WHY SO MANY MUTUAL FUNDS? Mutual fund families, market segmentation and financial performance. Massimo Massa INSEAD

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1 WHY SO Y UTUL FUDS? utual fund families, maret sementation and finanial performane. assimo assa SED JEL Classifiation:, 3 eywords: utual funds, finanial intermediation, maret struture, disrete hoie and performane. Finane Department, SED, Boulevard de Constane 77305, Fontainebleau Cedex, Frane. Tel: , Fax : than S.Berry, S.Das, J.Dermine, W.oetzmann,.rinblatt, J.eanaoplos,.Handel,.Shiller, C.Sims,.Stouhton, P.Tufano for helpful omments and advie. ll the remainin errors are mine.

2 WHY SO Y UTUL FUDS? utual funds, maret sementation and finanial performane. assimo assa SED bstrat: Why are there so many mutual funds around? What leads the industry to sement itself into an ever-inreasin number of ateories? What an be said about suh a maret onfiuration in terms of welfare? To address these uestions we model the proess that endoenously leads to maret sementation and to fund proliferation in the mutual fund industry. We arue that these phenomena an be seen as maretin strateies used by the manain ompanies to exploit investors heteroeneity. We explain ateory and fund proliferation providin an industry-speifi miro foundation on the basis of basis of the spillover that the perfomane of a fund provides to all the other funds belonin to the same family. We arue that maret fores may indue a sub-optimal number of mutual funds and ateories and identify the fators that determine suh ineffiieny. utual fund performane is endoenously derived as a funtion of investors and manain ompanies tastes and tehnoloy. This lets us shed new liht on the determinants of mutual fund performane and reonsider the traditional methods of testin fund effiieny. JEL Classifiation:, 3 eywords: utual funds, finanial intermediation, maret struture, disrete hoie and performane. Finane Department, SED, Boulevard de Constane 77305, Fontainebleau Cedex, Frane. Tel: , Fax :

3 ntrodution n the present paper we provide a framewor to explain maret sementation and produt proliferation in the mutual fund industry. n partiular, we as why there are so many mutual funds and what leads the industry to sement itself into an ever-inreasin number of ateories. We arue that maret fores may indue a sub-optimal number of mutual funds and ateories and identify the fators that determine suh ineffiieny. We show that, redefinin manaement ompany s investment deisions in terms of both numbers of funds to establish within a ateory and number of ateories to enter, we an reformulate performane as a funtion of the manain ompany maretin poliy, as well as of the maret struture the fund operates. strutural definition of performane is therefore provided, to be ompared with the redued form traditionally estimated in the empirial finane literature. The most larin stylized fat about the mutual fund industry is the existene of a very hih number of funds, differentiated into maret ateories and run by relatively few manain ompanies. owadays, the number of mutual funds in the US has already overtaen the number of stos traded on both the YSE and the EX added toether, reahin over 6,000 units. n partiular, in the period the number of mutual funds has rown from,37 to 6,778, while the number manain ompanies has only slihtly inreased, from 34 to 44. Durin the same time also the deree of sementation of the industry has rown, reahin around 4 different ateories in 997. This an hardly be explained in terms of the traditional finane literature, not only beause there already exists a number of seurities suffiient to pursue optimal investment strateies, but also beause maret sementation maes it harder to improve absolute performane. ndeed, sementation redues the sope and rane of ativity of the manaer and fores him to invest only in the assets speifi to the fund s ateory, possibly hamperin his maret timin sills. 3 seond important feature is the role inreasinly played by maretin and sementation strateies in an industry traditionally based on a homoeneous produt. The proliferation of the n the US maret the phenomenon is even more striin as the number of funds is now bier than the number of stos. For example, a Paifi Basin fund will find it diffiult to swith into British stos. 3 The existene of a mutual fund industry ould be justified if it helps to redue transation osts, thans to the possibility of ondutin lare-size transations at low osts. Even so, a limited number of mutual funds would be more than enouh to omplete the tas of mimiin the risy portfolios at low osts for most investors. 3

4 number of funds and ateories has been aompanied by an explosion in the amount of information athered by speialized maazines and publiations Barrons, BusinessWee, Consumer eports in response to investors need to evaluate funds. Between the two events there seems to be a stron two-way ausality: while the rise in the number of funds has inreased the publi desire for more information, more information has also made it more diffiult for mutual fund ompanies to differentiate themselves. This has led them to rely inreasinly on maretin strateies in order to raise the maret s heteroeneity and to mae inter-fund omparison harder, so as to affet the way investors pereive funds. From this perspetive, maret sementation and fund proliferation beome optimal strateies. n partiular, maret sementation inreases produt differentiation, limitin ompetition to the funds belonin to the same ateory, while fund proliferation inreases maret overae. t relies either on the reation of many funds in order to hide the poor performers merin them into the best ones, 4 or on the use of inubator fund strateies. 5 The issue of brand proliferation in the mutual fund industry has one unnotied in the finanial literature. Only reently horana and Servaes 999 empirially analyze the determinants of mutual fund starts, identifyin a series of fators that indue the manain ompany to set up new funds. mon the main determinants, they identify: the existene of eonomies of sale and sope, the fund family s prior performane, the deision by lare ompetin families to open similar funds in the previous year, the overall level of fund invested,.the novel ontribution of the present paper is to set up a model that, startin from these stylized fats, explains what determines the deision to set up new funds within existin ateories fund proliferation and to enter new ateories ateory proliferation. We arue that funds maretin strateies rely on investors heteroeneity and limited information, but that these fators by themselves are only neessary but not suffiient onditions to justify ateory proliferation. n partiular, we identify three ompetin fators affetin manain ompanies hoie between fund and ateory proliferation: sinallin externality, ris hedin externality and learnin-by-doin externality. The main impliation of this approah is that performane itself beomes an endoenous funtion of investors and manain ompanies harateristis. We, therefore, provide a strutural 4 brand proliferation stratey would lead a manain ompany to inrease its number of funds in order to steal ompetitors maret share and deter entry by rivals. This would allow it to sustain exess profits. 4

5 speifiation of performane, in ontrast to the traditional redued form usually estimated in the empirial finane literature. The paper is divided into two parts. n the first part we desribe the model, fousin on the determinants of number of funds, number of ateories and performane. n the seond part we deal with welfare issues. The basi framewor. eneral set-up anain ompanies maretin strateies hine upon investors limited information and heteroeneity. - Limited information implies that investors, not havin a full information set, are not apable of assessin the true uality of the produt they purhase mutual funds. Therefore, they use all the sinals they have available to evaluate the overall uality of the manain ompany that provides them. mon these sinals the performane of the other funds belonin to the same manain ompany plays a partiular role as investors impliitly assume that part of the superior manain sills displayed in the manaement of a partiular fund an also be onveyed to the manaement of all the other funds run by the same manaement ompany. This reates the possibility of a sinalin externality that stimulates ateory proliferation. ndeed, if investors pereive funds in different ateories as different produts, a new fund would not ompete diretly with any fund of the family in the already existin ateories. t the same time, thouh, a hih performane of a fund in a new ateory would send a positive sinal on the uality of the manain ompany that positively affets all its funds. The positive externality is, therefore, due to the fat that investors are heteroeneous and do not pereive funds in different ateories as lose substitutes for one another, while they use them as sinals of the manain ompany s overall ability. anain ompanies systematially exploit this externality by settin up flaship funds desined to provide exeptionally hih performane and distinuish themselves from ompetin ompanies. nedotal empirial evidene shows that often no manain fees are hared on the 5 This onsists of settin up many funds losed to outside investment. One some have established a ood history in terms of performane, the ompany would maret them as flaship or star funds of the family, losin down all the other the majority inubator funds that have performed poorly. 5

6 flaship funds. iven that funds performanes are evaluated net of manain fees, 6 this stratey boosts performane. The fat that manain ompanies are willin to tae losses on suh funds just to sinal their uality to the maret is a lear indiator of the type of effort the mutual fund industry is devotin to maretin ativities. - nvestors are defined as onsumers who pi the funds on the basis of the whole bundle of servies they provide, as opposed to mean-variane maximizers as it is usually the ase in the finane literature. To ater for investors desires, manain ompanies behave as multi-produt firms. They deide to start up a new fund within a ateory or to enter a new ateory on the basis of maret harateristis e.., the elastiity of the demand funtion, the deree of heteroeneity amon funds within the same ateory, the heteroeneity amon ateories as well as firm-speifi onsiderations. These inlude the ability to provide investors with an adeuate bundle of utility e.., performane, uality of servies provided, fees hared and the osts inurred to do that e.., start-up osts, and manain expenses. The existene of many produt lines funds offered in many setors ateories provides the manain ompany with a trade-off between the benefits provided by ris-hedin and the eonomies of sale enerated by learnin-by-doin. Cateory proliferation provides ris-hedin as it maes the overall portfolio of the fund family more diversified. Conversely, fund proliferation within a ateory allows the manain ompany to onentrate on ateories haraterized by better than averae returns. t also allows the ompany to better develop expertise of a partiular ateory, reapin a learnin-by-doin type of eonomies of sales. n order to be able to apture these effets and desribe the interation between fund manain ompanies and investors, we use an industrial oranization approah. utual fund ompanies are desribed as multi-produt firms, eah one runnin several funds, operatin in different maret sements and ompetin in many dimensions on the basis of the servies provided to the investors. nvestors are represented as onsumers who hoose one fund over alternative investment opportunities, assessin the whole bundle of servies the fund provides. ssumption : There are finanial maret roups or families of funds in the maret, eah run by a different manaement ompany. Eah fund is distinuishable aordin to the primary type of seurities in whih it invests i.e., euity funds, bond funds, balaned funds, money maret 6 For example in the Wiesenberer eport. 6

7 funds, and/or the investment style i.e., rowth fund, ash fund, inome fund,. These harateristis define the ateory i.e. maret sement the fund belons to. The manain ompany of eah of the roups must hoose the number of ateories it is simultaneously operatin and the number of funds it is manain in eah ateory i. sinle family an have more than one fund in a partiular ateory. Let s onsider, for example, the most a-priori homoeneous type of produt: the index fund ateory. The Fidelity roup has three S&P500 index funds: the Spartan US Euity ndex $.9 billion, the Spartan aret ndex $5.8 billion and the VP aret ndex $3.0 billion. They invest in the same type of asset S&P500, but differentiate themselves in terms of some other maretin harateristis. The Spartan US Euity ndex is a no-load fund, the Spartan aret ndex has a / % hare for redemption within 90 days, and the VP aret ndex is sold prinipally throuh insurane hannels. lso, there is a stron differene between the two Spartan indexes in terms of minimum initial investment that is eual to 0,000 dollars for the Spartan maret ndex / % load fund and 00,000 dollars for the Spartan US euity ndex noload fund.. Demand side ssumption : There is a ontinuum of heteroeneous investors, of total mass, with idiosynrati tastes that differ in terms of pereption and evaluation of the bundles of servies provided by the fund, as well as in terms of ris exposure. They pereive funds in different ateories and also funds within the same ateory as differentiated produts. That is, eah fund is a bundle of ost-benefit harateristis whih differentiate it from other funds and whih provide the basis for investors assessment and evaluation. Eah investor hooses a fund on the basis of the level of utility 7 the partiular fund offers him over the others. The net utility the ath investor derives from the th fund, operatin in the ith ateory and belonin to the th roup an be desribed as: U a i fund s harateristis onsumer stastes E pe i p i p ξ a i 7 We refer the reader to nderson, De Palma and Thisse 994 for a detailed desription of the disrete hoie models in an imperfet ompetition framewor. 7

8 That is, the utility the ath investor derives from the ith fund is a funtion of the fund s harateristis E pe p p, as well as of his own speifi idiosynrati tastes ξ a i i i. We assume that eah investor is pereivin and valuin performane differently, aordin to his own idiosynrati tastes where ξ ia aptures aents heteroeneity in terms of preferenes for ash flows pei and other additional benefits. The harateristis the investor taes into aount are defined in terms of the benefits he derives from investin in the fund i.e. performane and other benefits and the expenses he inurs to do so fees. There are two types of fees: load fees p whih are paid only one, at the entrane of the family of funds, and are assumed to be the same aross all funds of the same family, 8 and a manaement fee p i related to the size of the manaed portfolio. that varies for eah fund. The latter represents the part of prie diretly Here, E pe is the expeted performane of the th fund, belonin to the th roup and i operatin within the ith ateory, ross of the fees the manain ompany hares. iven that usually performane is reported net of manaement fees, investors deisions are based on net performane pe pe fee. We will, therefore, fous on net performane. i i i We define as additional benefits the ones not diretly expressible in terms of a traditional mean-variane framewor. They inlude the provision of hein failities, the linae to an insurane sheme, the possibility of usin the money deposited with the fund to ualify for a partiular minimum deposit reuirement with the ban of the roup, and so on. The investor weihts the uality and number of related servies provided by all funds belonin to the th roup and within the ith ateory. To be able to onentrate on the sinalin externality provided by performane, we assume that the additional benefits are the same for all the funds of the same family. s in the ase of performane, we will fous on the benefits net of load fees p. The advantae of this speifiation is that it avoids the limitations in representin the deision proess of the standard models. These models in eneral assume that the investor has perfet disriminatory power and unlimited information proessin apaity and that he is therefore 8 Sometime the fund also hares swith fees, that is, fees the investor has to pay to move his investments over two funds belonin to the same family. To avoid arbitrae opportunities, usually swith fees and load fees are strutured in a way that the investor is made to pay an investment fee uniform with all the funds of the same roup, and then he is free to move aross funds. 8

9 apable of ranin all the alternatives in a well-defined and onsistent manner. This is not always neessarily true as, when faed with a hoie amon several alternatives, people often experiene unertainty and inonsisteny. That is, people are not sure whih alternative they should selet, nor do they always tae the same hoie under seeminly idential onditions Tversy 969. Therefore, the parameter a ξ i an be interpreted as the random omponent of a stohasti deision proess in whih the alternatives are suessively eliminated until a sinle one remains Tversy 97, or, in other words, the random harateristis that differentiate otherwise deterministi utility funtions. lternatively, a ξi an be seen as the probabilisti omponent of investors stohasti utility funtions Fadden 98, that is, the idiosynrati taste differenes of investors. ll the investors differ one from another only with respet to some harateristis unobservable to the modeler, so that the modeler an at best predit individual hoie up to a probability funtion ansi 977. n both interpretations, the deision proess an be better represented as a probabilisti proess where all the alternatives are defined in terms of a finite set of harateristis. n individual miht foret to tae into aount some of the harateristis of an alternative or main an error evaluatin the importane of one or another harateristis assoiated with an alternative. This intuition fits very well our problem, for two orders of reasons. First this approah aommodates, at least partially, the objetion that the deision to invest in a mutual fund is an essentially irrational deision, not explainable in terms of the standard model of full rationality. Seond, this approah diretly tales investors heteroeneity. ntuitively we an say that investors are exposed to different idiosynrati endowment shos and use mutual funds to hede personal inome flutuations. t follows that the return on a mutual fund is differently evaluated aordin to its orrelation with investor s other soures of inome and his idiosynrati ash flow needs. iven that investors ris aversion hanes aross individuals and the desired portfolio is affeted by the orrelation with individual wealth and inome, whih are hihly variable aross individuals, a speifi desription of investors desired portfolios would be infeasible. ote that assumin linearity in utility does not neessarily imply ris neutrality. t an be shown 9 that if individual preferenes are drawn from the H lass, they form a Wilson 968 syndiate. n this ase, areatin prinipals is euivalent to havin a sinle prinipal with ris 9

10 tolerane eual to the sum of individual ris toleranes. Therefore, if individual ris toleranes are ept onstant, inreasin the number of aents inreases the areate level of ris tolerane. For a suffiiently hih number of individuals, the areate syndiate ris tolerane oes to zero. We an therefore assume that individuals are rouped in a ontinuum of syndiates, eah ris neutral and haraterized by different idiosynrati preferenes 0 for ash flows and other additional benefits. THEOE : f we assume that are {,...,,...} a ξ i is i.i.d. distributed as a double exponential and that there ompetin manain ompanies, { } {,..., },...,i,... maret ateories and,... i funds manaed within the same roup, we an express investor s demand for the th fund belonin to the th roup and operatin within the ith ateory as: P i Pi P i where: Si exp i exp i φi φ ih φih P i, P, i Si ln exp exp i i S j h h expφ exp exp ih φ jh h h j a i and φ i E pe p ξ, is the bundle of harateristis that suh a fund provides i i nderson, De Palma and Thisse 994. Here measures the deree of heteroeneity amon different ateories, while measures the deree of heteroeneity amon funds within the same ateory. 9 See Stouhton 993 for a detailed desription of the areation proess that leads to the syndiate in a ontrat theory set up. 0 We an also assume that investors already have a well-diversified portfolio so that E pei represents the expeted performane net of ris they reuire from a partiular fund. Both measures are bounded between zero when all the funds are pereived to be same and one when all the funds are pereived as ompletely different produts. 0

11 The beauty of this speifiation is that it allows us to rationalize the way investors deide to invest: first investors hoose the ateory whih best suits their investment needs P i. They then deide whih fund to invest in, piin it amon only funds of the seleted ateory P. i Furthermore, iven that we have a ontinuum of investors, defined in the unitary interval, the demand of eah one of them an be seen as the maret share of the investor with the partiular idiosynrati harateristis, or as his probability of purhase. n partiular, P i an be intuitively seen as the probability of hoosin the th fund within the ith ateory and belonin to the th roup. P i is the unonditional probability that the investor deides to invest in the ith ateory. For example, a ris averse individual with a short-term investment horizon would prefer the more liuid and less volatile ateory of money maret funds as opposed to euity funds. P i is the probability of hoosin the speifi th fund onditional upon the hoie of the ith ateory? For example, the ris averse individual who has deided to invest in money maret funds, pis the speifi th fund amon all the money maret funds aordin to its speifi harateristis. ssumption 4: nvestors do not now the performane of eah fund, but derive expetations of it by looin at some sinals suh as the past performane of the fund they are oin to invest and the performane of the other funds manaed by the same manain ompany Ψ aordin to: 3 E t pe i t peit Ψt,, where Ψ j pe jh j h. The rationale is that the mere fat that a fund of the same family has sored well in its own ateory stronly affets investor s demand for all funds run by the same manain ompany. This an be interpreted as a sinal released by the manain ompanies to distinuish themselves from ompetitors in the maret. nvestors reeive the sinal and interpret it, believin that part of the superior manain sills displayed in the manaement of the flaship fund will also be onveyed to the manaement of other funds run by the same manain ompany. We are not assumin that investors assess the ability of a manain ompany only on the basis of the ranin of its best performin manaed fund. But, iven the imperfet information, investors use the performane of the best fund within a family in addition to the other sinals suh as a fund s

12 speifi performane, uality of servies provided, et. to evaluate the overall ability of the family s manain ompany. We assume that the expetation is linear. From now on, for larity of exposition, we omit the time subsript. The model an be seen as intertemporal only inasmuh as investors expetations are shaped on the basis of atual performane..3 Supply Side ssumption 5: The manain ompany has to deide on the number of ateories to enter, the number of funds to establish within eah ateory and the performane and additional servies to provide. We adopt a profit-maximization approah that, in line with reent literature, defines as the main oal of a manain ompany the maximization of a fund s size in terms of manaed assets rather than the maximization of its performane. 3 n fat, iven that most of the fees paid by the unit holder are proportional to funds manaed assets, profits are maximised when net fund-raisin is maximised. t follows that better performane beomes instrumental in reater fund-raisin and the subseuent inrease in manaed assets beomes a sort of remuneration in an inentive ompatible ontrat sheme by whih the roup of investors-prinipals fores the manain ompany-aent to provide suh performane. Unlie this literature, however, we onsider performane as only one of the many dimensions in whih the funds an ompete to attrat investors. 4 The profit funtion an, therefore, be defined as: 4 π j j [ p p jh pe jh Pjh ] [ mu jh Pjh ] j h j h where is the number of potential investors and and pe i are, respetively, the osts inurred by the manain ompany for the provision of servies for eah fund unit and for the provision of a ertain level of performane pe i. We assume that the osts inurred to Chevalier and Ellison. 3 n this sheme, end-of-the-year window dressin operations enated by fund manaers an be rationalizable as a possible last moment effort aimed at inreasin performanes before investors judment. 4 t follows that in some irumstanes a manain ompany may optimally hoose not to see hiher performane, but to resort to other maretin tools in order to maximize assets.

13 provide servies derease with the amount provided < 0. That is, the roup enjoys eonomies of sale and learnin by doin in providin these servies. The variables and represent the fixed osts inurred, respetively, to start up a new fund and to enter a new ateory. Without loss of enerality, both are assumed to be fund and ateory invariant, i.e., to depend only on the ost struture of the manain ompany. One an thin of them as a learnin ost due to the different type of nowlede that the new area reuires 5 ompared to the others in whih the ompany is already operatin. The ompany hooses the level of performane net of manain fees for eah funds of the family, the uality and number of related servies offered at family level, net of load fees, the number of funds offered within eah ateory and the number of ateories to enter. The objetive of the ompany is to maximize profits and, therefore, to inrease both its marup mu i and its maret share P i. So far, the model is merely deterministi, but this is unrealisti in an industry haraterized by a hih deree of randomness in the main type of servie provided, i.e. performane. s a matter of fat, the manain ompany an not determine with ertainty the ontrol variable pe i, or, if it an, the ost inurred to do so is stohasti and not fully foreastable. ssumption 6: We assume the ost inurred to provide a ertain level of performane pe i to be stohasti 6 and eual to: 5 pei pei peiε i, with ε i i, σ i. i 5 For example, the nowlede reuired to deal with fixed inome assets an be very different from that neessary to invest into stos or derivatives. lso, sometimes maret loation and the type of tradin dramatially affet the inds of nowlede reuired. 6 We thin that the main performane-enhanin ativity is not researh, whih we an onsider as a fixed ost, but the fees and the performane bonuses paid to the fund manaers. These an be onsidered as a marinal ost, a ost that is variable and lined to the desired taret of performane under the assumption that hiher remuneration an buy better fund manaers. Furthermore, it is worth notin that we are onsiderin performane ross of expenses. f performane were post-expense, iven that expenses are passed throuh to the fund investors, it would turn out to be stohasti. 3

14 That is, the total ost pe i is made up of two different omponents: the first pe i is a deterministi funtion of the level of a fund s performane pe i : the hiher the performane the fund provides, the hiher the ost it inurs where is a proportionality oeffiient. ndeed, iven that a fund s ability to beat the maret depends on the uality of its manaers and the inentive provided, the ost involved in providin a better-than-averae performane rises with the level of suh performane. The ost of providin a ertain performane dereases with the number of funds the ompany has already set up in the same ateory. ndeed, the more speialized the ompany is i.e. the hiher the number of funds per ateory, the more it an benefit from some learnin-by-doin or sale effets due to the development of a partiular nowlede in a speifi ateory. The intuition is that the hiher the number of funds the manain ompany has within the same ateory, the hiher the expertise it has aumulated in the partiular area. The seond omponent pe ε is still related to the desired level of performane, but varies i i randomly over time and depends on the stohasti harateristis of the returns in the partiular ateory ε i. This affets the total portfolio risiness in two ways: on one side, the hiher the level of unertainty, the more diffiult it is for the manain ompany to reah the desired level of performane. On the other side, the manain ompany is already manain assets in other ateories, and enterin a new ateory provides an opportunity to hede overall exposure. Unlie the ase for the number of funds, havin entered more ateories does not neessarily imply a better expertise, as the ateories are very different in terms of the harateristis of the assets where they mainly invest. There is, therefore, a trade-off between the number of funds and the number of ateories similar to the one existin between apital deepenin and apital widenin. ssumption 7: The manain ompany is ris averse, and maximizes a funtion { } profits, where profits π are defined in euation 5. i V E π of eent empirial studies C. Beer, W. Ferson, D. yers and. Shill show that the mutual funds behave as ris averse 8 aents. is aversion inreases the inentive to set up new 7 Usin a sample of more than 400 US mutual funds for , Ferson et al estimate that mutual funds behave as ris averse, benhmar investors. 4

15 funds and enter additional ateories in order to hede both absolute and relative performane ris. ndeed, a ris averse ompany may wish to enter a new ateory or set up a new fund so as to hede not only the ris of a low performane, but also the ris that all funds will have a low performane relative to that of their ompetitors relative performane ris. ndeed, if investors demand for a sinle fund is affeted by the performane of all the other funds of the same family, a ris averse ompany may wish to redue both absolute performane ris and relative performane ris. 3 aret ompetitive euilibrium: number of funds, ateories and performane. Followin the standard literature, we hypothesize a two-stae ame: first the manain ompany deides simultaneously the number of ateories to enter and the number of funds for eah ateory. Then, onditional upon them, it selets the level of performane and the uality and number of servies to provide. To solve the model, we wor bawards startin from the manain ompany s hoie about performane-servies onditional upon the number of ateories entered and the number of funds already established within eah of them. Then, we solve for the number of funds and ateories. Finally, we loo at the free entry solution to determine the euilibrium number of funds and ompare suh euilibrium with the welfare optimum. THEOE 9 : Under the assumptions stated above, the model has a subame perfet ash euilibrium to the two-stae produt-line-setor ame. Suh euilibrium is defined in terms of nested loit demand, X P and the mar-up hared by the manain ompany is: i i 7 mui i for i,...,i,...,, where: P i i P i i i i E ε i i i ov π, ε i P Ω i, 8 possible justifiation of the empirially found ris aversion is the fat that fund-manaers are remunerated in terms of the overall performane of the family of funds they manae and that manain ompany ats in the best interest of its manaers who are ris averse and annot diversify suh ris. 5

16 Ω i Ψ, i E ε i, ov π i, ε maret i, i ov π i, ε i ε maret, pe i j φ jh V π exp E{ } j h π and π. j φ V π jh exp π i h ar-up an be interpreted as the return that the manaed assets provide to the manain ompany. n standard finane the reuired return on an asset is a funtion of investor s deree of ris aversion and on the ability of the asset to provide a hede aainst ris, that is the ovariane between investor s inome or onsumption and asset returns. naloously, the mar-up is positively related to the deree of ris aversion of the manaers and to the ovariane between the overall profits of the manain ompany inome and the random rate of return of the ateory that haraterises the returns of the new fund asset return. n partiular, the more ris averse manaers are, the more they have to be remunerated to inur the osts related to an unertain level of performane. n interestin insiht omes from the deomposition of the ovariane of profits with the returns of the ateory into two omponents: the ovariane between profits and the maret returns and the ovariane between profits and the differene between a ateory s i returns and maret returns i. The first omponent is diretly proportional to the number of ateories entered throuh the proportionality oeffiient i. The intuition is that inreasin the number of ateories helps the portfolio diversifiation of the manain ompany beause it maes the returns on its total manaed assets more in line with the maret portfolio. ndeed, eah ateory represents a part of the maret, and the hiher the number ateories the fund is already operatin, the more orrelated fund s returns are within the maret and the more the portfolio manaed by the fund approahes the maret portfolio. n partiular, the effet of enterin an additional ateory depends on the already existin portfolio omposition of the manain ompany and on the way the idiosynrati sho of the partiular ateory o-varies with it. Here i aptures the effet that enterin an additional ateory has on the orrelation between the returns on the assets manaed by the manain ompany and the returns on the additional ateory. The hiher i, the more the entrane into a new ateory ontributes to alin the 9 ll of the followin propositions will be proved in the ppendix. 6

17 omposition of the manaed assets to the maret portfolio. That is, the marinal value of a speifi fund inreases when the new fund provides a form of ris diversifiation for the manain ompany. The seond omponent, that is, the ovariane between profits and the differene between the ateory s returns and maret returns, aptures the idiosynrati sho of the partiular ateory, or the part of it in exess of the maret. t represents the benefits arued to a manain ompany from bein able to onentrate on ateories haraterized by better than averae returns. Finally, Ω aptures the externality effet that the performane of a family produes on all the funds of the same family. 0 THEOE 3: n a symmetri euilibrium, the optimal number of funds offered by eah family within a ateory is: 8 Ω Ω, while the optimal number of ateories entered by a family is: Ω Ω 9. Ω n partiular, assumin that > Ω, the partial derivatives of the supply of funds is: 0 < 0, > 0, > 0, > 0, > 0, < 0, > 0, < 0 < 0. f we also assume that is suffiiently reater than, that is the ost of settin up a new fund within an existin ateory is hiher than the ost of enterin a new ateory, we have we have: < 0, > 0, > 0. 0 The mar-up still depends on the number of funds within eah ateory and on the number of ateories entered. The hoie of these variables at the seond and third step of the ame will determine its absolute size. 7

18 lso, still assumin that > Ω, the partial derivatives of the supply of funds are: > 0, < 0, < 0, > 0, <> 0, < 0, < 0, < 0, > 0, > 0, > 0 > 0. That is, the deision to set up a new fund and the one to enter a new ateory are a funtion of some ost-performane harateristis of the manain ompany as well as of the maret struture. Some interestin onlusions are worth stressin. First of all, even thouh the model is expliitly desined to onsider produt differentiation, the deision to enter a ateory is not influened by the deree of heteroeneity amon ateories and is affeted, but not exlusively motivated, by the deree of heteroeneity within eah ateory.that is, onsumer heteroeneity is not suffiient for fund and ateory proliferation. There are other elements that play a major role in induin the manain firm to differentiate and provide inentives not onsidered by the traditional ost benefit analysis. Some of these elements an be onsidered as externalities as the benefits derived from settin-up a new fund do not arue only to the fund, but positively affets all the other funds of the family. These externalities are: sinalin externalities, ris-hedin externality and learnin-bydoin externality. Some other elements are related to the maret struture, suh as potential demand and number of ompetitors. We loo at these different omponents separately, startin from the standard ost benefit analysis to seein how muh they differ from it. From a pure standard ost benefit analysis, the benefits of enterin a new ateory are weihted aainst the ones of settin up of additional funds within the existin ateories on the basis of net payoffs returns minus osts inurred to provide them speifi to eah fund. first lane at euations 8 and 9 shows that, if the averae return within the eneri ateory is suffiiently hih, or the ost of eneratin it is suffiiently low, fund proliferation is better than ateory proliferation. f, on the ontrary, the payoff is low, or the osts of eneratin it are hih, it is worth oin for additional ateories. n partiular, if the rate of return of the averae ateory is hih and the manain ompany is already positioned in ateories providin a return hiher than the averae maret rate hih ov π, ε ε it pays off to inrease the number of funds within the existin maret 8

19 ateories, without venturin into new ones. naloously, it is preferable to establish additional funds within the existin ateories if the ost of either produin performane, or providin servies is low. Therefore, the inentive to enter a new ateory is a neative funtion of both the rate of return of the averae ateory and the deree of orrelation between the manain ompany s overall profit struture and the ateory s exess returns over maret returns. Conversely, it is positively related to the ost of eneratin performane and servies within eah ateory. The effet of fixed osts is different. Hiher osts to enter a new ateory tilt the hoie in favor of settin up new funds within the existin ateories as opposed to the one of enterin new ateories. Conversely, hiher osts of settin up a new fund indue the manain ompany hoose the entrane into new ateories. Enterin a new ateory also enerates a positive externality, as the performane in one ateory positively affets all the funds run by the same family in the other ateories. The reason why we all it an externality is that the benefits spill over all funds of the same family, reardless of the ateory they belon to. otie that we are not exoenously imposin that enterin a new ateory inreases the probability of ettin a better relative performane ranin. or are we assumin that the manain ompany is able to hide its bad performers and display the ood ones inubator funds stratey. This would only mae our results stroner. The positive externality is the endoenous result of the existene of heteroeneity amon investors in terms of tastes and investment deisions. t follows from the fat that aents preferenes are not onstrained as in the typial finanial models that assume perfet substitutability amon finanial produts. diret onseuene of the existene of finite and different elastiities of substitution amon funds within the same ateory and funds in different ateories is that, for a manain ompany, havin a well performin fund in a new ateory is worth more than havin a well performin fund within an existin ateory. The deree of heteroeneity plays a ruial role: the more heteroeneous the investors are, the more a ood performane of a fund positively reflets on all the funds of the same family, without providin diret ompetition to the other funds of the family. Hiher investors heteroeneity within a ateory favours the deision to set up a new fund as opposed to the one of enterin a new ateory. The intuition is that if investors are heteroeneous within a 9

20 ateory, the manain ompany does not have to enter new ateories to differentiate its funds from its ompetitors. Enterin new ateories helps also to diversify total ris. f the returns enerated from the assets in the new ateory are not perfetly orrelated with those of the already manaed ones, enterin a new ateory an redue the overall volatility and ris of the whole funds family ris-hedin externality. This effet is stroner the more the entrane in the new ateory shifts the portfolio omposition of the manaed assets toward the maret portfolio hiher. f, on the ontrary, the idiosynrati omponent of the new ateory is orrelated to the returns of the already manaed assets hih, enterin the new ateory would only inrease ris and, therefore, the inentive to enter the new ateory drops. otie that the diret effet of a hane in ris aversion is indeterminate, as it is diretly related to both and and it only plays a role of amplifyin them. n eneral, the benefits of ris-hedin are traded off aainst the benefits of speialization. iven that the ost of providin a ertain performane dereases with the number of funds the ompany has already set up in the same ateory, enterin an additional ateory means foreoin benefits in terms of learnin-by-doin that ome with a stratey of fund proliferation within the same ateory. Finally, both fund and ateory proliferation are positively affeted by the maret struture, defined in terms of potential demand. ndeed, a bier maret allows the ompany to expand the number of funds and enter new ateories without the need to redue the mar-up. Conversely, a hiher number of ompetitors redues the inentive to both enter new ateories and set up new funds within the existin ateories. Corollary : The euilibrium level of performane an be defined as: 3 Ω pe p pe This happens if ov π, <. ε maret 0

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