INIU O CONOMIC UDI aculy of social sciences of Charles Universiy uures Conracs Lecurer s Noes No. Course: inancial Marke Insrumens I eacher: Oldřich Dědek
I. BAIC AUR O ORWARD AND UUR CONRAC. Classificaion of conracs a) spo conrac is an agreemen beween wo counerparies abou immediae sale and delivery a a given marke price (called also spo price, promp price) he ransacion dae and he selemen dae (spo dae, value dae) should in principle coincide, in realiy a delay (up o wo days) is acceped b) forward conrac is and agreemen beween wo counerparies ha fixes he erms of an exchange ha will ake place a some fuure dae he conrac specifies: i) he size of an underlying asse ha will be exchanged ii) he price a which he exchange akes place (forward price) iii) he dae in he fuure a which he exchange akes place (delivery, mauriy dae) advanages: he conrac can be ailor-made (meeing requiremens of he wo counerparies by negoiaing deails of he conrac) disadvanages: - low markeabiliy (obligaions on one counerpary canno be easily ransferred o a hird pary, erminaion of he conrac is cosly) - a higher defaul risk (incenives for he losing counerpary o renege on obligaions ha were agreed a he ime of negoiaions) spo price > forvard price he buyer who akes delivery gains and he seller who makes delivery loses (and vice versa) c) fuures conrac is a sandardized agreemen ha fixes he erms of an exchange ha will ake place a some fuure dae (deails of a conracs are no negoiable) sandardized specificaions: - uni of rading (called underlying asse) commodiy fuures (aluminum, barley, cale, coffee, copper coon, crude oil, gold, palm oil, pork bellies, poaoes, rice silver, wool, zinc, ec.) financial fuures are conracs relaed o financial insrumens (bond, foreign currency, sock index, ec.)
- size of he conrac: an excessively large size discourages small invesors from rading while an excessively small size increases ransacion coss - delivery dae: usually conracs wih only four delivery daes are available (specific daes in March, une, epember and December) fuures conrac is referred by is delivery monhs - quoaion of price - ick size (minimum price movemen) advanages: fuures conracs are exremely liquid because i is possible o unwind a conrac a any ime by performing a reversing rade. Pracical arrangemens of rading wih fuures fuures are raded on specialized exchanges: Chicago Board of rade (CBO), London Inernaional inancial uures and Opions xchange (LI), urex, okyo uures xchange (), ingapore Inernaional Moneary xchange (IMX) a) clearing house clearing house keeps rack of all ransacions ha ake place in he exchange and becomes he formal counerpary o every ransacion his arrangemen eliminaes he problem of defaul risk direc rade clien A A s broker B s broker clien B (buyer) (seller) inerposiion of clearing house clien A A s broker CH B s broker clien B (buyer) (seller) (buyer) (seller) in case of he defaul of a clien s broker he clearing house honors all obligaions conneced wih agreed deals he arrangemen does no proec a clien agains he defaul of his broker b) operaion of margins 3
iniial margin is a deposi a he clearing house s margin accoun ha each counerpary mus make when he fuures posiion is opened a buyer of fuures conrac is said o have a long fuures posiion a seller of fuures conrac is said o have a shor fuures posiion he balance on margin accoun is used for covering daily losses semming from marking o marke (a safeguard agains defaul) iniial margin paymens are a small fracion of he value of he conrac ha gives access o gains caused by price changes a very limied coss mainenance margin is a paricular hreshold bellow which he balance on he margin accoun is no allowed o fall variaion margin is an addiional deposi of a counerpary on his margin accoun ha prevens he balance on he margin accoun o fall bellow he mainenance margin margin call is he noice of he clearing house o a counerpary o op up he balance on he margin accoun o he level of he mainenance margin if addiional funds are no provided he clearing house closes he counerpary s posiion by a reversing rade (he coss of unprofiable ransacions are born by a delinquen rader) c) marking o marke marking o marke is an accouning procedure by which he profis or losses of every day s rading accruing o he counerparies as a resul of ha day s change in he fuures price have o be received or paid by marking o marke he poenial loss from price changes is dilued ino limied day-oday price flucuaions and defaul risk is hus minimized... closing fuures price a he end of a day closing fuures price a he end of he nex day + + fuures price closes a a higher level: > + i) he long counerpary makes he money by holding he fuures conrac during he day + ; he could evenually close ou he posiion (sell he conrac) a a beer price) he long s margin accoun is credied wih he gain (he margin accoun s balance 4
increases by he amoun + ) ii) he shor counerpary loses he money by holding he fuures conrac during he day + ; he could evenually close ou he posiion (buy he conrac) a a worse price) he shor s margin accoun is debied wih he gain (he margin accoun s balance declines by he amoun + ) fuures price closes a a lower level: < + - an accouning loss for he long counerpary and an accouning gain for he shor counerpary - he long s margin accoun is debied and he shor s margin accoun is credied by he amoun + d) erminaion of fuures conrac i) closing ou he conrac by aking ou an offseing posiion before delivery day - seller of he conrac buys an equivalen conrac (a a given marke fuures price) - buyer of he conrac sells an equivalen conrac (a a given marke fuures price) he difference beween he price when he conrac was opened and he price when he conrac was closed is he rader s profi or loss a large majoriy of financial fuures are closed ou before mauriy ii) holding he open posiion o he delivery day - physical delivery: he long counerpary akes he underlying asse from he shor counerpary and pays for i a price - cash selemen: financial compensaion ha involves only ransfer of cash beween he long and he shor delivery process involves a sequence of seps ha mus be compleed in a specific order and a specific imes i is iniiaed by he buyer who is noifying he clearing house abou inenion o ake delivery he clearing house maches long and shor posiions and assigns a shor o make delivery o a long he shor delivers an eligible asse o he long and he long makes paymen o he shor a he invoice price 5
cheapes o deliver decision occurs when here is more han one ype of eligible asse ha can be delivered agains he conrac e) convergence of spo and fuures prices on he delivery dae he possibiliy of physical delivery of he underlying asse guaranees convergence of fuures and spo prices on he delivery dae basis spo price fuures price [beware of he alernaive convenion B ha is also used!] - basis is significanly negaive jus prior o he delivery dae (, 8) selling fuures conrac (+), buying he good in he spo marke and making delivery o he buyer of he fuures conrac ( 8) - basis is significanly posiive jus prior o he delivery dae ( 8, ) buying fuures conrac and aking delivery ( 8), selling he good in he spo marke (+) selling he good shor (+), buying fuures conrac ( 8) and giving he good back no arbirage condiion requires he base o be zero on he delivery dae oherwise subsanial arbirage profis can be made xchange Delivery elemen Price (DP) is a closing price on he las rading day DP is deermined by he clearing house according o an exac procedure DP formula is a pracical counerpar of he heoreical condiion of zero basis achieved on delivery dae Conango Backwardaion fuures price spo price spo price fuures price delivery delivery 6
conango is a siuaion of a negaive basis when he conrac is raded backwardaion is a siuaion of a posiive basis when he conrac is raded plus or negaive sign of he basis of a fuures conrac depends on he so-called cos of carry spread is he difference beween he wo fuures prices spread is usually defined as he fuures price of a shorer mauriy conrac minus he fuures price of a longer mauriy conrac (he opposie convenion is also used) shor selling is an arrangemen ha iniially involves he sale of a securiy ha is no currenly owned by he seller bu is borrowed from he seller s broker; a he end of he agreemen he securiy is bough in he open marke and handed over o he broker f) hedging propery of fuures conracs by holding a fuures conrac o mauriy he rader can lock in he price of he conrac s underlying asse a he level a which he conrac was opened gain/loss in he fuures conrac + paying a marke price in he spo marke paying an opening fuures price fuures price a which he long posiion was opened closing price a he end of a rading day selemen price of he fuures conrac spo price of he underlying asse when he fuures conrac maures a oal cash ouflow (gain/loss as a resul of marking o marke plus buying he asse a a spo price): marking o marke: ) + ( ) +. K + ( ) ( ) ( convergence of fuures and spo prices: oal cash ouflow ( ) hedging wih fuures conracs is possible only on delivery daes where he basis is zero, oherwise he rader is exposed o he basis risk oal cash ouflow + ( ) fuures price basis 7
g) pay-off from fuures conrac marke price of a fuures conrac a which he conrac was opened and held o mauriy marke price a which he asse can be bough/sold in he spo marke a mauriy of he fuures conrac gain/loss for he long gain/loss for he shor Pay-off from he long fuures Pay-off from he shor fuures h) quaniaive limis on rading daily price limis sipulae ha fuures prices mus flucuae wihin a band cenered on he selemen price a he close of he previous rading day fuures price rises above he upper limi (drops bellow he lower limi) he marke closes limi-up (limi-down), i.e. he rading is suspended for he remainder of he rading day or some shorer period he idea is o ensure an orderly marke by giving marke paricipans a cooling-off period wih a chance o reassess heir posiions posiion limis are a maximum number of conracs an individual rader or a group of raders acing in accord can open for speculaive purposes posiion limis should proec rading agains speculaive abuse of financial leverage of fuures conrac corning he marke occurs when a large number of long fuures posiions in combinaion wih conrolling supply of he underlying asse succeeds in shooing he price of he 8
underlying asse up (i becomes difficul for shors o close heir posiions and o deliver he asses for a reasonable price) Day Opening 3 4 5 Closing fuures price 5 95 9 Buyer (long) Marking o marke + + -5 - -5 Variaion margin 5 5 Margin accoun balance 3 4 5 oal gain/loss 5-5 - eller (shor) Marking o marke - - +5 + +5 Variaion margin Margin accoun balance 35 45 5 Margin call - - - oal gain/loss - - -5 5 Iniial margin Mainenance margin Opening price uppose ha he fuures conrac maures on he las 5 h day. he closing price is equal o ha day s marke spo price. A long counerpary can buy an underlying asse a a price: 9 (spo price) + (fuures loss) (opening fuures price) A shor counerpary can sell an underlying asse a a price: 9 (spo price) + (fuures gain) (opening fuures price) 9
II. XAMPL O INANCIAL UUR CONRAC four broad classes of financial fuures: i) shor-erm ineres rae fuures (UA is a shor-erm deposi) ii) long-erm ineres rae fuures (UA is a governmen bond) iii) currency fuures (UA is a foreign currency) iv) sock index fuures (UA is a sock index) exchanges where fuures conracs are raded are consanly inroducing new conracs, deleing conracs or changing specificaions of exising conracs. ock index fuures Name of he conrac LI index fuures ) Uni of rading Value of per one index poin ) Delivery monhs Las rading day Delivery day Quoaion ick size.5 3) ick value 5 March, une, epember, December :3, hird riday in delivery monh irs business day afer he las rading day Index poins Iniial margin 4) pread margin DP Average of he index every 5 seconds inclusively beween : and :3 (London ime) on he las rading day. Of he 8 measured values, he highes and lowes will be discarded and he remaining 57 will be used o calculae he DP. Calculaion will be rounded o he neares half index poin. ) index is an index of he share prices of Briain s larges companies by marke capializaion; compiled by he inancial imes; covers abou 74 % by value of all UK shares. ).g. value of he index of 58.5 represens he conrac value of 585. 3) wo icks are equivalen o one full index poin ( is rounded o he neares one place of decimal). 4) Iniial margin is abou.7 % of he value of he conrac.
erminaion of he conrac oday ive days laer Las rading day Buying conracs 585. elling conracs 5 58.5 DP 58.5 i) closing ou posiion ick change (58.5 585) - 5 oal loss number of conracs number of icks loss value per ick 5 5 5 he oal loss is he accumulaed sum of five daily marking o marke amouns ii) delivery he conrac has o be seled for cash because i is no possible o deliver a physical index he invoice amoun is deermined by he difference beween he DP and he selemen price on he day before he las rading day he sum is handed over by he long o he shor in final selemen invoice (58.5 58.5) 5-4. Currency fuures Name of he conrac CM uro currency fuures Uni of rading 5 raded agains he U dollar ) Delivery monhs Delivery day anuary, March, April, une, uly, epember, Ocober, December plus spo monhs wo business day afer he las rading day Quoaion U$ per (four decimal places) ) ick size. cens per,. $ per ick value $.5 3) Iniial margin??? DP he CM official closing price on he las rading day. ) he buyer of he conrac is expecing o receive 5 and o make paymen for his in
dollars. he seller is expecing o receive a sum of dollars and o pay 5 for his. ) he price is quoed in exacly he same way as foreign exchange is quoed (i.e..455 $ per ). 3) ick value ick size size of he conrac. 5 $.5. oday ive days laer Las rading day Buying conrac.3 elling conrac 5.383 DP.4 i) closing ou posiion ick change 383 3 7 oal gain number of conracs number of icks gain value per ick 7.5 $ 9 ii) physical delivery he invoice amoun is he price of he underlying asse a he DP exchange rae (spo exchange rae on he delivery day) invoice.4 5 $ 55.5 his amoun is paid by he long o he shor in exchange for receiving 5 locked in exchange rae UD/UR oal gain from marking o marke (4 3).5 9.5 $ 5 55.5 5 UD/UR. 3 (opening fuures price) 5.3 hor-erm ineres rae fuures Name of he conrac LI hree-monhs serling ineres rae fuures conrac (3) Uni of rading 5 ) Delivery monhs March, une, epember, December ) Las rading day Delivery day :, hird Wednesday of he delivery monh irs business day afer he las rading day Quoaion. minus rae of ineres (%) 3) ick size. % on annual basis
ick value.5 4) Iniial margin 5 pread margin 5 DP BBA LIBOR (Briish Bankers Associaion London Inerbank Offered Rae for hreemonh serling deposi a : on he las rading day (rounded o hree decimal places) ) In case of physical delivery he buyer of he conrac is expeced o allocae a 5 hree monh ime deposi a an eligible bank on he delivery dae, he faciliy being arranged by he conrac seller (a presen only he cash selemen is available) ) Along wih four basic delivery monhs oher serial monhs are added such ha delivery monhs are available for rading (deails are published in a rading calendar). 3) or example he fuures price 9.5 means ha an ineres rae of 9.5 % can be locked in for hree monhs on he delivery day. 4) ick value 5.¼.5 oday weny days laer Las rading day Buying conracs 9.5 elling conracs 9.55 DP 9.5 i) closing ou posiion ick change 955 95 5 oal gain number of conracs number of icks gain value per ick 5.5 $ 65. he long s margin accoun balance 5 (iniial margin) + 65 (marking o marke) effecive rae of reurn 5 + 65 / 365 365 / ( + r ) r.5 758 % 5 ii) cash selemen he invoice amoun is deermined by he difference beween he DP and he selemen price on he las rading day (7 icks) invoice (95 955).5 875 3
iii) physical delivery on he delivery dae he shor (seleced by he clearing house) arranges a 3M ime deposi of 5 a an eligible bank locked in 3M deposi rae DP (a marke rae) + implied fuures rae 6.5 + 875 365 8.75 % + 5 9 8.75 % +.75 % 9.5 % (as required) he long mus be also compensaed by he shor in case ha he acual deposi rae differs from he DP rae (he lock in rae mus be he rae a which he fuures conrac was opened).4 Long-erm ineres rae fuures Name of he conrac Uni of rading Delivery monhs Las rading day Delivery day Quoaion ick size. ick value ) Iniial margin pread margin 5 DP LI long gil fuures conrac Imaginary gil wih nominal value and YM of 7 % March, une, epember, December :, wo business days prior o he las business day in he delivery monh Any business day in delivery monh (a seller s choice) ) Per nominal he LI marke price a : on he las rading day ) Any gil on he Lis of Deliverable Gils may be delivered as published by he xchange on or before he enh business day prior o he firs noice day of he delivery monh (examples of no eligible bonds: index -linked, variable rae, parly redeemed, callable wihin a given period, converible, and ohers) ) ick value. /. 4
i) closing ou he posiion Buying a conrac 9.38 elling a conrac 9.3 ick change 93 938-5 oal loss 5 5 ii) physical selemen a) price (conversion) facor price facors pu eligible bonds (differen bond grades ) on roughly equal fooing fuures price of eligible bond fuures price price facor of he bond price facor is calculaed as he clean price (i.e. wihou accrued ineres) per nominal a which he bond has a yield o mauriy of 7 % (equal o he conrac s underlying imaginary gil) price facor is calculaed for he firs day of he delivery monhs and adjusmens are made for accrued ineres where he firs day of he delivery monh does no coincide wih a coupon paymen day (see example) price facors remain consan during he delivery cycle here are differen price facors for each eligible bond and for each delivery monh price facors are published by he xchange b) cheapes o delivery bond (he CD bond) he shor counerpary may choose any eligible bond (from a given lis) for delivery he cheapes o delivery is he bond ha gives he larges reurn from he so-called cashand-carry ransacion; he reurn iself is called implied repo rae he CD bond is a bond wih he larges implied repo rae cash-and-carry ransacion involves buying he bond wih borrowed funds in he spo marke and selling i ino he fuures marke cash ouflow: purchasing price of he bond + ineres cos of borrowing cash inflow: selling fuures price of he bond (fuures price price facor) + accrued ineres + reinvesed coupon implied repo rae cash inflow - cash ouflow cash ouflow 365 duraion of repo cash-and-carry ransacion is equivalen o a sell and repurchase agreemen (repo): he 5
shor buys a bond wih a provision o sell i back a a predeermined price and o receive a rae of ineres (he repo rae) c) choosing delivery dae d) invoice he shor can deliver he CD bond on any business day of he delivery monh bu in pracice only he firs and he las business days are ever used curren yield on he bond > money marke ineres rae he bond will be delivered on he las business day (he shor earns more by holding he bond hen by delivering i and invesing he proceeds in he money marke) curren yield on he bond < money marke ineres rae he bond will be delivered on he firs business day (he shor earns more by delivering he bond and invesing he proceeds in he money marke han by holding he bond) curren yield coupon clean price when he CD bond is delivered he long pays he shor an invoice amoun invoice DP Price facor Nominal value of conrac + Accrued ineres by paying his amoun he long ges a yield o mauriy of 7 % on his holding of he CD bond as required by he fuures conrac A bond wih nominal value of pays a semi-annual coupon of 4 (i.e. 8 % on annual basis). oday is April 9 and he las coupon paymen day was 8 November 8. On ha day he bond had years remaining o mauriy. he curren marke price (i.e. on April 9) of he bond is 99.3 and he une long gil fuures is rading a 88.9. Money marke ineres rae is 8 %. 8..8 Coupon day 34 days.4.9 oday 99.3 88.9 r 8 % 9 days 8.5.9 Coupon day 43 days 4 3.6.9 Delivery 7 days 8..9 Coupon day 6
i) compuing he price facor of he bond bond price on 8 November 9 a YM of 7 % (here are fuure semi-annual coupon paymen periods remaining o mauriy) 8 (8..9) + 7.5836.7.7 ( + ) ( + ) bond price a he beginning of delivery monh for which price facors are compued (here are 7 days beween firs day of delivery monh and nex coupon paymen dae) (.6.9).7 ( + ) 7 /8.5 ( 8 + 7.5836) 8. 646 deducion of accrued ineres beween previous coupon dae and he beginning of delivery monh because price facors are compued on clean price basis (here are 4 days beween las coupon paymen and firs day of delivery monh) 4 8 clean price (.6.9) 8.646 7.7578 8.5 price facor is scaled down o nominal value price facor 7.7578.77578 ii) choosing he delivery dae compuing he curren yield of he bond on April 9 (i is he day where he shor makes decision abou purchasing he bond) on April 9 he bond price includes 34 days accrued ineres from previous coupon paymen dae 34 8 clean price (.4.9) 99.3 96.93 8.5 8 curren yield.83 8.3 % 96.93 curren yield (8.3 %) exceeds money marke ineres rae (8 %) herefore he shor chooses he las business day of delivery monh as delivery dae iii) compuing implied repo rae cash ouflow from cos-of-carry ransacion purchasing price of bond on April 99.3 9 ineres cos on 9 days loan (from April o 3 une) 99.3.8 365.96 7
oal cash ouflow.9 cash inflow from cos-of-carry ransacion anicipaed selling price of he bond on delivery dae (curren fuures price price facor) 88.9.77578 95.3 coupon received on 8 May and reinvesed for 43 days (o delivery day) a money 43 marke rae (assumed equal o he curren rae) ( +.8 ) 4. 4 4 365 43 accrued ineres on 43 days (from coupon day o delivery day). 94 oal cash inflow...9 365 implied repo rae.433 4.3 %.9 9 4 8. 5 8
III. MODL CO-O-CARRY 3. Basic formula cos-of-carry (cash-and-carry) is a model ha deermines he fair price of forward/fuures conracs no-arbirage condiion posulaes ha a risk-free sraegy ha uses no individual s own wealh mus give zero profi model variables:... curren spo price of underlying asse... curren fuures price of underlying asse ime o mauriy of fuures conrac r... annual carries cos (cos incurred by holding he asse such as insurance, sorage, deerioraion and ineres on loan) d annual reurn from holding he asse (coupons, dividends, convenience yield) cash flow from he cos-of-carry rading sraegy now: + (aking a loan on buying he asse) (purchasing he asse in he spo marke) carrying period: (selling he asse in he fuures marke) d (earning any income from holding he asse) r (incurring any coss from holding he asse including ineres on borrowed funds) mauriy: + (making delivery ino he fuures marke) (repaying he loan) he sraegy involves no individual s own wealh and no risk, herefore no-arbirage condiion implies any profi from he ransacion ( + d ) ( + r ) cos-of-carry formula basis + carry ( r ( d r) d) 9
carry can be posiive (backwardaion) or negaive (conango) depending on he naure of he underlying fuures conrac alernaive derivaion now: + (borrowing he asse and selling i shor) (opening a ime deposi) carrying period: (purchasing he asse in he fuures marke) d (ransferring any income from holding he asse back o he owner) + r (earning ineres on ime deposi) mauriy: (aking delivery from he fuures marke) + (erminaing he ime deposi) he sraegy involves no individual s own wealh and no risk, herefore no-arbirage condiion implies any profi from he ransacion ( + r ) ( + d rearranging he equaion gives ) basis ( carry d r) any violaion of he cos-of-carry formula indicaes he exisence of arbirage profis ha could be capured by a well designed rading sraegy + d > r i is advanageous o ake a loan and buy he asse spo for + delivering i ino he fuures marke + d < r i is advanageous o sell he asse shor and buy i back in + he fuures marke cash flow from cos-of-carry sraegy involving wo fuures conracs curren price of fuures conrac wih a nearer delivery dae curren price of fuures conrac wih a farher delivery dae now: + (decision o ake a forward loan a on buying he asse)
(decision o buy he asse in he fuures marke wih delivery ) (decision o sell he asse in he fuures marke wih delivery ) carrying period: d ) (earning any income from holding he asse) ( r ( ) (incurring any coss from holding he asse including ineres on borrowed funds) mauriy: + (making delivery ino he fuures marke a ) no-arbirage condiion (repaying he loan a ) + d ( ) + r ( ) ( r d)( ) spread carry 3. Valuaion of open fuures conracs curren marke price of fuures conrac (he conrac promises o buy he underlying asse on delivery a a price ) f curren price of fuures conrac ha was opened a a price X (he conrac allows o buy he asse on delivery a a price X ) for he long he value of he fuures conrac is he presen value of he amoun by which he curren fuures price exceeds he delivery price X f + r applicaion of he cos-of-carry formula gives + ( r d) X f + r + r he value of he fuures conrac a ime i is firs enered ino is zero X f he price of fuures conrac in case of coninuous compounding + a a approximaions: & + ( a b); + a & e + b
+ ( r d) + r & r & e + r cos-of-carry formula f e d & + [( r d) r ] d Xe r r & e d 3.3 air pricing of currency fuures curren spo exchange rae UD/UR (spo price of one euro in dollar unis) curren forward exchange rae UD/UR wih delivery in monhs (fuures price of one euro in dollar unis) r $ curren -monh dollar ineres rae r curren -monh euro ineres rae base currency is he measured currency; i is represened by one uni in he exchange rae quoaion ( r B is he base currency s ineres rae) variable currency is he measuring currency; i is represened by a number of unis in he exchange rae quoaion ( r V is he variable currency s ineres rae) exchange raes are quoed in erms of a number of unis of he variable currency ha correspond o one uni of he base currency oday dollar 3 afer monhs 4 euro 5 cos-of-carry sraegy. borrowing one euro for monhs (he amoun due afer monhs is + ) ) ( r. exchanging he euro ino dollars a he spo exchange rae (afer conversion here are dollars available)
3. lending dollars for monhs a dollar ineres rae (afer monhs here will be ( r$ + ) available) 4. selling dollars forward for euros a forward rae (afer conversion here will be ( + r$ ) / euros available) 5. paying off euro loan wih euros purchased in forward ransacion no arbirage condiion cos-of carry formula covered ineres rae pariy + r$ ) / + ( r $ basis ( r r ) ( r r ) & $ cos of carry + r CIRP saes ha an exchange rae appreciaion/depreciaion implied by he fuures marke will equal he difference beween respecive ineres raes ( r r ) ( r r ) $ forward discoun/premium ineres rae differenial B V r > < r$ higher euro ineres rae mus be offse by a weaker forward euro exchange rae in erms of dollar forward discoun of euro agains dollar or forward premium of dollar agains euro r < > r$ forward premium of euro agains dollar or forward discoun of dollar agains euro uncovered ineres rae pariy UIRP saes ha an anicipaed exchange rae appreciaion/depreciaion will equal he difference beween respecive ineres raes ( ) & ( r r ) B V expecaions hypohesis forward exchange rae is he bes esimae of fuure spo exchange rae ( ) 3
non-deliverable forward ND is rading sraegy ha explois inconsisency beween covered and uncovered ineres rae pariy in he fixed exchange rae regime CZK/DM CZK is variable currency and DM is base currency r CZK > r DM CZK/DM > CZK/DM ( CZK/DM ) forward rae of he Czech koruna in erms of German marks mus be weaker expeced spo exchange rare equals curren spo exchange rae arbirage sraegy: selling one mark in he forward marke for korunas (one mark will be exchanged for korunas) immediae conversion of korunas ino marks in he spo marke ( korunas will be exchanged for / ( ) / > marks) a bank may arrange boh ransacions simulaneously and shares he arbirage profi wih he clien; ND is complicaed by a peg o a baske of currencies 3.4 air pricing of shor-erm ineres rae fuures in shor-erm ineres rare fuures conrac he long is expeced o open a ime a shor erm deposi mauring a ime delivery dae mauriy dae s... spo ineres rae for period (from now unil delivery dae of fuures conrac) s spo ineres rae for period (from now unil erminaion of deposi) f fuures ineres rae for period (over deposi s duraion) spo price of he underlying asse (a pay-off from he deposi obained a mauriy and discouned o he presen) + s 365 fuures price of he underlying asse (a pay-off from he deposi obained a mauriy 4
and discouned o he delivery dae of fuures conrac) + f 365 carrying cos of fuures conrac is he ineres rae over he period ( r s ) while he reurn componen is zero ( d ) cos-of-carry formula (for d ) + f 365 + s ( + r ) 365 ( + s ) cos-of-carry formula is equivalen o he ineres rae pariy equaion 365 ( + s ) ( + s ) ( + f ) 365 ineres rae pariy is a no-arbirage condiion beween wo invesmen sraegies a) invesing ino a ime deposi ha maures a ime 365 b) invesing ino a ime deposi ha maures a a shorer ime and reinvesing he accumulaed value of he deposi for he remaining period 365 3.5 air pricing of sock index fuures conracs cos-of-carry ransacion sraegy : now: purchasing all he shares in he sock index wih he appropriae weighs laer: selling he shares in he spo marke reurn ( ) + d (change in he value of he sock index + dividend yield) sraegy : now: invesing he amoun under sraegy a he money marke ineres rae and purchasing a fuures sock index conrac laer: erminaing he deposi reurn ( ) + ( + r) (difference beween he sock index fuure value and fuures price + earned ineres) boh sraegies have he same iniial invesmen, hey are idenical in erms of no final holding of shares and hey are exposed o he same risk from unknown fuure value of he sock index boh sraegies should have he same reurn ( ) + d ( ) + ( + r) 5
cos-of-carry formula ( + r d) pracical complicaions - ransacion coss involved in seing up he cash index under sraegy ; hey may become sufficienly high o make unfeasible he replicaion of reurn in sraegy ) - uncerain paern of paymens of dividends under sraegy (iming of dividends is irregular and unknown a he beginning of cash-and-carry ransacion) 6
IV. PCULAIV RADING RAGI WIH UUR 4. Key feaures of a speculaive rade speculaion is ineresed in aking eiher a shor or a long posiion in a paricular securiy in he hope of making a quick shor-erm profi (exended posiions have o be financed ha can be expensive) speculaors believe ha hey have beer informaion han oher marke paricipans abou - wheher a securiy is overpriced or underpriced compared wih is fair or fundamenal value (overpricing moivaes o ake shor posiions and underpricing moivaes o ake long posiions) - wheher he arrival of new informaion will cause he marke prices of a securiy o rise or fall imperfec raionaliy of financial markes means ha emporary mispricings are possible bu any price anomalies are supposed o be correced rapidly by efficien markes fundamenal analysis aims o idenify a fair (fundamenal) value of a securiy by relaing he behavior of prices o he underlying economic facors echnical analysis uses chars o deec paerns in he ime series of securiy prices and argues ha hese paerns repea hemselves over ime; once a paricular paern sars o form he chariss speculae on a given fuures course of prices a wide diversiy of views is essenial for he liquidiy of he marke araciveness of fuures conracs - a higher liquidiy compared wih spo markes (fuures posiions can be open and close rapidly hanks o a high sandardizaion of fuures conracs) - easy speculaion on falling prices by aking shor fuures posiions whereas shor selling on spo markes may be subjec o some regulaion (i.e. he up-ick rule on some xchanges) - subsanial financial leverage (iniial margin is a fracion of he nominal value of he fuures conrac) 7
Working of leverage Price P P ime P unlevered rae of reurn r U (he asse mus be purchased o ge access o an P expeced price change) P ru levered rae of reurn r L (access o un expeced price change is given a a αp α fracion of he asse s price ha magnifies he reurn on he ransacion) α % r 5 L r U 4. Open posiion rading open posiion rading (direcional rading) involves backing a view on he fuure direcion of price changes (high leverage of derivaives imply ha hose price changes can lead o magnified profis and losses) long open posiion rading involves backing a view ha prices are going o rise speculaors will wan o buy fuures conracs A speculaor believes ha shor-erm ineres raes are going o fall (fuures prices are going o rise). herefore he decides o buy on LI one December 3 conrac (hree-monh serling ineres rae fuures) a curren price 9.5. he conrac has iniial margin of 5, ick-value is.5. he following able shows he behavior of he conrac s fuures price during five rading days and he speculaor s resuling ne profi. Afer five days he fuures posiion is closed by selling he conrac. 8
Day Acion u. price Ini. margin Mark. o m. Ne gain Buy 9.5-5. -5. Hold 9. - -65. -5. 3 Hold 9.5 - +3.5-8.5 4 Hold 9.75 - +65. -87.5 5 ell 9.5 +55. +65. 937.5 he speculaor makes an overall profi of 937.5. Over he period of five days he marke wen up by only (9.5/9.5 -).83 %, bu he speculaor achieved a reurn of 937.5/5 88 %. shor open posiion rading involves backing a view ha prices are going o fall speculaors will wan o sell fuures conracs A speculaor believes ha share prices are going o collapse and herefore sells one December fuures conrac a he curren price 4. he conrac has iniial margin of and ick value (equal one half of one index poin) is 5. On day 3 he speculaor closes his posiion. Day Acion u. price Ini. margin Mark. o m. Ne gain ell 4 - - Hold - +4 +3 3 Buy 6 + +4 +8 Over he period of hree days he marke fell by (6/4 -) 33 %, bu he speculaor achieved he reurn of 8/ 8 % on employed capial. 9
4.3 pread rading spread rading involves he simulaneous purchase and sale of relaed financial fuures conracs (wo legs of he spread) speculaor is aking view on wha is going o happen o he difference beween wo fuures prices spread rading is likely o be less risky (because i is based on differences) and as a resul of lower risk he iniial margin requiremens are lower Developmen of spreads epember une spo price une ime epembe r he meaning of buying he spread depends on how he spread is defined spread ep une buying he spread involves he purchase of he disan conrac and he sale of he nearby conrac spead une ep buying he spread involves he sale of he disan conrac and he purchase of he nearby conrac speculaor will buy he spread if he hinks ha he spread will widen (becomes more posiive or less negaive) he meaning of selling he spread depends on how he spread is defined spread ep une selling he spread involves he purchase of he nearby conrac and he sale of he disan conrac spead une ep selling he spread involves he sale of he nearby conrac and he purchase of he disan conrac 3
speculaor will sell he spread if he hinks ha he spread will narrow (becomes less posiive or more negaive) expeced change in he size of he spread is relaed o he expeced change in he cosof-carry ( r d)( ) inra-conrac spread rade involves buying and selling differen mauriies of he same fuures conrac on he same exchange (e.g. involvemen of 3M December and 3M une ineres rae fuures conracs) iner-conrac spread rade involves buying and selling differen ypes of fuures conracs (e.g. involvemen of 3M une serling conrac and 3M epember urodollar conrac) A speculaor believes ha he cos-of-carry is going o rise and so decides o buy a spread on he hree-monh shor-erm serling fuures. He buys one epember fuures a 9. and sells one une fuures a 89.5. he conrac s ick is one basis poin and he ick s value is.5. he iniial margin is 5. he posiion has been wound up in he fifh day. buying he spread (day ): epember 9. (buy) une 89.5 (sell) spread 9. 89.5 5 icks selling he spread (day 5): epember 89. (sell) une 88.5 (buy) profi (75 5) icks.5 per ick 3.5 [(89 9) (885 895)].5 3.5 rae of reurn 39 % on employed capial 5 buerfly spread is composed of wo spreads (bull and bear spread) wih a common middle conrac his sraegy is used when he middle conrac looks ou of line wih he conracs on eiher side bu i is no clear which conrac will adjus in order o correc he anomaly 3
he hree mauriies of he serling currency fuures display following quoes among which he epember conrac looks relaively undervalued. he conrac s ick is. cens per and he ick s value is 6.5. une epember December.7.7.7 } icks } 8 icks he buerfly spread is 7 icks. he speculaor buys he buerfly spread ha means he buys he une-epember spread (which is expeced o widen) and sells he epember-december spread (which is expeced o narrow). On day 3 he fuures prices are following une epember December.75.7.74 } 5 icks } 4 icks he buerfly spread is - icks and he speculaor reverses he rade. profi [(5 ) (4 8)] icks 6.5 per ick 5 4.4 Basis rading basis rading involves buying or selling a fuures conrac and simulaneously execuing he opposie ransacion in he underlying cash securiy he meaning of buying he basis depends on how he basis is defined basis buying he basis involves being long in he fuures conrac and shor in he cash securiy basis buying he basis involves being long in he cash securiy and shor in he fuures conrac speculaor will buy he basis if he hinks ha he basis will widen (becomes more posiive or less negaive) he meaning of selling he basis depends on how he basis is defined basis selling he basis involves being shor in he fuures conrac and long in he cash securiy 3
basis selling he basis involves being shor in he cash securiy and long in he fuures conrac speculaor will sell he basis if he hinks ha he basis will narrow (becomes less posiive or more negaive) expeced change in he size of he basis is relaed o he expeced change in he cos-ofcarry d r) if a speculaor believes ha shor erm ineres raes are ( going o rise he will sell he basis (he buys fuures and sells shor he cash securiy) I is April and he une long gil fuures is rading a 9.38, while he CD long gil wih a price facor of.333 is rading a a diry price of 8 per nominal value. Money marke ineres raes are 8 %. uures conrac specificaions: nominal value, ick size bp, value of ick. Basis ( April) k 8.333 938. 39 A basis rader expecs he basis o narrow due o an expeced rise of money marke raes. He implemens he basis rade (he sells he basis, i.e. he buys he fuures and sells shor he bond) on million nominal of he CD bond. o be hedged agains absolue changes in he bond s price he rader opens a number of fuures conrac according o he formula face value of cash bond number of fuures conracs price facor face value of a fuures conrac.333 & 3 Afer days he rader observes ha he fuures conrac is rading a 9.88 and he CD bond is rading a 8.5. Basis ( April) k 8.5.333 988. 79 he basis has narrowed and he rader closes his posiion by selling he fuures and buying he bonds. profi on fuures number of conracs number of icks gained ick value 3 (988 938) 65 loss on bonds number of bonds loss per bond 33
(8.5 8) 5 ineres earned on proceeds from sale of bonds 8.8 573 365 oal profi from he basis rade 65 5 + 573 973 34
V. ARBIRAG RADING RAGI WIH UUR 5. Key feaures of an arbirage rade an arbirage rade aims o exploi pricing anomalies ha creae so called money machine (i.e. unlimied profis a almos no risk and no cos in case he arbirage opporuniy persiss for some ime and is no correced soon) boh arbirage and speculaive rade are designed o profi from perceived misvaluaions bu he arbirageur hopes o make profis a minimal risk while he speculaor hopes o make profis by aking on risk arbirage has a key role in deermining he fair values of securiies feaures of arbirage porfolio:. i uses no ne wealh. i is riskless 3. i generaes a real reurn in excess of he riskless rae of reurn a fair value of a securiy is deermined by he condiion ha an arbirage porfolio canno exis in an efficien marke In New York he euro vis-à-vis he dollar is raded a an exchange rae of UD/UR. (a number of dollars per one euro) while in London i is raded a UD/UR.3. he arbirage rade:. one euro borrowed and sold in London gives a revenue of.3 dollars. immediae sale of.3 dollars in New York gives a revenue of.3/..833 euros 3. repaymen of loan gives a profi of.833.833 per one euro invesed pracical obsacles o arbirage rades - arbirage rades canno be enirely riskless because during a delay beween individual legs of an arbirage sraegy, even if i is only a maer of seconds, pricing anomaly can be correced - arbirage profis can be eliminaed by echnicaliies of fuures rading such i) variaion margins due o marking o marke ha can arise a any ime before delivery of fuures conracs 35
ii) involvemen of rounded numbers of fuures conracs only - ransacion coss can subsanially reduce and even eliminae rading profis so in presence of ransacion coss here can be significan pricing anomalies uppose ha in he previous example forex dealers in New York and London charge a percen bid-ask spread. New York London bid price (buying for dollars).8.7 mid-poin..3 ask price (selling for dollars).3.43 he execuion of he previous rade would end up wih a loss:. one euro borrowed and sold o a London s dealer gives a revenue of.7 dollars. immediae sale of.7 dollars o a New York s dealer gives a revenue of.7/.3.8864 euros 3. repaymen of loan leads o a loss of.8864 -.36 per one euro invesed program rading denoes rading sraegies ha use compuer sofware which riggers immediae insrucions o buy or sell securiies afer a price anomaly or arbirage opporuniy has been spoed 5. Box arbirage box arbirage is he sraegy ha replicaes he scheme of he covered ineres rae pariy using fuures conracs only ransacions of covered ineres rae pariy oday $ 3 afer 3 monhs 4 36
. borrowing euro for 3 monhs a a euro ineres rae. exchanging euro ino dollars in he spo marke 3. invesing dollars a a dollar ineres rae 4. exchanging dollars back ino euros in he forward marke An arbirage opporuniy exiss if he forward ransacion leads o euro revenue in excess of he repaymen of he euro loan plus ineres ime-securiy grid horizonal righ-poining arrow invesing or going long horizonal lef-poining arrow borrowing or going shor verical arrows exchanging one securiy for anoher securiy ransacions of box arbirage un $ e 3 epembe r 4 curren dae is 5 April Mauriy uro-dollar currency fuures (C) hree-monh dollar ineres rae fuures (I$) hree-monh euro ineres rae fuures (I ) une.365 89.5 88.5 epember.3 89.5 88.5. selling une I conracs in une one euro will be borrowed a ( 88.5).75 % for hree monhs in epember he loan will have o be repaid, due amoun will be 3 +.75.94 UR. selling une C conracs in une one euro will be exchanged ino dollars a an exchange rae of.365 UD/UR 37
available dollar amoun will be.365 UD 3. buying une I$ conracs in une.365 dollars will be invesed a ( 89.5).5 % for hree monhs in epember he loan will be erminaed, amoun available will be 3.365 ( +.5).363 UD 4. buying epember C conracs in epember.363 dollars will be exchanged ino euros a an exchange rae of.3 UR/UD available euro amoun will be.363/.3. 375 UD box arbirage discovers a guaraneed profi of.375.94. 8 per one euro invesed 5.3 Conversion arbirage conversion arbirage involves combinaions of cash marke securiies and derivaive insrumens he mos imporan example is he cash-and-carry arbirage ha plays a key role in he deerminaion of fair fuures prices long cash-and-carry arbirage consiss of he following seps:. borrowing funds a money marke and using he proceeds o purchase an underlying asse. simulaneous sale of he fuures conrac o lock in he price a which he underlying asse can be sold a he conrac s mauriy 3. receiving he cash flow from he underlying asse 4. delivering he underlying asse ino he fuures conrac 5. repaying he loan wih ineres shor cash-and-carry arbirage consiss of he following seps:. shoring (i.e. borrowing and selling) he underlying asse and using he proceeds of shor sale o inves in he money marke. simulaneous purchase of fuures conrac o lock in he repurchase price of he underlying asse 3. ransferring he cash flow from he underlying asse o he asse s owner 38
4. obaining he underlying asse from he fuures conrac 5. delivering he underlying asse o is owner swich raio risk in shor cash-and-carry arbirage on he delivery of some fuures conracs (paricularly long-erm ineres rae fuures) he arbirageur will receive he cheapes-o-delivery asse which may no be he same as he asse he has sold shor he arbirageur will have o swich ou of he delivered asse ino he asse he has sold shor shor cash-and-carry arbirage can never be enirely riskless 39
VI. HDGING RADING RAGI WIH UUR 6. Basic conceps he aim of hedging is o be proeced agains uncerainy caused by adverse price movemens of a hedged asse he hedger is a person or a company ha ransfers all or a leas mos risk o anoher person or company (he counerpar of he hedger is a speculaor who akes on he risk) ypology of hedging forward hedge consiss in removing uncerainy by negoiaing a specific price for a given fuure ransacion (ypical feaure of forward conracs) fuures hedge consiss in he selecion of a suiable hedging insrumen whose price movemens mirror closely hose of he hedged insrumen (loss on he hedged asse is expeced o be offse by gain on he hedging asse) perfec hedge is one in which he price movemens of he hedging insrumen are perfecly negaively correlaed wih hose of he hedged insrumen (in pracice no easy o esablish) direc hedge is one in which he hedging insrumen involves he same underlying securiy as he securiy being hedged parial hedge is one in which he hedging insrumen is imperfecly negaively correlaed (i is ofen beer han no hedge a all) cross hedge is one in which he hedging insrumen involves a differen securiy as he securiy being hedged long hedge is one ha involves a long posiion in a fuures conrac a long posiion makes a profi when he fuures price declines and incurs a loss when he fuures price increases shor hedge in one ha involves a shor posiion in a fuures conrac a shor posiion makes a profi when he fuures price increases and incurs a loss when he fuures price declines naural hedge occurs when he overall level or risk has been reduced by an appropriae srucure of business operaions (i.e. eliminaion of mauriy mismaching beween ineres rae asses and liabiliies, producion of expor goods from impored supplies) 4
6. Hedging using spo ransacions forward and fuures conracs are no echnically indispensable for hedging because hedged posiions can be creaed by an appropriae design of spo ransacions disadvanages of spo hedges: - spo ransacions are of fixed erm and non-negoiable so i is impossible o unwind he hedge once i has been implemened - spo ransacions are generally more expensive o implemen han fuures hedges because a higher number of spo conracs involved increases ransacion coss - spo ransacions involve an expansion of he balance shee (by aking ou deb) ha may adversely affec he company s leverage A corporae reasure wishes o ake ou a hree-monh loan o begin in wo monh s ime and is concerned ha ineres raes will rise over he nex wo monhs. he reasure could borrow now for five monhs a he 5M ineres rae r 5 M and simulaneously deposi he money for wo monhs a he curren M ineres rae r. M now in monhs in 5 monhs wo monh s spo lending hree monh s forward borrowing five monh s spo borrowing he reasurer effecively locks in a borrowing rae for 3 monhs beginning in wo monhs ime denoed M f 5 M. 6 amoun available in monh s ime X + r ) ( 36 M 5 amoun due in 5 monh s ime X + r ) ( 36 5M implici hree-monh borrowing rae beginning in wo monh s ime 5 + 36 r5 M 5 M f M 6 + 36 r M 36 5 A reasurer expecs revenue denominaed in euros in one monh s ime and is concerned ha he will suffer a loss if he exchange rae CZK/UR depreciaes over he nex monh. 4
he reasurer could borrow euros now for one monh a a presen one moh euro ineres rae r and conver euros ino CZK a a presen spo exchange rae s /. By repaying he euro loan from fuure expor earnings he reasurer effecively ransfers he fuure conversion o he presen a a presen spo exchange rae. CZK UR CZ K U R now curren conversion of UR ino CZK one monh s spo UR borrowing in monh repaymen of UR loan from expor revenues expeced expor revenue X UR X borrowed euro amoun UR + r 3 36 3 amoun due in one monh s ime ( r ) X X + 3 36 r + 36 UR 6.3 Basic elemens of fuures hedging hedger s exposiion on he spo marke physical Q (size of he cash exposure is an amoun of a cash securiy hedged agains price flucuaions) nominal V Q (acual spo price hedged amoun) hedger s exposiion on he fuures marke physical q n (size of he posiion aken in fuures exposiion is he size of one fuures conrac number of conracs) nominal V q n (acual fuures price fuures exposiion) hedge raio size of h he posiion aken in fuures conracs size of he cash exposure q n Q number of fuures conracs size of he cash exposure n hedge raio h size of he fuures conrac Q q 4
i) locking in he fuure price if he hedging horizon coincides wih he ime o mauriy of he fuures conrac han a hedger can lock in an effecive fuure price obained or paid for he asse a) long hedge he hedger wishes o buy a securiy (he is shor in cash securiy) he hedger opens a long fuures posiion wih he hedge raio h when he fuures conrac maures he hedger buys he securiy on spo marke and collecs an amoun on his margin accoun oal cash flow a mauriy ( C, qn Q ) Q + q n ( ) Q Q ( ) Q he fuure cash flow is cerain because all of is componens are known a poin of ime when he hedge has been opened b) shor hedge he hedger wishes o buy a securiy (he is long in cash securiy) he hedger opens a shor fuures posiion wih he hedge raio h when he fuures conrac maures he hedger sells he securiy on spo marke and collecs an amoun on his margin accoun oal cash flow a mauriy ( C, qn Q ) Q q n ( ) + Q + Q ( ) + Q he fuure cash flow is cerain because all of is componens are known a poin of ime when he hedge has been opened pracical consideraions: by holding he fuures posiion during he delivery monh he hedger risks he physical delivery o happen; making or aking delivery can be a cosly and inconvenien o avoid physical delivery he hedger may close ou he fuures posiion earlier in he delivery monh or he can choose a conrac wih a laer delivery monh in boh cases he hedger is exposed o some uncerainy regarding he effecive price a which he asse is purchased or sold (known as basis risk) gain/loss on fuures posiion hedge efficiency gain/loss on spo exposure 43
ii) basis risk basis risk occurs whenever he basis (i.e. he difference beween he spo and fuures prices) will no be zero a he effecive dae of he hedge (when he hedging posiion is unwound) long hedge: C Q + q n ( ) Q Q ( ) Q Q B shor hedge: C Q q n ( ) + Q + Q ( ) + Q + Q B sources of basis risk: - he asse whose price is o be hedged may no be exacly he same as he asse underlying he fuures conrac - he hedger may be uncerain as o he exac dae when he cash asse will be bough or sold on spo marke - he hedger may require he fuures conrac o be closed ou well before is expiraion dae iii) esimaing he basis by linear inerpolaion if he convergence of he spo and fuures prices is assumed o be smooh and linear hen he hedger can esimaed he size of he basis using he mehod of similar riangles B B B B B B exposure dae of he hedge expiraion of fuures conrac 44
iv) inerpolaive hedge inerpolaive hedge is a mehod of hedging he basis risk; he sraegy can be used whenever he exposure dae of he hedge lies beween wo rading fuures conracs seps in using an inerpolaive hedge: a) esimaion of a curren price of a ficive fuures conrac whose mauriy dae coincides wih he exposure dae of he hedge (using linear inerpolaion),, curren prices of he une, epember and ficive fuures conracs,, ime o mauriy of he une, epember and ficive fuures conracs number of days he mehod of similar riangles gives + w + 45 b) compuaion of he opimal number of ficive fuures conracs Q n h q Q q he hedge raio can be se o because he ficive fuures conracs are held o mauriy and he hedger wans o lock in he fuure price of he underlying asse a a level of (a mauriy of ficive fuures conrac he ficive basis w is zero) c) he hedger mus reproduce he payoff of he ficive conracs wih he exising fuures conracs he hedger divides he number of ficive fuures conracs n ino n une conracs and n epember conracs in he proporion ha corresponds o he weighs of he inerpolaive hedge w and w
n n + n w n + w he sraegy of inerpolaive hedge hus involves selling n epember conracs n n une conracs and selling d) he inerpolaive hedge is no echnically feasible beyond he expiry dae of une conracs mainaining a given number of fuures conracs for hedging purposes n requires replacing mauring une conracs wih he same number of epember conracs he hedger buys back n une conracs and sells of epember conracs is hus n ) n epember conracs (oal number he hedger is now exposed o he basis risk a he end of he hedge horizon because he holds only epember conracs whose expire dae comes afer he end of he hedge horizon (he basis is no zero a ime ) e) he problem can be parially solved by buying n epember-december spreads ha will o some exen offse an opposie change in he spo-epember basis f) a he exposure dae of he hedge he hedging sraegy is unwound (all fuures conracs are closed up) ne balance of all cash flows from inerpolaive hedge available a he exposure dae : paying he marke price of he underlying asse n ) cash from une conracs sold a and bough a heir mauriy ( n ( ) cash from epember conracs sold a and bough a n ( ) cash from epember conracs ha replaced expiring une conracs; hey were sold a and bough back a n ( ) cash from epember conracs sold a as a par of he epember- December spread; hey were bough back a D D + n ( ) cash from December conracs bough a as a par of he epember-december spread; hey were sold back a he iems of he oal cash flow can be decomposed ino he following groups: cerain cash ouflow a : n + n 46
D wo offseing spreads a : n [( ) ( )] D parially offseing basis and spread a : ( n ) n ( ) exrapolaive hedge is used whenever he exposure dae of he hedge lies beyond he mos disan rading conracs v) minimum variance hedge raio he hedge raio of. is no necessary he opimal one when he end of he hedge horizon differs from he expiry dae of he fuures conrac he hedging hen consiss in exploiing muually offseing changes in he spo and fuures exposures (a long exposure on one marke is offse by a shor exposure on he oher marke) change in he spo exposure Q ) Q ( change in he fuures exposure q n ) q n ( change in value of he hedged porfolio Q qn Q qn Q Q ( h ) he opimal hedge raio is one ha minimizes he variance of he hedge porfolio v h var [Q( h )] Q ( σ + h σ h ρ σ σ ) min differeniaion wih respec o h and seing o zero gives vh δh Q (hσ opimal hedge raio h σ ρ σ ρ h is σ σ ) opimal number of fuures conracs Q n h q he opimal hedge raio raio of he sandard deviaions of esimaed from hisorical daa) n is h is he produc of correlaion beween and and and he (parameers ρ, σ, σ are usually 47
if he fuures price mirrors he spo price perfecly hen he hedge raion is ρ and σ h σ he higher is he volailiy of he fuures price relaive o he volailiy of he spo price, he lower is he number of fuures conracs ha is needed o hedge a given spo exposure he opimal hedge raio is he slope of he bes fi line when is regressed agains (he formula for he opimal hedge raio is idenical o he regression coefficien) 6.4 ome pracical examples of hedging using fuures a) hedging wih shor-erm ineres rae fuures A company is expecing a cash inflow of million pounds in wo monhs ime which will be invesed in a hree-monh deposi accoun. he company is concerned ha shor-erm ineres raes are going o decline and wans o be proeced agains his risk. Because he company is shor on he spo marke i pus on a long hedge by purchasing 3 une ineres rae fuures conracs. he hedge is going o be almos perfec (he underlying is idenical wih he cash invesmen) so he hedge raio is se o one. Q number of conracs h 4 q 5 he summary of marke daa: oday (5.4) In monhs (5.6) po marke 3M ineres rae is % Invesing mil in 3M deposi a an ineres rae of 9 % uures marke Purchasing 4 fuures 3 conracs a a price of 88.5 (.5 %) elling fuures conracs a a price of 9.5 (9.5 %) loss in he spo marke (he company is hi by he ineres rae drop) (..9) 3 gain in he fuures marke (he long posiion benefis from an increase in fuures price 4 conracs (9.5 88.5) icks.5 ick value hedge efficiency % 48
he fuures posiion showed a profi ha compleely compensaed he loss in he spo marke. b) hedging wih sock index fuures On April a pension fund manager is uncerain abou where he marke is going over he nex hree monhs and wishes o hedge million of his equiy porfolio which has a bea of.5. Because he fund manager is long in he cash marke he will need o be shor in he fuures marke. He decides o use LI sock index fuures. he index is currenly sanding a 4 and he value of he une fuures conrac is 3. i) deerminaion of he number of conracs V (decline in porfolio value) Z (gain on he fuures posiion) V β P (porfolio bea measures he responsiveness of he porfolio s V value o he change in he marke index) Z n q n q (n is he number of conracs, q ) k, k (he cos-of-carry is assumed o be a consan number) V Z Vβ P n q / / V q β P face value of he sock porfolio number of conracs porfolio bea face value of one fuures conrac.5 5 3 he porfolio manager sells 5 une sock index fuures conracs. ii) locking in he porfolio value a expiry of une fuures uppose ha on une 3 when he une fuures conrac maures here is a fall in he index o 7 (he sock marke fell by almos 3 %). erminal value of unhedged porfolio 49
5 7374.5 4 4 7 + + P V V β gain on fuures posiion ( 7 ) 3 3) (7 5 ( ) q n Z erminal value of hedged porfolio 374 3 7374 + + Z V hedge efficiency 4% 7374 3 η prediced value of he hedged porfolio (deermined from observable variables) a expiry dae) ( 48.5 3 3 4 ) ( ) ( B V V q q V V q n V Z V P P P P P P + + + + β β β β β β & a small deviaion from rue erminal value is caused by he applied approximaion (he approximaion is fairly good for small relaive changes) ) ln (ln ) ln (ln ln ln ln ln + + + & & iii) esimaing he basis If he fund manager wishes o hedge only unil une (and no unil expiry dae) hen here is some risk ha he fuures price will no move in line wih he spo price. He may lose on boh sides of he hedge (falling spo price and rising fuures price).
5 erminal value of hedged porfolio + + + + + + P P P P P P P P B B V V V q n V Z V β β β β β β β β ) ( & B is he basis fund manager canno lock in he porfolio value on une because he does no know he size of he basis; he can only esimae he basis (i.e. by linear inerpolaion) number of days from April (now) o 3 une (expiry dae of une fuures) 9 number of days from une (exposure dae of he hedge) o 3 une 3 3.6 9 3 3) (4 B B fund manager migh expec o lock in he hedged porfolio on une 349.5 3 3.6.5 4 3 4 + + + P P B B V Z V β β uppose ha on une he spo index had fallen o 9 and he fuures index had fallen o 95. acual erminal value of hedged porfolio on une 59.5 3 95 9.5 4 3 4 + + + P P B V Z V β β he hedge is worse han anicipaed bu i is sill much beer ha no hedging a all. iv) inerpolaive hedge uppose ha on April a fund manager wishes o pu a hedge on his million sock porfolio for Augus. his dae lies beween expiry daes of une and epember fuures conracs. he une conrac is momenarily rading a 3 and he epember conrac is rading a 365. he fund manager iniiaes he sraegy of an inerpolaive hedge. uppose ha
on 3 une when expiring une conracs are replaced by epember conracs and he fund manager buys epember-december spreads he cash index and une index are sanding a D 7, epember fuures is 74 and December fuures is 8. inally on Augus when he hedge is unwound, he spo index is 79, epember D fuures is rading a 85 and December fuures is rading a 87. sraegy for an inerpolaive hedge number of days from 3 une o 3 epember 9 5 number of days from Augus o 3 epember 6 number of days from 3 une o Augus 3 hypoheical curren price of a fuures conrac expiring on Augus 6 3 3 + 365 9 9 + anicipaed value of hedged porfolio on Augus V + Z V β P 4 3.6.5 5873 3.6 number of ficive Augus fuures conracs V n β P.5 49.5 & 5 q 3.6 ( 3.6 a expiry dae) decomposiion of ficive Augus conracs ino une and epember fuures conracs n 6 5 33, 9 n 5 33 7 ne balance of all cash flows a he exposure dae : value of he sock porfolio V V + β P 79 4 +.5 783984 4 profi on une conracs sold a and bough a heir mauriy dae n q( ) 33 (7 3) 98 profi on epember conracs sold a and bough a
n q( ) 7 (85 365) 98 loss on epember conracs ha replaced expiring une conracs; hey were sold a and bough back a n q( ) 33 (85 74) 85 loss on epember conracs sold a as a par of he epember-december spread; hey were bough back a n q( ) 33 (85 74) 85 profi on December conracs bough a as a par of he epember-december spread; hey were sold back a + n D D q( ) 33 (87 8) 3 oal value of hedged porfolio V Z 783984 + 98 + 98 85 85 + 3 4784 + c) hedging wih long-erm ineres rae fuures long-erm ineres rae fuures are a convenien insrumen for hedging he marke risk of bond porfolios (he value of bond porfolio will fall as a consequence of an ineres raes increase) spo marke alernaive: dispose of he bonds and repurchase hem afer ineres raes have risen (his would involve ransacion coss and here is no guaranee of being able o repurchase he same porfolio of bonds) hedging he CD bond cos-of-carry relaionship (ignoring he cos-of-carry componen) p P + P ( r d) & P CD CD CD CD pcd & P CD CD p is price facor of he CD bond P CD is marke value of he CD bond change in spo exposure V CD VCD P CD VCD p CD 53
change in fuures exposure V q n q is face value of one fuures conrac number of fuures conracs for hedging he spor exposure V CD V n q V CD p CD face value of spo exposure price facor of face value of one fuures conrac he CD hedging a non-cd bond change in spo exposure VH VH PH change in fuures exposure V duraion relaionship q q P n n p rp P D P + r P P H is he marke value of a hedged bond CD CD assuming parallel percenage yield curve movemens r + r CD CD rh + r number of conracs ( V V ) H H r P is he yield o mauriy of a given bond VH n q VH q p p CD CD P P D D H CD H CD VH q P P H CD p CD VH q D D p CD CD H P P DHR H CD rh + rh r + r CD CD he expression DHR is called he duraion hedge raio A fund managers wishes o hedge his million bond porfolio agains ineres rae decline by opening a long posiion in long-erm une ineres rae fuures conracs. One fuures conrac has a nominal value of. he CD bond o he conrac is a reasury whose 54
price facor is.333, he duraion is.6 years and he bond is currenly rading a 8 (per nominal). uppose ha he duraion of he porfolio is 4. years (value-weighed average of he duraions of he individual bonds) and he value-weighed average price (per nominal) is.5. How many fuures conracs he fund manager has o buy? VH DH PH n pcd q DCD PCD 4..5.333.6 8 48.88 & 49 he number of fuures conracs necessary o hedge he given porfolio is 49. 55