INVESTMENT GUARANTEES IN UNIT-LINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE

Transcription

1 INVESMEN UARANEES IN UNI-LINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN AND INSURANCE NOVEMBER 27

2 1 INVESMEN UARANEES IN UNI-LINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE Nadine azer Hao Schmeiser JEL-Classificaion: D81, 11, 13, 22 ABSRAC In his aricle we idenify risk and reurn profiles of wo ypes of invesmen guaranees in uni-linked life insurance producs: an ineres rae guaranee and a lookback guaranee. his is done by comparing guaranee coss and performance for he mauriy payoff and by esing he invesmen alernaives wih respec o firs, second, and hird order sochasic dominance. In addiion, we analyze he impac of he underlying fund s sraegies by comparing resuls for a convenional fund having average rae of reurn and sandard deviaion over he conrac erm wih a Consan Proporion Porfolio Insurance managed fund. We compare empirical resuls for differen µ-σ-efficien diversified porfolios based on sock, bond, real esae, and money marke indices. he aim of our invesigaion is o show he ineracion beween guaranee coss, underlying fund sraegies, and payoff disribuions a mauriy. 1. INRODUCION In recen years, here has been an increasing demand for invesmen producs wih financial guaranees. Subsanial growh raes can be observed, e.g., in he segmen of uni-linked life insurance producs. 1 hese conracs are in essence 1 Boh auhors are wih he Universiy of S. allen, Insiue of Insurance Economics, Kirchlisrasse 2, 91 S. allen, Swizerland. In he European life insurance marke, he share of uni-linked producs in he oal premium volume has increased since 23 from 21.8% o 24.2% in 25 afer a decrease from 2 o 23 due o he sock marke crash (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 has been mainly driven by an increase in uni-linked producs (CEA, 27, p.13). Concerning invesmen producs in general, e.g., erman invesmen funds currenly manage 126

3 2 muual funds wih invesmen guaranees ha addiionally offer erm insurance. hus, he mauriy payou srongly depends on he performance of he underlying fund. From he policyholders perspecive, hese producs are aracive due o he possibiliy o paricipae in posiive marke developmens combined wih a capial guaranee, even hough hey bear he main invesmen risk. In his paper, we compare risk and reurn profiles of wo invesmen guaranees in uni-linked life insurance producs: he firs conrac provides an ineres rae guaranee on he premiums paid ino he conrac. he second produc includes a lookback guaranee. In his case, he payoff is defined by he number of unis he clien acquired over he conrac erm muliplied by he highes value of he uni price ha was aained before mauriy. he payoff of he wo producs srongly depends on he underlying fund sraegy. In he case of a fund wih fixed average rae of reurn and sandard deviaion, guaranee coss can be derived. Alernaively, guaranees can generally be secured using a Consan Proporion Porfolio Insurance (CPPI) sraegy via coninuous dynamic reallocaion of he invesmen beween wo asse classes, namely, a risky and a riskless asse. o compare he wo producs, we focus solely on heir invesmen guaranees and he effec of he underlying fund and do no consider erm life insurance or oher opions possibly embedded in he conrac. hus, he analysis is generally valid for invesmen producs and muual funds wih financial guaranees. We invesigae he characerisics of he mauriy payoffs of hese muual funds wih invesmen guaranees by calculaing descripive saisics and by using hree performance measures (Sharpe raio, Omega, and Sorino raio) for he wo fund sraegies (CPPI as well as average reurn and sandard deviaion). In addiion we es for firs, second, and hird order sochasic dominance. Iniial guaranee coss are deermined using opion pricing heory in a Black/Scholes framework, which has been used in similar cases in Kling, Ruß, and Schmeiser (26) (for ineres rae guaranees) as well as in erber and Shiu (23a) (for pricing lookback opions). We furher derive empirical resuls for differen µ-σ-efficien diversified porfolios based on sock, bond, real esae, and money marke indices. guaranee funds (up 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 In he lieraure, Brennan and Schwarz (1976) and Boyle and Schwarz (1977) were he firs o invesigae asse guaranees in uni-linked life insurance producs. he surrender opion in uni-linked life insurance wih an ineres rae guaranee and erm insurance is sudied in Bacinello and Oru (1993) and Bacinello (25). erber and Shiu (23a) derive closed-form soluions for several exoic opions, including lookback opions and dynamic fund proecion wih deerminisic and sochasic guaraneed level. erber and Shiu (23b) rea dynamic fund proecion in he conex of equiy-indexed annuiies, i.e., for perpeual American opions. Kling, Ruß, and Schmeiser (26) analyze he paid-up opion in erman governmen-subsidized pension producs for a fund wih an ineres rae guaranee. A comparison of ineres rae and lookback guaranees for differen underlying funds and differen invesmen sraegies has no been conduced so far. Hence, his analysis should provide general informaion abou guaranee coss and performance o buyers of uni-linked life insurance producs wih ineres rae and lookback guaranees and, more generally, o invesors and providers of invesmen producs wih financial guaranees. Comparing producs wih differen guaranees is ofen difficul due o differen mauriy guaranees, a differen underlying, 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 exposure 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, a case is analyzed in which boh producs provide he same convenional underlying fund and he same implied guaranee coss. Even hough boh funds provide a quie similar expeced payoff in his case, he muual fund wih a lookback guaranee has roughly a 5% probabiliy of leading o a payoff below he mauriy guaranee promised by an invesmen fund wih ineres rae guaranee. Furhermore, we

5 4 find ha neiher invesmen alernaive dominaes he oher by firs, second, or hird degree. Overall, he resuls illusrae he srong effec of he 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 = 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 i 1, { 1,..., 1}. N = n i N = 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. hus, he guaraneed mauriy paymen resuls in

6 5 N 1 g( ) P e =. = For g =, his implies = N P and for g >, we obain g 1 e = P e 1 e g g. Similar invesmen guaranees in he conex of pension producs have been considered in, e.g., Lachance and Michell (23) and in Kling, Russ, and Schmeiser (26). 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, which consiss of he value of he invesmen in he underlying fund, which will be a leas he guaraneed paymen, i.e., N 1 N 1 S L = max ( F, ) = max P, P e = S = N 1 N 1 S g( ) = P max, e = P L. ɶ = S = ( ) g hus, he amoun of premium paymens only serves as a scalar of he acual pay- off. he payoff o he invesor in, L, can be wrien as he value of he underlying asses plus a pu opion on his value wih srike, such ha

7 6 ( ) ( ) L = max F, = F + max F,. (1) Muual fund wih lookback guaranee he fund wih he lookback opion guaranees a payoff of he highes value (or peak) H of he index ha has been aained during he policy erm (for he case of lookback guaranees in he conex of equiy-indexed annuies, see erber and Shiu, 23a and 23b; Lin and an, 23), 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 ɶ. 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 whole conrac erm, creaing a minimum rae of reurn g on premiums of %. As before, he exac amoun of premium paymens is relevan only as a scaling facor. 3. INVESMEN SRAEIES OF UNDERLYIN 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 (average rae of reurn and sandard deviaion) For he convenional fund, he uni price evolves according o a geomeric Brownian moion. Hence, i can be described by he sochasic differenial equaion ( µ σ ) ds = S d + dw, wih consan drif µ, volailiy σ, and a sandard Brownian moion W. 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. r = ln R is normally disribued Hence, he coninuous one-period reurn ( ) 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 Black and Jones (1987). CPPI secures he guaranees via coninuous dynamic

9 8 reallocaion of he invesmen beween wo asse classes, namely, a risky and a riskless asse. 2 he risky invesmen A evolves according o a geomeric Brownian moion ( µ σ ) da = A d + dw, and he riskless invesmen is a bond process B wih a consan riskless rae of reurn r resuling in db = B rd. 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 = + = In his seing, r = µ + σ Z denoes he coninuous one-period A A A 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 r ( ) 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 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( ) = P e. i i = he cushion C for he risky invesmen is hen given by he difference beween he curren fund value (including he curren premium paymen) and he presen value of he guaranee, giving ( ) C = F + P e. 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 m C 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 UARANEES AND PERFORMANCE MEASUREMEN Valuaion of he invesmen guaranee Prices for invesmen guaranees a ime = can be obained if he underlying is a convenional fund (i.e., wih given average rae of reurn and sandard derivaion). Using a risk-neural valuaion echnique, he drif of he uni price process under he risk-neural measure Q (see Harrison and Kreps, 1979) changes o he riskless rae of reurn r, leading o Q ( σ ) ds = S rd + dw,

11 1 where W Q is a sandard Q-Brownian moion. he ne presen value of he invesmen guaranee Π a ime = is hen 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. Hence, i is given by N 1 N 1 Q r r Q r Π = E ( e L ) P e = P E ( e L ) e = = ɶ 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 managemen 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)) ( ( )) r Q Π = e E max F,, which is he price of a European pu opion on he fund value a mauriy wih srike. Analysis of he mauriy payoff o analyze he mauriy payoff L, we calculae is expeced value ( ) = P E ( Lɶ ) and sandard deviaion σ ( L ) P σ ( L ) E L = ɶ. 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 payoff and he value of he premium paymens compounded o mauriy, divided by he sandard deviaion of he mauriy payoff: ( ) Sharpe raio L r( ) E L P e E Lɶ e ( ) = = σ ( L ) N r( ) ( ) N 1 1 = = σ ( Lɶ ).

12 11 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. he lower parial momen of order k is given as (see, e.g, Fishburn, 1977) ( ( ) ) ( ( )) ( ) LPM L, E L = E max E L L, k. k iven his definiion, he Omega (see Shadwick and Keaing, 22) and he Sorino measures (see Sorino and van der Meer, 1991) can be obained: ( ) Omega L = r( ) E L P e ( ) N 1 = 1, ( ( )) LPM L E L, ( ) Sorino raio L = r( ) E L P e ( ) N 1 = ( ( )) LPM L E L 2,. he probabiliies ψ ha he fund value a mauriy does no cover he promised guaranees are given by ( F ) Ψ = P >, 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

13 12 can be found in, e.g., Bawa (1975) and Levy (1992). Le F 1 denoe he cumulaive disribuion funcion of L and F 2 denoe he cumulaive disribuion funcion of L on he inerval [a, b]. hen H L dominaes ( ) ( ) [ ] F x F x all x a b ; 1 2, H L by he firs degree (FSD) if and only if L dominaes L by he second degree (SSD) if H x ( ) ( ) [ ] x F d F d all x a, b 1 ; 2 and only if H L dominaes L 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 lobal overnmen Bond, PR eneral PSI Europe, and he JPM Euro Cash 3, respecively. We exraced monhly reurns beween January 1994 and December 26 from he Daasream daabase o obain mean and sandard deviaion of he annualized reurns (able 1), and o calculae heir correlaions (able 2). o consider reurns from changes in prices and dividend paymens, we only look a performance indices. As he risk-free rae we use he reurn of he JPM Euro Cash 3 Monh (r = 3.95%).

14 13 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 (worldwide) Equiy Marke Proxy (S) 1.27% 16.16% Bonds (worldwide governmen bonds) Real esae (worldwide) Money (money marke in he JPM lobal ov. Bond (B) 5.26% 6.49% PR eneral PSI Europe (R) 9.47% 11.63% JPM Euro Cash 3 (M) 3.95% - EMU) Noes: JPM: J.P. Morgan, PR: lobal Propery Research, PSI: Propery Share Index, EMU: European Moneary Union 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. Furhermore, 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 Euro Cash 3 (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

15 14 CV 2 (invesmen in money marke is %) will serve as underlying for he convenional fund. Here, porfolios CV 2 and CP 2 coincide. ABLE 3: Efficien porfolios based on indices in able 1 Porfolio µ A σ A Share Share Share Share in (S) in (B) in (R) in (M) CPPI CP 1 7% 6.11% 13.44% 61.14% 25.42% - CP 2 9% 9.8% 25.3% 15.85% 59.12% - r 3.95% % Convenional CV 1 7% 5.74% 14.43% 3.63% 31.51% 23.43% CV 2 9% 9.8% 25.3% 15.85% 59.12% % 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 lasserman, 24) on he basis of 2, 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. 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 ) canno dominae F 2 ( H H L ) if he smalles value of L is larger han he smalles value of L. Hence, if hese condiions are violaed neiher alernaive dominaes he oher H (ha is if, e.g., E ( L ) E ( L ) smalles value of L ). H > bu he smalles value of L is lower han he

16 15 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, P N, is 12,. For all cases, he ess on sochasic dominance show ha neiher alernaive dominaes he oher by firs, second, and hird order. 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 he riskier porfolio CP 2 (for deails on he efficien porfolios, 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 volailiy of he mauriy payoff of he ineres rae guaranee is subsanially increased from 983 o 1,718, while he change in he volailiy of he lookback guaranee is more moderae.

17 16 FIURE 1: Cos and performance of ineres rae and lookback guaranees for CPPI managed underlying fund Descripive saisics and performance Ineres uaranee 12, (g = %) a) Porfolio CP 1: µ A = 7%, σ A = 6.11% Lookback 12, (g = %) ψ % % E ( L ) 16,63 15,344 σ ( L ) Sharpe raio Omega Sorino raio Descripive saisics and performance Ineres uaranee 12, (g = %) b) Porfolio CP 2: µ A = 9%, σ A = 9.8% Lookback 12, (g = %) ψ % % E ( L ) 17,15 15,819 σ ( L ) 1, Sharpe raio Omega Sorino raio F(x) F(x) Cumulaive disribuion funcions Cumulaive Disribuion Mauriy Payoff (CP 1) Lookback guaranee Ineres rae guaranee x x Cumulaive disribuion funcions Cumulaive Disribuion Mauriy Payoff (CP 2) 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. hese resuls are due o he possibiliy of a (on average) higher sock exposure 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 volailiy of he mauriy payoff of he fund wih ineres rae guaranee is concerned. he volailiy increases subsanially due o even higher shares in he risky invesmen. In conras, he disribuion of he lookback guaranee shows almos no changes.

18 17 For hese reasons, he fund wih he ineres rae guaranee has a considerably higher percenage of mauriy payoffs ha are above 15,, and also several above 2,, whereas he lookback guaranee payoffs peak ou a 18,. 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 (Par a) in Figure 2), he Sharpe raio is approximaely he same for boh producs; he Sorino raio is higher for he ineres rae guaranee, whereas he Omega performance measure is lower. his observaion illusraes ha i makes a difference how deviaions from he expeced value hrough lower parial momens are aken ino accoun. For he riskier porfolio CP 2 in Par b), however, all hree performance measures show higher values for he lookback guaranee, which is implied by he subsanially lower volailiy. 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 (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 invesmen guaranees expire if he invesor sops paying premiums (paid-up case) or if he invesor cancels he conrac before mauriy (surrender case). 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

19 18 erm, hus only he premiums would be paid back a mauriy. 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 = 3.35% in he case of porfolio CV 1 (Par a) in Figure 2), leading o coss of around 293. FIURE 2: Cos and performance of ineres rae and lookback guaranees for convenional underlying fund Descripive saisics and performance a) Porfolio CV 1: µ A = 7%, σ A = 5.74% Ineres Lookback Cos Π uaranee 14,274 12, (g = 3.35%) (g = %) ψ 3.62% 59.39% E ( L ) 17,614 17,743 σ ( L ) 1,995 1,925 Sharpe raio Omega Sorino raio Descripive saisics and performance b) Porfolio CV 2: µ A = 9%, σ A = 9.8% Ineres Lookback Cos Π uaranee 15,32 12, (g = 4.67%) (g = %) ψ 1.42% 67.92% E ( L ) 2,221 2,644 σ ( L ) 3,922 3,847 Sharpe raio Omega Sorino raio F(x) F(x) Cumulaive disribuion funcions Cumulaive Disribuion Mauriy Payoff (CV 1) Lookback guaranee Ineres rae guaranee x x 1 4 Cumulaive disribuion funcions Cumulaive Disribuion Mauriy Payoff (CV 2) 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.

20 19 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 H probabiliy is 3.62%; for he lookback guaranee, he probabiliy of { L F } > is much higher wih ψ = 59.39%. hese shorfall evens have o be 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 lower expeced mauriy payoff bu a higher sandard deviaion han he lookback guaranee. he high sandard deviaion of he ineres rae guaranee is an upurn deviaion, since he minimum mauriy payoff is se a 14,274. 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 is also confirmed by he cumulaive disribuion funcions of he mauriy payoffs. Sronger differences can be observed in he case of he riskier underlying porfolio CV 2 wih a volailiy of σ A = 9.8% (Par b) in Figure 2). Here, he coss for he lookback guaranee increase remendously o 978. his corresponds o 8.2% of he sum of premium paymens (compared o 2.4% for porfolio CV 1). o reach his value, he guaraneed ineres rae g needs o be raised considerably o 4.67%. he high guaranee coss are in line wih he probabiliies ψ, which increase o 1.42% compared o 3.62% for porfolio CV 1 (where g was 3.35%) for he fund wih ineres rae guaranee, and o 67.92% for he fund wih lookback guaranee. Compared o Par a), especially he volailiy of he mauriy payoffs is much higher, which is also illusraed by he cumulaive disribuion funcions.

21 2 According o hese curves, here is one obvious difference in he range of mauriy payoffs below 15,32 where he lookback guaranee exhibis nearly 5% of such realizaions. In conras, he payoff of he ineres rae guaranee does no fall below 15,32 due o he minimum guaranee of g = 4.67%. 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 oo expensive. For porfolio CV 1 wih µ A = 7% and a low volailiy of σ A = 5.74%, for insance, he lookback guaranee corresponds (in erms of guaranee coss) o a guaraneed ineres rae of 3.35%. his is considerable, given ha he riskfree rae r is 3.95%. For he riskier porfolio CV 2, he guaraneed rae mus even be raised o 4.76%. he differen cases compared in his secion provide insigh ino he differen forms of invesmen guaranees and he corresponding risk and reurn profiles regarding he payoff disribuion a mauriy, informaion of grea relevance o poenial invesors who have differen risk preferences. However, whenever he focused cash flows of he invesmen producs under consideraion are priced fairly in financial erms as is done in our analyses no addiional value can be generaed for invesors in erms of ne presen value. Hence, in his seing, all ypes of producs considered in his secion lead o a ne presen value of zero. herefore, none of hese producs can be found superior unil furher informaion abou he underlying ransacion coss of he producs is available o poenial invesors. 6. SUMMARY his paper empirically compared risk and reurn profiles of ineres rae and lookback guaranees in uni-linked life insurance producs, which essenially corresponds o a muual fund wih invesmen 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

22 21 differen diversified porfolios based on sock, bond, real esae, and money marke indices. We examined he characerisics of hese invesmen guaranees for he differenly managed underlying funds by calculaing descripive saisics of he mauriy payoffs (expeced value and sandard deviaion), he probabiliy ha he mauriy fund value is below he guaraneed paymen, hree performance measures (Sharpe raio, Omega, Sorino raio), and by esing for firs, second, and hird order degree sochasic dominance. In he 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 inensified when changing he underlying porfolio o a riskier 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 performance measures were higher for he fund wih lookback guaranee. 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. However, one difference was ha he lookback guaranee had an approximaely 5% probabiliy of having a mauriy payoff below he minimum guaraneed paymen of he produc wih 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. For all cases under consideraion, neiher invesmen alernaive dominaed he oher by firs, second, or hird order degree.

23 22 Overall, he oucomes showed ha especially he fund wih lookback guaranee reacs very sensiively wih respec o he underlying fund s volailiy. Volailiy should remain low; oherwise, he lookback guaranee becomes oo valuable and hus oo expensive. his is also confirmed in he case of he underlying CPPI sraegy, where he 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 volailiies and hus a higher upside poenial for invesors.

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