rates and expected short rates in the short end. For longer maturities, all markets have positive term premia, expressed as the difference between the forward rates and expected short rates. 4.2.2 Foreign exchange rate modeling The literature on empirical properties of foreign exchange rates provides evidence that (at least) two important properties should be considered for the purpose of foreign exchange risk management. First, as established in e.g. Westerﬁeld (1977), and later Osler and Savaser (2011), the return distribution of foreign exchange rates exhibit excess kurtosis, i.e. possess fat tails compared to the normal distribution. Second, the carry trade literature documents that the distributions of foreign exchange rates are conditionally skewed, with signiﬁcant neg- ative skewness for high-interest-rate currencies and positive skewness for low-interest-rate currencies (see e.g. Brunnermeier et al. 2008). Motivated by these empirical facts, we model the foreign exchange rates by discrete time Poisson jump-diffusion models with stochastic volatility. The dynamics of foreign exchange rate e ∈ E, f ,are givenby e,t √ √ f = f exp μ Δt + σ Δt + α σ z ξ Δt , e,t +Δt e,t e e,t e,t e e,t e,t e,t (15) e,t +Δt ln − γ Δt e,t 2 2 σ = β + β σ + β , e,0 e,1 e,2 e,t +Δt e,t Δt where the random variables ,ξ ∼ N (0, 1), z ∼ Po(λ Δt ),and α represents the e,t e,t e,t e e relative impact of a Poisson jump. The Poisson process describes the arrival of an uncorrelated normally distributed random variable with standard deviation α σ Δt, which creates fat e e,t tails. The skewness of returns derives from the asymmetric update of variance with γ Δt.The (conditional) cumulative distribution function (cdf ) for the logarithmic returns of exchange rate e ∈ E over [t, t + Δt ] is given by x − μ Δt (λ Δt ) e e F (x ) = F exp (−λ Δt ) , (16) e,t N e k! σ (1 + α k)Δt e,t k=0 where F is the standard normal cdf. We use maximum likelihood estimation to determine the parameters of the foreign exchange rate processes in (15). Due to non-convexities, we have solved the problem for a large number of starting solutions to obtain the parameter estimates in Table 3. An interest- ing point to take from the estimation results is that γ is negative for all e ∈ E. This implies that all exchange rates, expressed with SEK as the term currency, are estimated to have pos- itive skewness. The implication of this is that periods for which SEK is weakened coincides with an increased volatility. During stressed market conditions there tends to be an increased 123 52 Ann Oper Res (2018) 266:35–69 Table 3 Parameters for the Poisson jump-diffusion models in (15), which have been determined using maximum likelihood estimation on weekly data over the period from September 19th, 1997 to July 26th, β β β α λ μ γ e,0 e,1 e,2 e e e e EUR/SEK 0.000049 0.873 0.0637 1.385 14.06 0.01 −0.172 GBP/SEK 0.00042 0.825 0.0821 1.154 14.56 −0.006 −0.0846 USD/SEK 0.00035 0.915 0.0488 1.375 3.14 −0.023 −0.206 Historical log-returns Garch-Poisson transformed returns 6 4 0 0 -1 -2 -2 -4 -3 -6 -4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 Standard normal quantiles Standard normal quantiles (a) (b) Fig. 4 Quantile–quantile (QQ) plots constructed from weekly data for EUR/SEK over the period from Septem- ber 19th, 1997 to July 26th, 2013. a The QQ-plot for the historical log-returns. b Illustrates the ﬁt of the Poisson jump-diffusion model and is constructed by a transformation of historical log-returns using the Poisson jump- diffusion cdf in (16), followed by an inverse transformation using the standard normal cdf ﬂow of capital to large currencies (safe havens), which provides economic intuition for these results. The estimated values of α for EUR/SEK and USD/SEK in Table 3, imply that one Poisson jump corresponds to an increased volatility (on that particular day) of approximately 1 + 1.38 − 1 ≈ 70%. The parameters λ represent the expected number of jumps per year, which is approximately 14 for EUR/SEK and GBP/SEK, and just over 3 jumps per year for USD/SEK. From a quantile–quantile plot we can investigate how well the Poisson jump-diffusion model ﬁts historical data. Figure 4 illustrates the improved ﬁt when using the Poisson jump-diffusion model compared to a log-normal distribution, exempliﬁed by the EUR/SEK exchange rate. Panel (a) of Fig. 4 clearly illustrates the fat tails usually observed in foreign exchange rates, and we can see from panel (b) that the Poisson jump-diffusion model provides a good description of historical EUR/SEK movements. Based on quantile–quantile plots, the empirical distributions of GBP/SEK and USD/SEK are also signiﬁcantly better described by the Poisson jump-diffusion model than by the log-normal distribution. 4.2.3 Cash ﬂow modeling The mechanism underlying uncertainty in the cash ﬂows is the update of the order-size forecast provided by the customer. We model the change in the order-size forecast as a log- normally distributed random variable. This scales the cash ﬂow forecast in currency c ∈ C Data quantiles Data quantiles Ann Oper Res (2018) 266:35–69 53 for time τ ∈ T from project j = 1,..., M as σ √ τ, j C = C exp − Δt + σ Δt ξ (17) c,t +Δt,τ, j c,t,τ, j τ, j τ, j over [t, t + Δt ],where σ is the order-size volatility, and ξ ∼ N (0, 1). The present value τ, j τ, j of (expected) future cash ﬂows can be determined as b = d D + C SEK,t +Δt,τ SEK,t,τ SEK,t +Δt,τ τ ∈T + f d D + C , e,t +Δt c (e),t +Δt,τ c (e),t,τ c (e),t +Δt,τ b b b e∈E τ ∈T where D is the (deterministic) cash ﬂows resulting from previously traded hedging c (e),t,τ instruments. 5 Numerical results In this section we present numerical results based on the optimization framework in Fig. 2. We start by describing details of the problem instances studied, and then discuss properties of the optimal hedging decisions resulting from in-sample analysis. This is followed by an investigation of the out-of-sample performance for the model. 5.1 Problem details The aim of the numerical experiments is to investigate properties of the optimal hedge from the stochastic programming model in (3) given the alternative speciﬁcations of the objective function in (4), (5), and (7), and with variations in model parameters. We use the in-sample analysis to investigate the properties of the optimal hedge given different ﬁrm objectives, varying asset universe, and with and without uncertainty in the cash ﬂows. The aims of the out-of-sample analysis are to test if the results obtained in-sample can be validated, and to investigate the robustness of the model. For the in-sample as well as the out-of-sample analysis, we consider the following variations of model speciﬁcations and parameters: • objective function deﬁned in terms of variance, (4); expected shortfall (given 95% con- ﬁdence level), (5); and mean log project value given constraints on expected shortfall, (7); • a single project (M = 1), and order size volatility, σ = 0, or σ = 0.05 ∀τ ∈ T ; τ,1 τ,1 • asset universe with all assets (66 contracts), 1-year currency forwards (3 contracts), and 1-year currency forwards with 5-year IRSs (7 contracts). The in-sample analysis is based on optimal hedging decisions determined on July 26th, 2013, with a 1-week hedging horizon. The out-of-sample analysis starts with the optimal solutions from the in-sample analysis and runs over 191 weeks ending on March 27th, 2017. Parameters for the stochastic processes are estimated using weekly data from September 19th, 1997 to July 26th, 2013. Hence, parameter estimates are kept ﬁxed over the whole out-of-sample period. The only exceptions are the GARCH volatilities in the Poisson jump- diffusion models for the foreign exchange rates, which are updated on the basis of realized market returns. All trading in the hedging instruments induces transaction costs, modeled as a spread of 80 percentage in point (‘pip’) for the currency forwards, and 2 basis points for the interest 123 54 Ann Oper Res (2018) 266:35–69 rate swaps. We assume that the case company starts with unhedged cash ﬂows and no cash holdings. The initial project value, corresponding to the present value of expected future cash ﬂows, is 1334 million SEK (MSEK) at the date of the (ﬁrst) hedging decision. We use 10,000 scenarios in all studied hedging problems, which have been chosen to maintain simulation efﬁciency while implying in and out-of-sample stability as deﬁned in Kaut et al. (2007). For the purpose of reducing the variance in the scenarios, we sample with the latin hypercube technique and use antithetic variates (see e.g. Glasserman 2013). We use MATLAB for scenario generation, computations, and data handling, while the optimization problems are modeled in AMPL and solved with CPLEX and IPOPT. 5.2 In-sample analysis We begin the in-sample analysis by studying the situation faced by the risk manager. This is illustrated in Fig. 5. The blue-yellow bars in panel (e) [and (f)] show the expected cash ﬂows in the four currencies at the time of the hedging decision in units of respective currency. Panels (a–d) illustrate the distribution of the cash ﬂows over a 1-week horizon implied by uncertainty in exchange rates, interest rates, and cash ﬂows. The distributions are presented in terms of the present value in SEK and represent the set of scenarios used in the in-sample analysis. In the ﬁrst numerical experiment, we compare the optimal hedge portfolios resulting from the minimization of variance and expected shortfall, given an asset universe with all 66 hedging instruments, and with and without uncertainty in the project cash ﬂows. In this ﬁrst numerical test, we add position constraints which limit the risk manager from taking both long and short positions in the same contracts. The motivation for the position constraints is the fact that variance is location-invariant. This implies that the objective is indifferent to the level of the mean, and hence it may be optimal to “destroy” project value by taking long and short positions in the same contracts. On the other hand, the location-invariance makes variance a suitable benchmark measure in the model set-up with premia and transaction costs, as it is indifferent to the cost of hedging. To highlight structural differences in the two hedging strategies, we present the cash ﬂows resulting from the optimal hedge portfolios together with the project cash ﬂows in panels (e) and (f) of Fig. 5, for the case with deterministic and stochastic cash ﬂows respectively. We can see that for both cases, the variance strategy (black) implies (approximate) cash ﬂow matching, while the ES strategy (red) produces (partly) off-setting exposure only for a subset of the cash ﬂows. Another way to investigate the properties of the hedging problems studied is to analyze the distributions of the project values. These are presented in Fig. 6. Panels (a) and (c) One basis point is one hundredth of a percent, and one ‘pip’ is here used to refer to one unit of the fourth decimal point. In-sample stability means that the optimal solution does not vary over scenario sets of the same size, and out-of-sample stability is obtained if the optimal solutions evaluated on a benchmark distribution shows low variation. We have used a distribution with 100,000 scenarios as the benchmark distribution to test out-of- sample stability. With position constraints we allow for trading only in the direction that (directly) reduces the risk exposure I,S I,S I,S I,L (see cash ﬂow structure in panel (e) of Fig. 5), andsoweadd x = x = x = x = 0, EUR,k GBP,k SEK,k USD,k F,S F,S F,L k = 2,..., 10, and x = x = x = 0, k = 1,..., 10 to the optimization EUR/SEK,k GBP/SEK,k USD/SEK,k problem in (3). Note that, given non-zero transaction costs, long and short positions in currency forward contracts and interest rate swaps give rise to different cash ﬂows, and so the net-trade is generically not equivalent to the portfolio of long and short positions. 123 Ann Oper Res (2018) 266:35–69 55 GBP EUR -8 -12 -8.5 -13 -9 -14 -9.5 -15 -10 -16 -10.5 -17 -11 -18 123456789 10 123456789 10 Year Year (a) (b) SEK USD -50 -55 -60 -65 -70 123456789 10 123456789 10 Year Year (d) (c) Hedge - deterministic cash flows Hedge - stochastic cash flows 1y 2y 3y 4y 5y 6y 7y 8y 9y 10y 1y 2y 3y 4y 5y 6y 7y 8y 9y 10y -10 -10 -10 -10 -10 -10 -10 -10 EUR GBP SEK USD EUR GBP SEK USD (e) (f) Fig. 5 Project and hedge portfolio cash ﬂows. a–d The distributions of the present value of the cash ﬂows (in SEK) over a 1-week horizon given stochastic cash ﬂows. The central mark in the boxes is the median, the bottom and top edges represent the 25th and 75th percentiles, and the + signs beyond the outer marks represent ‘outliers’ (see MATLAB, boxplot()). e, f The expected cash ﬂows in the four currencies (blue- yellow bars) along with the cash ﬂows resulting from the optimal hedge portfolios (variance—black, expected shortfall—red), given deterministic and stochastic cash ﬂows respectively show the distributions for the unhedged project given deterministic and stochastic cash ﬂows respectively. We see that the risk for the project, if left unhedged, is signiﬁcant for both cases. The standard deviations are 2.64, and 2.73% over the 1-week horizon for the deterministic and Number of currency units MSEK MSEK Number of currency units MSEK MSEK 56 Ann Oper Res (2018) 266:35–69 Fig. 6 Distributions of unhedged and optimally hedged cash ﬂows. a, c The distributions of project value (present value of expected cash ﬂows in SEK) in 1 week given unhedged cash ﬂows, with and without uncertainty in the cash ﬂows respectively. b, d The distributions resulting from minimization of variance and expected shortfall stochastic cases respectively. We can also observe that the shape of the distribution changes only slightly as uncertainty (σ = 5%, yearly) is added to the project cash ﬂows. We next τ, j examine the distributions implied by the optimal hedge portfolios. These are presented in panels (b) and (d) for the cases with deterministic and stochastic cash ﬂows respectively. Starting with panel (b), we see that the variance strategy is able to eliminate all variation in project value. Given deterministic cash ﬂows, this result is expected as long as the asset universe is rich enough to allow perfect cash ﬂow matching. Examining the ES strategy, we see that the resulting distribution has a non-zero variance, but importantly, a higher project value than the variance strategy in all scenarios. For the case with stochastic cash ﬂows presented in panel (d), none of the strategies can eliminate all variation in project value. The distribution implied by the variance strategy shows a slightly lower dispersion in project values but is again shifted to the left, relative to the ES strategy, and hence produces a hedge with higher expected costs. The analysis of cash ﬂow structure and project value distribution undertaken so far suggests that the ES strategy is selective in the choice of hedging instruments, and that the expected cost is an important criterion in determining the optimal hedge. On the other hand, the variance strategy seems to produce hedge portfolios that are more costly, but as expected, imply lower variation in project value. 123 Ann Oper Res (2018) 266:35–69 57 Table 4 Risk exposure statistics for the unhedged (UH), variance (Var), and expected shortfall (ES) strategies Currency exposure Fisher–Weil duration NI’s ΔPV EUR GBP USD EUR GBP SEK USD Deterministic cash ﬂows UH −11.08 −15.08 332.95 5.29 5.25 5.25 5.11 – – Var 11.08 15.08 −332.80 5.29 5.25 5.15 5.13 66 0.84 ES 11.51 15.20 −334.38 6.05 5.40 5.48 4.98 9 9.03 Stochastic cash ﬂows UH −11.08 −15.08 332.95 5.29 5.25 5.25 5.11 – – Var 11.07 15.08 −332.78 5.28 5.25 5.16 5.13 66 1.01 ES 33.48 32.69 −365.75 8.83 8.00 2.15 4.58 7 14.13 The optimal hedge portfolios are determined using (3) with the objectives in (4)and (5), given deterministic and stochastic cash ﬂows. Currency exposure represent the present value of the project (UH) and hedge (Var, ES) cash ﬂows, and is expressed in millions of each currency. NI’s denotes the number of (unique) instruments traded, and ΔPV the increase in expected project value (MSEK) relative the unhedged case. Scenarios for the stochastic cash ﬂows have been generated with an order size volatility of 5% (yearly) To further investigate properties of the optimal hedge portfolios, we present in Table 4 information about risk exposure broken down into risk factors, the number of (unique) instru- ments in the hedge portfolios, and the change in project value implied by the optimal hedges. The hedgeable risk in the problems studied is composed of currency and interest rate risk. A majority of the interest rate risk can be attributed to shifts in the yield curve and is commonly measured as duration. The currency risk can be eliminated simply by matching the initial exposure, and the risk coming from shifts in the yield curve is hedged by creating a portfolio with off-setting duration. We present properties of the optimal hedge portfolios resulting from variance (Var), and ES minimization along with the unhedged case (UH), i.e. the project cash ﬂows, in Table 4. To capture the risk from shifts in the yield curve, we study the Fisher–Weil duration which is deﬁned as −y t i i D = t c e , i i i =1 where c is the cash ﬂow at time t , y the corresponding continuously compounded spot rate, i i i and P the present value of the set of cash ﬂows. Starting with the variance strategy, we see from Table 4 that: (i) the currency and duration risks are well hedged; (ii) all available contracts are entered; and (iii) the hedge is costly when compared to the ES strategy. On the other hand, the ES strategy leaves exposure to the major risks, enters only a subset of the available hedging instruments, and has low(er) expected costs. These observations conﬁrm the results obtained so far, but raise a new question; namely why it is optimal to remain exposed to the major risks under the ES strategy, speciﬁcally given non-zero cash ﬂow volatility. To help answer this question, we present supplementary results in “Appendix C”. Table 6 adds results for cash ﬂow volatility of 1, 3, and 10%, Table 7 presents the composition of the optimal hedge portfolios, and Table 8 shows the expected returns of the different hedging instruments. Based on these results the pattern is clear; with increased cash ﬂow volatility the ES strategy produces larger positions in contracts with It is also common to study the Macaulay duration, see e.g. Zheng (2007). 123 58 Ann Oper Res (2018) 266:35–69 (relatively) high expected returns, implying that the (expected) project value increases, but necessarily, also creates residual exposure to the major risks. To understand the rationale for this, recall ﬁrst that ES can be reduced by shifting the distribution. Note also that the cash ﬂows are assumed to be uncorrelated with the other risk factors. Hence, with increased cash ﬂow volatility, the marginal contribution to the total risk from foreign exchange and interest rates decreases. This effect allows larger positions in contracts with (relatively) high expected returns, that positively affect ES, to be entered. The optimal hedge portfolio captures the trade-off between the positive effect on ES from shifting the distribution, and the negative effect of more tail-events. Finally, we note that the relative deviation in currency exposure is signiﬁcantly larger in EUR and GBP than in USD. However, the relevant measure of the effect on project value, and thus on ES, is the absolute deviation. A lesson to learn from this numerical example is that variance and expected shortfall may produce structurally different hedge portfolios when the set of hedging instruments induces different costs. We have seen that the variance strategy eliminates the major risks, and that this is achieved by trading in a large set of hedging instruments, possibly at a (relatively) high cost. On the contrary, the ES strategy sets up a hedge that carefully considers the expected cost, and potentially leaves exposure to major risks. In the problem instance studied, the difference in expected costs between contracts comes from varying term premia, but may more generally be a consequence of e.g. non-homogeneous transaction costs or liquidity premia. Now we leave the comparison of the variance and ES strategies and go on to analyze optimal hedging given ﬁrm objective deﬁned in terms of mean log-project value, and with constraints on ES. We analyze the risk-return proﬁle resulting from this problem formulation, and in addition, we investigate the impact of the asset universe available to the risk manager. For this purpose, we study three different cases with asset universe containing; (i) 1-year currency forwards (3 assets), (ii) 1-year currency forwards and 5-year IRSs (7 assets), and (iii) 1–10 year forwards and 2–10 year IRSs (66 assets). The ﬁrst case allows the major risk, i.e. the currency risk, to be hedged, but gives little ﬂexibility in hedging interest rate risk. By adding 5 year IRS some, but not all of the interest rate risk can be hedged. Note also that with only one contract per market and asset type, there is little ﬂexibility for choosing hedging instruments based on their expected cost. The case with all assets is set-up to be ﬂexible enough to hedge the currency and interest rate risk while allowing for a selective choice of hedging instruments. Compared to the previous analysis, we here focus solely on the case with stochastic cash ﬂows (σ = 0.05) and we remove the position constraints that were used to handle the τ,1 location-invariance of the variance strategy. We study the maximization of mean log project value for different upper limits on ES and present this in terms of the equivalent Lagrangean formulation given in (7), with λ ∈[0, 1]. The extreme case with λ = 0 corresponds to the most restrictive limit on ES for which a feasible solution exists, and is equivalent to minimization of ES. The other extreme, namely λ = 1, corresponds to a limit on ES that is loose enough to make the constraint redundant, and is equivalent to maximization of the mean log project value. To illustrate the relation between risk and hedging costs for different limits on ES, we present the efﬁcient frontier as the trade-off between ES and the increase in (expected) project value. The results for the full asset universe and the two restricted versions, together with the unhedged case, are presented in Fig. 7. Removing cash ﬂow uncertainty does not alter the main conclusions of the presented analysis. 123 Ann Oper Res (2018) 266:35–69 59 All assets 1y forwards & 5y IRS 1y forwards 40 Unhedged 0 102030 -10 0 50 100 150 200 250 300 350 400 450 ES (MSEK) 0.95 Fig. 7 The efﬁcient frontier determined using (3), with objective function (7), for λ = 0, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.98, 1. Project cash ﬂow scenarios are generated using (17) with σ = 0.05. The magniﬁed area shows results for λ up to 0.9, 0.95, and 0.98 for the cases with all assets, τ,1 8 assets and 3 assets respectively. The increase in project value is calculated as the difference between the expected project value in 1 week given the optimal hedging decision, and the project value at the starting date, 1334 MSEK We see that the efﬁcient frontier generated from the case with access to the full asset universe clearly dominates the restricted cases. Focusing ﬁrst on the risk reduction poten- tial, we note that using all the hedging instruments available ES can be reduced by 95.3% compared to the unhedged case (from 76.8 to 3.6 MSEK). Comparing this to the restricted case with 1-year currency forwards only, the upper limit on risk reduction in terms of ES is 62% (to 29.2 MSEK). Adding 5-year IRSs to the menu of hedging instruments allows ES to be reduced by approximately 69% (to 24.2 MSEK). Examining next the risk-return proﬁles, we see that access to a richer asset universe offers additional ﬂexibility for the risk manager to choose a cost-efﬁcient hedge. This is illustrated by the increased slope of the efﬁcient frontier as more hedging instruments are added to the asset universe. Note however that the upper-right areas of the efﬁcient frontiers, corresponding to unconstrained maximization of mean log-project value, produce portfolios with large risk. For the case with access to the full asset universe, the resulting ES is almost 34% of the initial project value, which is a risk level that is unlikely to be acceptable to most corporate risk managers. From the perspective of practical risk management, we argue that the relevant formulations are the cases with (low) limits on ES. In the out-of-sample analysis that follows, we will therefore focus on the cases producing low risk in the project, but still investigate the impact of slightly looser limits on ES. 5.3 Out-of-sample analysis The in-sample analysis provided insights about properties of the optimal hedge under different model speciﬁcations. First, comparing the ES and variance strategy, we observed that the latter produced hedge portfolios containing more assets and with higher expected hedging costs. Increase in project value (MSEK) 60 Ann Oper Res (2018) 266:35–69 We also learned from the in-sample analysis that the asset universe made available to the risk manager can have a signiﬁcant impact on the potential to set up a cost efﬁcient hedge that reduces risk properly. Finally, we learned that the allowed level of risk can have a substantial impact on the expected hedging cost. With the out-of-sample analysis, we aim to test on out-of-sample data, (i) whether these ﬁndings can be validated, and (ii) if the proposed model is robust enough to be a candidate for practical decision support. The out-of-sample tests are based on the realized foreign exchange and interest rates over the studied period, along with simulated trajectories for the project cash ﬂows. The procedure starts with the optimal hedging decision determined on July 26th, 2013. We then move forward 1 week in time, and with reference to (3), update the pre-decision holdings in cash, h, and the cash ﬂow vector b. We handle hedging positions as the equivalent set of cash ﬂows and collect them in the vector b along with the expected project cash ﬂows. We assume that the asset universe available to the risk manager is the same in every period, which is a relevant assumption considering how OTC contracts are quoted in the market. Hence, previously entered contracts can not be traded. The project cash ﬂows are assumed to have ﬁxed dates, and so get closer for each week the clock advance. In the analysis that follows, we make comparisons with the in-sample results which are based on the objectives in (4), (5)and (7). As discussed in Sect. 5.2, the location-invariance property makes variance an improper objective in the studied environment with transaction costs, and given access to both long and short positions. As we move from a single period to sequential decisions, position constraints are no longer appropriate, as they limit the ﬂexibility needed to control the risk under changing market conditions. To handle the implications of the location-invariance while still allowing comparisons with the in-sample results, we study instead the mean squared deviation from the (current) project value capitalized by the risk free rate. As in the in-sample analysis, we begin by investigating ES and (modiﬁed) variance, and we focus ﬁrst on the case with deterministic cash ﬂows. Figure 8 shows the realized project value trajectories given the variants of the asset universe studied in the in-sample analysis. We also present the standard deviation, σ (yearly), of the relative changes in project value along each trajectory. We note ﬁrst that both strategies produce trajectories with (relatively) low variation in project value, which is a ﬁrst requirement for the proposed model to be considered as robust. As a comparison, the unhedged portfolio value has a volatility of 11.3% (yearly) over the same period. Examining next the impact of the asset universe, we see that the risk, in terms of variation in project value, decreases signiﬁcantly as the number of available hedging instruments is increased. This validates the results obtained in-sample of the importance of having access to a large-enough asset universe to properly control risk. Comparing the trajectories from the two strategies pairwise for each asset universe, the ES strategy produces higher project values in the ﬁnal period in all cases. Hence, over the studied period, the hedging cost resulting from using the ES strategy would have been lower relative the variance strategy, irrespective of the asset universe available for hedging. We note, however, that more data is needed to show statistically signiﬁcant differences in expected hedging costs. We generate trajectories of project cash ﬂows using (17). 19 N 2 With the notation in (3), the suggested objective is p (z − z R) ,where z is the pre-decision i i 0 i =1 project value in each period. The decline in project value around period 10–40 for the variance strategy with 1-year forwards and 5-year IRSs (blue trajectory), is (partly) explained by long and short positions in the same hedging instruments. This conﬁrms the potential problems of using objectives that does not properly capture the impact of expected returns in an environment with transactions costs and premia. 123 Ann Oper Res (2018) 266:35–69 61 Min ES Min var 1.44 1.45 All assets All assets 1y forwards & 5y IRS 1y forwards & 5y IRS 1y forwards 1.42 1y forwards 1.4 σ=3.83% 1.4 σ=3.84% 1.35 1.38 σ=0.084% σ=1.14% 1.36 1.3 σ=0.099% 1.34 1.25 σ=1.39% 1.32 1.2 0 50 100 150 200 1.3 0 20 40 60 80 100 120 140 160 180 200 Period (week) Period (week) (a) (b) Fig. 8 Out-of-sample trajectories of project value resulting from solving (3) sequentially over the out-of- sample period of 191 weeks, given minimization of a modiﬁed variance, and b ES, for different sets of hedging instruments, given deterministic cash ﬂows. The project value volatilities, σ (yearly), are the standard deviations of the relative change in project value over each trajectory Fig. 9 The nominal value of entered hedging instruments (in SEK) in period 1,..., 191, in the 66 hedging instruments (1–10 year currency forwards, F, and 2–10 year IRSs, I), given minimization of a variance, and b expected shortfall In Fig. 9, we show the absolute nominal values (in SEK) entered in the different hedging instruments over all out-of-sample periods for the case with the full asset universe. The nominal values correspond to the number of contracts entered in each hedging instrument times the prevailing foreign exchange rate in the respective market (with SEK as the term currency). Comparing the concentration of bars in panels (a) and (b) of Fig. 9, we see that the variance strategy implies that signiﬁcantly more instruments are entered to control the risk. The average number of contracts entered per period is 61.2 for the variance strategy while it is only 6.96 for the ES strategy. Hence, the observation in the in-sample analysis that the variance strategy trades in signiﬁcantly more instruments is validated in this ﬁrst out-of- sample experiment. As expected, the largest positions are entered in the ﬁrst period when the cash ﬂows are still unhedged. This is most pronounced for the ES strategy, for which the ﬁrst-period hedging constitutes 89.4% of the total nominal hedging value aggregated over all periods. In the second numerical experiment we add cash ﬂow uncertainty by generating scenario trajectories for the project cash ﬂows, and study these along with the realized market prices Project value (bnSEK) Project value (bnSEK) 62 Ann Oper Res (2018) 266:35–69 Table 5 Out-of-sample statistics given uncertain cash ﬂows Variance Expected shortfall Unhedged All assets 7 assets 3 assets All assets 7 assets 3 assets σ(%) 5.59 5.75 6.64 5.65 5.83 6.74 12.08 NI’s 41.22 4.63 2.76 6.21 2.38 1.35 – The volatility, σ (yearly), is the standard deviation of relative project value changes along trajectories. NI’s is the average number of traded instruments per period. The results are based on realized market prices on FX and interest rates over the out-of-sample period, along with 50 scenario trajectories for the project cash ﬂows (σ = 0.05). Results are presented for the cases with all assets, 1-year forwards and 5-year IRSs (7 assets), τ,1 1-year forwards (3 assets), and ﬁnally the unhedged case 2.5 1.35 λ=0 λ=0.2 1.348 λ=0.3 λ=0.4 1.346 λ=0.5 1.5 1.344 λ=0, σ=0.099% 1.342 λ=0.2, σ=0.0997% λ=0.3, σ=0.100% 1 λ=0.4, σ=0.102% 1.34 λ=0.5, σ=0.111% 1.338 0.5 1.336 1.334 1.332 -0.5 0 50 100 150 200 0 50 100 150 200 Period (week) Period (week) (a) (b) Fig. 10 Out-of-sample trajectories resulting from (3) with the objective function in (7). a The realized project value trajectories given λ = 0, 0.2, 0.3, 0.4, 0.5. b The performance of the different objectives relative the ES minimization case (λ = 0) on foreign exchange and interest rates. In Table 5, we present summary statistics for the variance and ES strategies given 50 scenario trajectories for the stochastic cash ﬂows. Consistent with the results obtained so far, we see that: (i) the proposed optimization model shows evidence for being robust in handling risk; (ii) hedging with the variance strategy induces trading in many more instruments than does the ES strategy; (iii) the number of hedging instruments made available to the optimization model has a clear impact on the potential to reduce risk. The ﬁnal numerical experiment studies the maximization of mean log project value given constraints on ES for the case with deterministic project cash ﬂows. Panel (a) of Fig. 10 presents the out-of-sample trajectories resulting from applying (3) with objective function (7), for different limits on ES. Panel (b) shows the differences in trajectory values relative the ES minimization case. We focus on the cases with relatively strict limits on ES, corresponding to λ ≤ 0.5. Although the project values resulting from looser limits on ES (λ> 0.5) proved to produce higher project values for the period studied, the resulting hedge implies increasingly riskier portfolios, and is unlikely to be acceptable to a risk manager. We recall from the in-sample analysis, that unconstrained maximization of mean log project value is associated with large risks. By comparing the trajectories in Fig. 10, we see from panel (a) that a looser constraint on ES produces a (slightly) higher volatility in project value. On the other hand, as highlighted in panel (b), the slightly increased risk implies a reduced hedging cost, which is consistent with the risk-return proﬁle from the in-sample analysis in Fig. 7.Wenoteagain,however, Project value (bnSEK) Difference in project value (MSEK) Ann Oper Res (2018) 266:35–69 63 that the amount of data is insufﬁcient to obtain statistically signiﬁcant differences in expected hedging costs. 6 Concluding remarks In this study, we have developed a framework for the optimal hedging of foreign exchange and interest rate risk given uncertain cash ﬂows in multiple currencies. We carefully consider the environment faced by the risk manager in that we include transaction costs, the empirically well documented term premia, non-normal foreign exchange rates, and trading at market prices. We study optimal hedging given three alternative objective functions: namely variance; ES; and mean log proﬁts with limit on ES. The numerical results show that (i) the choice of objective function can have signiﬁcant implications on the composition of the optimal hedge, the resulting risk, and the hedging costs, (ii) the size of the asset universe made available to the risk manager is important for the ﬂexibility to control risk, and to set up a cost-efﬁcient hedge, (iii) the expected cost of different hedging instruments, governed by term premia and transaction costs, is a fundamental determinant of the optimal hedge, and (iv) the model is robust when applied to out-of-sample data. The proposed stochastic programming framework offers important ﬂexibility, in that model components can be easily exchanged and modeling assumptions re-examined, in order to add even more realism to the model. As the framework provides optimal portfolio hold- ings as well as concrete measures of risk, it has the potential to serve as a useful and ﬂexible decision support tool for risk managers. Acknowledgements We thank Ou Tang, Pontus Söderbäck, Mathias Barkhagen, three anonymous reviewers, and seminar participants for helpful comments. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. A Instrument pricing For an exchange rate e ∈ E,welet thefunction c (e) determine the base currency and c (e) b t the term currency, e.g. for e = EUR/SEKwehavethat c (e) = EUR and c (e) = SEK. b t Also, let 0 denote the root node of the stochastic programming problem. Given the interest rate swap (IRS) implied spot rates, r (τ ), in currency c ∈ C, with time to maturity τ,and c,0 the spot exchange rates today, f , for exchange rate e ∈ E, the forward price for long and e,0 short positions are L F κ = f exp{ r (τ ) − r (τ ) τ}+ η , e,0 c (e),0 c (e),0 e,τ t b e S F κ = f exp{ r (τ ) − r (τ ) τ}− η , e,0 c (e),0 c (e),0 e,τ t b e where τ = 1,..., 10 and η is the transaction cost for a forward position in exchange rate e ∈ E. A long position is an agreement to buy the foreign currency c (e), whereas a short position is an agreement to sell the foreign currency. In scenario i, 1 week from today, the spot rate in currency c ∈ C, for time to maturity τ,is r (τ ). The values of long and a short c,i positions in currency forwards in scenario i are given by 123 64 Ann Oper Res (2018) 266:35–69 f exp{ r (τ − Δt ) − r (τ − Δt ) (τ − Δt )}− κ e,i c (e),i c (e),i F,L t b e,τ P = , e,τ,i exp{(τ − Δt )r (τ − Δt )} c (e),i f exp{ r (τ − Δt ) − r (τ − Δt ) (τ − Δt )}− κ e,i c (e),i c (e),i F,S t b e,τ P = , e,τ,i exp{(τ − Δt )r (τ − Δt )} c (e),i where τ = 1,..., 10, and Δt = 1/52. The price to enter a currency forward is zero. When modeling an IRS it is assumed that the ﬁxed leg is paid annually. The consistent bid and ask rates are −r (τ )τ −r (τ )τ c,0 c,0 1 − e 1 − e L I S I s = − ρ and s = + ρ , c,τ τ c c,τ τ c −r ( j ) j −r ( j ) j c,0 c,0 e e j =1 j =1 where ρ is the transaction cost in currency c ∈ C. To simplify modeling, the IRS is modeled I,L I,S as a coupon bond. The initial prices are P = P = f for c ∈{EUR, GBP, USD}, c,0 c,τ,0 c,τ,0 I,L I,S and P = P = 1. The coupon for a long position is s ,and theprice in SE K ,τ,0 SE K ,τ,0 c,τ scenario i for a Swedish IRS is I,L L −( j −Δt )r ( j −Δt ) −(τ −Δt )r (τ −Δt ) SEK,i SEK,i P = s e + e , SEK,τ SEK,τ,i j =1 for τ = 2,..., 10, and for IRSs in currency c (e), e ∈ E, ⎛ ⎞ I,L −( j −Δt )r ( j −Δt ) −(τ −Δt )r (τ −Δt ) c (e),i c (e),i ⎝ b b ⎠ P = f s e + e , e,i c (e),τ c (e),τ,i b j =1 for τ = 2,..., 10. For a short position the coupon paid is s , which gives the price for c,τ Swedish IRSs, I,S S −( j −Δt )r ( j −Δt ) −(τ −Δt )r (τ −Δt ) SEK,i SEK,i P = s e + e , SEK,τ,i SEK,τ j =1 for τ = 2,..., 10, and similarly for bonds in currency c (e), e ∈ E, ⎛ ⎞ I,S −( j −Δt )r ( j −Δt ) −(τ −Δt )r (τ −Δt ) c (e),i c (e),i ⎝ b b ⎠ P = f s e + e , e,i c (e),τ c (e),τ,i b b j =1 for τ = 2,..., 10. B Monte Carlo simulation c T ×3 Let X ∈ R be the T historical state variable observations for the interest rates in EUR GBP SEK USD currency c ∈ C.For X = (X , X , X , X ), the historical changes in the R (T −1)×12 state variables, X ∈ R , are calculated as X = X − X , t = 2,..., T and j = 1,..., 12. t, j t −1, j t, j e T ×1 Similarly, let F ∈ R , e ∈ E, be the historical foreign exchange rates. Deﬁne F = EUR/SEK GBP/SEK USD/SEK (F , F , F ), and let the transformed historical logarithmic R (T −1)×3 returns, F ∈ R , be calculated as R −1 F = F (F (ln(F /F ))), t = 2,..., T and j = 1,..., 3, e,t −1 t, j t −1, j t, j 123 Ann Oper Res (2018) 266:35–69 65 −1 where F is the inverse of the standard normal cdf,and F , e ∈ E,isthe cdf given e,t −1 in (16). Collect the returns, that after the transformation should be close to Gaussian, in G R R G (T −1)×15 R = (X , F ), R ∈ R . The (yearly) covariance matrix can be computed as G G C = R R − μμ Δt, (T − 2)Δt 15×1 where μ ∈ R is calculated as T −1 μ = R , j = 1,..., 15, and Δt = 1/52. t, j (T − 1)Δt t =1 By performing a Cholesky decomposition of the covariance matrix, C = LL , random numbers with preserved covariance are generated as ξ = LΞ ,where Ξ ∈ N (0, I ).Let the t t t random variables for each interest rate market and foreign exchange rate be collected in the vector f f f X X X X ξ ,ξ ,ξ ,ξ ,ξ ,ξ ,ξ = ξ , EUR,t GBP,t SEK,t USD,t EUR/SEK,t GBP/SEK,t USD/SEK,t X 1×3 where ξ ∈ R , c ∈ C,and ξ ∈ R, e ∈ E. Given the initial state x = e,t c,i,0 c,t c c c (X , X , X ) on market c ∈ C, for all scenarios i = 1,..., N, the evolution of T ,1 T ,2 T ,3 the state variables is modeled by a discretization of (10), c X x = x − K x Δt + ξ Δt , c,i c,0 c,0 where K is the feedback matrix in currency c ∈ C. Future states for the spot rates, r (τ ), c,i are completely determined by the scenarios for the state variables, x ,onmarket c ∈ C, c,i from (8)as r (τ ) = −A(τ ) + B(τ ) x , c,i c,i where i = 1,..., N. As a consequence of interest rate movements being driven by a limited number of state variables, the initial pricing errors, i.e. the difference between market and model implied prices, can be utilized by the model in creating ﬁctitious returns. By adding a random term for currency c ∈ C, and time to maturity τ , corresponding to the spot rates that are not c,τ perfectly observed, the noise in the imperfectly observed yields is also modeled. To determine a reasonable estimate of the covariation in the noise of the non-trusted spot rates we study the measurement error implied by the model, i.e. the difference between the market yields and the model implied yields. Note that these are zero by construction for the three yields that we trust, while they are (generically) non-zero for the non-trusted yields. Letting C denote the covariance matrix of the measurement errors, the noise terms, , are assumed to be c,τ multivariate normal distributed with zero mean and covariance matrix C . Finally, we describe the details in the modeling of foreign exchange rates. The variances, 2 2 2 σ ,σ ,σ , are updated according to EUR/SEK,t GBP/SEK,t USD/SEK,t e,t +1 ln − γ Δt e,t 2 2 σ = β + β σ + β , e,0 e,1 e,2 e,t +1 e,t Δt where f is the foreign exchange rate e ∈ E at time t. We model foreign exchange rates in e,t scenario i as −1 f = f exp μ Δt + σ ΔtF F (ξ ) . e,t +1,i e,t e e,t N e,t e,i 123 66 Ann Oper Res (2018) 266:35–69 C Complementary numerical results In Table 6, we present results corresponding to Table 4 for additional cash ﬂow volatilities. Table 7 shows the composition of the optimal hedge portfolios given the minimization of expected shortfall for different levels of cash ﬂow volatility. Note speciﬁcally that the two instruments with the lowest expected returns both leave the optimal portfolios immediately as cash ﬂow uncertainty is introduced. Table 6 Complementary results to Table 4 Currency exposure Fisher–Weil duration NI’s ΔPV EUR GBP USD EUR GBP SEK USD UH −11.08 −15.08 332.95 5.29 5.25 5.25 5.11 – – Deterministic cash ﬂows Var 11.08 15.08 −332.80 5.29 5.25 5.15 5.13 66 0.84 ES 11.51 15.20 −334.38 6.05 5.40 5.48 4.98 9 9.03 Stochastic cash ﬂows, σ = 0.01 Var 11.08 15.08 −332.80 5.29 5.25 5.15 5.13 66 0.85 ES 16.34 15.13 −338.64 8.02 8.00 4.75 4.96 8 10.22 Stochastic cash ﬂows, σ = 0.03 Var 11.08 15.08 −332.79 5.29 5.25 5.15 5.13 66 0.99 ES 25.45 23.84 −352.53 9.00 8.00 4.60 4.78 7 12.26 Stochastic cash ﬂows, σ = 0.05 Var 11.07 15.08 −332.78 5.28 5.25 5.16 5.13 66 1.01 ES 33.48 32.69 −365.75 8.83 8.00 2.15 4.58 7 14.13 Stochastic cash ﬂows, σ = 0.1 Var 11.04 15.09 −332.67 5.26 5.23 5.25 5.14 66 1.09 ES 53.53 55.18 −398.75 8.00 8.00 15.31 4.22 5 18.31 Table 7 Optimal hedge portfolios given different levels of cash ﬂow volatility, determined using (3)given objective deﬁned by (5) EU R EU R EU R GB P GB P USD USD USD USD USD F F F F F F F I I I 2 8 9 1 8 4 10 3 4 10 Deterministic cash ﬂows 3.79 8.93 – 5.67 11.34 −92.04 −40.41 −83.59 −83.38 −49.21 Stochastic cash ﬂows, σ = 0.01 – 18.36 0.43 – 17.97 −81.92 −52.28 −72.25 −112.96 −36.00 Stochastic cash ﬂows, σ = 0.03 – – 30.17 – 28.31 −58.23 −75.85 −110.55 −118.12 −11.42 Stochastic cash ﬂows, σ = 0.05 – 6.70 32.77 – 38.83 −47.18 −86.32 −165.47 −90.56 – Stochastic cash ﬂows, σ = 0.1 – 61.52 – – 65.54 −76.08 −79.89 −266.27 – – The ﬁgures represent the number of entered contracts (i millions). One forward contract, F, corresponds to exchanging one unit of the base currency, and the interest rate swaps, I , are deﬁned to have nominal value one in the base currency 123 Ann Oper Res (2018) 266:35–69 67 Table 8 Expected returns of 1–10 year currency forward contracts tabulated by base currency, and 2–10 year IRSs tabulated by market T Forwards Interest rate swaps EUR GBP USD EUR GBP SEK USD Long positions 1 −0.007 −0.007 −0.006 2 −0.004 −0.010 −0.017 −0.006 −0.014 −0.001 −0.018 3 −0.003 −0.018 −0.029 −0.007 −0.023 −0.001 −0.032 4 −0.004 −0.020 −0.028 −0.006 −0.023 −0.001 −0.029 5 0.000 −0.007 −0.012 −0.003 −0.012 −0.001 −0.015 6 0.012 0.021 0.010 −0.002 0.004 −0.002 −0.001 7 0.030 0.056 0.027 −0.004 0.017 −0.004 0.002 8 0.045 0.079 0.031 −0.009 0.019 −0.006 −0.008 9 0.046 0.068 0.016 −0.009 0.007 −0.007 −0.024 10 0.009 −0.010 −0.019 0.003 −0.017 −0.001 −0.026 Short positions 1 −0.001 −0.001 −0.002 2 −0.004 0.003 0.009 0.003 0.010 0.000 0.016 3 −0.005 0.010 0.022 0.001 0.017 0.000 0.028 4 −0.004 0.013 0.021 −0.001 0.016 −0.000 0.024 5 −0.008 0.000 0.005 −0.005 0.003 −0.000 0.009 6 −0.019 −0.028 −0.017 −0.008 −0.016 0.001 −0.006 7 −0.036 −0.062 −0.034 −0.007 −0.030 0.003 −0.011 8 −0.051 −0.085 −0.038 −0.004 −0.033 0.005 −0.002 9 −0.052 −0.074 −0.022 −0.006 −0.023 0.005 0.013 10 −0.015 0.003 0.013 −0.019 −0.001 −0.000 0.014 The expected returns for the forward contracts and IRSs are for one unit of base currency and nominal value one in the base currency respectively. 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Published: Nov 1, 2017
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