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

Long Term exchange raTe risk and hedging wiTh

QuanTiTy uncerTainTy in a markeT ThaT onLy

Provides shorT Term FuTures conTracTs

riesgo de TiPo de cambio de Largo PLazo

y coberTura con incerTidumbre de canTidad

Abstr Act

This paper analyzes the problem faced by an inves-tor expecting to receive an uncertain amount of cash flow in a foreign currency on a certain future date T. The investor is also assumed to be exposed to long-term exchange rate risk, and has access only to short-term futures contracts to hedge. A closed form solution for both the optimal hedg-ing strategy and the quality of the hedghedg-ing are identified. Next, we explored how those solutions depend on some key factors such as the volatility of the exchange rate, the volatility of the amount of foreign currency to be received and the degree of cor-relation between all the stochastic variables con-sidered.

Key words: Risk management, hedging, quantity uncertainty.

r esumen

Este artículo analiza el problema que enfrenta un inversionista que espera recibir una cantidad incierta de dinero en una moneda extranjera en una cierta fecha futura T. Se asume que el inversionista está expuesto a un riesgo de largo plazo, teniendo acceso solo a contratos futuros de corto plazo para efectuar cobertura. Se identifica una solución analítica tanto para la estrategia de cobertura óptima como para la calidad de la cobertura alcanzable. También se explora cómo la solución obtenida depende de cier-tos factores clave tales como la volatilidad del tipo de cambio, la volatilidad de la cantidad de moneda extranjera a recibir y la correlación entre las variables estocásticas consideradas.

Palabras clave: administración de riesgo, cobertura, incertidumbre de cantidad.

Augusto castillo

Universidad Adolfo Ibáñez, Santiago, Chile augusto.castillo@uai.cl

rafael Aguila

Pontificia Universidad Católica de Chile, Santiago, Chile

raguilab@uc.cl Jorge niño

Universidad Adolfo Ibáñez, Santiago, Chile jorge.nino@uai.cl

1. introduction

Firms that have their sales indexed to a for-eign currency and their expenses indexed to a local currency are exposed to exchange rate risk. Those investors could protect themselves by taking short positions in exchange rate for-ward or futures contracts. Normally, the hedg-ing would not improve the expected outcome in local currency, but would make the cash flows more certain and reduce exposure to risk.

Previous research on why firms hedge, such as those undertaken by Smith and Stultz (1985), Bessembinder (1991), Froot, Scharfstein and Stein (1993), and Mello and Parsons (1995) have identified the desire to minimize the variance of future cash flows, to reduce the volatility of taxable income, the desire to reduce disper-sion of accounting earnings, or even the hope of being able to avoid financial distress as the main reasons for hedging. Neuberger (1999) assumes that the desire for hedging comes from risk averse agents wishing to maximize expec-ted utility.

The use of forward or futures contracts to hedge against exchange rate risk works less than perfectly in the real world for several rea-sons. First, the exchange rate we want to hedge from may not be the same as the exchange rate considered in the futures contracts available in the market. In this case the quality of the hedge will critically depend on how closely correla-ted those two exchange rates are. This point has been developed in all the major derivatives and risk management textbooks. See for exam-ple Duffie (1989), Stulz (2003) or Hull (2008). A recent paper by Basak and Chabakauri (2012) gives new insights on how to solve this pro-blem through different techniques. Second, the date of expiration or maturity of the future or forward contracts available to perform the hed-ging may not coincide exactly with the particu-lar date in the future we will receive the foreign currency. This could happen for example if there are only short-term futures contracts available to hedge against long-term exchange rate exposure or if there are long-term futures contracts to hedge against short-term exchange rate exposure.

It has been proven that using short-term forward contracts to hedge against long-term exposure would allow us to reach perfect hed-ging if interest rates are non-stochastic and there is no uncertainty in quantity, and other conditions regarding availability of contracts are fulfilled. For example, Brennan and Crew (1997) present a model where a simple or tai-led stack and roll strategy allows us to reach perfect hedge when deterministic interest rates are assumed and when the “rollover gain” deri-ved from differences between prices of diffe-rent futures contracts at the maturity of some of them are ignored. Neuberger (1999) shows that even with deterministic interest rates, perfect hedging would not be reached unless forward prices can be predicted in advance and perfectly. He assumes that the price at which a contract first trades is a stochastic function of the prices of other contracts already trading in the market, and by assuming that the expected value of the opening price is a linear function of the prices of other contracts, the author pro-ves that there is a unique hedging strategy, independent of the agent’s utility function, that dominates all the other strategies in the sense of second order stochastic dominance. The methodology proposed by Neuberger allows us to remove 85% of the risk of a six-year oil supply commitment.

Schwartz (1997) compares three models of the stochastic behavior of commodity prices, taking mean reversion into account. He also analyzes the implications of those models for hedging risk exposure. Broll, Wahl and Zil-cha (1999) analyze hedging in a multiperiod framework for a risk averse exporting firm facing exchange rate uncertainty. Castillo and Lefort (2003) and Castillo (2003) show how a firm could obtain optimal hedging against exchange rate exposure using short-term futu-res contracts when it is known that a single amount of foreign currency will be received in a certain long term-future period T, and inter-est rates are stochastic. They present analytical solutions when feasible and simulation-ba-sed solutions otherwise. All the papers men-tioned in this paragraph assume the company is restricted to hedge in the long term through

the use of short term contracts and ignore the quantity uncertainty problem.

A third reason that hedging works less than perfectly will arise if there is uncertainty regar-ding the amount of foreign currency to be received (or in the number of units of a cer-tain good available for selling). This problem has been analyzed by, among others, Rolfo (1980), Newbery and Stiglitz (1981), kamgaing (1989), kerkvliet and Moffett (1991), Moschini and Lapan (1995), Wong (2003), Näsäkkälä and keppo (2005), Castillo and Aguila (2005), Cas-tillo and Aguila (2008), Frestad (2009), oum and orem (2010) and korn (2010).

Rolfo (1980) obtains an expression for the optimal hedging strategy under price and quantity uncertainty if the investor’s preferen-ces are properly represented by a logarithmic utility function. Newbery and Stiglitz (1981) solve a similar problem by maximizing a cons-tant risk aversion utility function. kerkvliet and Moffett (1991) reach an expression for the optimal hedging strategy when there is uncer-tainty in the amount of foreign currency to be received if the objective is to minimize the cash flow variance of local currency to be received. Moschini and Lapan (1995) solve for the opti-mal hedging strategy when facing quantity and price uncertainty and basis risk. They show that the quality of the hedging can be improved by using option contracts. kamgaing (1989) deri-ves an optimal hedging strategy under quantity, price and exchange rate uncertainty assuming that the producer is a mean-variance maximi-zer with an exponential utility function. Wong (2003) examines the optimal hedging deci-sion of a competitive exporting firm facing exchange rate risk and price risk and only with access to futures and option contracts on the exchange rate to hedge. The paper assumes the firm wants to maximize expected utility and concludes that hedging with options can be better than hedging with futures for some cases where the price is negatively correlated with the exchange rate. Näsäkkalä and keppo (2005) assume a firm facing price and quantity risk and wanting to minimize the variance of the cash flow. They also consider that the only tools available to perform the hedging are futu-res contracts. Due to transaction costs and

illi-quidity concerns they choose not to consider the possibility of using options to hedge.

Castillo and Aguila (2005) and Castillo and Aguila (2008) find the optimal hedging stra-tegy when facing price and cost risk and quan-tity uncertainty, if the company is trying to minimize cash flow variance, and only futu-res contracts are available to hedge. Ffutu-restad (2009) considers nonfinancial firms facing hedgeable price risk, unhedgeable quantity risk and the presence of financial contrac-ting costs. The paper concludes that variance-minimizing hedging strategies are very close in economic terms to optimal, value-maximi-zing hedging strategies for most firms and that the marginal gains from shifting to nonlinear hedging strategies are often small enough to be neglected. oum and orem (2010) show that when a firm faces a multiplicative risk of price and quantity, its profit is nonlinear in price, and it cannot be fully hedged by a forward or futu-res contract, which has a linear payoff struc-ture. korn (2010) shows that for some ranges of correlation between price and quantity hed-ging with futures can be optimal, but for some other ranges of correlation the quality of the hedging will improve if non linear instruments (options) are used. In general the papers cited in this paragraph assume that there is a pro-blem of uncertainty in quantity, but assume the availability of contracts with the proper time extension.

our study analyzes how to obtain the opti-mal hedging strategy if only short-term futures contracts are available to hedge an uncertain amount of foreign currency that is expected to be received on a long-term future date T. We are the first researchers to consider both pro-blems at the same time. We will assume that the absence of option contracts to hedge is due to two reasons. The first is that the impact of using options to hedge has already been stu-died. The second is that futures (or forward) contracts on exchange rates are available in most Latin American countries, but option contracts are either not available or highly illi-quid when available.1 _{In this paper we will solve} 1 In addition to those reasons we can add two more. First, according to Frestad (2009), the improvement in the qual-ity of the hedging when using options instead of futures

the problem assuming that interest rates are not stochastic (or that we are able to remove that uncertainty through interest rate forwards). In a second stage (in our next paper), we will allow interest rates to be stochastic. The quality of the hedging and the dependence of the quality of the hedging on some key parameters will also be analyzed. We will assume both the absence of transactions costs and that the futures con-tracts available are infinitely divisible. These are standard assumptions in the cited literature.

This paper is organized as follows. In section 2, the long term exchange rate risk hedging problem faced by the investor is described and the optimal analytical solution is presented, assuming that short term corresponds to one period and long term corresponds to T periods. Some particular cases are also presented. Sec-tion 3 reports the implementaSec-tion of the opti-mal hedging strategies described in section 2 to a particular case and explores how efficient those optimal hedging strategies are under a series of different scenarios. Section 4 presents the main conclusions of this paper.

2. The Long Term hedging model

Let us assume that there is an investor who is expecting to receive a certain amount of for-eign currency, T periods from now. For now we will suppose that the investor is facing only two sources of uncertainty, which are the exact amount of foreign currency to be received and the amount of cash to be received once he con-verts the foreign currency into local currency. If only one-period forward contracts are avail-able to hedge, he could hedge using those short-term forward contracts through the following procedure: He would have to take at each period

t, starting at t = 0, and until period t = T – 1, tht + 1

positions in forward contracts. Those positions, taken at each period t, would have to be rebal-anced at the expiration of the contracts, in each of the next periods. If we name the amount of foreign currency that we will receive in T, Q,

is in general not very significant. And the empirical evi-dence presented by, for example, gay, Nam and Turac (2002) and Huang, Ryan and Wiggins (2007) show that most companies hedge through the use of forwards or futures contracts.

and let ST represents the exchange rate at T,

the cash flow generated by the company if no forwards are used to hedge will be described by the following equation:

CFTNH= ⋅ Q ST (1)

If we assume that the firm hedges and if we suppose that all the gains or losses generated by the positions taken in one-period forward con-tracts over the periods from t = 0 to t = T – 1 are transformed to a cash flow in period T, the following expression represents the total cash flow that the company would generate at T:

CF Q S h S F r TH T t t t T t t t L T = ⋅ + ⋅

(

−

)

⋅ + + = − + + − −

∑

1 0 1 1 1 1 1 ( ) tt (2)

where tFt + 1 is the forward price fixed at t for

a contract expiring at t + 1; tht + 1 represents the

number of positions taken in those forward contracts in period t; and rL corresponds to the

risk-free interest rate in the local currency.2 Under deterministic interest rates it is possible to assume that tFt + 1 will be computed as:

t t t L F F S r r + = ⋅ + + 1 1 1 ( ) ( ) (3)

where rF corresponds to the risk-free

inter-est rate in the foreign currency. Replacing (3) in (2) we obtain the cash flow in T as a function of the value of the underlying asset S in each period t = 0 to t = T, as shown by the following equation: CF Q S S TH T T T T t t t t F t T h h h r = + + − +     ⋅ ₋ ⋅ ₋ + ⋅ = −

∑

1 1 1 1 1 1 1 S 1 1 0 1 0 +

(

)

− ⋅ ⋅ + + − ⋅ r h S r r L T t t L T F ( ) ( ) (4)

To find the optimal hedging strategy we have to assume that the investor is optimizing a certain function. Let’s assume he is trying to minimize the variance of the cash flow to be 2 on each period t this risk free interest rate could be

dif-ferent to the one observed the previous period but we will ignore that to avoid unnecessary complexity.

received in T. The following expression repre-sents the variance of the cash flow we want to minimize. var CF var Q S TH T T T TT t t t t h h h

( )

=

(

)

+ + − ⋅ ₋ ⋅ − + 1 2 1 ₁ 1  ++    

(

+

)

+ ⋅ − ⋅ = − ⋅ −

∑

_r r h F L T t t T tt T 2 2 1 1 1 1 2 ( )  TT T T t t t t F L T t t T h h r r ⋅ ⋅ ⋅ ₋ + ⋅ − ⋅ = −

(

)

+ − +    

(

+

)

∑

cov Q S S, 2 1 1 1 1 1 1 S Q S cov ,t T T T t T h ⋅ _{⋅ −} ⋅ = −

(

)

+2 1

∑

1 1 t t t t F L T t tT t t t t F h h r r h h r − + ⋅ − ⋅ ⋅ ₋ + − +    

(

+

)

+ − +   1 1 1 1 1 1 2 1   _ − +    ⋅ = + − = − − +

∑

∑

u t T t T u u u u F h h r 1 1 1 2 1 ₁ 1 ⋅

(

+

)

− − ⋅ 1 rL 2 T t u tu  (5)

once we minimize the variance of the cash flow we obtain the following optimal hedging strategy for the company:

j j i L T i F i j i j T h r r − − ⋅ − = = −

(

+

)

(

+

)

∑

1 1 1 * Σ Σ (6) j = 1, 2, 3, … T

where  represents the exchange rate varian-ce-covariance matrix and i represents a

matrix composed by the same components of  with the only exception of column i, originally a vector with Covariance(Si,Sj)1 x T which has been replaced by the Covariance(Sj,Q⋅ST)1 x T vector (where j = 1,…T).

once the company has implemented the opti-mal hedging strategy described here the level of maximum efficiency, defined as the proportion of the total cash flow variance that will be redu-ced by hedging, can be represented as seen in equation 7.

All the previous expressions for the general case will change to describe some particular possibilities that are considered as interesting. If all the exchange rates are independent, the following expressions represent the optimal hedging strategies and the maximum efficiency that can be reached:

j j T i ii L T i F i j i j T h r r − ⋅ − ⋅ − = = −

(

)

+

(

)

(

+

)

          

∑

1 1 1 * cov Q S S,     = − +

(

)

(

+

)

        − ⋅ − =

∑

QS S L T i F i j i j T T i r r / 1 1 (8) MaxEfficiency Ri i T = =

∑

2 1 (9)

where QS ST i/ represents the slope of an oLS

regression between Q⋅S_T and Si and where R2

i

represents the determination coefficient of the same oLS regression. The next particular case to be analyzed is when Q is independent from all the Sj (exchange rates). Under this scenario

the expressions that represent the optimal hed-ging strategies and the maximum efficiency to be reached will be:

j j F T j h E r j T − = −

( )

− +

(

)

= 1 1 1 2 3 * Q _{, , ,...,} (10) MaxEfficiency E TT T =

( )

₍

⋅

₎

⋅ 2 _Q Q S  var (11)

It is interesting to note that the optimal hed-ging strategy becomes a direct function of the expected amount of foreign currency to be received only after assuming independence between the amount of foreign currency to be received at T and the exchange rates. The last special case to be reviewed is when we assume

MaxEfficiency TH i T i i T i ii i T CF Q S S

( )

=

(

)

− − ⋅ ⋅ ⋅ ⋅ = ⋅ =

∑

∑

2 1 2 1 Σ Σ cov , Σ  22 1 1 1 2 ⋅ ⋅ ⋅ = + = − ⋅ ⋅

∑

∑

(

)

Σ Σ Σ i j ij j i T i T T  var Q S (7)

that Q is deterministic. Under those circums-tances the optimal hedging strategy and the maximum efficiency to be reached become the well known results:

j j F T j h Q r − = − − +

(

)

1 * 1 (12) MaxEfficiency =1 0. (13)

Here the optimal hedging strategy allows the variance of the cash flow at T to be comple-tely eliminated. As we said earlier, the solutions provided here assume that the interest rates are deterministic. The appendix at the end of this paper presents equations (1) to (13) for the par-ticular case when T = 3. In the next section we will implement the model and the solutions described here.

3. implementing the optimal hedging

strategy

In this section the model described previously is implemented. We assume T = 3 so that we are trying to hedge against the volatility of the cash flow of period three and we only have access to one period futures contracts.3 _Tables 1 to 4 show how the hedging strategies and the quality of the hedging change if only one of the parameters is changed and all the others are kept constant.4

Section A of Table 1 shows the inputs requi-red to implement the methodology, and Section B of Table 1 contains the results of applying the 3 The appendix shows the formulas that are used in this

section.

4 As shown by tables 1 to 4 we are not able to hold perfectly constant the other parameters. This is due to the use of simulations to generate the results.

Table 1. Changing the Correlation Between output and Exchange Rates

section a: inputs

inputs scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7

E(S1) 501 501 501 501 501 501 501 E(S2) 500 500 500 500 500 500 500 E(S3) 500 500 500 500 500 500 500 E(Q) 100 100 100 100 100 100 100 var(S1) 2.384 2.384 2.384 2.384 2.384 2.384 2.384 var(S2) 2.635 2.635 2.635 2.635 2.635 2.635 2.635 var(S3) 2.581 2.581 2.581 2.581 2.581 2.581 2.581 var(Q) 109 92 112 107 99 104 101 corr(S1;S2) 0,45 0,45 0,45 0,45 0,45 0,45 0,45 corr(S1;S3) 0,39 0,39 0,39 0,39 0,39 0,39 0,39 corr(S2;S3) 0,44 0,44 0,44 0,44 0,44 0,44 0,44 corr(Q;S1) 0,40 0,20 0,03 -0,20 -0,40 -0,60 -0,75 corr(Q;S2) 0,43 0,19 0,01 -0,23 -0,41 -0,62 -0,79 corr(Q;S3) 0,44 0,19 0,02 -0,29 -0,35 -0,59 -0,75 corr(QS3;S1) 0,46 0,38 0,29 0,16 0,00 -0,24 -0,50 corr(QS3;S2) 0,51 0,41 0,30 0,17 0,04 -0,20 -0,48 corr(QS3;S3) 0,84 0,79 0,70 0,59 0,58 0,45 0,38 section b: outputs 0h1 -168,80 -127,91 -101,33 -59,13 -38,42 -1,51 21,46 1h2 -149,55 -117,87 -98,71 -66,31 -62,32 -35,49 -17,84 2h3 -128,39 -110,67 -102,67 -77,03 -84,83 -68,06 -59,36 var(CFNH) 78.168.753 58.953.865 54.881.590 36.812.420 32.937.371 20.746.420 12.146.423 varoPT(CFH) 20.081.165 22.013.881 27.614.832 23.757.201 18.752.773 10.270.415 1.162.715 % Efficiency 74,3% 62,7% 49,7% 35,5% 43,1% 50,5% 90,4%

methodology.5 _{Section A of Table 1 presents 7} scenarios. The difference between them is the assumed correlation between the exchange rates St (with t = 1, 2, 3) and the amount of

foreign currency to be received at T = 3, QT. Section B of Table 1 presents the results of the optimal hedging policies and maximum effi-ciency to be reached under each scenario. The quality of the hedging policies increase as the correlation between the St and QT becomes either more positive or more negative. These results were expected because more correla-tion between St and QT means that we can more easily reduce volatility of the cash flow produ-ced by the volatility of QT through the use of futures contracts on the exchange rate St. The level and the trajectory of the hedging policies are also a function of the correlation between the St and QT. It is also important to remem-ber that negative positions represent short posi-tions in future contracts and positive posiposi-tions represent long positions in futures contracts.

For positive correlations between the St and QT, the hedging policies are all negative and hig-her (in absolute value) than E(QT). Those optimal hedging strategies are also decreasing in time (in absolute value), becoming closer to E(QT) as time passes. For negative correlations between the St and QT, the hedging policies are in general nega-tive (short positions) and smaller (in absolute value) than E(QT), and could even start as posi-tive (long positions) as shown in Table 1.6 _These results are important because they show that the strategy of hedging by a magnitude of E(QT) is not the best solution unless there is no correla-tion between St and QT, and it also shows how to adjust the optimal hedging strategy when we take in account those correlations. Figure 1 shows the trajectory of the optimal hedging stra-tegies for each of the seven scenarios considered in Table 1. Following the same procedure we could explore how changes in the volatilities of the St and QT, and how changes in the autocorre-lation of the exchange rates of different periods 5 All the results presented in this section are the result of generating sets of 100.000 random numbers with the required characteristics for each scenario.

6 Hedging a long position in the underlying asset with a long position in the futures contract is counterintuitive, but this result will appear under some scenarios of nega-tive correlation between prices and quantity.

affect not only the level and trajectories of the optimal hedging strategies, but also the quality of the hedge that could be reached.

-200 -150 -100 -50 0 50 H ed gi ng P os iti on s Scenario 7 Scenario 6 Scenario 5 Scenario 4 Scenario 3 Scenario 2 Scenario 1 0h1 1h2 2h3

Figure 1. Hedging Strategies for Different Degrees

of Correlation Between output and Exchange Rates

Table 2 shows the inputs and outputs of applying the methodology to 5 different scena-rios. As shown by section A of Table 2, the diffe-rence between those scenarios is the variance level of Q considered. Section B of Table 2 pre-sents the results of the optimal hedging policies and maximum efficiency to be reached under each scenario. The quality of the hedging poli-cies decreases as the variance of QT increases. Remember that in terms of our problem, we are dealing with two sources of uncertainty, the price of the exchange rate and the quantity of exchange rate. The futures contracts are a tool that allows us to eliminate the volatility caused by the volatility of the price, not a device to take care of the volatility of the quantity.7 _{The level} and the trajectory of the hedging policies are also a function of the variance of QT.

When the optimal hedging strategies are negative from the beginning (that happens for lower variances of Q) they always stay nega-tive and lower (in absolute terms) than –E(QT) and they increase over time, becoming closer to –E(QT) as time passes. When the optimal hedging strategies are positive at the beginning (and this happens for higher variances of Q), they become negative and closer to –E(QT) as time passes. Figure 2 shows the trajectory of the optimal hedging strategies for each of the 5 scenarios considered in Table 2.

7 The expected result under no volatility in quantity would be perfect hedging. This result is consistent with the efficiency reached in scenario 1, with almost no variance in Q.

Table 2. Changing the Variance of Q

section a: inputs

inputs _{nario 1}sce- _{nario 2}sce- _{nario 3}sce- _{nario 4}sce- _{nario 5}

sce-E(S1) 501 501 501 501 501 E(S2) 503 503 503 503 503 E(S3) 504 504 504 504 504 E(Q) 100 100 100 99 98 var(S1) 2.481 2.481 2.481 2.481 2.481 var(S2) 2.831 2.831 2.831 2.831 2.831 var(S3) 2.686 2.686 2.686 2.686 2.686 var(Q) 1 27 107 368 805 corr(S1;S2) 0,4 0,4 0,4 0,4 0,4 corr(S1;S3) 0,4 0,4 0,4 0,4 0,4 corr(S2;S3) 0,4 0,4 0,4 0,4 0,4 corr(Q;S1) -0,4 -0,4 -0,4 -0,4 -0,4 corr(Q;S2) -0,4 -0,4 -0,4 -0,4 -0,4 corr(Q;S3) -0,4 -0,4 -0,5 -0,4 -0,5 corr(QS3;S1) 0,4 0,3 0,0 -0,2 -0,3 corr(QS3;S2) 0,4 0,2 0,0 -0,2 -0,3 corr(QS3;S3) 1,0 0,9 0,5 0,1 -0,1 section b: outputs 0h1 -91,45 -63,91 -26,42 21,22 90,41 1h2 -94,37 -74,27 -50,60 -14,28 27,66 2h3 -97,67 -88,36 -72,17 -56,13 -18,94 var(CFNH) 24.847.284 22.227.358 28.956.973 76.470.145 155.765.646 varoPT(CFH) 161.677 4.859.988 18.557.520 66.471.214 136.515.418 % Efficiency 99,3% 78,1% 35,9% 13,1% 12,4% Table 3 shows the inputs and outputs of applying the methodology to four different sce-narios. As shown by section A of Table 3, the difference between those scenarios is the level of assumed correlation between the different exchange rates ST. Section B of Table 3 presents the results of the optimal hedging policies and maxi-mum efficiency to be reached under each scena-rio. The quality of the hedging policies decreases as the correlation between the pairs of exchange rates increases. This result goes against one’s intuition, but it is caused by the assumed nega-tive correlation between St and QT. If we assume no correlation between St and QT (these results are not reported here) we reach the expected result of a higher quality of hedging the hig-her the correlation between the St. The level and the trajectory of the hedging policies are also a function of the correlation between those pairs of exchange rates. -150 -100 -50 0 50 100 H ed gi ng P os iti on s Scenario 5 Scenario 4 Scenario 3 Scenario 2 Scenario 1 0h1 1h2 2h3

Figure 2. Hedging Strategies as a Function

of Variance of Q

When the optimal hedging strategies are negative from the beginning (that happens in all the scenarios with positive correlations bet-ween exchange rates) they always stay nega-tive and lower (in absolute terms) than E(QT) and they increase over time, becoming closer to E(QT) as time passes. When the optimal hedg-ing strategies are positive at the beginnhedg-ing (this

Table 3. Changing the Correlation

Between Exchange Rates

section a: inputs

inputs scenario 1scenario 2scenario 3scenario 4

E(S1) 498 499 499 498 E(S2) 502 499 498 495 E(S3) 502 500 499 493 E(Q) 100 100 100 100 var(S1) 2.565 2.366 2.249 2.236 var(S2) 2.262 2.265 2.358 2.449 var(S3) 2.449 2.487 2.281 2.415 var(Q) 91 91 91 91 corr(S1;S2) 0,0 0,2 0,4 0,6 corr(S1;S3) 0,0 0,2 0,4 0,6 corr(S2;S3) 0,0 0,2 0,4 0,6 corr(Q;S1) -0,4 -0,4 -0,4 -0,3 corr(Q;S2) -0,4 -0,4 -0,4 -0,3 corr(Q;S3) -0,4 -0,4 -0,4 -0,4 corr(QS3;S1) -0,3 -0,1 0,0 0,3 corr(QS3;S2) -0,4 -0,2 0,0 0,3 corr(QS3;S3) 0,6 0,6 0,6 0,6 section b: outputs 0h1 18,12 -17,37 -31,28 -58,72 1h2 -18,37 -41,69 -50,85 -63,36 2h3 -60,59 -71,87 -76,51 -77,78 var(CFNH) 27.482.344 27.109.652 25.922.319 29.834.804 varoPT(CFH) 11.053.222 14.309.143 15.478.571 18.880.460 % Efficiency 59,8% 47,2% 40,3% 36,7%

happens in the scenario with zero correlation between the exchange rates), they become neg-ative and closer (in absolute terms) to E(QT) as time passes. Figure 3 shows the trajectory of the optimal hedging strategies for each of the 4 scenarios considered in Table 3.

-100 -80 -60 -40 -20 0 20 40 H ed gi ng P os iti on s Scenario 1 Scenario 2 Scenario 3 Scenario 4 0h1 1h2 2h3

Figure 3. Hedging Strategies as a Function of the

Correlation Between the Exchange Rates

Table 4 shows the inputs and outputs of applying the methodology to 4 different sce-narios. As shown by section A of Table 4, the difference between those scenarios is the level of variance of the exchange rates St (for t = 1, 2, 3). Section B of Table 4 presents the results of the optimal hedging policies and maximum efficiency to be reached under each scenario. There is no linear relationship between the level of the variances of the St and the quality of the hedging policies.8 _{The quality of the} hedg-ing decreases with the increases in variance of exchange rates when we move from scenarios 1 to 2, but this inverse relationship becomes direct when we move from scenarios 2 to 3 and 3 to 4. The level and the trajectory of the hedg-ing policies are also a function of the variances of the exchange rates.

When the optimal hedging strategies are negative from the beginning (that happens in scenarios with high variances of exchange rates) they always stay negative and lower (in absolute terms) than E(QT) and they increase over time, becoming closer to E(QT) as time passes. When the optimal hedging strategies 8 The relationship would become linear if the correlation between the exchange rates and the output quantities were zero or positive. Under those scenarios we verified that more variance of the St allows for a higher quality of

hedging, as expected.

Table 4. Changing the Variance

of the Exchange Rates

section a: inputs

inputs scenario 1 scenario 2 scenario 3scenario 4

E(S1) 500 500 500 501 E(S2) 500 500 500 500 E(S3) 500 500 500 501 E(Q) 100 100 100 100 var(S1) 100 629 2.470 10.227 var(S2) 98 618 2.456 10.002 var(S3) 98 635 2.521 10.037 var(Q) 100 100 100 100 corr(S1;S2) 0,42 0,41 0,41 0,42 corr(S1;S3) 0,41 0,41 0,39 0,41 corr(S2;S3) 0,42 0,41 0,41 0,40 corr(Q;S1) -0,41 -0,42 -0,40 -0,41 corr(Q;S2) -0,40 -0,41 -0,41 -0,39 corr(Q;S3) -0,42 -0,41 -0,42 -0,41 corr(QS3;S1) -0,35 -0,23 -0,01 0,23 corr(QS3;S2) -0,34 -0,22 0,01 0,22 corr(QS3;S3) -0,24 0,10 0,54 0,86 section b: outputs 0h1 231,49 34,88 -31,08 -64,50 1h2 125,28 -10,85 -53,06 -76,11 2h3 21,20 -55,79 -75,05 -87,08 var(CFNH) 21.735.926 20.810.377 28.821.875 82.478.612 varoPT(CFH) 18.009.663 17.767.189 17.899.029 18.418.806 % Efficiency 17,1% 14,6% 37,9% 77,7% are positive at the beginning (this happens in the scenario with lower variances of exchange rates), they either become negative and closer (in absolute terms) to E(QT) as time passes, or they become closer to zero as we approach T. Figure 4 shows the trajectory of the optimal hedging strategies for each of the 4 scenarios considered in Table 4. -150 -100 -50 0 50 100 150 200 250 H ed gi ng P os iti on s Scenario 1 Scenario 2 Scenario 4 Scenario 3 0h1 1h2 2h3

Figure 4. Hedging Strategies as a Function of the

4. summary and conclusions

This study analyzes how to obtain an optimal hedging strategy if only short-term futures con-tracts are available to hedge an uncertain amount of foreign currency that it is expected to be received at a long-term future date T. This is the first paper to address both problems at the same time. We leave the possibility of using options to hedge because in many countries options are not available and only liquid forward contracts on the exchange rate can be used.

We find a closed form solution for both the optimal hedging policies trajectory and the level of maximum efficiency that could be reached. We also show how to identify the optimal hed-ging strategies through oLS regressions in the particular case that the exchange rates are inde-pendent from each other.

Those closed form solutions allow us to explore how the different parameters conside-red here, such as the volatilities of all the sto-chastic variables considered or the degree of correlation between them would impact both the optimal hedging policies and the quality of the optimal hedging solution. Section III presents an example of how to implement the methodology when T = 3 and explores how the solutions are affected by the parameters considered.

The example allows us to verify how the opti-mal hedging solution can deviate from the sim-ple hedging strategy of taking a number of short positions in futures contracts that mat-ches the expected amount of foreign currency we are expecting to receive at time T, and it also enables us to understand how and when the number of (short) optimal positions to be taken should become higher or lower (in absolute terms) than E(QT), and how and when those optimal positions could become long positions. This paper should be considered a contribution to how firms should select the optimal hedging position they should take under the particular conditions they are facing.

We should carefully consider some of the results from the sensitivity analysis in section III, since the conclusions regarding both the quality of the hedging and the evolution over time of the hedging strategies should not be considered as general rules. The interaction of the parameters

is not as straightforward as it seems. This point becomes clear when we try to use intuition to explain the results. We show how intuition can be sometimes misleading, as happens in the sce-narios presented in Table 3 and Table 4.

An extension of this work would be to solve the hedging problem considering stochastic interest rates, even though we will probably find no analytical solutions under that situa-tion. Another extension would be to include the possibility of using options to hedge, as propo-sed by Wong (2003) and korn (2010).

Augusto castillo

Es Ph. D. en Finanzas, máster en Economía y MBA de la Universidad de California de Los Ánge-les (UCLA), Estados Unidos e ingeniero comer-cial de la Pontificia Universidad Católica de Chile. Actualmente se desempeña como profesor e inves-tigador de la Escuela de Negocios de la Universi-dad Adolfo Ibáñez y como director del Máster en Finanzas 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

Es máster en Estadística Matemática del Cen-tro Interamericano de la Enseñanza de Estadís-tica (CIENES-oEA). Profesor de MatemáEstadís-ticas de la Pontificia Universidad Católica de Chile. Actualmente se desempeña como profesor e investigador de la Facultad de Economía y Administración de la Pontificia Universidad Católica de Chile. Consultor de empresas del sector privado nacional. Autor de varios artícu-los en diversas revistas académicas.

Jorge niño

Es doctor en Ciencias Empresariales de la Uni-versidad Autónoma de Madrid, España y tiene un MBA de la Universidad de Rochester, Esta-dos UniEsta-dos. Contador, auditor e ingeniero en información y control de gestión de la Universi-dad de Chile. Actualmente es profesor y director del Máster en gestión de Negocios de la Univer-sidad Adolfo Ibáñez, es consultor del Consejo Superior de Educación y del Banco

Interame-ricano de Desarrollo, y evaluador de Fondecyt. Autor de numerosos artículos académicos y del libro Contabilidad gerencial.

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appendix

optimal hedging: The particular case when T = 3

The cash flow generated by the company at

T = 3 if no forwards are used to hedge will be

described by the following equation:

CFNH Q S

3 = × 3 (A1)

If we assume that the firm hedges, the following expression represents the total cash flow that the company would generate at T = 3:

CF Q S S S 3 3 2 3 3 0 1 1 2 2 1 1 1 H F L h h h r r = + + -+ æ è çç çç ö ø ÷÷÷ ÷ +

(

)

+ × × × × _{11 2} 2 3 ₂ 0 1 0 1 2 1 1 1 h h r r h F r F L L -+ æ è çç çç ö ø ÷÷÷ ÷

(

+

)

-

(

+

)

× × × × S (A2)

Under deterministic interest rates it is possi-ble to assume that tFt + 1 will be computed as:

t t t L F F S r r + = × + + 1 ₍(1₁ )₎ (A3)

Replacing (A3) in (A2) we obtain the cash flow in T as a function of the value of the underlying asset S in each period t = 0 to t = T, as shown by the following equation:

CF Q S S S 3 3 2 3 3 0 1 1 2 2 1 1 H F L h h h r r = + + -+ æ è çç çç ö ø ÷÷÷ ÷ +

(

)

× × × × 11 1 2 2 3 2 0 1 0 3 1 1 1 1 + -+ æ è çç çç ö ø ÷÷÷ ÷

(

+

)

- + + × × × × h h r r h S r r F L L F S ( ) ( ) (A4)

The following expression represents the variance of the cash flow at T = 3 that we want to minimize: var CF3 var Q S3 2 32 33 0 1 1 2 2 1 1 H F L h h h r r

(

)

=

₍

₎

+ + -+ æ è çç çç ö ø ÷÷÷ ÷ + × × × 

((

)

+ -+ æ è çç çç ö ø ÷÷÷ ÷

(

+

)

+ × × × × × × 4 11 1 2 2 3 2 2 22 2 3 1 1 2   h h r r h F L cov Q S33 3 0 1 1 2 2 3 1 1 2 1 1 2 , cov , * S Q S S

(

)

+ -+ æ è çç çç ö ø ÷÷÷ ÷×

(

+

)

×

(

×

)

+ × h h rF rL hh h r r h h h F L 2 2 3 3 2 2 3 0 1 1 2 1 1 2 1 -+ æ è çç çç ö ø ÷÷÷ ÷

(

+

)

(

)

+ -× × × × × cov Q S S, + + æ è çç çç ö ø ÷÷÷ ÷

(

+

)

+ -+ æ è çç çç ö ø ÷÷÷ ÷ × × × × r r h h h r F L F 1 2 1 2 13 2 3 1 2 2 3  ×× × + × × +

(

)

-+ æ è çç çç ö ø ÷÷÷ ÷ - + æ è çç çç ö ø 1 2 1 1 23 0 1 1 2 1 2 2 3 r h h r h h r L F F  ÷÷÷÷ ÷×

(

1+

)

× 3 12 rL  (A5)

once we minimize the variance of the cash flow we obtain the following optimal hedging strategy for the company:

0 1 1 2 2 3 2 1 1 1 1 h*_=-

(

+rL

)

rL rF rF + +

(

)

(

+

)

+

₍

₊

₎

S S S S 1 2 2 3 1 1 h*_=- +rL rF + + S S S (A6) 2 3h*=- 3 S S

where  represents the exchange rate varian-ce-covariance matrix and i represents a matrix

composed by the same components of  with the only exception of column i, originally a vector with covariance (Si,Sj)1x3 which has been replaced by the covariance (Sj,Q⋅S3)1x3 vector (where j = 1,…3). The level of maximum effi-ciency to be reached through the optimal hed-ging solution, defined as the proportion of the total cash flow variance that will be reduced by hedging, can be represented as:

MaxEfficiency i i i i i ii i =

(

)

- -× × × × = × = × ×

å

å

2 3 2 1 3 ₂ 1 3 S S cov Q S S, S  S Sjj ij j i i × = + = × ×

å

å

(

)

 1 3 1 2 3 2 var Q S S (A7)

If all the exchange rates are independent of each other, the following expressions represent the optimal hedging strategies and the maxi-mum efficiency that can be reached:

2 3h QS S3 3 * / =- (A8.1) 1 2h ₁ 3_r2 ₁ 3_r3 QS S L QS S F *_=- / / + + + é ë ê êê ù û ú úú   (A8.2) 0 1 2 3 1 3 2 1 1 1 h r r r QS S L QS S L F Q *_=- / / +

(

)

+

₍

+

₎

₍

+

₎

é ë ê ê ê +    SS S F r 3 3 1 2 / +

(

)

ù û ú ú ú (A8.3)

MaxEfficiency =R12+R22+R32 (A9)

where QS S3/i represents the slope of an oLS

regression between Q⋅S₃ and Si and where R2 i represents the determination coefficient of the same oLS regression. When Q is independent from all the Sj (exchange rates), the expressions that represent the optimal hedging strategies and the maximum efficiency to be reached will be:

2 3h*=-E( )Q 1 2h ₁E _r F *_=- ( ) + Q (A10) 0 1 2 1 h E rF *_=-

( )

+

(

)

Q MaxEfficiency =E

( )

(

)

× × 2 33 3 Q Q S  var (A11)

The last special case occurs when we assume that Q is deterministic. Under those circums-tances the optimal hedging strategy and the maximum efficiency to be reached become the well known results:

2 3h*=-Q 1 2h ₁ Q rF * =-+ (A12) 0 1 2 1 h Q rF * =-+

(

)

MaxEfficiency =1 0. (A13)

Recepción del artículo: 23/03/2011 Envío evaluación: 10/04/2012 Recepción de correcciones: 27/04/2012 Aceptación del artículo: 11/06/2012