Speculation and Risk Sharing with New Financial Assets

Transcription

1 Speculaton and Rsk Sharng wth New Fnancal Assets Alp Smsek May 2, 212 Abstract Whle the tradtonal vew of nancal nnovaton emphaszes the rsk sharng role of new nancal assets, belef dsagreements about these assets naturally lead to speculaton, whch represents a powerful economc force n the opposte drecton. Ths paper nvestgates the e ect of nancal nnovaton on portfolo rsks n an economy when both the rsk sharng and the speculaton forces are present. Fnancal assets provde hedgng servces but they are also subject to speculaton because traders do not necessarly agree about ther payo s. I de ne the average varance of traders net worths as a measure of portfolo rsks for ths economy, and I decompose t nto two components: the unnsurable varance, de ned as the average varance that would obtan f there were no belef dsagreements, and the speculatve varance, de ned as the resdual varance that results from speculatve trades based on belef dsagreements. Fnancal nnovaton always decreases the unnsurable varance because new assets ncrease the possbltes for rsk sharng. My man result shows that nancal nnovaton also always ncreases the speculatve varance. Ths s true even f traders completely agree about the payo s of new assets. The ntuton behnd ths result s the hedge-more/bet-more e ect: Traders use new assets to hedge ther bets on exstng assets, whch n turn enables them to place larger bets and take on greater rsks. My second set of results concern endogenous nancal nnovaton. When belef dsagreements are su cently large, a pro t seekng market maker ntroduces assets that maxmze the average varance among all possble choces, completely dsregardng the rsk sharng motve for nancal nnovaton. My thrd set of results concern a calbraton of the model n the context of the GDP ndexed assets proposed by Athanasouls and Shller (21). When traders have reasonable amounts of belef dsagreements, these assets would have the unntended consequence of ncreasng ther consumpton rsks. JEL Class caton: G1, G11, D53, D61. Keywords: nancal nnovaton, rsk sharng, belef dsagreements, speculaton, portfolo rsks, unnsurable varance, speculatve varance, hedge-more/bet-more, heterogeneous prors, nancal nstablty, natonal ncome markets. Harvard Unversty and NBER (e-mal: asmsek@fas.harvard.edu). I am grateful to Daron Acemoglu for numerous helpful comments. I also thank Malcolm Baker, Eduardo Davla, Bengt Holmstrom, Sam Kruger, Andre Shlefer, Jeremy Sten, Muhamet Yldz and the semnar partcpants at Berkeley Unversty, Central European Unversty, Harvard Unversty, MIT, Penn State Unversty, Sabanc Unversty, Unversty of Houston, Unversty of Maryland, Unversty of Wsconsn-Madson, Yale Unversty; and conference partcpants at AEA and Koc Unversty for helpful comments. All remanng errors are mne.

2 1 Introducton Accordng to the tradtonal vew of nancal nnovaton, new nancal assets facltate the dvers caton and the sharng of rsks. 1 However, ths vew does not take nto account that new assets are often assocated wth much uncertanty, especally because they do not have a long track record. Belef dsagreements come as a natural by-product of ths uncertanty and change the mplcatons of rsk takng n these markets. In partcular, market partcpants dsagreements about how to value new assets naturally lead to speculaton, whch represents a powerful economc force that tends to ncrease rsks. An example s o ered by the recent crss. Assets backed by pools of subprme mortgages (e.g., subprme CDOs) became hghly popular n the run-up to the crss. One role of these assets s to allocate the rsks to market partcpants who are best able to bear them. The safer tranches are held by nvestors that are lookng for safety (or lqudty), whle the rsker tranches are held by nancal nsttutons who are wllng to hold these rsks at some prce. Whle these assets (and ther CDSs) should have served a stablzng role n theory, they became a major trgger of the crss n practce, when a fracton of nancal nsttutons realzed losses from ther postons. Importantly, the same set of assets also generated consderable pro ts for some market partcpants, 2 whch suggests that at least some of the trades on these assets were speculatve. What becomes of the rsk sharng role of new assets when market partcpants use them to speculate on ther d erent vews? To address ths queston, ths paper analyzes the e ect of nancal nnovaton on portfolo rsks n a model that features both the rsk sharng and the speculaton motves for trade. Traders wth ncome rsks take postons n a set of nancal assets, whch enables them to share and dversfy some of ther background rsks. However, traders have belef dsagreements about asset payo s, whch nduces them to take also speculatve postons on assets. I assume traders have mean-varance preferences over net worth. In ths settng, a natural measure of portfolo rsk for a trader s the varance of her net worth (calculated accordng to her own belefs). I de ne the average varance as an average of ths rsk measure across all traders. I further decompose the average varance nto two components: the unnsurable varance, de ned as the varance that would obtan f there were no belef dsagreements, and the speculatve varance, de ned as the resdual amount of varance that results from speculatve trades based on belef dsagreements. I model nancal nnovaton as an expanson of the set of assets avalable for trade. My man result characterzes the e ect of nancal nnovaton on each component of the average varance. In lne wth the tradtonal vew, nancal nnovaton always decreases the unnsurable varance because new assets ncrease the possbltes for rsk sharng. Theorem 1 shows that nancal nnovaton also always ncreases the speculatve varance. Moreover, there 1 Cochrane (21) summarzes ths vew as follows: Better rsk sharng s much of the force behnd nancal nnovaton. Many successful new securtes can be understood as devces to more wdely share rsks. 2 Lews (21) provdes a detaled descrpton of nvestors that took a short poston on housng related assets n the run-up to the recent crss. 1

3 exst economes n whch ths ncrease n the speculatve varance s su cently strong that nancal nnovaton ncreases the average varance (by an arbtrary amount). My analyss dent es two dstnct channels by whch nancal nnovaton ncreases the speculatve varance. Frst, new assets lead to new dsagreements because they are assocated wth new uncertantes. Second, and perhaps more mportantly, new assets also amplfy speculaton on exstng dsagreements. To llustrate the second channel, Theorem 1 shows that new assets ncrease the speculatve varance even f traders completely agree about ther payo s. Ths result s somewhat surprsng because traders use new assets to hedge some of the speculatve rsks they have been undertakng from ther bets on exstng assets. In vew of ths drect hedgng e ect, one could expect new assets (on whch there s complete agreement) to reduce the speculatve varance. Ths vew does not take nto account a powerful ampl caton mechansm, the hedge-more/bet-more e ect. To llustrate ths e ect, consder the followng example. Suppose two traders have d erng vews about the Swss Franc, whch s hghly correlated wth the Euro. The optmst beleves the Franc wll apprecate whle the pessmst beleves t wll deprecate. Traders do not dsagree about the Euro, perhaps because they dsagree about the prospects of the Swss economy but not about the Euro zone. Frst suppose traders can only take postons on the Franc and not the Euro. In ths case, traders postons n the Franc wll be determned by a standard rsk-return trade-o. Traders may not take too large postons on the Franc especally because the Franc s a ected by multple sources of rsks, e.g., the shocks that a ect the Swss economy as well as the shocks to the Euro zone. To bet on ther belef d erences, traders must bear all of these rsks, whch mght make them reluctant to take large postons. Suppose nstead the Euro s also ntroduced for trade. In ths case, traders wll complement ther postons n the Franc by takng the opposte postons n the Euro. Ths s because the complementary postons enable traders to hedge the rsks that also a ect the Euro, whch they don t dsagree about, and to take purer bets on the Franc. Wth purer bets, traders bear less rsk for each unt poston on the Franc, whch n turn enables them to take larger postons. Put d erently, when traders are able to hedge more, they are nduced to bet more. Theorem 1 shows that the hedge-more/bet-more e ect s su cently strong that the ntroducton of the Euro n ths example (and more generally, a new asset) ncreases the speculatve varance. Theorem 1 takes the new assets as exogenous and analyzes ther mpact on portfolo rsks. In practce, new nancal assets are endogenously ntroduced by economc agents wth pro t ncentves. A szeable lterature emphaszes rsk sharng as a major drvng force n endogenous nancal nnovaton [see, for example, Allen and Gale (1994) or Du e and Rah (1995)]. A natural queston s to what extent the rsk sharng motve for nancal nnovaton s robust to the presence of belef dsagreements. I address ths queston by ntroducng a pro t seekng market maker that nnovates new assets for whch t subsequently serves as the ntermedary. The market maker s expected pro ts are proportonal to traders perceved surplus from tradng new assets. Thus, traders speculatve tradng motve, as well as ther rsk sharng motve, 2

4 creates nnovaton ncentves for the market maker. In partcular, the optmal asset desgn (characterzed n Theorem 2) depends on the sze and the nature of belef dsagreements, n addton to the rsk sharng possbltes. When traders have common belefs, the market maker nnovates assets that mnmze the average varance, as n Demange and Laroque (1995) and Athanasouls and Shller (2). In contrast to these tradtonal results, Theorem 3 also characterzes the polar opposte case: When traders belef dsagreements are su cently large, the market maker nnovates assets whch maxmze the average varance among all possble choces, completely dsregardng the rsk sharng motve for nancal nnovaton. A natural queston s how large belef dsagreements must be to make these results practcally relevant. To address ths queston, I consder a calbraton of the model n the context of the natonal ncome markets proposed by Shller (1993), and analyzed n detal by Athanasouls and Shller (21). Assets whose payo s are lnked to (varous combnatons of) natonal ncomes could n prncple facltate the sharng of ncome rsks among d erent countres. Athanasouls and Shller (21) characterze the optmal desgn of such assets. They also calbrate ther model for G7 countres, and argue that the nnovaton of a couple of these assets would lead to large welfare gans n vew of the reducton n ndvduals consumpton rsks. I consder the e ect of belef dsagreements on ther results about consumpton rsks. Usng exactly ther data and calbraton, I nd that reasonable amounts of belef dsagreements about the GDP growth rates of G7 countres (much smaller than mpled by the Survey of Professonal Forecasters) mply that these new assets would actually ncrease average consumpton rsks. Intutvely, per-capta ncome rsks n developed countres s small relatve to ther percapta ncomes. Moreover, ncome rsks are correlated across developed countres. Thus, even f these rsks are perfectly dvers ed, the reducton n the standard devaton of consumpton amounts to a relatvely small fracton of ncome. In contrast, wth a typcal calbraton for the relatve rsk averson parameter, relatve = 3, traders are wllng to rsk a greater fracton of ther ncomes n ther pursut for speculatve gans. Consequently, a small amount of belef dsagreements s su cent to ensure that the ncrease n the speculatve varance domnates the relatvely small decrease n unnsurable varance. The rest of the paper s organzed as follows. The next subsecton dscusses the related lterature. Secton 2 ntroduces the basc envronment. Ths secton also uses smple examples to llustrate the two channels by whch new assets ncrease traders portfolo rsks. Secton 3 characterzes the equlbrum and decomposes traders portfolo rsks nto the unnsurable and the speculatve varance components. Secton 4 presents the man result, whch characterzes the e ect of nancal nnovaton on these two components. Secton 5 analyzes endogenous nancal nnovaton. Secton 6 dscusses the postve and the normatve mplcatons of the results. Secton 7 presents the calbraton exercse and Secton 8 concludes. Appendx A contans the results and proofs omtted from the man text. 3

5 1.1 Related Lterature As the above dscusson clar es, my paper belongs to a szeable lterature on nancal nnovaton and securty desgn [see also Van Horne (1985), Mller (1986), Ross (1988), Merton (1989, 1992), Du e and Jackson (1989), Cuny (1993), Tufano (23)]. Ths lterature, wth the excepton of a few recent papers (some of whch are dscussed below), has not explored the mplcatons of heterogenous belefs for securty desgn. For example, n ther survey of the lterature, Du e and Rah (1994) note that one theme of the lterature, gong back at least to Workng (1953) and evdent n the Mlgrom and Stokey (1982) no-trade theorem, s that an exchange would rarely nd t attractve to ntroduce a securty whose sole just caton s the opportunty for speculaton. The results of ths paper show that ths observaton does not apply f traders have heterogeneous pror belefs rather than heterogeneous nformaton. The observaton also does not apply f traders have heterogeneous nformaton but securty prces do not reveal nformaton fully due to the presence of nose traders. The analogues of my results can be derved for ths alternatve settng. The mportant economc ngredent s that traders contnue to have some dsagreements after observng asset prces. In addton, the quanttatve results of ths paper suggest that a relatvely small amount of belef dsagreements of ths type s su cent to ensure that speculaton s a sgn cant factor n nancal nnovaton. My paper s most closely related to the recent work of Brock, Hommes, and Wagener (29), who also emphasze the destablzng e ects of nancal nnovaton n the presence of belef dsagreements. The two papers are complementary n the sense that they use d erent ngredents, and they focus on d erent aspects of nstablty. Brock et al. s (29) man ngredent s renforcement learnng: That s, they assume traders choose ther belefs accordng to a tness measure, such as past pro ts made by the belef. They show that, wth renforcement learnng, new assets make the steady-state correspondng to the fundamental asset prce more lkely to be dynamcally unstable. In contrast, ths paper takes traders pror belefs as gven, and establshes a statc nstablty result (ncrease n speculatve varance) regardless of how those belefs are formed. In partcular, my results do not rely on renforcement learnng. My paper s part of a recent lterature whch emphaszes that ncreased tradng opportuntes mght ncrease portfolo rsks when traders have dstorted or d erent belefs. Weyl (27) notes that cross-market arbtrage mght create addtonal rsks when nvestors have mstaken belefs. Deckman (29) shows that rare-event nsurance can ncrease portfolo rsks when traders dsagree about the frequency of these events. The contrbuton of my paper s to systematcally characterze the e ect of nancal nnovaton on portfolo rsks for a general envronment wth belef dsagreements and mean-varance preferences. I show that new assets reduce unnsurable varance, whch captures the nsghts of the tradtonal lterature on nancal nnovaton, but that they also always ncrease the speculatve varance through two dstnct channels. I also characterze the endogenous asset desgn wth belef dsagreements, and show that t s partly drven by the speculaton motve for trade. My paper also contrbutes to a lterature whch analyzes the welfare mplcatons of nancal 4

6 nnovaton. A strand of the general equlbrum theory, e.g., Hart (1975) and Elul (1994), has shown that new assets that only partally complete the market may make all agents worse o n vew of general equlbrum prce e ects. Sten (1987) shows that speculaton drven by nancal nnovaton can reduce welfare through nformatonal externaltes. I abstract away from these channels by focusng on an economy wth sngle good (hence, no relatve prce e ects) and heterogeneous pror belefs (hence, no nformaton). In partcular, the compettve equlbrum n ths economy s constraned Pareto e cent f agents welfare s calculated accordng to ther own subjectve belefs. However, several economsts, e.g., Stgltz (1989), Mongn (1997), Glboa, Samet, and Schmedler (24) and Kreps (212), have noted that the Pareto crteron mght not be approprate when agents have d erent belefs. The key nsght s that agents belefs are nconsstent wth one another, whch creates a collectve rratonalty (see Secton 6 for detals). In recent work, Brunnermeer, Smsek, Xong (212) and Glboa and Schmedler (212) propose alternatve welfare crtera that could be used n envronments wth belef dsagreements. Applyng Brunnermeer, Smsek, Xong s (212) crteron n my paper detects nancal nnovaton as ne cent when new assets ncrease portfolo rsks. That sad, I do not take a strong normatve stance n ths paper. Instead, my prmary goal s to systematcally characterze the postve e ects of new assets on portfolo rsks. 3 My paper s also related to a large lterature that analyzes the mplcatons of belef dsagreements for asset prces or volume of trade. A very ncomplete lst ncludes Mller (1977), Harrson and Kreps (1978), Varan (1985, 1989), Harrs and Ravv (1993), Kandel and Pearson (1995), Zapatero (1998), Chen, Hong and Sten (23), Schenkman and Xong (23), Geanakoplos (29), Cao (21), Smsek (211). The man d erence of my paper from ths lterature s the focus on the e ect of belef dsagreements on the rskness of traders portfolos, rather than the rskness (and the level) of prces or the volume of trade. My model s closest to Lntner (1969), who generalzes the captal asset prcng model to an envronment n whch belefs and rsk averson coe cents are heterogeneous. However, Lntner (1969) does not analyze the rsk sharng mplcatons of ths theory. 2 Basc Envronment and Man Channels Consder an economy wth two dates, f; 1g, and a sngle consumpton good, whch wll be referred to as a dollar. There are a nte number of traders denoted by 2 I = f1; 2; ::; g. Each trader s endowed wth e dollars at date, whch s constant. Trader s also endowed 3 From the normatve pont of vew, my paper s also related to a recent lterature that analyzes the role of nancal nnovaton n generatng nancal nstablty. Gennaol, Shlefer and Vshny (21) nvestgate the neglected rsks assocated wth new nancal assets. Rajan (25) and Calomrs (28) emphasze the e ect of nancal nnovaton on agency problems. My paper dent es the ncrease n speculaton (and speculatve varance) as an addtonal channel by whch nancal nnovaton decreases nancal stablty. Posner and Weyl (212) push the normatve mplcatons of speculaton further by callng for a regulatory authorty, along the lnes of the FDA, whch approves nancal products based on whether they wll ncrease or decrease the portfolo rsks. The potentally destablzng role of speculaton s also dscussed n Stgltz (1989), Summers and Summers (1991), and Stout (1995). 5

7 wth w dollars at date 1, whch s a random varable that captures the trader s background rsks. Traders only consume at date 1, and they can transfer resources to date 1 by nvestng n one of two ways. They can nvest n cash whch yelds one dollar for each dollar nvested. Alternatvely, they can nvest n rsky assets denoted by j 2 J = f1; ::; jjjg. Asset j s n xed supply, normalzed to zero, and t pays a j dollars at date 1, whch s a random varable. Assets payo s and prces are respectvely denoted by jjj 1 column vectors a = a 1 ; ::; a jjj. The uncertanty n ths economy s captured by an jmj 1 random vector, v = (v 1 ; ::; v m ). Traders have potentally heterogeneous pror belefs about v. They agree about the varance of v, whch smpl es the analyss, but they mght dsagree about the mean of v. In partcular: Assumpton (A1). dstrbuton, N ( v ; v ), where v matrx wth full row rank. 4 Trader s belef for the random varable v s gven by the Normal Traders date 1 endowment can be wrtten n terms of v as: 2 R m s the mean vector and v s an m m covarance w = (W ) v, for an jmj 1 vector W 2 R m. Asset j s payo can also be wrtten n terms of v as: a j = A j v, for an jmj 1 vector A j 2 R m. The vectors, A j that the assets are not redundant., are lnearly ndependent, whch ensures j At date, the asset s traded n a compettve market at prce p j. Trader s poston n asset j s denoted by x j 2 R. Gven the jjj1 prce vector p = p1 ; ::; p jjj, each trader chooses, a jjj 1 asset portfolo, x = x 1 ; ::; xjjj and nvests the rest of her budget, e x p 2 R, n cash. 5 Wth these nvestment decsons, her net worth at date 1 s gven by: n = e x p + w + x a. (1) Trader maxmzes subjectve expected utlty over net worth at date 1. Her utlty functon takes the CARA form. Snce the asset payo s and endowment shocks are jontly Normally 4 Note that I have not spec ed the emprcal (or realzed) dstrbuton for v. Ths dstrbuton does not matter for much of the analyss n ths paper. In partcular, the mean of the emprcal dstrbuton does not play a role n any of the results. Ths s because the man goal of ths paper s to characterze traders portfolo rsks, for whch t s not necessary to take a poston on who s rght on average. In addton, the varance of the emprcal dstrbuton plays only a lmted role. Ths s because traders portfolo rsks could be de ned by usng ther perceved varance, v, wthout reference to the emprcal varance. Ths s the approach that wll be taken n the man text. Appendx A.1 generalzes the man results to the case n whch portfolo rsks are de ned wth the emprcal varance. 5 Note that traders are allowed to take unrestrcted negatve postons n rsky assets or cash, that s, both short sellng and leverage are allowed. Smlarly, the asset payo s can take negatve values because the envronment s frctonless. In partcular, there s no lmted lablty and repayment s enforced by contracts. 6

8 dstrbuted, the trader s optmzaton reduces to the usual mean-varance problem: 6 max x E [n ] 2 var [n ]. (2) Here, denotes the trader s absolute rsk averson coe cent, whle E [] and var [] respectvely denote the mean and the varance of the trader s portfolo accordng to her belefs. The equlbrum n ths economy s a collecton of asset prces, p, and portfolos, x 1 ; ::; x, such that each trader chooses her portfolo to solve problem (2) and prces clear asset markets, that s, X x j = for each j 2 J. I wll capture nancal nnovaton n ths economy as an expanson of the set of traded assets, J. Before I turn to the general characterzaton, I use a smple example to llustrate the e ects of nancal nnovaton on portfolo rsks. 2.1 An llustratve example Suppose there are two traders wth the same coe cent of rsk averson,.e., I = f1; 2g and 1 = 2. The fundamental rsk s captured by 2 uncorrelated random varables, v 1 ; v 2. Traders date 1 endowments are perfectly negatvely correlated: w 1 = v and w 2 = v, where v = v 1 + v 2. In partcular, traders endowments are a ected by both both sources of fundamental rsk, wth the weght,, capturng the relatve mportance of the second rsk. As a benchmark suppose traders have common belefs about v 1 and v 2 gven by N (; 1). In ths benchmark, rst consder the case n whch there are no assets,.e., J = ;. In ths case, there s no trade and traders net worths are gven by: n 1 = e + v and n 2 = e v. (3) Traders net worths are rsky because they are unable to hedge ther endowment rsks. Next suppose a new asset s ntroduced wth payo a 1 = v = v 1 + v 2 : 6 The only role of the CARA preferences and the Normalty assumpton s to generate the mean-varance optmzaton n (2). In partcular, the results of ths paper apply as long as traders portfolo choce can be reduced to the form n (2) over net worth. An mportant specal case s the contnuous-tme model n whch traders have tme-separable expected utlty preferences (whch are not necessarly CARA), and asset returns and background rsks follow d uson processes. In ths case, the optmzaton problem of a trader at any date can be reduced to the form n (2) (see Ingersoll, 1987). The only caveat s that the reduced form coe cent of absolute rsk averson,, s endogenous snce t depends on the trader s value functon. Thus, n the contnuous tradng envronment, the results of ths paper apply at a tradng date condtonal on traders coe cents of absolute rsk averson, f g. 7

9 In ths case, traders equlbrum portfolos are gven by: x 1 1 = 1 and x 1 2 = 1 (and the equlbrum prce s p 1 = ). Traders net worths are constant and gven by n 1 = n 2 = e: Thus, the benchmark analyss shows that, wth common belefs, nancal nnovaton enables traders to hedge and dversfy ther dosyncratc rsks. Next suppose traders have heterogeneous pror belefs for some of the fundamental rsk n ths economy. In partcular, traders have common belefs for v 2 gven by the dstrbuton, N (; 1). They also know that v 1 and v 2 are uncorrelated. However, they dsagree about the dstrbuton of v 1. Trader 1 s pror belef for v 1 s gven by N ("; 1) whle trader 2 s belef s gven by N ( "; 1). The parameter " captures the level of belef dsagreements. I next use ths spec caton to llustrate the two channels by whch new assets ncrease portfolo rsks. Channel 1: New assets generate new dsagreements Consder the case n whch asset 1 s avalable for trade. Snce traders dsagree about the mean of v 1, they also dsagree about the mean of the asset payo, a 1 = v 1 + v 2. In ths case, t s easy to check that the asset prce s p 1 = (by symmetry) and that traders portfolos are: x 1 1 = 1 + x S 1 and x 1 2 = 1 + x S 2, (4) where x S 1 = " and xs 2 = " Note that traders postons devate from the optmal rsk sharng benchmark n vew of ther belef dsagreements. I de ne the d erence as the traders speculatve portfolos and denote t by x S. Traders net worths can also be calculated as: n 1 = e + " v 1 + v and n 2 = e " v 1 + v (5) If " > 1 + 2, then traders net worths are rsker than the case n whch no new asset s ntroduced [cf. Eq. (3)]. Intutvely, trader 1 s so optmstc about the asset s payo that she takes a postve net poston, despte the fact that her endowment covares postvely wth the asset payo. Consequently, the new asset ncreases the rskness of her net worth. Hence, when traders dsagreements about the asset payo are su cently large, nancal nnovaton ncreases traders portfolo rsks. 8

10 Channel 2: New assets amplfy exstng dsagreements Next consder the ntroducton of a second asset wth payo : a 2 = v 2. Note that traders do not dsagree on the payo of ths new asset. Nonetheless, ths asset also ncreases traders portfolo rsks through a second channel: By amplfyng traders speculaton on exstng dsagreements. To see ths, rst consder traders equlbrum portfolos n ths case whch can be calculated as: " # " # " x = + x 2 1 x 1;S 1 " x 2;S 1 " # and As before, traders portfolos feature speculatve postons, " x 1 2 x 2 2 # = " (" 1 # x 1;S x 2;S + " #) x 1;S 2 " x 2;S 2 " #, (6), whch represent the devatons from the optmal rsk sharng benchmark. Gven these postons, traders net worths are gven by: n 1 = e + " v 1 and n 2 = e " v 1. (7) Note that the magntude of traders speculatve postons on asset 1 s greater than the earler settng n whch asset 2 was not avalable [cf. Eqs. (6) and (4)]. Importantly, traders net worths are also rsker [cf. Eqs. (7) and (5)]. Put d erently, the nnovaton of asset 2, about whch traders do not dsagree, enables traders to take greater speculatve postons on asset 1 and ncreases ther portfolo rsks. The ntuton for ths result s related to an mportant economc force: the hedge-more/betmore e ect. When only asset 1 s avalable, traders speculatve postons and portfolo rsks are decreasng n [cf. Eqs. (4) and (5)]. Intutvely, asset 1 provdes the traders wth only an mpure bet because ts payo also depends on the rsk, v 2, on whch traders do not dsagree. To take speculatve postons, traders must also hold these addtonal rsks, whch makes them reluctant to bet. When asset 2 s also avalable, traders complement ther speculatve postons n asset 1 by takng the opposte postons n asset 2 [Eq. (6)]. Ths enables them to take a purer bet on the rsk, v 1. When traders are able to take purer bets, they also take larger bets, whch n turn leads to greater portfolo rsks [Eq. (5)]. 3 Equlbrum and the Decomposton of Average Varance Ths secton characterzes the equlbrum, and decomposes traders portfolo rsks nto two components whch respectvely correspond to traders rsk sharng and speculatve motves for trade. The man result n the next secton characterzes the e ect of nancal nnovaton on the two components of portfolo rsks. Gven assumpton (A1), trader beleves the asset payo s are Normally dstrbuted, 9

11 N ( ; ), wth A v and A v A; where s a jjj 1 vector and s a jjj jjj matrx. In addton, trader beleves that her endowment s Normally dstrbuted, and that the covarance of her endowment wth the asset payo s s gven by: = A v W ; where s a jjj 1 matrx. Gven these belefs, traders portfolo demand [cf. problem (2)] can be solved n closed form. Aggregatng traders demands and usng market clearng, asset prces are gven by: p = 1 X 2I, (8) where P 2I 1 = 1 s the Harmonc mean of traders absolute rsk averson coe - cents. Intutvely, an asset commands a hgher prce f traders are on average optmstc about ts payo, or f t on average covares negatvely wth traders endowments. Usng the prce expresson (8), a trader s equlbrum portfolo can also be solved as: Here, the expresson x = x R + x S, where (9) x R = 1 ~ and x S = 1 ~. ~ = 1 denotes the relatve covarance of the trader s endowment, and X { (1) {2I 1 X ~ = {2I { (11) { denotes her relatve optmsm. Note that the trader s portfolo decomposes nto two components. The rst component, x R, s the portfolo that would obtan f there were no belef belef dsagreements (.e., f ~ = for each ). Hence, I refer to x R as the trader s rsk sharng portfolo. The optmal rsk sharng portfolo s determned by traders endowment rsks and ther rsk tolerances. The second component, x S, captures traders devatons from ths benchmark n vew of ther belef dsagreements, ~. Hence, I refer to x S as the speculatve portfolo of trader. Eqs. (8) (11) complete the characterzaton of equlbrum n ths economy. The man goal of ths paper s to analyze the e ect of nancal nnovaton on portfolo rsks. Gven the mean-varance framework, a natural measure of portfolo rsk for a trader s the varance of 1

12 her net worth, var (n ). I consder an average of ths measure across all traders, the average varance, de ned as follows: = 1 X var (n ) = 1 X 2I W 2I v W + 2x + x x. (12) A couple comments about ths de nton are n order. Frst, the portfolo rsk of a trader s calculated accordng to traders (common) belef for the varance v. Appendx A.1 generalzes the man results to the case n whch portfolo rsks are de ned wth the emprcal varance (that s, the varance that would be re ected ex-post n the data). Second, note that traders that are relatvely more rsk averse are gven a greater weght n the average,. I use as my man measure of average portfolo rsks for two reasons. Frst, Secton 6 shows that s a natural measure of welfare n ths economy (although t s not equvalent to welfare accordng to the Pareto crteron). The second just caton s provded by the followng lemma. Lemma 1. The rsk sharng portfolos, x R feasble portfolos: mn fx 2R jjj g, mnmze the average varance,, among all s.t. X x =. (13) When there are no belef dsagreements,.e., ~ v = for each, then the complete portfolos and the rsk sharng portfolos concde,.e., x = x R for each. Thus, Lemma 1 shows that s the measure of rsks that would be mnmzed n equlbrum absent belef dsagreements. Thus, t s natural to take as the measure of average rsks, and to characterze the extent to whch t devates from the mnmum benchmark n (13) when traders have belef dsagreements. To ths end, I let R denote the mnmum value of problem (13) and refer to t as the unnsurable varance. I also de ne S = R and refer to t as the speculatve varance. Ths provdes a decomposton of the average varance nto two components: = R + S : The man result n the next secton concerns the e ect of nancal nnovaton on R and S. The next lemma characterzes the two components of average varance n terms of the exogenous parameters of the model. The forms of R and S are ntutve. Eq. (14) llustrates that the unnsurable varance s lower when the assets provde better rsk sharng opportuntes, captured by larger ~. Smlarly, Eq. (15) llustrates that the speculatve varance s greater when the assets feature greater belef dsagreements, captured by larger ~. Lemma 2. The unnsurable varance s gven by: R = 1 X W 2I v W ~ 1 ~, (14) 11

13 and the speculatve varance s gven by: S 1 X ~ 1 ~. (15) 2I 4 Fnancal Innovaton and Portfolo Rsks I model nancal nnovaton as an expanson of the set of traded assets. For ths purpose, t s useful to de ne the notaton, z ^J, to refer to the equlbrum varable z when only a subset ^J J of the assets n J are traded. I next present the man result. Theorem 1 (Fnancal Innovaton and Portfolo Rsks). Suppose J conssts of a set of old assets, J O, and a set of new assets, J N (formally, J = J O [ J N where J O and J N are dsjont sets). () Fnancal nnovaton always reduces the unnsurable varance, that s: R (J O [ J N ) R (J O ). () Fnancal nnovaton always ncreases the speculatve varance, that s: S (J O [ J N ) S (J O ). The rst part of ths theorem s a corollary of Lemma 1, and t shows that nancal nnovaton always provdes some rsk sharng bene ts. Ths part formalzes the tradtonal vew of nancal nnovaton n the context of ths model. On the other hand, the second part of the theorem dent es a second force whch always operates n the opposte drecton. In partcular, when there are belef dsagreements, nancal nnovaton also always ncreases the speculatve varance. Hence, the net e ect of nancal nnovaton on average varance s ambguous, and t depends on the relatve strength of the two forces. Furthermore, t s easy to see that there exst economes n whch the ncrease n the speculatve varance s su cently large that nancal nnovaton ncreases the average varance (by an arbtrary amount). Most of the lterature on nancal nnovaton consders the specal case wthout belef dsagreements. Theorem 1 shows that the common-belefs assumpton s restrctve, as t shuts down an mportant economc force by whch nancal nnovaton has a postve e ect on portfolo rsks. It s also worth emphaszng the generalty of Theorem 1. The result apples for all sets of exstng and new assets, J O and J N, wth no restrctons on the jont dstrbuton of asset payo s or traders belefs for v [except for the relatvely mld Assumpton (A1)]. For example, Theorem 1 shows that nancal nnovaton ncreases the speculatve varance even f there are no belef dsagreements about new assets (as n Example 1). The rest of ths secton provdes a sketch proof and a complementary ntuton for the second part of Theorem 1. The proof proceeds n three steps. Frst, the form of x S n Eq. 12

14 (9) mples that the speculatve portfolo, x S, solves the followng verson of the mean-varance problem: max (~ ) x x 2R ^J 2 x x. (16) Moreover, the speculatve varance, S, s found by averagng the varance costs for each trader at the soluton to ths problem: S = 1 X x S x S. (17) Intutvely, problem (16) s the traders mean-varance problem n a hypothetcal economy that s dentcal except that traders have no background rsks (.e., W = for all 2 I), so that the only motve for trade s speculaton. The soluton to ths problem gves the speculatve portfolo n the actual economy, and also determnes the speculatve varance as captured by (17). Second, nancal nnovaton relaxes the constrant set of problem (16). That s, when the asset set s ^J = J O [ J N, traders are able to make all the speculatve trades they could make when the asset set s ^J = J O, and some more. Put d erently, new assets expand the bettng possbltes fronter for traders. Ths observaton renforces the nsghts of Example 1. In partcular, nancal nnovaton ncreases the bettng possbltes fronter through two dstnct channels: By creatng new dsagreements, and by enablng traders to take purer bets on exstng dsagreements. Thrd, when the constrant set of problem (16) s more relaxed, each trader obtans a greater certanty-equvalent payo from bettng. Moreover, snce the problem s a quadratc optmzaton, expected payo s at the optmum are proportonal to the expected varance of the payo s, that s: (~ ) x s = 2 2 (x s ) x s. Consequently, a relaxed constrant set also ncreases the expected varance, 2 (x s ) x s. Intutvely, at the optmal speculatve portfolo, hgher expected returns go hand-n-hand wth hgher rsks. These steps establsh that nancal nnovaton ncreases the speculatve varance of each trader. It follows that nancal nnovaton also ncreases the average speculatve varance n Eq. (17), completng the proof (see Appendx A.3 for an alternatve proof based on matrx algebra). A complementary ntuton for Theorem 1 can be provded by characterzng traders speculatve rsks n terms of the Sharpe ratos of ther speculatve portfolos. To ths end, consder the hypothetcal economy n the above proof n whch there are no background rsks. De ne the speculatve Sharpe rato of a trader as the Sharpe rato of her portfolo n ths hypothetcal 13

15 economy. 7 Usng Eqs. (8) (11), ths rato can be calculated as: Sharpe S = xs ( p) q = x S x S q ~ 1 ~. (18) Next consder the trader s portfolo return gven by n =e (where recall that e s the trader s ntal net worth). The standard devaton of ths return can also be calculated as: S 1 e q x S x S = 1 q ~ e 1 ~. (19) Note that the rato, e, provdes a measure of trader s coe cent of relatve rsk averson. Thus, combnng Eqs. (18) and (19) gves the famlar result that the standard devaton of the portfolo return s equal to the Sharpe rato of the optmal portfolo dvded by the coe cent of relatve rsk averson (see Campbell and Vcera, 22). Ths textbook result apples also n ths model when there are no background rsks. Theorem 1 can then be understood from the lenses of Eqs. (18) and (19). Fnancal nnovaton ncreases the traders speculatve Sharpe ratos by expandng the bettng possbltes fronter through the two dstnct channels emphaszed before. Once traders are able to obtan hgher Sharpe ratos, they also undertake greater speculatve rsks, provdng a complementary ntuton for the man result. 5 Endogenous Fnancal Innovaton The analyss so far has taken the set of new assets as exogenous. In practce, many nancal products are ntroduced endogenously by economc agents wth pro t ncentves. A large lterature emphaszes rsk sharng as a major drvng force for endogenous nancal nnovaton [see, for example, Allen and Gale (1994), Du e and Rah (1995), Demange and Laroque (1995), Athanasouls and Shller (2, 21)]. 8 A natural queston, n vew of the results n the earler sectons, s to what extent the rsk sharng motve for nancal nnovaton s robust to the presence of belef dsagreements. To address ths queston, ths secton endogenzes the asset desgn by ntroducng a pro t seekng market maker and obtans two man results. Frst, the optmal asset desgn depends on the sze and the nature of traders belef dsagreements, n addton to the possbltes for rsk sharng. Second, when traders belef dsagreements are su cently large, the market maker desgns assets that maxmze traders average portfolo rsks among all possble choces, completely dsregardng the rsk sharng motve for nancal 7 Recall that the Sharpe rato of a porto o s de ned as the expected portfolo return n excess of the rsk-free rate (whch s normalzed to n ths model) dvded by the standard devaton of the portfolo return. 8 Rsk sharng s one of several drvers of nancal nnovaton emphaszed by the prevous lterature. Other factors nclude mtgatng agency frctons, reducng asymmetrc nformaton, mnmzng transacton costs, and sdesteppng taxes and regulaton (see Tufano, 24, for a recent survey). These other factors, whle clearly mportant, are left out of the analyss n ths paper to focus on the e ect of belef dsagreements on the rsk sharng motve for nnovaton. 14

16 nnovaton. The man feature of the model n ths secton s that the assets, J, are ntroduced by a market maker. The market maker s constraned to choose jjj m assets, but s otherwse free to choose the asset desgn, A. Here, recall that the matrx, A = A 1 ; A 2 ; ::; A jjj, captures the asset payo s whch are gven by a j = A j v for each j. Thus, the market maker s choce of A a ects the belef dsagreements and the relatve covarances accordng to [cf. Eqs. (11) and (1)]: where the devaton terms are de ned as: ~ (A) = A ~ v and ~ (A) = A v ~W, ~ v = v 1 X {2I { v { and ~W = W 1 X W {. {2I Once the market maker chooses the asset desgn, A, the assets are traded n a compettve market smlar to the prevous sectons. The market maker ntermedates these trades whch enables t to extract some of the surplus from traders. To keep the analyss smple, suppose the market maker can extract the full surplus. 9 In partcular, the market maker sets a xed membershp fee,, for each trader and makes a take t or leave t o er. If trader accepts the o er, then she can trade the avalable assets n the compettve market. Otherwse, trader s out of the market, and her net worth s gven by her endowment, e + W v. The equlbrum of ths economy can be characterzed backwards. Frst consder the compettve equlbrum after the market maker has chosen A and traders decded whether or not to partcpate n the market. Assume that all traders have accepted the o er, whch wll be the case n equlbrum. In vew of the mean-varance framework, traders portfolo choces are ndependent of the xed fees they have pad. In partcular, the equlbrum s characterzed as n the earler sectons. Next consder the xed fees the market maker charges for a gven choce of A. If trader rejects the o er, she receves the certanty equvalent payo from her endowment. Otherwse, she receves the certanty equvalent payo from her equlbrum portfolo net of the xed fee, (A). The market maker sets (A) so that the trader s just nd erent to accept the o er. Straghtforward calculatons (relegated to Appendx A.4) show that the market maker s expected total pro ts are gven by: X (A) = X 2I ~ (A) 2 ~ (A) 1 ~ (A) ~ (A). (2) Ths expresson re ects the two motves for trade n ths economy. Traders are wllng to pay 9 The results below reman unchanged under the less extreme (reduced form) assumpton that the market maker extracts a constant fracton, 2 (; 1], of the surplus regardless of the choce of A. 15

17 to trade assets that facltate better rsk sharng [.e., larger ~ (A)], or to trade assets that generate greater belef dsagreements [.e., larger ~ (A)]. The market maker chooses an asset desgn, A, that maxmzes the expected pro ts n (2). Note that many choces of A represent the same tradng opportuntes over the space of the underlyng rsks, v (and thus, also generate the same pro ts). Thus, suppose wthout loss of generalty that the market maker s choce s subject to the followng normalzatons: = A v A = I jjj, and j ( v ) 1=2 A for each j 2 J. (21) 1 Here, ( v ) 1=2 denotes the unque postve de nte square root of the matrx, v. The rst condton n (21) normalzes the varance of assets to be the dentty matrx, I jjj. Ths condton determnes the column vectors of the matrx for normalzed asset payo s, ( v ) 1=2 A, up to a sgn. The second condton resolves the remanng ndetermnacy by adoptng a sgn conventon for these column vectors. Theorem 2 (Optmal Asset Desgn). Suppose the matrx 1 X ( v ) 1=2 ~ v ( v ) 1=2 ~W ( v ) 1=2 ~ v ( v ) 1=2 ~W (22) s non-sngular. Then, an asset desgn s optmal f and only f the columns of the matrx for normalzed asset payo s, ( v ) 1=2 A, correspond to the egenvectors correspondng to the jjj largest egenvalues of the matrx n (22). If the egenvalues are dstnct, then the asset desgn s unquely determned by ths condton along wth the normalzatons n (21). Otherwse, the asset desgn s determned up to a choce of the jjj largest egenvalues. Ths result generalzes the results n Demange and Laroque (1995) and Athanasouls and Shller (2) to the case wth belef dsagreements, ~ v 6=. Importantly, the expressons (2) and (22) show that nancal nnovaton s partly drven by the sze and the nature of traders belef dsagreements. The sze of the belef dsagreements, ( v ) 1=2 ~ v, (along wth the rsk averson coe cents, ) determne to what extent endogenous nnovaton s drven by the speculaton motve for trade as opposed to rsk sharng. Assumng that ths term s sgn cant, the nature of the belef dsagreements, ( v ) 1=2 ~ v jj( v ) 1=2 ~ v jj that maxmze the opportuntes for speculaton., bas the choce of assets towards those The next result characterzes the optmal asset desgn further n two extreme cases: when traders have common belefs, and when ther belef dsagreements are very large. Theorem 3 (Optmal Asset Desgn and Portfolo Rsks). Consder a collecton of economes whch are dentcal except for belefs gven by v ;K = Kv for all, where K s a parameter that scales belef dsagreements. Suppose the matrx n (22) s non-sngular wth dstnct egenvalues for each K. 1 Let K () denote the average varance and A K denote the 1 The assumpton of dstnct egenvalues can be relaxed at the expense of addtonal notaton. 16

18 optmal asset desgn (characterzed n Theorem 2) for each K.. () Wth no belef dsagreements,.e., K =, the market maker nnovates assets that mnmze the average varance: v { A 2 arg mn ^A ^A subject to (21). For the next part, suppose there exsts at least two traders wth d erent belefs,.e., v 6= for some ;{ 2 I. Let K (;) denote the average varance wthout any assets. () As K! 1, the market maker nnovates assets that maxmze the average varance. 1 In partcular, the lmts lm K!1 A K and lm K!1 K 2 K ^A exst and are nte, and they satsfy: 11 lm A 1 K 2 arg max lm K!1 ^A K!1 K 2 K ^A subject to (21). (23) Wthout belef dsagreements, the market maker nnovates assets that mnmze average portfolo rsks n ths economy, as llustrated by the rst part of the theorem. The second part provdes a sharp contrast to ths tradtonal vew. When traders belef dsagreements are large, the market maker nnovates assets that maxmze average portfolo rsks, completely dsregardng the rsk sharng motve for nnovaton. Thus, belef dsagreements change the nature of nancal nnovaton as well as ts mpact on portfolo rsks. Ths result complements the man result, Theorem 1, by dentfyng su cent condtons under whch assets that ncrease portfolo rsks endogenously emerge n ths economy. 6 Dscusson and Welfare Implcatons Whle Theorems 1 and 3 show that nancal nnovaton may ncrease portfolo rsks, they do not reach any welfare conclusons. In fact, t s easy to see that nancal nnovaton n ths economy results n a Pareto mprovement f traders welfare s calculated accordng to ther own belefs. Ths s because each trader perceves a large expected return from her speculatve postons n new assets, whch just es the addtonal rsks that she s takng. Ths observaton naturally rases concerns about the sgn cance of Theorems 1 and 3. Ths secton preempts these concerns, and clar es the postve and the normatve content of these results. Even absent any normatve consderatons, Theorems 1 and 3 are mportant because of ther postve mplcatons. A large lterature n nancal nnovaton has argued that nancal assets mprove welfare because they facltate rsk sharng (see, for example, Allen and Gale, 1994, or Du e and Rah, 1995). Theorem 1 shows that, whle new assets mght mprove welfare, they mght do so for qute d erent reasons than emphaszed n the prevous lterature. In partcular, when belef dsagreements are su cently large, traders welfare gans do not come from a The scale factor,, n (23) ensures that the lmt lm K 2 K!1 K 2 K ^A s nte, so that the optmzaton problem s well de ned. 17

19 decrease n ther rsks, but from an ncrease n ther perceved expected returns. Importantly, ths analyss also generates a new testable mplcaton: The ntroducton of new assets wll ncrease traders average portfolo rsks as long as traders belef dsagreements are su cently large. Theorem 3 generates a further testable mplcaton that pro t seekng market makers mght ntroduce new assets that wll ncrease rsks, as opposed to assets that wll facltate rsk sharng. Among other thngs, these results mght provde an explanaton for why most of the macro futures markets proposed by Shller (1993) have not been adopted n practce, despte the fact that they are n prncple very useful for rsk sharng purposes. Theorems 1 and 3 mght also have a normatve content dependng on the nterpretaton of belef dsagreements. There are two dstnct ways n whch traders mght come to have d erent (pror) belefs. On the one hand, these d erences mght re ect traders d erent subjectve pror belefs as n Savage (1954). Under ths nterpretaton, traders belefs are convenent representatons of ther preferences under uncertanty. The fact that they have d erent belefs s smply a re ecton of ther d erng personal experences (and possbly also ther lmted past opportuntes to learn). Consequently, the Pareto crteron s arguably the rght crteron despte the fact that traders have d erent belefs. On the other hand, belef d erences mght also emerge from overcon dence (whch s commonly observed n expermental studes) or other psychologcal bases. Under ths second nterpretaton, traders belef dsagreements represent belef dstortons, and the Pareto crteron s arguably no longer approprate. Whle all traders expect to receve a hgh perceved return, at most one of these expectatons can be correct. Put d erently, traders belefs under the second nterpretaton represent a collectve rratonalty. 12 The welfare crteron should deally correct for ths rratonalty. However, there s a practcal problem because the planner mght not know the nature of traders belef dstortons. In partcular, t s not clear whch trader s belef the planner should use to evaluate welfare. It s perhaps fortunate that speculaton s ne cent regardless of whose belef one uses to evaluate welfare (as long as that same belef s consstently appled to calculate each trader s welfare). In recent work, Brunnermeer, Smsek, and Xong (212) propose an alternatve welfare crteron whch s spec cally desgned to capture ths aspect of speculaton. They say that an allocaton s belef-neutral Pareto ne cent f t s Pareto ne cent accordng to any convex combnaton of agents belefs. To apply ths crteron n ths model, let subscrpt h = h 1 ; :::; h, wth h and P h = 1, denote an arbtrary convex combnaton of agents belefs. In partcular, the dstrbuton of v accordng to belef h s gven by N ( v h ; v ), where v h = P h v. Consder the sum of traders certanty equvalent net worths under belef h, 12 The second nterpretaton, and the assocated collectve rratonalty, s also supported by a large body of emprcal evdence whch shows that excessve tradng sgn cantly worsens portfolo performance of nonnsttutonal nvestors, e.g., Odean (1999), Barber and Odean (2), Barber, Lee, Lu, and Odean (29). To gve one example, Barber and Odean (2) analyze a large sample of brokerage accounts held by households n the US, and show that households that trade the most frequently underperform the market return by more than 5% per year. 18

20 gven by: N h = X 2I E h [n ] 2 var h (n ). Note that ths expresson s a measure of welfare under belef h, n the sense that any allocaton ~x that yelds a hgher N h than another allocaton x can also be made to Pareto domnate allocaton x (under belef h) after combnng t wth approprate ex-ante transfers. Usng the expressons (1) and (12), along wth the market clearng condton P 2I x =, ths welfare measure can further be smpl ed to: " # X N h = E h e + w 2. 2I In partcular, the welfare measure for ths economy has two components: An expected endowment component whch does not depend on traders portfolos, and the average varance component whch depends on the portfolos but whch s ndependent of belef h [cf. Eq. (12)]. Consequently, regardless of the belef h, an allocaton ~x yelds a hgher welfare than x f and only f t yelds a smaller average varance. Intutvely, the portfolo allocatons n ths economy do not generate expected net worth snce they smply redstrbute wealth across traders. Hence, the portfolos a ect the welfare measure, N h, only through ther e ect on traders portfolo rsks. It follows that the average varance,, emerges as a belef-neutral measure of welfare for ths economy. In partcular, nancal nnovaton s belef-neutral ne cent whenever the ntroducton of new assets ncreases the average varance. Despte ths observaton, I do not take a strong normatve stance n ths paper, and emphasze the postve mplcatons of Theorems 1 and 3. Ths s because there are some mportant ngredents mssng from ths model whch mght change the welfare arthmetc. Most mportantly, traders belefs n ths model contan no nformaton about asset prces. When traders have some nformaton, even pure speculaton can be bene cal by ncreasng the nformatonal e cency of prces. I leave a more complete analyss of welfare for future work. 7 A Quanttatve Exploraton The results n the prevous sectons have theoretcally establshed that belef dsagreements, when they are su cently large, change the nature of nancal nnovaton as well as ts e ect on portfolo rsks. A natural queston s how large belef dsagreements should be to make these results practcally relevant. To address ths queston, ths secton consders a calbraton of the model n the context of the natonal ncome markets analyzed by Athanasouls and Shller (AS, 21). AS argue that assets whose payo s are lnked to (varous combnatons of) natonal ncomes would lead to large welfare gans because they would enable ndvduals to nternatonally dversfy ther consumpton rsks. I rst replcate AS s emprcal results by 19

21 mappng ther model and calbraton to ths framework. I then show that, wth reasonable amounts of belef dsagreements, the new assets proposed by AS would have the unntended consequence of ncreasng ndvduals consumpton rsks. Replcatng Athanasouls and Shller (21) results n ths framework To replcate the AS results, consder a settng n whch the underlyng rsks correspond to the ncome shocks of G7 countres. More spec cally, let c 2 C = f1; ::; jcjg denote a G7 country, and suppose (as AS does) that the yearly ncome per-capta of country c s gven by: y (c) = y past (c) + past (c) + v (c). Here, y past (c) denotes last year s ncome per-capta, past (c) denotes the predetermned change n ncome per-capta, and v (c) denotes a zero-mean random varable whch captures the yearly shock to ncome per-capta. Let v = fv (c)g c2c denote the jcj 1 vector whch captures the underlyng uncertanty. Suppose v s Normally dstrbuted wth the jcj jcj varance matrx, v. A key challenge for emprcal analyss s to estmate v. AS specfy a spatal correlaton model for v, whch they estmate usng data from the Penn World Table over the years I use exactly the same spec caton to reconstruct ther estmate for v, whch facltates the comparson of results. However, as t wll become clear, the results are robust to reasonable varatons n v. To nterpret the numercal results below, t s useful to note that the standard devaton of v (c) s estmated to be the same for each country and gven by v(c) = $364 n 1985 dollars. The average per-capta ncome of G7 countres n 1992 s y = $14784 (n 1985 dollars), whch mples that the standard devaton of yearly ncome growth n a G7 country s about 2:46%. In ths settng, the ntroducton of assets lnked to country ncome shocks, v, can facltate nternatonal rsk sharng. To see ths, let I (c) denote the set of ndvduals n country c that wll actually trade the new assets, whch I take to be proportonal to the populaton of country c. 13 To focus on nternatonal rsk sharng, suppose (as AS do) that traders n the same country experence the same ncome shocks: That s, the ncome shock of trader (c) s gven by (W c ) v, where W c s a jcj 1 vector that has 1 n ts c th entry and everywhere else. To smplfy the analyss, suppose (as AS do) also that traders have the same rsk averson coe cents, (c) for each trader (c). Fnally, suppose that new assets j 2 J = f1; ::; jjjg are lnear combnatons of the countres ncome shocks, a j = A j v for each j 2 J. AS characterze the optmal desgn of these assets that maxmzes a socal welfare functon. The top panel of Fgure 1 llustrates the optmal desgn when only two assets can be ntroduced. The most 13 AS take I (c) to be equal to the populaton of country c. Ths assumpton s unreasonable snce t s well documented that a large fracton of households do not partcpate n rsky nancal markets at all (see Campbell, 26, for evdence from the US). I make the less strngent assumpton that I (c) s proportonal to the populaton of the country. Ths s wthout loss of generalty snce the level of partcpaton does not play an mportant role for the results n ths secton. 2

22 Fgure 1: The top table llustrates the asset desgn and the equlbrum portfolos for the benchmark wthout belef dsagremeents (wth two assets). The last two columns dsplay the asset desgn normalzed by the country populatons for comparson wth Table 1 of AS. The bottom table shows the e ect of nancal nnovaton on consumpton rsks n the US. The P columns dsplay the slope coe cents n the followng regresson, Consumpton = + 7 c=1 cv c, whch has a perfect t n the model. mportant asset to create resembles an ncome swap between the US and Japan. Intutvely, ths asset enables rsk sharng between the traders n the US and Japan. The model pcks the US and Japan because these are large countres whose ncome shocks are relatvely less correlated (snce they are geographcally far), whch ncreases the bene ts from rsk sharng. The second most mportant asset also resembles an ncome swap, ths tme between Japan and the core EU regon, for a smlar reason. When there are no belef dsagreements, my analyss n the earler sectons replcates the results n AS. 14 In partcular, traders rsky asset portfolos are also dentcal n both settngs. The top table of Fgure 1 llustrates these equlbrum portfolos (cf. Table 1 of AS). Note that much of the trade n the rst asset s among the traders n Japan and the US who take the opposte postons to dversfy ther ncome rsks. Moreover, traders consumpton rsks are also dentcal n both settngs. The second panel of Fgure 1 llustrates these rsks for a sample country, the US. Before nancal nnovaton, the consumpton of traders n the US has an exposure of one to the US ncome shock, v US, and an exposure of zero to the ncome shocks of other countres. The new assets enable the traders to reduce ther exposure to the 14 Ths mght seem surprsng snce AS consder a dynamc model, whereas ths paper consders a statc model. However, n vew of CARA preferences, the dynamc and the statc models are equvalent n several aspects. In partcular, the equlbrum portfolos of rsky assets are dentcal n both settngs. Moreover, the varance of an ndvdual s consumpton n the dynamc model s the same as the varance of her net worth n the statc model. It follows that the average varance,, accurately descrbes the average consumpton rsks n the AS model. 21

23 US ncome shock by takng on some exposure to the ncome shocks of other countres (n partcular, Japan). Consequently, traders are able to dversfy and reduce ther consumpton rsks. Wth two assets, the standard devaton of ther consumpton declnes from about $364 (2:46% of average ncome) to $315:4 (2:13% of average ncome). Introducng addtonal assets reduces rsks further but there are dmnshng returns. E ect of belef dsagreements on consumpton rsks I next consder the robustness of the AS results to the presence of belef dsagreements. Modelng belef dsagreements represents two addtonal emprcal challenges. The rst challenge s to calbrate traders belef dsagreements. In lne wth assumpton (A1), I assume that traders know the varance matrx, v, but they dsagree about the mean of ncome shocks, v. To keep the analyss smple, I assume that a trader s belef for the mean of a country s ncome shock s an..d. draw from a Normal random varable, v (c) N ; ( ) 2 : (24) The assumpton that a trader has uncorrelated belefs for varous countres s admttedly arbtrary. However, the qualtatve conclusons n ths secton are robust to varatons of the structure of a trader s belefs, as long as there are dsagreements across traders. The spec caton n (24) has the bene t of capturng dsagreements wth a sngle parameter. In partcular, let v = denote the cross-sectonal standard devaton of traders belefs for v ncome shocks relatve the standard devaton of the same shock. Note that v represents a measure of belef dsagreements whch s ndependent of lnear transformatons of v (c). AS results correspond to the case n whch v =. I next show that, wth the spec caton n (24), v = :2 s su cent to overturn ther results about consumpton rsks. Dsagreements of ths sze do not seem unreasonable. Accordng to the Phladelpha Fed s Survey of Professonal Forecasters (SPF) on macroeconomc forecasts n the US, the nterquartle range of forecasts of US yearly GDP growth s on average gven by :7% between the rst quarter of 1992 and the thrd quarter of 211. Over the same perod, the hstorcal standard devaton of the US yearly GDP growth s 2:8%. Ths suggests v = :25, whch s an order of magntude larger than the parameter that I consder, v = :2. 15 The second challenge for emprcal analyss wth belef dsagreements s to calbrate the rsk averson coe cent. Ths coe cent plays an mportant role because t a ects the sze of traders speculatve portfolos, but not ther rsk sharng portfolos [cf. Eq. (9)]. As a benchmark, I follow AS and calbrate the relatve rsk averson coe cent as relatve = 3, as representng a consensus by many who work n ths lterature as a reasonable value to assume. Ths mples 15 A caveat s order at ths pont. Note that the traders belef dsagreements n the model are nonnformatonal,.e., they agree to dsagree. However, the v observed n the SPF mght re ect to some extent forecasters prvate nformaton, whch creates an upward bas n the calbraton. Whle t s d cult to adjust v for ths bas, the fact that the unadjusted v s an order of magntude larger than necessary suggests that the results would contnue to hold for reasonable amounts of prvate nformaton. 22

24 an absolute rsk averson coe cent = relatve y = :2. Usng a larger calbraton for relatve makes the results less stark but t does not overturn the qualtatve conclusons n ths secton. 16 To llustrate the equlbrum wth belef dsagreements, I start by consderng the portfolo allocatons for a few traders whose belefs are spec ed exactly and who are representatve of a larger class of traders wth smlar belefs. Let a moderate be someone whose belef for the ncome shocks of all countres s the same as the mean belef. In contrast, de ne a c - optmst as someone whose belef for the ncome shock of country c s exactly one standard devaton above the mean belef, and whose belef for all other ncome shocks s equal to the mean belef. The top table of Fgure 2 shows the portfolo allocatons for moderates and c - optmsts when there are two new assets. A moderate holds exactly the rsk sharng portfolo n her respectve country, whch s llustrated n the thrd and the fourth columns. Thus, her portfolo s una ected by the presence of belef dsagreements. In contrast, a c -optmst who lves n country c combnes the rsk sharng portfolo for country c wth her speculatve portfolo, whch s llustrated n the last two columns. Note that, for a US-optmsts and a Japan-optmst, the speculatve portfolo s comparable n magntude to (and often larger than) the rsk sharng portfolos. Consequently, the complete portfolo of a US-optmst or a Japan-optmst s sgn cantly n uenced by the speculaton motve for trade. The bottom table n Fgure 2 shows the consumpton rsks for moderates and US-optmsts who lve n the US. Moderates contnue to dversfy ther rsks n ths case, as llustrated by the second column. However, the thrd column llustrates that a US-optmst does not dversfy her rsks. The rsk sharng consderatons would requre ths ndvdual to take a short poston n the rst asset. However, her optmsm about the US nduces her to take a long poston. When there are two assets, the two forces almost perfectly balance for ths ndvdual (cf. the top table), who remans exposed to the US ncome shock (cf. the bottom table). The last column llustrates the case wth complete markets, n whch case the speculaton motve for trade domnates for a US-optmst. In ths case, ths ndvdual has a greater exposure to the US ncome shock, and consequently greater consumpton rsks, than before nancal nnovaton. Ths analyss llustrates that, wth belef dsagreements, nancal nnovaton has a d erent qualtatve e ect on the ncome rsks of moderates and optmsts. Guded by the earler analyss, I assess the overall e ect by consderng p, whch provdes a quadratc average of the standard devaton of consumpton over traders n G7 countres. The left table n Fgure 3 shows that nancal nnovaton ncreases ths average for each country, p c. The rght panel of Fgure 3 plots the world average standard devaton of consumpton normalzed normalzed by average ncome, p =y. In contrast wth the case wthout belef dsagreements (the dashed lne), nancal nnovaton ncreases p =y (the sold lne). 16 For an extreme case, suppose relatve = 5 whch s the level of the relatve rsk averson parameter that s necessary to ratonalze the equty premum puzzle n a CRRA envronment, but whch s also consdered to be mplausbly hgh (Campbell, 2). In ths case, the AS results are overturned as long as belef dsagreements are as hgh as mpled by the SPF data, v = :25. 23

25 Fgure 2: The top table llustrates the equlbrum portfolos wth belef dsagreements ( v = :2) and wth two assets. The thrd and the fourth columns llustrate the rsk sharng portfolos n country c. These are also the complete portfolos of moderates, who have the mean belef for each country. The last two columns llustrate the speculatve portfolos of c -optmsts, whose belefs for country c are one standard devaton above the mean and whose belefs for all other countres are equal to the mean. The bottom table llustrates the consumpton rsks n the US for moderates and US-optmsts (see Fgure 1 for a detaled explanaton). 4% 3.5% 3% 2.5% 2% Fgure 3: The panels llustrate the e ect of nancal nnovaton on average consumpton rsks n each country (left) and n all G7 countres (rght) wth belef dsagreements parameterzed by v = :2. 24