Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 Aymerc Kalfe (France), Ludovc Goudenege (France), aad Mou (France) Managng gap rss n CPPI for lfe nsurance companes: a rs reurn cos analyss Absrac Indvdualzed consan proporon porfolo Insurance (CPPI) producs are aracve alernaves o radonal un lned producs offerng a guaraneed mnmum reurn, such as varable annues. hey offer hgh poenal reurns whls lmng he downsde rs by mplemenng a dynamc allocaon sraegy beween rsy and rs-free asses alored o he rs aude of he benefcary. Bu performance evaluaon of CPPI producs should no rely on he unrealsc assumpons of connuous mare prce varaons and connuous rebalancng of asse allocaons. We adop a more general and realsc prce jump model and examne several dynamc sraeges as well as gap pu opons o mgae he rs ha he value of he produc falls below he guaraneed mnmum. Key words: CPPI, dynamc mulpler, jump processes and gap rs, vanlla and gap opons Inroducon Increased mare volaly and fallng neres raes rggered by he 2008-9 fnancal crss reduced he performance of radonal long-erm nvesmen producs, ncreased her rss and, where applcable, her capal requremens. In hs conex he new CPPI producs provde an aracve alernave o many radonal long-erm nvesmen producs offerng a guaraneed mnmum reurn, such as varable annues, for several reasons: lower exposure o unceran volales and exreme mare prce movemens, lower coss, and lower regulaory capal requremens, o name a few. Already, wh rsng lfe expecances, curren provsons for reremen may no be suffcen for many people o secure accepable lfe sandards afer reremen. o acheve suffcenly hgh nvesmen reurns ogeher wh low rss over he long erm, funds should reman nvesed n socs and oher rsy asses as well as n he safer bonds over an exended perod well no reremen. he desgn of long-erm nvesmen producs should also reflec he requremens and rs audes of ndvdual nvesors. Consan proporon porfolo nsurance (CPPI) s he name gven o an nvesmen sraegy ha provdes a mnmum guaraneed reurn, he floor (usually defned as he dscouned value of a fnal capal guaranee) and ams o manan a all mes an exposure o a rsy asse equal o a consan mulple of he cushon defned as he excess value of he fund above he floor. he fnal capal guaranee and he mulpler are chosen o sasfy he rs aude of he nvesor. Varous auhors, among whch Perold (986), Meron (97) and Blac and Jones (987), proved ha, assumng a geomerc Brownan process for he rsy asse prce dynamcs, a consan rae of reurn for he rs-free asse, and connuous rebalancng a no cos beween he wo asses,.e, under Blac-choles condons, he CPPI payoff Aymerc Kalfe, 204. 24 s opmal for an nvesor wh a coeffcen of rs olerance varyng lnearly wh wealh. pecfcally, he CPPI payoff s equal o he chosen floor plus a cushon value proporonal o a power of he rsy asse prce equal o he chosen mulpler; he floor and he mulpler are chosen accordng o he wo parameers of a HARA uly funcon so as o maxmze he expeced uly of he nvesor. Addonal advanages offered by CPPI sraeges over more radonal nvesmens wh mnmum guaraneed reurns are: prce ransparency, open me-horzon, no early redempon penaly, wde range of alernave nvesmens for he rsy asse, and flexbly o add oher guaranees such as raches (see II..4). CPPI s a CPPI sraegy adaped o evolvng ndvdual needs and mare condons. he floor and he mulpler are modfed accordngly. hus CPPI may combne mos of he advanages of CPPI wh he need for flexbly and enhanced rs managemen. However he provder of an CPPI produc (ypcally, an nsurance company) faces many challenges n he mplemenaon of he dynamc sraegy ha replcaes he guaraneed payoff. he rebalancng of he rsy asse/rs-free asse allocaon can only be made a dscree mes, here are ransacon coss, and rsy asse prces may jump. hus here s lely o be a dfference beween he realzed compared o he heorecal value of an CPPI sraegy under hypohecal condons of connuous prce movemens, unfeered zero-cos radng, and connuous rebalancng. In parcular, here s a fne probably for he value of he fund o fall below he guaraneed floor. We call such shorfall he gap rs. Managng or nsurng he gap rs may be delegaed o a hrd pary (e.g. a ban). he analyss of he gap rs has ofen been lmed o smple condons o preserve analycal racably: Unrealsc modellng of he rsy asse prce mare ncludng connuous prce dynamcs, zero-cos radng and unlmed lqudy.
mple parameerzaon of he CPPI sraegy such as consan capal guaranee and mulpler. mplsc rebalancng sraeges such as consan frequency. As a resul, he CPPI offers a mechansm ha aes advanage of he specfc advanages of boh socs and bonds, whle complyng wh growng needs of flexbly as experenced by polcyholders. However, he mplemenaon of CPPIs a nsurance companes levels suffers from a number of operaonal consrans on he asse managemen: he rebalancng occurs hrough regular albe dsconnuous (a mos daly) checs beween he nsurance company and a ban; dependng on he desgn of he CPPI and he dsconnuous rebalancng frequency, he magnude of he earnngs a exreme rs may requre he exernalzaon of he gap rs managemen o he ban. As a resul he man ssue experenced by he nsurance company remans o mnmze he downsde rs and eep conrol of he gap rs, whch nvolves hree man challenges: hs arcle exends prevous analyses of he gap rs by nroducng: Prce jump dynamcs. A dynamcally adjused mulpler. Advanced rebalancng sraeges, vanlla and gap pu opons o mgae he gap rs. Revew of CPPI mechansm bascs. Consder a me a rsy asse (e.g., a share) wh prce and a rs-free asse (e.g., a reasury bond) wh prce B reurnng a consan rae r. he CPPI fund s nvesed no hese wo asses so ha par of s value, called he floor F, s guaraneed whls he excess value above he floor, called he cushon C = V F, remans exposed o he rsy asse prce flucuaons. A any me, he exposure o he rsy asse, e, s ep a a consan mulple, m, of he cushon, ha s: e = mc. he res of he value of he fund s nvesed (or, f negave, borrowed) a he rs-free rae (oe ha he exposure e may be acqured a no cos f usng an offbalance shee nsrumen such as a fuure, whch may be advanageous because of lqudy and low ransacon coss). he floor s ofen chosen o ncrease over me a he rs-free rae ( could no be made o ncrease faser ndefnely), ha s: F Fe () r 0. In heory, when he rsy asse prce follows a geomerc Brownan moon, and wh connuous, zero- Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 m cos rebalancng (Blac-choles condons), he value of he cushon s pah ndependen and proporonal o In oher words, s he value of a power opon. I s convex when m > (le a long call opon), lnear when m =, and concave when m <, le a shor pu opon. Bu unle sandard call and pu opons here s no need o fx an expry dae, a CPPI sraegy s open-ended. Under he above assumpons, he value of he cushon would never fall o zero; n pracce, f does fall o zero or below zero (e.g., because of a prce jump or of dscree rebalancng), he enre fund s monezed,.e., s enrely nvesed n he rs-free asse, and he produc provder mus mae up he shorfall o delver he floor value. In pracce here may also be oher consrans such as no borrowng or addonal feaures such as racheng up he floor. In hose cases, he pah ndependency and open-endedness of he produc are los and he payoff profle becomes more complex. 2. Mehodolody and resuls 2.. CPPI n heory and pracce.... Connuous-me framewor. he rsy asse s defned by he dffuson equaon d[ d dw ] where W s a sandard Brownan moon. he prevous hypohess for he rs-free asse s ep. In such conex, and assumng connuous me CPPI, he cushon s log-normally dsrbued wh drf ( - r) + r and volaly m: 2 2 m C = C0 exp m( r) r m W. (2) 2 and he porfolo value V has he pah ndependen expresson: V F V F m r r m W 2 2 2 m = ( 0 0)exp ( ). (3) However, such assumpons are unrealsc and no conssen wh mare pracce. o remedy hese unrealsc hypohess, wo alernaves are suded: modelng n a dscree-me framewor and n a Lévy framewor. 2..2. Dscree-me CPPI. In pracce he CPPI s rebalanced n dscree me, where he shorfall probably s no longer equal o 0, whch mples o moneze more ofen. able. Fnal value mercs: buy & hold sraegy vs CPPI wh m = 3 vs CPPI wh m = 6 Buy & hold sraegy CPPI wh m = 3 CPPI wh m = 6 Daly Weely Monhly Daly Weely Monhly Mean 26.97 23.3 22.39 9.75 24.0 24.87 25.0 d-dev 7.8 3.58 32.66 36.86 42.62 43.88 48.0 95% quanle 6.90 00.48 99.98 97.0 99.99 99.3 89.69 25
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 able (con.). Fnal value mercs: buy & hold sraegy vs CPPI wh m = 3 vs CPPI wh m = 6 Buy & hold sraegy CPPI wh m = 3 CPPI wh m = 6 99.5% quanle 3.42 00.02 99.88 9.47 99.98 95.20 74.28 5% quanle 40.2 94.37 95.23 97.94 26.5 28.50 225.46 0.5% quanle 50.63 266.47 284.07 282.58 29.49 293.75 3.46 Rebalancng cos 0.0 0.9 0.44 0.26 0.78 0.46 0.3 pbh 0 0.008 0.0947 0.5289 0.206 0.5730 0.6555 A sequence of equdsan refnemens of he nerval [0, ] s defned: 0.... (4) 0 where for = 0,. he number of shares s consan on he nervals ], + ]. Le mn 0 s V F. he frs me he porfolo value ouches he floor. he dscree-me cushon follows. mn s, ( mn, ) r s r ( ) ( ). 0 0 C e V F m m e (5) or recursevely: C r C ( ) m m e f C >0 =. r C 0 e f C (6) V s gven hrough he relaon V = C + F. In order o comply wh he CPPI algorhm and respec praccal consrans, he number of shares of he rsy and safe asses ( and ) are as follows: mc V mn max,0,. V B. (7) When addng ransacon coss, hese are aen as a proporon of he change n he rsy exposure (.e Proporonal cos ( ) ). o a me, he number of shares of he rsy asse wll be reduced: ~ nb of bps. (8) he CPPI capal guaranee s ensured as long as he bond floor s no breached hrough, enablng o fully nves he porfolo no he non rsy asses. he probably of breachng he floor s defned as he probably ha he porfolo value falls below he BF P := V G = 0, : V F floor (.e. he local shorfall probably s he condonal probably defned as: LBF P, = V > F V F. he = wo are relaed as follows P BF LBF = ( P, ). = hs probably whch was equal o zero n he connuous Blac-choles model, s now greaer han zero. Assumng he porfolo hasn breached he floor up o me, he probably of breachng he floor a me + s ha of a downsde jump n he rsy asse of more han abou /m. Is mahemacal expresson s: r LF m P, :. e (9) m where he evoluon of he rs-free par wh rae r s aen no accoun. he bacesng s based on he perod Q-2006 o Q4-200 on &P500 ndex. mulang pahs ( = 0,000) n he Blac & choles model s made usng he 3-monh realzed volaly based on he sandard devaon (see Fgure ), a consan asse reurn m = 8%. he rae of he rs-free asse s r = 4%. hree rebalancng frequences are beng compared regardng he dsrbuon of he fnal porfolo value (daly, weely and monhly), wh he followng assumpons: Inal nvesmen/guaranee: $00, and $00 Duraon: 5 years ransacon coss: 0 bps. he CPPI sraegy under daly rebalancng performs beer agans a bear mare han he weely and monhly ones due o s reacveness o decrease he rsy exposure whenever needed. Wh such frequency, he guaranee s almos ensured; he less frequen we rebalance he more we are exposed o breachng he floor (as llusraed by faer lef als (see Fgure 2 boom, rgh). he bacesng (Fgure 2 op) and able 2 llusrae he followng remars: In perods of mld mare condons, ransacon coss negavely affec he performance of a daly rebalancng, alhough no o a sgnfcan exen. Durng a mare crash, he hree sraeges moneze, wh he daly rebalancng havng less losses han he wo ohers. he emprcal probably of breachng he floor decreases when he rebalancng frequency ncreases. 26
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 he cos of rebalancng ncreases wh he frequency and wh he mulpler. However, n our resuls, he cos of daly rebalancng for m = 6 s lower han he one wh m = 3. hs s explaned by he fac ha such a hgh mulpler allows for a oal rsy exposure and hus no rebalancng reducng he cos. When comparng dfferen sraeges (buy & hold, CPPI wh m = 3 and m = 6), we have he followng resuls: he Buy & Hold sraegy has hgher expecaon and lower sandard devaon (able 2). hs s manly due o he low exposure o he rsy asse. Is performance s hghly correlaed o he non-rsy reurn (chosen o be 4%). he 5% and 0.5% quanles show ha he CPPI wh m = 6 has a larger rgh al and hus, performs beer han he wo ohers n bullsh mare. hs remar s also llusraed n Fgure 8. Daly rebalancng almos prevens he bond floor from beng breached, whch ensures he capal guaranee a maury. However, consan volaly and log-normal dsrbuon modelng are no conssen wh emprcally observed jumps durng exreme mare moves lely o breach he bond floor. In order o relax hese unrealsc assumpons, jumps are hus added hrough Lévy processes as developed n he nex secon. 2..3. Addng jumps. We assume ha he process of he rsy asse follows a Lévy process: d dz, (0) where Z s a Lévy process. he rs-free asse F s sll deermnsc. nf : he me where he porfolo value s fully nvesed n he rs-free asse. Unl * C he acualzed cushon ( C ) s as follows: F * * C C0 ( ml), where denong he sochasc exponenal: Le V B Z < > 2 Z Z s 0 s, Z 0 s s ( Z ) = Z e ( Z ) e,() whch gves us he porfolo value: V 0 V ( ml) V = F0. r( ) Ve f > (2) he probably of breachng he floor can be expressed as: BF = [0, ], = P V B, L <. (3) m / m BF P =exp ( dx), (4) whch s llusraed by he fac ha he number of ownsde jumps of sze more han follows a Posson dsrbuon wh nensy v(-, /m). m For compuaon racably, we choose he double exponenal Kou model (see Kou [2002]). Under he rs neural probably, he rsy asse s modeled as follows: d Y = d dw d e =, (5) where W s a sandard brownan moon, s a posson process wh rae A, he consans and > 0 are drf and volaly of he dffuson par and he jump szes {Y, Y 2,...} are.d.d random varables wh a common asymmerc double exponenal dsrbuon of densy: ( )=( ) y y f y p e p e. (6) Y y0 y<0 + s nensy of posve jumps whle and p are he nensy of negave jumps and he probably of her occurrence. Under hs jump model, and assumng a connuous rebalancng frequency, he probably of breachng he floor aes he followng form: BF P =exp p ( ). (7) m In hs secon, he CPPI sraegy eeps he same characerscs excep for he rsy asse whch s modeled hrough a Kou process calbraed on mpled volaly smle (beween 2006 and 20 on a -monh mpled volaly on a weely ). We carred ou he calbraon by mnmzng he quadrac 9 error Mare Kou C (, K) C (, K,, p,,,) 2, (8) = where s -monh maury, K sres from 80 o 0 and ( p,,,, ) are he jump parameers (more deals abou). We gve dfferen sascs for hese parameers n he able below: Average 5% Quanle d-dev p 0.64 0.84 0.24 + 0.6 0.28 0.06-0.5 0.28 0.07 0.62 2.44 0.2 8.29% 29.64% 0.08 27
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 In order o avod nsably n parameers, we chose several sarng pons and se boundary condons. An example of he resul on he calbraon s shown n Fgure 4. A few remars on he calbraon can be made: nce he upward-slopng par of he smle s very small, he posve jumps are hardly calbraed n a relable manner. However, he prcng of he gap opon (secon II.2.2) only needs he negave jumps nensy (.e he downwardslopng par of he smle). he calbraon s beer on close-o-maury opons (as menoned n anov, 200). I allows a beer capure of nsananeous jump. he calbraed parameers wll be used for hedgng gap rs n he las secon Fgure 5 compares dfferen dscree rebalancng frequences wh a jump modelng: Even for daly rebalancng, breachng he floor s unavodable wh he same probably as he wo oher frequences. he hree rebalancng frequences gve smlar resuls when ang ransacon coss no accoun. Kou model Daly Weely Monhly Mean 46.28 47.0 47.57 d-dev 52.84 52.93 53. 95% quanle 92.9 92.2 92.03 99.5% quanle 59.38 59.08 59.23 5% quanle 238.3 238.67 239.4 0.5% quanle 349.4 350.92 350.37 Rebalancng cos 0.92 0.45 0.26 he prevous llusraons show ha boh he frequency of he rebalancng and he modelng affec he fnal value. he wo mercs prevously defned for dfferen modelng assumpons he local probably of breachng he floor: P := V F V > F. (9) LBF, he overall probably of breachng he floor: BF P := [0, ]: V F = V F. (20) P LBF, For B& model n dscree-me rebalancng: m 2 log( ) ( r) = m 2 (2) BF LBF P = P. (22) and, =0 BF P For Kou jump process n connuous me: =exp p ( ). (23) m Resuls depend on he model parameers and dscrezaon me sep: Gap rs goes o 0 as he rebalancng ends o be more frequen. When consderng a dsconnuous pah (jump models), even n connuous rebalancng he gap rs value > 0. Model Frequency P BF B& Monhly Weely Daly 9. 07 0-5.2 0-0 Kou Connuous 0.0040 Consder he soppng me as he frs me he porfolo value breaches he floor whch does no depend on he bond floor level. he dsrbuon of s he same n case of addng he rache,.e. he probably of breachng he floor s no usually affeced by he rache feaure n heory. However, n our smulaons, hs probably n hgher for he monhly rebalancng. hs mgh be. 2..4. Impac of he rache feaure. he rache feaure s used by nsurance companes o arac nvesors as perodcally locs n prof (see Brgo and Mercuro [2006] and Andersen and Perbarg, 200 for more deals): a annversary daes he guaranee s se o he hghes value so far. he guaranee G becomes a me dependen funcon. V f =0 G G V f 0 * = max( *, * ) =. * * G * f [, ] r ( ) he bond floor s hen defned as F = Ge. (24) hs feaure has advanages and drawbacs. Locngn he cash wll ensure a hgher guaranee bu also reduces he cushon, he rsy exposure and hus he upsde poenal rs. he man resuls from Fgure 6 are: he mean and sandard devaon of he fnal value ncrease wh he rebalancng frequency (see able 2). hs s jusfed by he pah dependency of he guaranee whch has a larger dsrbuon wh hgher rebalancng frequency. he quanles on he wo als of he fnal value dsrbuon ncrease wh he rebalancng frequency, whle he dsrbuon s shfed o he rgh wh narrower body. 28
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 2.2. Mgang downsde rs: Prevenng from breachng he floor. 2.2.. Adjusng he mulpler o mare condons. By focusng on managng reurns n downsde mares, CPPI effecvely manages porfolo volaly. Over he 5-year daa (whch ncluded one bullsh mare, one bear mare and a recovery), he CPPI sraegy resuled n a slghly lower reurn bu also a sgnfcanly lower volaly. Addonally, he wors one year reurn for he CPPI sraegy was sgnfcanly less han ha of he ndex porfolo. able 2. Fnal value mercs: comparson beween a CPPI whou and wh he rache feaure Whou rache Wh rache Daly Weely Monhly Daly Weely Monhly Mean 23.82 24.26 24.7 45.46 43.0 34.03 d-dev 4.96 43.29 47.25 00.08 8.75 45.60 % quanle 99.99 99.68 90.88 00.6 00.53 99.99.5% quanle 99.99 97.57 77.84 99.99 99.94 98.27 % quanle 24.8 26.42 222.23 268.74 26.73 29.52.5% quanle 289.5 292.33 34.58 700.97 603.28 359.59 P BF 0. 0.47 0.64 0. 0.48 0.84 he manager usually ses he mulpler a he begnnng of he perod. he rsy exposure depends hen on he evolvng cushon. As he probably of breachng he floor may surge n mare crash, or he manager mgh mss he subsequen mare recovery, he mulpler needs o be adjused accordngly wh he mare condons. A frs approach o defne a dynamc mulpler s he choce of he opmal m, deduced from he closed form soluons for opmal payoffs, and opmal cerany equvalen reurns (CERs) usng HARA ules and log-normal dsrbuon (see Pezer, 20). he auhors * 2 gve he followng formula m = ( r)/ ( here s he nvesor s sensvy of rs olerance o wealh). A parcular case s he growh opmal leverage wh = whch s resuled n opmzng he growh rae of he leveraged sraegy (cushon). An alernave o he opmal mulpler s a value-ars based mulpler where nvesors choose he confdence level accordng o her rs olerance as well bu focused on al rss. Based on he wegh w of he value-a-rs based porfolo nsurance (VBPI) nroduced by Jang e al. (2009), and he expresson of he rsy exposure n boh sraeges E = = R mc wv he expresson for he mulpler a me s: m =. 2 exp ( r )( ) zp 2 R (25) As he dynamc mulpler depends on boh volaly and reurn esmaes, n order o mprove s effcency, and can be made me dependen. However, snce he esmaon of he drf s hardly accurae for a shor wndow, we wll resran he me dependency o he volaly. I wll be reesmaed hrough a 3-monh sldng wndow o ae no accoun dfferen mare regmes. he wo approaches offer an neresng alernave o he consan mulpler whch lacs flexbly dependng on mare condon. he comparson beween hese wo approaches hrough a bacesng from 2006 o 20 s llusraed n Fgure 7. he focus on wo perods (2006-2007 and pos 2008 crss) (Fgure 8) llusraes ha he VaR-based mulpler can perform beer han he opmal one n bullsh mare and recovery (e.g 8% reurn Q2-2009 unl Q-20 vs % n he pos 2008 crss). In conras, durng bear mare, usng he opmal mulpler (hrough m ) helps eep a relavely hgher cushon bu msses he recovery as doesn allow a hgh leverage. In order o allow o parcpae n he mare recovery o a greaer exen, he mulpler s adjused wh a modfed volaly esmaor, eher hrough a shorerm exponenally weghed movng average (EWMA wh =0.94) realzed volaly or an esmaor based on mpled volaly (of he sre conssen wh he laes mare reurns). For example, f he underlyng jumped 5% downward, he mpled volaly wh sre 95% wll be chosen. For unavalable sres, we use a lnear nerpolaon. hs sraegy sars renvesng no he rsy asse as soon as Q3 2009, resulng n a hgher performance by allowng he porfolo o capure more of he upsde reurn when mares rebound. he bacesng n Fgure 0 llusraes ha he new mulpler s more reacve when adjusng wh he mpled volaly esmaor. However, he 3-monh realzed volaly provdes a hgher mulpler and, when consderng ransacon coss, leads a lower cos of managemen. Fnally, he fxed frequency rebalancng s swched o a rgger rebalancng whch occurs when he mulpler s ou of a specfc range chosen by he porfolo manager. In our case, on average he rebalancng frequency becomes every oher day, whch s conssen wh he usual pracce n CPPI asse managemen. A he same me, he cos of rebalancng 29
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 s cu by half n comparson o a daly rebalancng (.e. as low as a weely or monhly rebalancng). Fgure llusraes he ncreasng performance specfcally under a range-bound hgh volaly regme, e.g. Q- Q3 2008. Adjusng he mulpler dynamcally allows o be more reacve o mare condons and explcly dependen on he nvesor s rs averson. However, does no oally annhlae he downsde rs n case of sudden jumps, where opons may be useful o hedge hose gap rss 2.2.2. Hedgng gap rss. he CPPI mehodology wll no necessarly proec he porfolo agans a blac swan even (such as a mare crash of 20% n one day). o he exen ha asse allocaon shfs are mplemened va underlyng funds, he rebalancng rade can only occur a he end-of-day AV. Even f fuures are used o mplemen shfs nra-day, here can be gap movemens n he fuure mares. hs s where a small gap rs proecon sleeve can add value o he porfolo. o proec agans such a blac swan even, s mporan o already have pu opons on mare ndces n he porfolo. 2.3. Vanlla Pu opon. A smple hedgng sraegy for he CPPI hrough embedded opon can be consruced usng shor maury pu opons. ouchng he bond floor s mahemacally equvalen o he cushon becomng negave. Assumng he even hasn occurred up o me, usng equaon (), we have: C m m e <0 r ( ) <0. (26) Hedgng hs rs s equvalen o forcng hs quany o be posve. hs can be done by buyng a pu opon a each of he CPPI rebalancng perod wh sre r ( ) e and as a maury he CPPI rebalancng m frequency. o hedge he whole porfolo he manager C needs a number of m pus, whch s he rsy asse exposure. he dscouned payoff n hs case s r r (( ) e C m e m ). he hedgng cos a me can be wren as: C r Cos = ( ) m e. (27) m wo approaches can be consdered: he hedgng coss (pu prces) are deduced only aferwards from he porfolo value (whch allows an esmaon of how much he hedge would cos). In hs case, he cushon follows he recursve relaon: = ( ) r. C C m m e (28) he cos of hedgng can be compued as he sum of all pu opons prces necessary for he hedgng: n C r C = m e. (29) =0 m In pracce, he prce of he pus used for he hedge wll be deduced from he porfolo value a each sep. hs s ranslaed n he second approach where he cushon dynamcs follows he recursve equaon: r m ( m) e r = C e C. r m ( m) e (30) We compare he effecs of he pu hedgng n Fgure 3. We can see he followng remars: he guaranee s ensured and he manager no longer holds he rs of breachng he floor. However, once he pu s exercsed and he floor recovered, he manager needs o moneze n order o eep he guaranee unl maury. In erms of dsrbuons, he CPPI dsrbuon wh a pu hedgng s a runcaon of he classcal CPPI where losses are cu (lef al lmed by he guaranee). 2.4. Gap pu opon. An alernave rs mgang acon les n he use of Gap Opons whch allow for a proecon agans sudden sgnfcan and perssen downsde mare moves: f a gap even occurs beween wo consecuve daes, he buyer receves he dfference beween he performance of he rsy asse a gap r = and he hreshold J. In case of he CPPI, he proposed soluon s a gap pu opon whose noonal s he rsy exposure wh sre J = /m, where m s he mulpler. 2.5. Defnon and properes. (see anov, 200 for deals). uppose ha he me o maury of a gap opon s subdvded ono perods of lengh h (e.g. days): h =. he reurn of he h perod wll be denoed by R = h / ( ) h. Le denoe he reurn level whch rggers he gap even and * 30
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 be he me of frs gap expressed n he uns of h: * h := nf{ : R }. he gap opon s an opon h whch pays o s holder he amoun f ( R * ) a me * h, f * and nohng oherwse. Assumng a deermnsc neres rae r and an..d h log reurns ( R ) = and denoe he dsrbuon of h log ( R ) by ph ( dx ). hen he prce of a gap opon s gven by: r e ( ph ( dx)) rh x Gh = e f ( e ) ph( dx), r e p ( dx) wh := log( ) < 0. h (3) Obanng numercal resuls usng hs formula s complcaed n he general case. An approxmae formula s used. X Assume = 0e, where X s a Lévy process. 4 Consderng he hypohess rh 0 and h 0, he followng formula s obaned: r e ( h ( )) x p dx G h f ( e ), (32) r e ( dx) Assumng a Kou model (for s racably and smplcy n negraon) and consderng he pu payoff (.e f ( x)=( K x) ). he prce hen becomes: ( / r pe ) p / e G h K. / (33) r pe wh p he probably ha a gven jump s negave, s nensy and he Posson process rae. Moreover, for he CPPI we are neresed n he payoff rh h (( m ) e m ) whch s equvalen o ( ) h m rh ( e x ) and hus, K = ( /m)e -rh. m able 3. Fnal value mercs: comparson beween dfferen hedgng sraeges Vanlla Pu Hedge Hedgng sraeges Vanlla Pu Hedge 2 Gap Opon Mean 36.97 33.35 34.98 % quanle 28.70 25.40 27.00 5% quanlee 277.2 273.53 275.22 Hedgng cos.c 2.26.08 he gap pu opon allows o cu he loss compensaes for he loss as he porfolo value breaches he bond floor. However, nsurance nvesors holdng a CPPI who wan o hedge wh gap opon may face he followng ssues: he prce of he gap opon s usually sold hgher han s heorecal cos for several reasons: he cos of he hedgng he gap opon for he ban may be que hgher because of he llqudy of deep ou of he money opons ha replcae. he replcang formula s rcy o mplemen and nerpre, as sgnfcanly model dependen (jumps mulple parameers, lac of robusness). Acually, he gap opon proposed by he ban mgh have a dfferen desgn and payoff from he one consdered for he hedge. he ban usually hedges he gap up o he frs order only. he gap rs s borne by he ban only f here s some reconclaon by he nsurance company whn 24/48 hours, ou of whch he nsurer bears oneself he gap rs. As a resul, operaonal rss are sgnfcan and represen a major par of he economc capal requremens (e.g. under olvency II framewor). Concluson In hs arcle we have presened a sudy of he CPPI as an nsurance conrac, a revew of s heory and pracce as well as s modelng and hedgng ssues for a rs/reurn/cos perspecve. he man conclusons are: Connuous CPPI s only heorecal: gven mare frcons and he probably of no ensurng he guaranee, all he more ha jumps occur more han no. As a resul, jump processes are a valuable npu for he CPPI modelng: hey allow o cach a probably of breachng he floor dfferen han zero (even n he connuous-me framewor; Garca and Goosens, 2009 and Garca e al., 2008) came up wh he same concluson) and herefore, deec, defne and hedge gap rs. Correcly choosng and adjusng he mulpler dynamcally sgnfcanly reduce he downsde rs accordng o a Value-A-Rs ndcaor: he mulpler decreases n perod of urmols reducng he rsy exposure and ncreases bac durng mare recovery. Hedgng he gap rs s possble hrough wo ypes of opons: vanlla pus and gap pu opons. he frs one s more common due o lqud asses, bu he hedgng cos may urn ou o be oo expensve and he maury oo lmed. he second ype of opons s less lqud (bough only hrough an agreemen) bu s cheap. 3
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 References. L. Andersen and V. Perbarg. (200). Ineres rae modelng, Alanc Fnancal Press. 2. F. Blac and R. Jones. (987). mplfyng porfolo nsurance, Journal of Porfolo Managemen, 4 () pp. 48-5. 3. Damano, Brgo and Fabo. Mercuro. (2006). Ineres rae models heory and pracce: wh smle, nflaon and cred. prnger Fnance. 4. J. Garca,. Goosens, and W. chouens. (2008). Le s jump ogeher: Prcng cred dervaves. Rs, pp. 30-33, epember. 5. Joao, Garca and erge, Goosens. (2009).he ar of cred dervaves: Demysfyng he blac swan. he Wley Fnance eres. 6. C. Jang, Y. Ma and Y. An. (2009). he effecveness of he var-based porfolo nsurance sraegy: An emprcal analyss. Inernaonal Revew of Fnancal Analyss, 8, pp. 85-97. 7..G. Kou. (2002). A jump-dffuson model for opon prcng. Managemen cence, 48 (8), pp. 086-0, Augus. 8. R.C. Meron. (97). Opmum consumpon and porfolo rules n a connuous-me model, Journal of Economc heory, 3, pp. 373-43. 9. F. Perold. (986). Consan porfolo nsurance. Harvard Busness chool. 0. Jacques Pézer. (20). Raonalzaon of nvesmen preference crera, ICMA Cenre Dscusson Papers n Fnance DP20-2.. P. anov. (200). Prcng and hedgng gap rs. Journal of Compuaonal Fnance, 3 (3), pp. 33-59, prng. Appendx Fg.. Evoluon of an nvesmen n he &P500 for he perod Q 2006 o Q4 200 32 Fg. 2. Bacesng and dsrbuon of he hree varous rebalancng frequences under B& model.
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 Fg. 3. Comparson beween he Buy & Hold sraegy, CPPI wh m = 3 and CPPI wh = 6 hrough bacesng (&P500) Fg. 4. Calbraon of he Kou model usng -monh maury call opons prce on he &P500 Fg. 5. mulaon and dsrbuon of he hree varous rebalancng frequences under Kou model. 33
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 Fg. 6. he fgure on he op s a bacesng on he prevous se of daa o compare a classcal CPPI and one wh he rache feaure. he hree hsograms on he boom are hose of he fnal value dsrbuon for he hree dfferen rebalancng frequences on he CPPI wh rache Fg. 7. Comparson beween dfferen mulplers (VaR-based wh p = 99.5% and he opmal one wh rs olerance = 0.2, 0.4 and ) based on Realzed Volaly 34
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 Fg. 8. Focus on wo bullsh mare perods where he CPPI wh he VaR-based m performs beer han he opmal one Fg. 9. Comparson beween dynamc mulpler based on RV and on IV hrough bacesng Fg. 0. Comparson beween dynamc mulpler based on RV and on EWMA hrough bacesng 35
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 Fg.. Comparson beween rgger rebalancng vs fxed frequency rebalancng Fg. 2. Comparson beween he dynamc mulpler and an adjused one based on a manager decson dependng on mare recovery 36
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 Fg. 3. Comparson beween no hedgng and pu hedgng n s wo approaches Fg. 4. Comparson beween a vanlla and a gap opon hedgng 37