Research on Optimization Strategy of CPPI

Transcription

1 MPRA Munich Personal RePEc Archive Research on Opimizaion Sraegy of CPPI Baoyuan HU and Dima TAWIL and Xiyang LIU Rennes schcool of business, Rennes school of business, XI Dian Universiy 18 Online a hps://mpra.ub.uni-muenchen.de/8756/ MPRA Paper No. 8756, posed 7 June 18 9:4 UTC

2 Research on Opimizaion Sraegy of CPPI Auhors: Baoyuan Hu Dima TAWIL Xiyang LIU

3 Absrac Economic globalizaion and financial marke inegraion have increased flucuaion in financial markes. Invesors expec an invesmen and wealh managemen produc ha can mainain he minimum proecion when he price decreases, while i brings profis when he marke grows. So he porfolio insurance sraegy jus mees he invesors' requiremens. The research analyzes he operaion mechanism of CPPI, and hen empirically ess he performance and risk of CPPI sraegy based on he back-esing mehod. Keywords: CPPI; operaing mechanism; invesmen and wealh managemen Inroducion Invesors encouner a variey of risks when invesing in he financial markes, he key of risk conrol is ha seeks ways o quanify risks. In oher words, he meaning is how o quanify he risk, so ha venure capial invesors can more clearly undersand he marke hey face. The rapid developmen of new financial derivaives makes he financial marke more difficul o conrol, resuling in a sharp improvemen in risk. I is imporan o invesors and marke regulaors ha grow he supervision of marke risks and ensure he sable developmen of he marke. CPPI's operaing mechanism The core idea of CPPI is ha invesors se a minimum accepable level of porfolio value, o calculae he safey margin ha goes beyond he boom line and inves in risky asses wih some amplificaion muliplier (Dichl and Drobez, 1). In order o implemen he CPPI sraegy of porfolio insurance, invesors mus confirm he capial preservaion boom line F firsly a any momen during he capial preservaion cycle. This boom line adds value o he risk-free reurn on asses and guaranees capial preservaion afer mauriy (Dichl and Drobez, 1). CPPI sraegy porfolio consiss of risk-free capial preservaion asses C such as governmen bonds and risky asses R like equiies. The margin of safey N is beween he porfolio valuev and he boom line F. CPPI sraegy decision rule is ha in he porfolio, he risk asses R and safey margin N mainain a consan raio m greaer han 1, also known as amplificaion muliplier. Then consiue he following model: V = R + C=N + F (1) R =m ( V ~ F) () The curren capial preservaion boom line, which is based on he final capial reserve arge for he porfolio (which is a percenage of he principal level, such as 9%), and a reasonable discoun rae (according o he bond yield o mauriy Kobe) can also be referred o as he value reference line(louis and Xavier, 1). For ensuring ha he porfolio value is no lower han he capial preservaion arge due o frequen flucuaions, a risk ma is furher designed on he basis of he capial

4 preservaion. The risk ma is a cerain proporion of he invesmen principal, which is generally wihin 5%. The minimum arge value line for he porfolio is he sum of he capial preservaion boom line and he risk cushion, and he safey margin is he curren porfolio value minus he porfolio minimum arge value line, which represens he asse porfolio's abiliy o wihsand impairmen when he marke flucuaes (Chen and Liao, 7). Moreover, he Porfolio Insurance CPPI sraegy requires ensuring ha he amplificaion muliplier is wihin he arge's monioring range. However, oo frequen adjusmens also imply huge ransacion coss, for which reason dynamic adjusmen rules of risk asses mus be deermined on a regular basis. For convenience of explanaion, he following symbols are defined: T is he CPPI sraegy o proec he deadline; R is he value of a risky asse a ime ( [, T ]); N is he safey offse of ; m is he calculaed amplificaion muliplier for, m d is he lower bound for he calculaion of he amplificaion muliplier, mu is he upper bound for he calculaion of he amplificaion muliplier, m is he sandard amplificaion muliplier, andc is he capial preservaion value of (If C is a negaive cash posiion when he CPPI sraegy allows borrowing, furher assume he larges cash lending posiion is C d ). During CPPI sraegy [, T ], on a cerain dynamic day, he dynamic adjusmen rules of risky asses raio can be divided ino he following four cases: 1. When N >, and md m mu, i indicaes ha he proporion of risk asses wihin he scope of monioring will no be adjused. The value of he risky asse configured a his ime is R= N m.. When N >, and m> mu, i indicaes ha he proporion of risky asses is oo large. A his poin, he raio of risky asses mus be adjused downwards o he sandard amplificaion muliplier m on he period of regular adjusmen, and he adjused risk asse value is R= N m. 3. When N >, and m <md, A his poin, if C> Cd, should furher reduce he capial preservaion of asses and increase he risk-holding asses; if C= C, ha has reached he maximum amoun of borrowing, mainaining he number of original risk asses, will no be adjused. 4. When N, he safey offse is or negaive, which is called a cash inciden. d

5 The occurrence of cash evens is generally due o he sudden and drasic decline of he marke, which canno resul in he imely liquidaion of risky asses. For ensuring ha he porfolio is above he capial preservaion level a he end of he capial preservaion period, when he safey margin is reduced o N =, he risk asses in he invesmen porfolio mus be cleared and allocaed o he capial-guaraneed asses, wih R =. In summary, he following proposiion can be drawn. Proposiion 1: When he decrease of marke causes he calculaion of amplificaion muliplier o exceed he upper limi of he monioring raio, he CPPI sraegy requires he reducion of holdings of risky asses and he preservaion of asses; When calculaed muliplier fall o he lower limi of he monioring raio because of he improvemen of marke, he CPPI sraegy requires he holdings of risky asses o be capial-proeced up o he maximum borrowing limi. Therefore, in essence, he porfolio insurance CPPI is a" more and more o buy, more or sell more" invesmen sraegy. CPPI's paymen funcion Assume ha a CPPI porfolio can inves in wo ypes of asses: one is a money marke accoun, denoed by B ; he oher is a radable porfolio of risk asses, denoed by S, such as a composie sock index. The invesmen period is [, T ] and each invesmen sraegy is self-financing. The value of B risk-free asse obeys db = B rd. Among hem, r is he risk-free ineres rae of coninuous compound ineres. The marke value of a risk porfolio obeys he following classic sochasic diffusion process(joossens and Schouens, 15): d S =( ud + dw ) (3) A any momen, he CPPI sraegy mus ensure ha he porfolio value is above he boom line value F, F provides he dynamically variable mauriies, apparenly F obeys he following: df F rd = (4) According o he operaing mechanism of he porfolio CPPI sraegy, V = C + R ; where C is capial preservaion, R is risky asse, and R= mn. Due o he

6 Taylor formula, he CPPI porfolio value a ime can be expressed as: df d V ) + ds V R =( - R (5) F S And because of V = F + N, according o ype (3), ype (4) and ype (5), he safey offse mees he following random process: dn = d( V F ) df =( V - R) F =N( [ m( u r) + r) d + m dw ] ds + R S df =( N + F mn) + mn df F df (6) Obviously, F mus be less hanv, and he difference beween he wo V F is denoed by N. Based on formula (6): m N= Nexp( [ m( u r) + r ) + m W ] (7) From (3), he sochasic differenial equaion S= Sexp[ W + ( u ) ] of risk asse S can be drawn as: 1 S 1 W ln ) = ( ( u ) (8) S Subsiuing formula (8) ino formula (7): S N=N( S m ) exp{ [ r m( r ) m ] } (9) N Leing =( ) exp[ ], =[ r m( r ) m ], according ov m = F + N, S from (4) and (9), he value of CPPI sraegy porfolio a any ime in he invesmen inerval [, T ] is a funcion of he amplificaion mulipliers m and S as follows: m S r V( m, S )=F e + (1) Generally, he porfolio insurance CPPI sraegy has a capial preservaion cycle of 1 year, a risk asse iniial price of 1, a volailiy of %, a capial preservaion arge of 1% of he principal, a risk-free ineres rae of 5%, and he range of change in

7 mauriy asse prices is ~. And CPPI sandard amplificaion mulipliers m respecively is, 4, 6 and 8(Joossens and Schouens, 1). Proposiion : Porfolio Insurance CPPI sraegy effecively conrols he downside risk in he marke while also giving invesors an opporuniy o paricipae in an upside marke. Proposiion 3: The paymen curves of any wo CPPI sraegies inersec, and he paymen curve of one CPPI sraegy will no be compleely above anoher curve. Therefore, here is no dominan CPPI sraegy. CPPI sraegy and is opimizaion of empirical research The following repor empirically ess CPPI sraegy performance and is risk based on back-esing. A presen, 55 surviving capial preservaion funds have relaively shor erm of proecion, usually 3 years. In order o examining he impac of he capial preservaion erm on he CPPI sraegy, his paper selecs wo capial preservaion periods: he firs, he long-erm porfolio, April 3, 9 ~ December 4, 14; he second, he shor-erm porfolio, 14 July 1 ~ June 8, 17 This paper selecs he and he as risk asses for long-erm and shor-erm porfolios respecively. The main parameers of he rerospecive es are as follows: (1) Guaraneed level, is generally 1%; () Risk ma, is generally 5%;(3) arge muliplicaion facor, wih normally 5 imes, in order o reducing he cos of frequen adjusmens o he asse porfolio, se he lower limi of amplificaion muliplier and calculae he upper limi of amplificaion muliplier respecively. There is no adjusmen when he calculaion of amplificaion muliplier is wihin his range, if no, adjus o he arge amplificaion muliplier; (4) According o he relevan policies of margin financing and lending, he maximum loan raio is se o 5%; (5) The dynamic adjusmen inerval is wo weeks, which is every Friday of he second week. Dynamic monioring and calculaion of daily amplificaion muliplier, if he 5 consecuive working days o adjus he marke o achieve a subsanial rigger condiions for cash evens, dynamic

8 adjusmen mechanism can be acivaed in advance; (6) The CPPI sraegy coss are as follows: he managemen fee is 1% and he hosing fee is.5%; (7) The main ineres raes are as follows: he ineres rae of shor-erm funds is.5%, and he erm srucure of capial preservaion asses refers o Ineres rae level of he iner-bank bond marke; (8) Mandaory dividend or spli clause. If he asse porfolio share is more han 1.5 yuan for wo consecuive dynamic adjusmen days, he manager can choose o dividend or divide o reurn he share face value back o $ 1 and sar a new CPPI simulaion; (9) Afer he cash inciden, invesors furher have he following opions: Fixed-income securiies held wih he remaining duraion of he guaraneed period achieves he capial preservaion arge, Or he combined share of asses afer he cash inciden causes he share of he value of he resoraion of 1 yuan and a new round of CPPI sraegy simulaion. Table 1 shows he resuls of he shor-cycle CPPI sraegy back-esing. Among hem, he arge amplificaion mulipliers are se o 3, 4, and 5 respecively. The risk mas for sraegy 1 o sraegy 3 are 3.9% and he level of principal proecion is se o 1%; while he risk ma of Sraegy 4 ~ Sraegy 6 is, and he guaraneed level is se as 9%. During he sample period, he flucuaed from 56 poins o 1 poins wih a gain of -1.59%. However, sraegies 1 ~ 3 wih relaively low risk levels all achieved he capial preservaion successfully, while sraegies 4 ~ 6 wih relaively high risk levels also achieved 9% of he capial preservaion arges. Table 1 Shor-cycle CPPI Sraegy Back-esing Resuls ( ~ ) Sraegy 1 Sraegy Sraegy 3 Sraegy 4 Sraegy 5 Sraegy 6 Risk asses Risk mas 3.9% 3.9% 3.9% Targe magnifica ion

9 muliplier Amplify he lower limi of he muliplier Enlarge he muliplier limi Guaranee 1% 1% 1% 91% 91% 91% d level CPPI 4.% -3.34% -5.1% -7.5% revenue CPPI 6.99% 8.91% 1.95% 1.9% 15.17% 17.11% Volailiy -1.59% -1.59% -1.59% -1.59% -1.59% -1.59% earnings.61%.61%.61%.61%.61%.61% volailiy As he sock marke coninued o fall, Sraegy, Sraegy 3, had a cash inciden on January, 17. Compared wih he.61% volailiy of he, he volailiy of all CPPI sraegies is significanly reduced. In paricular, he volailiy of Sraegy 1 is only 6.99%(Schöle and Werner, 15). The long-period CPPI sraegy reurns and volailiy characerisics are shown in Table, in which he arge amplificaion muliplier is se as 4, he upper and lower limi of amplificaion mulipliers are calculaed as 4.5 and 3.5, he risk cushion is 4% and he

10 guaraneed level is 1%. During he sample period, he index rose 41.17%, bu i experienced a surge and hen a sharp decline. This paper inroduces he cash lending mechanism, mandaory dividend or disagreemen clause and invesors' cash even opion mechanism while examine he impac of CPPI sraegy opimizaion on performance and risk characerisics. Among hem, Sraegy 1 is he sandard CPPI sraegy; Sraegies and 5 inroduce invesors cash even opions; Based on his, sraegies 3 and 6 also inroduce a mandaory dividend or spli clause sraegy; he maximum loan raio of sraegy 4 o sraegy 6 is 5% (Schöle and Werner, 15). Table : Long-cycle CPPI Sraegy Back-esing Resuls (9-4-3 ~ ) Sraegy 1 Sraegy Sraegy 3 Sraegy 4 Sraegy 5 Sraegy 6 Risk asses index index index index index index Risk mas 4.% 4.% 4.% Targe magnifica ion muliplier Amplify he lower limi of he muliplier Enlarge he muliplier limi Guaranee 1% 1% 1% 1% 1% 1%

11 d level The 5% 5% 5% maximum loan raio Mandaor No No Yes No No Yes y dividend or spli Cash No Yes Yes No Yes Yes inciden opion CPPI 4.5% % 46.4% 188.5% revenue CPPI 6.54% 7.1% 15.74% 6.54% 31.14% 17.3% Volailiy 41.17% 41.17% 41.17% 41.17% 41.17% 41.17% earnings 9.41% 9.41% 9.41% 9.41% 9.41% 9.41% volailiy Table shows ha Sraegy 1 had a cash even on January 18, 1 wih a simulaed reurn of zero. Sraegy ses he invesor opion afer a cash inciden bu does no se a mandaory dividend or spli clause. For example, a cash even occurred on January 18, 1 wih simulaed earnings of 4.5%. Sraegy 3 no only has se a mandaory dividend or spli clause, bu also se he invesor cash even opion mechanism. There were four mandaory dividends or spli evens, including: Ocober, 1, July 5, 1, January 31, 13, Ocober 17, 13, and a cash inciden occurred on February 13, 14. The simulaed revenue increased significanly o %. Compared wih

12 CPPI sraegies 1 ~ 3, afer he inroducion of he lending mechanism, he simulaion benefis of sraegies 4 ~ 6 have been improved, a he same ime he volailiy has also increased accordingly. Back-esing resuls show ha CPPI sraegy volailiy is significanly lower relaive o he and he, excep for he long-erm CPPI sraegy 5. In addiion, opimizing he CPPI sraegy hrough innovaive mechanisms, such as he inroducion of cash even opion mechanism, can make full use of possible reversal opporuniies afer he marke falls sharply (Moresi, e al. 11); In a significanly rising marke, he inroducion of a mandaory dividend or spli mechanism can lock in huge income earlier; The inroducion of a cash lending mechanism will help o capure he higher reurns from he marke as i suffers from a higher level of risk. Conclusion Porfolio Insighs CPPI sraegies no only enjoy high reurns in he bull marke cycle, bu also achieve capial preservaion goals in he bear marke cycle where he marke is eiher sharply or declining. CPPI sraegy is a "more and more o buy, more or sell more" sraegy in he magnificaion of he monioring range relaive o he risk of asses. Relaive o he major invesmen benchmark of marke, he improved CPPI sraegy ouperformed he passive index invesmen sraegy during he simulaion period, wih less volailiy. The correcive measures adoped in his paper include: 1.compulsory dividend or spli clause. In he increasing bull marke, he proporion of invesmen in risky asses should be reduced;. Design invesors' opions afer he cash-in even. I is he chance o capure a rebound afer he marke has dropped sharply; 3. Esablish a cash lending mechanism. The reason is ha improve he effeciveness of CPPI's sraegy under greaer risk. Reference [1]Louis P., Xavier L.(1) One-Dimensional Pricing of CPPI[J]. Social Science

13 Elecronic Publishing, 1, 18(3):7-5. []Chen J S, Liao B P. (7). Nonlinear Goal-Direced CPPI Sraegy[M]// Compuaional Inelligence in Economics and Finance. Springer Berlin Heidelberg, 7: [3]Joossens E, Schouens(15) W. PORTFOLIO INSURANCES, CPPI AND CPDO, TRUTH OR ILLUSION?[J]. Alernaive Invesmens & Sraegies, 15. [4]Joossens E, Schouens W. (15) PORTFOLIO INSURANCES, CPPI AND CPDO, TRUTH OR ILLUSION?[J]. Alernaive Invesmens & Sraegies, 15. [5]Chen J S, Liao B P. (7) Piecewise nonlinear goal-direced CPPI sraegy[j]. Exper Sysems wih Applicaions, 7, 33(4): [6]Moresi S, Reiman D, Salop S C, e al. (11) Gauging Parallel Accommodaing Conduc Concerns wih he CPPI[J]. Social Science Elecronic Publishing, 11. [7]Null N. (9) Efficien Pricing of CPPI using Markov Operaors[J].. [8]Joossens E, Schouens W.(1) PORTFOLIO INSURANCES, CPPI AND CPDO, TRUTH OR ILLUSION?[M]// Alernaive Invesmens And Sraegies. 1: [9]Schöle K, Werner R.(15) ON THE BENEFITS OF ROBUST ASSET ALLOCATION FOR CPPI STRATEGIES[M]// Alernaive Invesmens And Sraegies. 15: [1]Dichl H, Drobez W.(1) On he Populariy of he CPPI Sraegy: A Behavioral-Finance-Based Explanaion and Design Recommendaions[J]. Journal of Wealh Managemen, 1, 13():41-54.