Financial Planning and Risk-return profiles

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1 Financial Planning and Risk-etun pofiles Stefan Gaf, Alexande Kling und Jochen Russ Pepint Seies: Fakultät fü Mathematik und Witschaftswissenschaften UNIERSITÄT ULM

2 Financial Planning and Risk-etun pofiles Stefan Gaf * Ph. D. student, Univesity of Ulm Helmholtzstaße 22, Ulm, Gemany Phone: , fax: Alexande Kling Institut fü Finanz- und Aktuawissenschaften Helmholtzstaße 22, Ulm, Gemany Phone: , fax: Jochen Russ Institut fü Finanz- und Aktuawissenschaften Helmholtzstaße 22, Ulm, Gemany Phone: +49 (731) , Fax: +49 (731) This vesion: 13 Decembe 2010 Abstact The impotance of funded pivate o occupational old age povision will incease due to demogaphic changes and the esulting poblems fo govenment-un pay-as-you-go systems. Clients and advisos theefoe need eliable methodologies to match offeed poducts and clients needs and isk appetite. In this pape, we analyze existing appoaches such as sample illustations and histoical backtesting that ae often used fo compaing and explaining poducts. We find that the infomation povided is often insufficient o even misleading. We intoduce an altenative methodology based on isk-etun pofiles, i.e. the (fowad-looking) pobability distibution of benefits. In a model with stochastic inteest ates and equity etuns including stochastic equity volatility, we deive isk-etun pofiles fo vaious types of existing unit-linked/equity-linked poducts with and without embedded guaantees. We highlight the diffeences between actual poduct chaacteistics and the impession geneated by existing appoaches and explain the esulting misleading incentives fo poduct developes and financial advisos. * Coesponding autho

3 FINANCIAL PLANNING AND RISK-RETURN PROFILES 1 1 Intoduction The demogaphic tansition esulting fom a continuing incease in life expectancy combined with athe low fetility ates constitutes a sevee challenge fo govenment-un pay-as-yougo pension systems in many counties. Theefoe, the impotance of funded pivate and/o occupational old age povision has been inceasing and will accoding to industy suveys continue to incease. Competing fo clients money, povides of old age povision poducts (e.g. life insues and asset manages) have come up with a vaiety of poducts that claim to be suitable packaged poducts fo old age povision. Often, equity investments equipped with cetain guaantees ae maketed as optimal solutions combining the upside potential of stock makets with the secuity of cetain guaantees, e.g. money back guaantees o guaanteed etuns. Howeve, clients as well as financial advisos face a sevee poblem: The infomation that is (o legally has to be) povided by poduct povides is usually not sufficient to assess and to compae the isk-etun pofile of diffeent poducts since impotant poduct chaacteistics ae ignoed. Thus, poduct chaacteistics cannot be matched with the client s isk avesion. Infomation povided to clients about a poduct s pefomance potential and investment isk often falls in one of the following two categoies: Deteministic sample illustations o histoical backtesting. A deteministic sample illustation gives the hypothetical development and matuity benefit of a poduct, assuming that the undelying investment pefoms steadily with some constant etun evey yea. Diffeent poducts ae then anked by compaing thei pefomance assuming the undelying investment vehicles pefom at this ate of etun. Often, equity funds ae pojected with the same constant annual ate as balanced funds, money maket funds, o guaanteed funds that ae managed accoding to some stategy, e.g. CPPI. In contast, a histoical backtesting shows how the poduct would have pefomed, had it been offeed in the past, o at seveal times in the past. Both, sample illustations and backtestings ae obviously of vey limited explanatoy value fo futue etun potential and isk of a poduct. Futhemoe, backtestings lead to a cetain cyclical behavio of financial advisos and clients because poducts that would have pefomed well in the immediate past ae typically pefeed. Thee ae two main steams of liteatue in the field of individual financial planning. On the one hand, thee exists a lage body of liteatue on the question whethe and what potion of wealth people should annuitize when eaching etiement age, e.g. a seminal pape by Yaai (1965) o moe ecent studies by Milevsky (1998), Milevsky et al. (2005) o Gead et al. (2010). 1 On the othe hand, some authos deal with finding optimal investment stategies fo accumulating money fo etiement. This eseach is usually concened with deiving the optimal (often dynamic) allocation among given asset classes by maximizing expected utility. 1 Fo a moe detailed liteatue oveview on this topic, we efe to the efeences theein.

4 FINANCIAL PLANNING AND RISK-RETURN PROFILES 2 Accodingly, e.g. Cains et al. (2006) find optimal asset allocations (called stochastic lifestyling ) egading a defined contibution plan, wheeas e.g. Boyle and Tian (2009) deive optimal paametes of an equity indexed annuity (a poduct consisting of some minimum guaantee, a pefomance cap and an index paticipation ate) such that the investo s expected utility is maximized given cetain constaints. Howeve, these impotant theoetical esults ae often too complex fo clients and/o advisos fo a pactical use by clients and/o advisos and highly depend on the paticula choice of a utility function. Futhe, a pactical implementation of such appoaches would often equie an ongoing management of clients accounts which is often too complex, not feasible fo athe small contact volumes o might esult in tax disadvantages upon each tansaction. Theefoe, in pactice often so-called packaged poducts whee cetain stategies ae implemented that do not equie any action on the client s side duing the tem of the poduct, ae offeed and many financial advisos ty to find the most suitable poduct out of a vaiety of such poducts fo each client. We ae not awae of any liteatue dealing with the pactical issue of how to compae existing packaged poducts and how the elevant infomation necessay fo a client-individual poduct selection can be deived fom infomation povided. In the pesent pape, we theefoe deal with the question of how clients should select thei best choice fom (a limited numbe of) diffeent old-age povision poducts offeed in the maket. We explain the weaknesses of poduct illustations and compaisons that ae pedominant in many counties and popose a new methodology based on a compaison of fowad-looking isk-etun pofiles. By this, we mean the pobability distibution of the (matuity) benefit calculated within a stochastic model fo the financial maket. Using isketun pofiles, poducts that match a client s isk avesion can be identified. Besides pesenting this methodology, we deive numeical esults compaing basic poduct categoies that can be found in many counties. In paticula, we focus on compaing diffeent ways of geneating investment guaantees such as dynamic stategies like CPPI o static option-based stategies. We explain why typical poduct infomation used hitheto is not only insufficient but often even misleading. The emainde of this pape is oganized as follows: In Section 2, we pesent the poducts consideed in ou analysis. Section 3 then shows how these poducts ae typically explained using sample illustations and histoical backtestings. Section 4 intoduces the financial model used fo picing the deivatives embedded in some poducts in Section 5 and fo the deivation of fowad-looking isk-etun pofiles in Section 6. Finally, Section 7 concludes. 2 Consideed poducts In ou analyses, we conside seveal poduct types that ae (sometimes in diffeent vaiants) common in etiement planning and offeed by vaious financial institutions such as banks, insues o asset manages in many counties. We distinguish poducts with and without embedded investment guaantees and conside poducts with single pemium and with egula monthly pemium payment P and a tem of T yeas. Futhe, fo the sake of

5 FINANCIAL PLANNING AND RISK-RETURN PROFILES 3 simplicity and to focus on the main effects, the same (athe simple) fee stuctue is applied to all diffeent packaged poducts: Pemium popotional chages β P educe the amount invested to ( 1 β ) P (this amount is also efeed to as savings pemium ). Account popotional chages γ quoted as an annual fee ae deducted on a monthly basis fom the client s account. Additionally, fund management chages c (also quoted as an annual chage but deducted daily) ae applied within mutual funds if such funds ae used in the packaged poduct. Additional fees fo the guaantee may apply fo the poducts with guaantee (see below). The consideed poducts ae displayed in Table 1: Poducts without embedded guaantee Poducts with embedded guaantee Investment in mutual funds o equity fund o balanced fund o lifecycle managed funds Money back guaantee Static option based guaantee ( undelying plus put o zeo plus call ) Semi-static guaanteed poduct of type zeo plus undelying Dynamic CPPI stategy used on a client by client individual basis Savings pemium guaantee Dynamic CPPI stategy implemented in a mutual fund and thus managed on a collective basis (so-called CPPI guaanteed fund with high watemak guaantee) Table 1: Consideed poducts In poducts without embedded guaantee, the savings pemium is contibuted to a mutual fund investing in equities o bonds depending on the fund s pofile. In ou analysis we conside equity, balanced and lifecycle funds. In ou numeical analyses in Sections 5 and 6, we assume that an equity fund is completely invested in the modelled equity-pocess, wheeas a balanced fund has a constant potion [ 0,1 ] x (in ou numeical analyses: 50%) of its capital invested in equity shaes and the emaining pat in bonds. We assume that the bond investment has some taget duation d and theefoe model the bond potfolio by a simple zeo-bond with time to matuity d = 5. In addition, a lifecycle managed fund applies a time-dependent (howeve not path-dependen asset allocation, whee ( ) [ 0,1] S x models S, t t the equity potion at time t. We assume that the lifecycle fund stats with 100% equity

6 FINANCIAL PLANNING AND RISK-RETURN PROFILES 4 investment and then linealy deceases the equity potion each yea to 0% in the last yea. Finally, the cuent asset allocation of all consideed funds is ebalanced daily to match equied potions and the zeo-bond s time to matuity espectively. Poducts equipped with money back guaantee guaantee that at least the client s contibutions ae paid back at matuity. Howeve, the way of geneating this guaantee vaies thoughout the consideed poducts: We let A t denote the client s account value at time t and G t the guaanteed amount at time t to be paid at the contact s matuity T, i.e. the sum of paid pemiums up to time t in ou analysis. The static option based stategy invests the savings pemiums into the intoduced equity fund. Each month additionally to the fees intoduced above an account popotional guaantee fee 2 g quoted as an annual chage is deducted fom the client s account and invested in some deivative secuity (o hedge potfolio) on the intoduced equity fund deliveing max{ AT,0} G at contact s matuity. Theefoe fom a client s point of view T the poduct coesponds to an equity investment plus a put option on the equity fund. Note that the focus of ou analyses is not on the hedging and its pefomance but athe on the option s payoff fom client s pespective at matuity. The zeo plus undelying type poduct consists of a iskless asset (e.g. a zeo-bond 3 ) and a isky financial instument (e.g. an equity fund). The potion invested to the isk-fee asset is detemined such that its payoff at the contact s matuity T coincides with the guaantee If a single pemium is paid at contact s outset, the guaantee s pesent value deived by discounting with the zeo-bond s isk-fee ate of etun less account popotional fees is invested in the iskless asset which then geneates the guaantee at the contact s matuity. The emaining pat is invested in the undelying equity fund. Regading egula contibutions, the analogous (semi-static) asset allocation is pefomed as follows: Each month 4 say at time t the client s account value pesent value the so-called floo G T. A t is split up into both assets. Fist, the guaantee s F t (calculated analogously as in the single pemium case) is invested the iskless asset which then exactly ensues the issued guaantee at matuity. Second, the emaining cushion C t = A F is invested in the isky asset which povides t t 2 Note g is fixed thoughout the contact s tem at outset and not adjusted late on. 3 We ignoe potential default isk in ou appoach. 4 In geneal, a eadjustment of the undelying potfolio is only necessay at time of new contibutions. Since, we conside single pemium and egula payment we howeve pefom monthly ebalancing pemanently.

7 FINANCIAL PLANNING AND RISK-RETURN PROFILES 5 futhe upside potential. Even in case of total loss in equities, the undelying concept ensues the guaantee by espective zeo-bond investment. Constant popotion potfolio insuance (CPPI 5 ) essentially futhe develops the idea of ceating guaantees by investing in iskless and isky assets. Howeve, hee the asset allocation dynamically changes ove time. At each ebalancing time t (usually and also in ou numeical analyses daily) the povide detemines the optimal asset allocation fo the client s account. Howeve, in contast to the zeo plus undelying stategy, not only the cushion Ct = At Ft, but a cetain multiple theeof, i.e. max( m C t, A t ), is invested in the isky asset. This is built on the assumption that the isky asset will not loose moe than a wost-case-pefomance of 1 until the next ebalancing occus. Theefoe, the client s m account will not loose moe than ( t t ) t 1 max m C, A C and thus the account value will m not dop below the floo. Obviously, the poduct povide faces two souces of isk within a CPPI stuctue: Fist, the isky asset might loose moe than 1 duing one peiod (this isk m is often efeed to as gap isk o ovenight isk) and second, the floo might have changed within one peiod due to inteest ate fluctuations. Most of the povides hedge the fist isk by puchasing so-called cash-potection puts (essentially 1 -out-of-the-money-puts) but do m not hedge the second isk. Tankov (2010) povides useful insights in picing and hedging elated cash-potection-puts in a Lévy-type famewok. In ou appoach we will howeve skip valuation of the cash-puts and athe assume a flat additional chage of 0.2% p.a. on the isky asset within CPPI poducts. We distinguish a CPPI stategy managed individually fo each client (so-called individual CPPI o icppi) whee a cetain minimum value of each client s account is guaanteed and CPPI managed within a mutual fund whee a cetain guaanteed matuity value pe unit of the fund is guaanteed. Such fixed-tem funds ae vey popula in seveal Euopean counties, e.g. in Gemany. They essentially pomise a money back guaantee at matuity fo all contibutions eve made to the fund (which is typically less than the contibutions made to the poduct that invests in the fund). In ode to guaantee all contibutions made to the fund by diffeent clients (which may stat thei contacts at diffeent times, have diffeent pemium pattens like single pemium vs. monthly pemium, o constant pemiums vs. inceasing pemiums), the guaanteed matuity value of the fund has to be inceased wheneve the cuent net asset value (and thus the pice at which new money is invested in the fund) exceeds the pevious guaanteed value (high watemak guaantee). Futhe note, only contibutions kept within the fund until matuity ae guaanteed. Theefoe if any chages ae 5 Cf. Black and Peold (1992).

8 FINANCIAL PLANNING AND RISK-RETURN PROFILES 6 deducted thoughout the poduct s lifetime by selling pats of the puchased fund units even less than the savings pemiums might be guaanteed at contact s matuity. 3 Poduct compaisons Existing appoaches We now summaize appoaches typically used fo compaing the chaacteistics of the consideed poducts. We stat with a methodology vey common e.g. in Gemany o Austia based on deteministic sample illustations and continue with an appoach using histoical data which is often used e.g. in the United Kingdom and ecently gains populaity also in othe pats of Euope. To the best of ou knowledge, while thee ae diffeences in details, most existing appoaches woldwide to compae the payoff chaacteistics of old age povision poducts ae eithe based on the assumption of some hypothetic capital maket scenaio 6 o on histoical data. Thee ae only vey few fowad-looking stochastic appoaches fo analysing packaged poducts, e.g. in Italy whee the egulato demands calculating cetain shotfall pobabilities and pecentiles of etun distibutions 7. Howeve, these pobabilities have to be computed using isk-neutal pobabilities which to ou opinion significantly educes the explanatoy value. Sections 3.1 and 3.2 show how the consideed poducts ae chaacteized using existing methodologies. Thoughout the following sections we apply chages as displayed in Table 2. Chage alue Pemium popotional chage β = 5% Account popotional chage γ = 0.5% p. a. Management fee within the funds c = 1.3% p. a. Guaantee fee fo option based poduct As piced in Section 5 Cash-put potection fee 0.2% p.a. pe unit of exposue Table 2: Geneal chages The multiplie within the CPPI constucts is set to 4 coesponding to an assumed wost case of 25% within one ebalancing peiod. 3.1 Sample illustations In this section, we investigate deteministic sample calculations. Unde this appoach, poducts ae typically compaed based on thei pojected matuity benefit deived by 6 Benad et al. (2009) shows impessive examples of quite optimistic scenaios used by povides of equity indexed annuities in the US. 7 Minenna et al. (2009) povides a detailed desciption of the undelying methodology.

9 FINANCIAL PLANNING AND RISK-RETURN PROFILES 7 assuming that the poduct s undelying investment steadily pefoms accoding to some deteministic illustation ate. This appoach does typically not distinguish funds accoding to thei chaacteistics such as the invested asset classes o any special investment mechanisms. In consequence, e.g. equity funds ae pojected at the same ate of etun as balanced funds. Futhe, path-dependent effects such as an ongoing ebalancing of iskless and isky assets (that happens e.g. in CPPI stategies) ae not evealed by this appoach since the consideed sample paths exhibit no volatility. To make things even wose, typical illustations in Gemany o Austia assume that the consideed fund (and not the fund s undelying) pefoms at the consideed ate. Theefoe all chages that occu inside funds ae neglected. Fo instance, diffeent funds investing in the same undelying but having diffeent chages (such as management fees o cash-potection puts) will esult in the same matuity benefit. Figue 1 shows the development of an equity fund investment (uppe) and the account value of an icppi type poduct espectively (lowe) egading a poduct with a single pemium of P = 100, 000 and 12 yea tem. We assumed constant equity fund pefomances of 0%, 3%, 6% and 9% and a flat inteest ate cuve of 4.5% p.a. 8. Equity 250, , , ,000 50, % 3% 6% 9% 8 In Gemany ates of 0%, 3%, 6% and 9% ae commonly used.

10 FINANCIAL PLANNING AND RISK-RETURN PROFILES 8 icppi 250, , , ,000 50, % 3% 6% 9% Figue 1: Sample Illustation of equity fund investment and icppi Due to the absence of volatility in the consideed scenaios, the icppi poduct looks almost identical to the pue equity fund investment with a slightly lowe pojected benefit consideing the stictly positive illustation ates esulting fom the additionally chaged cash-potection fees 9 and a slightly highe pojected benefit consideing 0% equity fund pefomance due to the guaantee mechanism. Note that e.g. an investment into a high watemak CPPI fund would show exactly the same pojected benefit as the pue equity fund although ou analyses in Section 6 will show that the high watemak guaantee fund constitutes a vey diffeent investment with significantly lowe expected etun and of couse also with lowe isk. Pathdependent effects ae completely neglected and theefoe the impession is geneated that the icppi s (o high watemak CPPI fund s) guaantee has vey little (completely no) effect on the etun potential. Futhe, obviously by using only non-negative illustation ates the downside-isk appeas smalle than it is, in paticula since clients with a athe low financial education might peceive the lowest illustation as some kind of wost case. Section 6 eveals the temendous diffeences between a deteministic sample illustation and fowadlooking stochastic analyses of consideed poducts and shows that the pobability of getting the benefit pojected in a sample illustation may be vey small fo some concepts. Even these simple examples ae sufficient to conclude that deteministic sample calculations ae not suitable to asses and compae the isk-etun pofile of diffeent poduct types and that moe meaningful infomation is equied fo pactical use in financial planning. 9 Note that we assume that these fees ae chaged at the level of the packaged poduct. If they wee chaged within the equity fund, they would be neglected in this compaison.

11 FINANCIAL PLANNING AND RISK-RETURN PROFILES Backtesting We have seen that a majo disadvantage of sample illustations is thei lack of volatility in the undelying assumptions. In consequence, anothe wide-spead methodology uses ealized histoical scenaios of equity and inteest ate makets that is one ealized sample path and then deives the poduct s payoff assuming the poduct had been offeed in the past. The following examples use histoical data fom 01 Januay 1973 to 31 Decembe 2009 of the MSCI Wold equity index and of Geman govenment bonds 10. Figue 2 shows the backtested account value of the same poducts that have aleady been analyzed in Section 3.1 assuming they have been issued at 01 Januay Equity 250, , , ,000 50,000 0 exposue 10 We use monthly data of govenment bonds with time-to-matuities 0.5, 1, 2 15 yeas obtained fom Geman Fedeal Bank and calculate bond pices needed in the consideed poducts fom these inteest ates.

12 FINANCIAL PLANNING AND RISK-RETURN PROFILES 10 icppi 250, , , ,000 50,000 0 Bond exposue exposue Figue 2: Backtesting of equity fund investment and icppi incepted on 01 Januay 1973 In contast to a sample illustation (Figue 1), a histoical backtesting uncoves the diffeent mechanics between the pue equity linked and the path-dependent icppi poduct. Wheeas a deteministic sample illustation completely neglects volatility and hence the downside isk of the isky asset, the backtesting shows the essential featue of CPPI type poducts, namely an ongoing eallocation between isky and iskless asset to assue the given guaantee. When equity values declined in the consideed time-fame, a significant pat of the client s account value had to be moved to the iskless asset to secue the guaantee. In consequence, the client could not paticipate in the following ecovey of equity pices which esulted in a lowe matuity benefit. We also see that, although the icppi poduct was at some stage almost completely invested in the isk-fee asset, at matuity, the poduct was completely invested in the isky asset, due to leveaging the isky exposue with a multiplie of 4. Summaizing, compaed to a sample illustation, the key chaacteistics and behaviou of diffeent poducts can be uncoveed and shown tanspaently using histoical data. Howeve, just one (no matte how ealistic ) scenaio cannot explain the (stochastic) up- and downside potential of such a poduct popely. Futhemoe, esults ae highly dependent on the chosen histoical time inteval. To illustate this effect, Figue 3 gives simila computations assuming the poducts commenced at 01 Januay 1998.

13 FINANCIAL PLANNING AND RISK-RETURN PROFILES 11 Equity 250, , , ,000 50,000 0 exposue icppi 250, , , ,000 50,000 0 Bond exposue exposue Figue 3: Backtesting of equity fund investment and icppi incepted on 01 Januay 1998 Wheeas Figue 2 showed the equity fund investment esulting in a highe matuity benefit than the icppi poduct, Figue 3 shows a backtesting whee the icppi-poduct yields a highe etun than an equity fund investment. So while backtestings ae somewhat supeio to sample illustations intoduced in Section 3.1 in that they help explaining the poduct s chaacteistics, a majo disadvantage is that the esults in paticula the elation between poducts highly depends on the chosen time inteval. In paticula, when using backtestings, thee is a stong incentive to design poducts that would have pefomed paticulaly well in the immediate past thus esulting in good backtestings.

14 FINANCIAL PLANNING AND RISK-RETURN PROFILES 12 In addition, we can conclude that financial advisos elying on backtesting as a decision tool might act cyclically. This is illustated by Figue 4. Pocyclic selling 500,000 2, , ,000 2, , , , , , ,000 50, Backtest 12y Equity Initial Investment MSCI Wold Figue 4: Backtesting fooled by andomness 11 The dotted black line gives the MSCI Wold s development fom 01 Januay 1973 to 01 Januay The oange line accodingly shows the final value of the consideed 12-yea equity fund investment when taken out at some time t. Fo example if the contact stated at the beginning of 1973, the matuity benefit would have amounted to oughly 182,000. The light blue line depicts the esult of a backtesting pefomed at the same time, e.g. at the beginning of 1985 the coesponding matuity benefit obtained by backtesting equals 182,000. 1,500 1, In 1998, both, immediate backtesting and so-called epeated backtesting (cf. Section 3.3) showed vey attactive esults. Howeve, an investo who invested in 1998 actually lost money, an event that was labelled as only theoetically possible based on backtesting. 3.3 Repeated backtesting Figue 2 and Figue 3 showed that fo assessing the poduct s isk-etun pofile just investigating one o seveal sample paths is not sufficient. Theefoe, a common appoach is to pefom histoical backtestings using many diffeent histoical scenaios. The esulting fequency distibution is then often used as a poxy fo the pobability distibution of the poduct s payoff and fo assessing a poduct s likelihood to achieve a cetain etun. Since 11 Taleb (2005) compaes designing poducts that show good backtestings to thowing monkeys on typewites, without specifying what book you want the monkey to wite and hence hitting on hypothetical gold somewhee.

15 FINANCIAL PLANNING AND RISK-RETURN PROFILES 13 old-age-povision contacts geneally have a athe long tem to matuity, a lage numbe of histoical scenaios can usually only be geneated by using ovelapping time intevals stemming fom the same histoical time seies at diffeent stating points. We have calculated the esulting distibutions of vaious poducts 12 matuity benefits and thei coesponding intenal ates of etun using all possible 12-yea time-intevals stating on the fist of each month within the undelying time seies fom 01 Januay 1973 to 31 Decembe Hence, the fist obseved matuity benefit is deived fom a contact stating at 01 Januay 1973, the second obsevation is deduced fom a contact stating at 01 Febuay 1973, and so on. Cetain pecentiles and the expected value of the distibution of the matuity benefits (uppe) and the coesponding intenal ates of etun (lowe) ae displayed in Figue 5. Along with the 5-, 25-, 75- and 95-pecentile the obseved minimum (tuquoise do, the median (oange diamond) and the mean (ed ba) ae displayed. Futhe, the dak blue (light blue) aea contains 90% (50%) of the obsevations. The coesponding numbes fo the poducts intenal ates of etun ae given in Table 3 whee the expected etun is detemined as intenal ate of etun of the expected matuity benefit. Matuity benefit 500, , , , , , , , ,000 50,000 0 Equity icppi Zeo + Undelying Balanced Life Cycle CPPI high watemak 50% 90% Pemium Paid Median Mean Minimum Sample Illustation 12 We omit the option based poduct in these calculations.

16 FINANCIAL PLANNING AND RISK-RETURN PROFILES 14 Intenal ate of etun 16% 14% 12% 10% 8% 6% 4% 2% 0% -2% -4% Equity icppi Zeo + Undelying Balanced Life Cycle CPPI high watemak 50% 90% Median Expected etun Minimum Figue 5: Distibutions esulting fom epeated backtesting 12 yea single pemium Equity icppi Zeo + Undelying Balanced Life Cycle CPPI high watemak Minimum 5.15% 2.60% 5.92% 6.48% 1.39% 0.88% 5% 6.73% 6.47% 6.71% 6.76% 4.87% 3.13% 25% 8.53% 8.05% 7.41% 7.52% 6.48% 4.87% Median 10.38% 10.19% 8.93% 8.56% 8.28% 8.83% 75% 12.18% 11.98% 10.32% 9.42% 9.80% 10.65% 95% 13.76% 13.55% 11.37% 10.31% 11.45% 12.46% Maximum 14.66% 14.43% 12.51% 11.01% 12.16% 13.32% Expected etun 10.49% 10.21% 9.04% 8.55% 8.39% 8.52% Table 3: Distibutions esulting fom epeated backtesting 12 yea single pemium The equity fund investment shows the highest upside potential. Futhe, egading the equity fund s downside isk, the minimum etun is found as 5.15% p.a. within the consideed timepeiod. This may be peceived by less educated clients as some kind of wost case esulting in a wong assessment of the poduct s actual isk. In addition, consideing the displayed pecentiles, the equity fund appeas to dominate the lifecycle stategy, a esults that is clealy contadicted by the poducts isk-etun pofiles (cf. Section 6). Futhe, all poducts without embedded guaantees (equity fund, balanced fund and lifecycle fund) display a vey low downside isk. Negative etuns did not occu in the consideed timepeiod. Hence, the deived distibutions suggest that thee is no need fo guaantees, since shotfall is only theoetically possible. Even vey isk avese clients might conclude fom such esults that poducts with guaantees ae not desiable since they educe the upside potential in tun fo potection against what seems to be a vey unlikely event.

17 FINANCIAL PLANNING AND RISK-RETURN PROFILES 15 Consideing the poducts with embedded guaantees, the icppi stuctue displays almost the same upside potential as the pue equity investment and additionally povides capital potection (that is howeve not tiggeed). The zeo plus undelying type poduct induces less skewness as compaed to icppi still offeing faily high upside potential and athe stable etuns. The high-watemak-cppi stategy displays the smallest minimum etun of all consideed poducts with guaantees and is dominated by the individual CPPI. It is intuitively clea, that majo weakness of epeated backtestings stems fom the fact that the time intevals used fo the diffeent backtestings show a significant ovelap. Fo instance the fist and the second backtesting fom the esults above (stating at 01 Januay 1973 and 01 Febuay 1973 espectively) ae two 12-yea intevals that have 143 out of 144 months in common. To quantify this effect, Figue 6 shows the autocoelation function of the undelying time-seies of pojected matuity benefits egading the equity fund investment. Autocoelation function (ACF) % 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% Time lag Figue 6: Autocoelation (ACF) of equity fund s time seies As expected, massive autocoelation is detected in the time-seies. Theefoe, afte having obseved a good pefomance of a poduct in one time inteval, it is vey likely to obseve a good pefomance in the next inteval, as well. Hence, the diffeent scenaios ae not independent and theefoe might povide questionable incentives to financial plannes o poduct developes elying on such methodologies. Summaizing, the histoical backtesting appoach is in many ways supeio to a deteministic sample illustation e.g. because it can help evealing chaacteistics (in paticula of pathdependent poducts) moe tanspaently. Howeve, by solely elying on past scenaios inappopiate incentives may be set. Distibutions geneated fom epeated backtesting show too little vaiability because the undelying scenaios have a vey high auto-coelation (compaed to Figue 7).

18 FINANCIAL PLANNING AND RISK-RETURN PROFILES 16 Theefoe, a fowad-looking stochastic methodology pojecting the poduct s benefits and deiving thei isk-etun pofile unde independent scenaios using appopiate capital maket models and paametes appeas desiable. Wheeas the appoaches in Sections 3.1, 3.2 and 3.3 eithe elied on deteministic o (massively coelated) histoical scenaios, appopiate isk-etun pofiles have to be deived using stochastic modelling of futue capital maket scenaios. In Section 4 we fist pesent the financial model used in ou analyses wheeas Section 6 quantitatively investigates esulting isk-etun pofiles. We futhe show how the pobability distibution can be educed to cetain key figues (suitable fo pactical use) descibing up- and especially downside potential. In pactical applications, in paticula fo analyzing individual poducts (as opposed to poduct types which is the pupose of this pape) all poduct specific infomation e.g. about chages has to be consideed, especially undelying guaantee fees in option based poducts. In this pape howeve, Section 5 detemines fai pices of the guaantee ide that ae consistent with the model used to compae the poducts. 4 Financial model We stat with an intoduction of the eal-wold asset model used fo ou analysis and then pefom a change of measue to a isk-neutal picing measue necessay fo valuation of the option based poduct s guaantee fee g. The consideed financial poducts (cf. Section 2) invest in stocks and zeo-bonds. Hence, we need a model fo equity and inteest ates. Due to the long-tem natue of the consideed poducts modelling stochastic volatility also seems appopiate. Thus, we use a slightly modified vesion of the Heston model (cf. Heston (1993)) fo stock makets and the Cox- Ingesoll-Ross model (cf. Cox et al. (1985)) fo inteest ate makets denoted as Heston-CIR hybid model by Gzelak and Oostelee (2010) and fist studied by Bakshi et al. (2000) in a moe geneal setup. Theefoe, let ( Ω,F,F,P) be a filteed pobability space equipped with the natual filtation F = ( F t ) t = ( σ (( W1( s), W2 ( s), W3 ( s) ), s t )) t geneated by P Bownian Motions W 1(, W ( ) and W ( ). Futhe let (t ) denote the shot-ate and S (t ) denote the equity s spot 2 t 3 t pice at time t, espectively. The (eal-wold) asset model is then summaized with the dynamics whee d( = κ ds( = S( d( = κ ( θ ( ) dt + σ ( dw1( ( ( + λs ) dt + ( dw2( ) ( θ ( ) dt + σ ( dw ( 3 P λ S denotes the equity isk pemium. We futhe assume dw dw ( = dw ( dw ( 0 and dw dw ( = ρdt 1 ( = 2 ( 3 with [ 1,1 ] ρ denoting the coefficient of coelation between the equity value and its instantaneous vaiance ( t ).

19 FINANCIAL PLANNING AND RISK-RETURN PROFILES 17 Fo picing deivatives such as zeo-bonds o equity deivatives, we futhe need to specify some picing measue Q. The undelying asset model lacks completeness due to the nontadable shot ate and the non-tadable stochastic vaiance. Hence, in contast to a conventional Black-Scholes model, the isk-neutal measue is not uniquely defined. Thee exists a whole set of pobability measues that ensue discounted (taded) assets being matingales and theefoe povide abitage fee deivative pices instead. Hence, let λ and λ denote the maket pice of inteest ate and volatility isk, espectively. Risk-neutal dynamics can then be intoduced as 13 d( = ~ κ ds( = S( d( = ~ κ ~ ~ ( θ ( ) dt + σ ( dw1( ~ ( ( ) dt + ( dw2( ) ~ ~ ( θ ( ) dt + σ ( dw ( 3 with ~ κ = κ + λ σ, ~ κ = κ + λ σ, ~ ~ Motions W (, W ( ) and W ~ ( ). 1 2 t 3 t ~ θ κ θ ~ =, θ κ + λ σ κθ = κ + λ σ and Q Bownian Within this setting, zeo-bond pices with time to matuity d at time t ae given by 14 ( B( d) ( )) P( t, d) = A( d)exp t with B( d) = 2 ( exp( h d ) 1) ( ~ κ + h)( exp( h d ) 1) + 2 h 2 h exp( ( ~ κ + h) d / 2) ( ~ κ + h) ( exp( h d ) 1) + 2 ~ 2 ~ κθ 2 σ A( d) =, h 2 2 whee h = ~ κ + 2 σ. In ou following quantitative analyses we assume an equity isk pemium of λ = 3% accoding to quantitative eseach by the Euopean Cental Bank (cf. Capiello et al. (2008)). Estimating the inteest ate paametes especially in a P Q setting is athe cucial and heavily depending on data and methodology chosen. The elated liteatue often poposes a two-step calibation 15 : Fist, eal-wold paametes ae estimated using ovenight ates as poxy fo the shot ate. Second, afte having fixed the eal-wold paametes one then calibates the maket pice of isk to fit obseved yield cuves. Howeve, egading the fist step, ovenight ates often lack obustness and possible outlies due to liquidity issues may distot the estimation esults. Theefoe, estimates based on this appoximation ae likely to be numeically instable. Futhemoe, Benashi et S 13 Fo technical details on espective measue change, efe to Cox et al. (1985), Wong and Heide (1996) and Paulsen et al. (2009). 14 Cf. e.g. Bingham and Kiesel (2004). 15 Cf. e.g. Benashi et al. (2007).

20 FINANCIAL PLANNING AND RISK-RETURN PROFILES 18 al. (2007) show the (faily high) vaiability and numeical instability of estimates deived by above appoaches ove time. Theefoe, we calibated the inteest ate paametes to match level and vaiability of obseved Geman 1-yea and 10-yea govenment bonds 16 and to obtain easonable (eal-wold) shot ates 17 by setting κ θ σ λ (0) 20% 4.5% 7.5% 0% 4.5% Table 4: Inteest ate paametes (base case) Regading stochastic volatility (moe pecisely: vaiance), we use paametes deived by Eake (2004) and stated in annualized fom by Paulsen et al. (2009) as κ θ σ ρ (0) 475% (22%) 2 55% -56.9% ( 22 %) 2 Table 5: olatility paametes (base case) Finally, we assume isk-neutality with espect to volatility isk and theefoe set the maket pice of volatility to λ = 0%. 5 Option picing Abitage-fee deivative pices ae usually eithe calculated by means of Fouie invesion techniques 18 o based on Monte-Calo simulation 19. Regading Fouie invesion, Gzelak and Oostelee (2010) povide appoximations to the necessay chaacteistic function of the undelying hybid model which is then used to pice vanilla options. Howeve, the option-based poduct s deivative (cf. Section 2) depends on the pemium payment mode, the included chages and the fund investment chosen and is theefoe of athe exotic natue. Futhe as descibed in Section 2, the client does not puchase an option on top of his contibution, instead some pat of his account value is used to finance the deivative thoughout the contact s tem. In consequence, a lowe (highe) guaantee fee g equies potentially less (moe) payments aising fom the deivative. Hence, one needs to calibate the guaantee fee such that the fai values of guaantee fees and the deivative s benefits comply. Theefoe, due to its flexibility we choose a Monte-Calo picing 16 Accoding time seies was obtained fom Geman Fedeal Bank ( 17 Note, the isky asset s expected daily etun is detemined by the eal-wold shot ate in ou modelling appoach. 18 Cf. e.g. Ca and Madan (1999). 19 Cf. e.g. Fishman (1996).

21 FINANCIAL PLANNING AND RISK-RETURN PROFILES 19 appoach on the basis of Baue et al. (2008) using 50,000 tajectoies to deive the fai guaantee fee 20. In the following we will analyze fou diffeent model points with egula monthly pemium payment of P = 100 and single pemium payment of P = 100, 000 espectively and distinguish a 12- and 30-yea tem. Table 6 gives the option based poduct s fai guaantee fee g fo these fou model points assuming capital maket paametes explained above. Tem single pemium payment Pemium payment mode egula payment % p.a. 3.15% p.a % p.a. 0.81% p.a. Table 6: Fai guaantee fee (quoted annually) base case Fist, the fai value of the guaantee ide heavily depends on the contact s tem, i.e. the highe the tem the lowe the equied annual guaantee fee. Second, the guaantee is less costly fo single-pemium contacts than fo poducts with egula pemium payment. Consideing egula contibutions the deivative s payoff can oughly be chaacteized as fowad stating put on futue contibutions. Theefoe, the capital tied up in the contact is geneally lowe than compaed to the single pemium payment and hence especially guaanteeing the late contibutions is quite costly. 6 Risk-etun pofiles We now intoduce the concept of isk-etun pofiles fo the consideed poducts. Simila with the appoach in Section 5, we pefom 50,000 tajectoies (but unde the pobability measue P ) to deive estimates of the matuity benefits distibutions. 6.1 Shot tem-contacts We stat by analyzing the isk-etun pofile of a 12-yea single pemium ( P = 100, 000 ) contact. We deive the distibution of matuity benefits and compute thei intenal ates of etun (IRR) accodingly. Figue 7 depicts a gaphical illustation of the esulting empiical distibutions and futhe shows esults of coesponding sample illustation assuming 6% and 9% p.a. fund pefomance and 4.5% inteest ates flat. 20 In pactice povides do not need to pice the deivative fai, but may account fo additional isk o pofit magins.

22 FINANCIAL PLANNING AND RISK-RETURN PROFILES 20 Matuity benefit 600, , , , , ,000 0 Equity icppi Option Based Poduct Zeo + Undelying Balanced Life Cycle CPPI high watemak 50% 90% Pemium Paid Median Mean Sample Illustation (6%, 9%) Intenal ate of etun 20% 15% 10% 5% 0% -5% -10% -15% Equity icppi Option Based Poduct Zeo + Undelying Balanced Life Cycle CPPI high watemak 50% 90% Median Expected etun Sample Illustation (6%, 9%) Figue 7: Risk-etun pofile 12 yea single pemium payment Clealy, using the methodology intoduced in Section 3.1 all consideed mutual fund poducts esult in exactly the same sample illustation esult since the asset allocation within the fund is completely ignoed by assuming any constant illustation ate. Howeve, the stochastic analysis eveals significant diffeences in the isk-etun pofile of espective funds. Fo an investment in a pue equity fund, fo example, moe than 25% of the scenaios esult in negative etuns. Even the moe consevative balanced o lifecycle funds and also the guaanteed fund (CPPI high watemak) may esult in nominal losses. Fo all fund investments, in moe than 50% of obseved scenaios the illustated etun is not

23 FINANCIAL PLANNING AND RISK-RETURN PROFILES 21 achieved, i.e. the distibution s median is below the sample illustation of 6%. Fo the CPPI high watemak fund, even the 75 th pecentile is below the sample illustation, i.e. the pobability to meet the sample illustation esult is lowe than 25%. At the same time, the CPPI high watemak poduct de facto shows the least expected etun of all consideed poducts, a quite alaming esult given that the sample illustation shows one of the highest value implying to financial advisos, that this is a poduct with a athe high upside potential. The lifecycle fund and the balanced fund show a vey simila isk-etun pofile. Due to a slightly highe aveage equity allocation within the lifecycle fund, it shows a slightly highe etun potential but also moe isk (in tems of downside isk) than the simple balanced fund. Consideing poducts with embedded guaantees, the icppi poduct shows simila upside potential as the pue equity investment. Howeve, the tade-off between lage upside potential and the money back guaantee is eflected in a quite low median of 0.35% p.a., i.e. 50% of the obseved scenaios povide a matuity benefit vey close to the single pemium paid. In contast, although the option based poduct s upside potential is slightly less ponounced, its median equals some modeate etun of 1.24% p.a. and is hence significantly highe than compaed to the icppi poduct. The semi-static potfolio insuance povided by the zeo + undelying poduct shows the least vaiability in etuns obtained. An analysis of the isk-etun pofiles eveals the misleading infomation povided by simple appoaches such as deteministic sample calculations. In paticula, the analyses show (as expected) that thee is no best poduct dominating all othes (in tems of fist ode dominance). Instead, thee ae poducts that fit a given client s equiements bette than othes. While deteministic sample illustations as well as backtestings simply ty to ank diffeent concepts and ceate the impession that the concept with the highest illustated benefit is supeio, isk-etun pofiles can help advisos and clients undestand the upside potential and the isk embedded in a poduct and match this with the client s pefeences. Howeve, the complete distibution of matuity benefits might be too complex fo advisos and/o clients to be undestood popely. Theefoe, key statistics summaizing the up- and downside potential of consideed poducts can and pobably should be employed instead. The expected etun o cetain pecentiles such as the 75 th o 95 th pecentile of the intenal ate of etun may be used to asses the upside potential of the undelying poducts. With espect to the expected etun, we howeve note that although, thee is a one-to-one elationship between matuity benefits and coesponding intenal ates of etun, caution is equied when deducing futhe statistics, such as mean o standad deviation, fom the distibution of intenal ates of etun. In ou numeical analysis above, the mean of the intenal ate of etun s distibution is calculated as 3.12% p.a. and 3.37% p.a. egading the equity fund investment and icppi poduct, espectively. Hence, although the equity fund shows a highe aveage matuity benefit than the icppi managed poduct, its coesponding IRR s mean is lowe. The eason is that the intenal ate of etun is deived as aveage annual etun on the contibutions made such that some matuity benefit is achieved, i.e. detemined by a function ϕ on the matuity benefit. Since fo an abitay andom vaiable X the equation E[ ( X )] ϕ( E[ X ]) ϕ = does not hold in geneal, the intenal ate of etun of the

24 FINANCIAL PLANNING AND RISK-RETURN PROFILES 22 benefit s expectation does not necessaily coincide with the expectation of the IRRs and typically diffes due to Jensen s inequality. Theefoe, the IRR coesponding to the expected matuity benefit denominated as expected etun in the following is a moe meaningful numbe than the expected value of the pobability distibution of the andom vaiable IRR. Table 7 summaizes a few (downside) isk measues: Besides seveal shotfall pobabilities, we give the expected shotfall as pecentage of the pemium paid (given a etun lowe than 0%) and a CTE measue. Analogously to the calculation of the expected etun easoned with above thoughts, the conditional tail expectation at level 95% (CTE 95) gives the intenal ate of etun coesponding to the aveage matuity benefit of the lowest 5% of the matuity benefits. Equity icppi Option Based Poduct Zeo + Undelying Balanced Life Cycle CPPI high watemak P(IRR<0%) 32.44% 0.00% 0.00% 0.00% 17.16% 22.78% 17.31% P(IRR<0.01%) 32.49% 20.98% 42.71% 0.00% 17.25% 22.85% 17.47% P(IRR<2%) 43.37% 62.86% 54.40% 23.11% 35.19% 39.36% 48.66% Expected shotfall 37.65% 0.00% 0.00% 0.00% 19.82% 24.58% 5.96% CTE % 0.00% 0.00% 0.68% -4.02% -5.77% -0.86% Table 7: Risk measues 12 yea single pemium payment Figue 7 aleady depicted the athe significant vaiability and the potential downside isk of the lifecycle managed fund compaed to the balanced fund. The lifecycle managed fund indeed shows a highe isk at all consideed isk-measues. Out of the guaanteed poducts, icppi and option based poducts have the lagest upside potential (measued by the 95 th pecentile). Howeve, the pobability fo low etuns (i.e. a benefit at o close to the guaanteed benefi is petty lage: The pobability of getting exactly the guaanteed value (appoximated by the shotfall pobability at level 0.01%) amounts to 21% fo the icppi) and 43% fo the option based poduct. Also, the poducts have a athe high pobability of etuns below 2% (63 and 54%, espectively). It is also woth noting that an investment into the CPPI high watemak fund leads to a negative CTE 95 and a elatively high pobability fo negative etuns of about 17%. Finally, the semi-static poduct shows the lowest isk among the consideed poducts unde all consideed isk measues. Next, we investigate the isk-etun pofiles consideing poducts with monthly pemium payment of P = 100 summaized by the empiical distibution of thei intenal ates of etun (cf. Figue 8).

25 FINANCIAL PLANNING AND RISK-RETURN PROFILES 23 Intenal ate of etun 20% 15% 10% 5% 0% -5% -10% -15% Equity icppi Option Based Poduct Zeo + Undelying Balanced Life Cycle CPPI high watemak 50% 90% Median Expected etun Sample Illustation (6%, 9%) Figue 8: Risk-etun pofile 12 yea egula payment The elationship between the diffeent poducts looks almost the same as in the single pemium case. The only elationship that significantly changes is between the lifecycle fund and the balanced fund. While in the single pemium case the lifecycle fund showed moe upside potential and moe downside isk than the coesponding balanced fund investment, in the case of egula contibutions the lifecycle fund shows a moe consevative isk-etun pofile, i.e. modeate upside potential and less downside isk (cf. Table 8). The eason fo this effect obviously is the time-dependent change of the fund s investment stategy (i.e. deceasing equity atio) combined with the inceasing account value of a egula-pemium contact: Equity exposue is only high in the ealy yeas, whee the account value is still low and theefoe the aveage equity exposue ove the tem of the contact is lowe than in the single pemium case. Fo the balanced fund, howeve, the equity exposue is not timedependent and thus the aveage equity exposue of the egula-pemium contact is the same as with the single pemium contact. A simila effect can be obseved fo the CPPI high watemak fund whee also the equity atio is typically deceasing ove time and thus the isk-etun pofile becomes moe consevative in the case of egula pemium payments. Consideing poducts with money back guaantee, an essential featue of the icppi poduct is evealed. New contibutions made to the poduct help inceasing the amount invested in the isky asset due to leveaging the new cushion geneated by the new contibution (cf. Section 2). 21 This, of couse, inceases the etun potential of the poduct. Consequently, the 21 Thus, a egula-pemium icppi-poduct can build new equity exposue afte an event that is typically efeed to as cash-lock, i.e. an event whee the equity exposue has dopped to zeo. This is not the case fo single-pemium icppi poducts and this is also not the case fo a high watemak CPPI fund.

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