PRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES

Transcription

1 PRICING AND PERFORMANCE OF MUUAL FUNDS: LOOKBACK VERSUS INERES RAE GUARANEES NADINE GAZER HAO SCHMEISER WORKING PAPERS ON RISK MANAGEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAGEMEN AND INSURANCE APRIL 28

2 1 PRICING AND PERFORMANCE OF MUUAL FUNDS: LOOKBACK VERSUS INERES RAE GUARANEES Nadine Gazer Hao Schmeiser JEL-Classificaion: D81, G11, G13, G22 ABSRAC he aim of his paper is o compare pricing and performance of muual funds wih wo ypes of guaranees: a lookback guaranee and an ineres rae guaranee. In a simulaion analysis of differen porfolios based on sock, bond, real esae, and money marke indices, we firs calibrae guaranee coss o be he same for boh invesmen guaranee funds. Second, heir performance is conrased, measured wih he Sharpe raio, Omega, and Sorino raio, and a es wih respec o firs, second, and hird order sochasic dominance is provided. We furher invesigae he impac of he underlying fund s sraegy, firs looking a a convenional fund having a consan average rae of reurn and sandard deviaion over he conrac erm, and hen a a Consan Proporion Porfolio Insurance managed fund. his analysis is inended o provide insighs for invesors wih differen risk-reurn preferences regarding he ineracion of guaranee coss and he performance of differen muual funds wih embedded invesmen guaranees. 1. INRODUCION In recen years, here has been an increasing demand for invesmen producs wih financial guaranees. For example, sales of uni-linked life insurance producs have seen subsanial growh. 1 hese conracs are ypically muual 1 Boh auhors are wih he Universiy of S. Gallen, Insiue of Insurance Economics, Kirchlisrasse 2, 91 S. Gallen, Swizerland. In he European life insurance marke, he share of uni-linked producs in oal premium volume has increased from 21.8% in 23 o 24.2% in 25 (CEA, 27, p. 11). For insance, in France, he second larges life insurance marke in Europe, he growh rae of 12.4% in premium income in 25 was mainly driven by an increase in sales of uni-linked producs (CEA, 27, p.13). Invesmen producs in general have been enoying a growh surge. For example, German invesmen funds currenly manage 126 guaranee funds (up

3 2 funds wih invesmen guaranees ha addiionally offer erm insurance. hus, he mauriy payou depends on he performance of he underlying fund. From he invesors perspecive, hese muual fund producs are generally aracive due o he possibiliy of paricipaing in posiive marke developmens combined wih a guaraneed minimum payoff a mauriy. Brennan and Schwarz (1976) and Boyle and Schwarz (1977) were he firs o invesigae asse guaranees in uni-linked life insurance producs. Goldman, Sosin, and Gao (1979), Conze and Viswanahan (1991), Gerber and Shiu (23a), and Lin and an (23) derive closed-form soluions for he valuaion of differen exoic opions, including lookback opions and dynamic fund proecion wih boh deerminisic and sochasic guaraneed levels. Gerber and Shiu (23b) rea dynamic fund proecion in he conex of equiy-indexed annuiies, i.e., for perpeual American opions. Lachance and Michell (23) and Kling, Ruß, and Schmeiser (26) analyze he value of ineres rae guaranees in governmen-subsidized pension producs in a Black/Scholes framework. However, o dae, here has been no comparison of ineres rae and lookback guaranees for differen underlying funds and differen invesmen sraegies wih respec o pricing and performance, even hough his informaion is an imporan prerequisie for decision making. he presen analysis inends o fill his gap by providing his informaion o invesors wih differen risk-reurn preferences. In his paper, we compare pricing and performance of wo muual funds wih differen invesmen guaranees: he firs conrac provides an ineres rae guaranee on he premiums paid ino he conrac. he second produc includes a lookback guaranee, under which he payoff is defined by he number of unis he clien acquired over he conrac erm muliplied by he highes value of uni price achieved before mauriy. he payoff for each produc is highly dependen on he underlying fund sraegy. In he case of a convenional fund wih fixed average rae of reurn and sandard deviaion, guaranee coss can be derived. Alernaively, guaranees can be secured using a Consan Proporion Porfolio from 81 a he end of 25) wih a fund asse value of billion Euros (up from 1.1 billion Euros a he end of 25) (

4 3 Insurance (CPPI) sraegy via dynamic reallocaion of he invesmen in risky and riskless asses. Iniial guaranee coss are deermined using opion pricing heory in a Black/Scholes framework. his pricing approach assumes he replicabiliy of cash flows, which is a realisic assumpion for produc providers, bu no usually feasible for invesors. hus, a comparaive assessmen of wo invesmen alernaives (i.e., a muual fund wih eiher a lookback guaranee or an ineres rae guaranee) will ypically depend on risk-reurn preferences and can be based on performance measures. Furher, if invesors pay he same premium for eiher ype of conrac, only he risk-reurn profile of he mauriy payou maers in he performance measuremen. o accoun for hese issues and o obain a comprehensive picure of he characerisics of muual funds wih invesmen guaranees, we employ he following procedure. We firs calibrae guaranee coss o be he same for boh guaranee producs. Nex, we invesigae he characerisics of he mauriy payoffs of hese producs by calculaing descripive saisics and by using hree performance measures (Sharpe raio, Omega, and Sorino raio) for he wo fund sraegies (CPPI and average reurn and sandard deviaion). We also es for firs, second, and hird order sochasic dominance. Empirical resuls are derived for differen μ-σ-efficien diversified porfolios based on sock, bond, real esae, and money marke indices. Comparing producs wih differen guaranees is ofen difficul due o differen mauriy guaranees, differen underlyings, and differen paymens by he clien (caused by differen guaranee coss). Hence, in order o ensure comparabiliy, he premium paymen is assumed o be he same for all cases under consideraion. We firs compare he siuaion where boh invesmen funds provide a minimum ineres rae guaranee of % (i.e., a money-back guaranee) and boh funds underlying is managed on he basis of a CPPI sraegy. Because of he possibiliy of a (on average) higher sock porion in he case of an invesmen fund wih an ineres rae guaranee in a CPPI framework, we find a considerably higher expeced payoff and sandard deviaion of he mauriy payoff compared o he siuaion involving he lookback guaranee. Second, we

5 4 analyze a case in which boh producs provide he same convenional underlying fund and he same implied guaranee coss. Even hough boh funds have quie similar expeced payoffs in his case, he muual fund wih a lookback guaranee has roughly a 2.5% probabiliy of resuling in a payoff below he minimum mauriy guaranee promised by an ineres rae guaranee. Furhermore, we find ha neiher invesmen alernaive dominaes he oher by firs, second, or hird degree. Overall, he resuls illusrae he srong effec of fund volailiy on he lookback guaranee, which can rapidly become very expensive compared o he ineres rae guaranee. he remainder of he paper is organized as follows. In Secion 2, he model framework for he wo differen guaranee ypes is inroduced. In Secion 3, wo differen invesmen sraegies concerning he underlying funds are derived. Secion 4 provides he valuaion of he implied guaranees and an analysis of he mauriy payoff using descripive saisics and differen performance measures. Several numerical examples based on a Mone Carlo simulaion are provided in Secion 5. Secion 6 concludes. 2. MODEL FRAMEWORK We assume ha boh producs under consideraion have a erm of years wih consan monhly premium paymens P a ime =, 1,, N-1 (wih Δ = 1 = 1/12 ). he premiums are invesed in a raded muual fund and yield a sochasic payoff in N =. he muual fund is spli ino unis, where S( i ) denoes he uni price of he fund a ime i. Hence, he number of unis acquired a ime i is given by he premium paymen divided by he uni price, i.e., P n =, i {,..., N 1}, i S i and he oal number of unis a ime i before paying he (i + 1) s premium is i 1 = { } N = n, i 1,..., N 1. i

6 5 Muual fund wih ineres rae guaranee A fund wih an ineres rae guaranee provides he invesor a minimum ineres rae guaranee g on he premiums paid ino he conrac. hus, he guaraneed mauriy paymen resuls in N 1 g( ) G = P e. = For g =, his implies G = N P and for g >, we obain G g 1 e = P e 1 e g gδ. he value of he invesmen in, F, is given by he number of acquired unis N imes he value of a uni, S, leading o S N 1 = =, = S F N S P or, equivalenly, a ime S = +. ( ) F F 1 P S 1 A mauriy, he invesor receives he erminal payoff L G, which consiss of he value of he invesmen in he underlying fund, which will be a leas he guaraneed paymen G, i.e., N 1 N 1 G S L = max ( F, G) = max P, P e = S = N 1 N 1 S g ( ) G = P max, e = P L. = S = ( ) g

7 6 hus, he amoun of premium paymens only serves as a scalar of he acual payoff. he payoff o he invesor in, L G, can be wrien as he value of he underlying asses plus a pu opion on his value wih srike G, such ha ( ) ( ) G L = max F, G = F + max G F,. (1) Muual fund wih lookback guaranee he fund wih he lookback feaure guaranees a payoff of he highes value (or peak) H of he index ha has been aained during he policy erm, where H = max S. {,..., N 1} hus, he payoff in depends on he previous N 1 uni prices and can be wrien as max S L N H P P L N 1 H = = = = S {,..., N 1 } H. he lookback guaranee s mauriy payoff benefis from ups and downs in uni price. he wors case for he invesor would be if he uni price of he underlying fund does no move a all, bu remains consan over he conrac erm. As before, he exac amoun of premium paymens only serves as a scaling facor. 3. INVESMEN SRAEGIES OF UNDERLYING FUNDS In he following, we compare wo invesmen sraegies: firs, we model he underlying asses of a fund wih fixed average rae of reurn and sandard deviaion during he policy erm (he convenional fund ). he second case involves an underlying fund ha uilizes a Consan Proporion Porfolio Insurance (CPPI) sraegy.

8 7 Convenional fund Le ( W ),, be a sandard Brownian moion on a probabiliy space (Ω, F, P) and (F ),, be he filraion generaed by he Brownian moion. In he sandard Black/Scholes framework, for he convenional fund, he uni price evolves according o a geomeric Brownian moion. Hence, i can be described by he sochasic differenial equaion (under he obecive measure P ) ( μ σ ) ds = S d + dw, wih consan drif μ, volailiy σ, and a sandard P -Brownian moion W, assuming a complee, perfec, and fricionless marke. he sochasic differenial equaion is solved by (see, e.g., Börk, 24) 2 ( μ σ /2) ( 1) + σ ( 1)( W W 1 ) S = S e 1 2 ( μ σ /2)( 1) + σ ( 1) = S e 1 = S R 1, Z where Z are independen sandard normally disribued random variables. Hence, he coninuous one-period reurn r ln ( ) = R is normally disribued wih an expeced value of μ σ 2 /2 and sandard deviaion σ. Consan Proporion Porfolio Insurance (CPPI) managed fund In case of a convenional fund, guaranees have o be secured using risk managemen measures like, e.g., hedging, reinsurance, or equiy capial. Insead of invesing in risk managemen measures, guaranees can be secured using porfolio insurance sraegies, which dynamically reallocae he invesmen porfolio so as o reach he mauriy guaranee and, also, paricipae in rising markes (see O Brien, 1988). Porfolio insurance was developed by Leland (198) and Rubinsein and Leland (1981). In his conex, Perold and Sharpe (1988) showed ha hese payoff sraegies have o be convex, i.e., an increasing porion invesed in sock when sock prices go up, and vice versa. CPPI was firs inroduced by

9 8 Black and Jones (1987). CPPI secures he guaranees via coninuous dynamic reallocaion of he invesmen beween wo asse classes, namely, a risky and a riskless asse. 2 Under he obecive measure P, he risky invesmen A evolves according o a geomeric Brownian moion da = A( μd + σdw), and he riskless invesmen is a bond process B wih a consan riskless rae of reurn r resuling in db = Brd. For a discree (monhly) adusmen of he share in boh asse classes, he evoluion of he underlying fund is given by A A B r r S S α 1 1 ( 1 1) 1 1 ( 1 S e 1) e A α B α α Δ = + = A In his seing, r = μa Δ + σ A Δ Z denoes he coninuous one-period reurn of he risky invesmen wih yearly expeced value μ A and yearly sandard deviaion σ A. he value of he accumulaed invesmen in he muual fund S in i before paying he i-h premium is given by rδ ( ) S F = F + P = F + P e + 1 e, A ( ) i r 1 ( 1 ) α 1 ( α 1) S 1 i i i i F =. he guaranee o be secured in he case of he lookback guaranee is G H i i k i k =, P = max S S and in he case of he ineres rae guaranee by 2 However, porfolio insurance programs may fail in case of high ransacion coss, due o marke liquidiy risk, disconinuous price process (including ump componens), or unexpeced changes in he volailiy of he underlying socks (Rubinsein and Leland, 1981, p. 66). In he sock marke crash in 1987 he wo imporan precondiions marke liquidiy and coninuous price processes were violaed a he same ime (Rubinsein, 1988, p. 39). In his case, an invesor may no be able o adus is sock posiions in he asse porfolio o he degree demanded by he underlying rading sraegy.

10 9 G g( ) G = P e. i i = he cushion C for he risky invesmen resuls from he difference beween he curren fund value (including he curren premium paymen) and he presen value of he guaranee G, giving ( ) C = F + P e G. r ( i ) i i i he sock exposure in period [ i, i+1 ) is limied by he facor α and can be calculaed as he produc of he muliplier (or leverage) m and he cushion C, i.e., α i mc i = min max,, α. F i he muliplier m corresponds o he invesor s risk aversion. A high muliplier implies heavy paricipaion in posiive marke developmens hrough a high exposure in he risky invesmen. A low muliplier reduces he shorfall probabiliy of he CPPI sraegy. 4. VALUAION OF HE INVESMEN GUARANEES AND PERFORMANCE MEASUREMEN Ne presen value calculaions are based on he replicabiliy of he conrac s cash flows wih asses raded on he capial marke. An individual ha is able o replicae a cash flow will decide in favor of a conrac if is ne presen value is posiive. Hence, his decision does no depend on, e.g., he individual s degree of risk aversion. Even hough he replicabiliy of he conrac s cash flow can be regarded as pracicable for produc providers, we hink i is usually no feasible for buyers of muual funds as, e.g., shor selling of asses is in general required. hus, a comparaive assessmen of wo invesmen alernaives (here: a muual fund wih a lookback or an ineres rae guaranee) from he viewpoin of a produc buyer will ypically depend on risk preferences.

11 1 One common way o proceed in his siuaion is o compare he expeced discouned value of he conrac s cash flows given a ime separable uiliy funcion. In our case, he same sequence of premiums is paid ino boh conracs and hence, a preference dependen valuaion of he mauriy payoff is sufficien for comparison. Insead of adoping specific uiliy funcions, risk reurn models can be used, which form he basis for performance measures. hese models have he advanage of being easier o handle and only require explici measures of risk and reurn as well as a funcional relaionship beween risk and reurn. In wha follows, we focus on hree performance measurers, namely he Sharpe raio, he Omega, and he Sorino raio. he form of uiliy funcions ha makes a decision based on he Sharpe raio, Omega or he Sorino raio consisen wih he concep of expeced uiliy maximizaion is shown in, e.g., Fishburn (1977), Sarin and Weber (1993), Farinelli and ibilei (28). Valuaion of he invesmen guaranee In he case of a convenional fund (i.e., wih given average rae of reurn and sandard derivaion for he conrac erm), prices for invesmen guaranees a ime = will be obained using risk-neural valuaion echnique. Under he unique equivalen maringale measure Q (see Harrison and Kreps, 1979), he drif of he uni price process changes o he riskless rae of reurn r, leading o ( ) ds S rd dw = +σ Q, where W Q is a sandard Q -Brownian moion. he ne presen value of he invesmen guaranee Π a ime = is given as he difference beween he expeced presen value of he conrac s payoff under he risk-neural measure Q and he presen value of he premiums paid, discouned wih he riskless ineres rae r: N 1 N 1 Q r r Q r Π = E ( e L) P e = P E ( e L ) e = = r. he guaranee coss mus be paid by he invesor a ime = in addiion o he ongoing premium paymens and he provider mus inves hem in risk manage-

12 11 men measures such as hedging sraegies, equiy capial, or reinsurance. In he case of a muual fund wih an ineres rae guaranee, Π can also be wrien as (see Equaion (1)) ( ( )) G r Q Π = e E max G F,, which is he price of a European pu opion on he fund value a mauriy wih srike G. Analysis of he mauriy payoff o analyze he mauriy payoff L, we calculae is expeced value ( ) = P E( L ) and sandard deviaion σ ( L ) P σ ( L ) E L = under he obecive measure P. Furhermore, hese figures can be used for performance measuremen by way of he Sharpe raio (see Sharpe, 1966). As a performance measure, he Sharpe raio (SR) akes risk and reurn ino accoun. For our case, we define he Sharpe raio as he difference beween he conrac s expeced payoff E( L ) and he value of he premium paymens compounded o mauriy N 1 r ( ) Y = P e, divided by he sandard deviaion of he mauriy payoff ( ) = ( ) L σ : ( ) Sharpe raio L = ( ) σ ( L ) E L Y. In addiion o he Sharpe raio, wo oher common performance measures he Omega and he Sorino raio are employed, which use lower parial momens as he relevan risk measure. Lower parial momens belong o he class of downside-risk measures ha describe he lower par of a densiy funcion; hence only negaive deviaions are aken ino accoun (see, for example, Fishburn (1977), Sorino and van der Meer (1991)). he lower parial momen of order k is given as ( ) ( ) ( ) LPM L, Y = E max Y L, k. k

13 12 For decision making, he degree of risk aversion can be conrolled by varying he power k. For k =, only he number of shorfall occurrences is couned; for k = 1, all deviaions are weighed equally. Hence, he Omega (see Shadwick and Keaing, 22) and he Sorino measures (see Sorino and van der Meer, 1991) can be obained by ( ) Omega L ( max (,) ) LPM1 ( L, Y) E L Y =, ( ) Sorinoraio L ( max (,) ) LPM ( L, Y ) E L Y =. 2 he probabiliies ψ ha he fund value a mauriy does no cover he promised guaranees are given by ( F ) G Ψ = PG >, and H ( F ) H Ψ = P L > for he funds wih ineres rae guaranee and lookback guaranee, respecively. he performance measures, as well as he probabiliies ψ, do no depend on he amoun of premiums paid ino he conrac. Sochasic dominance he wo alernaive invesmens are furher esed for sochasic dominance. A discussion of he relaion beween sochasic dominance crieria and uiliy heory can be found in, e.g., Bawa (1975) and Levy (1992). Le F 1 denoe he cumulaive disribuion funcion of L G and F 2 denoe he cumulaive disribuion funcion of L H on he inerval [a, b]. hen

14 13 G H L dominaes L by he firs degree (FSD) if and only if F x F x all x a b ; ( ) ( ) [ ] 1 2, G L dominaes L H by he second degree (SSD) if and only if x x F ( ) d F ( ) d all x [ a, b] 1 ; 2 G L dominaes L H by he hird degree (SD) if and only if x v x v b b F ( ) ddv F ( ) d dv all x [ a, b] 1 and ( ) ( ) 2 F d F d 1 2 (e.g., Aboudi and hon, 1994). All hree cases require sric inequaliy for a leas one x. 5. SIMULAION ANALYSES Inpu parameers Providers of invesmen producs wih guaranees ypically hold a worldwide diversified porfolio of socks, bonds, real esae, and money marke insrumens. In our analysis, we include one marke index for each asse class: For socks, we use he Equiy Marke Proxy used in Fama and French (1993) and Carhar (1997), which is a value-weighed porfolio of all NYSE, Amex, and Nasdaq socks. Bonds, real esae and money marke indices are given by JPM Global Governmen Bond, GPR General PSI Global, and he JPM US Cash 3 Monh (as a proxy for he risk-free rae of reurn), respecively. We exraced monhly reurns beween January 1994 and December 25 from he Daasream daabase o obain mean and sandard deviaion of he annualized reurns (able 1), and o calculae heir correlaions (able 2).

15 14 ABLE 1: Expeced value μ A and sandard deviaion σ A of annualized reurn for seleced indices Asse class Index Abbreviaion μ A σ A Socks Equiy Marke Proxy (S) 1.27% 16.16% Bonds JPM Global Govermen Bond (B) 5.26% 6.49% Real esae GPR General PSI Global (R) 9.47% 11.63% Money Marke JPM US Cash 3 Monh (M) 3.57% - Noes: JPM: JPMorgan Chase & Co., GPR: Global Propery Research, PSI: Propery Share Index ABLE 2: Correlaion marix for seleced indices in able 1 Index (S) (B) (R) (S) (B) (R) he seleced marke indices can usually be acquired over index funds wih low ransacion coss. In addiion, hey are broadly diversified so ha hey are generally well suied for performance measuremen (for crieria o selec represenaive benchmark indices, see, e.g., Sharpe, 1992). Based on he indices reurns and heir correlaion, we calculae μ-σ-efficien porfolios under shorselling resricions. able 3 ses ou efficien porfolios ha will be used as he basis for he underlying funds in he simulaion analysis. he money marke index JPM US Cash 3 Monh (M) represens he riskless invesmen for he CPPI managed fund (firs par in able 3). o obain inpu daa for he risky invesmen, efficien porfolios CP 1 and CP 2 were calculaed based on sock (S), bond (B), and real esae (R) indices, wihou he money marke index. For he convenional fund (second par in able 3), all four indices were included in he calculaion of efficien porfolios. hus, porfolios CV 1 and CV 2 will serve as underlying for he convenional fund.

16 15 ABLE 3: Efficien porfolios based on indices in able 1 Porfolio μ A σ A Share in Share Share in Share (S) in (B) (R) in (M) CPPI CP 1 6.% 5.8% 7.65% 83.78% 8.58% - CP 2 7.5% 6.75% 16.34% 49.82% 33.85% - r 3.57% % Convenional CV 1 6.% 4.16% 1.3% 26.6% 21.91% 41.2% CV 2 7.5% 6.74% 16.65% 43.2% 35.43% 4.9% he ime o mauriy of boh producs is given wih = 1 years wih monhly premium paymens P = 1. Concerning he CPPI sraegy, he muliplier is fixed a m = 2 and ensures, in he numerical examples provided in he following secion, he adherence of he invesmen guaranee. For he produc wih underlying CPPI sraegy, he sock exposure α is iniially limied o α = 5%. Because of pah dependence, here is generally no closed-form soluion for he payoff srucure of he conracs. Hence, numerical valuaion is conduced using Mone Carlo simulaion (see Glasserman, 24) on he basis of 1, simulaion runs wih monhly reallocaion of he porfolio when using CPPI sraegy. o ensure ha he simulaion resuls for he wo producs are comparable, we used he same sequence of random numbers for all simulaions. 3 o es for firs, second, and hird order sochasic dominance he mehod proposed in Porer, War, and Ferguson (1973) is implemened. he algorihm makes furher use of wo properies of he ordering rule ha simplify he es procedure. Firs, a larger mean is a necessary condiion for dominance; second, F 1 ( L G ) canno dominae F 2 ( L H ) if he smalles value of L H is larger han he smalles value of L G. Hence, if hese condiions are violaed neiher alernaive dominaes he oher G H (ha is if, e.g., E( L ) > E( L ) bu he smalles value of L G is lower han he smalles value of L ). H 3 In he following calculaions, he sandard error for E ( L ) is beween.26% and.48%.

17 16 wo differen cases are analyzed. Firs, we compare he wo invesmen guaranees when boh producs are managed wih CPPI and have he same minimum rae of reurn g of %. Second, boh producs are assumed o have he same underlying convenional fund wih fixed parameers over he conrac erm. o increase comparabiliy and be able o analyze he pure impac of he invesmen guaranees, we calibrae he guaranee such ha he producs have idenical guaranee coss. We furher analyze resuls for differen efficien porfolios wih differen expeced value and sandard deviaion of he reurn. In boh cases under consideraion, he sum of premiums paid ino he conrac is 12,. Invesmen guaranees wih CPPI managed underlying fund Firs, we sudy he case where he funds of boh invesmen producs are managed wih CPPI and have a minimum rae of reurn of g = %. Descripive saisics, performance, and cumulaive disribuion funcions for ineres rae and lookback guaranee are displayed in Figure 1. In Par a), he fund is given by porfolio CP 1; Par b) conains oucomes based on porfolio CP 2 wih higher volailiy (see able 3). Due o using CPPI, guaranee coss are implicily conained in he conrac s payoff. his also implies ha he CPPI sraegy leads o a zero probabiliy ψ. Figure 1 illusraes ha he fund wih an ineres rae guaranee has a higher expeced value and a much higher sandard deviaion han he fund wih he lookback guaranee, despie a minimum rae of reurn of %. When changing he underlying fund o porfolio CP 2 (Par b)), he mauriy payoff s sandard deviaion in case of he ineres rae guaranee is increased from 868 o 1,13, while he change in he sandard deviaion of he lookback guaranee is more moderae.

18 17 FIGURE 1: Resuls for ineres rae and lookback guaranees for CPPI managed underlying fund a) Porfolio CP 1: μ A = 6.%, σ A = 5.8% Descripive saisics and performance Ineres Guaranee 12, (g = %) Lookback 12, (g = %) ψ % % E( L ) 15,421 14,866 σ ( L ) Sharpe raio Omega Sorino raio FSD, SSD, SD none none Descripive saisics and performance Ineres Guaranee 12, (g = %) F(x) Cumulaive disribuion funcions Cumulaive Disribuion Mauriy Payoff Lookback guaranee Ineres rae guaranee x x 1 4 b) Porfolio CP 2: μ A = 7.5%, σ A = 6.75% Lookback 12, (g = %) ψ % % E ( L ) 16,19 15,157 σ ( L ) 1, Sharpe raio Omega Sorino raio FSD, SSD, SD none none F(x) Cumulaive disribuion funcions Cumulaive Disribuion Mauriy Payoff Lookback guaranee Ineres rae guaranee x x 1 4 Noes: g = minimum rae of reurn; ψ = probabiliy ha value of fund a mauriy is lower han guaraneed payoff; L = conrac s payoff a mauriy ; FSD, SSD, SD = Firs, Second, hird Order Sochasic Dominance. hese resuls are due o he possibiliy of a (on average) higher sock porion α when using he CPPI sraegy in he muual funds wih an ineres rae guaranee compared o he case of a lookback guaranee. Furher analysis showed ha when raising he maximum share in he risky invesmen from α = 5% o α = 1%, mainly he sandard deviaion of he mauriy payoff of he fund wih ineres rae guaranee is concerned. he sandard deviaion increases subsanially due o

19 18 even higher shares in he risky invesmen. In conras, he disribuion of he lookback guaranee shows almos no changes. For hese reasons, he fund wih he ineres rae guaranee has a considerably higher percenage of mauriy payoffs ha are above 14,5, and also several above 18,, whereas he lookback guaranee payoffs peak ou a 16,5. hese resuls are evidence ha a larger sandard deviaion combined wih a minimum rae of reurn allows paricipaion in posiive marke developmens, hus leading o a higher probabiliy of receiving a high payoff a mauriy and, a he same ime, be assured of receiving a leas he guaraneed payoff. Furhermore, he performance of he wo producs depends on he ype of measure chosen. Specifically, for porfolio CP 1 and CP 2 in Figure 1, he Sharpe raio is slighly higher for he lookback guaranee; he Omega and he Sorino raio are higher for he ineres rae guaranee. his observaion illusraes ha i makes a difference how deviaions from he expeced value are aken ino accoun. he Sharpe raio uses he sandard deviaion as a measure of risk, which also includes upside deviaions and hus chances. his leads o beer resuls for he muual fund wih lookback guaranee. In conras, he Omega and Sorino raio solely evaluae downside risk using lower parial momens. Under hese risk preferences, an invesor would choose he fund wih ineres rae guaranee in his example. Invesmen guaranees wih convenional underlying fund Second, resuls for he wo guaranee producs for a convenional underlying fund are compared. Figure 2 ses ou resuls for ineres rae guaranee and lookback guaranee. Par a) shows resuls for porfolio CV 1; Par b) is based on porfolio CV 2 (see able 3 for deails on he porfolios). o ensure he guaraneed paymens a mauriy, boh producs require guaranee coss Π ha are due a he conrac s incepion, in addiion o he monhly premium paymens (see Figure 2, firs row in lis of descripive saisics). Oher financial guaranees are no embedded in he conracs under consideraion and hence do no influence he guaranee coss. In paricular, we assume ha he in-

20 19 vesmen guaranees expire if he invesor sops paying premiums (paid-up case) or if he invesor cancels he conrac before mauriy (surrender case). FIGURE 2: Resuls for ineres rae and lookback guaranees for convenional underlying fund a) Porfolio CV 1: μ A = 6.%, σ A = 4.16% Descripive saisics and performance Ineres Lookback Π Guaranee 13,865 12, (g = 2.8%) (g = %) ψ 1.57% 54.16% E ( L ) 16,562 16,612 σ ( L ) 1,35 1,298 Sharpe raio Omega Sorino raio FSD, SSD, SD none none Descripive saisics and performance b) Porfolio CV 2: μ A = 7.5%, σ A = 6.74% Ineres Lookback Π Guaranee 14,374 12, (g = 3.48%) (g = %) ψ 4.63% 62.57% E ( L ) 18,181 18,384 σ ( L ) 2,41 2,323 Sharpe raio Omega Sorino raio 1, FSD, SSD, SD none none F(x) F(x) Cumulaive disribuion funcions Cumulaive Disribuion Mauriy Payoff Lookback guaranee Ineres rae guaranee x x Cumulaive disribuion funcions Cumulaive Disribuion Mauriy Payoff Lookback guaranee Ineres rae guaranee x x 1 4 Noes: Π = value of he guaranee in = ; g = minimum rae of reurn; ψ = probabiliy ha value of fund a mauriy is lower han guaraneed payoff; L = conrac s payoff a mauriy ; FSD, SSD, SD = Firs, Second, hird Order Sochasic Dominance.

21 2 he lookback guaranee has a minimum rae of %, which would occur only if he uni price does no change a all and remains consan over he whole conrac erm. o make he conracs comparable, we calibrae he minimum rae of reurn g for he fund wih an ineres rae guaranee such ha he iniial guaranee coss are he same as for he lookback guaranee. his is achieved by seing he guaraneed ineres rae o g = 2.8% in he case of porfolio CV 1 (Par a) in Figure 2), leading o guaranee coss of around 149. he guaranee coss Π need o be paid only in he case of a convenional underlying fund, if securiizaion is no achieved wih CPPI. his implies ha he guaranees have a posiive probabiliy ψ ha he fund value is below he promised guaraneed mauriy paymen. For he ineres rae guaranee, his probabiliy H is 1.57%; for he lookback guaranee, he probabiliy of { L > F} is much higher wih ψ = 54.16%. hese shorfall evens are secured hrough he iniial up-fron paymen Π. In conras o he case wih he CPPI managed underlying fund, he descripive saisics show ha he ineres rae guaranee has a slighly lower expeced mauriy payoff bu a higher sandard deviaion han he lookback guaranee. he sandard deviaion of he ineres rae guaranee is mainly an upurn deviaion, since he minimum mauriy payoff is se a 13,865. hese characerisics of he payoff disribuion affec he hree performance measures (he Sharpe raio, he Omega, and he Sorino raio), which are displayed in he lower par of he able. he performance measures ypically reac very sensiively wih respec o he risk of he payoff disribuion. Hence, all hree performance measures lead o lower resuls for he fund wih an ineres rae guaranee compared o he resuls for he fund wih a lookback guaranee. Overall, he resuls are quie similar for boh producs, which are also confirmed by he cumulaive disribuion funcions of he mauriy payoffs. Sronger differences can be observed in he case of he underlying porfolio CV 2 wih a volailiy of σ A = 6.74% (Par b) in Figure 2). Here, he coss for he lookback guaranee increase o 491. his corresponds o 3.4% of he sum of premium paymens (compared o 1.3% for porfolio CV 1). o reach his value, he

22 21 guaraneed ineres rae g needs o be raised considerably o 3.48%. A he same ime, he probabiliy ψ increases o 4.63% compared o 1.57% for porfolio CV 1 (where g was 2.8%) for he fund wih ineres rae guaranee, and o 62.57% for he fund wih lookback guaranee. Compared o Par a), especially he sandard deviaion of he mauriy payoffs is much higher, which is also illusraed by he cumulaive disribuion funcions. According o hese curves, here is one difference in he range of mauriy payoffs below 14,374 where he lookback guaranee exhibis nearly 2.5% of such realizaions. In conras, he payoff of he ineres rae guaranee does no fall below 14,374 due o he minimum guaranee of g = 3.48%. A he same ime, he lookback guaranee has a slighly higher probabiliy for larger payoffs. hese oucomes illusrae ha in he case of he lookback guaranee, he characerisics of he underlying fund play a cenral role. In paricular, he underlying fund s volailiy should be moderae, since oherwise, he lookback guaranee becomes very expensive. For porfolio CV 1 wih μ A = 6.% and a low volailiy of σ A = 4.16%, for insance, he lookback guaranee corresponds (in erms of guaranee coss) o a guaraneed ineres rae of 2.8%. his is considerable, given ha he riskfree rae r is 3.57%. For porfolio CV 2, he guaraneed rae mus even be raised o 3.48%. 6. SUMMARY his paper compares he pricing and performance of muual funds having ineres rae guaranees wih hose having lookback guaranees. he impac of he underlying fund wih respec o he embedded guaranees was analyzed by comparing a convenional fund wih consan parameers over he conrac erm and a CPPI managed fund. Resuls were derived in a simulaion analysis for differen diversified porfolios based on sock, bond, real esae, and money marke indices. Privae invesors ypically make a decision on which of wo producs o purchase based on risk preferences. Hence, o compare he muual fund wih ineres rae guaranee o he muual fund wih lookback guaranee, we examined he

23 22 characerisics of hese invesmen guaranees for differenly managed underlying funds by calculaing descripive saisics of he mauriy payoffs (expeced value and sandard deviaion) and hree performance measures ha reflec differen risk preferences (Sharpe raio, Omega, Sorino raio). We furher deermined he probabiliy ha he mauriy fund value is below he guaraneed paymen, and esed for firs, second, and hird order sochasic dominance. Premium paymens were assumed o be he same for all cases under consideraion. In our numerical analysis, we firs analyzed he case where boh producs have a minimum ineres rae guaranee of % (money-back guaranee) and an underlying fund ha is CPPI managed. In conras o a convenional underlying, securiizaion was achieved hrough CPPI and hus is coss are implicily conained in he payoff, wihou he invesor needing o make an addiional paymen a incepion of he conrac. In his example, he fund wih an ineres rae guaranee had a higher expeced mauriy payoff, a much higher sandard deviaion, and a higher probabiliy of large mauriy payoffs. hese resuls were even sronger when he underlying porfolio was changed o one wih a higher sandard deviaion. hese oucomes are due o he possibiliy of an on average higher sock exposure in he case of he muual fund wih an ineres rae guaranee for he CPPI managed underlying. Due o he high volailiy of he ineres rae guaranee produc, he Sharpe raio was slighly higher for he fund wih he lookback guaranee. he Omega and Sorino raio were higher for he fund wih he ineres rae guaranee due o using lower parial momens and hus downside risk measures. Second, we considered he case where boh invesmen guaranees have a convenional underlying fund and he same guaranee coss. he resuls were very similar for boh producs. One difference was ha he lookback guaranee had an approximaely 2.5% probabiliy of having a mauriy payoff below he minimum guaraneed paymen of he produc wih an ineres rae guaranee, bu a higher probabiliy for larger mauriy payous. Due o he higher sandard deviaion of he ineres rae guaranee payoff, all hree performance measures were higher for he lookback guaranee, a resul ha changes when he underlying fund s volailiy increases. In his scenario, he Omega and Sorino raio led o higher values for he ineres guaranee due o he limiaion of

24 23 downside risk. For all cases under consideraion, neiher invesmen alernaive dominaed he oher by firs, second, or hird order. Our resuls show ha he mauriy payou for funds wih a lookback guaranee is very sensiive o he underlying fund s volailiy. Unless he volailiy is kep low, he lookback guaranee becomes very expensive. his is also confirmed in he case of he underlying CPPI sraegy, where volailiy is subsanially reduced due o a higher share in he riskless invesmen. his is differen from he ineres rae guaranee produc, which allows for higher volailiy when managed via CPPI and hus a higher upside poenial for invesors. In our examples, invesors wih a risk-reurn profile refleced in he Omega and Sorino raio performance measures (boh based on lower parial momens) would prefer he muual fund wih an ineres rae guaranee in he case of a CPPI managed underlying or in he case of a convenional underlying fund wih higher volailiy. If an invesor s decisions are based on he Sharpe raio, risk is aken ino accoun using he payoff s sandard deviaion, which includes upside deviaions as well. Given his crierion, for his invesor, he lookback guaranee would be preferable over he ineres rae guaranee. Overall, he differen cases examined in his paper provide insigh ino wo differen forms of invesmen guaranees and heir corresponding risk and reurn profiles regarding he payoff disribuion a mauriy, informaion of grea relevance o poenial invesors having differen risk-reurn preferences.

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