«A Time-varying Proportion Portfolio Insurance Strategy based on a CAViaR Approach»

Transcription

1 «A Time-varying Proporion Porfolio Insurance Sraegy based on a CAViaR Approach» Benjamin Hamidi Berrand Maille Jean-Luc Prigen Preliminary Draf - March Absrac Among he mos popular echniques for porfolio insurance sraegies ha are used nowadays, he so-called Consan Proporion Porfolio Insurance (CPPI) allocaion simply consiss in reallocaing he risky par of a porfolio according o he marke condiions. This general mehod crucially depends upon a parameer - called he muliple - guaraneeing a predeermined floor whaever he plausible marke evoluions. However, he uncondiional muliple is defined once and for all in he radiional CPPI seing; we propose in his aricle an alernaive o he sandard CPPI mehod, based on he deerminaion of a condiional muliple. In his ime-varying framework, he muliple is condiionally deermined in order he risk exposure o remain consan, bu depending on marke condiions. We hus propose o define he muliplier as a funcion of an Exended Expeced Value-a-Risk. Afer briefly recalling he porfolio insurance principles, he CPPI framework and he main properies of he condiional or uncondiional muliples, we presen and compare several models for he condiional muliple and esimae hem using parameric, semi-parameric and non-parameric mehods. We hen compare he condiional muliple models when i is used in insured porfolio sraegies and inroduce an original ime-varying sraegy - called Time-varying Proporion Porfolio Insurance (TPPI) - whose aim is o adap he curren exposiion o marke condiions following a radiional risk managemen philosophy. Finally, we use an opion valuaion approach for measuring he gap risk in boh condiional and uncondiional approaches. Key Words: CPPI, Porfolio Insurance, VaR, CAViaR, Quanile Regression, Dynamic Quanile Model, Expeced Shorfall, Exreme Value. JEL Classificaion: G, C3, C4, C22, C32. We hank Chrisophe Boucher, Thierry Chauveau, Jean-Philippe Médecin, Paul Merlin and Thierry Michel for heir kind help and advices. We also wish o address special hanks o Emmanuel Jurczenko, who largely conribued o firs earlier versions of our research in he field of porfolio insurance. The second auhor hanks he Europlace Insiue of Finance for financial suppor. Preliminary version: do no quoe or diffuse wihou permission. The usual disclaimer applies. A.A.Advisors-QCG (ABN AMRO), Variances and Universiy of Paris- (CES/CNRS). benjamin.hamidi@gmail.com. A.A.Advisors-QCG (ABN AMRO), Variances and Universiy of Paris- (CES/CNRS and EIF). Correspondence: Dr. B. Maille, CES/CNRS, MSE, 06 bv de l'hôpial F Paris cedex 3. Tel: (fac). bmaille@univparis.fr. Universiy of Cergy-Ponoise (THEMA). jean-luc.prigen@u-cergy.fr.

2 «A Time-varying Proporion Porfolio Insurance Sraegy based on a CAViaR Approach» Preliminary Draf - March Absrac Among he mos popular echniques for porfolio insurance sraegies in use nowadays, he socalled Consan Proporion Porfolio Insurance (CPPI) allocaion simply consiss in reallocaing he risky par of a porfolio according o he marke condiions. This general mehod crucially depends upon a parameer - called he muliple - guaraneeing a predeermined floor whaever he plausible marke evoluions. However, he uncondiional muliple is defined once and for all in he radiional CPPI seing; we propose in his aricle an alernaive o he sandard CPPI mehod, based on he deerminaion of a condiional muliple. In his ime-varying framework, he muliple is condiionally deermined in order he risk exposure o remain consan, bu depending on marke condiions. We hus propose o define he muliplier as a funcion of an Exended Expeced Value-a-Risk. Afer briefly recalling he porfolio insurance principles, he CPPI framework and he main properies of he condiional or uncondiional muliples, we presen and compare several models for he condiional muliple and esimae hem using parameric, semi-parameric and non-parameric mehods. We hen compare he condiional muliple models when i is used in insured porfolio sraegies and inroduce an original ime-varying sraegy - called Time-varying Proporion Porfolio Insurance (TPPI) - whose aim is o adap he curren exposiion o marke condiions following a radiional risk managemen philosophy. Finally, we use an opion valuaion approach for measuring he gap risk in boh condiional and uncondiional approaches. Key Words: CPPI, Porfolio Insurance, VaR, CAViaR, Quanile Regression, Dynamic Quanile Model, Expeced Shorfall, Exreme Value. JEL Classificaion: G, C3, C4, C22, C32. 2

3 «A Time-varying Proporion Porfolio Insurance Sraegy based on a CAViaR Approach». Inroducion Leland and Rubinsein (976) firs show ha an opional asymmeric performance srucure can be reached using some porfolio insurance sraegies. Thanks o dynamic allocaion sraegies, insured porfolios are proeced agains large falls by a conracually guaraneed predeermined floor and hey ake parially advanage of marke performances. A porfolio insurance rading sraegy is defined o guaranee a minimum level of wealh a a specified ime horizon, and o paricipae in he poenial gains of a reference porfolio (Grossman and Villa, 989; Basak, 2002). Thus he invesor reduces his downside risk and paricipaes o marke rallies. The mos prominen examples of dynamic versions are he Consan Proporion Porfolio Insurance (CPPI) sraegies and Opion Based Porfolio Insurance (OBPI) sraegies wih synheic pus.. On a micro-economic perspecive, such sraegies using insurance properies are hus raionally preferred by individuals ha are specially concerned by exreme losses and compleely risk averse for values below he guaranee (or floor). We propose in his paper a new applied financial sraegy ha helps he invesor in realizing his objecives for mos of he marke condiions. The Consan Proporion Porfolio Insurance (CPPI) is inroduced by Perold (986) on fixed income asses. Black and Jones (987) exend his mehod by using equiy based underlying asses. In ha case, he CPPI is invesed in various proporions in a risky asse and in a non-risky one o keep he risk exposure consan. CPPI sraegies are very popular: hey are commonly used in hedge funds, reail producs or life-insurance producs. The main difficuly of he CPPI Opion-Based Porfolio Insurance (OBPI) wih synheic pus is inroduced in Leland and Rubinsein (976). Synheic is undersood here in he sense of a rading sraegy in raded asses ha creae he pu. In a complee financial marke model, a perfec hedge exiss (i.e. a self financing and duplicaing sraegy). In conras, he inroducion of marke incompleeness impedes he concep of perfec hedging. 3

4 sraegy is o deermine he parameer defining he porfolio risk exposure, known as he muliple. Banks direcly bear he risk of he guaraneed porfolios hey sell: a mauriy, if he guaraneed floor is no reached, he banks have o compensae he loss wih heir own capial. The sharpes deerminaion of he muliple is he main acual challenge of he CPPI sraegy. Uncondiional muliple deerminaion mehods have been developed in he lieraure such as he exreme value approach o he CPPI mehod (see Berrand and Prigen, 2002 and 2005; Prigen and Tahar, 2005). Bu all hese uncondiional seing mehods of he muliple reduce he risk exposure o he risky asse exposure. Thus hese radiional uncondiional mehods do no ake ino accoun ha he risk of he risky underlying asse changes according, for insance, o marke condiions. We develop here a seing mehod of condiional muliple o he CPPI sraegy o keep a consan risk exposure. The aim of his paper is o model and compare differen ways for esimaing he muliple, condiionally o marke evoluions, when keeping a consan risk exposure defined by he Valuea-Risk (VaR) or by he Expeced Shorfall. The condiional muliple is esimaed using parameric, non-parameric and semi-parameric mehods. Several dynamic quanile models are used as exensions of CAViaR models (see Engle and Manganelli, 2004; Gouriéroux and Jasiak, 2006) based on quanile regression esimaions (see Koenker and Basse, 978; Mukherjee, 999). Afer having briefly recalled porfolio insurance principles, he CPPI mehod and properies ha condiional or uncondiional muliple have o follow (secion 2), we jusify, describe and esimae he condiional muliple model when keeping a consan exposiion o he risk using parameric, semi-parameric and non-parameric mehods (secion 3 and 4). We illusrae hen he Time-varying Proporion Porfolio Insurance in secion 5. The inroducion of marke incompleeness and model risk may impede he concep of dynamic porfolio insurance. Measuring he risk ha he value of a CPPI sraegy is less han he floor (or guaraneed amoun) is herefore of pracical imporance. For example, he inroducion of rading resricions (liquidiy problem ) is one jusificaion o ake ino accoun he gap risk in he sense ha a CPPI 4

5 sraegy canno be adjused adequaely. Thus, an addiional opion is ofen wrien. The opion is exercised if he value of he CPPI sraegy is below he floor. Using his hedging approach, we finally evaluae he gap risk beween condiional and uncondiional approaches in a proporion porfolio insurance framework according o differen opion valuaion model (secion 6). 2. Cushioned Porfolio Evidences Porfolio insurance is defined o allow invesors o recover, a mauriy, a given proporion of heir iniial capial. One of he sandard porfolio insurance mehods is he Consan Proporion Porfolio Insurance (CPPI). This sraegy is based on a specific dynamic allocaion on a risky asse and on a riskless one o guaranee a predeermined floor. The properies of CPPI sraegies are sudied exensively in he lieraure, (Booksaber and Langsam, 2000; Black and Perold, 992). A comparison of OBPI and CPPI is given in Berrand and Prigen (2005). The lieraure also deals wih he effecs of jump processes, sochasic volailiy models and exreme value approaches on he CPPI mehod (Berrand and Prigen, 2002 and 2003). The managemen of cushioned porfolio follows a dynamic sraegic porfolio allocaion. The floor, denoed F, is he minimum value of he porfolio, which is accepable for an invesor a mauriy. The value of he covered porfolio, denoed V, is invesed in a risky asse denoed by S and a non-risky asse denoed by B. The proporion invesed in he risky asse varies relaively o he amoun invesed in he non-risky asse, in order o insure a any ime he guaraneed floor. Hence, even if he marke is downward sloping a he invesmen horizon T, he porfolio will keep a mauriy a value equal o he floor, (i.e. a predeermined percenage of he capial deposi a he beginning of he managemen period). A mauriy, he heoreical guaraneed value canno be obviously higher han he value iniially invesed and capialized a he non-risky rae B r, denoed by r B T e V 0. 5

6 The cushion, denoed by c, is defined as he spread (which can vary across ime) beween he porfolio value and he value of he guaraneed floor. Therefore, i saisfies: c = V F () Thus, he cushion is he maximal heoreical amoun, which we can loose over a period wihou reducing he guaraneed capial. The raio beween he risky asse and he cushion corresponds, a each ime, o he arge muliple, denoed by m. The muliple reflecs he maximal exposure of he porfolio. The cushioned managemen sraegy aims a keep a consan proporion of risk exposure. The posiion in he risky asse, denoed e, has o be proporional o he cushion. Thus, we have a any ime: e = m c (2) I means ha he amoun invesed in he risky asse is deermined by muliplying he cushion by he muliple. A any ime, he flucuaing muliple moves away from is arge value. This is he reason why a hird parameer is inroduced, he olerance, denoed by τ, o deermine when he porfolio has o be rebalanced. If afer he flucuaion of he risky asse, he remaining muliple, denoed by * m, moves away from is arge value of a superior percenage of he olerance, here will exis adjusmens (hus ransacion fees). Therefore, we have: * [ 0, K,T ], [ m ( τ ), m ( +τ )] m (3) The problem of he cushion managemen is he deerminaion of he arge muliple. For insance, if he risky asse drops, he value of he cushion mus remain (by definiion) superior or equal o zero. Therefore, he porfolio based on he cushion mehod will have (heoreically) a value superior or equal o he floor. However, in case of a drop of he risky underlying asse, he higher he muliple, he higher he cushion and he exposure ends rapidly o zero. Neverheless, 6

7 before he manager can rebalance his porfolio, he cushion allows absorbing a shock inferior or equal o he inverse of he superior limi of he muliple. 3. Quanile Models for Cushioned Porfolios The main parameer of a CPPI dynamic sraegy is he muliple. The firs purpose of his paper is o deermine condiional muliple models and o sudy hem using several esimaion mehods. The firs model is based on Value-a-Risk (VaR). Nowadays, VaR is considered as he sandard measure o quanify marke risk. Thus, i seems appropriae o use i for modelling he condiional muliple, which allows he hedged porfolio o keep a consan exposure o risk. In his secion, he inuiion of using VaR o model he condiional muliple has o be jusified. This firs condiional muliple model based on VaR is hen deailed, before reviewing he main quanile models and esimaion mehods used in his paper. The opimaliy of an invesmen sraegy depends on he risk profile of he invesor. In order o deermine he opimal rule, one has o decide wha sraegy o adop according o he expeced uiliy crierion. Thus, porfolio insurers can be modelled by uiliy maximizers where he opimizaion problem is given under he addiional consrain ha he value of he sraegy is above a specified wealh level. In a complee marke, he CPPI can be characerized as expeced uiliy maximizing when he uiliy funcion is piecewise HARA and he guaraneed level is growing wih he riskless ineres rae (Kingson, 989). This argumen can be no more valid if addiional fricions are inroduced such as for example rading resricions. Wihou posulaing compleeness, we refer o he works of Cox and Huang (989), Brennan and Schwarz (989), Grossman and Villa (989), Grossman and Zhou (993, 996), Basak (995), Cvianic and Karazas (995 and 999), Browne (999), Tepla (2000 and 200) and El Karoui e al (2005). Mosly, he soluion of he maximizaion problem is given by he unconsrained problem including a pu opion. Obviously, his is in he spiri of he OPBI mehod. Bu, he inroducion 7

8 of various sources of marke incompleeness in erms of sochasic volailiy and rading resricions makes he deerminaion of an opimal invesmen rule under minimum wealh consrains quie difficul (if no impossible) 2. For example, if he payoff of a pu (or call) opion is no aainable, he sandard OBPI approach is no more viable since a dynamic opion replicaion mus be inroduced. I explains why he CPPI mehod has become so popular among praciioners. In incomplee markes, hedging sraegies depend on some dynamic risk measure (Schweizer, 200). In his framework, he use of he reurn quanile inverse o deermine he condiional muliple is jusified by is precise definiion of he expeced maximum loss. The cenile approach can also direcly be inerpreed as a VaR, which is considered nowadays as a sandard measure o quanify marke risk. Moreover, he VaR is now direcly used in profi and loss paradigm. Thus, in porfolio selecion model under shorfall consrains inroduced in he work by Roy (952) on safey-firs heory and developed by Lucas and Klaassen (998), he shorfall consrain is defined such ha he probabiliy of he porfolio value falling below a specified disaser level is limied o a specified disaser probabiliy. Campbell e al (200) have used he VaR o define he shorfall consrain in order o develop a marke equilibrium model for porfolio selecion. Following his work, he risky asse exposiion is driven by a condiional muliple deermined by he inverse of a shorfall consrain quanified by he VaR. The hedging depends in fac on his risk measure. The condiional muliple allows he hedged porfolio o keep a consan exposure o risk defined by he VaR. To be proeced in a CPPI seing, he muliple of he insured porfolio has o say smaller han he inverse of he underlying asse maximum drawdown, unil he porfolio manager can rebalance his posiion. Addiionally, o obain a convex cash flow wih respec o he risky asse 2 The marke imperfecion can be caused by rading resricions. The price process of he risky asse, is driven by a coninuous-ime process, while rading is resriced o discree ime. Therefore, he effeciveness of he OBPI approach is given by he effeciveness of a discree-ime opion hedge. The error of ime-discreizing a coninuousime hedging sraegy for a pu (or call) is exensively sudied in he lieraure (Cf. Boyle and Emanuel, 980; Russel and Schacher, 994; Bersimas e al, 998; Mahayni, 2003; Talay and Zheng, 2003; Hayashi and Mykland, 2005). 8

9 reurn, he invesor has o require a muliple higher han one 3. Moreover, o keep a porfolio value superior o is floor, he cushion has o be posiive. Thus, we have: [ 0,, T ] K c / c 0 X + ( m ) r / m B + + and: ( ) where c is he cushion value a ime, S r ( X Δ S S ), m is he muliple and / + + c / 0 c X m + has o fulfil he following condiion, a any given ime : wih max{ } 0 X is he opposie of he underlying asse rae of reurn B r is he risk free rae. To be proeced, a CPPI [ X ] m (4) X = X s < s, is he maximum of he realizaions of he random variable for beween 0 and T. For a capial guaranee consrain a a significance level α %, he muliple mus be lower han he inverse of he α condiional-quanile of he asse reurn disribuion, ha is: wih: [ 0] α ( + τ) { } [ 0,..., ] P c c m Q T + α, + { ( ) α} Q =, inf r F r α r + IR m ( τ ) (5) where T is he erminal dae, Q α, + is he α h quanile of he condiional disribuion denoed (). S F of he risky periodic asse reurns and r + is a fuure periodic reurn. Thus, he arge muliple can be inerpreed as he inverse of he maximum loss ha can bear he cushioned porfolio before he re-balancing of is risky componen, a a given confidence level. For a given parameric model, explici soluions for he upper bound on he muliple can be provided from Relaion 5. (see Appendix for an example when he risky asse 3 By definiion, he muliple is sricly posiive. Wih a muliple equal o, he proecion is absolue: he risky asse exposiion is hen equal o he value of he cushion. A muliple inferior o one is herefore irraional. Neverheless, even if he muliple was smaller han one, (-m ).r B is neglecible relaive o m and his propery would be furher verified. 9

10 price follows a general marked poin process). However, wihou specific assumpions on he marke dynamics, upper bounds on he muliple can also be deermined from marke condiions. Our firs aim is o compare several condiional esimaion mehods of he arge muliple wihin he cushioned porfolio framework. We use a quanile hedging approach based on he condiional VaR. Wihin his framework, he muliple can be modelled by he inverse of he firs percenile condiional on he disribuion of he asse reurn augmened of he excess reurn comparaively a he cenile, D ( θ ) where ( S ; ) ( S ; ) ( ) h θ. The arge muliple can be wrien as: quanile of he m = C r β + D θ (6) C r β is he firs percenile of he condiional disribuion of reurns of he underlying asse S r, corresponding o he periodic reurn of he risky par of he porfolio covered, β is he vecor of unknown parameers of he condiional percenile funcion, and ( θ ) he quanile of he reurn in excess observed in case of overaking he condiional cenile. D represens The maximum of poenial drops in he risky asse value is esimaed a each period adding a he firs condiional percenile esimaed, he quanile of he addiional reurn observed in case of overaking. This las erm whose esimaion denoed by Dˆ ( θ ) is calculaed over he period by using he relaion: S [ 0, K,T ] ( ) ( ) where ( S ; ) D ˆ max ;,0 θ n s Cs rs β r s (7) C r β is he cenile of he risky asse reurn esimaed by he condiional model considered above and n is he oal number of observaions before. The muliple can be modelled by he inverse of he firs condiional percenile of he risky asse reurn disribuion augmened by he θ h quanile of he excess reurn relaively o his cenile. The muliple is modelled as a funcion of cenile models, which is he VaR a a 99% probabiliy confidence level (denoed VaR 99% hereafer). We presen below he main VaR models and esimaion mehod used in his paper for esimaing his firs model of muliple. 0

11 The financial regulaion has generaed a quie impressive lieraure abou Value-a-Risk (VaR) calculaion, which has emerged o provide an accurae assessmen of he exposure o marke risk of porfolio. Nowadays, VaR can be considered as he sandard approach o quanify marke risk. Differen approaches have been proposed for esimaing condiional ail quaniles of financial reurns. I measures he poenial loss of a given porfolio over a prescribed holding period a a given confidence level. The VaR is a quanile funcion, so an inverse of he cumulaive disribuion funcion. The common approach o model dynamic quanile is o specify a condiional quanile a a risk level α as a funcion of condiioning variables known a a given ime : f ( α ) = Q( x, β ) wih β a vecor of some real parameers. The quanile funcion (, ) Q x β is o be an increasing funcion of he risk level α. The quanile esimaors have o fulfil he monooniciy propery wih α. Thus, a well specified quanile model is expeced o provide esimaors ha behave like rue quaniles and o increase for any values of parameers and condiioning variables according o he risk level α. Quanile funcions follow oher properies as for insance he following examples. ) If f is a quanile funcion defined on [0, ] hen: Q a ( α ) = f ( α ) and Q( α ) = f ( α ) wih a>0 are also quanile funcions, denoed Q ( α ). f,..., 2) If k =,2 n are quanile funcions wih idenical real domains, hen: where ak are posiive is also a quanile funcion. Q K ( α ) = ( α ) k = a k f k These properies can be used for derive new quanile funcions from an exising one.

12 We review below dynamic quanile models ha have already been developed in he lieraure. Engle and Manganelli (2004) classify VaR calculaion mehods ino hree differen caegories: parameric, semi-parameric and non-parameric approaches. For esimaing a predeermined quanile, he mos widely used non-parameric mehod is based on so-called hisorical simulaions. These do no require any specific disribuional assumpions and lead o esimae he VaR as he quanile of he empirical disribuion of hisorical reurns from a moving window of he mos recen period. The essenial problem is o deermine he widh of his window: including oo few observaions will lead o large sampling error, while using oo many observaions will resul in esimaes ha are slow o reac o changes in he rue disribuion of financial reurns. Oher mehods involve allocaing o he sample of reurns exponenially decreasing weighs (which sum o one). The reurns are hen ordered in ascending order and saring a he lowes reurn he weighs are summed unil he given confidence rae is reached; he condiional quanile esimae is se as he reurn ha corresponds o he final weigh used in he previous summaion. The forecas is hen consruced from an exponenially weighed average of pas observaions. If he disribuion of reurns is changing relaively quickly over ime, a relaively fas exponenial decay is needed o ensure coheren adaping. These exponenial smoohing mehods are simple and popular approaches. Parameric approaches of quanile esimaion involve a parameerizaion of he sock prices behavior. Condiional quaniles are esimaed using a condiional volailiy forecas wih an assumpion for he shape of he disribuion, such as a Suden-. A GARCH model can be used for example o forecas he volailiy (see Poon and Granger, 2003), hough GARCH misspecificaion issues are well known. Semi-parameric VaR approaches are based on Exreme Value Theory and Quanile Regression mehods. Quanile regression esimaion mehods do no need any disribuional assumpions. Condiional AuoRegressive VaR (CAViaR) inroduced by Engle and Manganelli (2004) is one of hem. They have defined direcly he dynamics of risk by means of an auo 2

13 regression involving he lagged-value a Risk (VaR) and he lagged value of endogenous variable called CAViaR. They presen four CAViaR specificaions: model wih a symmeric absolue value, an asymmeric slope, indirec GARCH(,) and an adapive form, denoed respecively: C ( r ;β), C ( r ;β), C ( r ;β), ( r ;β) SVA PA IG C where: A C C C C SVA PA IG A ( r ; β) = β + β 2 C ( ) SVA r ; β + β 3 r ( r ; β) = β + β 2 C PA ( r ; β) + β 3 max( 0, r ) + β 4 [ min( 0, r )] 2 2 ( r ; β) = {[ β + β 2 [ C IG ( )] + ]} r ; β β 3 r ( r ; β) = C ( r ; β) + β + exp{ 0,5 r C ( r ; β) } 0.0 A [ [ ] ] A (8) and where β i are parameers o esimae and r is he risky asse reurn a ime. Discussing he general CAViaR model wihou he auoregressive componen, he condiional quanile funcion is well defined if parameers can be considered as quanile funcions oo. In fac, CAViaR models weigh differen baseline quanile funcions a each dae and can be herefore considered as quanile funcions. Adding a non-negaive auoregressive componen of VaR, he CAViaR condiional quanile funcion becomes a linear combinaion of quanile funcions weighed by non-negaive coefficiens. Thus CAViaR model saisfies he properies of a quanile funcion, even if he indirec GARCH(,) or adapive specificaions do no saisfy he monooniciy propery. CAViaR model parameers are esimaed using he quanile regression minimizaion (denoes QR Sum) presened by Koenker and Basse (978): wih ( ) β x { [ ]} y < Q [ y Q ( α )] α I ( α ) min { } (9) β Q α = where x is a vecor of regressors, β is a vecor of parameers and I {} is an Heavyside funcion. When he quanile model is linear, his minimizaion can be formulaed as a linear program for which he dual problem is convenienly solved. Koenker and Basse (978) show ha he resuling quanile esimaor, Qˆ ( α ) proporion less han he corresponding quanile esimae is α., essenially pariions he y observaions so ha he 3

14 The procedure proposed by Engle and Manganelli (2004) o esimae heir CAViaR models is o generae vecors of parameers from a uniform random number generaor beween zero and one, or beween minus one and zero (depending on he appropriae sign of he parameers. For each of he vecors hen evaluaed he QR Sum. The en vecors ha produced he lowes values for he funcion are hen used as iniial values in a quasi Newon algorihm. The QR sum is hen calculaed for each of he en resuling vecors, and he vecor producing he lowes value of he QR Sum is o be chosen as he final parameer vecor. Gouriéroux and Jasiak (2006) have inroduced dynamic quanile models (DAQ). This class of dynamic quanile models is defined by: Q K (, β ) = a ( y, β ) Q ( α, β ) a ( y β ) α (0) k= k k k k + 0, where Q ( α, β ) are pah-independen quanile funcions and a ( β ) k k funcions of he pas reurns and oher exogenous variables. A linear dynamic quanile model is linear in he parameers, hen: p ( α β ) = β ( α ) k = k k, 0, k y k are non-negaive Q, Q () Thus DAQ model can use differen quanile funcions o model a given quanile. To increase he accuracy of he model of he muliple, we can also combine differen quanile funcions, in a muli-quanile mehod. Acually, previous quanile funcions can be exended o define a simple class of parameric dynamic quanile models. 4

15 4. The Time-varying Proporion Porfolio Insurance Approach: Some Empirical Evidences In his secion, we describe implemenaion mehods for he condiional muliple model, presened in secion 3, hen we compare he performances of cushioned porfolio using several VaR esimaion mehods o compue he condiional muliple. 4. Condiional Muliples: Empirical Evidences The sample period used in our sudy consised of 29 years of daily daa of he Dow Jones Index, from 2 January 987 o 20 may This period delivered 4,64 reurns. We have used a rolling window of 3,033 reurns o esimae dynamically mehod parameers. The pos sample period consised of,608 reurns. Figure presens he daa ha are used for he empirical analysis. Figure : Dow Jones Daily Reurns Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005; compuaions by he auhors. The firs model of condiional muliple can be inerpreed as: 99% m = VaR + d (4) where VaR 99% is he reurn firs cenile 4, and d is a consan 5. 4 The probabiliy of % associaed o he VaR was chosen in order for focusing on exremes and having a he same ime enough poins for backing-ou good esimaions see below. I is also a sandard in Risk Managemen for defining exreme loss. We esed several higher probabiliy levels in he following; in hese cases, he muliple is always flaer and is evoluion is mosly explained by he variaion of d. In ha sense, he inroducion of d has he 5

16 The muliple is hen given a each dae by he inverse of he VaR 99% augmened by he exceeding maximum reurn during he esimaion period. Thus he parameer d allows aking ino accoun he risky asse dispersion of reurn in he ail of he disribuion, and especially for fa ailed disribuion of he risky asse. Using his model he porfolio VaR 99% is conrolled and exreme reurns are aken ino consideraion. To compue his condiional muliple, we use main mehods of VaR esimaion exposed in he lieraure or used by operaional. Eigh mehods of VaR calculaion are presened: one nonparameric mehod using he naive hisorical simulaion approach, hree mehods based on disribuional assumpions, and he four CAViaR specificaions. The VaR 99% based on hisorical (or naive ) simulaion is denoed H_VaR 99%, i represens he empirical cenile of in sample reurns. VaR 99% based on disribuional assumpions are he normal VaR 99% (denoed Normal_VaR 99% ), he Risk merics VaR 99% (denoed RM_VaR 99% ), and a GARCH(,) VaR 99% (denoed GARCH_VaR 99% ). The normal VaR is buil under assumpion of normaliy of reurns. I s compued wih he empirical mean and sandard deviaion of he reurns of he in sample period. The Risk merics VaR is a sandard for praciioners: an exponenial moving-average is used o forecas he volailiy, and o compue he quanile assuming a Gaussian disribuion. Anoher very popular mehod based on volailiy forecass is presened as he GARCH VaR. To compue i, we implemen a GARCH(,) model. Our choice of he (,) specificaion was based on he analysis of he iniial in-sample period of 3,033 reurns and on he populariy of his order for GARCH models. We have derived he model parameers using maximum likelihood based on a Suden- disribuion and he empirical disribuion of effec of reducing he aggressiviy of he sraegy. Moreover, since he consan d represens he highes failure of he model, corresponding o one of he highes negaive reurn in he sample, he combinaion of he VaR and of d is closely linked o a measure of he expeced shorfall (see also conclusion). 5 If we do no inroduce he parameer d, he probabiliy of violaing he floor would have been equal o %. Working on a daily frequency, his probabiliy would hus have been oo high for describing a realisic invesor s demand (muliple ofen equal o 30 or so). However, ess wih or wihou d show no significan impac on he resuling global performance of he sraegy on he sample. 6

17 sandardized residuals. Finally, we presen he muliple analysis, using he four CAViaR models (he Symmeric Absolue Value CAViaR, he Asymmeric Slope CAViaR, he IGARCH(,) CAViaR and he Adapive CAViaR denoed respecively, SAV_CAViaR 99%, AS_CAViaR 99%, IGARCH_CAViaR 99%, and Adapive_CAViaR 99% ) using he procedure used by Engle and Manganelli (2004), he implemenaion mehod was already presened (see secion 3). Compuing several VaR 99% mehods (exposed above) for he,608 pos sample period, we can see, on Table 3 and Figures 2, ha excep for he hisorical VaR 99% (using naive mehod and denoed H_VaR 99% ), and he VaR assuming a Gaussian disribuion of he reurns (denoed Normal_VaR 99% ) 6, he six oher mehods have a hi raio no significanly differen from %. According o his crierion, he condiional cenile should be well modelled by hese differen mehods. We can noice ha he CAViaR asymmeric model has a hi raio exacly equal o %, on he pos sample period. Figures 2: VaR 99% Esimaes Fig.2.: Hisorical VaR 99% Fig.2.2: Risk merics_var 99 Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, VaR are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. Black lines represen he VaR evoluion; he blue poins represen he Dow Jones daily reurns. 6 To evaluae he pos sample condiional quanile esimaes, we use he hi raio. The hi raio is he percenage of observaions falling below he esimaor. Ideally for a VaR 99% he percenage should be %. Here we have a sufficienly large sample, so he significance es can be performed on he percenage using a Gaussian disribuion and he sandard error formula for a proporion. 7

18 Fig.2.3: Normal_VaR 99% Fig.2.4: GARCH(,)_VaR 99% Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, VaR are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. Black lines represen he VaR evoluion; he blue poins represen he Dow Jones daily reurns. Fig.2.5: Symmeric Abs. Value CAViaR 99% Fig.2.6: Asymmeric Slope CAViaR 99% Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, VaR are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. Black lines represen he VaR evoluion; he blue poins represen he Dow Jones daily reurns. Fig.2.7: IGARCH CAViaR 99% Fig.2.8: Adapive CAViaR 99% Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, VaR are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. Black lines represen he VaR evoluion; he blue poins represen he Dow Jones daily reurns. 8

19 Table : Hi Percenage of % Condiional Quanile H VaR 99% RM VaR 99% Normal VaR 99% GARCH VaR 99% SAV CAViaR 99% AS CAViaR 99% IGARCH CAViaR 99% Adapive CAViaR 99% Hi raio.49%*.24% 2.49%* 0.87% 0.87%.00%.2%.06% Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, VaR are dynamically esimaed for,608 pos-sample periods. The sars indicae hi raios significanly differen from %; compuaions by he auhors. H VaR, RM VaR, Normal VaR, GARCH VaR, SAV VaR, AS CAViaR, IGARCH CAViaR, Adapive CAViaR sand respecively for Hisorical VaR, Risk merics VaR, he Gaussian VaR, he GARCH (,) VaR, he Symmeric Absolue Value CAViaR, he Asymmeric Slope CAViaR, he IGARCH CAViaR and he Adapive CAViaR. Afer having evaluaed hese VaR99% muliples according o he firs model presened above: models, we can use hem for esimaing condiional m = VaR + d,99% where VaR,99% is he reurn firs cenile, and d is a consan. We use he same noaions, which were used for VaR 99% models. Thus for example, he condiional muliple esimaed using he GARCH(,) VaR 99% (denoed GARCH_VaR 99% ) is denoed m_garch. Condiional muliple esimaions over he,608 pos sample periods are represened on figures 3. We noe ha, in Ocober 999, every condiional muliple fligh from a level around.5, o 4. I is due o he exi of Ocober 987 crisis from he esimaion period. The esimaions of condiional muliples spread beween.5, and 6, which are compaible wih muliple values used by praciioners on his marke (beween 3 and 8). The muliple is he parameer which deermines he exposiion of he cushioned porfolio. To guaranee a predeermined floor, he muliple has o be inferior o he inverse of he poenial loss ha he risky asse could reach before he porfolio manager rebalances is posiion. If we assume ha he porfolio manager can rebalance oally his posiion during one day, all esimaions of condiional muliple wih his firs model allow guaraneeing he predeermined floor defined by he invesor. 9

20 Figures 3: Condiional Muliple based on VaR 99% Esimaes Fig.3.: Muliple based on he Hisorical VaR 99% Fig.3.2: Muliple based on he Risk merics_var 99% Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. Fig.3.3: Muliple based on he Normal VaR 99% Fig.3.4: Muliple based on he GARCH(,)VaR 99% Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. Fig. 3.5: Muliple based on he Symmeric Absolue Value_CAViaR 99% Fig. 3.6: Muliple based on he Asymmeric Slope CAViaR 99% Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. 20

21 Fig. 3.7: Muliple based on he IGARCH_CAViaR 99% Fig. 3.8: Muliple based on he Adapive_CAViaR 99% Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. Under esimaions Table 2: Under-esimaion of he Max Drawdown by Condiional Muliples M_H m_rm m_normal m_garch m_sav m_as m_igarch m_adapive never never never never never never never never Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. The noaions m_h, m_rm, m_normal, m_garch, m_sav, m_as, m_igarch, m_adapive sand respecively for muliple based on Hisorical VaR, Risk merics VaR, he Gaussian VaR, he GARCH (,) VaR, he Symmeric Absolue Value CAViaR, he Asymmeric Slope CAViaR, he IGARCH CAViaR and he Adapive CAViaR.. If we analyze he disribuions of condiional muliple (see figure 4), we observe ha for every condiional muliple esimaion mehods we obain wo modes. One mode is around a low value of he muliple (beween and 2 depending of esimaion mehods) associaed wih a more defensive behavior of he cushioned porfolio during urbulen period, and anoher mode around 4.5 associaed o classical and realisic value of he muliple. On one hand, densiies of condiional muliple using Asymmeric Slope, IGARCH(,) and Symmeric Value of he muliple are very similar, and on he oher hand naive hisorical, adapive and Gaussian condiional muliples densiies have comparable characerisics. For he sake of clariy, we have hus only represened on figure 4 he densiies of he asymmeric slope model and for he naive hisorical esimaion of he muliple. 2

22 Figure 4: Condiional Muliples Densiies based on VaR 99% Esimaes Naive Muliple AS Muliple Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 pos-sample periods; compuaions by he auhors. The noaions Naive Muliple and AS Muliple sand respecively for muliple based on Hisorical VaR and he Asymmeric Slope CAViaR. Kernel of condiional muliples densiies are esimaed using cross-validaion crierion (Cf. Silverman, 986); compuaions by he auhors Condiional versus Uncondiional Sraegy Comparison Evidences Values of cushioned porfolio are pah dependen. The performances of hese guaraneed porfolios, deermined by using differen esimaion mehods of he condiional muliple, are no easy o compare. Acually, he performance of he insured porfolio depends more on is sar dae, and on he invesmen horizon han of he esimaion of he condiional muliple. To invesigae wheher hese differen esimaions mehods lead o significanly differen performances (for a long erm analysis), we propose o inroduce an original muli-sar analysis. The muli-sar analysis consiss for a fixed invesmen horizon (here one year) in compuing every value of insured porfolios beginning a every momen of he pos sample period, according o is condiional muliple esimaions. An illusraion of he mehod is presened on figure 5. Table 3 repors he main characerisics of he reurns of cushioned porfolio using he muli-sar analysis. 22

23 Figure 5: Illusraion of he Various One-year Covered Porfolios launched a Differen Saring Daes Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 possample periods; compuaions by he auhors. Coloured lines represen every one year covered porfolios evoluion in a muli-sar framework: every day a new covered porfolio is launched for on year. Table 3: Performance Analysis using he Muli-sar Framework Covered Porfolio lead by: m_h m_rm m_normal m_garch m_sav m_as m_igarch m_adapive Mean Reurn 2.47% 2.45% 2.45% 2.46% 2.47% 2.43% 2.48% 2.5% Sandard Deviaion 0.08% 0.08% 0.08% 0.08% 0.09% 0.09% 0.09% 0.07% Kurosis Skewness Sharpe Raio Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 possample periods; compuaions by he auhors. On he conrary of wha we observe in a classical analysis wih a predefine sar and end of analysis, he characerisics of covered porfolios reurns are no significanly differen. Bu if we choose a specific sar and end dae, hen covered porfolio reurns using his condiional muliple can be more differen. In following secions, he analysis is limied o he Asymmeric Slope muliple model, for he sake of simpliciy and because of he similariy of he behavior of his muliple wih he one of he bes muliple esimaion mehods. The Asymmeric Slope muliple based Time-varying 23

24 Proporion Porfolio Insurance was dynamically back-esed from 937: he floor was violaed only one ime (during he 987 crash), loosing 0.6% of is guaraneed value. Table 4: Exended Backes of he Asymmeric Slope TVPPI Muliple from 937 on he US Marke Ampliude of Annualized Violaing Floor (dae) Floor Violaion Excess Reurn Condiional Asymmeric Slope Muliple TVPPI Volailiy One ime (20/0/987) 0.6%.7% 5.58% CPPI Muliple Never % 0.34% CPPI Muliple 2 Never - 0.9% 0.68% CPPI Muliple 3 Never - 0.3%.05% CPPI Muliple 4 One ime (20/0/987) 0.05% 0.45%.48% CPPI Muliple 5 One ime (20/0/987) 0.47% 0.60%.97% CPPI Muliple 6 One ime (20/0/987) 0.78% 0.77% 2.53% CPPI Muliple 7 One ime (20/0/987) 0.97% 0.96% 3.8% CPPI Muliple 8 One ime (20/0/987).05%.6% 3.90% Source: Bloomberg, Daily Reurns of he Dow Jones Index from 0/02/928 o 2/2/2005, condiional muliple is every week dynamically esimaed; compuaions by he auhors. Excess reurns are calculaed agains he risk free rae. A successive one year invesmen capial guaranee horizon was used. 5. Abou he Gap Risk Esimaion We have used a probabilisic approach o model he condiional muliple. By consrucion, he guaranee is associaed o a iny level of probabiliy bu i is no an absolue guaranee. The risk of violaing he floor proecion is called gap risk. A any rae, even in an uncondiional muliple CPPI framework, he guaranee depends on he esimaion of he maximum poenial loss ha he risky asse can reach before he porfolio manager is able o rebalance his posiion. Acually, in coninuous ime, he CPPI sraegies provide a value above a floor level unless he price dynamic of he risky asse has jumps. In pracice, i is caused by liquidiy consrains and price jumps. Boh can be modelled in a seup where he price dynamic of he risky asse is described by a coninuous ime sochasic process bu rading is resriced o discree ime. If he poenial loss is underesimaed, he predefined guaranee of he porfolio is no anymore insured. The only way o be sure o reach a perfec guaranee is o choice an uncondiional muliple equal o one (he poenial loss is hen 24

25 supposed o be 00%). For all oher cases, we should esimae he gap risk beween a perfec insurance and he insurance proposed assuming an esimaion of he poenial loss. We propose here a way o esimae his gap risk, and we recommend o add i as an addiional performance crierion of insured porfolios. To esimae he gap risk in a CPPI framework, we use he muliple a any ime o ge he assumed esimaion of he maximum poenial loss. To reach a perfec guaranee assuming no maximum poenial loss scenario, we will hedge he gap risk buying a pu whose mauriy will be he rebalance frequency. The srike will be defined each day hanks o he assumed (or modelled) maximal poenial loss a T. The pu price is compued using several pricing mehods for European opions and realisic ransacion coss wih he daa used in previous secion (for he Asymmeric Slope model). Table 5: Annual Coss of Gap Risk Esimaion using Classical Opions Pricing Models for he Asymmeric Slope Model of he Muliple as funcions of he Fees Opions Model used o esimae he Gap Risk: Black and Scholes Model Cox, Ross, Rubinsein s Binomial Tree Boyle s Trinomial Tree Meron s Jump Diffusion Model Fees (in basis poins): Annual Gap Risk Coss (in basis poins) Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 pos-sample periods ; compuaions by he auhors. No difference can be made beween he hree firs opion models when pricing he insured porfolios gap risk from 0/02/987 o 05/25/

26 Figure 7: Esimaion of he Gap Risk using a Jump Diffusion Model for he Asymmeric Slope Model of he Muliple Source: Daasream, Daily Reurns of he Dow Jones Index from 0/02/987 o 05/25/2005, condiional muliples are dynamically esimaed for,608 pos-sample periods; compuaion by he auhors. The insured porfolios gap risk is esimaed from 0/02/987 o 05/25/2005 using he Asymmeric Slope Model of he Muliple. 6. Preliminary Conclusions The model proposed in his paper for he condiional muliple, allows guaraneeing he insured floor according o marke evoluions, on a large pos sample period. This mehod provides a rigorous framework o deermine he muliple, which is he main parameer of cushioned porfolio preserving a consan exposiion o risk. To compue he condiional muliple, according o his model, an appropriae mehod o esimae he cenile of he risky asse reurn has o be chosen. If he cenile is well modelled (hi raio no significanly differen from % and no cluser of exceedances), he guaranee is insured. Moreover, even if huge differences of characerisics exis beween he reurns of covered porfolio using his model and differen 26

27 cenile specificaions for specified period, no significan differences on long-erm performance can be underlined over he large pos sample period, by he muli-sar analysis. We model in he paper he muliple as a funcion of he condiional cenile. A firs naural exension of our work will consis in replacing his reference by he Expeced Shorfall expressed in a quanile regression condiional seing. This would have he advanages of dealing wih a more robus and flexible muliple cenile esimaions a he same ime, in a coheren risk measure framework. Moreover, his paper has inroduced he use of an original muli-sar/horizon performance comparison framework, which seems more adaped o pah and sar dependen financial produc such as he consan proporion porfolio insurance porfolio. This comparison framework will be hus second furher exended wih he inroducion of he Muli-sar Timevarying Proporion Porfolio Insurance Approach (MTPPI) based on aggregaion of various muli-sar TPPI. Thirdly, we have also proposed an empirical way for esimaing he gap risk beween heoreical perfec porfolio insurance and he insurance proposed wih ransacion coss. Addiional performance crieria will also be inroduced o examine he adequacy of such insured porfolios ogeher wih he muli-quanile muli-sar approach. 27

28 References Arzner P., F. Delbaen, J. Eber and D. Heah, (999), Coheren Measures of Risk, Mahemaical Finance 9, Basak S., (2002), A Comparaive Sudy of Porfolio Insurance, Journal of Economic Dynamics and Conrol 26, Beirlan J., Y. Goegebeur, J. Sergers and J. Teugels, (2004), Saisics of Exremes, John Wiley, 478 pages. Ben Hameur H. and J.-L. Prigen, (2007), Porfolio Insurance: Deerminaion of a Dynamic CPPI Muliple as Funcion of Sae Variables, THEMA Working Paper, Universiy of Cergy- Ponoise, 22 pages. Berrand P. and J.-L. Prigen, (2002), Porfolio Insurance: he Exreme Value Approach o he CPPI Mehod, Finance 23, Berrand P. and J.-L. Prigen., (2003), Porfolio Insurance Sraegies: A Comparison of Sandard Mehods when he Volailiy of he Sock is Sochasic, Inernaional Journal of Business 8, 5 3. Berrand P. and J.-L. Prigen, (2005), Porfolio Insurances Sraegies: OBPI versus CPPI, Finance 26, Brennan M. and E. Schwarz, (989), Porfolio Insurance and Financial Marke Equilibrium, Journal of Business 62, Browne S., (999), Beaing a Moving Targe: Opimal Porfolio Sraegies for Ouperforming a Sochasic Benchmark, Finance and Sochasics 3, Black F. and R. Jones, (987), Simplifying Porfolio Insurance, Journal of Porfolio Managemen, Fall, Black F. and A. Perold, (992), Theory of Consan Proporion Porfolio Insurance, Journal of Economic Dynamics and Conrol 6, Booksaber R. and J. Langsam, (2000), Porfolio Insurance Trading Rules, The Journal of Fuures Markes 8, 5 3. Buchinsky M. and J. Hahn, (998), An Alernaive Esimaor for he Censored Regression Model, Economerica 66, Campbell R., R. Huisman e K. Koedijk, (200), Opimal Porfolio Selecion in a Value-a-Risk Framework, Journal of Banking and Finance 25, Cappiello L., B. Gérard and S. Manganelli, (2005), Measuring Comovemens by Regression Quaniles, ECB Working Paper 50, 48 pages. Chernozhukov V. and L. Umansev, (200), Condiional Value-a-Risk: Aspecs of Modelling and Esimaion, Empirical Economics 26, Chernozhukov V., (2005), Exremal Quanile Regression, Annals of Saisics 33 (2),

29 Cox J. and C. Huang, (989), Opimal Consumpion and Porfolio Policies when he Asse Price follows a Diffusion Process, Journal of Economic Theory 49, Cvianic J. and I. Karazas, (996), On Porfolio Opimizaion under Drawdown Consrains, Mahemaical Finance 65, Cvianic J. and I. Karazas., (999), On Dynamic Measures of Risk, Finance and Sochasics 3, El Karoui N., M. Jeanblanc and V. Lacose, (2005), Opimal Porfolio Managemen wih American Capial Guaranee, Journal of Economic Dynamics and Conrol 29, Embrechs P., C. Kluppelberg and T. Mikosch, (2000), Modelling Exremal Evens, Springer-Verlag, 645 pages. Engle R. and T. Bollerslev, (986), Modelling he Persisence of Condiional Variances, Economeric Reviews 5, -87. Engle R. and S. Manganelli, (2004), CAViaR: Condiional Auo Regressive Value-a-Risk by Regression Quaniles, Journal of Business and Economic Saisics 22, Esep T. and M. Krizman, (988), Time Invarian Porfolio Proecion: Insurance wihou Complexiy, Journal of Porfolio Managemen Summer, Föllmer H. and P. Leuker, (999), Quanile Hedging, Finance and Sochasics 3, Gouriéroux C. and J. Jasiak, (2006), Dynamic Quanile Models, Les cahiers du CREF 06-2, 46 pages. Grossman S. and J. Villa, (989), Porfolio Insurance in Complee Markes: A Noe, Journal of Business 62, Grossman S. and J. Zhou, (993), Opimal Invesmen Sraegies for Conrolling Drawdowns, Mahemaical Finance 3, Judd K., (999), Numerical Mehods in Economics, MIT Press, 633 pages. Karvanen J., (2006), «Esimaion of Quanile Mixures via L-momens and Trimmed L- momens», Compuaional Saisics & Daa Analysis 5 (2), Kingson G., (989), Theoreical Foundaions of Consan Proporion Porfolio Insurance, Economics Leers 29(4), Koz S. and S. Nadarajah, (2000), Exreme Value Disribuions, Imperial College Press, 87 pages. Koenker R., (2005), Quanile Regression, Economeric Sociey Monographs, 350 pages. Koenker R. and G. Basse, (978), Regression Quaniles, Economerica 46, Koenker R. and J. Machado, (999), Goodness-of-Fi and Relaed Inferences Processes for Quanile Regression, Journal of he American Saisical Associaion 94, Koenker R. and K. Hallock, (200), Quanile Regression, Journal of Economic Perspecives 5,