1 YEN FUTURES: EXAMINING HEDGING EFFECTIVENESS BIAS AND CROSS-CURRENCY HEDGING RESULTS ROBERT T. DAIGLER FLORIDA INTERNATIONAL UNIVERSITY SUBMITTED FOR THE FIRST ANNUAL PACIFIC-BASIN FINANCE CONFERENCE The auhor wishes o hank Luisa Chong for compuer assisance.
3 3 YEN FUTURES: EXAMINING HEDGING EFFECTIVENESS BIAS AND CROSS-CURRENCY HEDGING RESULTS ABSTRACT The usefulness of yen fuures for hedging purposes is examined in several unique ways. The wo key conceps are: (1) o deermine he exen of he hedging effeciveness bias ha is inheren in he porfolio/regression model of hedging, and (2) o examine he hedging effeciveness and he poenial insabiliy of he hedge raios for cross-currency hedging beween yen fuures and cash currencies of indusrialized and lesser developed counries. The analysis also provides evidence on he subperiod yen fuures hedging resuls, he effec of he lengh of he ime inerval on he hedging effeciveness, and he effec of changing he iming difference beween he fuures and cash daa. NOTE: Since DRI (Daa Resources Inc.) had considerable difficuly in providing he correc daa for he analysis conemplaed above, he resuls given here are incomplee. Therefore, he resuls of he curren draf of his paper only reflecs an examinaion of one week hedges for he yen fuures versus yen cash and cross-currency hedges for indusrial counries plus an examinaion of he associaed bias; however, he inroducion o he paper noes oher areas ha will be explored, including: 1) Daa analysis for ) Cross currency resuls for lesser developed counries 3) Two week and one monh hedges (each period will be lenghened)
4 4 4) Using he open fuures price o deermine he effec of iming differences.
5 5 YEN FUTURES: EXAMINING HEDGING EFFECTIVENESS BIAS AND CROSS-CURRENCY HEDGING RESULTS INTRODUCTION Hedging sudies for currency fuures and oher ypes of fuures conracs have concenraed on deermining he hedge raios and hedging effeciveness for a given ime period by using he porfolio/regression approach (see Ederingon (1979)). Hill and Schneeweis (1982) use his model wih currency fuures and associaed cash daa for , deermining ha he yen fuures only possess a hedging effeciveness of 16.0% and 14.8% for one week and wo week hedges when he nearby conrac is employed. Oher currencies provide effeciveness resuls ranging from 46.1% o 67.4% for one week hedges and 74.4% o 83.1% for wo week hedges. The firs objecive of his paper is o deermine he hedging effeciveness of he yen fuures for he ime period and is subperiods o examine he abiliy of he yen fuures conrac o achieve is objecive of providing an effecive means of hedging cash yen currency flucuaions during he volaile ime period of he 1980's. These resuls will be compared o a recompuaion of he daa o examine wha differences exis. The effec of he ime inerval used o measure hedging effeciveness and he iming issue of maching he cash and fuures quoaions also will be explored. The second objecive of his paper is o examine he cross-currency hedging effeciveness and hedge raio sabiliy of yen fuures wih European/indusrialized counries and wih lesser developed counries. The hird and mos imporan objecive of his paper
6 6 is o develop models ha examine he effec of unsable hedge raios on hedging effeciveness; hese models are hen esed on he yen fuures conrac. The paper is organized as follows: he wo models which measure he amoun of bias in he ypical ex-pos hedging effeciveness value are developed, he daa and resuls relaing o yen fuures hedging are examined, and hen conclusions and implicaions are given. THE MODELS FOR HEDGING EFFECTIVENESS BIAS Two models are developed o show he effec of an unsable hedge raio on hedging effeciveness. The firs model assumes ha one wishes o hedge agains all price changes excep changes due o convergence. This simplificaion provides a sraighforward resul ha is easy o calculae. The second model is based on he desire o hedge agains all price changes. This model is more complicaed in form bu heoreically will be more accurae, especially for markes wih rend changes, a large convergence facor, or for cross-hedging siuaions which have deviaions beween he behavior of he fuures and cash markes. A Simplified Model The ypical ex-pos variance minimizing hedge raio for ime period +1 is designaed as b * +1 and is defined as: b * +1 = SF / 2 F (1) Where: SF = he covariance beween he spo (S) and fuures (F) price
7 7 F 2 changes during ime period +1 = he variance of he fuures price changes during ime period +1 The basis a a specific ime k wihin he ime inerval +1, as defined in erms of he ex-pos minimum variance hedge raio, is: (2) Where: H * +1 (k) = Basis = S +1 (k) - b* +1 F +1 (k) H * (k) = he basis a ime k wihin ime inerval +1, as +1 deermined by using he ex-pos hedge raio b * +1 S (k) = spo price a ime k wihin inerval F (k) = fuures price a ime k wihin inerval Similarly, we define he change in he basis from ime k o ime k+1 wihin ime period +1 as: H * +1 (k,k+1) = S +1 (k,k+1) - b* +1 F +1 (k,k+1) (3) If one wishes o hedge agains all price changes oher han hose due o convergence or o he average change in he basis over he period, hen he variabiliy of he basis change during ime period +1 can be deermined by: (4) var( H * +1 ) = 2 * 2 2 * + b F - 2 b S SF Where: S 2 = he variance of spo price changes during period +1 When an unsable minimum variance hedge raio exiss beween ime period "" and ime period "+1" hen b * +1 can be defined in erms of b* and he change in he hedge raio from "" o "+1":
8 8 b * +1 = b* + b (5) Where: b * = he minimum variance hedge raio over he ime period b = he change in he hedge raio from ime period o ime period +1 Consequenly, he change in he basis beween ime k and ime k+1 wihin ime inerval +1 can be redefined o consider he effec of employing he previous period's minimum variance hedge raio b * as an esimae of he rue curren period's minimum variance hedge raio. Thus, if b * + b from (5) is subsiued for b * +1 in (3) we have: (6) H * +1 (k,k+1) = S +1 (k,k+1) - (b* + b ) F +1 (k,k+1) The resulan equaion for he variabiliy in he basis change is: (7) var( H * +1 ) = 2 + (b S * + b )2 2 * - 2 (b F + b ) SF Likewise, if a he beginning of ime period +1 one uses he minimum variance hedge raio b * as he bes esimae of b*, hen one may +1 deermine wha he variabiliy of he basis change would be during +1 by using b * : (8) var( H +1 ) = 2 * 2 2 * + b F - 2 b S SF Where: var( H ) = he variance of he change in he basis during ime +1 period +1 as deermined by using he previous period's
9 9 minimum variance hedge raio b *. Subracing (7) from (8) we can deermine he addiional basis risk from using b * as an esimae of b* +1 when he minimum variance hedge raio changes over ime: (9) var( H +1 ) - var( H* +1 ) = - b* 2 2 * F - 2 b b + 2 b F2 SF = 2 b ( - b * SF ) - b F F F 2 = 2 b F 2 ( SF / F 2 - b * F2 / F 2 ) - b Since from (5): 2 * = 2 b (b F +1 - b* ) - b 2 2 F b = b * +1 - b* we deermine ha: (10) var( H +1 ) - var( H* +1 ) = b 2 F 2 > 0 Using E * +1 = R +12 as he ypical measure of he minimum variance hedging effeciveness for period +1, equaion (11) saes his definiion in erms of he variabiliy in he basis change by employing he minimum variance hedged posiion ( H * ) and he variabiliy of he changes in he +1 unhedged or cash ( S +1 ) posiion: (11) Where: E * +1 = R 2 * = 1 - var( H )/var( S +1 ) E * +1 = he hedging effeciveness for period +1 by using he minimum variance hedge raio b * +1 The upward bias in he +1 minimum variance hedging effeciveness value when b * is used as an esimae of b* +1 can be deermined by using
10 10 (10): E * +1 - E +1 = 1 - var( H* +1 )/ 2 - [1 - var( H S +1 )/ 2 ] S (12) = b 2 [ F 2 / S 2 ] Where: E * +1 = he minimum variance hedging effeciveness measure when he ex-pos hedge raio b * +1 is employed during ime period +1 E +1 = he hedging effeciveness when he ex-ane hedge raio b * from period is employed during ime period +1 Equaion (12) deermines he upward bias inheren in E * +1 when he ex-pos minimum variance hedge raio b * +1 is employed o deermine he hedging effeciveness and he hedge raio is no sable over ime. Equaion (12) shows ha his bias is relaed o he size of he change in he hedge raio squared, b 2, and he volailiy scale facor F 2 / S 2. Including he Average Change in he Basis in he Model Anoher model of he effec of unsable hedge raios on he ex-pos hedging effeciveness can be deermined by including he effec of he average change in he basis during ime period +1. Since he ypical variance model employed in (12) above deermines he variabiliy around he mean of he disribuion, any rend or convergence in he daa ha shows up as an average change in he basis will no be considered as variabiliy by he model derived above. However, if we assume ha he hedger wishes
11 11 o minimize variabiliy abou a zero change in he basis, hen he following model is appropriae o deermine he exen of he bias in he hedging effeciveness measure. Equaions (1) hrough (3), (5), and (6) define basis and he change in he basis in erms of b * +1, b*, and he change in hese hedge raios from o +1, b. If we use he regression mehodology o define he change in he cash price beween inervals k and k+1 during period +1 we have: (13) S +1 (k,k+1) = a * +1 + b* +1 F +1 (k,k+1) + e* +1 (k,k+1) Where: a * +1 = he y-inercep for he minimum variance hedge raio regression equaion during period +1 e * +1 (k,k+1) = he error erm for he minimum variance hedge raio regression equaion during period +1, for he price change occurring during he ime inerval k o k+1 Then subsiuing ino equaion (3) we obain: H * +1 (k,k+1) = [a* +1 + b* +1 F +1 (k,k+1) + e* +1 (k,k+1)] - b* +1 F +1 (k,k+1) (14) = a * +1 + e* +1 (k,k+1) Squaring each change in he basis and summing over all of he ime inervals k in period +1, one obains he oal variabiliy in he basis during period +1: (15) ( H * +1 )2 = (a * +1 + e* +1 )2 k k
12 12 Alernaively, if one employs he previous period's minimum variance hedge raio b * during ime period +1 hen he change in he basis for a given ime inerval is: H +1 (k,k+1) = S +1 - b* F +1 F +1 (k,k+1) (16) = [a * +1 + b* +1 F +1 (k,k+1) + e* +1 (k,k+1)] - b* Subsiuing from (5), b * +1 = b* + b, squaring each basis change, and summing over k we obain: (17) ( H +1 )2 = (a * +1 + e* +1 + b F +1 )2 k k The following formulas employ he squared variabiliies being summed over he ime inervals k during ime period +1 o define he hedging effeciveness measures: (18) E * +1 = R 2 = 1 - ( H +1 *+1 )2 / ( S ) 2 +1 and E +1 = 1 - ( H +1 )2 / ( S ) 2 +1 k k k k Noe ha he summaion of he variabiliy of H is he oal basis variabiliy of he hedged posiion. This oal basis variabiliy depends on wheher b * +1 or b* is employed as he hedge raio during period +1 o deermine E * +1 and E +1, respecively. The upward bias in he minimum variance hedging effeciveness measure E * +1 when here exiss an insabiliy in he hedge raio from periods o +1 is: E * +1 ( H +1 )2 / ( S +1 ) 2 ] - E +1 = 1 - ( H * +1 )2 / ( S +1 ) 2 - [1 -
13 13 (19) = [ ( H +1 )2 - ( H * +1 )2 ]/ ( S +1 ) 2 Subsiuing equaions (15) and (17) ino (19), combining erms, rearranging, and noing ha e = 0: (20) E * +1 - E +1 = b 2 2 * F+1 + 2a +1 b F +1 Now, since: and hus (21) F 2 = F 2 /N - F 2 F 2 = N F 2 + N F 2 Where: F 2 = he variance of F over ime period +1 F = he mean of F over ime period +1 and similarly for S 2, upon summing and subsiuing (21) ino (20) we obain: (22) E * E +1 = [ b * + S ] F + b F + 2a +1 b F]/[ S Where: a * +1 = he average per period change in he basis during period +1 Inerpreing he Models The models in he previous secions show ha using he variance minimizing hedge raio echnique when hedge raios are unsable over ime resuls in an upward biased value for he hedging effeciveness measure.
14 14 Concepually, if b * +1 is he minimum variance hedge raio during ime +1 using regression, hen any oher hedge raio b ha differs from b * +1 will have a larger sum of squared errors han b * +1 and hus possess a lower R 2 or E value. Model (1) is based on he concep ha one wishes o minimize he variance of he price changes around he average change in he basis. Hence, he assumpion is made ha a sysemaic change in he basis due o convergence or oher exernal economic facors can no be hedged away. This resuls in he conclusion ha he bias in he hedging effeciveness wih an unsable hedge raio is deermined by (12): (23) E * +1 - E +1 = b 2 [ F 2 / S 2 ] Model (2) is based on he desire o minimize he variance of all price changes, i.e. o hedge agains any change in he basis, including any sysemaic change in he basis. Equaion (22) shows he bias in hedging effeciveness for model(2): (24) E * +1 - E +1 = [ b 2 F 2 + b 2 F 2 + 2a * b F]/[ S 2 + S 2 ] The implicaions of hese models for he hedger of using minimum variance hedging effeciveness measures from period +1 as an esimae of he acual effeciveness value for +1 are obvious: if here is a large change in he hedge raio or a large average change in he basis hen he minimum variance effeciveness measure may conain a significan upward bias. Thus, unsable hedge raios increase he basis risk of he hedge compared o he ypical R 2 hedging effeciveness resuls. Since he minimum variance E * +1 = R +12 values have been employed in
15 15 mos of he previous research o deermine hedging effeciveness, and since unsable hedge raios affec he more realisic E +1 values, he empirical implicaions of he above resul need o be examined. Specifically, o wha exen do unsable hedge raios affec he hedging effeciveness of he model? The nex secion explores his quesion. DATA AND RESULTS Daa Cash and fuures yen currency values are employed from o deermine he hedge raios and hedging effeciveness values for weekly, biweekly, and monhly inervals. Weekly daa for 26 weeks are used for each ime period, while he biweekly resuls are based on annual periods, and he monhly inervals use wo years o form one ime period. Each observaion is aken as of he Wednesday of he week; Wednesday was chosen o avoid anomalies which may occur when raders close posiions on Friday and o provide a more exensive daabase for cross-currency raes. The cash currency values are based on lae afernoon prices from The Bank of American in London; he daa was obained from Daa Resources Inc. Fuures values used in he analysis are he close and he open raes from he Chicago Mercanile Exchange. The close daa ofen are used by oher researches examining hedging effeciveness and ypically provide significan liquidiy, especially for he nearby conrac. Open daa more closely correspond in ime o he lae afernoon London cash prices, given he six hour difference beween Chicago and London. Cash currency values are employed for he yen, European/indusrialized counries, and lesser developed counries. The cross-currency daa allows an examinaion of cross hedging for currencies ha has previously no been
16 16 explored. Cash and fuures currency values are convered o percenage changes o execue he regression hedging model. 1 Subperiod resuls allow for he examinaion of poenial insabiliy of he hedge raios and he effec on he hedging effeciveness via he models developed earlier in his paper. Resuls Tables I o IV presen he resuls of using he porfolio/regression mehodology o obain hedge raios and hedging effeciveness measures. Tables I o III show he minimum variance hedge raio for each period, b *, he absolue value of he change in he minimum variance hedge raio +1 from he previous period, b, he hedging effeciveness value for period +1, E * +1 = R* +12, and he bias in he hedging effeciveness ha exiss when he hedge raio is unsable over ime. The resuls are based on using weekly inervals over 26 week periods and herefore are designaed in erms of he firs and second half of he year. Table I shows he resuls for he yen fuures versus he yen cash. The hedging effeciveness measures for he yen fuures/cash relaionships range from 66% for he period o 93%; hese are respecable effeciveness measures and are much higher han indicaed by Hill and Schneeweis (1982) for he period. The changes in he hedge raios are generally small for he yen fuures/cash relaionships, causing only small biases in he hedging effeciveness measures, wih mos of he individual biases being 3% or less TABLE I ABOUT HERE
17 17 -- Table II presens he relevan resuls for he yen fuures/ausralian dollar cash relaionships. The hedging effeciveness measure for he period is essenially 0%, and five periods have effeciveness measures below 10%. The period provides poor hedging resuls for all of he yen fuures/cash relaionships summarized in Table IV; during his period he yen experienced several weeks of exremely large changes. The oher periods for he yen/ausralian dollar comparisons provided effeciveness measures up o 59%. While 8 of he 13 periods produced insignifican hedging effeciveness biases, he oher five periods possessed large changes in he hedge raios, causing biases as large as 33%. Table III shows somewha beer resuls for he yen fuures/french franc comparisons: effeciveness measures here are above 10% for all bu he period. However, he effeciveness biases are above 10% for five of he periods. In paricular, noe ha he period had a bias of 68%; his indicaes ha using he previous period's hedge raio would creae a variabiliy which is 68% larger han if no hedge was underaken (he E * measure was 0%). Such disressing resuls ypically occurred for several periods for each of he fuures/cash relaionships summarized in Table IV TABLES II AND III ABOUT HERE Table IV provides summary resuls for he periods for he yen fuures/cash associaions and for six European/indusrialized counries.
18 18 This able saes he averages of he per period resuls for he same saisics given in ables I o III. The able shows ha he average hedge raio varies among counries jus as he previous able showed ha i varied among periods for a given currency, wih he cross currency hedge raio being significanly lower han he yen fuures/cash hedge raio. The average absolue change in he hedge raio is small for he yen only relaionship bu large for he cross currencies, especially when he average change is compared o is average hedge raio. The hedging effeciveness measures are much lower han he yen fuures/cash value, bu hese effeciveness values are sill respecable for cross hedging associaions. However, he hedging biases for hese cross currency resuls average 9% o 24% per period, while he bias for he yen fuures/cash hedging resuls averages only abou 1%. Noe ha Model (2) does resul in slighly higher biases, alhough he difference only amouns o several percen per period TABLE IV ABOUT HERE IMPLICATIONS AND CONCLUSIONS This paper examines wo issues. Firs, i derives wo models which deermine he exen of he bias in he R 2 values when hedge raios are unsable over ime and he previous period's minimum variance hedge raio is employed as he esimae of he curren period's hedge raio. Empirical
19 19 resuls showing he size of his bias are provided for hedges involving yen fuures and cash currencies from six European/indusrialized counries. Second, he paper shows he minimum variance hedge raios and hedging effeciveness values for hese cross currency hedges. The same period hedging effeciveness values for he cross currency hedges are reasonable; however, if one assumes imperfec knowledge abou he fuures minimum variance hedge raio and uses he previous period's hedge raio as he bes forecas hen he resuling bias in he cross currency hedge effeciveness measures range from 9% o 24% per period. Moreover, mos currency resuls possess several periods where he variabiliy in price changes are acually increased by using he previous period's hedge raio as compared o using a no hedge sraegy. The imporance and implicaions o he hedger of unsable hedge raios and he resulan effec on hedging effeciveness is obvious, namely: he use of pas daa o forecas fuure hedge raios and hedging effeciveness mus be underaken wih greaer care for cross hedging. On he oher hand, when he hedger uses he same cash and fuures insrumen o creae a hedge hen he biases resuling from unsable hedge raios end o be negligible overall. Previous research using he minimum variance hedge raio approach implicily assumed ha he hedger possessed ex-pos daa o deermine he hedging effeciveness, wheher a hedge should be employed, and he resulan consequences of he proposed hedge posiion. This assumpion needs o be reevaluaed. These resuls also sugges ha addiional fuures conracs would be desirable for hose who inend o hedge cash insrumens ha are "significanly differen" from currenly raded fuures conracs. Addiional research is needed o idenify hose cash insrumens ha
20 20 possess a large degree of bias and which would have sufficien liquidiy o jusify a fuures conrac.
21 21 FOOTNOTES 1 Technically, price changes raher han percenage changes are ypically employed in he regression model. Percenage changes are used here in order o provide a sraighforward comparison of he size and variabiliy of he hedge raios across currencies. Using percenage changes does no affec he hedging effeciveness measures and one may easily conver he hedge raios o correspond o price changes by muliplying by a scale facor. Rollovers for he fuures conracs are conduced during he monh of expiraion of he fuures; he appropriae percenage change is employed in he analysis, i.e. all percenage changes used o compue he hedge raios are compleed beween like-mauriy conracs.
22 22 REFERENCES Bell, David E. and William S. Krasker. "Esimaing Hedge Raios," Financial Managemen, 1986, Vol. 15 No. 2, pp Ederingon, Louis H. "The Hedging Performance of he New Fuures Markes," The Journal of Finance, 1979, Vol. 34 No. 1, pp Gjerde, Oysein. "Measuring Hedging Effeciveness in a Tradiional One-Periodic Porfolio Framework," The Journal of Fuures Markes, 1987, Vol. 7 No. 6, pp Hill, Joanne and Thomas Schneeweis, "Forecasing and Hedging Effeciveness of Pound and Mark Forward and Fuures Markes," Managemen Inernaional Review, 1982, pp Hill, Joanne and Thomas Schneeweis, "The Hedging Effeciveness of Foreign Currency Fuures", The Journal of Financial Research, Spring 1982, Vol. 5 No. 1, pp Hill, Joanne and Thomas Schneeweis, "A Noe on he Hedging Effeciveness of Foreign Currency Fuures," The Journal of Fuures Markes, Winer 1981, Vol. 1 No. 1, pp Lasser, Dennis, "A Measure of Ex-Ane Hedging Effeciveness for he Treasury Bill and Treasury Bond Fuures Markes," Review of Fuures Markes, 1987, Vol. 6 No. 2, pp
23 23 TABLE I HEDGING EFFECTIVENESS BIASES: YEN FUTURES VERSUS CASH Hedging Effeciveness Bias Period b * b E * Model (1) * Model (2) ** _ _ * * Model (1): E +1 - E +1 = b [ F / S ] ** * Model (2): E +1 - E +1 = [ b * 2 F + b F + 2a b F]/[ + S S 2 ]
24 24 TABLE II HEDGING EFFECTIVENESS BIAS: YEN FUTURES VERSUS AUSTRALIAN DOLLAR CASH Hedging Effeciveness Bias Period b * b E * Model (1) * Model (2) ** _ _ * * Model (1): E +1 - E +1 = b [ F / S ] ** * Model (2): E +1 - E +1 = [ b * 2 F + b F + 2a b F]/[ + S S 2 ]
25 25 TABLE III HEDGING EFFECTIVENESS BIAS: YEN FUTURES VERSUS FRENCH FRANC CASH Hedging Effeciveness Bias Period b * b E * Model (1) * Model (2) ** _ _ * * Model (1): E +1 - E +1 = b [ F / S ] ** * Model (2): E +1 - E +1 = [ b * 2 F + b F + 2a b F]/[ + S S 2 ]
26 26 TABLE IV HEDGING EFFECTIVENESS BIAS: SUMMARY OF YEN AND CROSS HEDGING RESULTS Average Per Period Resuls Hedging Effeciveness Bias Yen fuures vs. Counry b * b E * Model (1) * Model (2) ** _ Japan Ausralia Belgium Ialy Neherlands Spain France _ * * Model (1): E +1 - E +1 = b [ F / S ] ** * Model (2): E +1 - E +1 = [ b * 2 F + b F + 2a b F]/[ + S S 2 ]