Optimal Foreign Exchange Risk Hedging : Closed Form Solutions Maximizing Leontief Utility Function

In this paper, we extend Kim (2013) [9] for the optimal foreign exchange (FX) risk hedging solution to the multiple FX rates and suggest its application method. First, the generalized optimal hedging method of selling/buying of multiple foreign currencies is introduced. Second, the cost of handling forward contracts is included. Third, as a criterion of hedging performance evaluation, there is consideration of the Leontief utility function, which represents the risk averseness of a hedger. Fourth, specific steps are introduced about what is needed to proceed with hedging. There is a computation of the weighting ratios of the optimal combinations of three conventional hedging vehicles, i.e., call/put currency options, forward contracts, and leaving the position open. The closed form solution of mathematical optimization may achieve a lower level of foreign exchange risk for a specified level of expected return. Furthermore, there is also a suggestion provided about a procedure that may be conducted in the business fields by means of Excel.


Introduction
Recently, foreign currency fluctuations are one of the key sources of risk in multinational business/investment operations because of the widespread adoption of the floating exchange rate regime in many countries after the breakdown of the Bretton Woods system. 1 The U.S. Department of Commerce has also warned that "The volatile nature of the FX market poses a great risk of sudden and drastic 1   The number of countries with floating and free floating arrangements are 36 and 29 by 2014, respectively, according to IMF (https://www.imf.org/external/pubs/nft/2014/areaers/ar2014.pdf). of the specific characteristics of a hedging tool, by relying on a questionnaire survey.Bodie, et al. (2002)  [2] and Nancy (2004) [1] illustrate the technique of computerized optimization and simulation modeling to manage foreign exchange risk.However, their techniques are not a closed form optimal hedging solution that requires additional computational burden.So its application is limited in the real business world.In this regard, Kim (2013)  [9] introduced the optimal foreign exchange risk hedging solution by exploiting a standard portfolio theory. 3 Hsiao (2017)  [7] applies the framework of Kim (2013)  [9] to investigate the effects of foreign exchange exposures on the performance of Taiwan hospitality industry and try to propose some hedging strategies and strengthen their corporate risk management.
In this paper, we extend Kim (2013) [9] for the optimal single FX risk hedging solution and theory to the multiple FX rates and suggest its application method in the business fields.First, the generalized optimal hedging method of selling/buying of multiple foreign currencies is introduced.Second, the cost of handling forward contracts is included.Third, as a criterion of hedging performance evaluation, we consider the Leontief utility (or profit for a firm) function, which represents the risk averseness of a hedger.Fourth, steps are introduced about what is needed to proceed with hedging.There is a computation of the weighting ratios of the optimal combinations of three conventional hedging vehicles, i.e., call/put currency options, forward contracts, and leaving the position open.As in the standard portfolio theory, the closed form solution of mathematical optimization may achieve a lower level of foreign exchange risk for a specified level of expected return.There is also a suggestion provided for a procedure that may be conducted in the business fields by means of Excel. 4 The rest of this paper is as follows.Section 2 derives the expected return and return variance of the hedging vehicles.Section 3 analyzes the optimal hedging selection.Section 4 is on application of developed method, and Section 5 is the conclusion.
1 m × at a future time T where i θ is represented by the unit of i-th currency.He is worrying about the foreign exchange risk of domestic currency (e.g., US dollar) term translated value of Γ and to hedge it optimally at time 0. The m-foreign exchange rates at time t in terms of domestic currency, is denoted as , , , S e e e ′ ≡  .For instance, it e is the dollar price of one euro or yen where the dollar is the domestic cur- rency.It is presupposed that there are three hedging tools, i.e., European currency put (or call) option, forward contracts, and leaving the position open. 5rthermore, there are the following definitions: a forward contract rate vector , , , F e e e ′ ≡  , a striking price vector ( ) , and its premium , , , m P p p p ′ ≡  at time t of a European put (or call) option with the common maturity T. 6 Finally, ( )  .Now we derive the return and its variance of different hedging tools, where the return is compared with the selling (or buying) a foreign currency (as a bench mark) by the spot rate 0 s .

FX Selling Case
First, we derive the expected return R n and its variance 2 n V of the non-hedging (leaving the position open), as follows.

5
It is a non-hedging and to buy the foreign currency at time T. 6   The value of the put option was derived by Garman and Kohlhagen (1983)  [5].Theorem 2.2.Suppose Assumption 2.1 holds.Then the expected return for non-hedging of FX asset Θ is R n = 0 and its variance during time T is 2 All proofs of the theorems are in the Appendix.
Second, we derive the expected return R f and its variance 2 f V of the forward contract as follows.
Theorem 2.3.Suppose Assumption 2.1 holds.Then the expected return of forward is , and Now we derive the expected return R p and its variance 2 p V of currency put option as follows.
Theorem 2.4.Suppose Assumption 2.1 holds.Then, (a) the expected return of currency put option is given as: and (b) its variance of currency put option is: where ( ) χ distri- bution with the degree of freedom q and: In the above Theorem 2.4, it was suggested that a form of  ) . Otherwise, there is a need for integration by a formula ( ) , which requires an additional burden.
Next, there is a derivation of the covariance among the three hedging tools.
Note the covariance of returns between non-hedging (or option) and forward is obviously zero since the forward return is not random.Then the covariance of returns between put option and non-hedging is given as follows.
Theorem 2.5.Suppose Assumption 2.1 holds.Then the covariance of returns between put option and non-hedging is:8

FX Buying Case
First V of forward contract as follows.
Theorem 2.6.Suppose Assumption 2.1 holds.Then the expected return of forward 9 is = − − and its variance is 2 0 f V = .Now we derive the expected return c R and its variance 2 c V of currency call option as follows.
Theorem 2.7.Suppose Assumption 2.1 holds.Then, (a) the expected return of currency call option is given as: and (b) its variance of currency call option is: The covariance of returns between call option and the non-hedging is given as follows.
Theorem 2.8.Suppose Assumption 2.1 holds.Then the covariance of returns between put option and non-hedging is:

Efficient Hedging Frontier Construction
Based upon above derivation of expected return (R) and return variance (V 2 ) structure, now we can derive the efficient hedging frontier.It is exactly matched with the portfolio possibilities curve in a standard portfolio theory (e.g., Elton, et al. ( 2007) [4]).
For this purpose, first, there is consideration of a portfolio composed of non-hedging and put in the option (for FX selling) that are all risky.Let the weight of non-hedging be as w and 1-w for the option where w is a real number.
Then, from the above derivation in Section 2, its expected return is defined as follows. 10) ( ) ( ) Buying the foreign exchange means outflow of domestic currency.So, a negative of the forward amount is taken.
Therefore note ( ) In this case, the return of forward has zero variance with the expected return, say, f R .Thus, it is regarded as a riskless asset in the standard portfolio theory.
Now the hedging allocation line (a line of R and V) 11 connecting the riskless forward contract and a combination of non-hedging and put option is defined as follows.
( ) ( ) where R denotes the return and V denotes the standard deviation of return (as a risk); is a constant slope for a given w where Then the efficient hedging allocation line12 is given by solving following problem: that is maximizing the slope of Equation (3.2) with the argument w.The problem (3.3) may be solved without restriction, according to Elton, et al. ([4]: pp.100-103), as follows.
where Finally, the efficient hedging frontier is given by: For the given efficient frontier in (3.5), the optimal hedging (cf., separation theorem) is conducted as follows.First, the hedging ratio between non-hedging and option are set as ( ) . See Figure 1.Second, ρ is set for the for- ward and 1 ρ − is set for the first combination of non-hedging and option.So if 1 ρ = , then the forward becomes the unique hedging tool.Y.-Y.Kim may be a rule to determine an optimal ρ .The following section suggests an op- timal hedging solution through determining an optimal ρ under the Leontief utility function.

Optimal Hedging under Leontief Utility Function
A Leontief utility (or profit for a firm) function is considered as a criterion for hedging performance evaluation where 0 β < .Note, for the maximization of a Leontief utility function under the efficient hedging frontier in Figure 1, a pair (V, R) should satisfy a line: .
To show it, let us derive an indifference curve.For this, suppose 0 0 (as in Figure 2).Then a utility of ( ) V R has the same utility with ( ) while the utility of ( ) larly, a utility of ( ) 0 , V R has the same utility with ( ) while the utility of ( ) the North-West direction indicates the increase of utility in a space of (V, R).
Later, the above Equation (4.1) will be called a utility maximizing locus (UML).The UML might be interpreted as that which denotes how V is transformed into R with the same utility.It also denotes a cost of the standard deviation (volatility) for a hedging portfolio.See Figure 2 where the cost for the volatility 0 V is evaluated as in terms of return.Note, the above Leontief utility function and conformable UML represent an extreme risk averseness.It is related to the marginal rate of substitution of the volatility to a return at the utility maximizing point along UML, which is +∞ , i.e., the marginal increase of V requires an infinite return increase (as compensation for augmented risk) for the same utility, whereas, a marginal decrease of V does not require any return to be at the same utility level.This assumption is not so unrealistic because this model is not designed for the speculator but for the hedger/firms in the real world of business who are concerned with the volatility of fund flow.Now to estimate α and β by an ordinary least square regression, we re- write Equation (4.1) as: where ( ) is a change rate of FX asset during a maturity from 0 to T, z is a sample average of Ti z , and Ti ε is assumed as a mean zero error term that is not correlated with Ti z .Now note the intersection of UML (4.1) and the efficient hedging frontier (3.5), which is given as follows.
( ) ( ) after solving two Equations (3.5) and (4.1) with two unknowns R and V when * 0 1 w ≤ ≤ .The above solution point ( )   helps to find the optimal weight for the riskless forward contract as . See Figure 3.
It is also equivalently written as from the property of the proportional triangular. Y.-Y.Kim

( ) ( )( )
becomes the optimal hedging ratio of forward, non-hedging, and put option using (4.3) for the vector Θ .So, for instance, the weight * ρ of Θ needs to be distributed to the forward.
Note, if the slope coefficient β as a marginal cost of volatility V is decreased to ( ) β β ′ < , then the new optimal weight for the forward contract (riskless) is decreased as . So more risk can be admitted because the marginal cost of volatility is decreased.See Figure 3 to see this change.
However, if V  is larger than ( ) * V w because β is sufficiently small, then the weight for the forward contract (remind 0 n R = ) may become zero. 14In this case, * w is not any further an optimal weight between the leaving open position and the option.Rather, we have to choose it from the intersection of UML and the locus of ( ) ( )  for the optimization are computed as follows. 15Theorem 4.1: Suppose a pair (V, R) satisfies a line (4.1).Then In Theorem 4.1, we may have two different solutions that need to be selected to maximize the utility.So, we need to select one R among them maximizing the utility and define a conformable optimal expected return as . See following Figure 4. Finally, the optimal hedging ratio of the forward, non-hedging, and put option becomes ( )  Finally, if 1 ρ ≥ , then a weighting vector (1, 0, 0) that is just selling the for- ward becomes the optimal hedging ratio.

Application Procedures
In application, suppose, at time 0, an investor hopes to sell one unit of foreign exchange at a future time T. Then following steps need to be carried out for hedging.
1) Select three vehicles of hedging as: forward contracts, leaving the position open (Selling foreign exchange case) and European currency put option.
2) Compute mean, variance, and covariance of each tool using the formula in Section 2.
3) Compute a weighting coefficient * w as in (3.4) or w as in (4.5) if V V <  or leaving the position open against the put option.4) Decide α and β using OLS regression as in (4.2). 5) Compute an optimal weighting coefficient for the forward against for the portfolio of option and leaving the position open ρ as in (4.3).6) Finally compute the optimal hedging ratio of the forward, non-hedging, and option.

( ) ( )( )
as in (4.4).Consequently, we summarize the optimal weighting vectors of forward, option, and non-hedging for optimal hedging, as shown in Table 1.
Then we apply the developed method for the exchange rate of the euro against the US dollar.The data frequency and period are presented on a monthly basis from January 1999 to March 2015.All data have been taken from FRED of FRB St. Louis.Thus we assume, at time 0, i.e., June   We assume 0.01 α = .See Figure 5 for the optimal weighting ratio in (4.4) change as β decreases 16 .Note, if β as a marginal cost of volatility V is de- creased, then the optimal weight for the forward contract (riskless) is decreased, as expected in the above theoretical explication (see Section 3).

Conclusions
This paper introduced the optimal foreign exchange risk hedging solution by exploiting a standard portfolio theory, thus extending Kim (2013) [8] in its following features.First, the case of the selling/buying of multiple foreign currencies is also considered.Second, the cost of handling forward contracts is included.Third, as a criterion of hedging performance evaluation, we consider the Leontief utility function, which represents the risk averseness of a hedger.Fourth, steps are introduced about what is needed to proceed with hedging.There is a computation of the weighting ratios of the optimal combinations of three conventional hedging vehicles, i.e., call/put currency options, forward contracts, and leaving the position open.The closed form solution of mathematical optimization may achieve a lower level of foreign exchange risk for a specified level of expected return.There is also a suggestion provided about a procedure that may be conducted in the business fields by means of Excel.
The structure may be extended to cover the futures and American options and it will be a future research topic for us.However, I hypothesize that a similar logic may be readily applied to these extensions applying developed method in this paper.Furthermore, a development of a convenient computer program for FX risk hedging users, based on above results, would be a useful project.

Appendix: Proofs of Theorems
Proof of Theorem 2.2: Note the return of non-hedging is approximately the value of following 17 : ( ) ( ) ( ) Proof of Theorem 2.3: Note the expected return for forward is the value of following: ( ) − − is small.Its variance is obviously zero since the return is not random.


Proof of Theorem 2.4: (a) Note the inflow of selling weighted put option at time T is given as Thus its return is given as following: for the third equality (5) from Greene ([6]: p. 759), and where (b) The return's variance of ( 4) is defined as: Note the second term of right hand side in ( 8) is derived from ( 5) directly.
Then the first term of right hand side in ( 8) is arranged as: where because, for the second term in last equation in (9), we may show that ( ) ( ) from the change of variable formula where g and G denote the density and dis- ) using the symmetry of normal distribution; solving following equation for ( ) for the final equality of (11).Further note from Marchand ([10]: p. 26 and Remark 4), where ]: p. 26 and Remark 4) where ( ) ( ) Plugging (15) into (11)   12) for the third equality, from (14) for the fourth equality and from (16) for the final equality.


Proof of Theorem 2.5: Note the covariance between non-hedging and put option conditional on Ω is defined as:  is constant conditional on Ω for the second equali- ty and the fourth equality holds from ( ) Now the claimed result is derived since T T from (10) and ( ) from ( 17) where   (b) The return's variance of call option conditional on Ω is given as: Note the second term of right hand side in (24) is derived from (21) directly. where as similarly in (10) and Case 1: From symmetry and T z has a standard normal distribution, from (12) for the second equality, from ( 11) and ( 16) for the final equality.

F
denotes the distribution function of central ( )2 q

χ
distribution for the computation of conditional expectation

(
the maximization problem (3.3) should be solved under the restriction [ ] 0,1 w∈ using a typical Kuhn-Tucker condition.
hedging ratio of the forward, non-hedging, and put option.Note the expected utility maximization 11 It is called as the capital allocation line in the portfolio theory.

Figure 3 .
Figure 3. Derivation of optimal weight for forward.
Then, under Assumption 2.1, the claimed results hold as: return conditional on Ω in (4) is computed as: negative for buying of foreign currency (also for the forward contract) because it means the outflow of domestic currency. 18


Its variance is obviously zero since the return is not random.Proof of Theorem 2.7: (a) Note the outflow of buying call option at time T + .Thus its return normalized by 0 S is given as the negative value of following: return conditional on Ω is value of following:

8 :
Note the covariance between non-hedging and call option conditional on Ω is defined as: since the fourth equality holds from ( )0 T E x Ω = .Proof of Theorem 4.1: To get such a solution point, we solve following three equations: , note we have the same expected return 0
σ =$/€, a hedger hopes to sell one unit of foreign exchange at a Y.-Y.Kim )