A note on the preferred hedge instrument

Transcription

1 ingnan University Digital ingnan University ong ong Institute o Business tudies Working aer eries ong ong Institute o Business tudies 香港商學研究所 6-5 A note on the reerred hedge instrument Arthur AU Follow this and additional works at: htt://commons.ln.edu.hk/hkibsw art o the Finance and Financial Management Commons Recommended Citation au, A. (5). A note on the reerred hedge instrument (IB Working aer eries 59-45). Retrieved rom ingnan University website: htt://commons.ln.edu.hk/hkibsw/95 This aer eries is brought to you or ree and oen access by the ong ong Institute o Business tudies 香港商學研究所 at Digital ingnan University. It has been acceted or inclusion in ong ong Institute o Business tudies Working aer eries by an authorized administrator o Digital ingnan University.

2 A NOTE ON TE REFERRED EDGE INTRUMENT ABTRACT Contrary to Battermann et al.'s () claims, this aer shows that risk-averse exorters may roduce less with air commodity utures than with air ut otions; moreover, they may reer the latter instrument or hedging against its exchange rate risk. IB/W/59-45

3 . INTRODUCTION The behavior o a risk-averse irm under exchange rate or rice risk in the resence o commodity ut otions and/or utures has been analyzed in several recent studies. By assuming that ut otions are indivisible, Wong () shows that a risk-averse irm roduces less under uncertainty in the resence o air indivisible ut otions than under certainty (or than that o a risk-neutral counterart). urrisingly, Battermann et al. () show that the same result holds even when ut otions become divisible and that a risk-averse irm always reers using air commodity utures to using air divisible ut otions or hedging. This aer shows using a counter-examle that simle risk aversion does not guarantee Battermann et al.'s () conclusions. The over-hedging behavior o a risk-averse exorter using airly riced ut otions renders the ayo unction non-monotonic and convex in the exchange rate risk making Battermann et al.'s roos incorrect. It is shown that the monotonicity o the ayo unction in the risk concerned is a key reason why Battermann et al.'s results hold under Wong's () model with indivisible (as oosed to) ut otions.. ECANGE RATE RI AND EDGING UING UT OTION An exorting irm roduces a single outut that is sold at cometitive world rice denominated in a oreign currency. The irm aces a random oreign exchange rate ~ with distribution unction G, robability density unction g, mean F, realization, and suort[, ], where >. From now on, a ~ means that the variable is a random Battermann et al. () does not distinguish random variable ~ rom realization ; they use to IB/W/59-45

4 variable; the same variable without a ~ is its realization. The exorter's roduction cost C satisies C > and C >. The exorter is risk-averse with utility unction U satisying U > and U <. uose a commodity ut otion with strike rice becomes available. For each unit o the ut otion urchased, the irm receives i is at or above and receives i alls below. Following Battermann et al. (), assume that the ut otion is unbiased (or air) with unit rice ~ [ (,)] = ( ) dg( ) R = E max, () where E is the exectation oerator. Ater urchasing Z units o the ut otion, the roducer's roit given realization o ~ is Y = C C ( ) + Z ( ) Z R, i [, ] ( ) Z R, i [, ] () which is equivalently to Y ( ) + Z [ (, ) R] = C max. (3) The roducer's roblem is to choose (, Z ) to maximize its exected utility { C( ) + Z [ max(,) R] } dg( ). U (4) The irst-order conditions are given by : [ ] ( ) ~ ~ [ C ( )] U ( Y ) dg ( ) = E{ C ( ) U Y } = (5) Z : ( ) U ( Y ) ( ) dg RU ( Y ) dg( ) denote both. IB/W/59-45

5 {[ ( ~ ~ max,) R] U ( Y )} = = E (6) where and Y are the otimal values o and Y, resectively. From now on, a denotes the otimal value o a variable. As ointed out by Eeckhoudt et al. (99), even in a non-roduction model, the second-order condition or the otimal choice o deductible insurance (and similarly ut otion) with a kinked ayo unction is comlicated. Thereore, ollowing Battermann et al. (), it is assumed that the second-order condition or a maximum holds. Battermann et al. () have correctly roved the ollowing claim stating that a risk-averse exorting irm over-hedges by urchasing more ut otion units than what are needed or covering its oreign currency denominated revenue. Claim Z >. Notice careully that Claim imlies that otimal ayoy irst decreases in and then increases in with a kink at =. Clearly, Y is non-monotonic and convex in the exchange rate risk. This turns out to be very imortant to the orthcoming analysis. 3. RODUCTION AND EDGING DECIION IN TE REENCE OF CURRENCY FUTURE Now, suose airly riced currency utures, instead o ut otions, are available. The irm chooses outut level and Z units o currency utures at unbiased orward rice ~ ( ) F = E such that its roit at realization o ~ equals 3 IB/W/59-45

6 Y = C ( ) + Z ( F ) Battermann et al. () have correctly roved the ollowing claim seciying that a risk-averse exorter chooses to ully hedge against its exchange rate risk when airly riced currency utures are available: Claim Z =. With ull hedging using airly riced currency utures, the exorter's otimal roit ~ Y is non-random. Battermann et al. () rove the ollowing certainty-equivalence result stating that in the resence o air currency utures, a risk-averse exorter chooses its outut level as i it is under certainty with ixed exchange rate F : Claim 3 F = C ( ). 4. RODUCTION DECIION IN TE REENCE OF UT OTION Battermann et al. () show that all risk-averse exorting irms roduce less in the resence o air ut otions than in the resence o air currency utures, i.e., <. Unortunately, both their claim and its roo are incorrect. The ollowing examles show that a risk-averse exorter's otimal outut level may decrease or increase when air ut otions are relaced by air currency utures: Examle A case with >. uose U is quadratic with ( ) = + ξ U Y ξ Y (where ξ is suiciently large) and > > A more detailed set o comutation results or this examle is available on request. IB/W/

7 ( Y ) = < U ξ or all Y in the relevant range, g 4 =, =, =, g( ) = =, [ ] and 3 3, ( ) = =, [,].imle calculation gives R = ( ) The irst ste is to show that ( Z ) > integration by arts and the above assumtions, that. It can be checked rom (6), using ( ( ) ( Y = ) ( ) ( ) dg U Y = ) RdG O ( Z ) U ( Y ) ( R) dg( ) d µ µ + ( ) ( ) U Y RdG µ d = U ( Z ) ξ = ξ ξ +. (7) 36 4 Assume by contradiction that ( Z ) gives. ubstituting this andξ < into (7) = >. 4 ξ ξ (8) 7 A contradiction! Thereore, ( Z ) >. Next, it can be checked that = 5 gives imilarly, it can be checked that F. Denote Ω( ) = ( F ) dg( ) µ. imle calculation Ω( ) d = = = µ dµ d. (9) IB/W/59-45

8 Ω µ () ( ) d = dµ d = =. Using irst-order condition (5), integrating by arts, and realizing that ( Z ) > and ξ <, it can be checked that [ F C ( )] U ( Y ) dg( ) ( F) U ( Y ) ( ) [ ( )] dg C U ( Y ) dg( ) = ( F) U ( Y ) dg( ) ( ) ( ) Ω( ) Z U Y d U ( Y ) Ω( ) = = ( ) d Ω( ) ξ Ω d. > 5 ξ = + > 4 44 ξ. () 44 = d F and hence The inequality in () clearly imlies that C ( ) < > as ( ) F = C according to Claim 3 andc >. Examle A case with <. 3 uose U,, and ollow the assumtions in Examle. Assume now thatg is a, uniorm distribution with ( ) =, [,] g. imle calculation gives R = 8. Under the new set o assumtions, the irst equality in (7) now gives 3 [( Z )] ξ 5 = ξ ξ +. () uose by contradiction that ( Z ) () gives. ubstituting this and then ξ < into 3 A more detailed set o comutation results or this examle is available on request. IB/W/

9 = <. 64 ξ ξ (3) 96 A contradiction! Thereore, ( Z ). Next, it can be checked that F = giving rise to imilarly, Ω ( ) ( ) ( ) µ Ω = ( µ F ) dg( µ ) = =. (4) ( ) = ( F ) dg( µ ) (5) imlies that ( µ ) ( ) ( ) µ = = = Ω( ). (5) ( ) d = Ω( ) d = Ω( T ) d( T ) = Ω( ) Ω d <. (6) ubstituting (6) and ( Z ) < into the third equality in() gives [ C ( )] ( ) ( ) < Ω( ) Ω( ) U Y dg d d = F ξ. (7) The inequality in (7) imlies that and hence < as F C ( ) ( ) > F C, (8) = and C >. 5. NON-MONOTONIC, CONVE AYOFF AND OME INTUITION To understand why simle risk-aversion does not guarantee < and why Battermann et al.'s () roo is incorrect, notice that their roo relies critically on the ollowing inequality: 7 IB/W/59-45

10 ~ [ ( ) ~ Y, C ( )] < cov U. 4 To show that the above inequality holds, Battermann et al. (, ootnote 4) suggest that ~ ~ ~ ~ gn{ cov[ U ( Y ), C ]} = gn{ cov[ U ( Y ), ]} ~ ~ such that the roblem reduces to showing that cov[ ( Y ), ] > U. owever, in the roo o the last inequality, Battermann et al. (, Aendix) have only attemted to show that ~ ~ [ U ( ), ] cov > monotonically increasing in. In act, kink at. According to () and Claim, it can be checked that ~ ~ =. Clearly, gn cov U ( Y ), same in general given only U <. Y is not Y irst decreases in and then increases in with a ~ ~ { [ ]}, and gn { cov[ U ( ), ]} are not the Interestingly, it turns out that the non-monotonicity o the ayo unction in the resence o ut otions is a crucial eature that distinguishes the model in this aer rom that o Machnes (99) and Wong (). Wong () shows that simle risk aversion is suicient to guarantee that a cometitive exorting irm always roduces more under certainty than under outut rice risk when it is rovided with a airly riced indivisible commodity ut otion. 5 A close look at Wong's model suggests that his result relies on the act that the ayo unction is monotonically non-decreasing in outut rice risk. Translating Wong's model o outut rice risk to the model in this aer, it can be checked that in the resence o an indivisible ut otion with a ixed number o ut otion units, Z, equal to outut level, 4 Battermann et al. () write (5) as E U Y E C Clearly gn ~ ~ ~ [ ] + cov[ U ( Y ), C ( )] = [ ( )] ( ) [ ] [ ] ~ ~ ~ [ F C ( )] = gn{ E C ( )} = gn{ cov U ( Y ), C ( )}. 5 Wong's () roosition states that in the resence o a airly riced indivisible commodity ut otion, a risk-averse cometitive irm always roduces less comared to an otherwise identical risk-neutral irm. Clearly, the otimal outut level o a risk-neutral irm equals in the context o exchange rate risk. IB/W/

11 the exorter's roit Y ( ) + [ max(, ) R] C =. 6 (9) The irm's irst-order condition (o choosing x ) is given by { C ( ) + [ max(, ) R] } U ( Y ) dg( ) = which can be simliied to ~ cov ( ) [ + max( ~,), U ( Y )] = E[ U ( Y )] F C () Equation () is analogous to Wong's () equation 7. A simle insection o (9) suggests that Y is monotonically non-decreasing in. Thereore, the covariance term in () is negative givenu <, exactly as suggested by Wong (), such that ( ) > F C () and hence < according to Claim 3 asc >. The non-monotonicity oy as discussed in ection renders Wong's aroach not usable when indivisible ut otions becomes divisible and are reely chosen by the exorter. A simle exlanation o this is that the convexity o the ayo unction ossibly makes the overall utility o the exorter non-concave in the exchange rate risk even when its utility unction is concave. 6. TE COICE BETWEEN FAIR FUTURE AND FAIR UT OTION Battermann et al. () argue that risk aversion alone is suicient to guarantee that an exorter always reers using air commodity utures to using air ut otions or hedging against its exchange rate risk. Unortunately, their argument is based critically uon their incorrect claims that risk aversion alone guarantees that < and that (8) always holds. 6 Machnes' (99) and Wong's () aers actually deal with outut rice risk instead o exchange rate risk. Their model has been modiied so that Wong's result can be directly comared to that in this aer. 9 IB/W/59-45

12 ~ Taking exectation on both sides o (3) gives E( Y ) = F C( ) ~ Claim 3 imly that ( ) [ ( )] [ ] ( ) ~ ~ E Y su E Y = su Y > E Y ~ al.'s (,.88) argument to show that ~ ~ ~ [ U ( Y )] U[ E( Y )] < U[ E( Y )] E[ U ( Y )]. Inequality(8)and =. One can now use Battermann et E < =. () The irst inequality in () ollows rom Jensen's inequality. The second inequality is due to ( Y ~ ) E( ~ ) E <. Y Notice that inequality () always holds such that Battermann et al.'s () argument can be used to show that a risk-averse exorter reers air commodity utures to air indivisible ut otions or hedging against its exchange rate risk. On the contrary, (8) does not hold in general such that the second inequality in ()may be reversed. Thereore, the last conclusion may not hold when ut otions become divisible. Divisible ut otions are obviously more common than indivisible ut otions as suggested by Wong (). IB/W/59-45

13 REFERENCE Battermann,.., M. Braulke, U. Broll, and J. chimmelennig,, The reerred hedge instrument, Economics letters 66, Eeckhoudt,., C. Gollier, and. chlesinger, 99, Increases in risk and deductible insurance, Journal o economic theory 55, Machnes, Y., 99, roduction decisions in the resence o otions, International review o economics and inance, Wong,..,, roduction decisions in the resence o otions: a note,international review o economics and inance, 7-5. IB/W/59-45