journal of financial and quantitative analysis financial innovation, market participation, and...

JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL. 39, NO. 3, SEPTEMBER 2004COPYRIGHT 2004, SCHOOL OF BUSINESS ADMINISTRATION, UNIVERSITY OF WASHINGTON, SEATTLE, WA 98195

Financial Innovation, Market Participation, andAsset Prices

Laurent Calvet, Martın Gonzalez-Eiras, and Paolo Sodini�

Abstract

This paper investigates the pricing effects of financial innovation in an economy with en-dogenous participation and heterogeneous income risks. The introduction of non-redundantassets endogenously modifies the participation set, reduces the covariance between divi-dends and participants’ consumption and thus leads to lower risk premia. In multisectoreconomies, financial innovation spreads across markets through the diversified portfolioof new entrants, and has rich effects on the cross-section of expected returns. The pricechanges can also lead some investors to leave the markets and give rise to non-degenerateforms of participation turnover. The model is consistent with several features of financialmarkets over the past few decades: substantial innovation, higher participation, significantturnover in investor composition, improved risk management practices, a slight increase inreal interest rates, and a reduction in risk premia.

I. Introduction

The introduction of a vast array of non-redundant securities has transformedfinancial markets over the past few decades. Derivative contracts now allow firmsto routinely manage their exposure to commodity and financial prices.1 New in-surance and financial products provide additional diversification opportunities toindividual investors. Initial public offerings have enabled entrepreneurs to diver-sify some of their wealth away from the companies in which they had managerial

�Calvet, [email protected], Department of Economics, Harvard University, Cambridge, MA02138, Department of Finance, Stern School of Business, New York, NY 10012, and NBER;Gonzalez-Eiras, [email protected], Departamento de Economıa, Universidad de San Andres, VitoDumas 284 (1644), Victoria, Buenos Aires, Argentina; Sodini, [email protected], Department ofFinance, Stockholm School of Economics, Sveavagen 65, Box 6501, SE-113 83 Stockholm, Sweden.Earlier versions of this paper appear in the Ph.D. dissertations of M. Gonzalez-Eiras and P. Sodini.We received helpful comments from D. Acemoglu, F. Alvarez, S. Basak, O. Blanchard, R. Caballero,J. Campbell, J. Detemple, P. Diamond, A. Fisher, R. Giammarino, G. Mankiw, E. Maskin, M. Pagano,T. Persson, S. Ross, R. Uppal, D. Vayanos, T. Vuolteenaho, J. Wang, an anonymous referee, andseminar participants at Boston University, CEU, ESSEC, Gerzensee, Harvard, INSEAD, the LondonSchool of Economics, MIT, NYU, Salerno, San Andres, the Stockholm School of Economics, UBC,the University of Amsterdam, the University of Cyprus, UTDT, the University of Trieste, the RolfMantel Memorial Conference, the 2000 LACEA Meeting, the 2001 European Summer Meeting of theEconometric Society, and the 2002 NBER Meeting on Economic Fluctuations and Growth. The paperalso benefited from excellent research assistance by Charles P. Cohen.

1See, for instance, Ross (1976).

431

432 Journal of Financial and Quantitative Analysis

interests.2 Financial theory suggests that these innovations might have profoundimplications for portfolio choice, asset pricing, and the cross-sectional allocationof risk.3 For instance, new instruments allow agents to reduce their exposure toidiosyncratic or background shocks, which increases the demand for stocks andleads to a lower market premium (e.g., Kihlstrom, Romer, and Williams (1981),Nachman (1982), and Campbell and Viceira (2002)).

This paper shows that new mechanisms arise when participation is endoge-nized in an economy with heterogeneous income risks.4 The introduction of newsecurities changes the participation set, decreases the covariance between divi-dends and participants’ consumption, and thus amplifies the reduction in the mar-ket premium. In multifactor economies, financial innovation also spreads acrossmarkets through the diversified portfolio choices of new entrants and has rich ef-fects on the cross-section of expected returns. The price changes can also leadsome investors to leave financial markets. Innovation then induces the simultane-ous entry and exit of investors, giving rise to non-degenerate forms of participa-tion turnover.5

Our approach builds on the existence of fixed costs to participate in financialmarkets. Corporate hedging requires the employment of experts able to effec-tively reduce the firm’s risk exposure using existing financial assets. Investorshave to sustain learning efforts and expenses related to the opening and mainte-nance of accounts with an exchange or a brokerage firm. Statutory and govern-ment regulations often create costly barriers to the participation of institutionalinvestors in some markets.

We introduce a two-period economy with incomplete markets and endoge-nous participation. Agents can borrow or lend freely, but have to pay a fixed entrycost to invest in risky assets. They receive heterogeneous random incomes deter-mined by a finite number of macroeconomic risk factors. Security prices and theparticipation structure are jointly determined in equilibrium. The model is com-putationally tractable and leads to a closed-form system of equilibrium equationsin the CARA-normal case. As mentioned earlier, three new mechanisms arise inthis economy.

First, the introduction of non-redundant instruments encourages more in-vestors to participate in financial markets for hedging and diversification pur-poses. Under plausible conditions on the cross-sectional distribution of risk, thenew entrants reduce the covariance between stock returns and the mean consump-tion of participants, leading to a lower market premium. The model thus illus-trates the connection between the cross-sectional distribution of risk and the priceimpact of new assets. It is also consistent with features that have characterizedfinancial markets in the past few decades: substantial financial innovation, a sharp

2Shiller (2003) provides a recent review of these developments.3Theoretical investigations of financial innovation include Allen and Gale (1994a), (1994b), Cal-

vet (2001), Cass and Citanna (1998), Detemple and Selden (1991), Duffie and Jackson (1989), Gross-man (1989), Huang and Wang (1997), and Stein (1987). See Duffie and Rahi (1995) for a review.

4While the discussion focuses on the effect of an increase in spanning, the paper also analyzes theconsequences of a reduction in transactions costs.

5Hurst, Luoh, and Stafford (1998) and Vissing-Jørgensen (2002) report substantial rates of entryand exit in U.S. stock ownership.

Calvet, Gonzalez-Eiras, and Sodini 433

increase in investor participation, improvements in risk-management practices, aslight increase of real interest rates, and a reduction in the risk premium.6

Second, participation plays an important role in spreading the effects of in-novation across markets. When a common factor becomes tradable, some agentsdecide to pay the entry fee and trade the new contract in order to manage their riskexposure. The new participants achieve optimal diversification by trading otherassets, which modifies the risk premia even in sectors uncorrelated to the newcontract. Furthermore, the entrants tend to have a stronger impact on the averageexposure to the factor than on other sectors of the economy. As a result, pricechanges are more pronounced for assets correlated with the new contract. Finan-cial innovation thus differentially affects distinct sectors of the economy and hasa rich impact on the cross-section of expected returns.

Third, the price changes induced by financial innovation can give rise tonon-degenerate forms of participation turnover. For instance, the introduction ofa new asset leads some agents to enter financial markets to manage their incomerisk. Other agents, however, are adversely affected by the induced changes in riskpremia and may find it optimal to stop trading. Innovation can therefore inducesimultaneous entry and exit. The paper thus helps relate participation turnoverwith the cross-sectional distribution of risk in multisector economies.

Section II introduces a tractable asset pricing model with endogenous entry.Section III demonstrates the pricing and participation effects of financial inno-vation in a one-factor model of risk exposure. Multiple factors are consideredin Section IV. Simple simulations suggest that financial innovation substantiallyreduces the equity premium, differentially spreads across security markets, andeither increases or decreases the interest rate. All proofs are given in the Ap-pendixes.

A. Review of Previous Literature

This paper builds on two strands of the asset pricing literature that have es-sentially been developed separately. First, researchers have examined how lim-ited investor participation affects the prices of a fixed set of securities. Second,the price impact of financial innovation has been examined both empirically andtheoretically without consideration of participation. The novelty of this paper isto combine these two lines of research in a simple and tractable framework.

Empirical research on stock market participation was pioneered by Blume,Crockett, and Friend (1974), Blume and Friend (1978), and King and Leape(1984). Mankiw and Zeldes (1991) report that only 28% of households ownedstocks in 1984, and that 47% of households with liquid assets in excess of $100,000held any equity.7 The fraction of households owning stocks increases with incomeand education, implying that there could be fixed information costs to participatein financial markets. Stockholder consumption is also more highly correlatedwith the market portfolio than aggregate consumption. The distinction between

6The decrease in the risk premium is reported in Blanchard (1993), Campbell and Shiller (2003),Cochrane (1997), and Fama and French (2002). Similarly, Barro and Sala-i-Martin (1990) and Hono-han (2000) document a slight increase in real interest rates over the past three decades.

7The structure of stock ownership is further analyzed by Blume and Zeldes (1993) and Bertautand Haliassos (1995).


stockholders and non-stockholders therefore helps explain the equity premiumpuzzle. The empirical validity of this mechanism is further confirmed by Vissing-Jørgensen (1997). Poterba and Samwick (1995) and Vissing-Jørgensen (1997)also document the sharp increase of stock market participation in the U.S. since1945.

These empirical findings have prompted the development of theoretical mod-els that restrict participation exogenously. Saito (1995) and Basak and Cuoco(1998) thus consider two-asset exchange economies in which the risky securityspans individual income. At low participation levels, a small number of agentsbears in equilibrium the aggregate risk of the entire population. As a result, themarket premium is high and matches the historical average under reasonable lev-els of risk aversion. Heaton and Lucas (1999) extend the analysis by consider-ing heterogeneous incomes with a common non-marketable factor. In contrastto this earlier work, we consider multiple assets and factors, and endogenize theparticipation structure by considering fixed costs to trading in financial markets.The entry-cost approach has been widely used in finance to analyze issues suchas portfolio choice (Campbell, Cocco, Gomes, and Maenhout (2001)), volatility(Pagano (1989), Allen and Gale (1994b), and Orose (1998)), futures risk premia(Hirshleifer (1988)), market size (Allen and Gale (1990), Pagano, (1993)), andthe effect of social security reform on capital accumulation (Abel (2001)). Weuse this setup to analyze how financial innovation affects investor participationand asset prices.

The paper is also related to the literature that examines the price impact offinancial innovation without consideration of participation. Conrad (1989) andDetemple and Jorion (1990) find empirically that the introduction of new batchesof options had a substantial price impact between 1973 and 1986. The effect isstronger for underlying stocks, but can also be observed for an industry index thatexcludes the optioned stock as well as for the market index. Similar empiricalevidence is available for other countries and derivative markets (e.g., Jochum andKodres (1998)).

A rich theoretical literature has also explored the impact of innovation, andthree results are most directly related to our model. First, the introduction ofa new security permits agents to better share risk and thus weakens the precau-tionary supply of savings. This leads to a higher equilibrium interest rate, whichreduces the price of all assets in fixed-participation exchange economies (Weil(1992), Elul (1997)).8 In contrast, we will show that price changes induced by in-novation can cause some investors to exit the market. Lower participation then en-courages precautionary savings and reduces the equilibrium interest rate in someeconomies.

Second, financial innovation does not affect risk pricing in standard mean-variance settings with fixed participation (Oh (1996)). Specifically in quadraticor CARA-normal economies, the price of any risky security relative to the bondis unaffected by changes in the span. For this reason, researchers have made littleuse of mean-variance models to analyze the pricing implications of innovation.9

8See Angeletos and Calvet (2000) for the analysis of production economies.9Detemple and Selden (1991) is a notable exception.


We will show that CARA-normal economies can be quite useful when the partic-ipation structure is endogenized. New assets induce the entry of new investors,which lowers the systematic consumption risk of participants and therefore in-creases the price of preexisting securities.

Third, financial innovation allows agents to reduce their exposure to id-iosyncratic or background shocks, which stimulates the demand for stocks (e.g.,Kihlstrom, Romer, and Williams (1981), Nachman (1982), Gollier (2001), andCampbell and Viceira (2002)). As a result, equilibrium risk premia tend to belower after the introduction of new assets. In our model, however, agents haveCARA utilities and reductions in background risk do not affect the demand forrisky assets. Changes in the systematic consumption risk of participants are theonly contributing factors to lower premia in our setup, which will greatly simplifythe interpretation of the results. We anticipate that our mechanism complementsand amplifies the background risk channel when individual utilities are isoelasticor more generally exhibit decreasing absolute risk aversion.

II. A Model of Endogenous Market Participation

We examine an exchange economy with two periods �t � 0� 1� and a singleperishable good. The economy is deterministic at date t � 0 and stochastic att � 1. During her life, each agent h receives an exogenous random endowmenteh � �eh

0� eh�, which corresponds for instance to a stochastic labor income. Her

preference over consumption streams �ch0��ch� is represented by a utility function

Uh�ch0��ch�. We thus adopt the two-period setup that has widely been used in the

financial innovation literature for its tractability (e.g., Allen and Gale (1994a),(1994b)). We anticipate that our model provides useful insights on the propertiesof multiperiod economies with permanent shocks.10

This paper places no restriction on the set of agents H, which can be finite orinfinite. To provide a uniform treatment, we endow the space H with a measure �that satisfies ��H�� 1. This is equivalent to viewing each element of H as a type,and the measure � as a probability distribution over all possible types.

At date t � 0, agents can exchange a finite number of real securities, whoseinitial prices and final payoffs are denominated in units of the consumption good.The financial structure is exogenous and contains two types of assets. First, agentscan trade a riskless asset costing �0 � 1�R in date t� 0 and delivering one unit ofthe good with certainty at date t�1. Note that R is the gross interest rate. Second,there also exist J risky assets �j�1� � � � � J�with price �j and random payoff aj. Weassume for simplicity that all assets are in zero net supply.11 Investors can freelyoperate in the bond market but have to pay a fixed entry cost � in order to investin one or more risky assets. This assumption is consistent with complementarities

10A large body of research shows that transitory shocks can be easily self-insured while persistentshocks have profound pricing and welfare implications in dynamic economies with incomplete mar-kets (e.g., Bewley (1977), Telmer (1993), Constantinides and Duffie (1996), Storesletten, Telmer, andYaron (1996), and Levine and Zame (2002)). The extension of our framework to the dynamic case,following Calvet (2001), is a promising direction for future research.

11A positive supply of assets could be considered by redefining individual endowment as the sumof a labor income and an exogenous portfolio of securities. This is a standard convention in assetpricing theory, as discussed in Magill and Quinzii ((1996), ch. 3).


of learning in trading activities, and the results of the paper easily generalize tomore flexible specifications of the entry cost. Investors are price takers both intheir entry and portfolio decisions, and there are no constraints on short sales. Theprices and payoffs of risky securities are conveniently stacked into the vectors �and a. Let �h � �J denote the units of risky assets bought (or sold) by investor h.We also consider the dummy variable 1��h�� 0� equal to 1 if �h �� 0, and equal to 0otherwise. The agent is subject to the budget constraints,

ch0 +

�h0

R+ � � �h + �1��h�� 0� � eh

0�

ch � eh + �h0 + a � �h�

These equations are standard, except for the presence of the entry cost in the re-source constraint at date 0. We determine the optimal choice �ch

0� ch� �h

0� �h� by

calculating the consumption-portfolio decision under entry and non-entry. Com-paring the resulting utility levels yields the optimal participation decision.

Let e0 ��

H eh0d��h� and e �

�H ehd��h� respectively denote the average

income of the entire population in each period.

Definition (Equilibrium). A general equilibrium with endogenous participation(GEEP) consists of an interest rate R, a price vector �, and a collection of optimalplans �ch

0� ch� �h

0 � �h�h�H such that

i) The good market clears in every state:�

H�ch0 + �1��h�� 0��d��h� � e0, and�

H ch��d��h� � e�� for all � � ;

ii) The asset markets clear:�

H �hj d��h� � 0 for all j � �0� � � � � J�.

In the absence of entry fee �� 0�, the definition coincides with the traditionalconcept of general equilibrium under incomplete markets (GEI). When participa-tion is costly, a GEEP differs from a GEI through two different channels. First,agents endogenously decide whether to pay the entry fee in order to trade riskyassets. Second, trading activities use some of society’s resources and crowd outprivate consumption, as seen in the market clearing condition at date t � 0. Thisphenomenon, which we call the displacement effect, probably plays a minor rolein actual economies. Extensions of our model could transfer a fraction of tradingfees to some consumers (such as exchange owners), or seek to provide a moredetailed description of the financial industry.

The existence of equilibrium is proven in Appendix A. As in the GEI case,equilibrium allocations are usually Pareto inefficient because missing markets in-duce incomplete risk sharing. With two periods and a single good, however, GEIallocations are known to satisfy a limited or constrained form of efficiency. Nosocial planner can improve the utility of all agents when income transfers areconstrained to belong to the asset span. This limited form of efficiency easilygeneralizes to our setting when the planner pays the entry fee required to use therisky asset.

Definitions (Constrained Efficiency). An allocation �ch0� c

h�h�H is feasible if:

i) For all h, there exists ��h0 � �

h� � � � �J such that ch � eh + �h0 + a � �h.

ii)�

H�ch0 + �1��h�� 0��d��h� � e0, and

�H ch��d��h� � e�� for all � � .


A feasible allocation is called constrained Pareto-efficient if no other feasible al-location makes all agents strictly better off.

Appendix A shows that a GEEP allocation is constrained Pareto-efficient. As aresult, the introduction of a non-redundant asset cannot make all agents worse off.

In order to analyze the effect of financial innovation on participation andprices, we now specialize to a tractable class of CARA-normal economies. In-vestors have identical utility of the Epstein-Zin type:

U �c0��c� � �e��c0 � �� e��c

��

where � and � are positive coefficients. The agent maximizes �e��c0 � e��c1

when she reallocates through time a deterministic income flow. On the otherhand, atemporal risky choices only depend on � e��c . When future consumptionis normally distributed, we can rewrite the utility as�e��c0 �e��EE �c�� var��c��2�.The specification reduces to standard expected utility when �� .

Individual endowments and the payoffs of risky assets are jointly normal.The securities generate a linear subspace in the set L2�� of square-integrablerandom variables. We assume without loss of generality that risky assets are cen-tered and mutually independent: �a1� � � � � aJ� � � �0� I�. Let A denote the spanof the risky assets, and A� the subspace orthogonal to all securities (including thebond). Projections play an important role in the discussion, and xV convenientlydenotes the projection of a random variable x on a subspace V .

A. Individual Entry Decision

We solve the decision problem of a given trader h by calculating the consump-tion-portfolio choice under entry and non-entry. Consider the security mA ��R��

�Jj�1 �jaj, which is determined by risk aversion and market prices. We

can easily show the following proposition.

Proposition 1 (Optimal Portfolio of Participant). A market participant buys

�h�p0 �

R1 + R

�eh0 � � eh � �� h�p +

��R��

+�

2

�var

�ehA�

�+ var

mA��

units of the bond, and �h�pj � � cov�aj� eh� � R�j�� units of risky asset j. Con-

sumption is

�ch�p � � eh + �h�p0 + mA + ehA�(1)

in the second period.

We infer from �1� that the investor exchanges the marketable component ehA ofher income risk for the tradable portfolio mA, which provides the optimal mix ofrisk and return. Because markets are incomplete, she is also constrained to bearthe undiversifiable income risk ehA� .

Investment in the riskless asset is the sum of two components, which corre-spond to intertemporal smoothing and the precautionary motive. First, the agent


uses the riskless asset to reallocate her expected income stream between the twoperiods. Note that she compensates for any discrepancy between her subjec-tive discount factor and the interest rate. Second, she saves more when futureprospects are more uncertain. As will be seen in the next section, financial inno-vation affects the precautionary component by modifying the portfolio mA and byreducing the undiversifiable income risk ehA� .

The consumption of the non-participating investor is obtained from Proposi-tion 1 by setting A � �0� and �� 0. The agent saves

�h�n0 �

R1 + R

�eh0 � � eh +

1��R� +

�

2var

eh

(2)

in the first period, and consumes ch�n � eh + �h�n0 in the second. She thus bears the

entire endowment risk in her final consumption, and her precautionary demandfor the bond depends on the whole variance of future income.

The investor makes her participation decision by comparing utility under en-try and non-entry. In the CARA-normal case, this reduces to maximizing thecertainty equivalent � �ch � � var��ch��2. We can easily check that the benefit oftrading risky assets is � var

ehA � mA

�2, while the opportunity cost is �R, lead-

ing to Theorem 1.

Theorem 1 (Entry Condition). The investor trades risky assets when�

2var

ehA � mA

�R�(3)

and is indifferent between entry and non-entry if the relation holds as an equality.

Relation �3� has a simple geometric interpretation in L2��, which is illustratedin Figure 1. The agent trades risky assets if the distance between her income riskehA and her optimal portfolio mA is larger than

�2�R��. The trader pays the

entry fee only if her initial position is sufficiently different from the optimum, asis standard in decision problems with adjustment costs.

The theorem has a natural interpretation when all agents have a positive ex-posure to certain classes of risks. Participants with low exposure to marketableshocks buy the corresponding assets to earn a risk premium; these agents arecalled speculators or investors.12 On the other hand, agents with high exposurehedge by shorting the corresponding risky assets; these agents are hedgers or is-suers. The model thus closely matches the type of risk sharing examined in thefutures literature.

B. Equilibrium

Let H denote the set of participants in the risky asset markets. Whenthe class of indifferent agents has measure zero, we can write

��h � H : � var

ehA � �mA

�2 � �R

��(4)

12Massa and Simonov (2002) document empirically the substantial impact of labor income riskon individual participation. In particular, households are less likely to invest in financial marketswhen their income has a higher correlation with a diversified stock portfolio. These findings areconsistent with entry condition (3) since the participation benefit can be rewritten as ��2 cov�m A;eh� + var�e hA� + var�m A��2.


FIGURE 1

Geometry of the Entry Condition

Risky Asset Span

hAe�

he�

~m

A

~ ~e m

hA A−

An agent with random income eh participates in risky asset markets if the distance between the tradable component ofher endowment ehA and the tradable security mA is larger than

�2�R��.

Market participants can have different income risk characteristics than the entirepopulation. We will show in Sections III and IV that this difference is a drivingelement of the model.13 While e denotes the mean income in the population, wedefine the average endowment of participants as ep �

�� ehd�p�h�, where �p is

the conditional measure ��.14

In equilibrium, the common consumption risk mA coincides with the averagetradable income risk of participants,

mA � e pA�(5)

We also establish Theorem 2.

Theorem 2 (Asset Prices). In equilibrium, an asset a is worth

��a� � �� a� � cov �ep� a�� R�(6)

The interest rate satisfies

��R � ��R0 + ��

�� +

�

2

��

varehA � e pA

d�p�h�

�(7)

where ��R0 � ��1�� + �� e� e0�� 2��

H var�eh�d��h�.

13Heaton and Lucas (2000) show the empirical validity of this distinction.14More specifically, the conditional measure � p is �� if �� 0, and identically zero

otherwise.


The participation set and asset prices are jointly determined by (4)–(7).An asset is valuable if it provides a hedge against the consumption risk of

participants. Since entry is endogenous, financial innovation can change the mar-ket endowment ep, and therefore the price ��a��R�1 � � a � � cov�ep� a� of arisky asset relative to the bond. As noted by Oh (1996), such a channel does notexist in a CARA-normal economy with a fixed set of traders: an increase in thespan then has no effect on cov�ep� a� and thus on the price of a risky asset relativeto the bond. The possible effect of financial innovation on the risk premium thuscrucially relies on the endogeneity of participation in our model.

The equilibrium interest rate R is influenced by two economic effects corre-sponding to the last two terms of (7). First, the interest rate tends to be higherwhen more first period resources �� are absorbed in the entry process. Thesecond term of (7) corresponds to the precautionary motive. The variance of indi-vidual consumption is var�epA�+var�ehA�� if an agent participates, and var�ehA�+var�ehA�� otherwise. Entry thus reduces on average the variance of consumptionby �

�

varehAd�p�h�� var

e pA

�

��

varehA � e pA

d�p�h��(8)

This term is large when many agents participate or many hedging instruments areavailable. The financial markets then permit agents to greatly reduce their riskexposure, which dampens their precautionary motive, reduces the demand for theriskless asset, and leads to an increase in the equilibrium interest rate.15

The entry condition (3) suggests that a lower fee or improved spanning tendsto encourage entry. For instance, when the cost � is infinite, no agent trades riskyassets and the equilibrium interest rate equals R0. The equilibrium set of partici-pants, however, may not increase monotonically with the financial structure. Thisis because the entry condition (3) depends on the endogenous variables ep and R.When new assets are added, a participating agent h may leave the market becausethe diversification benefit � var�ehA � �epA��2 has dropped or the opportunity costhas increased. Sections III and IV will provide examples of such behaviors.

The effect of financial innovation on the interest rate is easily predicted whenthe average endowment of participants remains constant.

Proposition 2. Financial innovation leads to a higher interest rate when the meanendowment ep is unchanged.

The proof has a straightforward intuition. A lower interest rate would reinforcethe favorable effect of financial innovation on entry and lead, by (7), to a higherinterest rate—a contradiction. Thus, if the participants’ average endowment doesnot vary, existing asset prices necessarily decrease with financial innovation.Changes in ep thus play a crucial role in determining the impact of financial inno-vation on asset prices. To better understand this mechanism, we now introduce afactor model of risk exposure.

15This equation is thus consistent with the well-known effect that financial innovation increases theinterest rate when the participation structure is exogenous (Weil (1992), Elul (1997)).


III. Economies with a Unique Risk Factor

We consider in this section an economy with a unique factor � that linearlyaffects all incomes. The endowment of each investor h is specified as

eh � � eh + �h��(9)

where �h is the individual risk loading. The model is tractable when the shockand the assets are jointly normal. Without loss of generality, we assume that � hasa standard distribution� �0� 1� and generates a non-negative average loading � inthe population.16 The cross-sectional distribution of � is specified by a measure� on the real line. To clarify the exposition, we assume that the measure � has acontinuous density f ��, whose support is the non-negative interval �0��.17

When financial markets are incomplete, existing securities span only par-tially the common shock. The projection of � on the asset span, �A�

�Ji�1 cov��A��aj��aj, is the tradable component of the factor. The corresponding variance,

� � var�A�

quantifies the insurable fraction of the risk � and thus represents a useful index ofmarket completeness. Since � has unit variance, the index � is contained between0 and 1. The values��0 and ��1 respectively correspond to the absence of riskyassets �A � �0�� and the full marketability of the shock �� A�. Intermediatevalues of� arise when agents can only trade the bond and a risky asset imperfectlycorrelated with the aggregate shock.

The portfolio �A and the index � have direct empirical interpretations. Weobtain �A by regressing the factor � on the asset payoffs. The determination coef-ficient R2 then provides an estimate of the completeness index �. This approachis easily implemented when the factor represents GDP or an aggregate shock thatis not directly tradable on organized exchanges (Roll (1977)). The introduction ofa non-redundant asset increases the index � and helps market participants hedgemore closely the risk �. Since macroeconomic variables such as GDP are oftenobserved with measurement errors and lags, improvements in national accountingcan also lead to more precise hedging and thus a higher �.

A. Participation Structure

We infer from Section II the equilibrium of the one-factor economy. Let �p

denote the average loading of participants,

� p �

��

�d�p��(10)

The participation set contains agents whose loading � is sufficiently differentfrom � p,

� �� : �� p � �� (11)

16Purely idiosyncratic shocks are ruled out in this section for expositional simplicity.17The theorems of this section are proved in the Appendix for densities f �� with arbitrary un-

bounded supports.


where��

2�R��.18 We note that� is the half-width of the non-participationregion, and henceforth call it the non-participation parameter. � increases withthe entry cost � and the interest rate R, and decreases with the completeness index� and the risk aversion �. Thus financial innovation, either in the form of a lowerentry fee or an increase in spanning, tends to decrease �. The participation set isillustrated in Figure 2. Agents � � �p + � are hedgers who trade assets in orderto reduce their risk exposure. Conversely, agents with loadings � � �p � � arespeculators who increase their consumption risk in order to earn a higher return.

FIGURE 2

Effect on Participation of a Decrease in �

Lo

adin

g D

ensi

ty

new

entr

y

IN OUT IN

p

ϕ −Λp

ϕp

ϕ Λ+

( )f ϕ

new

entr

y

ϕ

Factor Loading

Financial innovation, either in the form of improved spanning or lower transaction costs, reduces the nonparticipationparameter � �

�2�R��. When the cross-sectional distributional of risk is skewed to the left, a majority of entrants

have low risk exposure. As a result, the participants’ average loading �p and the risk premium fall.

The equilibrium calculation is simplified by the following observation: �p

is both the average loading and the midpoint of the participation set.19 As shownin Appendix B, these restrictions impose that a unique midpoint �p is consistentwith a given half-width �.

Property 1 (Participation Set). For any non-participation parameter � � 0, thereexists a unique loading �p�� satisfying conditions (10) and (11). The corre-

18Agent h trades risky assets if the participation benefit is larger than the opportunity cost:�� h � � p�2�2 � �R, or equivalently �� h � � p� � �. When assets have no correlation with therisk factor �� 0�, the participation set is empty under costly entry �� 0� and indeterminate underfree entry �� 0�.

19The loading � p satisfies� �p

��

�� p�f ��d�+��

� p+�� p�f ��d��0, which implicitlydefines � p as a function of �.


sponding participation sets � � �� ; �p�� p�� + � ; +�� arenested and decreasing in � : �� for all � � ��.

When the midpoint �p is exogenously fixed, it is clear that the sets �� ; �p �� p + � ; +�� are decreasing with the non-participation parameter �. Weshow in the Appendix that the property also holds when the midpoint�p�� variesendogenously. Furthermore, since the sets � are nested, the half-width � pro-vides a precise ordering of the participation structure. A high � corresponds to asmall set � and thus a low participation rate ��.

The midpoint �p controls the pricing of risk. Participants have average in-come ep � � ep + �p�� A + �A�� and individual random consumption,

�ch � � �ch + �p �A + ehA� �

We note that the marketable consumption risk �p � A is identical for all traders.In equilibrium, the average loading �p determines the covariance between a mar-keted security�a and individual consumption: cov��ch��a�� p cov��a�, and there-fore the risk premium.

Financial innovation, either in the form of new assets or lower entry costs,affects �p through changes in �. The impact of innovation on the average loading�p and the risk premium therefore depends on the cross-sectional distribution ofrisk, as is now shown.

Property 2 (Monotonicity of Average Loading). When the loading density verifiesthe skewness condition,

f ��p � �� f ��p + ��(12)

the average loading �p locally increases with the non-participation parameter �.

Figure 2 illustrates the mechanism underlying this key result. When � decreases,the skewness of the loading density implies that more agents enter to the left(speculators) than to the right (hedgers) of �p, which pushes down the averageconsumption loading �p.

To develop intuition, we now analyze the effect of financial innovation onthe risk premium in an economy in which the interest rate is exogenously fixed.The entry condition � �

�2�R�� expresses the non-participation parame-

ter as a function of purely exogenous quantities. A higher completeness index� (or a lower transaction cost �) reduces the half-width � and thus increasesthe participation set �. The implied movement in �p controls changes in riskpricing. Consider an asset �a that has positive correlation with the factor, and let�Ra � �a��a� denote the random (gross) return. By Theorem 2, the relative riskpremium satisfies

� �Ra � RR

��p cov��a� ��

� �a� ��p cov��a� �� The higher � (or lower �� due to financial innovation reduces the participationparameter � and induces the entry of new agents. Under skewness condition(12), a majority of the new entrants has a low exposure and reduces the covariance


between the participants’ average loading and the asset. Financial innovation thussimultaneously induces higher participation and a reduction in the risk premium.These results illustrate the role of the loading density f �� for the comparativestatics of asset prices.

B. General Equilibrium

We now extend the analysis to the case of an endogenous interest rate. Theentry condition can be rewritten

R1�� 2��2��(13)

The equilibrium of the bond market implies

R2�� R0 �� + ��var�� 2��(14)

where var��

��p�2d�p�� denotes the variance of the participants’loadings. In Figure 3, we graph these functions in the ��R� plane. An equilib-rium corresponds to the intersection of these two curves. The quadratic functionR1�� is increasing, while R2�� monotonically decreases with �. This helpsestablish Theorem 3.

Theorem 3 (Existence and Uniqueness). There exists a unique equilibrium.

Figure 3 allows us to analyze the impact of financial innovation. An increasein � pushes up both curves in the figure, implying a higher interest rate and anambiguous change in the non-participation parameter �.

Proposition 3 (Impact of Financial Innovation). The riskless rate R increases withfinancial innovation. As the completeness index � increases from 0 to 1, the set ofparticipants has two possible behaviors. It is either monotonically increasing;or there exists �� 0� 1� such that increases on �0� �� and decreases on�� 1�.

The two behaviors are illustrated in Figure 4. The ambiguous effect of financialinnovation on market participation has a simple intuition. On one hand, a higher� increases the diversification benefit ��h ��p�2�2 of trading risky assets andencourages entry. On the other hand, new assets reduce the precautionary motiveand increase the interest rate, thus discouraging participation. In empirical set-tings, we expect that the favorable effect of improved diversification, which stemsfrom risk aversion, will tend to dominate the adverse effect of the precautionarymotive.20

20The global monotonicity results of Proposition 3 are proven in Appendix B using a single crossingargument. Additional insights can be gained by examining the local sensitivity of R 1 and R1 to thecompleteness index �. Let X�� d ��X�d �� denote the elasticity of an endogenous quantity X.We infer from (13) that �� R�� 1��2. Financial innovation increases the set of participants�� 0� if it has a weak impact on the interest rate � R�� 1�. Condition (13) also implies thatthe elasticity of R2�� with respect to � increases with the dispersion of the participants’ loadingsvar��. When traders have very heterogeneous incomes, financial innovation allows agents to greatlyreduce their average consumption risk, and thus has a strong impact on the interest rate. This explainswhy participation is non-monotonic in Figure 4 for the loading density with the highest variance.


FIGURE 3

Equilibrium of the One-Factor Economy

Inte

rest

Rate

Non-Participation Parameter

R1(Λ) - Entry DecisionR

Λ

E´

EBond Market

EquilibriumR2(Λ) -

The equilibrium E of the endogenous participation economy is determined by the intersection of the curves R1�� andR2��, which respectively express the entry condition and the equilibrium of the bond market. The introduction of a newasset pushes up both curves. In the new equilibrium E, the interest rate R is higher and the shift in the nonparticipationparameter � is generally ambiguous.

The one-factor model may help explain a number of features that have char-acterized financial markets in the past three decades. New financial instrumentsencouraged investors to participate in financial markets, which led to a reductionin the precautionary motive and in the covariance between stockholder consump-tion and the aggregate shock. These two effects in turn increased the interest rateand reduced the risk premium.21 Note that this argument is consistent with earlierempirical findings. Mankiw and Zeldes (1991) thus show that the consumption ofstockholders tends to be more correlated with the market than the consumption ofnon-stockholders. As financial innovation leads more people to enter the market,the risk premium falls. We leave the empirical exploration of this mechanism tofurther research.

In this section, financial innovation consists of providing a better hedgeagainst a common risk factor. In practice, however, households and firms facemultiple sources of income shocks, and innovation often permits them to hedgeclasses of risk that had been previously uninsurable. For this reason, we nextexamine a multifactor model of risk.

21As shown in Appendix B, a higher entry cost implies a higher risk premium under condition (12).Like models with exogenously restricted participation (e.g., Basak and Cuoco (1998)), our frameworkthus helps explain the equity premium puzzle.


FIGURE 4

Effect of Financial Innovation on Market ParticipationP

arti

cip

atio

n R

ate

α

σ = 0.8

μ(P)

σ = 1

Index of Market Completeness

0.4

0.2

0.3

0.2

0.1

0.4 0.6 0.8 1

We consider a one-factor economy with a lognormal cross-sectional distribution or risk: ln�� N�0� �2�. The solidcurve corresponds to � � 08, and the dashed curve to � � 1. The other parameters of the economy are: � � � 1,� � 1, and � � 1. When the completeness index � increases from 0 to 1, participation monotonically increases in oneeconomy ��08�, and reaches a maximum for an intermediate value of � in the other ��1�. Non-monotonicity arisesfrom the larger heterogeneity of individual incomes in the second economy. Financial innovation then induces a sharpreduction in the precautionary supply of savings, which leads to a strong increase of the interest rate and the exit of someparticipants.

IV. Multifactor Economies

We now investigate an economy with a finite number of risk factors ��1� � � � ��L�, which correspond to macroeconomic or sectoral shocks affecting individualincome. For instance, �1 could be an aggregate risk, and �2� � � � � �L industry orsector-specific shocks. We specify the income of each investor h as

eh � � eh +L�

��1

�h� ��(15)

and denote by �h � ��h1� � � � � �

hL� the vector of individual loadings. The model is

tractable when the risk factors and the asset payoffs are jointly normal. Withoutloss of generality, we normalize the factors to have unit variances and no mutualcorrelation: ��1� � � � � �L� � � �0� I�. The distribution of factor loadings in thepopulation is specified by a continuous density f �� on �L .

The factors may not be fully tradable when financial markets are incom-plete. As in the previous section, it is useful to consider their projections �A

� ��Jj�1 cov �� aj� aj on the asset span. We interpret �A

� as the marketable com-ponent of factor �, which can be estimated by regressing �� on asset payoffs. Weconveniently stack the projected factors in a vector �A�� A

1 � � � � � �AL �. The covari-


ance matrix �A � var��A� is a generalized index of market completeness, whosediagonal coefficients �� var��A

� � quantify the insurable fraction of each factor.We assume for simplicity that the projected factors are mutually uncorre-

lated: cov��A� � �

Ak �� 0 for all distinct � and k. In the next subsections, this hypoth-

esis will make it more striking that the improved marketability of factor � affectsthe risk premium on an uncorrelated component �A

k . The covariance matrix is thendiagonal,

�A �

�� 1

. . .�L

�� with coefficients �� var��A

� � contained between 0 and 1. We note that �A isequal to zero when there are no assets, and to the identity matrix when marketsare complete.

The equilibrium calculation follows directly from Section II. By (15), themean endowment of participants satisfies ep � � ep +

�L��1 �

p��, where �p

� rep-resents the traders’ average exposure to factor �. The equilibrium of financialmarkets implies the relations ��a� � �� a� � cov�ep� a��R and

��R � ��R0 + ��

�� +

�

2

L�i�1

�i var��i�

��(16)

where ��R0 � ��1�� + �� e � e0� � ��2��L

i�1 ��2i �. These equations

suggest that when the utility coefficients � and ��1 are large, financial innovationgenerates both substantial variations in the pricing of risk and small movementsin the interest rate.

By entry condition (3), the participation set depends on the completenessindex of each factor,

�

�� :

�

2

L��1

�� p��

2 � �R

��(17)

When all the coefficients �� are strictly positive, the participants are locatedoutside an ellipsoid centered at �p � ��p

1� � � � � �pL�.

22 The half-widths �� 2�R�� of the ellipsoid along each axis depend on the completeness in-

dex �� and the endogenous interest rate. The proof of Theorem 4 is shown inAppendix C.

Theorem 4 (Existence and Uniqueness). There exists a unique equilibrium.

The proof begins by establishing that the vector��1� � � � � �L� define a uniqueparticipation set �. In contrast to the one-factor case, however, � can movein more than one direction and thus need not be decreasing (as a set) in eachcomponent ��. The market clearing of the bond uniquely determines the interestrate R and the half-widths ��

�2�R��. The proof also provides a use-

ful algorithm for the numerical computation of equilibrium. We now examine

22The participants are located outside a cylinder when some coefficients � � are equal to zero.


the comparative statics of participation and asset prices with respect to financialinnovation.

A. Financial Innovation and the Risk Premium

The one-factor model shows that financial innovation can reduce the riskpremium of securities correlated with the aggregate shock. In a multifactor econ-omy, the improved marketability of a factor can also have pricing effects on un-correlated assets. Consider for instance an economy with two factors: ��1 is anaggregate risk to which all investors are positively exposed, and ��2 is purely id-iosyncratic or distributional. Let a � � �a + �A

1 , ��a� 0 denote an asset or stockthat is only correlated with the aggregate risk.23 By equation (6), the stock has therelative premium,

� �Ra � RR

��p

1�1

� �a� ��p1�1

�(18)

When participation is exogenously fixed, the consumption loading �p1 is constant

and the improved marketability of the idiosyncratic shock �2 does not affect thepremium (18). In our model, however, innovation can affect the set of participants,the consumption loading �p

1, and therefore the equity premium.To further illustrate this mechanism, assume that the stock a is the only asset

initially traded ��2 �0�. By (17), non-participants have loadings �1 that are closeto the market average: �1 � �p

1 ��

2�R��1��. When the index �2 increases,some agents become willing to pay the entry cost because their exposure �2 tothe idiosyncratic risk is sufficiently different from the average �p

2. These newparticipants also trade the stock a to achieve optimal diversification. The riskpremium on a declines if a majority of the new entrants have low exposure to theaggregate shock ��1 � �p

1� and increase the demand for the stock. We expect thislogic to hold when the distribution of �1 is skewed toward the origin, consistentwith the intuition developed in the one-factor case.

A simple simulation of the cross-sectoral effect is presented in Figure 5.We assume for simplicity that exposures to the aggregate and idiosyncratic risksare independent in the population. The cross-sectional loading density is thenf ��1� �2� � f1��1�f2��2�. This hypothesis makes it perhaps more surprising thatincreased marketability of the idiosyncratic risk modifies the equity premium.We specify f2��2� to be symmetric around zero, which implies that �p

2 � 0 inequilibrium. We discuss the choice of parameters in Appendix C. The averagepayoff of the stock is selected to obtain a risk premium ��Ra � R equal to 7%before the introduction of new contracts ��2 � 0�. In the absence of a futuresmarket, the net interest rate equals 1% and the standard deviation of the stock�var��Ra��

1�2 is 15%, implying an initial Sharpe ratio of about one-half.In Figures 5A–5D, the solid lines illustrate equilibrium in the endogenous

participation economy as �2 increases from 0 to 1. The dashed lines illustratethe corresponding outcomes when participation is exogenously fixed. For com-parison purposes, the participation sets in both economies coincide when �2 � 0.

23While the assets are assumed to be in zero net supply, we easily reinterpret the model in terms ofequity by viewing endowment as the sum of a labor income and an exogenous endowment of stocks.


FIGURE 5

Comparative Statics in a Two-Factor Economy

A. Equity Premium (%) B. Interest Rate (%)

C. Volatility of Stock Return (%) D. Participation Rate (%)

7

6

5

4

3

2

1

16

15

14

13

12

75

60

45

30

15

0.25 0.5 0.75 1

Completeness Index α2

0.25 0.5 0.75 1 0.25 0.5 0.75 1

0.25 0.5 0.75 1


Completeness Index α2Completeness Index α2

Individual labor income is exposed to an aggregate shock 1 and an idiosyncratic risk 2. The aggregate shock is partiallyinsurable ��1 � 05�. The idiosyncratic risk is uncorrelated to existing assets when �2 � 0 and is fully insurable when�2 � 1. The other parameters are � � 096, � � 10, � 05, � � 08%, �1 � 4%, and �2 � 10%. The solid linesillustrate equilibrium in the endogenous participation economy as the coefficient �2 varies from 0 to 1. The dashed linesare plotted assuming a fixed set of traders. For comparison purposes, the participation sets in both economies coincidewhen �2 � 0. The effect of innovation on the risk premium is substantially stronger under endogenous entry.

Under fixed participation, innovation only has a very modest impact on the equitypremium, the interest rate, and volatility.24 In contrast, substantial movementsare observed when entry is endogenous. The risk premium on the stock declinesfrom 7% to 4.5%. Providing insurance against the idiosyncratic shock thus sub-stantially decreases the risk premium through changes in participation. Consistentwith empirical evidence, the standard deviation of the stock return is almost con-stant at 15%.25 We observe that most of the decline in the risk premium occurswhen the hedging coefficient �2 increases from 0 to 0.5. The value �2 � 0�5 alsoyields values for participation (60%) and the real net interest rate (2.5%) that arereasonable for the current U.S. economy. This suggests that the cross-sectoral

24We know from equation (18) that financial innovation has no impact on the relative risk premium�� Ra � R��R. Figure 5 shows that the interest rate R and thus the risk premium � R a � R are alsoapproximately constant in the calibrated economy when the idiosyncratic risk becomes tradable.

25Campbell, Lettau, Malkiel, and Xu (2001) document that the volatility of stock market indiceshas been stationary over the past century.


effects induced by financial innovation may be quantitatively significant.26 Weleave to further research the full empirical assessment of this mechanism.

B. Differential Effects, Participation Turnover, and Interest Rate

We now explore three additional consequences of innovation in the multi-factor model: differential changes in sectoral risk premia, simultaneous entry andexit, and a possible reduction of the interest rate.

The previous simulations assumed that the loading density f2��2� is symmet-ric around zero. Participants insure the marketable component of the idiosyncraticshock at no cost, and an asset correlated only with ��2 yields no risk premium��p

2 � 0�. We now examine an economy in which the loading density f2��2� hasa positive support and is skewed toward the origin. The risks ��1 and ��2 are inde-pendent sources of aggregate uncertainty that yield positive and possibly distinctpremia. Financial innovation can differentially affect asset prices across sectors,and thus have rich effects on the cross-section of expected returns.

The comparative statics analysis of Figure 2 easily extends to the two-factorcase. We consider a financial structure with completeness indices �1 and �2, andassume for simplicity that interest rate R is exogenous. The ellipse delimiting theparticipation set is illustrated by a solid line in Figure 6, Graph A. It is centered at�p and has half-width��

�2�R�� along each axis. Consider an increase in

the second index from �2 to ��2. Since the interest is fixed, the limiting boundaryin the new equilibrium has the same horizontal �1 but a shorter vertical half-width ��2. We represent the intermediate ellipse centered at �p with parameters�1 and��2 in dotted lines. Agents in the shaded area have smaller average loadingsthan �p

1 and �p2, and thus tend to push the new equilibrium set toward the origin.

Because these agents are more spread out vertically than horizontally, the inducedmovement in �p tends to be stronger along the vertical axis, i.e., in the direction ofinnovation. The increased marketability of the shock ��2 may thus predominantlyinfluence the risk premium in the second sector.

The new set of participants is delimited by the ellipse centered at �pnew with

half-widths �1 and ��2, as illustrated in Figure 6, Graph B. Financial innovationinduces simultaneous entry and exit. Agents in the shaded area are initially outof the market. When the new asset is introduced, these agents face lower hedgingcosts and decide to participate. Agents in the dashed area, on the other hand, areinitially investing in financial assets. Lower risk premia reduce the profitabilityof their investments and result in their leaving the market. The possibility of si-multaneous entry and exit is thus an attractive feature of the multifactor model.27

26An increase in �2 reduces the idiosyncratic or background risk of all participants. When utilitiesare isoelastic (or more generally exhibit decreasing absolute risk aversion), the reduction in back-ground risk increases the demand for the stock and thus further reduces the risk premium. We an-ticipate that this additional channel substantially amplifies the pricing effect of financial innovationin more general setups. A similar argument is made by Heaton and Lucas ((1999), pp. 237–239).Their framework, however, only considers a unique risk factor and asset, and therefore does not per-mit the distinction between background risk and aggregate shock. We anticipate that their numericalresults would be strengthened by the simultaneous reduction of trader exposure to the aggregate andidiosyncratic risks considered in this paper.

27Participation turnover also arises when the distribution of � 2 is symmetric, as shown in the proofof Proposition 4.


FIGURE 6

Effect on Participation of an Increase in �2

entry

Graph B. New Participation Set

ϕ1

ϕ2

jp

new

.

exit

Graph A. Intermediate Set

ϕ1

ϕ2

jp.

jp

new

.

entry

entry

Initial participants are located outside the large ellipse of Graph A. When �2 increases, the boundary shrinks vertically(small ellipse), and new entrants move the average loadings from �p to �p

new. The new participation set is delimited bythe small ellipse of Graph B. The dotted area contains the new entrants, and the dashed area the agents who left themarkets.

In future work, this property may prove useful to relate the cross-sectional distri-bution of risk with patterns of entry and exit in U.S. stock ownership (e.g., Hurst,Luoh, and Stafford (1998), Vissing-Jørgensen (2002)).

The differential effect is illustrated in Figure 7 on a numerical example. Themarginal densities of the factor loadings are identical log-normals. The initialeconomy has hedging coefficients �1 � �2 � �. We assume that the interest rateis endogenous and consider two fixed assets �a� � x� + ��A

� �� 1� 2� with a risk


premium of 7%. The symmetry of the economy imposes that x1 � x2. As �2 in-creases from � to 1, both risk premia fall and the effect is stronger for the secondasset. The results of the figure are almost unchanged when the net interest rate isexogenously set at 2%. The differential effect is an important feature of the multi-factor economy. It distinguishes the introduction of sector-specific securities fromchanges affecting all security markets, such as a reduction in taxes or transactioncosts.

FIGURE 7

Differential Effects of Financial Innovation


Asset 2

Asset 1

Market Premium (%)

7.0

6.5

6.0

5.5

5.0

0.40.2 0.6 0.8 1

The cross-sectional loading density is the product: f��1� �2� � g��1�g��2�, where g is the density of a lognormalvariable Z : ln Z � N��35� 1�. Note that exposures to the aggregate and idiosyncratic risks are independent in thepopulation. The other parameters of the economy are: � � 10, � 05, � � 096, and � � 08�. The initial economyhas completeness indexes �1 ��2 � 02, and contains two fixed assets a� � x + A

�� 1� 2�. As �2 increases from

0.2 to 1, both risk premia fall. The effect is stronger for the second asset, which is in the sector experiencing financialinnovation.

Multifactor economies also imply novel results for the comparative staticsof the interest rate. As discussed in Section II, the introduction of new assets in-creases risk sharing opportunities and weakens the precautionary demand for sav-ings. In models with exogenous participation, this leads to a higher equilibriuminterest rate under many specifications, including CARA-normal (Weil (1992)and Elul (1997)). In contrast, Appendix C establishes that when participation isendogenous, we have Proposition 4.

Proposition 4. The interest rate locally decreases with financial innovation insome multifactor economies.

This result has a simple geometric intuition. When new assets are introduced, themovement of �p pushes the ellipse toward a region containing a large number ofparticipants. In some economies, this effect is sufficiently strong to reduce overallparticipation and the interest rate.


V. Conclusion

This paper develops a tractable asset pricing model with incomplete marketsand endogenous participation. Agents receive heterogeneous random incomesdetermined by a finite number of risk factors. They can borrow or lend freely,but must pay a fixed entry cost to invest in risky assets. Security prices andthe participation set are jointly determined in equilibrium. The introduction ofnon-redundant assets encourages investors to participate in financial markets forhedging and diversification purposes. Under plausible conditions on the cross-sectional distribution of risk, the new entrants reduce the covariance between div-idends and trader consumption, which induces a reduction in the risk premium.

This logic is easily demonstrated in a simple one-factor model. Financialinnovation also has cross-sectoral effects in economies with multiple sources ofrisk. When a factor becomes tradable, new agents are drawn to the market in orderto manage their risk exposure. Under complementarities in learning or increasingreturns to trading activities, the new agents also become active in preexisting mar-kets and can modify the risk premia of securities uncorrelated to the factor. Thismechanism differentially affects distinct sectors of the economy and thus mayhave a rich impact on the cross-section of expected returns. Simultaneous entryand exit is another attractive feature of the multifactor model, which could beuseful for the analysis of participation turnover in economies with heterogeneousincome risks.

This paper suggests several directions for empirical research. Future workcould assess the contribution of financial innovation to the decline of the equitypremium in the past few decades. Participation changes may also help explainthe pricing effects of new derivatives reported in the empirical literature. Froma policy perspective, the mechanisms examined in this paper provide useful in-sights on current debates in public and international economics. When countriesface fixed costs to financial integration, the model implies that the creation ofnew markets can have profound pricing, participation, and welfare consequences.An extension of this work could investigate the political economy of the macromarkets advocated by Shiller and others. Further research may also evaluate gov-ernment policies affecting asset creation and participation costs, such as financialregulation, taxes, and social security reform.

Appendix A: Existence and Constrained Efficiency

Existence

We prove existence for a convex economy with a finite state space ��1� � � � � S�. and non-negative consumption sets. The utility Uh of every agentis continuous, strongly monotonic, and strictly quasi-concave on �S+1

+ . At priceswhere agents are indifferent between entry ��h �� 0� and non-entry ��h � 0�, indi-vidual demand consists of two distinct points. We avoid discontinuities in aggre-gate demand by assuming that there is a finite number of types h� 1� � � � �H, anda continuum of agents in each type.


Given q � �p0� �0� ��, let �Bh�q� denote the set of plans �c0� �0� �� satisfyingthe budget constraints p0�c0 +�1��h�� 0��+�0�0 +� � � � p0eh

0 and eh + �0 + a � � �0. The no-arbitrage set is the open cone Q � ��p0� ��a0� � � � � ��aJ�; �p0� �� S+1

++ �. Given q � Q, we can calculate the optimal excess demands Zhp�q� ��chp

0 �q� + �� eh0� �

hp0 �q�� hp�q�� and Zhn�q� � �chn

0 �q�� eh0� �

hn0 �q�� 0� under entry

and non-entry. The excess demand correspondence Zh�q� is defined as Zhd�q�when decision d � �p� n� is strictly optimal, and as the interval �Zhp�q�� Zhn�q��when the agent is indifferent between entry and non-entry. Zh�q� is homogeneousof degree 0, upper hemi-continuous and satisfies Walras’ law. Consider a vectorq �� 0 on the frontier of Q, and a sequence �qn�n�1 of elements of Q convergingto q. Following Hens (1991), it is easy to show that ��z� ; z � Zh�qn�� and ��z� ; z � NZh�qn�� as n ��, where M � �a0� � � � � aJ� and

N �

�1

M

�

Since consumption is non-negative, the set NZh�qn� � �eh is also bounded below.We conclude by standard arguments that an equilibrium exists (Hens (1991)).

Constrained Efficiency of Equilibrium

We consider an equilibrium �R� �� ch0� c

h� �h0 � �

h�h�H�, and assume that thereexists a feasible allocation �dh

0��dh�h�H such that Uh�dh

0��dh� Uh�ch

0� ch� for all h.

For every agent, there exists ��h0 � �

h� such that �dh � eh + �h0 + a � �h. Since �dh

0 ��dh�

is strictly preferred to �ch0� c

h�, it must be that dh0 + �0�

h0 + � � �h + �1��h�� 0� eh

0.We infer

�H�d

h0 + �1��h�� 0��d��h� e0, which contradicts feasibility.

Appendix B: One-Factor Economies

Participation Set (Proof of Properties 1 and 2)

For every fixed� � 0� the function G��p�� p��

� ��p�d�+� +p+�

��p�d� is continuous and strictly decreasing on �. It also satisfies �� p� G�

��p� � +� and �� p+ G��p� ��. The equation G��p� � 0 has thus aunique solution �p��. By the implicit function theorem, the function �p�� hasderivative

d�p

d�� f ��p + �� f ��p � ��

f ��p + �� + f ��p � �� + ��

which is contained between�1 and 1. We conclude that �p�� and �p��+�are respectively decreasing and increasing in �, and thus that � is decreasing.

Equilibrium Properties (Proof of Theorem 3 and Proposition 3)

1. Existence and Uniqueness. Appendix A establishes the existence of equilib-rium in an economy with finite state space and non-negative consumption. We


now prove existence and uniqueness in the one-factor CARA-normal case. Con-sider H0�� and H1��var�� . The monotonicity of � im-

plies that H0�� is decreasing in �. Similarly, the function H1�� p��

��

�p�2f ��d� +

� +p+�� p�

2f ��d� has derivative H�1�� 2H�

0�� 0.Equilibrium is determined by �13�� 14�. The function R1 is strictly increasing,and R2 is decreasing. The difference function R1�� R2�� is therefore strictlyincreasing and maps �0�+�� onto ��R2�0��+��. We conclude that there exists aunique equilibrium.

2. Impact of Financial Innovation. Using Cramer’s rule, we check that financialinnovation increases the interest rate: dR�d� 0. On the other hand, d��d� hasthe same sign as G�� 2 � ��H1�� 2. Since G�0� � 0� the function�� is decreasing on a neighborhood of ��0. We also observe that if G��2,then G�� 0. The equations G�� 2 and d��d�� 0 thus have at most onesolution on �0� 1�.

3. Impact of Entry Fee. Cramer’s rule implies that d��d� 0. A higher entrycost � thus reduces the participation set �.

Appendix C: Multifactor Economies

Existence and Uniqueness of Equilibrium (Proof of Theorem 4)

The set ��p� �� :�L

i�1��i � �pi ��i�

2 � 1� is defined for any �p ��L and �� 1� � � � � �L� � �L

++ . We observe the following.

Fact C1. For any � � �L++ , the equation

��p;��

��p�d�� 0 has a uniquesolution �p

� � �L .

Proof. We rewrite the equation as a convex optimization problem. Considerk�� ; �p� ��

�Li�1��i � �p

i ��i�2 � 1� 1��p��, where 1��p�� denotes

the indicator function of ��p� ��. Since k�� ; �p� �� is convex in �p and �has an unbounded support, the function K��p� �� 1/2�

�RR L k�� ; �p� ��d�� is

strictly convex in �p. A vector �p thus minimizes K��p� �� on �L if and only if�K��p �

��p;��p�d�� 0. Since K diverges to +� as ��p� � +�,

there exists a unique minimizer �p.

For every R 0, let ��R� denote the vector with components ��R� ��2�R��1 � � � L�. It is convenient to define �p�R� � �p

��R�, the set

�R� � ��p��R�� R��, and the function z�R� � ��R0 + �

��R�� + ��2��L

i�1 �i��i � �pi �R��

2�d��. We check that z��R� � ��1 + R��d��R��dR�,which is weakly negative by Fact C2. We conclude that the equilibrium equationz�R� � ��R has thus a unique solution.

Fact C2. The mass of participants ��R�� is a decreasing function of R.

Proof. We first show that the property holds under complete markets: �� 1 forall �. For any R, the boundary of the participation set �R� is a sphere, whichis denoted S�R�. Given two positive numbers R and Æ, Æ � R, we seek to show


that ��R�� R � Æ��. The inequality is trivially satisfied when �R� �R � Æ�. When �R� � �R � Æ�, the intersection of the spheres S�R� andS�R�Æ� is contained in a hyperplane H. Consider axes such that H is described bythe equation �1 � 0, and the center of gravity �p�R�� x� 0� � � � � 0� has a positivefirst coordinate x. We check that �p�R � Æ� has coordinates �y� 0� � � � � 0�, wherey � x. When the interest rate moves from R to R�Æ, agents in+��R�Æ��R�enter, agents in � � �R��R � Æ� exit, and agents in C � �R� � �R �Æ� participate under both interest rates (Figure 8). Since �R� � � � C and�R � Æ� � + � C, we infer that

��

�1d�� +��C

�1d�� x ��R��,

and��+�1d�� +

��C

�1d�� y ��R � Æ��. Subtracting these equalitiesimplies �

�+

�1d��

�1d�� y ��R� Æ�� x ��R��

The left-hand side of the equation is positive because + is contained in the half-space �1 0 and � is contained in the half-space �1 � 0. Since x y, weinfer that x ��R�� y ��R � Æ�� x ��R � Æ��, and conclude that��R�� R� Æ�� holds under complete markets.

When markets are incomplete, consider the linear rescaling �� and the corresponding measure ��Æ��1. Note that these transforma-

tions are independent of R. For every R 0, the rescaled set ��R� � ��R��has a spherical boundary. We conclude that ��R�� R�� is decreasingin R.

Numerical Simulation

This subsection presents the microeconomic framework used in the simula-tion of Figure 5. Individual random income is specified as�eh�eh

0�1+�1��1�+�h2��2.

The individual loading �h1 � �1eh

0 0 is therefore proportional to expected in-come.28 The aggregate endowment in period 1 satisfies �e� e0�1 + �1��1�. Withoutloss of generality, mean income is normalized to unity: e0 � 1.

We specify the cross-sectional distribution of income to be lognormal: �� eh0 �

� ��z� �2z �. Since mean income is normalized to 1, �z and �2

z satisfy the restric-tion �z + �2

z �2 � 0. We choose �z � �0�25, which corresponds to a reasonableGini coefficient of 0�4. The standard deviation of aggregate income growth �1 isset at 0�04. Since �h

1 � �1eh0, the loading density f1��1� is now fully specified.

The loading density f2��2� is assumed to be a centered Gaussian � �0� �22�

with standard deviation �2 � 0�10. The discount factor is � 0�96. Since e0 � 1�the coefficients � and ��1 coincide with relative risk aversion and intertemporalelasticity at the mean endowment point. We choose � � 10 and ��1 � 2. Thefraction of mean income used in the entry process is set equal to �� 0�8%.

The aggregate shock is partially tradable. We choose �1 � 0�5, which isroughly consistent with the correlation between the NYSE value-weighted returnand the permanent aggregate labor income shock reported in Campbell, Cocco,Gomes, and Maenhout (2001). The stock is a traded asset of the form �a�x+��A

1 . We

28We assume for simplicity that there is no expected growth between the two periods.


FIGURE 8

Geometry of the Participation Sets

ϕ1

ϕp(R)

P-

H

xy

ϕp(R-d )

P+

Pc

When the interest rate falls from R to R � Æ, the average loading moves from �p�R� to �p�R � Æ�. Agents in �+ enterthe market, agents in �� exit, and agents in �c trade in both equilibria.

select x to obtain a risk premium ��Ra �R equal to 7% before financial innovation��2 � 0�.

Proof of Proposition 4

We provide an example in an economywith two uncorrelated factors ��1� �2�.Letting Æ � 0�01� we consider �A � ��2� 0�, �B � �1 + Æ� 0�, �C � �2� 0�, �� 0��1+Æ�, and �+��0� 1�Æ�, with respective weights mA�mB�1�5�mC�1�10�m+ � m� � 1�4. The other parameters of the economy are � � �� 0�7, �� 0�3,e0 � � �e � 1� �2 � 0�9.

A straightforward extension of Theorem 4 implies that a unique equilibriumexists for any �1. When �1 �0�55, we check that the participation set contains allthe agents of type A, C, +, and �, and the net interest rate is approximately 7�9%.On the other hand when �1 � 0�9, the participation set contains all the agents oftype A, B, and C, and the net interest rate has fallen to 5.7%.


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