Cash-Lock Comparison of Portfolio Insurance Strategies

Transcription

1 Cash-Lock Comparison of Porfolio Insurance Sraegies Sven Balder Anje B. Mahayni This version: May 3, 28 Deparmen of Banking and Finance, Universiy of Bonn, Adenauerallee 24 42, 533 Bonn. Mercaor School of Managemen, Universiy of Duisburg Essen, Loharsr. 65, 4757 Duisburg, Germany.

2 Cash-Lock Comparison of Porfolio Insurance Sraegies This version: May 3, 28 Absrac Porfolio insurance sraegies are designed o achieve a minimum level of wealh while a he same ime paricipaing in upward moving markes. The sraegies all have in common ha he fracion of wealh which is invesed in a risky asse is increasing in he asse price incremens. Therefore, in downward moving markes, he asse exposure is reduced. In he sric sense, a cash lock describes he even ha he asse exposure drops o zero and says here. Since a cash lock a an early ime prohibis any paricipaion in recovering markes, he cash lock cage is considered as a major problem wih respec o long invesmen horizons. We analyze a generalized form of a cash lock by focusing on he probabiliy ha he invesmen quoe recovers from small values. I urns ou ha, even in he case ha he dynamic versions of opion based sraegies and proporional porfolio insurance sraegies coincide in heir expeced reurn, hey give rise o a very differen cash lock behavior. In addiion, we poin ou ha, for comparabiliy reasons, i is necessary o disinguish beween sraegies where an invesmen quoe over one is admissible and such wih borrowing consrains. Keywords: CPPI, OBPI, porfolio insurance, cash lock, borrowing consrains, pah dependence. JEL: G, G2

3 . Inroducion Financial sraegies which are designed o limi downside risk and a he same ime o profi from rising markes are summarized in he class of porfolio insurance sraegies. Among ohers, Grossman and Villa 989 and Basak 22 define a porfolio insurance rading sraegy as a sraegy which guaranees a minimum level of wealh a a specified ime horizon, bu also paricipaes in he poenial gains of a reference porfolio. 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. Here, synheic is undersood in he sense of a rading sraegy in basic raded asses which creaes he pu. In a complee financial marke model, here exiss a perfec hedge, i.e. a self financing and duplicaing sraegy. In conras, he inroducion of marke incompleeness impedes he concep of perfec hedging. 2 In general, he opimaliy of an invesmen sraegy depends on he risk profile of he invesor. In order o deermine he opimal rule, one has o solve for he sraegy which maximizes he expeced uiliy. Thus, porfolio insurers can be modeled by uiliy maximizers where he maximizaion problem is given under he addiional consrain ha he value of he sraegy is above a specified wealh level. 3 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. A his poin, we also refer o Cox and Huang 998 and El Karoui, Jeanblanc and Lacose 25. In hese papers i is shown ha, under marke compleeness and various uiliy funcions, he opimal porfolio resembles an opion on a power of he underlying asse. However, i is well known ha in a complee marke, he classic CPPI sraegies are pah independen if no borrowing consrain are posed. In paricular, he porfolio value can be represened in erms of he floor plus a power claim. This implies a leas some similariies of boh mehods. In summary, here is he basic quesion which of he wo mehods, OBPI or CPPI is favorable. While he CPPI approach is appealing in is simpliciy and easiness o cusomize he sraegies o he preferences of he invesor, a major drawback is seen in he Opion based porfolio insurance OBPI wih synheic pus is inroduced in Leland and Rubinsein 976, consan proporion porfolio insurance CPPI in Black and Jones 987. For he basic procedure of he CPPI see also Meron One possible soluion is given by quanile and efficien hedging, c.f. Föllmer and Leuker 999 and Föllmer and Leuker 2. 3 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, 999, Browne 999, Tepla 2, 2 and El Karoui e al. 25.

4 so called cash lock cage. Recall ha porfolio insurance sraegies are designed o achieve a minimum level of wealh while a he same ime paricipaing in upward moving markes. The sraegies all have in common ha he fracion of wealh which is invesed in a risky asse is increasing in he asse price incremens. Therefore, in downward moving markes, he asse exposure is reduced. In he sric sense, a cash lock describes he even ha he asse exposure drops o zero and says here. Since a cash lock a an early ime prohibis any paricipaion in recovering markes, he cash lock cage is considered as a major problem wih respec o long invesmen horizons. The classic CPPI principle implies ha, once he invesmen quoe drops o zero, i says here wih probabiliy one. However, he cash lock probabiliy crucially depends on he borrowing consrains which are posed. In he following paper, we analyze a generalized form of a cash lock by focusing on he probabiliy ha he invesmen quoe recovers from small values. I urns ou ha, even in he case ha he dynamic versions of opion based sraegies and proporional porfolio insurance sraegies coincide in heir expeced reurn, hey give rise o a very differen cash lock behavior. In addiion, we poin ou ha, for comparabiliy reasons, i is necessary o disinguish beween sraegies where an invesmen quoe over one is admissible and such wih borrowing consrains. The whole analysis is carried ou in a simple Black Scholes ype framework. There are a few commens necessary concerning his approach. Obviously, a more realisic picure of cash lock probabiliies can only be gained wihin a more general model seup. However, he simple model seup used here is suiable in order o undersand he differences in he cash lock behavior of he wo approaches, OBPI and CPPI. 2 Wih respec o he lieraure, we resric ourselves o he srand focusing on sylized sraegies, i.e. OBPI, CPPI or a comparison of boh. Wihou posulaing compleeness, we refer o he following ones: 4 The properies of coninuous ime CPPI sraegies are sudied exensively, c.f. Booksaber and Langsam 2 or Black and Perold 992. A comparison of OBPI and CPPI in coninuous ime is given in Berrand and Prigen 22a. An empirical invesigaion of boh mehods can, for example, be found in Do 22 who simulaes he performance of hese sraegies using Ausralian daa. In Cesari and Cremonini 23, here is an exensive simulaion comparison of popular dynamic sraegies of asse allocaion, including CPPI sraegies. The lieraure also deals wih he effecs of jump processes, sochasic volailiy models and exreme value approaches on he CPPI mehod, c.f. Berrand and Prigen 22b, Berrand and Prigen 23. More recenly, Con and Tankov 27 and Balder e al. 28 analyze he risk profil 4 Basically, one can disinguish beween hree differen srands focusing on invesmen Sraegies wih guaranees, opimal invesmen decision based on uiliy and opimal insurance conrac design.

5 and gap risk of CPPI sraegies. 5 In conras o Con and Tankov 27 who place hemselves in a jump diffusion model seup, Balder e al. 28 inroduce he gap risk by he inroducion of rading resricions. 3 The ouline of he paper is as follows. In Sec. 2, we discuss he invesmen decision and he consrucion of porfolio insurance sraegies. Eiher, he consrucions is based on he invesmen quoe in erms of he fracion of wealh which is o be invesed in he risky asse. Alernaively, he sraegy can be defined via he number of asses. W.l.o.g. all sraegies under consideraion are self financing. In addiion, we we define a generalized cash-lock even. In Sec. 3, we focus on he disribuion and cash lock probabiliies which are associaed wih radiional porfolio insurance sraegies. While Sec. 3 considers he case where no exogenous resricion on he invesmen quoe is posed, he effecs of borrowing consrains are analyzed in Sec. 4. An illusraion and comparison is given in Sec. 5. In paricular, we focus on he difference of he cash lock behavior of CPPI and OBPI wih borrowing consrains. Sec. 6 concludes he paper. 2. Invesmen quoe vs number of risky asses Throughou he following, i is enough o consider he quesion which par of he wealh can be invesed in a risky asse and which par has o be placed in a risk free one. We consider wo invesmen possibiliies: a risky asse S and a riskless bond B which grows wih consan ineres rae r. The evoluion of he risky asse S, a sock or benchmark index, is given by a sochasic differenial equaion which, in he simples case, is a geomeric Brownian moion. I is worh menioning ha some of he following sraegies heir consrucion, respecively do and some do no depend on he specific assumpions on he sochasic process which is generaing he asse prices. 6 The firs case means ha, in order o deermine he number of asses and bonds which are o be bough or sold i is enough o observe he asse prices while i is no necessary o know he sochasic process which generaes hem. This is in spiri of he CPPI approach. In conras, he opion based approach wih synheic pu call, respecively is an example of a sraegy consrucion which explicily depends on he model assumpion. 7 5 If he CPPI mehod is applied in coninuous ime, he CPPI sraegies provide a value above a floor level unless he price dynamic of he risky asse permis jumps. The risk of violaing he floor proecion is called gap risk. In pracice, i is caused by liquidiy consrains and price jumps. 6 Obviously, he performance of he sraegies always depends on he daa generaing process. 7 For example, in he Black Scholes model, he volailiy of he underlying is one crucial inpu for he sraegy.

6 Firs, we define he general sraegy definiion in erms of he fracion of wealh invesed in he risky asse and in erms of he number of asses. Afer his, we give a generalized definiion of he cash lock probabiliy Dynamic rading sraegies. Along he lines of he lieraure on porfolio insurance, a coninuous ime invesmen sraegy or saving plan for he inerval [, T ] can be represened by a predicable process π T. π denoes he fracion of he porfolio value a ime which is invesed in he risky asse S. In he following, we will also refer o π as he invesmen quoe. Noice ha i is w.l.o.g. convenien o resric oneself o sraegies which are self financing, i.e. sraegies where money is neiher injeced nor wihdrawn during he rading period ], T [. Thus, he amoun which is invesed a dae in he riskless bond B is given in erms of he fracion π. Le V = V T denoe he porfolio value process which is associaed wih he sraegy π, i follows immediaely ha V π is he soluion of ds dv π = V π + π db, where V = x. S B Alernaively, a rading sraegy can be specified in erms of he numbers Φ of asses which are held, i.e. Φ = φ, φ where φ denoes he number of bonds and φ he number of he risky asse. In consequence, one obains V Φ = φ B + φ S where for a self financing sraegy φ i holds ha V φ is he soluion of dv φ = φ db + φ ds, where V = x. 2 Noice ha he second erminology is convenionally used in he lieraure abou arbirage free pricing and hedging of coningen claims. Wihou inroducing sources of marke incompleeness, boh approaches will give he same resul. In his case, we obviously have φ = πv S and φ = πv B. However, here are a few commens necessary. As indicaed above, radiionally a specificaion of he rading sraegy in erms of he invesmen quoe is closely linked o proporional porfolio insurance PPI while a specificaion of he oal shares held is used in he opion based approach OBPI. In a complee financial marke seing, boh approaches can be ranslaed, i.e. a dynamic OBPI sraegy can also be expressed as a PPI, c.f. Berrand and Prigen 22a. In conras, marke incompleeness and/or model risk give rise o differen resuls. Anoher example is given by rading resricions in he sense of discree ime rading. Here, a discree ime version can only be specified in erms of he numbers Φ as here is no rading possible beween wo rading daes. While he number of shares mus herefore be consan on each rading inerval, he invesmen quoe will obviously change if here are movemens in he asse price.

7 2.2. Cash Lock probabiliy. The following secion aims a a formal descripion of he invesmen quoe π. Recall ha he basic decision which he invesor faces is he division of her wealh beween he risky asse and he risk free one. In his sense, i is ineresing o measure he gains which are caused by improvemens of he asse price. This is easy in he case ha we have a pah independen sraegy. 8 Definiion 2. Paricipaion rae, level. The sensiiviy of he erminal sraegy value wih respec o a change in he erminal asse price is called paricipaion rae, i.e. p Φ S = S V Φ T S. 3 Le L Φ denoe he minimal asse price such ha he paricipaion rae larger han zero, i.e. In paricular, we call L Φ he paricipaion level. L Φ = inf S p Φ S > }. 4 Noice ha he above definiion is suiable wih respec o he porfolio insurance sraegies which are characerized by he paricipaion, if any, in upward moving markes. 9 However, more hings are o be considered in he case of pah dependen sraegies where he payoff he erminal value of he sraegy does no exclusively depend on he erminal asse price bu on he whole price pah. One migh argue ha he pah dependence causes an addiional source of uncerainy. On he oher hand, one migh argue agains he pah independency along he following lines. Recall ha a pah independen sraegy implies ha he value does only depend on he asse price incremen S S. Thus, in he case of a sudden drop in he asse price a here is no lock in of prior gains possible. However, he following concep of a cash lock measure does no depend on he pah dependency pah independency of he sraegy under consideraion. Definiion 2.2 Cash lock probabiliy. A [, T [, a sraegy is called α cashdominan iff i holds π α α [, ]. For he ime inerval [, τ] τ T, he α β Cash lock probabiliy P CL α,β,τ denoes he probabiliy ha he invesmen quoe a τ is less ha β given ha i is equal o α a, i.e. P CL α,β,τ := P [π V τ β π = α] 8 Noice ha a sraegy can be pah independen wih respec o one paricular model, bu fails o be pah independen wih respec o anoher model. 9 In paricular, he sraegies do no allow for shor posiions in he asse. Here he opposie definiion of he paricipaion level is convenien. An easy example is given by he sop loss sraegy where a barrier condiion implies eiher a paricipaion rae of one or zero. 5

8 In he nex secion we deermine he risk profile of classic porfolio insurance sraegies wihou posing borrowing consrains. Afer his, we consider he effecs of borrowing consrains which auomaically inroduce a pah dependency Cash Lock probabiliies and expeced values As a benchmark case, we consider he Black Scholes model, i.e. he dynamics of he risky asse S are given by he sochasic differenial equaion SDE d S = S µ d + dw, S = s, 5 where W = W T denoes a sandard Brownian moion wih respec o he real world measure P. µ and are consans and we assume ha µ > r and >. The following subsecions are all srucured analogously. Firs, we define he sraegy under consideraion in erms of heir heir associaed numbers of bonds and asses φ = φ, φ. Then we summarize he value of he sraegy, is disribuion and finally he cash lock probabiliy. 3.. Sop loss sraegy SL. Consider he sop loss sraegy SL where G := e rt G, τ := inf : V SL = G }. Wih respec o he number of bonds and asses which are held in he porfolio, he SL sraegy is hen defined by φ SL, = τ } G τ B τ = τ } G, φ SL, = τ>} V S 6 such ha he value V SL which is implied by he SL sraegy is V SL = V S S τ>} + G τ }. 7 In paricular, noice ha he SL sraegy is pah dependen in he sense ha is value is no exclusively specified in erms of he curren asse price bu does depend on he whole price pah unil. Concerning he disribuion of he value, one obains

9 Proposiion 3. Disribuion and densiy associaed wih SL. If he asse price dynamics are given by Equaion 5, hen i holds P [V SL w] = N P [V SL = G ] = N w V µ µ r V G wv + µ 2 N 2 G for w G 8 G V µ r µ r V + 2 G V N + µ r 2 2 G 9 P [V SL w] = for w < G. 7 In paricular, for w > G i holds P [V SL dw] = w w V φ µ µ r V 2 2 G wv + µ 2 φ 2 G dw. Proof: The proof is given in Appendix A. I is worh menioning o recall ha he SL sraegy is very aggressive in he sense ha i eiher implies an invesmen quoe π SL of one or zero. Therefore, i is enough o summarize he cash lock probabiliy wih respec o he cases π SL = and π SL =. Proposiion 3.2 Cash lock probabiliy of SL. Le β, hen i holds P [ π SL β π = ] = and P [ π SL β π = ] = β P [V SL = G ] β <. Proof: The proof follows immediaely wih Proposiion 3.. Concerning he expeced value, we have he following proposiion.

10 8.5 Expeced reurn SL parameer: V,Σ,Μ,r.,.2,.5,.5 Cash lock probabiliy parameer: V,Σ,Μ,r.,.2,.5, expeced reurn.3 G 5 G 8 G 9 cash lock probabiliiy G 5 G 8 G mauriy mauriy Figure. Expeced reurn and cash lock probabiliy Proposiion 3.3 Expeced value of SL. The expeced value of he sop loss sraegy is given by = G e r N G 2 µ r G e µ G 2 V N V µ r 2 2 G E[V SL ] 2 µ r + V e r G 2 V N V +µ r V e µ N G V Proof: The proof follows immediaely wih Proposiion 3.. V +µ r 2 2 G +µ r Generalized OBPI GO. The following resuls refer o he dynamic version of a generalized OBPI sraegy GO in he Black-Scholes model seup. If he asse price dynamics are given by Equaion, hen he GO sraegy φ GO = φ GO,, φ GO, is defined by φ GO, = G α β N d, S and φ GO, = αn d +, S where d ±, S := βs G T + r ± 2 2 T T. I is worh menioning ha, in conras o he SL sraegy and he following PI sraegies, he dynamic version of he OBPI iself depends on he assumpions on he asse price dynamics.

11 9 Paricipaion rae and level parameer: V,G,T,Σ,r., 9.,.,.3, normalized paricipaion rae Α V paricipaion levell G Β Figure 2. Paricipaion rae for varying paricipaion levels. I is sraighforward o show ha he value of he GO sraegy is given by 2 V GO = G + αn d +, S G α and V T φ GO = V GO T [ = G T + α S T G T β β N d, S for < T 3 ] +. 4 The above implies ha for an exogenously given iniial invesmen V, he paricipaion rae α and he paricipaion level G T β V G α Lemma 3.4 Paricipaion rae. Le mus saisfy he iniial condiion = Nd +, S G β Nd, S. α β := V G = Nd +, S G Nd β, S, hen i holds α β <. Proof: The proof follows immediaely by differeniaion. In paricular, α is inversely relaed o he price of a call opion wih srike K = G T β in he srike K. where he call price is decreasing 2 Noice ha he sraegy Φ BS = φ GO, G, φ GO, is he Black Scholes hedge for a call opion wih mauriy T and srike K = G β.

12 Noice ha he above proposiion saes ha a higher paricipaion rae α can only be achieved by an increase in he paricipaion level G T β effec is illusraed in Fig Proposiion 3.5 Disribuion and densiy associaed wih GO. P [VT GO v] = N v G T α P [V GO T = G T ] = N a reducion in β, respecively. This + G T β µ 2 2 T T G T β µ 2 2 T T for v G T 5 6 P [V GO T v] = for v < G T. 7 In paricular, for v > G T i holds P [V GO T dv] = v β α β G T T φ v G T α + G T β µ 2 2 T T. Proposiion 3.6 Cash lock probabiliy of GO. For he γ -γ 2 cash-lock probabiliy wih respec he ime poins, 2 of a GO sraegy wih ime o mauriy T i holds P [π GO 2 < γ 2 π GO = γ ] = N s,γ s 2,γ 2 µ where s, γ denoes he inverse mapping of π S = s, i.e. s, γ = s π S = s = γ. Proof: The proof follows immediaely from he assumpion ha he asse price incremens are independen and idenically disribued and P [π 2 < γ 2 π = γ ] = P [S 2 < s 2, γ 2 S < s, γ ].. Proposiion 3.7 Expeced value of GO. The expeced value of he dynamic opion based porfolio sraegy is given by E[VT GO ] = G T α β N d + αe µt N d+ where d ± := βs G T + µ ± 2 2 T. T

13 3.3. Classic CPPI CP. The following resuls refer o he classic CPPI version CP wih consan muliplier m and guaranee G = G expr}. Noice ha he classic version of he CPPI sraegy relies on a guaranee also called floor which is growing wih he risk free rae r. However, o specify he sraegy, i is no necessary o define a fixed ime horizon T. The numbers of asses and bonds are given by φ,cp = V CP mc CP, φ,cp B = mccp 8 S where C CP := V CP G. 9 Proposiion 3.8 Dynamics of cushion. If he asse price dynamics are lognormal as described by Equaion 5, he cushion process C CP of a classic CPPI is lognormal, T oo. I holds dc CP = C CP r + mµ r d + m dw. Proof:. The proof follows immediaely wih C CP := V CP G and Equaion 5 and Proposiion 3.9 Value and expeced value of CP. The value of he a simple CPPI wih parameer m and G is V CP In paricular, he expeced value is m = G + C CP e r S e r+ m S E [ ] V CP = G + V CP G exp r + mµ r } 2 Proof: For example, c.f. Black and Perold 992 and Berrand and Prigen 22a. Proposiion 3.. For α, β, he α-β-cash-lock-probabiliy is given by P [ π CP T P [ π CP T β π CP = α ] = for α =, 22 β π CP = α ] βm α m αm β = N µ r m 2 2 T for α >.23 T Proof: The proof is dedicaed o he appendix, cf. Appendix B PPI wih consan floor CFP. The CFP sraegy can be inerpreed as a modified version of he classic CPPI principle: The asse exposure equals a consan muliple imes he cushion and he cushion is given by he difference of he porfolio value and he floor. However, while he classic version is based on a floor which is growing according o

14 he risk free rae r, he CFP sraegy is based on a consan floor G. In paricular, he sraegy is defined by φ,cfp = V CFP mc CFP, φ,cp B 2 = mccfp 24 S where C CFP := V CFP G. 25 Proposiion 3. Dynamics of cushion. If he asse price dynamics are lognormal as described by Equaion 5, he cushion process C CFP of a CFP is lognormal, oo. T I holds C CFP = e mµ m r 2 m2 2 +mw C CFP + r G m = C CFP e r S e r+ m r G S Proof: The proof is given in he appendix, cf. Appendix C. e mµ m r 2 m2 2 s mw s ds26 m e r s S e r+ m s 2 2 ds 27 S s Corollary 3.2 Expeced value of CFP. For mµ m r, he expeced value of a CFP sraegy wih parameers m and G is given by E [ ] V CFP = G + C CFP e mµ m r rg + e mµ m r. 28 mµ m r Proof: 3.. The above resul follows immediaely wih V CFP = G + C CFP and Proposiion 4. Effecs of borrowing consrains Before we compare he performance of he porfolio insurance sraegies under consideraion, we consider he effecs of borrowing consrains. To our opinion, i seems o be unfair o compare sraegies which allow for an invesmen quoe which is higher han one wih sraegies which are consruced such ha he invesmen quoe is less or equal o one. Before we illusrae he performance of he consrained and unconsrained sraegies, we consider he effecs of borrowing consrains in his secion. 4.. Sop loss sraegy SL. Since he invesmen quoe of he SL sraegy is eiher zero or one, here are no effecs of borrowing consrains Generalized OBPI GO wih borrowing consrains, classic OBPI OB. In he case of borrowing consrains, no all sraegies belonging o he class of generalized

15 opion based porfolio insurance sraegies, c.f. definiion of he GO sraegies in subsecion 3.2, are admissible. In consequence, here is a resricion on he sraegy parameers α and β. 3 Lemma 4. Borrowing consrains. Resricing he invesmen quoe π GO o he value w w, i.e. π GO w, hen a necessary and sufficien condiion is given by wαn d +, S S wg α β N d, S for all S. In paricular, for w =, i.e. a maximal invesmen quoe of, he condiion simplifies o α. 29 β Proof: Wih π GO = φgo, S V GO w, Equaion 3 i follows immediaely π GO wαn d +, S S wg α β N d, S for all S. 3 Lemma 4.2 Classic OBPI. For α = β, i holds π GO. In paricular, we call a generalized OBPI sraegy GO sraegy wih α = β classic OBPI CO. Proof: The above lemma is an immediae consequence of Lemma 4.. In paricular, he classic opion based porfolio insurance sraegy CO is a meaningful version of he generalized one GO Capped CPPI CCP. Posing borrowing consrains on he classic CPPI sraegy sraighforwardly resuls in he capped CPPI CCP. Definiion 4.3. The capped CPPI sraegy CCP is defined by φ,ccp = V CCP where C CCP := V CCP G. mc CCP, φ,ccp B = max ωv CCP S, mc CCP Again, G = G e r, m m 2 is a consan and w w denoes he resricion on he invesmen quoe. Proposiion 4.4 Value and cushion of CCP. Le dx = ΘX d + dw 3 Recall ha α gives he paricipaion rae and G T β definiion 2.. gives he paricipaion level of he sraegy, cf.

16 4 where Θx = µ r 2 m x µ r 2 x > 3 and X = m V mg mc V m C. 3 m G mc < V If he asse price dynamics are lognormal as described by Equaion 5, he value process V CCP and cushion process C CCP T are given by T and V = G m m ex X + m emx X < m m C = G ex X m emx X <. 32 Proof: The proof is dedicaed o he appendix. Proposiion 4.5 Densiy of CCP. Le X be defined as in Proposiion 4.4, hen i holds P [ V CCP dv ] = p v m v mg p mv G dv v m G m dv v < m G m m v G G m where px := P X X dx. Proof: Cf. Appendix. D. Proposiion 4.6 Disribuion of CCP. For mv G V i holds P [V CCP T v] = L λ,t v G T K2λ K λ m v GT m K 2 λ G T K4λ K 3 λ m v K 4 λ mg T + K 5λ m v K 6 λ mg T + K 5λ+e 2x θ 2+2λ K 3 λ m v λ K 6 λ mg T K6λ K6λ G T < v m G T m m G T m < v V e rt V e rt < v

17 5 and for mv G < V we have P [VT CCP v] = L λ,t v G T K 5 λ+e 2x θ 2+2λ K 6 λ K3 λ m v GT m K 6 λ G T K K 4 λ 3 λ m v GT m λ K 4 λ G T K 5 λ m v GT K 6 λ G T K 2 λ m v mg T K λ λ K 2 λ K 6 λ m G T < v V e rt V e rt < v mg T m mg T m < v where L λ,t denoes he inverse Laplace ransform. Wih respec o he funcions K i λ and K i we refer o he appendix, cf. Theorem E.. Proof: Cf. Appendix E.. Noice ha he cash lock probabiliy follows immediaely wih he disribuion given in Proposiion 4.6. Finally, consider he momens of he CCP. Proposiion 4.7 Momens of CCP. For mv G V, i holds E[ V CCP n ] = e rt n V T L n λ,t λ nµ r nn 2 2 n n K λ i n + G n m m i im + K 2 λ K3 λ m n + K 4 λ + K } 5λ n + K 6 λ i= For mv G < V, we have E[ V CCP n ] = e n rt n n T L λ,t i i= n n + G n K3 λ i i= G n i V G i λ imµ r ii m2 2 i K5 m im K 4 λ + λ i m im K 6 λ n m K λ m n K 2 } Proof: The proof is given in Appendix E.2. Inuiively, i is clear ha he invesmen quoe of he CCP version of a CP sraegy is almos surely less han he one of he underlying CP sraegy. Assuming µ > r, his immediaely explains ha he CCP sraegy is due o a lower expeced reurn and, a he same ime, gives a lower variance. The effecs of borrowing consrains are summarized in Table which describes he disribuion of he final values of classic CPPI and he version which resuls from he differen borrowing consrains w in erms of he expeced value, he sandard deviaion, he skewness and kurosis. The iniial invesmen is V =,

18 6 m ω expeced value sandarddev. skewness kurosis CP CCP CP CCP CP CCP CP CCP , SL Table. Momens of classic CPPI CP and is capped version CCP wih respec o differen borrowing consrains w and muliplier is m. The parameer consellaion is summarized by V =, G = 8, µ =.85%, r =.5, =.2 and T = 2. he iniial guaranee G = 8. The Black Scholes parameers are se o µ =.85%, r =.5 and =.2. The ime horizon is given by T = 2 years. 5. Comparison of capped CPPI CCP and classic OBPI CO According o he previous secions, i is ineresing o compare he performance of he capped CPPI wih he one of a classic OBPI. Unless saed oherwise, we consider an iniial invesmen of V =, an iniial guaranee of G = 8 and a muliplier of m = 5. Wih respec o he asse price dynamics, we assume ha he volailiy is =.2 and µ =.5. The risk free ineres rae is given by r =.5. Fig. 3 illusraes he densiy of he erminal value for capped CPPI CCP, classic OBPI CO and sop loss SL. The ime horizon is 2 years years, respecively. Fig. 4 shows he expeced reurn of sraegies under borrowing consrain for varying imes o mauriy.

19 parameer:8v,g,t,s,m,r,m<=8., 8., 2.,.2,.5,.5, 5.< Capped CPPI Floor Adjused OBPI Sop-Loss.5.25 parameer:8v,g,t,s,m,r,m<=8., 8.,.,.2,.5,.5, 5.< Capped CPPI Floor Adjused OBPI Sop-Loss.2. densiy.5 densiy erminal value erminal value Figure 3. Densiy of he erminal value for capped CPPI CCP, classic OBPI CO and sop loss SL. The ime horizon is 2 years lef figure and en years righ figure. Finally, Fig. 5 illusraes he differen cash lock behavior. 6. conclusion Recenly, here is a growing lieraure concerning he opic of porfolio insurance. The revival of OBPI and CPPI sraegies is caused by he growing populariy of guaranees which is in paricular observable wih respec o reail producs. In falling markes, he basic proecion principle affords a reducion of he fracion of wealh which is invesed in risky asses. On he oher hand, here is he demand owards upside paricipaion in rising markes such ha a high paricipaion level is waned. One problem which is associaed wih dynamic versions of porfolio insurance sraegies is caused by he so called gap risk, i.e. he risk ha he guaranee is no honored wih probabiliy one. This risk is due o various sources of marke incompleeness. For

20 8 parameer:8v,g,t,s,m,r,m<=8., 8., 25.,.2,.5,.5, 5.< parameer:8v,g,t,s,m,r,m<=8., 9., 25.,.5,.5,.5,.< Capped CPPI Floor Adjused OBPI Sop-Loss. Capped CPPI Floor Adjused OBPI Sop-Loss ime o mauriy ime o mauriy Figure 4. Expeced reurn of sraegies under borrowing consrain for varying imes o mauriy. example, rading resricions, price jumps impeded he concep of perfec hedging as well as he porfolio proecion feaure of a CPPI sraegy. However, his paper aims a a differen problem, he cash lock cage. In he sric sense, a cash lock describes he even ha he asse exposure drops o zero and says here. Since a cash lock a an early ime prohibis any paricipaion in recovering markes, he cash lock cage is considered as a major problem wih respec o long invesmen horizons. We give a formal analysis of a generalized cash lock measure, i.e. we focus on he probabiliy ha he invesmen quoe recovers from small values. I urns ou ha, even in he case ha he dynamic versions of opion based sraegies and proporional porfolio insurance sraegies coincide in heir expeced reurn, he sraegies give rise o a very differen cash lock behavior. In addiion, we poin ou ha, for comparabiliy reasons, i is necessary o disinguish beween sraegies where an invesmen quoe over one is admissible and such wih borrowing consrains.

21 9 parameer:8v,g,t,s,m,r,m,b<=8., 8., 25.,.2,.5,.5, 5.,.5< parameer:8v,g,t,s,m,r,m,b<=8., 8., 25.,.5,.5,.5, 5.,.5< probabiliy.4 Capped CPPI Floor Adjused OBPI Sop-Loss probabiliy.3.2 Capped CPPI Floor Adjused OBPI Sop-Loss D D Figure 5. β =.5 Cash lock probabiliies of sraegies under borrowing consrain. Noice ha P [V SL Appendix A. proof of Proposiion 3. w] = P [V SL Observe ha he hiing even can be expressed as s V SL = G, τ ] + P [V SL w, τ > ]. 33 τ } = min s /G s } = min W s + µ r } 2 2 s s G } V Le M := max s W s and m := min s W s where W denoes a sandard Brownian moion. Accordingly, we define M ν := max s W ν s and m ν := min s W ν s where W denoes a Brownian moion wih drif ν. In paricular, we have τ } = m ν G } V where ν := µ r 2 2

22 2 Using well known resuls gives 4 P [V SL = G ] = P [τ ] = P = P [ M ν [ m ν G V ] V G G V = N µ r µ r V + 2 G N G ] V + µ r and P [V SL w, τ > ] = P = P [ W ν w r, m ν V G V [ M ν ] [ V G P W ν ] V w + r, M ν ] V G Wih Equaion 34, Equaion 33 yields [ P [V SL w] = P W ν V w + r, M ν w V = N µ µ r V + G ] V G 2 N 2 G wv + µ 2 2 Finally, one obains he densiy by differeniaing Equaions 35 and 8.. Appendix B. proof of Proposiion 3. The sraegy definiion implies ha he invesmen quoe equals π = mc V. Therefore, P [ πt β π = α ] = P [m βc T βg T m αc = αg ] m = P [C e µ r m 2 2 T +W T W β m β G C = α ] m α G [ α ] m = P e µ r m 2 2 T +W T W β m α m β [ = P W T W βm α m αm β µ r m ] 2 2 T. 4 In paricular i holds P [M b] = N b N b, c.f....

23 2 Appendix C. proof of Proposiion 3. Noice ha dc CFP = dv CFP Wih Equaion 5 and db B dc CFP = V CFP = C CFP mccfp V CFP m ds S + = r d i follows = mµ m r = B A A + CCFP ds + S mccfp V CFP G m C CFP rg mµ m r + CCFP db B db B. d + C CFP mdw 36 d + ςc CFP dw 37 where A := mµ m r and B := rg. Along he lines of Kloeden and Plaen 999, Kapiel 4, he above can be classified as an inhonogenous linear sochasic differenial equaion. Muliplying Equaion 37 wih e A 2 ς2 ςw yields [ d C CFP e A 2 ς2 ςw = dc CFP A 2 ς2 C CFP d ςc CFP dw + ] 2 ς2 C CFP d W ς d W, C exp A 2 ς2 ςw = exp A 2 ς2 ςw B d. Noice ha he righ hand side of he above Equaion does no depend on C CFP. Appendix D. Proof of Proposiion 4.5 Le fz denoe he densiy funcion of he random variable Z and le h : R R denoe a monoonously increasing funcion wih inverse h. Then, a well known is given by P [hz dz] = h z fh z. An applicaion of he above wih respec o he value process according o Equaion 32 and he random variable X gives in he case ha X > hx = mg m ex, h x = For X, we haven hx = G + m emx, h x = m x mg, m x G G, m h x = x. h x = mx G.

24 22 The res of he proof is sraighforward. Appendix E. Proof of Proposiions 4.6 and 4.7 Theorem E. Benes e al. 98. Le pλ, x, z, θ, θ denoe he Laplace-ransform of P x X dz wih respec o, i.e. pλ, x, z, θ, θ = e λ P x X dzd Dann is pλ, x, z, θ, θ. Then, for x i holds K λe zk 2λ z < < x pλ, x, z, θ, θ = K 3 λe zk4λ + K 5 λe zk 6λ < z < x K 5 λ + e 2x θ 2+2λ K 3 λe zk 6λ fr < x < z. 38 For x < i holds pλ, x, z, θ, θ = pλ, x, z, θ, θ. The fiuncions K i are given as follows K λ = K 2 λ = θ + 2e x θ + θ 2+2λ θ θ + θ 2 + 2λ + θ 2 + 2λ = θ 2 + 2λ θ 2 +2λ K 3 λ = e x θ + θ 2 + 2λ K 4 λ = θ + θ 2 + 2λ θ 2 +2λ K 5 λ = e x θ + θ 2 + 2λ = 2e x K 4 λ K 4 λ K 6 λ 2e xk4λ K 4 λ K 8 λ θ θ + θ 2 + 2λ θ 2 + 2λ θ θ + θ 2 + 2λ + θ 2 + 2λ 2K 8 λ K 6 λe x K 4 λ = K 4 λ K 6 λk 4 λ K 8 λ K 6 λ = θ θ 2 + 2λ K 8 λ = θ θ 2 + 2λ For x <, he associaed K i are denoed by K i. Wih respec o he following proofs, i is useful o noice ha he deerminaion of he inverse Laplace ransform can be achieved by he soluion of he Bromwich-inegrals L λ, F λ} = 2πi γ+i γ i e γ F λdλ

25 23 where γ > max k Rz k and z k denoes he singulaiy of F. 5 E.. Proof of 4.6. Firs, consider he case mc > V X >. Togeher wih Theorem E. and 4.4 i follows ha for v mg m P [V v] = P [G + m emx m m v G G = L λ, ] [ v = P X K e xk 2 dx } = L λ, m vv G ] G K K 2 G m v G } K 2 m. For m G m < v V e r < x x i holds [ < X m v mg m P [ m G < V v] = P } m v mg K 3 e xk 4 + K 5 e xk 6 dx =L λ, =L λ, K 3 K 4 m v mg K 4 + K 5 K 6 m v mg ] } K 6 K 3 K 5. K 4 K 6 Noice ha K K 2 K 3 K 4 K 5 K 6 =. In paricular, for m P [V v] = L λ, K 3 K 4 m v mg G m < v V e r i follows K 4 } K 5 m v K 6 +. K 6 mg Finally, consider he case v > V e r x > x. Here, [ P [V e r < V v] = P x < X ] m v mg } m v = L mg λ, K 5 + e 2x θ 2+2λ K 3 e xk 6 dx = L λ, x K 5 + e 2x K 6 θ 2 +2λ K 3 Using K 3 K 4 e x K 4 K 3 K 6 e 2x θ 2+2λ+x K 6 = gives λ P [V v] = λ + K 5 + e 2x θ 2+2λ K 3 K 6 m v mg m v mg K 6 e x K 6 } K 6. 5 In paricular, his implies ha, for a suiable λ, he order of inegraion can be changed. In he following, his applies since K 2 λ and K 4 λ are, wihou limi, monoonously increasing and K 6 λ and K 8 λ are, wihou limi, monoonously decreasing in λ. Therefor, he Bromwich-inegral is well defined.in he following, we omi he respresenaion of he funcional dependence beween K i and λ. In paricular, K i λ is simply denoed by K i..

26 24 Similar reasoning gives for mc < V v < V e r : P [V v] = L λ, v [V e r m, G m ] : P [V e r < V v] = L λ, v > m G m : P [ m m G < V v] = + K 5 K 5 +e 2x θ 2+2λ K3 K 6 K 3 K 4 e x K 4 e K x K 6 6 L λ, K K 2 K 6 m v G m G K 4 m v G m G K 6 m v G m G m v mg } K 2. } } Summing up gives he resulz. E.2. Proof of Proposiion 4.7. Le mc V. Proposiion 4.5 and Theorem E. imply E[VT n ] = m m G T v n P [V dv] G } T } =:A V e rt + m v n P [V dv] m G T } } =:B + v n P [V dv] V e } rt } =:C By insering he densiy funcion and changing he order of inegraion, A, B and C can be deermined as an inverse Laplace ransform. In paricular, i holds B = L λ, = L λ, A = L λ, = L λ, = L λ, = L λ, V e rt m m G T K 3 K 4 + n G T m n n i i= n n i i= n n i i= v + G T n K } m K 2 m v dv mv G T G n i T G n i T G n T K m K m G T m K 2 m G T m K 2 m m G T K K 2 + im m m K 2 + im } i v K 2 +im m dv v n K 3 m K 4 } K 5 m K 4 v + v dv v mg T v mg T m mg T K 4 v n+ K 4 V e rt v= m m G T + K 5 K 6 + n } K i+ 2 } GT m m m mg T K 6 v n+ K 6 V e rt v= m m G T }

27 C = L λ, = L λ, v n K 5 + K 3 e 2x V e v rt K 5 + K 3 e 2x K 6 + n θ 2 +2λ θ 2 +2λ Analogously o he proof of Proposiion 4.6, cf. differen K i gives he resul for mc V, i.e. 4.6 n m B + C = m G K3 T K 4 + n + K 5 K 6 + n K4 m mg V + K 3 V e rt n K 4 + n e2x } m K 6 v dv mg T m K 6 v V e } rt n+ K 6 mg T. 25 E., using he relaion beween he θ 2 +2λ m K 6 + n K6 mg V n m = m G K3 T K 4 + n + K 5 2V e rt n K 6 + n K 4 + nk 6 + n n m = m G K3 T K 4 + n + K 5 V e rt n + K 6 + n λ µ rn. nn 2 2 In he case ha mc < V, i follows V e rt E[VT n ] = v n P [V dv] + G } T } =:à where à = L λ, B = L λ, C = L λ, n i= n G n i T i n n G n i T i i= K n K G n T 2 m m G T K 5 + K 3 e 2x θ 2+2λ im K 6 K 3 m G T v n P [V dv] V e } rt } =: B K 4 m im K 4 m } n m v i K 4 m G T m + K5 + m v n P [V dv] m G T } } =: C K6 m K6 V e rt i m G T m G T K 6 m im K 6 v i K 6 m m G T v=e r V G

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