Hedging Portfolios with Short ETFs

Transcription

1 Hedging Pofolios wih Sho EFs hosen Michalik, Deusche Bank AG Leo Schube, Consance Univesiy of Applied Sciences Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

2 Absac Fund Managemen oday uses he acive and passive way o consuc a pofolio. Exchange aded Funds (EFs) ae cheap insumens o cove he passive managed pa of he invesmen. EFs exis fo sock-, bond- and commodiy makes. In mos cases he undelying of an EF is an Index. Besides he invesmen in EFs, fo some makes, sho EFs ae lised. Sho EFs allow funds manage o ean in beaish makes and heefoe, sho EFs offe a compeiive hedging possibiliy. o ge some insighs in he value of sho EF as insumen fo pefec hedging, empiical daa of he Geman sock index DAX ae used. Obviously, using sho EF fo hedging canno compleely neualize losses of he undelying insumen. he coss hedge of an individual pofolio by ShoDAX EF depiced a song isk educion. As isk measues, he vaiance, he absolue deviaion and some diffeen age-shofall pobabiliies ae applied. o find efficien pofolios fo he coss hedge, wo algoihms wee developed, which need no linea o mixed inege opimizaion sofwae. Resumen Hoy en día, la gesión de fondos emplea el méodo acivo y pasivo paa la adminisación de caeas. Los fondos coizados en bolsa (EF po sus siglas en inglés) son insumenos muy económicos que pemien cubi la pae pasiva de la invesión. Esos fondos exisen paa acciones, bonos y mecados de maeias pimas. En la mayoía de los casos, el valo subyacene de un EF es un Índice. Además de las invesions en EF, algunos mecados uilizan ambién EF invesos (o sho), que aumenan las ganancias de los gesoes en mecados a la baja y po ano, ofecen una posibilidad de cobeua muy compeiiva. Paa compende mejo el valo de esos EF invesos como insumenos de una cobeua pefeca, uilizamos daos empíicos del índice DAX de la Bolsa alemana. Obviamene, uiliza EF invesos paa la cobeua no puede neualiza compleamene las pédidas del insumeno subyacene. La cobeua cuzada de una caea individual empleando el ShoDAX EF mosó una gan educción del iesgo. Como medidas de iesgo, usamos la vaianza, la desviación absolua y algunas Pobabilidades Objeivo-Fallo, es deci, de obene menos que el objeivo espeado (age Shofall Pobabiliy, SP). Con el fin de encona caeas eficienes paa la cobeua cuzada, desaollamos dos algoimos que no pecisan de sofwae de opimización lineal y lineal enea mixa. Key-Wods: Pofolio Opimizaion, Hedging, Insuance and Immunizaion of pofolios, sho Exchange aded Funds (EFs), Mean Absolue deviaion Pofolios, Mean - age-shofall- Pobabiliy Pofolios. Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

3 . Inoducion Global financial makes seem o offe many insumens o ceae a well divesified pofolio in imes of gowing volailiy. o paicipae in hese makes i is necessay o have acual infomaion abou he companies o secos. o ge hem always in he igh momen is ime consuming. heefoe i is difficuly o manage a global divesified sock pofolio in an acive way and in many cases his pofolios wee less successful han he make index. o bea he make index as benchmak, fund manage oday manage only 0-40% of he budge in an acive way and he main pa by passive pofolio managemen. his combinaion is denoed as Coe-Saellie pofolio managemen 3. Insead of index acking, like some yeas ago, fund manages oday buy he index of a make o seco. Exchange aded Funds (EF) is a financial insumen which offes one of he efficien ways o do his so called indexing. As hey ae aded evey day a he exchange boad and poduce low managemen fees, hey ae an ineesing invesmen possibiliy fo pivae invesos, oo. Concening he isk, EF can be denoes as safe as a single sock of his make. An EF is a econsuced index o, if in some egions o secos no index exiss, of an adequae pofolio whose sucue is publicized. EF exis as long EF, which can be used e.g. o buy an index, o as sho EF. he sho vesion is consuced o incease in is value in imes when he value of he undelying is deceasing (a deailed discussion of his elaionship will be done lae). heefoe sho EF can be used fo hedging a pofolio. his insumen has some emakable advanages besides is hedging capaciy: hee is no duaion like in he case of opions o fuues. In he case of fuues, he inveso has o pay he so called combi -pice fo evey polongaion o olling he hedge fowad 4. his pice includes he buying of he old pu and selling one wih a lae expiaion dae. he sho EF do no consiue a conacual claim like OC opions, fowads and sucued poducs. heefoe EFs ae no in he bankup esae in he case of bankupcy of he financial insiue which issued he insumen. Sho EF as invesmen insumen exis wih and wihou leveage effec. In many counies, he insuance fund of insuance companies mus no be invesed in insumens wih such leveage effecs which offe he use of sho fuues o opions. he managemen fee of an EF is low compaed wih ohe insumens fo indexing. his paymen is fo a long EF abou 0.5% - 0.6% pe annum and fo a sho EF abou 0.3% - 0.8%. Fo some EFs exiss a spead beween he pice o buy and o sell. hese EFs should be used fo long em invesmens. In he following he diffeen hedging appoaches ae depiced befoe he sho EF and is hedging chaace will be exploed.. Hedging, Insuance and Immunizaion o poec a pofolio agains losses, diffeen financial insumens wee developed. Opions and fuues ae some old and famous examples. he pocess of hedging can be defined in a vey geneal way as he empoay compensaion of losses of one o moe asses by he pofi of one o some ohe invesmens. In a well divesified pofolio he included asses have o do his compensaion, bu no in a empoay way. heefoe he consucion of a pofolio wih minimized eun vaiance is denoed as divesificaion and no as hedging (alhough i educes isk by compensaion). he empiical measued negaive coelaion of sock euns nomally does no shofall Many pais of asses have such weak compensaion poenial. In he popely sense, hedging insumens fo an asse, index o pofolio, should have a moe negaive coelaion wih he eun of his asse, index o pofolio. he coelaion of he euns of sho selling o a sho fuue wih he eun of he undelying is exacly o vey closed o -. hese hedging insumens offe a pefec hedge o compensaion of losses and on he ohe side of pofis, oo. hey immunize he value of he undelying asse agains any developmen in is pice. heefoe, his hedging ype is called immunizaion 5. In he case of an individual pofolio, he undelying of he hedging insumen in 3 Singleon C., (004). 4 See Hull, J., (99), pp. 03f. 5 Immunizaion is ofen associaed wih he hedge of bond pofolios agains changes in he yield o make (see Faell J. L., (997), p.46), bu i can also be applied fo he pefec hedge of sock pofolios. 3 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

4 mos cases won be conguen wih his pofolio. heefoe he opimal quaniy x* of he hedging insumens fo his pofolio mus be deemined by opimizaion. he case of inconguen 6 hedging insumens is called coss hedge. If an inveso likes o hedge his pofolio agains losses bu believes ha he make could also be bullish, he will use only x < x* of he hedging insumen. his case is called nomal hedge. When he believes moe in a beaish make, he will apply x > x* o hedge his asses. his poecion is denoed as evesed hedge 7. As shown in Figue, he quaniy x has an influence o he pofis, oo. When he poecion should esic he losses while he chance o achieve pofis should emain, his ype of hedge is denoed as pofolio insuance 8. his hedge exiss e.g. when an inveso buys a pu opion o hedge he value of an asse. If he elaionship of he pu opion and he undelying asse which should be hedged is fixed by e.g. : i is called fix hedge 9. If he elaionship is changing ove he ime, a dynamic way of hedging is applied. An example fo a dynamic hedge is he dela hedge. his means, ha he numbe of pus will be changed duing he duaion 0. he aim is, o have always exacly he numbe of pus which ae necessay o compensae he change of he asse value by he change of his numbe of pu pice. A consequence of his saegy is ha pofis ae handled in he same way. heefoe, dela hedging can be denoed as a kind of immunizaion. Ohe dynamically saegies which ae founded on he changing of he volume of socks and of cash ae someimes called pofolio insuance-elaed. In Figue he elaionship of he diffeen hedging appoaches and he value of an asse (pofolio o index) which should be poeced is shown as linea alhough he elaionships has in some cases weak nonlinea defomaions. hedged asse value insuance pefec hedge fix hedge evesed hedge (x>x*) coss hedge (x=x*) nomal hedge (x<x*) asse unhedged Figue : Hedging-appoaches immunizaion asse value Hedge-EF-.dsf 0709 he hedging capaciy of a sho EF was no included in he discussion of he adiional hedging classes. his will be done moe deailed in he following. As example, he Geman Sock index DAX will be hedged by he ShoDAX EF o ge some insighs o he behavio of sho EFs. 6 he inconguence can exis due o diffeen duaions, qualiies and he quaniies (as deivaive insumens a he exchange boads mus be bough as packages, he inveso has o decide, how much packages he needs fo an opimal coss hedge). 7 See Spemann, K., (006), p See Elon E. J., Gube M. J., (99), pp. 598,599 o Spemann, K., (006), pp. 6, he adequae pendan of fixed hedging would be vaiable hedging insead of dynamic hedging. heefoe, he duo e.g. saic and dynamic would be moe consequen. 0 Alhough he dela of he Fuues pice is quie closed o, some invesos may use fuues o ge a dela hedge (see Hull J., (99), p. 88). Fuhemoe, a pofolio of diffeen Opions can be managed by he dela o hedge is value dynamically (see Hull J., (99), p. 88f). Faell, J. L., (997), pp. 94f. 4 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

5 3. Sho Exchange aded Funds he sho EF called ShoDAX was inoduced in he Geman exchange boads a he 6h of June 007. I is linked o he Geman Index DAX. he DAX conains socks of he 30 bigges companies of he couny whose socks ae lised in exchange boads. o buy his sho EF, he inveso has o pay is pice which will be denoed by ShoDAX. he financial insiuion which issued he ShoDAX now has o make a sho selling of he index DAX o ceae an invese pefomance wih a beaish chaace. By doing his he financial insiuion eceives a second ime he same amoun, he inveso paid o i. heefoe, he double value of he ShoDAX can be invesed by he financial insiuion o ean ineess fo he inveso. When he las ading day of he EF was δ days befoe, he will ge wice he inees ae i fo he ime ineval (-δ). Duing he week, δ = and a he weekend δ = o 3. he annual inees ae i which consiues he inees em of he equaion () is he Euopean Ovenigh Inees Aveage (EONIA) inees ae which is linea adaped in () o δ days. Besides he inees em, he value ShoDAX in peiod depends on he developmen of he value of he index DAX in he ineval -δ. his pa is called leveage em. When he DAX -δ will have a decease of 0% in -δ he ShoDAX -δ will incease by 0%. he faco of he leveage em would be: (-90/00) = (-0.9) =.0. When he DAX -δ will have an incease of 0% in -δ he evese will happen o he ShoDAX -δ. Fo he leveage faco of -, he ShoDAX -δ will ise o he double value, if he pice DAX -δ is falling beween -δ and o he value of zeo 3. On he ohe side, in his em he ShoDAX -δ will fall o he value zeo (in he Leveage em in ()), if he pice DAX -δ is ising o he double pice beween -δ and. he sum of he leveage em and he inees em consiue he value ShoDAX a ime (see equaion ()) 4. ShoDAX DAX i ShoDAX -δ + ShoDAX δ δ DAX. () δ 360 Leveage em Inees em = In he following, he value of he sho EF will be denoed by S and he index by I and fuhe moe he ime sep δ is supposed o be δ =. hen he value in of he sho EF fo one ime sep is ( - i /80) S = S - + () In equaion () is he eun of he index achieved beween and he las ading day -. In he example above, his ae was = If he iniial value is S 0, equaion () can be used o compue S, S ec. Afe days, he value of he sho EF is S = S 0 = i +. (3) 80 he value of he Index which sas wih I 0 will be in I ( + ) = I0. (4) = he EONIA is fixed by he Euopean Cenal Bank since I is he aveage of all ovenigh unsecued lending ansacions in he inebank make. 3 Highe leveage facos (e.g. -) can be found fo EFs (e.g. LevDAX (see Deusche Böse, 007, p. 3)) bu also fo Sho EFs (e.g. Ulasho Dow30 Poshaes (see Mogan Sanley, 007 o Fei, R. A., (008), pp. 6f)). he equaion () fo highe leveage facos can be found in he appendix III. 4 See Deusche Böse, 007, pp. 4f. 5 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

6 4. Hedging wih Sho EF 4. Immunizaion by a Pefec Hedge he value of an asse can be hedged by a sho fuue. If his asse is idenical concening volume, quaniy and qualiy wih he undelying of he fuue, a pefec hedge is possible. hen a isk fee payoff is possible like shown by a hoizonal line in Figue. he payoff level depends on he basis as he diffeence of he spo pice p Spo and he fuues pice p Fuue is called. When insead of sho fuues he way of sho selling was used fo hedging he asse, he payoff is he diffeence beween he ineess which can be eaned ill and he fee fo boowing he asse. In boh cases he pefec hedge can ceae a negaive payoff, oo. payoff asse long isk fee payoff p Spo p Fuue asse pice a mauiy fuues sho Hedge-EF-.dsf 0709 Figue : Pefec Hedge using fuues Sho EF as hedging insumen can poduce a simila fixed payoff, bu only, when =. hen he payoff is he inees paymen i /360. his pefec hedge h is he sum of is elemens I and S which wee in he beginning equal weighed especive ge he same pa of he budge (I 0 = S 0 ). Wih he equaions (3) and (4) he esul of he pefec hedge is i h = I + S = I0 ( + ) + S0 +. (5) 80 As he sho EF has he chaace of an invesmen insumen, 50% of he budge will be invesed in I 0 and 50% in S 0 o immunize he invesmen agains changing index pices. Now, equaion (5) can be educed o i i = + h = (6) Fom a heoeical poin of view (especive in absence of ansacion coss) his immunisaion can be applied fo >. o be evey ading day pefec hedged, he esul of h mus be ebalanced o 50% of he acual budge o I and 50% fo S. hen he equaion (6) mus be compleed by - inees facos o h i (7) = 360 = In equaion (7) he value of he hedge is only dependen on he inees paymens and no on he value of he index DAX. Bu besides he weak simplificaion ha δ = a songe one was supposed o ge his esul. his songe simplificaion ignoes he fac of ansacion coss. 6 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

7 In he following he hedge of he sho EF will depiced by empiical daa. As menioned above, he ShoDAX was inoduced a he exchange boad a he 6h June 007. o ge values of he las decade (s of Januay 000 o he 4h of Apil 009), he eun daa of he hedge wee deemined by he values of he DAX and he EONIA. he annual managemen fee of 0.5% was no inegaed in he calculaion of he value of he ShoDAX.,0 Relaiv value of he ShoDAX EF and he DAX Disanz of ading day; iniial value:.0,05,00 --ShoDAX,95,90,85,90,95,00,05,0,5 --DAX File: EF-SPSS-Gafiken Figue 3: Value of he ShoDAX dependen on he DAX wih = he immunizaion fo = is depiced in Figue 3. Obviously he value of he ShoDAX depends on he value of he DAX like in he Figue he sho fuues on he asse value. Fo evey new ading day, he sum of he value of he DAX and he ShoDAX wee se o especive 00%. heefoe Figue 3 shows he %-movemen of boh insumens. No visible is he small shif of he inees paymen. In Figue 4 his paymen was depiced fo he ime,000 Reun of he Hedge and he value of he DAX Disanz of ading day (>= calenda days); eun,0008, Reun-hedged,0004,000 0,0000,9,0,0,,, --DAX File: EF-SPSS-Gaf iken Figue 4: Reun of he ShoDAX and he elaive value of he DAX wihin one ading day (=) peiod of one ading day. Compaed wih Figue, he isk fee paymen is no a hoizonal line. Due o diffeen EONIA inees aes i and values δ, his paymen is a cloud of poins and no a line. Bu as inees paymens ae posiive, he isk fee esul is i oo. As he ime sep δ is no always one, highe euns occued. he highes one of efes o δ =5 when he EONIA was abou 360( /5) =5.75% (a 3 h of Apil 00). Fo a longe invesmen o hedging ime >, his immunizaion of he DAX by he ShoDAX becomes impefec. Like he Figues 5 shows, he eun of he hedge h = I + S wih I and S like in (4) especive (3) was in he las decade fo in he ange of -6% o +3% while he eun of he DAX was beween -57% and +8%. he esuls depend on he size of he ime ineval and on he sandad deviaion of he eun of he index (see below). he ineval efes in he compuaion of Figues 5a o 5e o calenda days and no o ading days. he posiive as he negaive euns of he hedge ae 7 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

8 gowing by. he menioned exeme values can be found in Figue 5e when he ime ineval was se o =300. In a shoe ime ineval e.g. =50, he hedge h ceaes euns beween -3% and +8%. he plos of Figue 5 ae sickle-shaped. his means, exeme posiive euns as well as exeme negaive euns of he DAX offe he possibiliy o ge posiive euns by he hedge. Supising ae he minima of hese sickle shaped clouds. While he minima of he uppe side of he cloud seem o be a he poin when he DAX eun is zeo, he minima a he lowe side has is posiion in evey Figue when he DAX poduces a negaive esul beween 0 and -0%. his can be inepeed as an asymmeic volailiy of he hedge: Posiive euns of he DAX seem o poduce a eun of he hedge which has a smalle volailiy han negaive euns of his index.,5,0 Reun of he Fix-Hedge and he DAX Disanz of =0 days; eun,5 0--Reun-Hedged,0,05 0,00 -,05 -,0 -,6 -,5 -,4 -,3 -, -, -,0,,,3,4,5,6,7,8,9 0--DAX-Reun File: EF-SPSS-Gafiken Figue 5a: Reun of he DAX and he hedge h wih = 0,5,0 Reun of he Hedge and he DAX Disanz of =50 days; eun,5 50--Reun-Hedge,0,05 0,00 -,05 -,0 -,6 -,5 -,4 -,3 -, -, -,0,,,3,4,5,6,7,8,9 50--DAX-Reun File: EF-SPSS-Gafiken Figue 5b: Reun of he DAX and he hedge h wih = 50 8 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

9 ,5,0 Reun of he Hedge and he DAX Disanz of =00 days; eun,5 00--Reun-Hedge,0,05 0,00 -,05 -,0 -,6 -,5 -,4 -,3 -, -, -,0,,,3,4,5,6,7,8,9 00--DAX-Reun File: EF-SPSS-Gafiken Figue 5c: Reun of he DAX and he hedge h wih = 00,5,0 Reun of he Hedge and he DAX Disanz of =00 days; eun,5 00--Reun-hedged,0,05 0,00 -,05 -,0 -,6 -,5 -,4 -,3 -, -, -,0,,,3,4,5,6,7,8,9 00--DAX-Reun File: EF-SPSS-Gafiken Figue 5d: Reun of he DAX and he hedge h wih = 00,5,0 Reun of he Hedge and he DAX Disanz of =300 days; eun, Reun-Hedged,0,05 0,00 -,05 -,0 -,6 -,5 -,4 -,3 -, -, -,0,,,3,4,5,6,7,8, Dax-Rendie File: EF-SPSS-Gaf iken Figue 5e: Reun of he DAX and he hedge h wih = 300 Alhough hedging insumens wee used empoay, he plo was also ceaed fo a longe ineval of =500. In Figue 5f he shape of his plo is no like a sickle. he shape becomes fo his he fom of a descending funcion. I mus be emembeed, ha in Figues 5f (like in Figue 5a-5e) he eun of he DAX is compaed wih he eun of he hedge. Obviously he hedge ceaes in he 9 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

10 exeme posiions less posiive (hedge: 90%, DAX: 30) and songe negaive euns (hedge: -70%, DAX: -55%) han he DAX do.,00,80,60 Reun of he Hedge and he DAX Disanz of =500 days; eun 500--Reun-Hedged,40,0 0,00 -,0 -,40 -,60 -,80 -,00 -,50 0,00,50,00,50,00, DAX-Reun File: EF-SPSS-Gafiken Figue 5f: Reun of he DAX and he hedge h wih = 500 Fuhemoe, he obsevaion ha he hedge has posiive euns when he DAX has exeme posiive o negaive euns can no be confimed. he esul is fa away of a hedge which can be called immunized. he hedge looses by inceasing his chaace and becomes he conay of pefec. o invesigae he pah dependen behavio of he hedge, an expeimen wih aificial pahs of an index was used. his aificial index cha uses only wo euns and which wee sysemaically changed ove a ime ineval =00. As he hedge funcion exiss of muliplies, ohe cha-pahs ae included as hese depiced in Figue 6. he wo euns and may no be alenaing like in Figue 6. he hedge funcion would have he same value when fo = o 50 he eun was used and fo =5 o 00 eun. he aificial paen shows by diffeen and a pemanen up and down in he cha. mean (eun pe day) Aificial paen sd.dev. Figue 6: Aificial paen Hedge-EF-.dsf 0709 Fo he aificial paen, he mean of he eun pe day of he index and is sandad deviaion s can easy be compued (see able ). Fo he hedge funcion h, he inees i of he sho EF was fixed ove he ime ineval o i=0% o o i=5%. 0 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

11 Aificial paen = 0.5 ( + ) (8a) ( 0. 5 ( )) = (8b) s h = 0.5 I S (8c) wih I 0 = S 0 = 00, / / ( + ) ( ) ) I = I +, 0 S = S 0 i + 80 / i + 80 / able : Reun, sandad deviaion s and he value of he hedge h of he aificial paen he diffeen eun levels of he hedge of he aificial paen is shown in he wo maps of Figues 7a and 7b which axis ae buil by he mean eun pe day of he index and is sandad deviaion s. he whie coloed cicle in Figue 7a maks he aea whee mos of he eal DAX values would be inside due o hei and s. In his example, he inees ae was se o i=0%. Accoding o fomula (8a) he mean eun is zeo, if =-. his mean, ha he hedge becomes negaive, if he cha moves aside. he Figue 7a does no show he exac posiion o pah of he minimum eun which exacly is below he zeo eun line. Fo he eun =0.0, i=0% and =00, he funcion (8c) has is minimum a = (see appendix IV-4). Wih (8a) esuls wih = a posiion below he zeo eun line. Fo highe hese diffeence is shinking o = 0.0. An elevaed inees ae i can move he posiion of he minimum above he zeo eun line (see fomula (IV-4) especive (IV-5)). Besides he confimaion, ha aside movemens of he index poduce negaive hedge euns, he simple model shows, ha he hedge will achieve he highes euns when he DAX has exeme high posiive o exeme negaive euns. he eun levels of his model wihin he whie cycle ae moe o less like in he Figue 5c beween -4% and %. In Figue 7b, he inees ae was se o 5%. his educes he aea of losses wihin he whie cicle. Figue 7a: Reun-level of he hedge h o design paen I (i=0%, = 00) he obseved minima of he sickle-shaped plo in Figue 5c occued when he DAX cha was deceasing some pecen wihin =00 days wih high volailiy in he aside movemen. his occued in he ime ineval fom he 0 h of Ocobe 008 o he 3 d of Januay 009, like in Figue 8 illusaed. he DAX was loosing 8.44% and he hedge 3.44%. he eun pe day of he index was = and he volailiy s = hese chaaceisics of he hedge minima in he las decade ae signed in he Figue 7a and 7b by a yellow poin. While unde he condiion i = 0% he hedge eun unde he aificial condiions would be abou 5%, fo i =5%, he eun would be below 5% like in ealiy. Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

12 Figue 7b: Reun-level of he hedge h o design paen I (i=5%, = 00) DAX Cha DAX-value DAX: -8.04%; Hedge: -3.44% Dae: File: EF-SPSS-Gafiken Figue 8: DAX-Cha wih high negaive hedge h eun ( = 00) Finally Figues 7a and 7b depic ha losses canno be explained sufficien only by he ime, he inees i and he sandad deviaion s. Aenion mus be paid o he eun and pah of he index, oo. 4. Coss Hedge of he undelying he immunizaion wih sho EF seem no o be pefec like e.g. wih sho fuues. Sho EF ae o diffeen compaed wih is undelying especially, when i is used fo a long em hedge. A coss hedge can seve a save soluion by minimizing he isk of he hedge. Bu his should be done fo he planned hedge peiod. As isk measue ofen he vaiance of he eun is used. he opimal hedge soluion 5 x * is x * s cov(, ) =. (9) s + s cov(, ) In fomula (9) he vaiance of he eun of he index especive he sho EF is denoed by s especive s and he covaiance by cov(, ). he weighing of he sho EF is x * = - x *. he eun of he index and he sho index is always elaed o a peiod wihin he index should be hedged. Fo = 0, 50, 00, 00 and 300, he skewness of he eun of he sho EF was 0.77,.08, 0.94, 0.85 and Fo vey small pofolios wih song skewed euns ohe isk measues like e.g. absolue deviaion achieve bee esuls 6. heefoe he mean absolue deviaion appoach 7 and 5 See e.g. Gundmann W., Ludee B., (003), p See Schube, L., (005). Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

13 he mean - age-shofall-pobabiliy model 8 wee egaded as alenaive possibiliies o ge he opimal hedge-aio. he age-shofall-pobabiliy (SP) is a isk measue which can be undesood inuiively by evey inveso. As daa fo he following coss hedge he DAX and he ShoDAX wee used. he ShoDAX was compued like above (wih he EONIA inees ae and wihou managemen fee) fo he ime peiod fom s of Januay 000 o he 4h of Apil 009. While he minimal mean - vaiance pofolio was compued by (9), he mean-absolue deviaion pofolios and mean age-shofall-pobabiliy pofolios wee compued by algoihms which do no need linea opimizaion o mixed-inege pogamming knowledge 9. As moving eun compuaion was used (if possible, fo evey ading day, he eun of an invesmen wih he duaion of days wee deemined), he oal numbe of euns was dependen on. Fo = exis 379 and fo =300 only 69 eun cases. he able depics he opimal hedge aios fo he diffeen isk measues. x DAX x ShoDAX co isk-measue minimum isk sandad deviaion 0, , absolue deviaion (aveage deviaion) SP: age: 0.00 SP: age: -.0 SP: age: -.0 SP: age: -.03 SP: age: -.04 mean able : Coss hedge soluions wih diffeen isk measues Fuhe moe, he expeced esul of he minimum sandad deviaion soluion fo = confims he equaion (6). Fo his ime peiod immunizaion of he index values seem o be possible. he coelaion co = and he minimum isk s = heefoe he weighing in able is abou x* = (0.5, 0.5). Highe values (up o =300) ecommend a geae weighing fo he index. Naually he sandad deviaion and he mean ae ising by highe.,03 Mean-age-Shofall Pofolios =00, EONIA,0,0 Reun-00 0,00 -,0 -, 0,0,,,3,4 age,0000 -,0000-4,0000,5 SP File: DAX-Coss-H Figue 0: Coss hedge wih SP 7 See Konno, H., Yamazaki, H., (99), Feinsein, C. D., hapa, M. N.,(993). 8 See e.g.: Kaduff J. V., Spemann, K. (996), Leibowiz, M. L., Kogelmann, S., (99), Roy, A. D., (95), Schube, L., (00). 9 See appendix. 3 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

14 Fo =00 he minimum absolue deviaion soluion x DAX * =0.508 was compued. he esul in able shows ha he diffeence o he isk measue sandad deviaion (x DAX * =0.5098) is no vey high. he skewness of he sho EF wih =00 was By he hid isk measue SP diffeen ages soluions wee deemined fo he case =00. While he soluion in he case of τ =0.0 was beween he opima of he ohe isk measues, o avoid negaive esuls (τ =-.0 o τ =-.04) lowe weighings ( ) fo he index wee ecommended (see able ). Obviously in he example a geae pa of sho EF is necessay o educe negaive euns of he hedge. Fo he age τ =-0.04 he opimal SP is zeo. Fo hee ages he mean-sp pofolios ae depiced in Figue 0. he cuve of he age τ =0.0 shows, ha changing he opimal soluion will cause a song elevaion of he SP. 4.3 Coss hedge of a pofolio In he following example a coss hedge of a small pofolio wih 5 socks abiay seleced ou of he Geman HDAX (Adidas-Salomon AG, BASF AG, Aixon AG, Allianz AG, Acando AG) is depiced. hese n =5 socks wee equal weighed mixed a he beginning 4 h Januay 000. he ShoDAX EF was applied fo hedging he value of his pofolio. he ShoDAX EF was aificial geneaed like in he coss hedge above. he opimal hedge was deemined fo he isk measues vaiance, absolue deviaion and SP. As moving eun compuaion was used wih =360, he oal numbe of daa cases was 4. x Pofolio x ShoDAX isk-measue minimum isk mean sandad deviaion absolue deviaion (aveage deviaion) SP: age: 0.00 SP: age: SP: age: able 3: Coss hedge soluions wih diffeen isk measue Fo he deeminaion of he SP hee diffeen ages wee applied: τ =0, τ =-5 and τ =-0 o ge he Mean-SP pofolios of he hedge. he minimal isk soluion of he hee isk measues ae depiced in he able 3. he skewness of he eun of he ShoDAX was he coelaion beween he euns of he small pofolio and he eun of he shodax-ef is he opimal hedge in he case of he sandad deviaion as isk measue is diffeen compaed wih he soluion of he absolue deviaion o SP. o avoid negaive esuls (τ =0.0), he alenaive isk measues ecommend highe weighings x ShoDAX. o poec he hedge agains losses below -0%, he SP (τ =-0.0) poposes x ShoDAX = Fo his age, he SP would be Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

15 ,08,07,06,05 Mean - SP - Pofolios Small Pofolio (n=5) and ShoDAX EF =360, EONIA,04 au-value Mean,03,0,0 0,00 0,0,,,3,4,5 au= 0.00 au= au= -0.0,6 SP File: EF-SPSS-Gafiken Figue : Mean SP pofolios wih =360 Due o he ime ineval =360 and he abiay seleced pofolio which is no idenical wih he undelying of he sho EF wih 5 socks, he pobabiliy o ceae losses is highe han by he hedge of he DAX. In he example above, his pobabiliy was 0. and fo he hedge of he pofolio 0.9. In his decade, he sock make was boken down wice. Unde hese cicumsances, he educions of losses which show he SP seem o be an accepable esul. 5. Conclusion he discussion of EF showed ha his insumen can seve a pefec hedge in makes wihou ansacion coss. In his siuaion he inveso has o ebalance evey day his pofolio. Anohe possibiliy fo pefec hedging is given, when he inveso needs he poecion of his pofolio fo only one day ( =). Fo > when high ansacion coss make he ebalancing oo expensive, he hedge becomes impefec. As empiical daa illusae, he payoff of he hedge is sickle shaped. High posiive as well as high negaive euns of he index cause a posiive eun of he hedge. he sickle shaped payoff was obseved fo Fo lage ime peiods (e.g. =500) he hedge eun ansfoms ino a descending funcion which can poduce highe losses han he unpoeced index was able o do. heefoe, like menioned above, sho EFs should be employed as hedging insumen only fo a empoay compensaion of losses. Fo ime peiods up o one yea, sho EFs can ceae a kind of pofolio insuance. he maximum losses in he case of =300 did no exceed -6% while he chance o achieve a eun of 5% o 5% was possible. he eun disibuion of he sho EF seems o have a emakable posiive skewness. heefoe, fo he coss hedge, he absolue deviaion and he age shofall pobabiliy wee poposed as isk measues besides he adiional vaiance. Fo he compuaion of he efficien fonie wo algoihms wee ceaed which do no need pofound opimizaion knowledge o -sofwae. he use of sho EFs wih highe leveage facos was no discussed. o ge fuhe insighs in he possibiliies and esicions of sho EF in he pocess of hedging, he eseach should be exended o hese highe facos. Financial insiuions ecommend vey sho ime peiods (e.g. =0) when an EF wih highe leveage faco of λ = o λ =3 is applied. 5 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

16 6. Refeences [] Cama, Y., Schyns, M., (003), Simulaed annealing fo complex pofolio opimizaion poblems, Euopean Jounal fo Opeaion Reseach 50, pp [] Deusche Böse, (007): Leifaden zu den Saegieindizes de Deuschen Böse, Ve..0, July 007. [3] Elon, E. J., Gube, M. J., (99), Moden Pofolio heoy and Invesmen Analysis, 4 h ediion, Wiley, New Yok. [4] Faell J. L., (997), Pofolio-Managemen, Mc Gaw Hill, New Yok. [5] Feinsein, C. D., hapa, M. N., (993), A Refomaion of a Mean-Absolue Deviaion Pofolio Opimizaion Model, Managemen Science, Vol. 39, pp [6] Fei, R. A., (008), he EF Book, John Wiley and Sons. [7] Gundmann, W., Ludee B., (003), Fomelsammlung Finanzmahemaik, Vesicheungsmahemaik, Wepapieanalyse, eubne. [8] Hehn, E. (ed.), (005), Exchange aded Funds, Sucue, Regulaion and Applicaion of a New Fund Class, Spinge. [9] Hull, J., (99), Inoducion o Fuues an Opions Makes, Penice-Hall. [0] Kaduff, J. V., Spemann, K., (996), Sichehei und Divesikaion bei Shofall-Risk, zfbf 48, Sepembe, pp [] Konno, H., Yamazaki, H., (99), Mean Absolue Deviaion Pofolio Opimizaion Model and is Applicaions o okyo Sock Makes, Managemen Science, Vol. 37, May, pp [] Leibowiz, M. L., Kogelmann, S., (99), Asse allocaion unde shofall consains, he Jounal of Pofoliomanagemen, pp 8-3. [3] Mogan Sanley, (007), Exchange-ades Funds US Equiy Doubling Up o Doubling down wih Leveaged EFs, Okobe. [4] Roy, A. D., (95), Safey Fis and he Holding of Asses, Economeica, Vol. 0, pp [5] Rudolf, M., (994), Efficien Foniee and Shofall Risk, Finanzmak und Pofolio Managemen, Ausgabe, pp [6] Schube, L., (00), Pofolio Opimizaion wih age-shofall-pobabiliy-veco, Economics Analysis Woking Papes, Vol. No 3, 00, ISSN [7] Schube, L., (005), Pefomance of linea pofolio opimizaion, Univesidad de Cosa Rica Ceno de Invesigación en Maemáica Pua y Aplicada, CIMPA-0-005, ISSN , Mayo 005. [8] Singleon, C., (004), Coe-Saellie Pofolio Managemen: A Moden Appoach o Pofessionally Managed Funds, McGaw-Hill. [9] Spemann, K., (006), Pofoliomanagemen, Oldenboug, München. [0] Wiand, J., McClachy, W., (00), Exchange aded Funds, John Wiley and Sons. 6 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

17 Appendix he compuaion on opimal hedge aios is concenaed o he isk measue vaiance, alhough he absolue deviaion and he age-shofall-pobabiliy (SP) offe in he case of skewed disibuions bee esimao. he compued sho EFs had always emakable high posiive skewness. heefoe, he use of ohe isk measues han he vaiance is ecommended. he use of diffeen ages offes addiional infomaion abou he SP isk of he hedge. o enable he compuaion of mean absolue-deviaion pofolios o of he mean- SP pofolios wihou linea opimizaion knowledge o mixed inege pogamming expeiences wo algoihms wee ceaed o deemine he opimal hedge (see appendix I and II). In appendix III, he compuaion of evese EFs wih highe leveage faco is depiced. he minimal hedge of he aificial funcion (8c) of able is compued in appendix VI. I) Algoihm fo hedging a pofolio wih minimal absolue-deviaion he eun of a pofolio especive of he hedging-insumen (e.g. ShoDAX) is denoed by especive ove =,, m ime inevals. o ge he minimal absolue-deviaion pofolio he following sochasic pogamming algoihm wih 3 seps can be used. he fis deemines he absolue deviaion o especive o in each ime ineval. hese values ae used in he nex sep o compue he weighings q of he pofolio (especive -q of he hedging insumen) when he sign of he absolue deviaion is changing. Addiional, he diecion of his change is deemined by p. o do his compuaions, he diffeen ime inevals have o be classified ino ses O ++,, O -- accoding o he sign of he absolue deviaions o and o. In he hid sep, he ime inevals and coesponding daa wee compleed by q 0 =0 and q m+ = and anked o ge q 0 q q m q m+. he absolue deviaion of he hedge in can also be wien as x o + (-x) o = x (o o ) + o wih x = q. he second fom was applied o ge q in sep no. and o compue he absolue deviaion AbsDev(x)= x S () + S (). he vaiable S () epesens he sum of he vaiable pa and S () he sum of he fix pa. Boh vaiables ae deemined by a ecusive way. Fo he weighing x =q =0, only he fix pa is impoan fo he compuaion of he AbsDev(0). heefoe, in equaion (I-3) he deeminaion of S 0 () (wih q 0 = 0) would no be necessay bu of S () (if q > 0). Saing by x =q 0 =0 he AbsDev(x) can be compued unil x = o ge he complee fonie of possible hedges o only unil he minimum hedge is found like i is poposed in he hid sep. In he following he algoihm o ge a minimal absolue deviaion pofolio is illusaed:.) Compue he deviaions of he mean o i = i - µ i fo =,, m and i=,. ++.) Define he following ses: = { o 0 o 0} O /, = { / o < 0 o < 0} + + O = { / o o 0}, O = { / o < o 0} 0 < O, 0 and compue heshold values fo =,, m o + if O o o ++ q = 0 if { O O } (I-) o + if O o + o Se q 0 = 0 and q m+ =. Le he value p (=0,, m+) be + if O + p = + if O (I-) 0 else. 3.) Build a ank ode of he heshold values q ha q 0 q q m q m+. Deemine he sa values ( ) ( ) S0 = S = (o o ) (o o ) and (I-3) ++ + { O O } + { O O } 7 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

18 ( ) ( ) S0 = S = o o { O O } { O O } and compue he absolue deviaion fo x = q (fo =0,, m+) by ( ) ( ) AbsDev( x = q ) = x S + S (I-4) ( ) ( ) ( ) ( ) wih S + = S + p (o o ) and S + = S + p o. eminae if AbsDev(x=q - ) < AbsDev(x=q ) wih x* = (x, -x)* = (q -, -q - ) o if AbsDev(x=q m ) > AbsDev(x=q m+ ) wih x* = (x, -x)* = (, 0). he advanages of he algoihm ae: No linea opimize o opimizaion knowledge is necessay; he numbe of peiods m nealy does no esic he algoihm; he values x =q offe he compuaion of he exac cones of he efficien cuve and heefoe he complee infomaion abou is pah (see lef side of Figue IIb). In conas he linea opimizaion appoach 0 only alenaes he mean o ge an appoximaion of he efficien fonie. In he compuaion of he following example, he paamee is used, o build he index of he ank ode ove he m=0 ime peiods accoding o he diffeen hesholds q. he example in able I shows he eun of wo socks (e.g. a pofolio and of a hedging insumen). he ank ode of o he heshold values (see I-) is used (see q especive ank ). heefoe, he ime column is no inceasing. he diffeen ses O ++ ec. can be ecognised in he columns O. Fo example in ime peiod 8 (esp. =0), he value o = -8.7 < 0 and 3,5 Mean - Absolue-Deviaion Pofolios Example wih m=0 3,0,5,0,5 Mean,0, Absolue Deviaion File: EF-SPSS-Gafiken Figue I: Example fo Mean - Absolue-Deviaion Pofolios wih m=0 0 See Konno, H., Yamazaki, H., (99), Feinsein, C. D., hapa, M. N., (993). 8 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

19 ime o o O q p S 0 S 0 S S AbsDev Mean Mean able I: Example fo he algoihm fo hedging a pofolio by minimal absolue deviaion o =6.3 >0, heefoe his peiod paicipaes o he se O -+. Fo his se, he heshold q 0 = 6.3/( ) = 0.4 and he value p = - (see equaliy (I-) and I-)). he values ( )=-5 and 6.3 (see equaion (I-3) fo se O -+ ) ae he conibuions of his peiod o he sa values S 0 () =-6.0 and S 0 () = Fo =0 he inclinaion S 0 () =-50.9 of he absolue deviaion and S 0 () = his values mus be acualized fo = accoding equaion (I-4) wih q = 0.4 o S () = (-) ( ))= -0.9 and S () = (-)6.3=8.3. Now he absolue deviaion AbsDev(x=q ) = 0.45 (-0.9) =9.50 when q is used wihou ounding eo. Fo = he minimal absolue deviaion pofolio was found wih x * = (0.45, ). he expeced eun of his pofolio is =.74. II) Algoihm fo hedging a pofolio wih minimal age-shofall-pobabiliy (SP) he eun of a pofolio especive of he hedging-insumen (e.g. ShoDAX) is denoed by especive ove =,, m ime inevals. As isk measue, he SP α wih P( < τ) α is used. he eun of he pofolio is signed by and he eun age byτ. o deemine he minimal SP pofolio o he SP of each pofolio (see e.g. Figue IIa), he following sochasic pogamming algoihm wih hee seps can be applied. he fis sep compues he shofall o deviaion of he age o especive o fo each ime ineval. Accoding o he sign of hese deviaions, fou ses O ++,, O -- ae buil in he second sep o compue he weighings q of he pofolio (especive -q of he hedging insumen) when he sign of he absolue deviaion of he hedge is changing. hese weighings ae a his sep called hesholds. A he heshold q, hee is no fuhe shofall, bu a q +ε. he value ε is a vey small numbe. Addiional, in his sep he value p is deemined. his value shows a x = q, whehe he sign of he deviaion of he hedge x o + (-x) o = x (o o ) + o changes fom posiive o negaive (p = +) o evese (p = -). he value p = 0 is fixed, if he sign do no change fo x [0, ]. In he hid sep, he ime inevals and coesponding daa wee compleed by q 0 =0 and q m+ = and anked o ge q 0 q q m q m+. As he deviaion of he hedge efeing he age has a vaiable pa x (o o ) and a fix pa o fo x =q 0 =0 only he fix pa is esponsible fo shofalls. 9 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

20 hese occu in ime inevals of he ses O -- and O +-. Fo x = q >0 his numbe of cases of age shofalls S(x) mus be inceased o educed by he coesponding p of he nex heshold. he SP(x) is compued by he aio S/m. o ge he complee fonie of possible hedges his compuaion begins a x = q m+ = and ends a q 0 =0. A minimal hedge is he soluion wih minimal SP which may no be unique. In he following he hee seps o find a minimal SP pofolio is depiced:.) Compue he deviaions of he age o i = i - τ fo =,, m and i=,. ++.) Define he following ses: O = { / o 0 o 0}, = { / o < 0 o < 0} + + O = { / o o 0}, O = { / o < o 0} 0 < O, 0 le he heshold values q 0 = 0 and q m+ = and compue he ohe hesholds by o + if O o o ++ q = 0 if { O O } (II-) o + + ε if O. o + o In (II-) ε is a vey small numbe. Le he value p (=0,, m+) be + + if O + p = if O 0 else. 3.) Build a ank ode of he heshold values q ha q 0 q q m q m+. Compue he ageshofalls S(x) by + { O O } if = 0 S( x = q ) = (II-) = + S(q + if,...,m ) p and he age-shofall pobabiliy beginning wih SP(x=)=S(q m+ )/m S(q ) / m if q q + SP ( x = q ) = SP(q + ) if q = q + he minimal SP pofolio is no always found if SP(x=q - ) < SP(x=q ) wih he soluion x * = (q -, -q - ) *. heefoe, all SP(q ) values mus be consideed o ge he opimal soluion x *. he advanages of he algoihm ae: No Banch and Bound opimize and mixed inege opimisaion knowledge ae necessay; he numbe of peiod m nealy does no esic he algoihm; Moe diffeen ages can be egaded, oo; he exacly eun ineval fo each single SP can be deemined (see veical lines on he igh side of Figue IIb). Fo wo hesholds q q + he eun ineval is he se { / = x +(-x) ; q x < q + }. In able II he eun daa of he example of able I wih m =0 wee used o illusae he applicaion of he algoihm. As age τ = 0 was seleced. he hee seps deemine he soluions fo x =q which ae in he wo columns a he igh side of able II. he age shofall vaiable S(q 0 )=0 sas wih he sum of he columns O -- and O +- and accoding o (II-) S(q 4 )=. he value S(q 5 )= S(q 4 )+p 5 =+(-)=0 wih q 5 = Due o he small 0 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

21 ime o o O ++ O -- O +- O -+ q p S(q ) SP Mean τ=0 τ=0 0 0, ,50 0, Mean able II: Example fo he algoihm fo hedging a pofolio by minimal age shofall pobabiliy daabase of m =0, he minimal SP =0.0/0 can be found fo 0.38 x < he eun ineval fo he minimal SP is.63 <.. In Figue IIa only he lowe posiion if his ineval is depiced. In his Figue he shape of he efficien fonie is convex and no concave as usual. his chaaceisic of Mean SP pofolios ecommends selecing invesmens wih highe isk. he connecion of wo poins of he efficien fonie is shown a he lef side of Figue IIb fo absolue deviaion and on he igh side fo SP. his connecion is no always linea like someimes supposed o like i is when absolue deviaion is used as isk measue. In he case of he SP i is no coninuous. Fo SP he diec connecion of wo poins does no exis. Insead of his veical lines can be found (see igh side of Figue IIb) which ae he eun inevals menioned in he example above. See Rudolf, M., (994). Cama, Y., Schyns, M., (003) use he sandad deviaion as isk measue. heefoe, he line beween wo poins had o be nonlinea. Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

22 3,5 Mean - age-shofall-pobabiliy Pofolios Example wih m= 0 and age = 0 3,0,5,0,5 Mean,0,5 -, 0,0,,,3,4,5,6 age-shofall-pobabiliy File: EF-SPSS-Gafiken Figue IIa: Example fo Mean SP pofolios wih m =0 eun eun. absolue deviaion age-shofall pobabily Figue IIb: Efficien fonie and isk measue Hedge-EF-.dsf 0609 III) Sho EFs wih highe leveage faco Sho EF exis wih diffeen leveage faco λ, oo. In he value of a sho EF is S I i S- (λ ) λ + (λ + ) S δ I δ +. (III-) δ 360 Leveage em Inees em = δ he leveage faco λ is wihou sign especive λ =, o 3. Wih highe leveage facos λ = o λ = 3 he isk o loose ises, oo. o ge he highe leveage he issue of he sho EF has o sell wice o hee imes he undelying index. heefoe he accuing of ineess ises by he same faco. he use of he eun pe day of he index offes a fom of (III-): S ( -λ + ( λ + ) ( i / ) δ) = S (III-) IV) Minimal hedge eun o find he minimal hedge eun of he funcion h of (8c) wih i =0%, he funcion has o be diffeeniaed: dh = ( + ) / / / / ( + ) + ( ) ( ) ( ) = 0. (IV-) d dh = ( + ) / ( + ) / / / + ( ) ( ) ( ) = 0. (IV-) d Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9

23 Documenos de abajo en Análisis Económico.- Volumen 8 - Númeo 9 3 he diffeenial (IV-) can be ansfomed o / / + = + - (IV-3) and solved fo he vaiable + + = / / / /. (IV-4) o show he chaace of a minimum he second diffeeniaion of funcion h mus be posiive: ( ) ( ) ( ) ( ) = / / / / d h d his condiion is saisfied if > and if and ae in he ange of [-00%, +00%]. he ansfomaion (IV-3) and (IV-4) can be applied analogous o equaion (IV-).