Page1 C H A P T E R 2 FORWARD AND FUTURES CONTRACTS 2.1 INTRODUCTION The main purpose of forward and fuures conracs is he managemen of risk. The exposure o risk as a resul of ransacing in he spo marke can be eliminaed (o he larges exen possible), miigaed or enhanced wih forwards or fuures. As he mos basic of derivaive securiies, forwards and fuures fix prospecive prices or raes and values and are he primary ools for hedging. Fuures provide gearing as a resul of he margining sysem whereby a small amoun of money (i.e., he iniial margin) conrols a large underlying exposure and forwards, as cusom-made insrumens, can serve he same purpose. In conras, opions provide leverage via he premium or cos paid o ransac and conrol a much larger exposure. Boh forwards/fuures and opions can proec he value of an underlying exposure, bu go abou i differenly as forwards/fuures provide securiy while opions provide insurance. The workings of forwards and fuures are sraighforward and easy o comprehend. 2.2 FORWARD AND FUTURES BASICS Holding a posiion in he marke exposes he invesor o he risk of adverse marke movemens, namely marke risk. Marke risk encompasses equiy risk, ineres rae risk, currency risk, and commodiy risk. Equiy risk, he risk ha share prices will change, is quanified in erms of a sandard deviaion (absolue measure) or bea (relaive o he marke). Equiy risk exposures, eiher long (owning a share) or shor (inending o own a share), can be eliminaed (zero bea, i.e. hedged) or adjused (reducing or increasing bea) by aking a posiion in an equiy fuures conrac. Bond and ineres rae fuures proec agains changing bond prices and ineres raes, currency fuures can accoun for flucuaions in foreign exchange raes, while commodiy fuures provide securiy for grains and meals raders. 2.2.1 Definiions and comparison A forward conrac is an agreemen beween wo paries in which one pary, he buyer, agrees o buy from he oher pary, he seller, an underlying asse a a fuure dae a a price esablished oday. The conrac is cusomised and each pary is subjec o he possibiliy ha he oher pary will defaul. A fuures conrac is a variaion of a forward conrac ha has essenially he same basic definiion, bu some clearly disinguishable addiional feaures, he mos imporan being ha i is no a privae and cusomised ransacion i is a public, sandardised ransacion ha akes place on a fuures exchange. Anoher imporan disincion is ha fuures conracs are subjec o margining ha is, he marking-o-marke (m--m) of a conrac hereby reseing is value o zero and virually removing he possibiliy of defaul.
Page2 FINANCIAL DERIVATIVES A summary of he differences beween forward and fuures conracs: Forwards Over he couner Privae Cusomized unique Defaul risk No marked o marke Held unil expiraion No liquid ailor made Unregulaed Fuures Fuures exchange Public Sandardized Defaul free Marked o marke Offse possible Liquid radeable Regulaed In addiion specific o fuures conracs: Like forwards, fuures have no value a iniiaion Unlike forwards, fuures do no accumulae any value Value always zero afer adjusing for day s gain or loss (m--m) Value differen from zero only during m--m inervals Fuures value equals he curren price minus price a las m--m Value of long posiion increases as forward/fuures price increases Value of shor posiion increases as forward/fuures price decreases Value se back o zero by he end-of-day mark o marke A fuures posiion can be closed ou prior o expiraion by an opposie (offseing) ransacion. Selemen can also occur by he long pary aking delivery shor pary delivers he underlying on a cerain dae and a a specified locaion. The cash selemen of a conrac (feaure of some conrac i.e., sock index fuures) negaes he need o close ou wih he posiion lef open and he final gain or loss marked o marke. An exchange for physicals is a variaion on delivery wih he counerparies arranging alernaive daes and/or locaions. A conrac can also be rolled over (exended) ino anoher idenical conrac bu wih a laer delivery dae. Fuures conracs available for rade: Shor-erm ineres rae fuures Treasury bill (T-bill) fuures and LIBOR (Euodollar) or JIBAR fuures which are sandardised forward rae agreemens. Inermediae- and long-erm ineres rae fuures Treasury noe (T-noe: 2-1 years) fuures and Treasury bond (T-bond: exceeding 1 years) fuures. Single sock and sock index fuures wih he underlying an individual share or share index (e.g., S&P5 Sock Index fuure) where he conrac price is equal o he fuures price imes a muliplier and cash seled. In Souh Africa he JSE Limied (SAFEX) liss conracs on he ALSI, INDI, FINI ec. (R1 x Index level). Currency fuures each have a designaed size and quoaion uni and in Souh Africa i is 1, of he underlying currency, quoed in Rand per foreign currency. USD/Rand, Euro/Rand, Serling/Rand and AUD/Rand conracs lised. Similar cusomized forward conracs also raded in he over-he-couner marke.
Page3 Accoun balance Variaion margin Variaion margin FORWARD AND FUTURES CONTRACTS 2 2.2.2 Marking o marke The iniial margin deermines he level of gearing obained from enering ino a fuures conrac and forms par of marking a conrac o marke. This procedure revaluaes a fuures conrac on a daily basis and adjuss he open posiion o reflec profis and losses resuling from price movemens ha occurred during he las rading session. I is equivalen o closing he conrac a he end of each day, seling gains and losses, and opening a new conrac. The amoun by which an accoun changes on any given day is referred o as he selemen variaion and should his cause he accoun value o fall below he mainenance margin (minimum accoun balance), he invesor receives a margin call and has o deposi an addiional amoun (variaion margin) o bring he accoun balance o he iniial margin. The mark-o-marke process ensures ha he counerparies all mee heir obligaions in erms of he fuures conrac. Failure o comply wih a margin call will resul in he close ou and financial selemen of he posiion. Iniial margin Mainenance margin Figure 2.1 Time Margining Margin calls There are hree ypes of margin as shown in Figure 2.1, namely he iniial, mainenance and variaion margin. The iniial margin is fairly low, equalling abou one day s maximum price flucuaion, and mus be posed before any rading akes place. Wih he daily selemen process called marking-heaccoun-o-marke, any losses for he day are removed from he rader s accoun and any gains are added o he rader s accoun. Boh he long and shor counerpary have o open and mainain a margin accoun ha is, deposi money ino he accoun when enering ino a conrac and susaining a minimum balance (op up as and when required) as deermined by he size of he fuures posiion. Example 2.1: A porfolio manager bough fify ALSI index-fuures on Monday morning wih he index a 3,. The nominal value of his purchase was (5 x R1 x 3,) which amouned o R15,,. The iniial margin was 5 x R18, (6% margin requiremen per conrac) oalling R9,. The selemen value of he index on Monday was 29,9 which generaed a selemen variaion of 5
Page4 FINANCIAL DERIVATIVES x R1 x (3, 29,9) oalling R5,. This amoun was removed from he accoun leaving a balance of R85,. Since he balance on his accoun exceeded he mainenance margin, [5 x 16, (mainenance margin per conrac) = R8,], no paymen ino he margin accoun was required. The selemen value on Tuesday was 29,75 and his generaed a selemen variaion of 5 x R1 x (29,9 29,75) oalling R75,. When his amoun was removed from he margin accoun, he remaining balance of R775, was below he mainenance margin requiremen of R8,. Therefore, he porfolio manager received a margin call and had o pay in an addiional R125, (variaion margin) o resore he margin accoun o he iniial level of R9,. Should he marke move in he porfolio manager s favour, any amoun in excess of he iniial margin can be wihdrawn. Example 2.2: Iniial fuures price = R1; Iniial margin = R5; Mainenance margin = R3 Accoun: Long posiion in 1 conracs Day BB Deposi SP Change Gain/Loss EB 5 1. 5 1 5 99.2 -.8-8 42 2 42 96. -3.2-32 1 3 1 4 11. 5. 5 1 4 1 13.5 2.5 25 125 5 125 13. -.5-5 12 6 12 14. 1. 1 13 Accoun: Shor posiion in 1 conracs Day BB Deposi SP Change Gain/Loss EB 5 1. 5 1 5 99.2 -.8 8 58 2 58 96. -3.2 32 9 3 9 11. 5. -5 4 4 4 13.5 2.5-25 15 5 15 35 13. -.5 5 55 6 55 14. 1. -1 45 The opposie effec of any change in he fuures price (SP selemen price) on he long and shor posiions is eviden from he wo accouns. The beginning (BB) and ending (EB) balances vary as money is deposied or removed. Noe ha he long posiion deposied R9 in oal and ended up wih a final balance of R13 o gain R4 from he conrac. The shor posiion deposied R85 in oal and recorded an ending-balance of R45 o loose R4 from he conrac. Whaever he underlying asse, aking an opposie fuures marke posiion in an opimal number of suiable conracs will neuralise a spo marke posiion. The following secion illusraes he ineracion beween spo and fuures prices and he maching bu opposie oucomes due o changing spo marke prices.
Page5 FORWARD AND FUTURES CONTRACTS 2 2.2.3 Hedge profiles The principle of hedging (fixing a purchase or sales price) is illusraed in Figure 2.2, showing ha any change in he value of an asse, owing o a changing spo price, is counered by an opposie change in he value of a fuures conrac. A shor posiion in he spo marke refers o he anicipaed purchase of an asse no currenly owned. A hypoheical loss would resul from any increase in he price of his asse as he invesor will have o pay his increased price once he asse is acually bough. Enering ino a long fuures conrac as shown in Figure 2.2 (long hedge), any loss in he spo marke is cancelled ou by an equivalen gain in he fuures marke, hereby locking in he price as sipulaed in he fuures conrac (inersecion of shor asse and long fuures profiles). + Long hedge Long fuures + Shor hedge Long asse Spo price (S) Spo price (S) Delivery price (K) Shor asse Delivery price (K) Shor fuures Figure 2.2 Hedge profiles A long posiion in he spo marke refers o he anicipaed sale of an asse currenly owned. A loss would resul from any decrease in he price of his asse as he invesor will have o accep his reduced price once he asse is acually sold. Enering ino a shor fuures conrac as shown in Figure 2.2 (shor hedge), any loss in he spo marke is cancelled ou by an equivalen gain in he fuures marke, hereby locking in he price as sipulaed in he fuures conrac. The oucomes in he spo marke and he fuures marke should be deermined independenly and need o arrive a he combined or final oucome. Example 2.3: An invesor bough 1, Dimension Daa (DDT) shares in he beginning of a year a R5.5 per share. A he ime of buying hese shares he also shored 1 March cash-seled single sock fuures conracs (DDQQ) a R6.2. A he expiraion of he SSF conrac, he spo price of DiDaa closed a R6.5. This hedge realised an assured profi of R7, (ignoring ransacion and opporuniy coss) compared o a likely unhedged profi of R1,.
Page6 FINANCIAL DERIVATIVES Spo marke oucome 1, x (6.5 5.5) R1, Fuures marke oucome 1 x 1 x (6.2 6.5) - R3, Ne resul (R6.2 per share locked in) R7, However, had he spo price declined o R5. (or any price below R6.2), he invesor sill would have profied R7, due o he shored fuures conrac. The shaded areas in Figure 2.2 represen he gains or losses from a fuures posiion. I should be eviden ha a loss in he spo marke affecs a gain in he fuures marke. Likewise, any gain in he spo marke is offse by a corresponding loss in he fuures marke. The fuures price conraced a is herefore he price effecively received or paid for a specific asse, regardless of wha ranspires in he spo marke. The invesor foregoes any poenial spo marke profis along wih he losses (ulimaely he raionale for hedging) and his fac should be acknowledged and acceped when conracing in he fuures marke. 2.2.4 Basis risk Saing ha an opposie fuures marke posiion in an opimal number of suiable conracs will neuralise a spo marke posiion, assumes ha he fuures conrac is derived direcly from he underlying being hedged. A perfec hedge will resul in a zero combined gain or loss o he invesor as he fuures price moves in unison (equal bu opposie changes) wih he spo price. However, should a subsiue fuures conrac (proxy) be used due o he unavailabiliy of any conracs based specifically on he underlying o be hedged, basis risk becomes an issue. Uncerainy regarding he exac dae he underlying will be bough or sold, and he close ou of a fuures conrac before expiraion also affec he final oucome. The basis is he amoun by which a spo price differs form he fuures price (b = S F) and basis risk is he uncerainy as o his difference beween he spo and fuures price. A consan difference, all oher facors equal or unchanged, will resul in a perfec hedge. The posiion of a shor hedger (long asse; shor fuures) improves as he basis increases or widens (spo price increases more han fuures price), and weakens when he basis decreases. The opposie is rue for long hedger. Example 2.4: The DiDaa-SSF case was an example of a near perfec hedge since he fuures conrac was derived from he underlying hedged. However, using an index fuures conrac o hedge he same DiDaa porfolio would inroduce basis risk and may resul in a possible unexpeced gain or loss o he invesor.
Page7 FORWARD AND FUTURES CONTRACTS 2 Assume he DiDaa porfolio of R55, (1, shares) can be hedged by selling weny-five ALMI (Mini ALSI) fuures conracs wih he Top4 Index (ALSI) level a 22,. Afer hree monhs and wih DiDaa a R6.5 he index level sood a 22,8. Spo marke oucome 1, x (6.5 5.5) R1, Fuures marke oucome 25 x (2,2 2,28) - R2, Ne resul R8, This hedge effecively resuled in a selling price of R6.3 per share. An ending index level of 23,2 would have resuled in an effecive price of R6.2 per share (same as SSF hedge). The spo price increased more (in relaive erms) han he index level (i.e., a srenghened basis) improving he posiion of he hedge (smaller loss). The reverse (weakened basis) would cause a larger fuures loss and smaller ne profi. Noes: Each single sock fuures conrac comprises one-hundred (1) underlying shares Each ALSI fuures is en (1) imes he index level; each ALMI conrac is one-enh (1/1) of he index Margin accouns have o be opened and mainained wih all fuures marke ransacions Examples exclude any commissions, brokerage, clearing fees and addiional margin percenages se by marke makers 2.2.5 Hedge raio Cross hedging occurs when he asse underlying he fuures conrac is differen o he asse whose price is being hedged (e.g., sock index fuures used o hedge a porfolio of socks). The hedge raio is he raio of he size of he fuures posiion o he size of he exposure and is he sandard deviaion of changes in he spo price (S) divided by he sandard deviaion of changes in he fuures price (F), muliplied by heir correlaion coefficien (). h S F (2.1) Example 2.5: A porfolio manager decides o hedge his equiy porfolio wih an index fuures conrac. The correlaion beween he porfolio value and fuures price is.8. The sandard deviaion of he porfolio value is 4 percen and ha of he fuures price is 38 percen per annum. 4 h.8.84 38 The porfolio manager should hedge 84 percen of he porfolio.
Page8 FINANCIAL DERIVATIVES 2.3 PRICING FORWARD AND FUTURES CONTRACTS The price (F) of forward and fuures conracs is largely deermined by he underlying spo price and he prevailing risk-free rae. Oher facors, namely cash flows (dividends or coupons), foreign risk-free raes (currency fuures), sorage coss (commodiies) and ransacion coss also play a role and require adjusmens o he basic pricing formula. Ineres rae forwards and fuures are priced differenly no according o he basic formula wih he underlying being a variable reference rae. 2.3.1 Basic pricing formula The forward or fuures price of a non-cash-flow domesic securiy (excluding ineres rae insrumens) is he spo price (S) adjused for he risk-free rae (r) and he period o expiry or delivery (T). Simply saed, i is he compounded spo price. The iniial price (F) is also referred o as he delivery price (K). rt F Se K (2.2) A any poin in ime afer iniiaion and before expiraion (), he conrac price is calculaed wih he hen curren spo price (S ) and he remaining period o expiraion (T ). Remaining period (T ) S S T F/K or F rt F S e (2.3) Any subsequen conrac price (P) deermines he value of ha conrac a ha poin in ime (applicable o forward conracs only). Should he acual or marke price deviae from his calculaed or heoreical price, an arbirage opporuniy cash and carry arbirage may be available (see Secion 2.5.1). Example 2.6: Suppose a nine-monh fuures (or forward) conrac on a non-dividend paying sock wih he share price a R6 and a 1% per annum (coninuously compounded) risk-free rae of ineres. The conrac price is:.19 12 F 6e R64.67 K
Page9 FORWARD AND FUTURES CONTRACTS 2 2.3.2 Equiy conracs A conrac on a sock paying a known cash dividend is priced afer adjusing for he dividend by subracing he presen value (PV) of any expeced dividends from he spo price before compounding a he relevan rae for ha period. F S PV D e rt (2.4) The same resul can be arrived a by subracing he fuure value (FV) from he compounded spo price. rt F Se FV D A conrac on a sock index and a porfolio of socks reurning a dividend yield (q) or assuming a dividend yield for a paricular sock sees he risk-free rae being reduced by he yield, resuling in a lower forward/fuures price. r qt Se F (2.5) Dividends cause he forward or fuures price o be lower compensaing for he fac ha he conrac holder, unlike he holder of he underlying sock, is no eniled o hese dividends. Example 2.7: Consider a sock priced a R6 which pays a dividend of R5 per share in wo monhs. The risk-free rae is 1 percen. A forward/fuures conrac expiring in nine monhs would have a price of:.12 12.19 12 F 6 5e e R59.37 Or alernaively:.1 9 12.17 12 F 6e 5e R59.37 Noe he differen ime periods used when in urn calculaing he presen value (2 monhs) or fuure value (7 monhs) of he dividends. Example 2.8: Consider a forward conrac on a sock index. The index level is on 5, he dividend yield is 2 percen and he risk-free rae is 8 percen. Wih he conrac life a wo years, he conrac price is:.8.22 F 5e 5637.48
Page1 FINANCIAL DERIVATIVES A coninuous dividend yield is commonly used when he underlying is a porfolio of socks or a sock index. 2.3.3 Fixed-income (Bond) conracs Coupon bonds issued in he capial marke (coupon bonds) rade a eiher a premium or discoun o par depending on he coupon rae versus marke rae relaion. Forward and fuures prices are derived from he underlying bond s price. A forward is based on a paricular bond, while fuures conracs have several qualifying bonds available for delivery he main disincion beween forward and fuures conracs on bonds. 1 Pricing fixed-income conracs is similar o pricing equiy conracs in ha he cash flows fixed coupon paymens in his insance have o be accommodaed in he calculaion. Accrued ineres complicaes he pricing slighly and a disincion is also made beween forward conracs and fuures conracs as here is a small variaion in pricing. C C T Cash price = quoed price + accrued ineres When a bond is sold in-beween coupon paymens (C), he seller is eniled o a pro-raa porion of he ineres o be received, in full, by he new owner. This pro-raa porion, called accrued ineres, is included in he selling price of he bond. The quoed price, also referred o as he clean price, is no he same as he cash price (which includes he accrued ineres), also referred o as he diry price. The forward or fuures conrac price is derived from he cash price (B). 2 Fixed-income: Timeline Y B B T F or F T+Y Mauriy of Bond T Mauriy of forward/fuures conrac Y Days remaining afer conrac expires Dae of valuaion T+Y Once he bond s cash price (iniial or inerim) is known, only he conrac period ( T) is significan o calculaing a forward/fuures price or forward value. Only coupons paid in his period are aken up in he conrac price and value. 1 In Souh Africa here is no such disincion as he JSE Limied (YieldX) only offers bond fuures on he following underlying Treasury Bonds (T-bonds) he R153, R157, R186, R21, R23, R24, R26, R28 and R29. 2 Prices are ypically quoed wihou he ineres ha has accrued since he las coupon dae, bu relevan examples in his chaper will always use he cash price.
Page11 FORWARD AND FUTURES CONTRACTS 2 Forward conracs A forward conrac on a coupon-paying bond is priced afer adjusing for he coupon by subracing he presen value (PV) of any coupons paid during he conrac period from he cash price before compounding a he relevan rae for ha period. F B PV C e rt (2.6) Any subsequen conrac price (F), accouning for any coupons in he period remaining, deermines he value of ha conrac a ha poin in ime. Example 2.9: Consider a bond wih semi-annual coupons. The bond has a curren mauriy of 583 days and pays four coupons, each six monhs apar. The nex coupon is paid in 37 days, followed by coupons in 219 days, 41 days, and 583 days, a which ime he principal is also reurned. Suppose ha his 4 percen coupon (R2 semi-annual paymens) bond s cash price is R984.45 and ha he risk-free ineres rae is 6 percen. Assume ha he forward conrac expires in 31 days and herefore ha he bond has 273 days remaining afer he forward conrac expires. 273 2 31 2 583 37 219 31 Noe ha only he firs wo coupons occur during he life of he forward conrac wih he coupons oal presen value being:.6 37 365.6 219 365 PV C 2e 2e R39.17 The forward price a iniiaion is:.6 31 365 F 984.45 39.17 e R994.7 Also noe ha he period remaining afer he conrac expires is no relevan o he calculaion. Only he forward conrac period and he coupons paid during ha period (and when) are of concern.
Page12 FINANCIAL DERIVATIVES Fuures conracs Even hough he Treasury bond fuures conrac is priced in similar fashion, he price needs o be aligned o he cheapes-o-deliver bond and his necessiaes he addiion of a conversion facor (CF). The pary wih he shor posiion can choose o deliver any bond ha has a mauriy as sipulaed in he fuures conrac. When his paricular bond is delivered, a parameer known as is conversion facor ses he price received for he bond. Once he shor pary has decided o deliver, he will choose he cheapes among he available bonds deermined by he mos recen selemen price and he conversion facor. This is in conras o a forward conrac which is based on a specific underlying bond. B PV C e rt (2.7) F Example 2.1: CF Calculae he price of a 1.2 year fuures conrac on a 7% coupon T-bond wih exacly 1 years o mauriy and a cash price of R1,4. The risk-free rae is 5 percen. Assume he cheapes o deliver bond has a conversion facor of 1.13. The semi-annual coupon is R35. A bondholder will receive wo coupons during he conrac erm ha is, a paymen.5 years and 1 year from now..5.5.5 1 PV C 35e 35e R67.43.5 14 67.43 e 1.2 F R913.9 1.13 2.3.4 Currency conracs Pricing (seing an exchange rae) a currency conrac is similar o a forward/fuures conrac on a porfolio of socks or sock index assuming a dividend yield. The foreign risk-free rae (rf) subsiue for he dividend yield (q) and is used in conjuncion wih he domesic risk-free rae (rd). F r r T d f Se (2.8) Should he foreign ineres rae exceed he domesic ineres rae, he forward exchange rae will be less han he spo exchange rae ha is, he domesic currency will appreciae agains he foreign currency and vice versa. Raes are quoed in erms of unis of he domesic currency per uni of he foreign currency. Example 2.11: Suppose he domesic currency is he Souh African Rand and he foreign currency is he U.S. dollar. Le he spo exchange rae be R7 per $, he Souh
Page13 FORWARD AND FUTURES CONTRACTS 2 African ineres rae be 6 percen, and he U.S. ineres rae be 3 percen. These raes are coninuously compounded and assumed o be fixed over he life of he conrac. The forward price (or rae) of a six-monh conrac would be:.6.36 12 F 7e R7.158 The forward rae indicaes a depreciaing domesic currency. Should he acual or marke price deviae from his calculaed or heoreical price, an arbirage opporuniy covered ineres arbirage may be available (see Secion 2.5.2). 2.3.5 Ineres rae conracs A forward ineres rae represens he cos of lending or borrowing money over a fuure period for which he rae of ineres is se oday. There are wo main derivaives of he forward ineres rae, namely he forward rae agreemen and shor-erm ineres rae fuures. Treasury bills are money marke (mauriies of less han one year) discoun insrumens. 9-day T-bill and LIBOR (or JIBAR) raes underlie ineres rae fuures conracs. However, T-bill fuures are no acively raded (even inernaionally) and he focus is on Eurodollar 3 fuures and JIBAR fuures. These fuures are in essence sandardised (9-day versions) forward rae agreemens (FRAs). Forward conracs A forward conrac on ineres raes is called a forward rae agreemen (FRA). A pary conracs o borrow or lend money a a cerain rae and on some fuure dae. A FRA enables a borrower or lender o fix an ineres rae which applies over a fuure period. There is no direc link beween he FRA and he underlying loan and his means ha a FRA canno be used o raise deb as he principal amoun is purely noional and never exchanged. Selemen is agains he relevan underlying reference rae (LIBOR or JIBAR) 4 and occurs a he sar of he fuure loan period when he FRA rae is compared o he underlying rae. A discouned cash selemen is made on any difference beween hese raes. Forward rae agreemens are usually used o proec he borrower agains rising ineres raes and o gain ineres rae cerainy on a porfolio of loans. A FRA may be used as a hedging insrumen o fix an ineres rae on a fuure loan. Assume ha a firm needs o borrow a cerain amoun in hree monhs ime for a six-monh period. The firm buys a 3x6 FRA o cover he ineres charged for his fuure period from a financial insiuion ha guaranees an ineres-rae cos of 5 percen for he six monhs of he loan. If in hree monh s ime he 6-monh JIBAR rae is more han 5 percen, he firm receives compensaion for he addiional ineres cos. If i is less han 5 percen, he firm mus pay he financial insiuion compensaion. The loan and FRA can be obained from he same or 3 LIBOR reflecs he rae on a Eurodollar deposi. 4 London Iner-Bank Offered Rae (LIBOR) variable rae a which inernaional banks borrow money from (lend money o) each oher. In Souh Africa he equivalen rae is referred o as he Johannesburg Iner-Bank Agreed Rae (JIBAR).
Page14 LIBOR R FINANCIAL DERIVATIVES differen insiuions ha s irrelevan as he compensaion paid or received effecively ensures an ineres-rae cos of 5 percen. Also used o ransform a curren obligaion from floaing o fixed or fixed o floaing (raes expeced o fall). A curren variable rae loan, for example, can be convered o a fixedrae loan a a fuure poin in ime and for a cerain period by buying a FRA. The borrower pays a fixed ineres rae (FRA rae) and receives a floaing ineres rae equal o a reference rae (underlying rae). Effecively swapping a floaing rae for a fixed rae an ineres rae swap is a series of FRAs over and exended period. 5 FRA rae (fixed) Borrower BANK LIBOR (floaing) Loan amoun A borrower previously enered ino a floaing-rae loan. In order o fix an ineres rae on ha loan for a cerain period, a long FRA is enered ino ha maures on a fuure dae. On ha dae, he payoff on he FRA (posiive or negaive) would effecively have convered he floaingrae loan o a fixed-rae loan a he FRA rae. Forward rae agreemens are efficien rading insrumens wih which o ake ineres rae risk as here is no requiremen o have an underlying exposure before a FRA can be raded. Banks use FRAs o ake direcional views on ineres raes and are buyers of FRAs if raes are anicipaed o rise and sellers of FRAs if raes are expeced o fall. I is a poen rading ool as here is no exchange of principal, and defaul risk failure of counerpary o honour obligaion is herefore limied o he selemen amoun. Forward rae agreemens are ofen used in he arbirage of pricing differences beween FRAs, shor-erm ineres rae fuures and forward ineres raes as hey all are essenially he same insrumen, based on he same underlying. The noaion applicable o forward rae agreemens: Noaion Selemen dae Mauriy dae Underlying rae 1x3 1 monh 3 monhs 2-monh JIBAR 1x7 1 monh 7 monhs 6-monh JIBAR 3x6 3 monhs 6 monhs 3-monh JIBAR 3x9 3 monhs 9 monhs 6-monh JIBAR 6x12 6 monhs 12 monhs 6-monh JIBAR 12x18 12 monhs 18 monhs 6-monh JIBAR 5 Refer o Chaper 6 Swaps Applicaion and Design for a deailed descripion on ineres rae swaps.
Page15 FORWARD AND FUTURES CONTRACTS 2 Any difference beween he FRA rae and he underlying rae accrue over a period saring on he dae he FRA maures (beginning of loan period) and alhough he paymen should only be made a he end of he loan period, i is already known a he beginning and herefore discouned o obain is curren payoff value. The payoff a expiraion is calculaed wih he numeraor being he end-of-period paymen and he denominaor a discoun facor: Udays Urae FRArae 36 FRApayoff NP Udays 1 Urae 36 Where: NP Noional principal FRArae Forward rae Urae Underlying marke rae (LIBOR) Udays Number of days in underlying (e.g., 9-day LIBOR) (2.9) Noe: The forward and underlying raes are measured wih a compounding frequency reflecing he lengh of he period hey apply o, corresponding o marke pracices for FRAs, and depars from he usual assumpion of coninuous compounding. Example 2.12: A borrower knows i will be borrowing R1 million in hree monhs and wishes o fix a borrowing rae oday as expecaions are ha raes are going o increase. Suppose a 3x6 FRA is bough a 11% and ha JIBAR is 12% a selemen. 9.12.11 36 FRApayoff 1 mil R24, 271.84 9 1.12 36 In hree monh s ime he company will borrow a he prevailing rae of 12 percen, bu he paymen received from he selemen will effecively reduce is borrowing cos o 11 percen. The FRA price or rae is calculaed as follows: h m 1 L 36 36 FRArae 1 h m 1 L 36 (2.1)
Page16 FINANCIAL DERIVATIVES FRA: Timeline m g h Iniiaion of conrac h+m h Day on which FRA expires m Number of days in underlying g Dae of valuaion Example 2.13: Consider a 3x9 FRA expires in hree monhs (9 days) and is based on 18- day JIBAR (9 3 = 6 monhs). The curren 9-day rae is 5.6 percen and he curren 27-day rae is quoed a 6 percen. Therefore, if he FRA is enered ino on day, he FRA rae (price) would be: 9 18 1.6 36 36 FRArae 1.611 9 18 1.56 36 Based on curren forward raes, a FRA rae of 6.11 percen would be available o poenial buyers or sellers adjused for and quoed as a spread. 6 Fuures conracs A 9-day ineres rae (LIBOR or JIBAR) fuures price is quoed as 1 minus he rae (%) priced ino he conrac by he fuures marke. This rae, 1 Rae is known inernaionally as he IMM (Inernaional Moneary Marke) Index. The LIBOR fuures conrac is based on $1 million noional principal and he underlying is he rae on a 9-day dollar-denominaed ime deposi issued by a bank in London (Eurodollar deposi). Each basis poin move is equivalen o $25. The 3 monh JIBAR fuures is based on R1, noional principal and 9-day JIBAR wih each basis poin equivalen o R2.5. Quoed price 1 Rae Rae 9 Acual pricep% NP 1 1 36 (2.11) Example 2.14: If he rae priced ino a JIBAR fuures conrac is 6.25 percen, calculae he quoed price and he acual fuures price. Also, calculae he fuures price should he rae go up o 6.5 percen. 6 Bid-ask spread he difference beween he price a which a marke maker is willing o buy a securiy (bid), and he price a which i is willing o sell i (ask).
Page17 FORWARD AND FUTURES CONTRACTS 2 Quoed price 1 6.25 93.75 9 P6.25% 1, 1.625 R98, 437.5 36 9 P6.5% 1, 1.65 R98, 375. 36 When he rae goes o 6.5 percen he quoed price decline o 93.5 and he acual fuures price drops o R98,375. Thus, he fuures price decreased by R(98,437.5 98,375) = R62.5. A rader who is long would have a loss of R62.5 and a rader who is shor would have a gain of R62.5. This change can be calculaed more direcly by noing ha each basis poin move is equivalen o R2.5. The rae increased by (65 625)bp = 25 basis poins and he resulan decrease in price is herefore R(25 x 2.5) = R62.5. 2.4 VALUING FORWARD CONTRACTS The value (f) of a forward conrac is deermined in relaion o he iniial forward price (F) or delivery price (K) as calculaed. Similar adjusmens o he basic valuaion formula are required o accoun for cash flows and foreign ineres raes. Recall ha, unlike a forward conrac, a fuures conrac has no accumulaed value during he conrac period due o he marking o marke process, reseing he conrac value o zero a he end of each rading day. 2.4.1 Basic valuaion formula The forward conrac value (f) a any poin in ime afer iniiaion (and before expiraion) is he hen curren spo price minus he discouned delivery price (F or K). The delivery price is discouned a he rae and for he period remaining a ha ime. rt f S Ke (2.12) Same resul obained by discouning he difference beween he iniial forward price (F or K) and any subsequen forward price (F) before expiry. f F K e r T This value can be posiive or negaive depending on he posiion (long or shor) aken in he fuures conrac wih any close ou or reversal requiring he opposie ransacion (i.e., sell when previously bough or vice versa). The iniial value (a iniiaion of he conrac) is se o be zero while he ending value is he difference beween he delivery price (F or K) and he spo price a expiry as he forward price converges o and finally equals he spo price a expiry. f S K (2.13) T T
Page18 FINANCIAL DERIVATIVES Example 2.15: Suppose a nine-monh forward conrac on a non-dividend paying sock when he share price is R6 and he risk-free ineres rae is 1% per annum. An invesor has jus aken a shor posiion in his conrac. Three monhs laer, he price of he share is R55 and he risk-free ineres rae is sill 1% per annum. The value of his forward conrac is calculaed as follows: The iniial forward price was deermined o be R64.67. Iniial value = (se o be zero a he iniiaion of he conrac) Curren forward price (3 monhs ino he conrac wih 6 monhs remaining):.16 12 F 55e R57.82 The value of his fuures conrac o he shor posiion holder:.16 12 f 55 64.67e R6.52 Or alernaively:.1 6 12 f 57.82 64.67 e R6.52 The iniial conrac was sold a R64.67 and can be reversed (closed ou) by buying i back a R57.82, represening a gain of R6.52 in presen value erms. A long posiion in he conrac would have realised a loss of R6.52. 2.4.2 Equiy conracs The value of a forward conrac on a sock paying a known cash dividend is he difference beween he curren spo price (adjused for any dividends) and he discouned iniial fuures price. rt f S PV D F e (2.14) The delivery price (F) is discouned back o ime dae of valuaion. Any dividends falling in he remaining period (T ) is accouned for in he spo price. Example 2.16: Consider a sock priced a R6 which pays a dividend of R5 per share in wo monhs. The risk-free rae is 1 percen. A forward conrac expiring in nine monhs was priced a R59.37. One monh laer he spo price is R62 and he risk-free rae is sill 1 percen. Wha are he forward price and he value of he conrac a his sage?
Page19 F S PV D e rt.1112.18 12 62 5e e R6.97.1112.18 12 f 62 5e 59.37e R1.5 FORWARD AND FUTURES CONTRACTS 2 The value of his conrac is R1.5 (posiive o he long and negaive o he shor). Noe ha one monh ino he conrac he dividend is expeced in one monh s ime and ha he remaining period is eigh monhs. A forward conrac on a porfolio of socks or sock index paying a dividend yield is calculaed in similar fashion wih he spo discouned a he dividend yield (q) and he delivery price a he risk-free rae (r). Example 2.17: qt rt f S e F e (2.15) Consider a forward conrac on a sock index. The index level is on 5, he dividend yield is 2 percen and he risk-free rae is 8 percen. Wih he conrac life a wo years, he conrac price was deermined o be 5637.48. Afer nine monhs, he index level is a 48 (dividend yield sill 2%) and he risk-free rae increased o 8.5 percen. Wha are he forward price and he value of he conrac a his sage? F S e rqt.85.2 15 12 48e 526.28.215 12.8515 12 f 48e 5637.48e 387.73 The value of his conrac is 387.73 (negaive o he long and posiive o he shor). Noe ha all levels and values should be muliplied by he relevan index muliplier o be saed in Rand erms. 2.4.3 Fixed-income (Bond) conracs The value of a forward conrac on a coupon-paying bond is he difference beween he curren bond cash price (adjused for he coupons) and he discouned iniial fuures price. The delivery price (F) is discouned back o ime dae of valuaion. Any coupons falling in he remaining period (T ) is accouned for in he cash price. rt f B PV C F e (2.16)
Page2 FINANCIAL DERIVATIVES Example 2.18: Consider a bond wih semi-annual coupons. The bond has a curren mauriy of 583 days and pays four coupons, each six monh apar. The nex coupon is paid in 37 days, followed by coupons in 219 days, 41 days, and 583 days, a which ime he principal is also reurned. Suppose ha his 4 percen coupon (R2 semi-annual paymens) bond s cash price is R984.45 and ha he risk-free ineres rae is 6 percen. Assume ha he forward conrac expires in 31 days and herefore ha he bond has 273 days remaining afer he forward conrac expires. Fify days laer, wih he iniial forward price as calculaed on R994.7 and he cash price a R971.36, wha are he forward price and he value of his conrac? Noe ha afer fify days only he second coupon remains during he life of he conrac. The forward price and value a ime is: F B PV C e rt.6169 365.626 365 971.36 2e e R993.47.6169 365.626 365 f 971.36 2e 994.7e R1.17 The value of his conrac is R1.17 (posiive o he shor pary). 2.4.4 Currency conracs A forward conrac on a currency is calculaed similarly o ha of a porfolio of socks or sock index wih he foreign risk-free (rf) rae subsiuing for he dividend yield (q). The spo rae is discouned a he foreign risk-free rae and he iniial forward rae a he domesic risk-free rae (rd). Example 2.19: rf T rd T f S e F e (2.17) Suppose he domesic currency is he Souh African Rand and he foreign currency is he U.S. dollar. Le he spo exchange rae be R7 per $, he Souh African ineres rae be 6 percen, and he U.S. ineres rae be 3 percen. These raes are coninuously compounded and assumed o be fixed over he life of he conrac. The forward price (or rae) of a six-monh conrac is R7.158. I is now four monhs laer and he spo rae is R7.2 wih ineres raes fixed. Calculae he forward rae and conrac value a his ime. F S e rdr ft.6.32 12 7.2e R7.2361
Page21 FORWARD AND FUTURES CONTRACTS 2.32 12.62 12 f 7.2e 7.158e R.129 The value of his conrac is R.129 (posiive o he long pary). 2.4.5 Ineres rae conracs The value of a forward rae agreemen (FRA) a a specific poin in ime (day g) beween iniiaion (day ) and expiraion (day h) is he presen value of R1 o be received a expiraion minus he presen value of 1 plus he FRA rae o be received a he mauriy dae of he JIBAR (or LIBOR) rae on which he FRA is based, wih appropriae period adjusmens. FRA: Timeline m g h Iniiaion of conrac h+m h Day on which FRA expires m Number of days in underlying g Dae of valuaion Calculaed as follows: m 1 FRArae 1 36 Vg h g h m g 1 L g 1 L g 36 36 (2.18) Example 2.2: Consider a 3x9 FRA expires in hree monhs (9 days) and is based on 18- day JIBAR (9 3 = 6 monhs). The curren 9-day rae is 5.6 percen and he 27-day rae is quoed a 6 percen. The price (FRA rae) was deermined o be 6.11 percen. Suppose one goes long he FRA (day ) and afer 25 days he 65-day rae is 5.9 percen, and he 245-day rae is 6.5 percen. Based on he new erm srucure, he value of his FRA is: 18 1.611 1 36 Vg.26 9 25 9 18 25 1.59 1.65 36 36 A ime g (25 days ino he FRA conrac) he value of his FRA is R.26 per R1 noional principal. The acual principal is muliplied by he FRA value o obain he full marke value of he FRA (e.g., R1 million x.26 = R26,).
Page22 FINANCIAL DERIVATIVES 2.5 ARBITRAGE Arbirage is he simulaneous purchase and sale of idenical or equivalen financial asses in order o benefi from a discrepancy in heir price relaionship. Cash-and-carry arbirage involves simulaneously ransacing in he fuures marke and spo marke, while covered ineres arbirage explois deviaions from ineres rae pariy beween currencies. 2.5.1 Cash-and-carry arbirage Any deviaion from he heoreical or fair value as calculaed my lead o a specific arbirage sraegy exploiing and profiing from his discrepancy. The basic principle is illusraed keeping in mind ha any acual arbirage opporuniy is subjec o ransacion coss, resricions on shor selling, and differenial borrowing and lending raes. Wih a marke price (P) exceeding he heoreical price (F), cash-and-carry arbirage is execued. Cash and carry arbirage (P > F): Sell he fuures conrac a he quoed marke price (P) Borrow money a he risk-free rae for he period unil expiry Buy he underlying a he spo price Wih a marke price (P) lower han he heoreical price (F), reverse cash-andcarry arbirage is execued. Reverse cash and carry arbirage (F > P): Buy he fuures conrac a he quoed marke price (P) Sell he underlying a he spo price Inves or lend he money a he risk-free rae for he period unil expiry The principle of buying low and selling high applies. Wih cash-and-carry arbirage he overpriced (oo expensive) fuures conrac is sold and he underlying asse bough wih borrowed funds. Reverse cash-and-carry involves buying he underpriced (inexpensive) fuures conrac, selling he underlying and invesing he proceeds. Example 2.21: Suppose a nine-monh fuures conrac on a non-dividend paying share wih a share price of R6 and a risk-free ineres rae of 1% per annum (as above). The fuures price (F) was calculaed o be R64.67 (heoreical or fair value). Assume ha he acual fuures price (P) available in he marke is (i) R65 and (ii) R64. Ignoring all ransacion and oher relevan coss, he appropriae arbirage sraegy and profi are deermined as follows:
Page23 (i) Cash and carry arbirage (65 > 64.67): FORWARD AND FUTURES CONTRACTS 2 Sell he fuures conrac a R65 Borrow R6 a 1% for he nine monhs (owes R64.67) Buy he underlying share a R6 Afer nine monhs, repay he loan and deliver he share under he conrac, receiving R65 and realising a profi of R.33 (difference beween acual and fair price). (ii) Reverse cash and carry arbirage (64.67 > 64): Buy he fuures conrac a R64 Sell he underlying share a R6 Inves R6 (proceeds) a 1% for nine monhs (owns R64.67) Afer nine monhs, wihdraw he invesmen and ake delivery of he share under he conrac, paying R64 and realising a profi of R.67 (fair price minus acual price). Covered ineres arbirage explois deviaions from ineres rae pariy beween he currencies of differen counries. Local ineres raes deermine he exchange rae and purchasing power of currencies. 2.5.2 Covered ineres arbirage Covered ineres arbirage is performed o benefi from a mismach in currency exchange raes due o he violaion of ineres rae pariy he relaionship beween exchange raes of wo counries and heir local ineres raes. According o his concep, he difference beween he marke ineres raes in any wo counries is abou he same as he difference beween he forward and he spo exchange raes of heir respecive currencies The ineres rae pariy (IRP) equaion as previously: F r r T d f Se Rearranged in order o deermine any arbirage opporuniy: e F e S rt d rt f (2.19) The domesic risk-free rae mus equal o he hedged foreign risk-free rae o preven any arbirage. Any discrepancy will be exploied via a specific series of ransacions deermined by he relaionship beween he unhedged (domesic) and hedged ineres raes. The low-rae currency is borrowed, convered o he high-rae currency and invesed.
Page24 FINANCIAL DERIVATIVES Example 2.22: 1-year forward rae (USD/GBP) $1.58/ Spo rae (USD/GBP) $1.6/ U.S. 1-year rae (C1) 6% U.K. 1-year rae (C2) 1% Use he daa as given and idenify wheher or no an arbirage opporuniy exiss. If one does exis, consruc he appropriae sraegy and compue he arbirage profis (le C1 be he domesic currency). Sep 1: Idenify he exisence of an arbirage opporuniy..6 1.1 1 1.58 e e 1.6 1.618 1.914 Borrow dollars (C1) Lend pounds (C2) Sep 2: Idenify and arrange he appropriae sraegy The USD ineres rae is less han he USD-adjused GBP rae (hedged foreign rae). One herefore should borrow USD (a he low rae); conver o GBP and inves he GBP (a he high rae); and a he end of he year, conver back o USD a he forward rae, and pay off he loan wih he proceeds. Sep 3: Consruc he rades necessary o execue he arbirage sraegy Borrow $1 for one year a 6% and repay $1.618 in one year Exchange $1 for.625 (1/1.6) a he curren exchange rae Lend pounds for one year a 1 percen and receive.625e (.1x1) =.697 in one year Ener ino a forward agreemen o exchange.697 for dollars a he forward rae of $1.58/ Wai one year and collec.697 x 1.58 = $1.914 o repay he $1.618 loan amoun Sep 4: Compue he arbirage profis Arbirage profis are $(1.914 1.618) = $.296 2.6 SUMMARY This chaper covered he pricing (forwards and fuures) and valuing (forwards only) of he mos basic derivaive insrumens on equiy, bonds, ineres raes and currencies. The chaper o follow looks a he acual applicaion of hese insrumens in alering risk and spo marke exposure.
Page25 FORWARD AND FUTURES CONTRACTS 2 Pracice Quesions 2.1 Assume ha you own a dividend-paying sock currenly worh R15. You plan o sell he sock in 25 days. In order o hedge agains a possible price decline, you wish o ake a shor posiion in a forward conrac ha expires in 25 days. The risk-free rae is 5 percen. Over he nex 25 days, he sock will pay dividends according o he following schedule: Days o nex dividend Dividends per share 3 R1.25 12 R1.25 21 R1.25 (i) Calculae he forward price of a conrac esablished oday and expiring in 25 days. (ii) I is now 1 days since you enered he forward conrac. The sock price is R115. Calculae he value of he forward conrac a his poin (gain or loss?). (iii) A expiraion, he price of he sock is R13. Calculae he value of he conrac a expiraion (gain or loss?). 2.2 An invesor purchased a newly issued bond wih a mauriy of 1 years 2 days ago. The bond carries a coupon rae of 9 percen paid semi-annually and has a face value of R1,. The price of he bond wih accrued ineres is currenly R1,146.92. The invesor plans o sell he bond 365 days from now. The schedule of coupon paymens over he firs wo years, from he dae of he purchase, is as follows: Coupon Days afer Purchase Amoun Firs 181 R45 Second 365 R45 Third 547 R45 Fourh 73 R45 (i) Calculae he no-arbirage price a which he invesor should ener he forward conrac. Assume ha he risk-free rae is 6 percen. (ii) The forward conrac is now 18 days old. Ineres raes have fallen sharply, and he risk-free rae is 4 percen. The price of he bond wih accrued ineres is now R1,32.26. Deermine he value of he forward now and indicae wheher he invesor has accrued a gain or loss on his posiion. 2.3 Suppose ha a pary waned o ener ino a forward rae agreemen (FRA) ha expires in 42 days and is based on 137-day JIBAR. The dealer quoes a rae of 4.25 percen on his FRA. Assume ha a expiraion, he 137-day JIBAR is 5 percen and he noional principal is R2,,. Calculae he FRA payoff o a long posion.
Page26 FINANCIAL DERIVATIVES Soluions 2.1 Forward conrac on a dividend-paying sock (i) Iniial forward price.5 3 365.5 12 365.5 21 365 PV D 1.25e 1.25e 1.25e R3.69 rt.5 25 365 F S PV D e 15 3.69 e R151.41 K (ii) Value of he forward conrac.5 2 365.5 11 365 PV D 1.25e 1.25e R2.48 rt.5 15 365 f S PV D F e 115 2.48 151.41e R35.81 Negaive value gain o he shor (iii) Value of he conrac a expiraion ft ST F 13 151.41 $21.41 Conrac expires wih a negaive value gain o he shor 2.2 Forward conrac on a coupon bond B B 45 45 f F 181 365 547 73 165 days 167 days Afer 2 days Afer 38 days Afer 565 days (i) Iniial forward price.6 165 365.6 347 365 PV C 45e 45e R86.3 rt.6 365 365 F B PV C e 1,146.92 86.3 e R1,126.21
Page27 FORWARD AND FUTURES CONTRACTS 2 (ii) Value of he forward conrac.4167 365 PV C 45e R44.18 rt f B PV C F e 1, 32.26 44.18 1126.21e R154.47 Posiive value loss o shor posiion 2.3 Forward rae agreemen.4185 365 FRA payoff NP Udays Urae FRArae 36 Udays 1 Urae 36 137.5.425 36 2 million R56, 17.45 137 1.5 36 Posiive value gain o he long posiion