Insurance: Mathematics and Economics

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1 Insurnce: Mthemtics nd Economics ) Contents lists vilble t ScienceDirect Insurnce: Mthemtics nd Economics journl homepge: he design of equity-indexed nnuities Phelim Boyle, Weidong in b, Wilfrid Lurier University, Ontrio, Cnd NL 3C5 b University of Wterloo, Ontrio, Cnd NL 3G1 r t i c l e i n f o b s t r c t Article history: Received My 008 Accepted 4 My 008 JEL clssifiction: G1 G13 Keywords: Equity-indexed nnuities Equity-lined contrcts Structured products Optiml design Optiml portfolio selection here is rich vriety of tilored investment products vilble to the retil investor in every developed economy. hese contrcts combine upside prticiption in bull mrets with downside protection in ber mrets. Exmples include equity-lined contrcts nd other types of structured products. his pper nlyzes these contrcts from the investor s perspective rther thn the issuer s using concepts nd tools from finncil economics. We nlyze nd critique their current design nd exmine their vlution from the investor s perspective. We propose generliztion of the conventionl design tht hs some interesting fetures. he generlized contrct specifictions re obtined by ssuming tht the investor wishes to mximize end of period expected utility of welth subject to certin constrints. he first constrint is gurnteed minimum rte of return which is common feture of conventionl contrcts. he second constrint is new. It provides the investor with the opportunity to outperform benchmr portfolio with some probbility. We present the explicit form of the optiml contrct ssuming both constrints pply nd we illustrte the nture of the solution using specific exmples. he pper focusses on equity-indexed nnuities s representtive type of such contrcts but our pproch is pplicble to other types of equity-lined contrcts nd structured products. 008 Elsevier B.V. All rights reserved. 1. Introduction his pper discusses nd critiques equity-lined contrcts. hese products re importnt since they constitute populr clss of investment contrcts for retil investors. We nlyze them from the investor s perspective. his viewpoint contrsts with much of the cturil literture which is bsed on the issuer s perspective nd focusses on the pricing, hedging nd ris mngement of these contrcts. We discuss the optimlity of the conventionl design of these contrcts from the consumer s viewpoint nd note tht in generl the design is inefficient. We propose new type of product which we cll the generlized contrct. Under generlized contrct n investor hs the opportunity to bet benchmr index with some probbility nd lso hs gurnteed minimum rte of return, irrespective of mret performnce. We give the optiml contrct design of this generlized contrct nd explore its properties using specific exmples. At this stge, brief description of equity-indexed nnuities my be helpful. In single premium contrct, the customerinvestor) pys n initil mount the premium) to the insurer. Suppose tht the contrct mtures in sy five yers. At mturity the pyoff is bsed on the performnce of some reference index which could be, for exmple, stoc mret index. he contrct prticiptes in the gins if ny) in the reference portfolio during this period. he detiled rrngements of how this prticiption is clculted vry, but usully the prticiption hs some cll option fetures. In ddition these contrcts provide floor of protection if the mret does poorly. For exmple the gurnteed floor my be return of the initil investment which mens tht the investor hs n embedded put option. In some contrcts there is mximum or cp on the investor s return. he cp cn limit the return over the life of the contrct or it my operte period by period. In some circumstnces the investor hs n erly redemption option to cncel or surrender the contrct. In this pper we use the equity-indexed nnuity EIA) s representtive exmple of this clss of contrcts in our nlysis. hese represent very populr 1 clss of structured products sold by insurnce compnies in USA with rich vriety of design fetures. At first sight the EIA ppers to deliver the best of both worlds: n investment tht goes up when the stoc mret goes up but provides gurnteed floor if the mret collpse. Corresponding ddress: University of Wterloo, Deprtment of Sttistics nd Acturil Science, 00 University Avenue West, NL 3G1 Ontrio, Cnd. el.: E-mil ddresses: pboyle@wlu.c P. Boyle), wdtin@uwterloo.c W. in). 1 According to the industry group LIMRA, sles of EIA were round 4 billion US dollrs in 004, up from $ 14.4 billion in 003, nd the sles volume is even bigger in /$ see front mtter 008 Elsevier B.V. All rights reserved. doi: /j.insmtheco

2 304 P. Boyle, W. in / Insurnce: Mthemtics nd Economics ) However, these contrcts hve some fetures tht my reduce their ttrctiveness to retil investor. First, mny contrcts hve high initil commissions pyble to the gent who sells the contrct to the investor. Second, the contrct design is often complicted ming it difficult for the consumer to understnd the product. hird the design my not relly suit the consumer s needs. One of the min ims of this pper is to ddress this lst feture. Specificlly, we discuss how to design contrct tht is optiml for the customers under certin ssumptions. Our proposed contrct is generlly more complicted thn existing products. However this optiml contrct provides useful benchmr. here is one spect of our nlysis tht deserves some discussion. his rises from our use of frmewor where finncil institutions such s insurnce compnies exist nd the extent to which we cn simultneously use the ssumptions of norbitrge nd complete mrets in the sme frmewor. Strictly speing, in world of perfect informtion, frictionless mrets nd complete set of mreted securities there is no role for finncil intermediries. In this world consumers cn use the existing securities to mximize their expected utility. However, in prctice individul consumers fce significnt trnsction costs nd informtionl costs if they wnt to trde directly to form their optiml portfolio. Merton hs discussed the role of finncil intermediries in this setting nd noted tht individuls use finncil institutions which fce lower trnsction costs to perform these functions. Indeed s Merton nd Bodie 005) note in modern well developed finncil system the lowest cost trnsctor my hve mrginl trding costs close to zero nd cn trde lmost continuously. Our pproch uses similr perspective to Merton s. Finncil institutions sell contrcts to their customers. he institutions re ble to replicte pyoffs with essentilly no trnsction costs. However n individul customer my be quite willing to py higher premium for the contrct thn its strict norbitrge vlue becuse heshe) fces significnt trnsction costs. here is n extensive cturil literture on the pricing, hedging nd ris mngement of these contrcts. See for exmple Biffis nd Millossovich 006), Nielsen 006), Hrdy 003) nd iong 000). his pproch is termed the fir premium pproch nd it cn be trced bc to Blc nd Scholes 1973), Merton 1973), Brennn nd Schwrtz 1976), Boyle nd Schwrtz 1977). Since the present vlue of the clim is often clculted by employing the no-rbitrge principle, we rgue tht the no-rbitrge) present vlue of the pyoff is just the first step in investigting this ind of contrct from the consumer s perspective. his follows from the discussion in the previous prgrph. he issuer will be ble to chrge the customer more thn the no-rbitrge vlue of the contrct. he im of this pper is to critique the conventionl structure nd design of EIAs nd suggest possible generliztion. In the spirit of the ppers of Arrow 1974) nd Rviv 1979), we investigte the optiml design of n EIA under the expected utility frmewor. 3 he expected utility frmewor is nturl for this nlysis becuse the contrct is n investment vehicle nd it is not resonble to ssume tht investors re ris-neutrl. Moore nd Young 005) lso use the expected utility pproch in the context of the design of perpetul equity-indexed nnuity. In conventionl EIA, both the gurnteed return nd the equity-indexed return re incorported into one gurnteed pyoff. herefore, the optiml design problem for n EIA becomes stndrd optiml portfolio selection problem See Cox nd Hung 1989), Merton 1971) nd Plis 1986)) subject to miniml gurnteed on the terminl welth. he optiml pyoff under the EIA is the optiml terminl welth in this optiml portfolio selection problem, nd it is derived explicitly in heorem 3.1 of Section 3.4. We show tht the current pyoff structure of the conventionl EIA is not optiml for most investors. 4 We re lso ble to investigte the impct of different investor ris preferences on the return to the investor. In the second prt of the pper we introduce new type of equity-lined product which we cll generlized equity index nnuity. Boyle nd in in press) hve derived the explicit solution for the optiml contrct in this cse under firly generl conditions. Here we develop the nlysis using the equity-indexed nnuity s the prototype but our results hold for other types of structured products with pproprite modifictions. We ssume tht the preferred contrct will mximize the investor s expected utility of terminl welth subject to two constrints. he first constrint is tht there is minimum gurnteed rte of return on the contrct. his constrint is very pervsive feture in these types of tilored retil products. he second constrint gives the investor n opportunity to outperform some benchmr with certin probbility α. We provide justifiction for this type of constrint lter but, for now, we note tht in the specil cse when α = 1, this spect of the contrct corresponds to the equity prticiption feture of existing contrcts. Since we do not ccount for the mortlity nd surrender fetures, the contrcts considered in this pper re very similr to structured products 5 which re sold by bns. Under some plusible ssumptions we re ble to solve for the optiml contrct design of the generlized equity-indexed nnuity. he explicit solution depends on the investor s utility function, the chosen benchmr nd the confidence level s well s certin cpitl mret prmeters. he optiml pyoff t mturity displys n interesting dependence on the level of the underlying index. In generl there will be discontinuities nd pyoff need not be n incresing function of the index. he discontinuity rises from the existence of the probbilistic constrint nd Bs nd Shpiro 001) document the sme type of behvior in their study of VR. he rest of the pper is orgnized s follows. In Section, we discuss in some detil populr type of EIA nown s point-topoint EIA. We exmine the vlution nd design of this contrct from the investor s perspective. We extend the nlysis to other types of EIAs in Section 3. Section 4 describes our proposls for generlized EIA. We lso exhibit nd discuss the solution to the optiml design of the generlized contrct. Section 5 describes the explicit construction of the optiml design for specific type of generlized contrct nd gives severl numericl exmples to help describe its min fetures. Section 6 concludes the pper. Some technicl formule re given in the Appendices A nd B. Merton nd Bodie 005) stte: But in the presence of substntil informtionl nd trnsctionl costs it is not relistic to posit tht the only process for individuls to estblish their optiml portfolios is to trde ech seprte security for themselves directly in the mrets. Insted individuls re liely to turn to finncil orgniztions such s mutul funds nd pension funds tht cn provide pooled portfolio mngement services t much lower cost thn individuls cn provide for themselves. 3 here is lso some reserch on the optiml design without using the utility frmewor. See Doherty nd Eechoudt 1995), Doherty nd Schlesinger 1983), Gollier nd Schlesinger 1996) nd Schlesinger 1997). 4 his is not surprising tht these insurnce contrcts re not optiml or Pretoefficient. For instnce, Brennn 1993) developed nice nlysis of the nonoptimlity of some insurnce contrcts. Since we explore the investor s perspective by including ris preferences, our nlysis is different from Brennn s 1993). 5 Generlly speing, structured product is often bsed on complicted underlying index, while n EIA my hve complicted formul for computing the investor s return. In USA structured products re registered under the Securities Act s securities nd most EIAs re registered s insurnce products. See Frncis et l. 000) for discussion of structured products.

3 P. Boyle, W. in / Insurnce: Mthemtics nd Economics ) Anlysis of point-to-point EIAs In this section, we consider simple exmple of populr EIA contrct nown s point-to-point EIA. We will often refer to this contrct throughout the pper. In stndrd point-to-point EIA, the buyer pys single premium t the beginning. he contrct typiclly credits return tht is lined to n externl reference portfolio which is often n equity index. he seller lso gurntees tht the contrct will py t lest minimum rte of interest on the investment. In this wy the purchser hs floor gurntee. In ddition the purchser prticiptes in the upside growth of the reference index. More formlly, let g denote the minimum gurnteed rte nd the prticiption rte. Assume tht the initil premium pid by the customer is x 0, nd the contrct hs term of yers. At contrct mturity the investor will receive the following pyoff 6 : ) } Γ = mx x 0 e g S, x 0. he no-rbitrge vlue of this pyoff is given by E[ξ Γ ] = y 0.1) where ξ is the stte-price density in this mret. Since the premium must be t lest s lrge s the no-rbitrge vlue of the pyoff Γ, we hve x 0 y 0..) We will focus on the cse tht x 0 > y 0 becuse the insurer chrges higher premium thn the no-rbitrge vlue of the pyoff. 7 Before proceeding we discuss the trde-off between the prticiption rte nd the minimum gurnteed rte g. For this purpose, we first imgine hypotheticl bre even point-to-point EIA contrct for which x 0 = y 0 holds. his condition will imply reltionship between the contrct prmeters nd g. Let us denote the members of this set by ˆ, ĝ}..3) We cn derive the fesible vlues of ˆ nd ĝ by finding the prmeter combintions for which the no-rbitrge vlue of the pyoff under the EIA is x 0. In other words ˆ nd ĝ stisfy x 0 = E[ξ ˆΓ ],.4) where ˆΓ = mx x 0 eĝ, x 0 S )ˆ}. By using.4), we cn djust either the prticiption rte, or the gurnteed rte g or both so tht the no-rbitrge vlue of the pyoff is less thn tht of the single premium x 0. In prctice this mens tht we pic < ˆ or g < ĝ or both together. 6 Returns re quoted s continuously compounded rtes in this pper. he return of EIA is 1 log Γ x 0 ) which is greter thn or equl to the product of prticiption rte times the return of the index, tht is 1 log S ), where Γ denotes the terminl pyoff of EIA. hen X x 0 S ). 7 Plmer 006) indictes tht in USA, the verge first yer commission rtes for EIAs in the lst few yers is bout 8% of the premium. he loded premium x 0 is often chrged by insurer to ginst non-hedgeble riss such s mortlity nd surrender options, nd generte business profit. A full equilibrium discussion of the reltionship between the no-rbitrge vlue y 0 of the pyoff Γ nd the premium x 0 is beyond the scope of this rticle. We cn illustrte this point in the cse where S hs lognorml distribution. Under this ssumption there is closed form expression for y 0. We ssume tht ds S = µdt + σ dwt).5) where µ is the drift, σ is the diffusion nd Wt) is stndrd Brownin motion under the rel world mesure P. We ssume tht µ > r where r is the ris-free rte. In this cse.1) becomes y 0 = e g r) x 0 Φα) + x 0 e 1)r+ 1 1)σ Φ α + σ ).6) where α = g r 1 σ ) σ nd Φx) is the cumultive norml distribution function. he condition y 0 < x 0 becomes e g 1 r+ Φα) + e σ ) Φ α + σ ) < e r..7) Hence for our specimen EIA, where the index hs log-norml distribution, Eq..4) becomes eĝ Φα) + eˆr+ ˆ 1 σ ) Φ α + ˆσ ) = e r..8) his lst eqution determines the reltionship between ˆ nd ĝ. We cn solve it numericlly to compute ˆ nd ĝ. Pnel A) in Fig. 1 displys the trde-off between the contrct prmeters ˆ nd ĝ for representtive set of prmeters. For instnce, when ĝ = %, the highest prticiption rte ˆ = 60.%; if ĝ = 3%, then ˆ is 48.6%. We now explore numericlly how the mgnitude of the difference x 0 y 0 ) is relted to the reduction in the prticiption rte, ˆ ). o mesure the loss to the investors, we use the rtio of x 0 y 0 to x 0, which is 1 y 0 x 0.9) to denote the loss percentge. Pnel B) in Fig. 1 displys the loss percentge s function of nd g where the prticiption rte equls 0.90ˆ. ˆ is determined by Eq..8) where g = ĝ. he loss percentge vries from round.% when g = 0% to 0.4% when g = 3.75%. 3. Improving the design of existing EIA s In this section we discuss how we cn improve the design of n EIA from the investor s viewpoint. We use the expected utility frmewor to compre different pyoffs. First we briefly review the clssic Merton portfolio selection problem. hen we will show how the design of the conventionl EIA cn be improved from the investor s perspective. We consider n investor who wnts to mximize expected utility of terminl welth nd in ddition hve pyoff t lest s good s tht of n EIA. We find the optiml contrct in this setting. his contrct mximizes the investor s expected utility subject to the pyoff being t lest s good s tht of the selected EIA. his section is in four prts. First we review the Merton model for optiml portfolio selection. hen we show how to modify the design of the trditionl point-to-point EIA to better incorporte the investor s preferences. he pyoff on the point-to-point EIA is ten to be the benchmr. In the third prt we discuss some other exmples of conventionl EIAs nd in the lst prt we discuss how to enhnce the design of ny conventionl EIA by ting into ccount the investor s preferences regrding the bsic EIA pyoff s the gurnteed benchmr.

4 306 P. Boyle, W. in / Insurnce: Mthemtics nd Economics ) A) rde-off between ĝ nd ˆ. B) Loss of percentge. Fig. 1. rde-off nd loss of percentge. Index is log-normlly distributed. Prmeters: contrct mturity is 5 yers, ris-free rte is 4%, index voltility is 0%. Pnel A) reports the trde-off between ˆ nd ĝ. In Pnel B), we choose = 0.9ˆ, nd ˆ is displyed in Pnel A) when g = ĝ. he loss of percentge, which is defined in.9), is displyed in Pnel B). Note tht both µ nd x 0 re not involved in both Pnel A) nd Pnel B) he Merton solution We first consider the generl problem of how n investor selects n optiml portfolio. his is clssicl problem in finncil economics nd it ws solved 8 by Robert Merton in Merton s solution nd some of the lter extensions will be useful in providing frmewor for nlyzing the design of equity-lined contrcts. We will show how the solution is modified when there is gurntee. However in this subsection we do not directly discuss equity-indexed nnuities. We focus on the solution to the optiml portfolio problem nd the structure of the pyoff ssuming tht the investor mes the optiml decision. We consider n investor with initil welth x 0 nd utility function given by u.), where u is strictly incresing, strictly concve nd twice differentible. he vilble ssets consist of risy sset S nd the ris-free sset. We ssume no-rbitrge, no frictions nd tht the mret is complete. Given these ssumptions there exists unique stte-price process ξ t }. We ssume tht the investor wishes to mximize the expected utility of terminl welth, denoted by X, over time horizon. he investor s optiml terminl welth is given by X = Iλm ξ ), 3.10) where the multiplier, λ m > 0, solves E[ξ Iλ m ξ )] = x 0, nd I.) is the inverse of the investor s mrginl utility function u.). We refer to the pyoff in formul 3.10) s the Merton solution. Now we modify the investor s objective function by including gurntee t mturity. In this cse the investor wishes to mximize the expected utility of terminl welth X subject to the constrint X x 0 e g, 3.11) where g is the gurnteed rte. his is nturl constrint given how often it occurs in prctice. We cn lso find the solution 9 to this problem. In this cse X = mxiλg ξ ), x 0 e g }, 3.1) 8 his mteril is well nown but it is briefly reviewed here since the pproch plys ey role in our lter nlysis. Merton ws ble to derive the optiml solution under firly generl conditions. See Merton 1971), Cox nd Hung 1989) nd Plis 1986). 9 Both this solution nd the solution of exmple in Section 3. follow from heorem 3.1 in Section 3.4. where E[ξ mxiλ g ξ ), x 0 e g }] = x 0. his lst solution gives the optiml pyoff when there is gurntee nd the new Lgrnge multiplier λ g will differ from the Merton one, λ m. In fct λ g > λ m. It turns out tht the investor s expected utility is reduced when we impose the constrint. his result is evident from inspection of the two pyoffs. However it is noteworthy given the prevlence of such gurntees tht their inclusion serves to reduce the investor s expected utility reltive to the unconstrined problem. We cn exmine the difference between these two solutions in the cse where the risy sset hs log-norml distribution since this cse is nlyticlly very trctble. In this cse ξ lso hs lognorml distribution, nd is given by ) b S ξ =, 3.13) where θ = exp µ 1 ) σ σ r + 1 ) } θ, b = θ σ, θ = µ r. σ We ssume tht the investor hs log utility function nd initil welth x 0. In this cse the optiml welth under the Merton solution is ) X = ξ Iλm ξ ) = I = x ) b 0 S. 3.14) x 0 From Eq. 3.1), the corresponding optiml terminl welth when there is gurntee is ) } b X = mxiλg ξ ), x 0 e g 1 S } = mx, x λ g 0 e g. 3.15) We use simple numericl exmple to compre these two solutions. his comprison is displyed in the left figure of Pnel A) in Fig.. Assume tht x 0 = 1, = 1, = 5, µ = 6%, σ = 0%, g = %, r = 4%. In this cse b =.5 nd = 1 so tht the optiml terminl welth in the Merton solution becomes X = S. 3.16)

5 P. Boyle, W. in / Insurnce: Mthemtics nd Economics ) A) Optiml welth. B) Effect of ris version prmeter γ. Fig.. Optiml welth nd ris version. In Pnel A) the investor hs log utility nd the index is log-normlly distributed. Prmeters: x 0 = 1, = 1. = 5, µ = 6%, σ = 0%, g = %, r = 4%. he left figure in Pnel A) displys Merton s optiml welth, which is determined in Eq. 3.10), nd the optiml welth with gurnteed % return, determined by Eq. 3.15). In the right figure in Pnel A), the prticiption rte = he optiml welth in this cse is displyed s well s the pyoff Γ. Pnel B) reports the loss of terminl welth for different levels of ris version. We ssume tht the investor hs CRRA utility with ris version prmeter γ, where γ = 1, nd γ = 3. When there is gurntee rte g =.0, we find tht λ g = , nd the correspond optiml welth in Eq. 3.1) is given by X = mx1.105, S }. 3.17) Compring expressions 3.16) nd 3.17) we see tht for lrge vlues of S the Merton unconstrined solution, 3.16), will give higher terminl welth thn 3.17). For low vlues of S the welth under the constrined problem is greter thn the welth under the Merton solution becuse of the gurntee. 3.. Optiml design of point-to-point EIA his discussion of the Merton optiml solution sets us up nicely to consider point-to-point EIA in the sme frmewor. o be specific consider point-to-point EIA with fixed gurntee of x 0 e g t mturity nd prticiption rte. We now use the expected utility frmewor to nlyze how to design the optiml contrct for n investor who desires pyoff tht is similr to point-to-point EIA. More precisely we ssume tht the investor wishes to hve pyoff tht t lest mtches the pyoff under given point-to-point EIA. We ssume tht the investor wishes to mximize expected utility of terminl welth with the constrint tht the pyoff hs to be t lest s good s tht under the point-to-point EIA. his nlysis is similr to tht of the lst subsection except tht the investor s optiml pyoff will now contin the pyoff of the pointto-point EIA s the gurnteed pyoff. We ssume tht the investor hs utility function u.) nd tht the terminl welth under the EIA is Γ where ) } Γ = mx x 0 e g S, x 0. Hence the optiml terminl welth of this investor is equivlent to the solution of the following mximiztion problem mx E[uX )] subject to the constrint tht the investor s terminl welth X stisfies ) } X mx x 0 e g S, x 0, 3.18) where x 0 is the initil welth. he optiml form of X cn be derived using the generl theory of portfolio selection in the Merton frmewor. It is similr to the optiml design we considered in the lst subsection. he optiml pyoff t time is given by ) } X = mx Iλ g ξ ), x 0 e g S, x 0, 3.19)

6 308 P. Boyle, W. in / Insurnce: Mthemtics nd Economics ) where λ g is positive number stisfying [ ) }] E ξ mx Iλ g ξ ), x 0 e g S, x 0 = x 0 3.0) nd I.) is the inverse function of u.). he clim ) } mx Iλ g ξ ), x 0 e g S, x 0 is the optiml pyoff for n investor with utility function u.). Eq. 3.0) mens tht the no-rbitrge vlue of the optiml pyoff is equl to x 0. Of course this is the idel contrct from the consumer s viewpoint nd there is no llownce for trnsction costs. In prctice the premium x 0 will exceed the no-rbitrge vlue of the contrct. We now consider numericl exmple when Γ is the pyoff under specific point-to-point EIA. We ssume the sme numericl prmeters s in the lst subsection, but in this cse the prticiption rte = 45%. For this exmple g =.0 = ĝ nd = 45% is less thn ˆ = 60.% so x0 > y 0. In fct, y 0 = he gurnteed contrct pyoff, Γ, provides the greter of gurnteed return %, nd prticiption in the mret with = 45%. he optiml terminl welth for this cse is ) ) } b X = mx x 0 e g S 1 S, x 0, λ g = mx1.105,.45, S }, 3.1) nd ) } Γ = mx x 0 e g S, x 0 = mx1.105,.45 }. he right figure of Pnel A) in Fig. displys the optiml welth X nd the gurnteed pyoff Γ. As one cn see from Fig., the pyoff under this optiml contrct is greter thn Γ for wider rnge of vlues of the mret index S. It is cler tht the contrct pyoff Γ is not the optiml terminl welth. 10 If = ˆ, the optiml welth in 3.19) is X = mx x 0 e g S )ˆ}, x 0. 3.) his fct follows esily from heorem 3.1 in Section 3.4. Formlly, let X denote the optiml welth in this cse when the gurnteed pyment is Γ. It is well nown tht either directly from its construction, or heorem 3.1), the constrint E[ξ X ] = x 0 is binding. As we hve noted the current no-rbitrge vlue of Γ is y 0 < x 0. Hence the loss to the investor in terms of current vlue is x 0 y 0 ). his does not depend on the investor s utility function. At time the loss to the investor is X Γ ) which will depend on the investor s utility function becuse X depends on the investor s utility function. We now illustrte how the investor s loss t mturity depends on her utility function. We compre the optiml terminl welth with the benchmr Γ for three different levels of investor ris version. Pnel B) in Fig. displys the loss of terminl welth 10 Boyle nd in in press) show tht, in the CRRA utility function fmily, there is just single vlue for the ris version prmeter tht could me the pyoff Γ optiml for the investor. An dditionl necessry condition is tht y 0 = x 0 otherwise Γ could not be optiml. In generl, it is nown tht ny given investment pln is only optiml for very specil clss of utility functions. See Blc nd Perold 199), Cox nd Lelnd 000), He nd Lelnd 1993) nd He nd Hung 1994). corresponding to the CRRA utility function ux) = x1 γ 1 γ, γ > 0 for γ = 1,, 3. Note tht the log-utility function corresponds to the cse γ = 1. When the ris version prmeter γ = 1, the loss percentge is firly smll. When the ris version prmeter γ increses, the loss increses. If γ =, the loss could be s high s 0% of the initil investment mount when the index S moves from 1$ to $ fter five yers. he loss could rech 45% when γ = 3. his implies tht the more ris verse the investor, the higher the loss. It is interesting to note tht the previling wisdom seems to be tht n EIA might be more pproprite for conservtive investors. In fct, s Wchter 003) showed tht the best strtegy for the most conservtive investor is to buy nd hold the risfree bond. However our findings is intuitive. For conservtive investor, the percentge investment in the index is smll. Hence if the index moves up substntilly the conservtive investor will hve foregone profitble investment opportunity. We cn use the nlysis of this section to discuss implictions for the design of EIAs. It is cler from our discussion tht investors with different levels of ris version would prefer different contrct designs. We sw tht the optiml X vries with the investor s utility function. However in prctice the investor in n EIA receives pyoff Γ which is offered by the issuer. Even in this cse there is scope for vrition in contrct design. If we ssume point-to-point EIA with prmeters, g}, then the issuer cn design rnge of contrcts with current no-rbitrge vlue, y 0 < x 0, by vrying nd g. Investors will not be indifferent mong these contrcts. ypiclly there will be preferred contrct in this set. Suppose we lbel the set of fesible pirs, g} by the set A. For A the corresponding EIA is Γ. hen n investor with utility function u.) will hve preferred EIA obtined s follows mx E[uΓ )]. A his suggests tht insurers should offer severl EIAs with different contrct prmeters to ppel to different investor clienteles. Anecdotl evidence suggests tht this hppens in prctice Exmples of conventionl style EIAs So fr we hve discussed only the plin vnill point-to-point EIA. he point-to-point EIA is one of the simplest types of EIA designs in the mret. he return on this type of contrct cn be modified in vrious wys. In this subsection we describe other EIA contrcts which cn be nlyzed within the frmewor used in the lst section. Here re some exmples. Exmple 1 Interest Rte Cp nd Floor). In some EIA policies, the index-lined interest rte is cpped t specific level nown s the cp rte. In the simplest cse the totl return is cpped. In this cse the pyoff is ) }} Γ = x 0 mx e g S, min, e c. 3.1) However in some designs the interest rte is cpped monthly. Assume tht 0 = 0 < 1 < < N = is the term over which the rte is cpped. he return of this EIA pyoff, 1 log Γ x 0 ), is given by mx g, N min i=1 1 i i 1 log where c is the monthly cp. ) Si ), c} }, 3.) S i 1 )

7 P. Boyle, W. in / Insurnce: Mthemtics nd Economics ) Exmple High Wter Mret). In this cse, the index-lined rte is bsed on the increse in index vlue from the index level t the beginning of the term to the highest index vlue during the contrct s term. he pyoff is ) mx St Γ = x 0, 3.3) where mx S t denotes the highest level long the pth mx S t = mxs0), S 1 ),..., S N 1 ), S N )}. Exmple 3 Rtchet EIA). his is lso nown s the cliquet design. In this design the contrct will ern, t the end of every period, the return on the index over tht period with prticiption rte, or the minimum gurnteed return g, whichever is higher. he benchmr becomes n ) } Γ = x 0 mx e g Sti,. 3.4) i=1 S ti 1 his list of EIAs, of course, is fr from complete. An EIA might combine more thn one feture in the list. For instnce the high wter mr feture could lso hve n interest rte cp. We refer to Streiff nd DiBise 1999) nd Hrdy 003) for more extensive description of the different designs. We cn summrize the different design fetures in terms of the pyoff s follows. he pyoff on the EIA is non-negtive rndom vrible Γ which is function of the reference index nd the contrct prmeters. Note tht surrender option is often embedded inside EIA. he surrender option gives policyholders to give up the contrct nd receive certin surrender vlue. he discussion on the optiml design of EIA with surrender option dds technicl complictions nd is not pursued in this pper Improving the design of conventionl EIA In this section we discuss how to improve the design of ny type of conventionl EIA. We ssume tht the bsic EIA contrct hs pyoff of Γ t the end of the term. We use the sme frmewor s before. he bsic EIA pyoff is ten to be the benchmr. In this cse the optiml design cn be recst s the following optimiztion problem mx E[uX x 0,π )] where x 0 is the initil investment, π is self-finncing strtegy such tht the terminl welth X x 0,π is subject to the constrint tht X x 0,π Γ. 3.5) o derive n explicit expression for the optiml contrct we ssume tht the finncil mret is complete. According to Hrrison nd Kreps 1979) nd Hrrison nd Plis 1981), there exists unique stte-price density process ξ }. For technicl resons we ssume tht ξ hs finite moments, positive or negtive. In ddition we will lso me use of the following ssumption on the continuity nd rnge of the stte-price density. Assumption 1. he distribution function of ξ is continuous nd Pξ < x) > 0 for every x > 0, > 0. he following theorem 11 solves the optimiztion problem. heorem ) here exists self-finncing trding strtegy π such tht X x 0,π Γ if nd only if E[ξ Γ ] x 0. ) If the gurnteed benchmr, Γ stisfies E[ξ Γ ] < x 0, then there exists unique optiml fesible terminl welth, nmely X ei mxiλ g ξ ), Γ } such tht the budget constrint is binding. More precisely, E[ξ mxiλ g ξ ), Γ }] = x 0. Furthermore, λ g is unique by virtue of Assumption 1. 3) If E[ξ Γ ] = x 0, then the only fesible terminl welth is Γ,.s. herefore, for ny given EIA with pyoff Γ, the optiml design or pyoff for n investor with utility function u.) should be mxiλ g ξ ), Γ }. he loss profit) of the terminl welth for the investor issuer) is mxiλ g ξ ), Γ } Γ. By heorem 3.1, the present vlue of the loss of the terminl welth E [ ξ mxiλ g ξ ), Γ } Γ ) ] = x 0 y ) x 0 y 0 is the initil loss for ny investor who buys the EIA nd this is independent of the investor s ris preference. However, the loss of the terminl welth will be different for different investors. he contrct design in heorem 3.1 is more widely nown in the finnce literture s portfolio insurnce since Γ represents the insurnce level for the portfolio s terminl welth. 1 he pyoff Γ is lso referred s the benchmr in the reminder of this pper. Using heorem 3.1 we re ble to design n optiml EIA type contrct with pyoff which is t lest equl to Γ. 4. he generlized EIA In this section we expnd the EIA contrct to include new design feture nd we refer to this new design s generlized EIA. his new design extends the concept of conventionl EIA in significnt wy. he contrct retins some of the fetures of conventionl EIA. Boyle nd in in press) hve obtined n explicit solution for the optiml design of this generlized EIA under firly generl conditions. In this pper we will quote their results without proofs. his section is orgnized s follows. In Section 4.1 we discuss the motivtion for introducing the generlized EIA contrct. We then explin how the optiml design of the generlized EIA cn be viewed s non-convex) optimiztion problem. Next we give the explicit optiml design of the generlized EIA contrct. Lstly we discuss severl extensions of the generlized EIA Motivtion In prctice, mny investors require sfety of their principl but they lso hner fter high returns. We hve seen trde-off between these requirements in point-to-point EIA in Section. he gurntee hs cost. he higher the gurntee rte, the lower the prticiption rte in point-to-point EIA. We cn twe the design of the EIA but the no-rbitrge principle limits wht cn be ccomplished. However it is possible to redesign the contrct to me it better suit investor preferences. We now describe how this cn be done. he distinguishing feture of generlized EIA contrct is tht it gives the investor n opportunity to bet or mtch) the performnce of some selected benchmr index t some confidence level. For exmple, under generlized EIA n investor might ern 15% of the index return with n 80% confidence level. 11 his theorem is well nown in the finnce literture nd occurs in different plces in different versions. We refer to Dybvig et l. 1999) nd Boyle nd in 007) for its proof in the current context. 1 See Blc nd Perold 199), Brennn nd Schwrtz 1988), Brennn nd Solni 1988), Bs 1995), Benning nd Blume 1985), Grossmn nd Zhou 1996), Lelnd 1980), nd epl 001).

8 310 P. Boyle, W. in / Insurnce: Mthemtics nd Economics ) his generlized EIA is similr to n event-driven EIA in some sense but the design ide is very different. o illustrte this point we consider two physicl) events s follows: A = S e }, B = mx S t L} 4.1) 0 t where > 0, L > 0 re determined by PA) = PB) = ) he corresponding event-driven EIAs hve pyoffs ) } 1.5 Γ 1 = x 0 mx e g S, 1 A, Γ = x 0 mx e g, S ) B }, 4.3) respectively. he optiml designs of the EIAs Γ 1 or Γ could be determined by heorem 3.1. he buyer of either one of these EIAs hs t lest n 80% chnce of obtining 15% of the index return but the set of events which coincide with the higher return is different in ech cse. For the EIA with pyoff Γ 1, the event is triggered if the finl index price S is high enough, while for the EIA with pyoff Γ, the event is depends on the pth of the index price process S t : 0 t }. Assuming now the investor is either not sure or not interested in specific event. We now show how the concept of generlized EIA cn be introduced using these exmples to build new type of contrct. In this new contrct, the ris for the investor is determined solely by the probbility of chieving the desired return. he benchmr return is bsed on the relized index return. here re severl justifictions to support this type of probbility constrints. First, from the no-rbitrge principle, if n investor wnts to bet the ris-free rte she must ber some ris. Second, mny investors do seem to cre bout beting benchmr. hird, there is often uncertinty bout the model nd its prmeters. Whenever there is model ris nd estimtion ris, probbility constrints provide wy to cope with model misspecifiction error. We illustrte the design of the generlized EIA using the pointto-point structure s n exmple. Assume tht g is the gurnteed rte, the benchmr is Γ nd tht the confidence level is α. he pyoff, X, of the generlized EIA, stisfies two constrints: ) ) X x 0 e g S, P X x 0 α. 4.4) here is n importnt difference between conventionl EIA nd generlized EIA, in terms of the pyoff structure. he issuer nd the investor hve very different perspectives. he issuer would lie to provide the desired pyoff s economiclly s possible. In ddition the issuer my hve other objectives s well. For exmple, one possible objective for the issuer is to minimize the expected return of the pyoff: min E[VX x 0,π )] where V.) denotes the utility function of the issuer, the terminl welth X x 0,π is subject to 4.4) nd π is self-finncing strtegy. 13 In contrst, the investor wishes to mximize the expected utility of terminl welth subject to the constrints 4.4). We focus on the investor s perspective in this pper. he optiml design for the investor cn be recst s the following optimiztion problem mx E[uX x 0,π )] where π is self-finncing strtegy such tht the terminl welth X x 0,π is subject to the constrints in 4.4). Fig. 3. rde-off between ˆα nd the prticiption rte. Investor hs log utility nd the index is log-normlly distributed. Prmeters: = 5, µ = 6%, r = 4%, σ = 0%. he gurnteed rte g = % is given. he mximum probbility ˆα is defined in 4.5), s function of the prticiption rte. We clculte ˆα by using the nlysis in Spiv nd Cvitnić 1999), Boyle nd in in press). he point on the curve where ˆα = 1 corresponds to ˆ = 0.60, which is determined by Eq..8) where ĝ = g = %. he mximum probbility ˆα is decresing with respect to the prticiption rte. 4.. Optiml design of the generlized EIA contrct In this subsection we present the optiml design of the generlized EIA. We ssume tht g is the gurnteed rte nd f = e g. We ssume tht Γ is positive.s.) rndom vrible nd tht the pyoff X stisfies PX Γ ) α, X fx 0 where α 0, 1) is the specified physicl probbility. We hve noted tht there is precise trde-off between ĝ nd ˆ in Section for point-to-point EIA. For generlized point-topoint EIA, the prticiption rte could be incresed substntilly with the sme gurnteed rte g. o illustrte the impct of the confidence level α, we define ) ) ˆα := mx P X x 0,π S x 0 4.5) π where π denotes self-finncing strtegy, nd X x 0,π is the terminl welth by following the strtegy π, strting from the initil mount x 0, nd the terminl welth stisfies X x 0,π fx 0. Given prticiption rte, ˆα is the mximum possible probbility tht the terminl welth bets x 0 S ). Conversely, given ˆα, we cn determine the mximum possible prticiption rte such tht the terminl welth bets x 0 S ) with confidence level ˆα. In prticulr, when ˆα = 1, the mximum possible prticiption rte is ˆ in Eq..4). 14 Fig. 3 displys the trde-off between the prticiption rte nd the mximum probbility ˆα, when the gurnteed rte is g. We now stte the generl result for the optiml generlized EIA contrct. For the complete technicl detils nd proofs we refer to Boyle nd in in press). o stte the result we need certin definitions nd technicl detils. We do not discuss these definitions or ttempt to motivte them in this section since our 13 his optiml design problem for the issuer is studied in Bernrd et l. submitted for publiction). 14 For complete discussion of ˆα we refer to Spiv nd Cvitnić 1999) nd Boyle nd in in press).

9 P. Boyle, W. in / Insurnce: Mthemtics nd Economics ) focus here is on pplictions rther thn mthemticl proofs. We ssume tht Γ is generl benchmr index. It is convenient to define the following functions for ny λ > 0, x 0, Gλ, x) = PmxIλξ ), fx 0 } < Γ, hλ, ξ ) x), 4.6) nd Hλ) = PmxIλξ ), fx 0 } < Γ ), λ > 0 4.7) nd let λ α := Supλ : Hλ) < 1 α}, 4.8) where hλ, ξ ) = umxiλξ ), fx 0 }) λξ mxiλξ ), fx 0 } + λξ Γ uγ ). 4.9) Under Assumption 1, there exists unique positive λ g such tht E[ξ mxiλ g ξ ), fx 0, Γ }] = x ) We lwys ssume tht λ α < λ g. Otherwise, this generlized EIA contrct reduces to the clssicl conventionl) EIA, nd heorem 3.1 presents the optiml design lredy. Define G 1 λ, 1 α) = x > 0 : Gλ, x) := 1 α}, λ > ) We impose one more technicl ssumption tht ensures the continuity of certin functions. Assumption. Gλ, x) is jointly continuous with respect to both λ nd x. Moreover, for ny λ λ α, there exists t most one member in G 1 λ, 1 α). Boyle nd in in press) show tht Assumption implies the existence of unique positive number dλ, α) such tht PmxIλξ ), fx 0 } < Γ, hλ, ξ ) dλ, α)) = 1 α. 4.1) Define X λ,α ) s follows: Iλξ ), if Iλξ ) Γ > fx 0 mxiλξ ), fx 0 }, if mxiλξ ), fx 0 } < Γ, hλ, ξ X λ,α ) = ) dλ, α) Γ, if mxiλξ ), fx 0 } < Γ, hλ, ξ ) < dλ, α) mxiλξ ), fx 0 }, if Γ fx 0. We re now in position to present the generl result which is proved in Boyle nd in in press). 15 heorem 4.1. Under Assumptions 1 nd, nd ssuming tht lim E[ξ fx 0 1 Γ >fx0,hλ,ξ λ ) dλ,α)}] + E[ξ Γ 1 Γ >fx0,hλ,ξ )<dλ,α)}] + E[ξ fx 0 1 Γ fx0 }]} < x ) then there exists self-finncing process π t with terminl welth X geqp ) such tht X geqp ) fx 0, PX geqp ) Γ ) α. And E[uX geqp ))] E[uX x 0,π ))] for ny self-finncing process π t whose terminl welth is subject to the constrint conditions: X x 0,π ) fx 0, PX x 0,π ) Γ ) α. Moreover, we cn choose X geqp ) = X λ,α) for some positive rel number λ > λ α. In the next section we will illustrte how to me use of heorem 4.1 to explicitly construct the optiml pyoff terminl welth) of the generlized EIA. he optiml pyoff of EIA in heorem 4.1 is designed from investor s perspective, but it is helpful for both investor nd issuer. If the issuer were to dynmiclly replicte this optiml pyoff, hedging rises importnt prcticl 16 issues tht we do not explore in this pper Extensions In the generlized EIA contrct, only gurnteed rte is imposed. Actully it is possible to embed conventionl EIA into generlized EIA. We explin the min points in this subsection. Given n EIA contrct with pyoff structure Γ 0 t the end of the term. We ssume tht Γ is benchmr nd α is specific probbility. In the generlized EIA, the pyoff X stisfies X Γ 0, PX Γ ) α. he optiml design of the generlized EIA is reduced to mximum expected utility problem subject to the lst set of constrints on the terminl welth X. heorem 4.1 cn be extended esily to this cse where fx 0 is replced by Γ 0. We ssumed constnt ris-free interest rte in this pper for convenience. We cn redily extend the results to deterministic term structure of interest rtes. Moreover, it is lso possible to use stochstic interest rte model in the nlysis. 5. Exmple of generlized EIA contrct In this section we illustrte the detils of our optiml solution for the generlized EIA by using specific exmple. he exmple is instructive since it revels number of interesting properties of the optiml contrct. We ssume tht the benchmr is Γ = x 0 S ) nd tht the gurnteed is g. We lso ssume tht ux) = logx) to simplify some technicl issues. Let us briefly recll the steps involved in deriving the optiml solution. We bse the construction on the generl solution given in heorem 4.1. We consider clss of fesible terminl welths X λ,α, nd the optiml X geqp ) will belong to this clss. he optiml terminl welth corresponds to the pyoff for the optiml generlized EIA. he construction of X λ,α, by definition, depends on the properties of some uxiliry functions hλ, ξ ), Gλ, x) nd Hλ). For instnce, using Hλ) we cn determine specil λ which we cll λ α. From Gλ, x) we determine dλ, α). Finlly, we cn find the optiml λ λ α from the binding constrint tht the norbitrge vlue of the clim X λ,α is x 0. o determine the exct functionl form of hλ, ξ ), we consider two regions. Over the region Iλξ ) fx 0 }, we hve hλ, ξ ) = h 1 λ, ξ ); over the region Iλξ ) < fx 0 }, hλ, ξ ) = h λ, ξ ), where ) h 1 λ, ξ ) = logλx 0 b ) 1 + b 1 logξ ) nd + λx 0 b ξ 1 b, 5.1) h λ, ξ ) = logf b ) λfx0 ξ + b logξ ) + λx 0 b ξ 1 b. 5.) 15 here is mjor obstcle in tcling this optimiztion problem under this nonconvex constrint. he solution given here solves the problem for the complete mret frmewor. he corresponding optiml design problem in n incomplete mret is not solved yet. 16 here re importnt technicl issues in hedging such s trding restrictions, trnsction costs, nd the discontinuity of the pyoff. We do not ddress the hedging problem in this pper but refer to Boyle nd Vorst 199), Edirisinghe et l. 1993), Ni nd Uppl 1994), nd Lelnd 1985).

10 31 P. Boyle, W. in / Insurnce: Mthemtics nd Economics ) We cn determine dλ, α) explicitly from the two functions hλ, ξ ) nd Hλ). he explicit construction of X λ,α ) will be given shortly. At this stge we mention tht it is esy to chec the vlidity of both Assumptions 1 nd when b. Since the ssumptions in heorem 4.1 re verified, the optiml terminl welth is determined by heorem 4.1. We now give the explicit expression of X λ,α ). he form of this expression depends on the reltive mgnitudes of nd b. he cses < b, = b nd > b re ll different nd we use different symbols for the optiml terminl welth in these three cses. In ech cse we hve fmily of the optiml) terminl welth, where ech fmily is indexed by positive number λ λ α. We use the following nottion for these fmilies in the three different cses. 1. When > b, there re two subcses. he terminl welth is either Z λ,α ) or V λ,α ), corresponding to different vlues of λ.. When < b, the terminl welth is Y λ,α ). 3. When = b, the terminl welth is W λ,α ). he explicit expressions for Z λ,α ), V λ,α ), Y λ,α ), W λ,α ) re quite complicted nd re given in Appendix A. In the rest of this section, we discuss this point-to-point generlized EIA contrct for the three cses > b, < b or = b respectively Cse one: > b As we explined erlier, given confidence level α, there exists mximum prticiption rte. his trde-off is cptured by the eqution: E[ξ fx 0 1 ξ <ξ 1 α } ] + E[x 0 b ξ 1 b 1 ξ 1 α ξ <c} ] + E[ξ fx 0 1 ξ c}] x 0 5.3) where both prmeter c nd ξ 1 α re defined in formul A.1), A.) of Appendix A, respectively. hen there exists fesible solution which stisfies the two constrints in the generlized EIA contrct; nmely chieving t lest the gurnteed return nd the probbility of beting the benchmr Γ. Furthermore the solution for the optiml contrct optiml welth) exists in this cse. Note tht λ α = 1 ξ b b 1 α x b, 5.4) 0 nd we ssume tht λ α < λ g. he implementtion procedure is s follows: Step 1. Chec whether the trde-off condition 5.3) holds or not. If this condition is stisfied, go to the next step. Otherwise, stop. Step. Chec whether E[ξ Z λβ,α)] x 0. If so, there exists λ [λ α, λ β ] such tht E[ξ Z λ,α)] = x 0. We solve for the optiml λ which is the solution of the eqution E[ξ Z λ,α)] = x 0. If not, go to the next step. Step 3. here exists λ > λ β such tht E[ξ V λ,α)] = x 0. Solve for the optiml λ in the eqution E[ξ V λ,α)] = x 0. he optiml terminl welth Z λ,α ) nd V λ,α ) re displyed when > b in Fig. 4. Assuming first = In this cse, λ α = 1.058, λ β = nd λ g = Since E[ξ Z λβ,α)] = x 0, there exists one optiml welth with the form Z λ,α ) for some λ λ α, λ β ]. Pnel A) of Fig. 4 displys the optiml welth Z λ,α) for λ = λ α, λ β ). 17 When = 0.75 nd other contrct prmeters re ept the sme, since E[ξ Z λβ,α)] = > x 0, then there exists one optiml welth with the form V λ,α ) for some λ > λ β. Pnel B) in Fig. 4 displys the optiml welth V λ,α) for λ = > λ β in the cse of = It is instructive to compre this generlized EIA with conventionl EIA. From Eq..8) we now tht the mximum prticiption rte ˆ =.60. In the bove generlized EIAs the prticiption rte is either 0.65 or 0.75 both of which re greter thn ˆ. he return of the generlized EIA cn be higher thn the return of EIA in some scenrios. he ris for the generlized EIA, however, is tht the higher prticiption rte is relized only with probbility of 85%. he optiml welth of the EIA with prticiption rte 0.60 is displyed in both Pnel A) nd Pnel B) of Fig. 4. In Pnel A) we see tht for some index movements the optiml pyoff under the generlized EIA bets the optiml pyoff under the stndrd EIA. On the other hnd, the optiml pyoff of the stndrd EIA bets the generlized EIA when the index price moves up significntly. his feture seems surprising becuse the generlized EIA with higher prticiption rte seems better thn the EIA with smller prticiption rte. However it is intuitive from the motivtion of the generlized EIA. he generlized EIA hs the potentil to hve higher return with probbility α. herefore, the ris of hving lower return hs probbility of 1 α. hus the optiml pyoff under the generlized EIA hs quite different profile from the optiml pyoff under conventionl EIA. he mjor point is tht except for smll probbility smller thn 1 α = 15%), the optiml pyoff under the generlized EIA bets the optiml pyoff under the conventionl EIA. We cn observe similr pttern from Pnel B) for = Cse two: < b his solution here is more complicted thn in the previous cse. he trde-off between the prticiption rte nd the probbility α is cptured by the following condition: lim E[ξ Γ 1 λ λ ξ yλ} y λ ξ <c}] + E[fx 0 ξ 1 y λ ξ y λ } ξ c}] < x ) he rest of the procedure is similr to the first cse > b with some technicl differences. In the current cse, we need to solve for three vribles y λ, y λ, λ} simultneously. hey re determined by the following three equtions: Py λ < ξ < y λ ) = 1 α, 5.6) nd h 1 λ, y λ ) = h λ, y λ ), 5.7) nd E[ξ Y λ,α )] = x ) he procedure is then similr to the first cse nd we omit the detils Cse three: = b he lst cse is specil but it is the most chllenging one from theoreticl perspective. he reson is s follows. Becuse = b, Assumption does not hold nd heorem 4.1 cnnot be used directly. he trde-off between nd α is cptured by the following conditions: fx 0 ξ 1 α E[1 ξ ξ 1 α } ] + x 0E[1 ξ 1 α <ξ < f }] 17 he clcultion of λ in this pper is bsed on formule in Appendix B. + fx 0 E[ξ 1 ξ f } ] > x 0, 5.9)