Hedging with Forwards and Futures

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1 Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures conrac for 10,000 barrels of oil wih a mauriy of six monhs, you can eliminae he price risk. Alernaively, you are a U dollar based firm and you have a conrac from which you will receive 500,000 yen in four monhs. You plan o sell yen and buy dollars. The exchange rae risk can be eliminae by aking he sell-side of fuures conrac wih a mauriy of four monh o exchange 500,000 yen for U$ dollars a he fuures/forward rae. In each of hese examples, he price or exchange rae risk is eliminaed wih he use of a fuures conrac. The six-monh oil fuures conrac will lock in he price of he oil and he four monh yen/u$ fuures conrac will lock in he exchange rae. A mauriy he mos likely scenario will be ha in neiher case will anyone acually ake delivery of he underlying asse. or example, in he case of he oil, a mauriy he oil hedger will buy he oil on he spo marke a - T, and close ou he fuures posiion realizing a payoff of +( T T ). The resul of he hedge is a cos of - T + ( T T ) = - T. If T > T, he posiive inflow from he fuures posiion will offse par of he cos. If T < T, hen he hedger will have o pay he difference and again he ne cos of purchasing he oil will be T. In he above examples, he hedging was one for one and he mauriy of he fuures conrac exacly mached he iming of he ransacion. Ofen imes he hedging approach is no as clear as i is in hese examples. or example, he iming of he mauriy of he available fuures conracs may no be he same as he iming of he obligaion. uppose for he 10,000 barrels of oil he only fuures conrac available was for a mauriy of eigh monhs, T. If we use his o hedge our six monhs obligaion,, in six monhs we buy he oil a he spo, -, and offse he original fuures posiion by aking he sell side of he same conrac which will yield +( T ). The fuures price a ime,, will be, = e ( c y ) ( T ), This noe was prepared by Professor Rober M. Conroy. Copyrigh 003 by he Universiy of Virginia Darden chool oundaion, Charloesville, VA. All righs reserved. To order copies, send an o No par of his publicaion may be reproduced, sored in a rerieval sysem, used in a spreadshee, or ransmied in any form or by any means elecronic, mechanical, phoocopying, recording, or oherwise wihou he permission of he Darden chool oundaion.

2 where c is cos of carry and y is convenience yield 1 a ime. As such, he ne resul of buying he oil a spo and hedge a ime is + ( ) = T T + ( e ( c y ) ( T ) 1) T. The resul is ha he hedge is no perfec. I will depend on wha is he cos of carry and convenience yield a ime and he resul would be cerainly differen from he fixed cos of T ha we had when he mauriy of he fuures conrac exacly mached he obligaion. Hence, his mismach in mauriies creaes no quie he prefec hedge. The resuling difference from having an exac mach of mauriies is referred o a basis risk. A poenially more significan basis risk comes from a siuaion where an invesor mus use fuures conracs on a differen asse o hedge anoher asse. or example, airlines ofen wish o hedge heir je fuel coss. They sell ickes well in advance bu he acual cos of delivering he fligh will depend largely on he cos of je fuel on he dae of he ravel. Airlines can eliminae his risk by using fuures. However, hey face a problem in hedging je fuel. There are no fuures conracs raded on je fuel. The neares subsiue is heaing fuel oil. Thus, an airline could aemp o hedge heir fuel cos exposure using Heaing Oil fuures conracs. However, hey do face some risk ha he changes in he Heaing Oil fuures conracs will no exacly mach he changes in he price of Je uel. The difference beween he price of Je uel and he price of heaing Oil fuures a he dae ha he je fuel is purchased is also referred o as basis risk. As an example, Exhibi 1 shows he spo prices for je fuel and for heaing (fuel) oil from 1985 o 001. The price movemens are similar bu no quie he same. Heaing oil prices are lower and appear o be less volaile. Exhibi 1. Je uel vs. Heaing (uel) Oil Price per on Apr-85 Apr-86 Apr-87 Apr-88 Apr-89 Apr-90 Apr-91 Apr-9 Apr-93 Apr-94 Apr-95 Apr-96 Apr-97 Apr-98 Apr-99 Apr-00 Apr-01 dae Je uel ($/on -,40 lbs.) uel Oil ($/meric onne-,05lbs.) 1 Please see orward and uures noe page 7. Also Hull (5 h ediion), Chaper 3 page 60. Page

3 Hence using heaing oil fuures conracs on a one-o-one basis may no provide a good hedge for je fuel. or an ideal hedge, over our ime horizon we would like he change in he fuures price o exacly mach he change in he value of he asse we wish o hedge, i.e., po = uures Exhibi shows he spo prices for je fuel and for hea oil 90-day fuures and 60-day fuures. Le s assume ha an airline wishes o hedge je fuel 30 days forward in ime and he only conracs available are 90-day fuures conracs for heaing oil. The change in he spo price for je fuel over a monh is jus he price a he end of he monh less he price a he beginning of he monh. The change in he value of a fuures conrac is slighly differen. A 90-day conrac a he beginning of he monh is a 60-day conrac a he end of he monh. Hence if we use a 90-day conrac o hedge for 30 days he change in he price is he difference beween he fuures price for a 60 day conrac a he end of he monh less he fuures price for 90 day conrac a he beginning of he monh. rom exhibi, i is clear ha he price changes of he spo je fuel prices and heaing oil fuures are no he same. This raises he quesion of wheher we can use a hedge raio, h, differen from 1.0 o hedge he je fuel prices or po = h uures. Bu how do we choose he bes h? The usual soluion is o choose h such ha i minimizes he following: Min h E [( h ) ]. This resuls in a value of h ha minimizes he squared differences beween he price changes. Anoher way of saing he same hing is o choose h such ha i minimizes he variance of he hedge. In choosing h, i places a big penaly on big differences beween The minimizaion can be rewrien as E [( h ) ] = E[ ] + h E[ ] h E[ ] Assuming E[ ]=0, and E[ ]=0, hen Cov = E = E [ ], [ ], (, ) = E[ ] = ρ, ubsiuing back in he original problem resuls yields [( h ) ] = + h h ρ E, Page 3

4 and. Noe ha we could have chosen a very differen objecive funcion. However, his paricular objecive funcion happens o be very convenien in a number of ways. The acual soluion 3, ĥ, o his formulaion is fairly sraighforward. ˆ =, s h ρ, where is he sandard deviaion of he spo price changes, is he sandard deviaion of he fuures price changes and ρ, is he correlaion beween he spo price changes and he fuures price changes. Exhibi 3 shows he calculaion of he opimal hedge using he hisorical daa in Exhibi. The basic saisics 4 are esimaed as follows: Means: = and = = 1 1 = = 1 andard Deviaions: ( ) = and ( ) = 1 Covariance: Cov(, ) = ( ) ( ) 1 = 1 1 = 1 Correlaion: ρ, Cov = (, ) 3 The soluion o he minimizaion problem is o ake he firs derivaive of he hedge variance wih respec o h, se i equal o zero, and solve for h. E( h ) h = 0 ( + h h ρ h hˆ = ρ, ρ, = 0, ) h = 0 4 The saisics shown below are based on he populaion. If everyhing was recalculaed on a sample basis he esimaed hedge raio would be he same. Be careful using saisical funcions in excel. You need o make sure ha he esimaes of sandard deviaions and correlaions have he same basis, populaion or sample. Page 4

5 Hedge: ˆ s h = ρ, I is also possible o esimae he opimal hedge using regression analysis. The basic equaion is = α + h ince he basic OL regression for his equaion esimaes he value of ĥ as ˆ =, s h ρ, we can use OL regression. This is he soluion o he minimizing he original objecive funcion. Hence, his is one of he reasons ha he objecive funcion of minimizing he squared differences is so appealing. Exhibi 4 shows he oupu of an Excel regression using he daa in Exhibi 3. Noe ha he resuls are he same. The opimal hedge raio 5 is This is very close o a value of 1.00, which is wha we would expec for wo very similar commodiies where he prices would end o move ogeher. I is useful o noe ha he regression analysis also provides us wih some informaion as o how good a hedge we are creaing. The r-square 6 of he regression ells how much of he variance in he change in spo price is explained by he variance in he change of he fuures price. In his case he r-squared saisic is.443 or 44.3%. A good hedge migh resul in an r-square value of.80. Hence, in his case, while he opimal hedge raio is close o 1.00, he hedge iself migh no be ha effecive. There is he poenial here for a lo of basis risk. Noneheless, he appropriae hedge is heaing oil fuures conracs for each on 7 of je fuel. I have one commen on he analysis presened in his secion. Here we used he price changes in he fuures conrac for Heaing Oil. Acually, for mos pracical purposes we could have used simply he changes in he spo prices of Heaing Oil o calculae he opimal hedge. I is ofen very difficul o ge a good consisen hisoric series of fuures prices. Equiy Porfolio Hedging Hedging porfolios is he same as hedging commodiies. Consider a porfolio wih a value oday of $5,345,456. We wish o hedge his porfolio using &P 500 fuures 5 I used he excel regression funcion wih as he y variable and as he x variable. 6 The r-square of he regression is esimaed as he square of he correlaion coefficien beween and. rom exhibi 3, he correlaion coefficien is.666. quaring his yields Noe ha since we calculaed he opimal hedge raio based on price changes, he difference in he onnage beween he long on (,40 lbs.) for je fuel and he meric onne for heaing oil was accouned for in he analysis. Page 5

6 conracs. While our porfolio is similar o he &P 500, i is no he same. If we follow wha we did above, he opimal hedge is h ˆ P = ρ p, & P. & P or equiy porfolios he opimal hedge is in erms of reurns. or example, assume we have a porfolio wih a curren value of $10, 968,000. You wish o use &P 500 fuures conracs o hedge he risk over he nex monh. Exhibi 5 shows he monhly values for he porfolio and he index for he las four years. In his case, insead of using price changes we will calculae he opimal hedge raio using monhly reurns 8. rom exhibi 5, he opimal hedge raio is Exhibi 6 shows he esimae of he hedge raio using regression analysis. Noe ha he regression model is R = α + β. P R & P This regression model is also a way o esimae Bea for a porfolio using he &P 500 porfolio as a proxy for he marke porfolio. Hence, in his conex one inerpreaion of he opimal hedge raio is Bea. ince he $ value of each &P 500 fuures conrac is he index value imes $50, he acual number of &P 500 fuures conracs o be wrien is deermined by aking he hedge raio imes he raio of he porfolio $ value divided by he curren value of he index underlying he fuures conrac, he &P 500 in his case. or he example, Number of Conracs 9 = approximaely conracs. $10,968,000 h ˆ = =.16 conracs or ( $50) 8 We use monhly reurns because he scale differences in he value of he porfolio and he value of he index. This much easier o scale each of he series and use reurns. 9 ince we calculaed he hedge raio using percenage reurns, he hedge raio does no accoun for he size differenial beween he porfolio and he index. Hence we need o ake his ino accoun when we esimae he number of conracs required. Page 6

7 Exhibi Je uel and Heaing Oil uures Prices Je uel $/on -,40 lbs. uel Oil 90 day fuures $/meric onne-,05lbs. uel Oil 60 day fuures $/meric onne-,05lbs. Je uel $/on -,40 lbs. uel Oil uures $/meric onne-,05lbs. Price Price Price Price Change Price Change** Jun Jul Aug ep Oc Nov Dec Jan eb Mar Apr May Jun Jul Aug ep Oc Nov Dec Jan eb Mar Apr May Jun Jul Aug ep Oc Nov Dec Jan eb Mar Apr May Jun Jul Aug ep Oc Nov Dec Jan eb Mar Apr May Jun ** Price change for fuures compares he 60 day price a ime o he 90 day price in ime -1. Page 7

8 Exhibi 3 Opimal Hedge Raio Je uel and Heaing Oil uures Prices uel Oil Je uel uures $/on -,40 lbs. $/meric onne-,05lbs. Price Change Price Change** (PJ-MJ)(PH-MH) Jun-97 Jul Aug (1.31) ep (38.) Oc Nov Dec Jan eb Mar (50.71) Apr May Jun Jul (5.85) Aug ep Oc Nov Dec Jan eb Mar Apr May Jun Jul Aug ep Oc Nov (.75) Dec Jan (353.68) eb (18.57) Mar (0.50) Apr (8.18) May Jun Jul Aug ep , Oc Nov Dec (0.8) Jan ,91.39 eb (11.1) Mar Apr May Jun (13.6) Mean andard Deviaion Covariance Correlaion Hedge Page 8

9 Exhibi 4 Opimal Hedge Raio Using Regression Analysis UMMARY OUTPUT Regression aisics Muliple R R quare Adjused R quare andard Error Observaions ANOVA Df M ignificance Regression E-07 Residual Toal Coefficiens andard Error a P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Inercep X Variable E Page 9

10 Exhibi 5 Equiy Porfolio Hedging Porfolio &P 500 Porfolio &P 500 Reurn index Monhly reurn Monhly reurn (Rp-MP)*(Rs&p-Ms&p) Jan-99 $ 9,78, eb-99 9,34, Mar-99 8,789, Apr-99 8,957, May-99 10,000, Jun-99 10,044, Jul-99 10,530, Aug-99 10,39, ep-99 9,96, Oc-99 9,574, Nov-99 9,649, Dec-99 9,859, Jan-00 10,105, eb-00 9,960, Mar-00 9,90, Apr-00 10,615, May-00 10,793, Jun-00 10,700, Jul-00 10,88, Aug-00 10,835, ep-00 11,678, Oc-00 11,198, Nov-00 11,80, Dec-00 10,706, Jan-01 11,47, eb-01 1,363, Mar-01 11,709, Apr-01 10,87, May-01 1,, Jun-01 1,638, Jul-01 1,8, Aug-01 1,343, ep-01 11,79, Oc-01 9,949, Nov-01 10,95, Dec-01 11,739, Jan-0 1,471, eb-0 1,99, Mar-0 1,356, Apr-0 13,085, May-0 1,896, Jun-0 1,168, Jul-0 11,80, Aug-0 10,007, ep-0 10,7, Oc-0 9,407, Nov-0 10,080, Dec-0 10,968, Mean and. Deviaion Covariance Correlaion Hedge (Bea) Page 10

11 Exhibi 6 Regression Resuls for Equiy Porfolio Hedging Regression aisics Muliple R R quare Adjused R quare andard Error Observaions 47 ANOVA df M ignificance Regression E-15 Residual Toal andard Upper Lower Upper Coefficiens Error a P-value Lower 95% 95% 95.0% 95.0% Inercep Bea Page 11

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