Long Term Exchange Rate Risk and Hedging with. Provides Short Term Futures Contracts

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1 Academia, Revista atinoamericana de Administración, 50, 0, Copyrigt 0 de Cladea, ttp://revistaacademia.cladea.org ong erm Excange Rate Risk and edging wit Quantity Uncertainty in a Market tat Only Provides ort erm utures Contracts Riesgo de tipo de cambio de largo plazo y cobertura con incertidumbre de cantidad Augusto Castillo Universidad Adolfo Ibáñez, antiago, Cile Rafael Aguila Pontificia Universidad Católica de Cile, antiago, Cile Jorge Niño Universidad Adolfo Ibáñez, antiago, Cile Abstract is paper analyzes te problem faced by an investor expecting to receive an uncertain amount of cas flow in a foreign currency on a certain future date. e investor is also assumed to be exposed to long-term excange rate risk, and as access only to sort-term futures contracts to edge. A closed form solution for bot te optimal edging strategy and te quality of te edging are identified. Next, we explored ow tose solutions depend on some key factors suc as te volatility of te excange rate, te volatility of te amount of foreign currency to be received and te degree of correlation between all te stocastic variables considered. Key words: Risk management, edging, quantity uncertainty. Resumen Este artículo analiza el problema que enfrenta un inversionista que espera recibir una cantidad incierta de dinero en una moneda extranera en una cierta feca futura. e asume que el inversionista está expuesto a un riesgo de largo plazo, teniendo acceso solo a contratos futuros de corto plazo para efectuar cobertura. e identifica una solución analítica tanto para la estrategia de cobertura óptima como para la calidad de la cobertura alcanzable. ambién se explora cómo la solución obtenida depende de ciertos factores clave tales como la volatilidad del tipo de cambio, la volatilidad de la cantidad de moneda extranera a recibir y la correlación entre las variables estocásticas consideradas. Palabras clave: administración de riesgo, cobertura, incertidumbre de cantidad. 66 Academia, revista latinoamericana de administración, 49, 0

2 Castillo, Aguila y Niño. Introduction irms tat ave teir sales indexed to a foreign currency and teir expenses indexed to a local currency are exposed to excange rate risk. ose investors could protect temselves by taking sort positions in excange rate forward or futures contracts. Normally, te edging would not improve te expected outcome in local currency, but would make te cas flows more certain and reduce exposure to risk. Previous researc on wy firms edge, suc as tose undertaken by mit and tultz (985), Bessembinder (99), root, carfstein and tein (99), and Mello and Parsons (995) ave identified te desire to minimize te variance of future cas flows, to reduce te volatility of taxable income, te desire to reduce dispersion of accounting earnings, or even te ope of being able to avoid financial distress as te main reasons for edging. Neuberger (999) assumes tat te desire for edging comes from risk averse agents wising to maximize expected utility. e use of forward or futures contracts to edge against excange rate risk works less tan perfectly in te real world for several reasons. irst, te excange rate we want to edge from may not be te same as te excange rate considered in te futures contracts available in te market. In tis case te quality of te edge will critically depend on ow closely correlated tose two excange rates are. is point as been developed in all te maor derivatives and risk management textbooks. ee for example Duffie (989), tulz (00) or ull (008). A recent paper by Basak and Cabakauri (0) gives new insigts on ow to solve tis problem troug different tecniques. econd, te date of expiration or maturity of te future or forward contracts available to perform te edging may not coincide exactly wit te particular date in te future we will receive te foreign currency. is could appen for example if tere are only sort-term futures contracts available to edge against long-term excange rate exposure or if tere are long-term futures contracts to edge against sort-term excange rate exposure. It as been proven tat using sort-term forward contracts to edge against long-term exposure would allow us to reac perfect edging if interest rates are non-stocastic and tere is no uncertainty in quantity, and oter conditions regarding availability of contracts are fulfilled. or example, Brennan and Crew (997) present a model were a simple or tailed stack and roll strategy allows us to reac perfect edge wen deterministic interest rates are assumed and wen te rollover gain derived from differences between prices of different futures contracts at te maturity of some of tem are ignored. Neuberger (999) sows tat even wit deterministic interest rates, perfect edging would not be reaced unless forward prices can be predicted in advance and perfectly. e assumes tat te price at wic a contract first trades is a stocastic function of te prices of oter contracts already trading in te market, and by assuming tat te expected value of te opening price is a linear function of te prices of oter contracts, te autor proves tat tere is a unique edging strategy, independent of te agent s utility function, tat dominates all te oter strategies in te sense of second order stocastic dominance. e metodology proposed by Neuberger allows us to remove 85% of te risk of a six-year oil supply commitment. cwartz (997) compares tree models of te stocastic beavior of commodity prices, taking mean reversion into account. e also analyzes te implications of tose models for edging risk exposure. Broll, Wal and Zilca (999) analyze edging in a multiperiod framework for a risk averse exporting firm facing excange rate uncertainty. Castillo and efort (00) and Castillo (00) sow ow a firm could obtain optimal edging against excange rate exposure using sort-term futures contracts wen it is known tat a single amount of foreign currency will be received in a certain long term-future period, and interest rates are stocastic. ey present analytical solutions wen feasible and simulation-based solutions oterwise. All te papers mentioned in tis paragrap assume te company is restricted to edge in te long term troug conseo latinoamericano de escuelas de administración, cladea 67

3 ong erm Excange Rate Risk and edging wit Quantity Uncertainty in a Market te use of sort term contracts and ignore te quantity uncertainty problem. A tird reason tat edging works less tan perfectly will arise if tere is uncertainty regarding te amount of foreign currency to be received (or in te number of units of a certain good available for selling). is problem as been analyzed by, among oters, Rolfo (980), Newbery and tiglitz (98), Kamgaing (989), Kerkvliet and Moffett (99), Moscini and apan (995), Wong (00), Näsäkkälä and Keppo (005), Castillo and Aguila (005), Castillo and Aguila (008), restad (009), Oum and Orem (00) and Korn (00). Rolfo (980) obtains an expression for te optimal edging strategy under price and quantity uncertainty if te investor s preferences are properly represented by a logaritmic utility function. Newbery and tiglitz (98) solve a similar problem by maximizing a constant risk aversion utility function. Kerkvliet and Moffett (99) reac an expression for te optimal edging strategy wen tere is uncertainty in te amount of foreign currency to be received if te obective is to minimize te cas flow variance of local currency to be received. Moscini and apan (995) solve for te optimal edging strategy wen facing quantity and price uncertainty and basis risk. ey sow tat te quality of te edging can be improved by using option contracts. Kamgaing (989) derives an optimal edging strategy under quantity, price and excange rate uncertainty assuming tat te producer is a mean-variance maximizer wit an exponential utility function. Wong (00) examines te optimal edging decision of a competitive exporting firm facing excange rate risk and price risk and only wit access to futures and option contracts on te excange rate to edge. e paper assumes te firm wants to maximize expected utility and concludes tat edging wit options can be better tan edging wit futures for some cases were te price is negatively correlated wit te excange rate. Näsäkkalä and Keppo (005) assume a firm facing price and quantity risk and wanting to minimize te variance of te cas flow. ey also consider tat te only tools available to perform te edging are futures contracts. Due to transaction costs and illiquidity concerns tey coose not to consider te possibility of using options to edge. Castillo and Aguila (005) and Castillo and Aguila (008) find te optimal edging strategy wen facing price and cost risk and quantity uncertainty, if te company is trying to minimize cas flow variance, and only futures contracts are available to edge. restad (009) considers nonfinancial firms facing edgeable price risk, unedgeable quantity risk and te presence of financial contracting costs. e paper concludes tat varianceminimizing edging strategies are very close in economic terms to optimal, value-maximizing edging strategies for most firms and tat te marginal gains from sifting to nonlinear edging strategies are often small enoug to be neglected. Oum and Orem (00) sow tat wen a firm faces a multiplicative risk of price and quantity, its profit is nonlinear in price, and it cannot be fully edged by a forward or futures contract, wic as a linear payoff structure. Korn (00) sows tat for some ranges of correlation between price and quantity edging wit futures can be optimal, but for some oter ranges of correlation te quality of te edging will improve if non linear instruments (options) are used. In general te papers cited in tis paragrap assume tat tere is a problem of uncertainty in quantity, but assume te availability of contracts wit te proper time extension. Our study analyzes ow to obtain te optimal edging strategy if only sort-term futures contracts are available to edge an uncertain amount of foreign currency tat is expected to be received on a long-term future date. We are te first researcers to consider bot problems at te same time. We will assume tat te absence of option contracts to edge is due to two reasons. e first is tat te impact of using options to edge as already been studied. e second is tat futures (or forward) contracts on excange rates are available in most atin American countries, but option contracts are eiter not available or igly illiquid wen available. In tis paper we will solve In addition to tose reasons we can add two more. irst, according to restad (009), te improvement in te quality of te edging wen using options instead of futures 68 Academia, revista latinoamericana de administración, 50, 0

4 Castillo, Aguila y Niño te problem assuming tat interest rates are not stocastic (or tat we are able to remove tat uncertainty troug interest rate forwards). In a second stage (in our next paper), we will allow interest rates to be stocastic. e quality of te edging and te dependence of te quality of te edging on some key parameters will also be analyzed. We will assume bot te absence of transactions costs and tat te futures contracts available are infinitely divisible. ese are standard assumptions in te cited literature. is paper is organized as follows. In section, te long term excange rate risk edging problem faced by te investor is described and te optimal analytical solution is presented, assuming tat sort term corresponds to one period and long term corresponds to periods. ome particular cases are also presented. ection reports te implementation of te optimal edging strategies described in section to a particular case and explores ow efficient tose optimal edging strategies are under a series of different scenarios. ection 4 presents te main conclusions of tis paper.. e ong erm edging Model et us assume tat tere is an investor wo is expecting to receive a certain amount of foreign currency, periods from now. or now we will suppose tat te investor is facing only two sources of uncertainty, wic are te exact amount of foreign currency to be received and te amount of cas to be received once e converts te foreign currency into local currency. If only one-period forward contracts are available to edge, e could edge using tose sortterm forward contracts troug te following procedure: e would ave to take at eac period t, starting at t = 0, and until period t =, t t positions in forward contracts. ose positions, taken at eac period t, would ave to be rebalanced at te expiration of te contracts, in eac of te next periods. If we name te amount of foreign currency tat we will receive in, Q, is in general not very significant. And te empirical evidence presented by, for example, Gay, Nam and urac (00) and uang, Ryan and Wiggins (007) sow tat most companies edge troug te use of forwards or futures contracts. and let represents te excange rate at, te cas flow generated by te company if no forwards are used to edge will be described by te following equation: C N = Q () If we assume tat te firm edges and if we suppose tat all te gains or losses generated by te positions taken in one-period forward contracts over te periods from t = 0 to t = are transformed to a cas flow in period, te following expression represents te total cas flow tat te company would generate at : C = Q ( ) t t t = 0 t t t t () were t t is te forward price fixed at t for a contract expiring at t ; t t represents te number of positions taken in tose forward contracts in period t; and r corresponds to te risk-free interest rate in te local currency. Under deterministic interest rates it is possible to assume tat t t will be computed as: = r ) t t t () were r corresponds to te risk-free interest rate in te foreign currency. Replacing () in () we obtain te cas flow in as a function of te value of te underlying asset in eac period t = 0 to t =, as sown by te following equation: t t C = Q t t t r = (4) t r ) r ) t 0 0 o find te optimal edging strategy we ave to assume tat te investor is optimizing a certain function. et s assume e is trying to minimize te variance of te cas flow to be On eac period t tis risk free interest rate could be different to te one observed te previous period but we will ignore tat to avoid unnecessary complexity. conseo latinoamericano de escuelas de administración, cladea 69

5 ong erm Excange Rate Risk and edging wit Quantity Uncertainty in a Market received in. e following expression represents te variance of te cas flow we want to minimize. var C ( )= var ( Q ) t t ( t) t t r ) t = r tt cov ( Q, ) t t t t t = r cov, Q t t r t t = t t r r t t t t t t = u= t r u u uu r t t t u r tu (5) Once we minimize te variance of te cas flow we obtain te following optimal edging strategy for te company: = =,,, Σ i i r ) r Σ i (6) were represents te excange rate variance-covariance matrix and i represents a matrix composed by te same components of wit te only exception of column i, originally a vector wit Covariance( i, ) x wic as been replaced by te Covariance(,Q ) x vector (were =, ). Once te company as implemented te optimal edging strategy described ere te level of maximum efficiency, defined as te proportion of te total cas flow variance tat will be reduced by edging, can be represented as seen in equation 7. Max Efficiency ( C )= i i All te previous expressions for te general case will cange to describe some particular possibilities tat are considered as interesting. If all te excange rates are independent, te following expressions represent te optimal edging strategies and te maximum efficiency tat can be reaced: Max = = Efficiency ( Q i) cov, ii i r Q / i i r = R i i i (8) (9) were Q represents te slope of an O / i regression between Q and i and were R i represents te determination coefficient of te same O regression. e next particular case to be analyzed is wen Q is independent from all te (excange rates). Under tis scenario te expressions tat represent te optimal edging strategies and te maximum efficiency to be reaced will be: Max E Q = Efficiency Σ Σ cov ( Q, ) Σ E = var ( Q ) Q =,,,..., (0) () It is interesting to note tat te optimal edging strategy becomes a direct function of te expected amount of foreign currency to be received only after assuming independence between te amount of foreign currency to be received at and te excange rates. e last special case to be reviewed is wen we assume i ( ) var Q Σ ii Σ Σ i i = i (7) 70 Academia, revista latinoamericana de administración, 50, 0

6 Castillo, Aguila y Niño tat Q is deterministic. Under tose circumstances te optimal edging strategy and te maximum efficiency to be reaced become te well known results: Q = r () Max Efficiency =0. () ere te optimal edging strategy allows te variance of te cas flow at to be completely eliminated. As we said earlier, te solutions provided ere assume tat te interest rates are deterministic. e appendix at te end of tis paper presents equations () to () for te particular case wen =. In te next section we will implement te model and te solutions described ere.. Implementing te Optimal edging trategy In tis section te model described previously is implemented. We assume = so tat we are trying to edge against te volatility of te cas flow of period tree and we only ave access to one period futures contracts. ables to 4 sow ow te edging strategies and te quality of te edging cange if only one of te parameters is canged and all te oters are kept constant. 4 ection A of able sows te inputs required to implement te metodology, and ection B of able contains te results of applying te e appendix sows te formulas tat are used in tis section. 4 As sown by tables to 4 we are not able to old perfectly constant te oter parameters. is is due to te use of simulations to generate te results. able. Canging te Correlation Between Output and Excange Rates ection A: Inputs Inputs cenario cenario cenario cenario 4 cenario 5 cenario 6 cenario 7 E() E() E() E(Q) var() var() var() var(q) corr(;) 0,45 0,45 0,45 0,45 0,45 0,45 0,45 corr(;) 0,9 0,9 0,9 0,9 0,9 0,9 0,9 corr(;) 0,44 0,44 0,44 0,44 0,44 0,44 0,44 corr(q;) 0,40 0,0 0,0-0,0-0,40-0,60-0,75 corr(q;) 0,4 0,9 0,0-0, -0,4-0,6-0,79 corr(q;) 0,44 0,9 0,0-0,9-0,5-0,59-0,75 corr(q;) 0,46 0,8 0,9 0,6 0,00-0,4-0,50 corr(q;) 0,5 0,4 0,0 0,7 0,04-0,0-0,48 corr(q;) 0,84 0,79 0,70 0,59 0,58 0,45 0,8 ection B: Outputs 0-68,80-7,9-0, -59, -8,4 -,5,46-49,55-7,87-98,7-66, -6, -5,49-7,84-8,9-0,67-0,67-77,0-84,8-68,06-59,6 var(cn) varop(c) % Efficiency 74,% 6,7% 49,7% 5,5% 4,% 50,5% 90,4% conseo latinoamericano de escuelas de administración, cladea 7

7 ong erm Excange Rate Risk and edging wit Quantity Uncertainty in a Market metodology. 5 ection A of able presents 7 scenarios. e difference between tem is te assumed correlation between te excange rates t (wit t =,, ) and te amount of foreign currency to be received at =, Q. ection B of able presents te results of te optimal edging policies and maximum efficiency to be reaced under eac scenario. e quality of te edging policies increase as te correlation between te t and Q becomes eiter more positive or more negative. ese results were expected because more correlation between t and Q means tat we can more easily reduce volatility of te cas flow produced by te volatility of Q troug te use of futures contracts on te excange rate t. e level and te traectory of te edging policies are also a function of te correlation between te t and Q. It is also important to remember tat negative positions represent sort positions in future contracts and positive positions represent long positions in futures contracts. or positive correlations between te t and Q, te edging policies are all negative and iger (in absolute value) tan E(Q ). ose optimal edging strategies are also decreasing in time (in absolute value), becoming closer to E(Q ) as time passes. or negative correlations between te t and Q, te edging policies are in general negative (sort positions) and smaller (in absolute value) tan E(Q ), and could even start as positive (long positions) as sown in able. 6 ese results are important because tey sow tat te strategy of edging by a magnitude of E(Q ) is not te best solution unless tere is no correlation between t and Q, and it also sows ow to adust te optimal edging strategy wen we take in account tose correlations. igure sows te traectory of te optimal edging strategies for eac of te seven scenarios considered in able. ollowing te same procedure we could explore ow canges in te volatilities of te t and Q, and ow canges in te autocorrelation of te excange rates of different periods 5 All te results presented in tis section are te result of generating sets of random numbers wit te required caracteristics for eac scenario. 6 edging a long position in te underlying asset wit a long position in te futures contract is counterintuitive, but tis result will appear under some scenarios of negative correlation between prices and quantity. affect not only te level and traectories of te optimal edging strategies, but also te quality of te edge tat could be reaced. edging Positions cenario 7 cenario 6 cenario 5 cenario 4 cenario cenario cenario 0 igure. edging trategies for Different Degrees of Correlation Between Output and Excange Rates able sows te inputs and outputs of applying te metodology to 5 different scenarios. As sown by section A of able, te difference between tose scenarios is te variance level of Q considered. ection B of able presents te results of te optimal edging policies and maximum efficiency to be reaced under eac scenario. e quality of te edging policies decreases as te variance of Q increases. Remember tat in terms of our problem, we are dealing wit two sources of uncertainty, te price of te excange rate and te quantity of excange rate. e futures contracts are a tool tat allows us to eliminate te volatility caused by te volatility of te price, not a device to take care of te volatility of te quantity. 7 e level and te traectory of te edging policies are also a function of te variance of Q. Wen te optimal edging strategies are negative from te beginning (tat appens for lower variances of Q) tey always stay negative and lower (in absolute terms) tan E(Q ) and tey increase over time, becoming closer to E(Q ) as time passes. Wen te optimal edging strategies are positive at te beginning (and tis appens for iger variances of Q), tey become negative and closer to E(Q ) as time passes. igure sows te traectory of te optimal edging strategies for eac of te 5 scenarios considered in able. 7 e expected result under no volatility in quantity would be perfect edging. is result is consistent wit te efficiency reaced in scenario, wit almost no variance in Q. 7 Academia, revista latinoamericana de administración, 50, 0

8 Castillo, Aguila y Niño able. Canging te Variance of Q 00 cenario 5 Inputs cenario ection A: Inputs cenario cenario cenario 4 cenario 5 E() E() E() E(Q) var() var() var() var(q) corr(;) 0,4 0,4 0,4 0,4 0,4 corr(;) 0,4 0,4 0,4 0,4 0,4 corr(;) 0,4 0,4 0,4 0,4 0,4 corr(q;) -0,4-0,4-0,4-0,4-0,4 corr(q;) -0,4-0,4-0,4-0,4-0,4 corr(q;) -0,4-0,4-0,5-0,4-0,5 corr(q;) 0,4 0, 0,0-0, -0, corr(q;) 0,4 0, 0,0-0, -0, corr(q;),0 0,9 0,5 0, -0, ection B: Outputs 0-9,45-6,9-6,4, 90,4-94,7-74,7-50,60-4,8 7,66-97,67-88,6-7,7-56, -8,94 var(cn) varop(c) % Efficiency 99,% 78,% 5,9%,%,4% able sows te inputs and outputs of applying te metodology to four different scenarios. As sown by section A of able, te difference between tose scenarios is te level of assumed correlation between te different excange rates. ection B of able presents te results of te optimal edging policies and maximum efficiency to be reaced under eac scenario. e quality of te edging policies decreases as te correlation between te pairs of excange rates increases. is result goes against one s intuition, but it is caused by te assumed negative correlation between t and Q. If we assume no correlation between t and Q (tese results are not reported ere) we reac te expected result of a iger quality of edging te iger te correlation between te t. e level and te traectory of te edging policies are also a function of te correlation between tose pairs of excange rates. edging Positions cenario 4 cenario cenario cenario 0 igure. edging trategies as a unction of Variance of Q Wen te optimal edging strategies are negative from te beginning (tat appens in all te scenarios wit positive correlations between excange rates) tey always stay negative and lower (in absolute terms) tan E(Q ) and tey increase over time, becoming closer to E(Q ) as time passes. Wen te optimal edging strategies are positive at te beginning (tis able. Canging te Correlation Between Excange Rates ection A: Inputs Inputs cenario cenario cenario cenario 4 E() E() E() E(Q) var() var() var() var(q) corr(;) 0,0 0, 0,4 0,6 corr(;) 0,0 0, 0,4 0,6 corr(;) 0,0 0, 0,4 0,6 corr(q;) -0,4-0,4-0,4-0, corr(q;) -0,4-0,4-0,4-0, corr(q;) -0,4-0,4-0,4-0,4 corr(q;) -0, -0, 0,0 0, corr(q;) -0,4-0, 0,0 0, corr(q;) 0,6 0,6 0,6 0,6 ection B: Outputs 0 8, -7,7 -,8-58,7-8,7-4,69-50,85-6,6-60,59-7,87-76,5-77,78 var(cn) varop(c) % Efficiency 59,8% 47,% 40,% 6,7% conseo latinoamericano de escuelas de administración, cladea 7

9 ong erm Excange Rate Risk and edging wit Quantity Uncertainty in a Market appens in te scenario wit zero correlation between te excange rates), tey become negative and closer (in absolute terms) to E(Q ) as time passes. igure sows te traectory of te optimal edging strategies for eac of te 4 scenarios considered in able. edging Positions cenario cenario cenario cenario 4 0 igure. edging trategies as a unction of te Correlation Between te Excange Rates able 4 sows te inputs and outputs of applying te metodology to 4 different scenarios. As sown by section A of able 4, te difference between tose scenarios is te level of variance of te excange rates t (for t =,, ). ection B of able 4 presents te results of te optimal edging policies and maximum efficiency to be reaced under eac scenario. ere is no linear relationsip between te level of te variances of te t and te quality of te edging policies. 8 e quality of te edging decreases wit te increases in variance of excange rates wen we move from scenarios to, but tis inverse relationsip becomes direct wen we move from scenarios to and to 4. e level and te traectory of te edging policies are also a function of te variances of te excange rates. Wen te optimal edging strategies are negative from te beginning (tat appens in scenarios wit ig variances of excange rates) tey always stay negative and lower (in absolute terms) tan E(Q ) and tey increase over time, becoming closer to E(Q ) as time passes. Wen te optimal edging strategies 8 e relationsip would become linear if te correlation between te excange rates and te output quantities were zero or positive. Under tose scenarios we verified tat more variance of te t allows for a iger quality of edging, as expected. able 4. Canging te Variance of te Excange Rates ection A: Inputs Inputs cenario cenario cenario cenario 4 E() E() E() E(Q) var() var() var() var(q) corr(;) 0,4 0,4 0,4 0,4 corr(;) 0,4 0,4 0,9 0,4 corr(;) 0,4 0,4 0,4 0,40 corr(q;) -0,4-0,4-0,40-0,4 corr(q;) -0,40-0,4-0,4-0,9 corr(q;) -0,4-0,4-0,4-0,4 corr(q;) -0,5-0, -0,0 0, corr(q;) -0,4-0, 0,0 0, corr(q;) -0,4 0,0 0,54 0,86 ection B: Outputs 0,49 4,88 -,08-64,50 5,8-0,85-5,06-76,,0-55,79-75,05-87,08 var(c N ) varop(c ) % Efficiency 7,% 4,6% 7,9% 77,7% are positive at te beginning (tis appens in te scenario wit lower variances of excange rates), tey eiter become negative and closer (in absolute terms) to E(Q ) as time passes, or tey become closer to zero as we approac. igure 4 sows te traectory of te optimal edging strategies for eac of te 4 scenarios considered in able 4. edging Positions cenario cenario cenario cenario igure 4. edging trategies as a unction of te Variance of te Excange Rates 74 Academia, revista latinoamericana de administración, 50, 0

10 Castillo, Aguila y Niño 4. ummary and Conclusions is study analyzes ow to obtain an optimal edging strategy if only sort-term futures contracts are available to edge an uncertain amount of foreign currency tat it is expected to be received at a long-term future date. is is te first paper to address bot problems at te same time. We leave te possibility of using options to edge because in many countries options are not available and only liquid forward contracts on te excange rate can be used. We find a closed form solution for bot te optimal edging policies traectory and te level of maximum efficiency tat could be reaced. We also sow ow to identify te optimal edging strategies troug O regressions in te particular case tat te excange rates are independent from eac oter. ose closed form solutions allow us to explore ow te different parameters considered ere, suc as te volatilities of all te stocastic variables considered or te degree of correlation between tem would impact bot te optimal edging policies and te quality of te optimal edging solution. ection III presents an example of ow to implement te metodology wen = and explores ow te solutions are affected by te parameters considered. e example allows us to verify ow te optimal edging solution can deviate from te simple edging strategy of taking a number of sort positions in futures contracts tat matces te expected amount of foreign currency we are expecting to receive at time, and it also enables us to understand ow and wen te number of (sort) optimal positions to be taken sould become iger or lower (in absolute terms) tan E(Q ), and ow and wen tose optimal positions could become long positions. is paper sould be considered a contribution to ow firms sould select te optimal edging position tey sould take under te particular conditions tey are facing. We sould carefully consider some of te results from te sensitivity analysis in section III, since te conclusions regarding bot te quality of te edging and te evolution over time of te edging strategies sould not be considered as general rules. e interaction of te parameters is not as straigtforward as it seems. is point becomes clear wen we try to use intuition to explain te results. We sow ow intuition can be sometimes misleading, as appens in te scenarios presented in able and able 4. An extension of tis work would be to solve te edging problem considering stocastic interest rates, even toug we will probably find no analytical solutions under tat situation. Anoter extension would be to include te possibility of using options to edge, as proposed by Wong (00) and Korn (00). Augusto Castillo Es P. D. en inanzas, máster en Economía y MBA de la Universidad de California de os Ángeles (UCA), Estados Unidos e ingeniero comercial de la Pontificia Universidad Católica de Cile. Actualmente se desempeña como profesor e investigador de la Escuela de Negocios de la Universidad Adolfo Ibáñez y como director del Máster en inanzas de la misma universidad. Consultor de empresas y de organismos tanto nacionales como internacionales. Autor de numerosos artículos y editor asociado en varias revistas académicas. Rafael Aguila Jorge Niño Es máster en Estadística Matemática del Centro Interamericano de la Enseñanza de Estadística (CIENE-OEA). Profesor de Matemáticas de la Pontificia Universidad Católica de Cile. Actualmente se desempeña como profesor e investigador de la acultad de Economía y Administración de la Pontificia Universidad Católica de Cile. Consultor de empresas del sector privado nacional. Autor de varios artículos en diversas revistas académicas. Es doctor en Ciencias Empresariales de la Universidad Autónoma de Madrid, España y tiene un MBA de la Universidad de Rocester, Estados Unidos. Contador, auditor e ingeniero en información y control de gestión de la Universidad de Cile. Actualmente es profesor y director del Máster en Gestión de Negocios de la Universidad Adolfo Ibáñez, es consultor del Conseo uperior de Educación y del Banco Interame- conseo latinoamericano de escuelas de administración, cladea 75

11 ong erm Excange Rate Risk and edging wit Quantity Uncertainty in a Market ricano de Desarrollo, y evaluador de ondecyt. Autor de numerosos artículos académicos y del libro Contabilidad gerencial. References Basak,., & Cabakauri, G. (0). Dynamic edging in incomplete markets: a simple solution. Review of inancial tudies, on-line version, -5. ttp://ssrn. com/abstract=978. Bessembinder,. (99). orward contracts and firm value. Journal of inancial and Quantitative Analysis, 7, Brennan, M. J., & Crew, N. (997). edging long maturity commodity commitments wit sort dated futures contracts. in Matematics of derivative securities (pp ). Cambridge, Mass.: Cambridge University Press. Broll, U., Wal, J. E., & Zilca, I. (999) edging excange rate risk: te multiperiod case. Researc in Economics, 5, Castillo, A. (00). Excange rate exposure and optimal edging strategies wen interest rates are stocastic: a simulation-based approac. Estudios de Administración, 0(), -. Castillo, A., & efort,. (00). Protección contra la exposición del tipo de cambio a largo plazo con contratos de futuros a corto plazo: el caso de los contratos forward en uf cilenas/dólares. El rimestre Económico, 70(79), Castillo, A., & Aguila, R. (005). Estrategias óptimas de cobertura en presencia de incertidumbre en costos y cantidad. Abante, 8(), Castillo, A., & Aguila, R. (008). Cobertura óptima de riesgos de mercado en presencia de riesgos de cantidad y de costos de producción. El rimestre Económico, XXV(99), Duffie, D. (989). utures markets. Englewood Cliffs, N.J.: Prentice-all. restad, D. (009). Wy most firms coose linear edging strategies. Journal of inancial Researc, XXXII(), root, K. A., carfstein, D.., & tein, J. C. (99). Risk management: Coordinating corporate investment and risk management policies. Journal of inance, 48, Gay, G. D., Nam, J., & urac, M. (00). ow firms manage risk: te optimal mix of linear and nonlinear derivatives. Journal of Applied Corporate inance, 4, 8-9. uang, P., Ryan,. E., & Wiggins, R. A. (007). e influence of firm and CEO- pecific caracteristics on te use of non linear derivative instruments. Journal of inancial Researc, 0(), ull, J. C. (008). Options, futures and oter derivatives (7t ed.). Upper addle River, N. J.: Prentice all. Kamgaing, M. C. (989). Optimal edging under price, quantity and excange rate uncertainty. African Development Review,, Kerkvliet, J., & Moffett, M.. (99). e edging of an uncertain future foreign currency cas flow. Journal of inancial and Quantitative Analysis, 6(4), Korn, O. (00). ow firms sould edge: an extension. e Journal of utures Markets, 0(9), Mello, A., & Parsons, J. (995). Maturity structure of a edge matters: lessons from te metallgesellscaft debacle. Journal of Applied Corporate inance, 8, Moscini, G., & apan,. (995). e edging role of options and futures under oint price, basis and production risks. International Economic Review, 6(4), Näsäkkälä, E., & Keppo, J. (005). Electricity load pattern edging wit static forward strategies. Managerial inance, (6), 6-7. Neuberger, A. (999). edging long term exposures wit multiple sort term futures contracts. e Review of inancial tudies,, Newbery, D., & tiglitz, J. (98). te teory of commodity price stabilization: a study in te economics of risk. Claredon: Oxford Press. Oum, Y., & Orem,.. (00). optimal static edging of volumetric risk in a competitive wolesale electricity market. Decision Analysis, 7(), 07-. Rolfo, J. (980). optimal edging under price and quantity uncertainty: te case of a cocoa producer. Journal of Political Economy, 88(), cwartz, E.. (997). Presidential address: e stocastic beavior of commodity prices: implications for valuation and edging. Journal of inance, 5, mit, C. W., & tulz, R. M. (985). e determinants of firm s edging policies. Journal of inancial and Quantitative Analysis, 0, tulz, R. M. (00). Risk management and derivatives (st ed.). Mason, O: out Western College Publising. Wong, K. P. (00). Currency edging wit options and futures. European Economic Review, 47, Academia, revista latinoamericana de administración, 50, 0

12 Castillo, Aguila y Niño Appendix Optimal edging: te particular case wen = e cas flow generated by te company at = if no forwards are used to edge will be described by te following equation: C = Q (A) N If we assume tat te firm edges, te following expression represents te total cas flow tat te company would generate at = : ö C = Q 0 - r ø ö r ) - r r ø - r 0 0 (A) Under deterministic interest rates it is possible to assume tat t t will be computed as: = r ) t t t (A) Replacing (A) in (A) we obtain te cas flow in as a function of te value of te underlying asset in eac period t = 0 to t =, as sown by te following equation: ö C = Q 0 - r ø r ) ö - r r ø (A4) e following expression represents te variance of te cas flow at = tat we want to minimize: var( C )= var( Q ) ö 0- r ø 4 ö - r ) r ø cov( Q, ) ö 0- r ) cov ( Q, ) r ø ö - r ) cov ( Q, ) r ø ö 0- r ) r ø ö - r ) r ø ö ö r ø r ø (A5) Once we minimize te variance of te cas flow we obtain te following optimal edging strategy for te company: =- 0 ( r r ) ( ) r ) r r =- =- (A6) were represents te excange rate variance-covariance matrix and i represents a matrix composed by te same components of wit te only exception of column i, originally a vector wit covariance ( i, ) x wic as been replaced by te covariance (,Q ) x vector (were =, ). e level of maximum efficiency to be reaced troug te optimal edging solution, defined as te proportion of te total cas flow variance tat will be reduced by edging, can be represented as: conseo latinoamericano de escuelas de administración, cladea 77

13 ong erm Excange Rate Risk and edging wit Quantity Uncertainty in a Market Max Efficiency å å å å i cov ( Q i)- i ii -, i i = i = var Q (A7) If all te excange rates are independent of eac oter, te following expressions represent te optimal edging strategies and te maximum efficiency tat can be reaced: Q/ =- (A8.) é ê ëê Q/ Q/ =- ê r r é ê ë Q/ Q/ 0 =- ê ( ) ù ûú r r r Q / r ù û ú Max = Efficiency R R R ( ) (A8.) (A8.) (A9) E Max = ( Q ) Efficiency var Q (A) e last special case occurs wen we assume tat Q is deterministic. Under tose circumstances te optimal edging strategy and te maximum efficiency to be reaced become te well known results: =-Q Q =- r (A) Q =- r 0 Max Efficiency =0. (A) were Q / i represents te slope of an O regression between Q and i and were R i represents te determination coefficient of te same O regression. Wen Q is independent from all te (excange rates), te expressions tat represent te optimal edging strategies and te maximum efficiency to be reaced will be: =-E( Q) E( Q) =- r E =- ( Q ) 0 r (A0) Recepción del artículo: /0/0 Envío evaluación: 0/04/0 Recepción de correcciones: 7/04/0 Aceptación del artículo: /06/0 78 Academia, revista latinoamericana de administración, 50, 0

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