Optimal Foreign Exchange Risk Hedging: a Mean Variance Portfolio Approach

Theoretical Economics Letters, 2013, 3, 1-6 http://dx.doi.org/10.4236/tel.2013.31001 Published Online February 2013 (http://www.scirp.org/journal/tel)

Optimal Foreign Exchange Risk Hedging: A Mean Variance Portfolio Approach

Yun-Yeong Kim Department of International Trade, Dankook University, Yong-In, Korea Email: [email protected]

Received October 7, 2012; revised November 10, 2012; accepted December 12, 2012

ABSTRACT This paper introduces the optimal foreign exchange risk hedging model following a standard portfolio theory. The results indicate that a lower level of risk can be achieved, given a specified level of expected return, from using optimization modeling. In the paper the expected hedging return is defined from the expected cost of the foreign currency using a specified hedging strategy minus the expected cost of the foreign currency when it is purchased form the spot market. The focal point of the technique is its ability to identify optimal combinations of hedging vehicles, those are currency options, forward contracts, leaving the position open (foreign exchange risk hedging tools suggested by the US. De- partment of Commerce) in a closed form.

Keywords: Foreign Exchange; Risk; Optimal Hedging; Closed Form

1. Introduction Beginning in the early 1970s, floating foreign exchange (FX) rates became more common, among the major cur- rencies. Now the recent global financial crisis including euro zone instability have clearly illustrated the critical importance of hedging for risks in foreign exchange rate. See following figures of monthly Euro/Dollar and Yen/ Dollar foreign exchange rates during 1999.1-2011.7, where both FX rates are fluctuating especially after global financial crisis.1

So foreign currency fluctuations are one of the key sources of risk in multinational operations. The various tools which have emerged to deal with foreign exchange risk have been treated extensively in the finance litera- ture. The nature, uses, and efficiency of their markets are quite well understood today (See [1]). The US Depart- ment of Commerce is also warning that “The volatile nature of the FX market poses a great risk of sudden and drastic FX rate movements, which may cause signifi- cantly damaging financial losses from otherwise profit- able export sales (Trade finance guide, http://trade.gov /publications/pdfs/tfg2008ch12.pdf).” The same guide suggests three FX risk management techniques consid- 1The axsis border graphs are kernel density functions. ered suitable for new-to-export US small and medium-

sized enterprises companies as non-hedging FX risk currency. For instance, St is the dollar price of one euro management techniques,2 FX forward hedges3 and FX where the dollar is the domestic currency. Further we options hedges.4 suppose that there are three hedging tools, i.e., European However, what has been ignored, as correctly pointed currency call option, forward contracts and leaving the 6 out [2], are the factors an investor should consider when position open. Define a forward contract rate Ft , a choosing from among the various available hedging tools striking price K and its premium P at time t of Euro- to reduce the risk resulting from a certain type of expo- pean call option with the maturity T , respectively.7 sure to foreign exchange risk for a given expected return. Now we would like to construct the efficient hedging [2] gauges the preferences of finance officers in terms frontier composed of expected return and variance of of the specific characteristics of a hedging tool relying on each hedging vehicle. So, it is exactly matched with the a questionnaire survey. [3,4] illustrate the technique of portfolio possibilities curve. An optimal combination of computerized optimization and simulation modeling to hedging vehicles is one, which maximizes the expected manage foreign exchange risk. However they did not return given a desired level of risk. derive the closed form optimal hedging solution analyti- Before proceeding, we assume the logarithm of ex- cally and thus it obviously requires the additional com- change rate follows a random walk following [5]: putational burden. Assumption 2.1. We suppose

So this paper introduces the optimal foreign exchange stt11 su t risk hedging model following a standard portfolio theory. where s  ln S and u is independent, identi- The results indicate that a lower level of risk can be t t  t  cally and normally distributed sequence with the mean achieved, given a specified level of expected return, from zero and variance  2 . using optimization modeling. In the context of this paper Above Assumption 2.1 represents the efficient market the expected hedging return is defined from the expected hypothesis for the foreign exchange rate. Now we derive cost of the foreign currency using a specified hedging the return and its variance of different hedging tools, strategy minus the expected cost of the foreign currency where the return is computed based upon the purchasing when it is purchased form the spot market. The focal a foreign currency by the spot rate S . The expected point of the technique is its ability to identify optimal 0 return is defined from the conditional expectation8 based combinations of most frequently using hedging vehicles, on the information of past exchange rates those are (European) currency options, forward contracts, 9 SS,, . leaving the position open (foreign exchange risk hedging 01 tools suggested by the US Department of Commerce) in 2.1. Derivation of Mean and Variance a closed form.5 The rest of this paper proceeds as follows. Section 2 At first, we derive the expected return [Rn] and its vari- 2 derives the expected return and variance of hedging ve- ance [Vn ] of non-hedging (leaving the position open) as: hicles. Section 3 analyzes the optimal hedging selection. Proposition 2.2. Suppose Assumption 2.1 holds. Then

Section 4 concludes. the expected return for non-hedging is Rn  0 and its 22 variance is Vn  T . 2. Moments of Triple Hedging Tools’ Proof. Note the return of non-hedging is the negative10 Returns value of following:11

Assume, at time 0, an investor hopes to buy one unit of 6It is a non-hedging and to buy the foreign currency at time T . 7 foreign exchange at a future time T . Denotes S as the The value of call option was derived by [6]. t 8Note the conditional expectation is the optimal forecasting of return foreign exchange rate at time t in terms of domestic which minimizes the mean squared forecasting error. 9 2The exporter can avoid FX exposure by using the simplest non-edging So an investor exploits the information  when she determines the technique: price the sale in a foreign currency. The exporter can then optimal hedging tool. 10 demand cash in advance, and the current spot market rate will deterne It is negative because buying of foreign currency means the outflow the US dollar value of the foreign proceeds. of domestic currency. 11 3The most direct method of hedging FX risk is a forward contract, We do not consider the cost-of-carry for the spot purchasing of for- which enables the exporter to sell a set amount of foreign currency at a eign currency since it is applied for the other hedging vehicles by the pre-agreed exchange rate with a delivery date from three days to one same amount. In particular, suppose an interest r is required to bor- year into the future. row the S0 till time T and then the non-hedging return Formula (1) 4 Under an FX option, the exporter or the option holder acquires the may be revised as; right, but not the obligation, to deliver an agreed amount of foreign SrS1  1 SS r currency to the lender in exchange for dollars at a specified rate on or T 0 T 0 11 rS r S1 r before the expiration date of the option. As opposed to a forward con- 00 tract, an FX option has an explicit fee, which is similar to a premium Thus it does not matter even if we just focus on the SSST  00 paid for an insurance policy. 5At least author’s knowledge, there are not any the same researches in since the other terms 11  r and rr1  are same for the other this direction. hedging tools.

SS where kK log , pPS , x ks, zx  T T 0 s s (1) 0 0000 T 0 and zz  z where  and  are the S0  00    0 standard normal density and distribution function re- assuming SST  0 is small. Then, under Assumption 2.1, the claimed results hold as: spectively and q denotes the distribution function of  2 density function with the degree of freedom q . q Es T  s0  0 Proof. 1) Note the outflow of call option at time T is and given as minSKT ,   P where P is the option premium. Thus its return is the negative value of following: Es s2 T 2 . T 0 minSKT ,   PS0 At second, we derive the expected return R and f S0 its variance V 2 of forward contract as: f SSKSP Proposition 2.3. Suppose Assumption 2.1 holds. Then minT 00 , SSS the expected return of forward is Rsff 0 T and its 000 2 variance is V  0 where f  log F . f TT minssksp , Proof: Note the expected return for forward is the T 00 negative value of following: assuming SST  0 and K  S0 are small. Now the expected return conditional on  is the FST  0 fT s0 negative value of following: S0

Esskspmin T 00 ,  assuming FT  S0 is small. Its variance is obviously  zero since the return is not random. ExxpminT , 0 Above forward contract may dominate the non-hedging if its expected return is positive, which is riskless.  x001zTzz  00   p Such dominance may be closely related with the interest where x  ss , since rates whenever the covered interest parity holds. See fol- TT0 lowing result. Exxmin T , 0   Corollary 2.4. Suppose Assumption 2.1 holds and  rr . Then the forward contracts dominates the non- ExxxxxxminTT ,00 , Pr  T 0  hedging where r and r denote the domestic and for- Exxxxxxmin , , Pr  (2) eign risk free interest rates respectively. TT00 T 0  Proof. From the covered interest parity, note rr = x00PrxxTTTT Exxx 0 , Pr xx0 f  s . So if fs0 or r r , then there is posi- T 0 T 0 xzTzz1    tive expected return without risk. In this case, the for- 00 00 ward contract dominates the non-hedging case. from the definition of conditional expectation, where Above result also implies that if the domestic interest 2 xNTT ~0,   from Assumption 2.1 and rate is higher than the foreign interest rate, then the 12 non-hedging may better than the forward contract. Ex TT x x00,   T z (3) Now we derive the expected return [Ro] and its vari- 2 for the Equality (2) from [7] (p. 759), and ance [Vo ] of currency call option as: Proposition 2.5. Suppose Assumption 2.1 holds. Then xT x0 1) the expected return of currency call option is given PrxxT 0 Pr  as: TT PrzzT 00 z Rxo 001  z  Tzz 00   p where zx  T. and 2) its variance is TT 2) The return’s variance of call option conditional on 22 230z  is defined as: Vxo 001  z T  z 0  z 10 2 2 ExxExxminTT ,00 min  ,  xzTzz001   00  (4)  2 2 ExxExxmin , min ,  12This result well explains why a lot of firms of the developing coun-  TT00  tries with the high interest rate do not use the forward contract. The problem is the cost to use the forward is bigger than the non-hedging if Note the second term of (4) is derived from (2) di- fT  s0 even though there is the risk difference. rectly. Then the first term of (4) is arranged as:

2 Exxmin ,  hz0,3,0 30 30 T 0  2 in [8] (Remark 3) where 00  0. Exxxxxxmin , , Pr  13 TT00  T 0 Finally if we plug (6) into (5), then we get the claimed 2 result as: Exxxxxxmin , , Pr  (5) TT00  T0 2 Exxmin ,  22 T 0  x00PrxxTTTT Exxx 0 , Pr xx0

22 2230z xzExxxz001, TT  00 x001 zT  z0 10z from the definition of conditional expectation for the first equality. 2.2. Derivation of Covariances However we may show At second, we derive the covariance among three hedging tools. Note the covariance of returns between non- 2230z ExTT x x0 T (6)  z hedging (or option) and forward is obviously zero since 10 the forward return is not random. Then the covariance of from following facts (b-i) and (b-ii): returns between option and non-hedging is given as: Proposition 2.7. Suppose Assumption 2.1 holds. Then 22gx T  the covariance of returns between option and non-hedg- ExTT x x0 x Td x T xxT  0 GxT  x0 ing is

gTz Covon x00Tz1   z0 22T    TzTT Tdz (b-i) zzT  0 GzT  z0 2 30z Tz 0 . 22 10z TEzzz TT 0 Proof. Note the covariance between non-hedging and gTz T option conditional on  is defined as: since  T is the truncated density function Gz z T 0 ExxpExxpx min , min ,    TT00   T of variable zT since  ExxExxxminTTT ,00 min  , gTz  gxT T 1dxTT Tzd  ExxxExxExmin ,  min ,   xxTT00 zz TT00 T T GxTT x00 Gz z Exxxmin , from the change of variable formula where g and G TT0 denote the density and distribution functions of xT re- since the fourth equality holds from Ex T  0. spectively, and  TzddT xT since zxTT  T by Now the claimed result is derived since definition. Exxxmin TT , 0   Ez2 z z TT 0 ExxxxxxxminTTT ,00 , Pr  T 0 3 Emin xxxxx , , Pr  xx  (b-ii) TTT00 T0 2 30zz 30 30 2 xE00 xTT x x,Pr x T x 0 1 10zz 10 10  2 2 ExxxTT00,Pr xx T from [8] (Remark 3), where xE000 xTT x x,1 z 2 1 31 ExTT x x00,  z  π and  π , 2 22 2 30z x00Tz1.  z 0 T   z0 where 10z

hz0,1,0 10  10 from (3) and (6) for the last equation and and 1300  0.

3. Exact Efficient Hedging Frontier Finally, the efficient hedging frontier is given by Construction RR w f Based upon above derivation of return structure, now we RRf  Vof the left of R , V V ww may derive the efficient hedging frontier. It is exactly w (10) matched with the portfolio possibilities curve in a stan-  RVww, of the right of  R , V dard portfolio theory (see [9] for a nice introduction). ww For this purpose, first of all, we consider a portfolio where 01 w  . composed of non-hedging and call option that are all For the given efficient frontier in (10), optimal hedg- risky. Let the weight of non-hedging be as w and 1 − w ing (c.f., separation theorem) is conducted as follows. At for the option where w is a number. Then, from the above first, the hedging ratio between non-hedging and option derivation, its expected return is defined as: re set as ww ,1  . At second,   rho is set for the

RwRwRwn1  o forward and 1   is set for the first combination of non-hedging and option. Expected utility maximization and its variance is given as: may be a rule to determine a  . Finally VwVwVww222121C2 2 ov ,1ww ,11 becomes the optimal hed- wn o on    ging ratio of the forward, non-hedging and call option where Covon denotes a covariance between the returns of non-hedging and call option. with a sum as one. See following Figure 1. In our case, the return of forward has the zero variance Now we suggest an example that shows how above result may be applied in the field. with the expected return is R f . Thus it is regarded as the riskless asset in the standard portfolio theory. Now the Example 3.1. Above result is applied to the dollar as hedging allocation line13 connecting the riskless forward domestic currency and the yen as the foreign currency. contract and a combination of non-hedging and call op- To compute the efficient hedging frontier in (10), we let tion is defined as: T  3 months, S0  0.8547 (August 18, 2010), F3  0.9000 , K  0.9000 , P  0.0100 dollar/100yen RR and an estimator of  2  0.0888 (during 2005.1- RRwf V (7) f  2009.12). Vw Then, at first, we get R f 0.1106 , V f  0 , Rn  0 , where R denotes the return and V denotes the stan- Vn  0.0025 , R0  0.0816 and V0  0.0008 from the above results. Then we obtain the return R 0.0679 dard deviation; RRVwf w is a slope. w Then the efficient hedging allocation line14 is given by and variance V  0.0037 of the portfolio non-hedging w solving following problem: and option where w  0.1678 . RR From this result, the Equation (10) in the efficient max wf (8) w hedging frontier becomes: Vw RV 0.1106 11.62 that is maximizing the slope of Equation (7) with the of the left of  0.0679,0.0037 (11) argument w. The problem (8) may be solved without restriction by Suppose an extremely risk-averse investor maximizes [9] (pp. 100-103) as:

m 0 w  1 (9) mm utility function non‐hedging standardvariance 12 deviationof where w* determination of returnreturn

1 2 RR rho determination m1 VnonCov nf  . m Cov V 2 RR 2 on o of forward option If w  0 or 1  w , then the maximization problem

(8) should be solved under the restriction 01 w  ‐1 using a typical Kuhn-Tucker condition. mean of return

13It is called as the capital allocation line in the portfolio theory. 14It is called as the capital market line in the portfolio theory. Figure 1. Efficient hedging frontier.

Copyright © 2013 SciRes. TEL 6 Y.-Y. KIM a utility function uR2000 V2 subject to (11).15 The cinnati, 1995. resultant portfolio induces V  0.0029 and R  0.0769. [2] S. Khoury and K. Chan, “Hedging Foreign Exchange It implies  0.0029 0.0037 0.7837 where the util- Risk: Selecting the Optimal Tool,” Midland Corporate ity is maximized with the constraint (11). Thus the for- Finance Journal, Vol. 5, 1988, pp. 40-52. ward, non-hedging and option are finally selected as [3] N. Beneda, “Optimal Hedging and Foreign Exchange Risk,” Credit and Financial Management Review, 2004. ,1ww ,11   [4] Z. Bodie, A. Kane and A. Marcus, “Investments,” Mc-  0.7837,0.0362,0.1800 Graw Hill, New York, 2002. [5] F. X. Diebold and J. A. Nason, “Nonparametric Exchange Rate Prediction?” Journal of International Economics, 4. Conclusion Vol. 28, No. 3-4, 1990, pp. 315-332. doi:10.1016/0022-1996(90)90006-8 We introduced the optimal foreign exchange risk hedging model following a standard portfolio theory. The results [6] M. Garman and S. Kohlhagen, “Foreign Currency Option indicate that a lower level of risk can be achieved, given Values,” Journal of International Money and Finance, Vol. 2, No. 3, 1983, pp. 231-238. a specified level of expected return, from using optimiza- doi:10.1016/S0261-5606(83)80001-1 tion modeling. The structure may be extended to cover the futures and American options and we will take it as a [7] W. Greene, “Econometric Analysis,” Pearson Education, future research topic. However I am sure the similar Upper Saddle River, 2003. logic may be readily applied to these extensions. Further [8] E. Marchand, “Computing the Moments of a Truncated a development of convenient computer program for FX Noncentral Ch-Square Distribution,” Journal of Statisti- risk hedging users based on above results might be a cal Computation and Simulation, Vol. 55, No. 4, 1996, pp. 23-29. doi:10.1080/00949659608811746 useful project. [9] E. Elton, M. Gruber, S. Brown and W. Goetzmann, “Modern Portfolio Theory and Investment Analysis,” REFERENCES Wiley, New York, 2007. [1] P. Sercu and R. Uppal, “International Financial Markets and the Firm,” South-Western College Publishing, Cin-

15This utility function largely penalizes the risk and reflects deep risk aversions (e.g., under recent financial crisis). Of course other utility functions may be readily applied.