SPECULATION AND RISK SHARING WITH NEW

FINANCIAL ASSETS

Alp Simsek

March 13, 2013

Abstract

I investigate the eect of nancial innovation on portfolio risks when traders have belief disagree-

ments. I decompose tradersaverage portfolio risks into two components: the uninsurable variance,

dened as portfolio risks that would obtain without belief disagreements, and the speculative variance,

dened as portfolio risks that result from speculation. My main result shows that nancial innovation

always increases the speculative variance through two distinct channels: by generating new bets and

by amplifying tradersexisting bets. When disagreements are large, these eects are su ciently strong

that nancial innovation increases average portfolio risks, decreases average portfolio comovements, and

generates greater speculative trading volume relative to risk sharing volume. Moreover, a prot seeking

market maker endogenously introduces speculative assets that increase average portfolio risks. JEL

Codes: G11, G12, D53

Keywords: nancial innovation, risk sharing, belief disagreements, speculation, portfolio risks, port-

folio comovements, trading volume, hedge-more/bet-more

Massachusetts Institute of Technology and NBER (e-mail: asimsek@mit.edu). I am grateful to DaronAcemoglu, Andrei Shleifer, Jeremy Stein and ve anonymous referees for numerous helpful comments. I alsothank Malcolm Baker, John Campbell, Eduardo Davila, Darrell Du e, Douglas Gale, Lars Hansen, BengtHolmstrom, Sam Kruger, Muhamet Yildiz and the seminar participants at Berkeley HAAS, Central EuropeanUniversity, Dartmouth Tuck, Harvard University, MIT, Penn State University, Sabanci University, University ofHouston, University of Maryland, University of Wisconsin-Madison, Yale SOM; and conference participants atAEA meetings, Becker Friedman Institute, Julis-Rabinowitz Center, Koc University, and New York Universityfor helpful comments.

1 Introduction

Financial markets in recent years have seen a proliferation of new nancial assets such as dier-

ent types of futures, swaps, options, and more exotic derivatives. According to the traditional

view of nancial innovation, these assets facilitate the diversication and the sharing of risks.1

However, this view does not take into account that market participants might naturally dis-

agree about how to value nancial assets. The thesis of this paper is that belief disagreements

change the implications of nancial innovation for portfolio risks. In particular, market par-

ticipantsdisagreements naturally lead to speculation, which represents a powerful economic

force by which nancial innovation increases portfolio risks.

An example is oered by the recent crisis. Assets backed by pools of subprime mortgages

(e.g., subprime CDOs) became highly popular in the run-up to the crisis. One role of these

assets is to allocate the risks to market participants who are best able to bear them. The

safer tranches are held by investors that are looking for safety (or liquidity), while the riskier

tranches are held by nancial institutions who are willing to hold these risks at some price.

While these assets (and their CDSs) should have served a stabilizing role in theory, they became

a major trigger of the crisis in practice, when a fraction of nancial institutions realized losses

from their positions. Importantly, the same set of assets also generated considerable prots for

some market participants, which suggests that at least some of the trades on these assets were

speculative.2 What becomes of the risk sharing role of new assets when market participants

use them to speculate on their dierent views?

To address this question, I analyze the eect of nancial innovation on portfolio risks in

a model that features both the risk sharing and the speculation motives for trade. Traders

with income risks take positions in a set of nancial assets, which enables them to share and

diversify some of their background risks. However, traders have belief disagreements about

asset payos, which induces them to take also speculative positions. I assume traders have

mean-variance preferences over net worth. In this setting, a natural measure of portfolio risk

for a trader is the variance of her net worth. I dene the average variance as an average

1Cochrane (2001) summarizes this view as follows: Better risk sharing is much of the force behind nancialinnovation. Many successful new securities can be understood as devices to more widely share risks.

2Lewis (2010) provides a detailed description of investors that took a short position on housing related assetsin the run-up to the recent crisis.

1

of this risk measure across all traders. I further decompose the average variance into two

components: the uninsurable variance, dened as the variance that would obtain if there were

no belief disagreements, and the speculative variance, dened as the residual amount of variance

that results from speculation. I model nancial innovation as an expansion of the set of assets

available for trade. My main result characterizes the eect of nancial innovation on each

component of the average variance. In line with the traditional view, nancial innovation

always decreases the uninsurable variance because new assets increase the possibilities for

risk sharing. Theorem 1 shows that nancial innovation also always increases the speculative

variance. Moreover, when belief disagreements are su ciently large, this eect is su ciently

strong that nancial innovation increases the average variance.

My analysis identies two distinct channels by which nancial innovation increases portfolio

risks. First, new assets generate new risky bets because traders might disagree about their

payos. Second, and more subtly, new assets also amplify tradersrisky bets on existing assets.

The intuition behind this channel is the hedge-more/bet-more eect. To illustrate this eect,

consider the following example of two currency traders taking positions in the Swiss Franc and

the Euro. Traders have dierent views about the Franc but not the Euro, perhaps because they

disagree about the prospects of the Swiss economy but not the Euro zone. First suppose traders

can only take positions on the Franc. In this case, traders do not take too large speculative

positions because the Franc is aected by several sources of risks some of which they dont

disagree about. Traders must bear all of these risks which makes them reluctant to speculate.

Next suppose the Euro is also introduced for trade. In this case, traders complement their

positions in the Franc by taking opposite positions in the Euro. By doing so, traders hedge

the risks that also aect the Euro, which enables them to take purer bets on the Franc. When

traders are able to take purer bets, they also take larger and riskier bets. Consequently, the

introduction of the Euro in this example increases portfolio risks even though traders do not

disagree about its payo.

My analysis also has implications for portfolio comovements and trading volume. Risk

sharing typically increases portfolio comovements (see, for instance, Townsend 1994). In con-

trast, speculation tends to reduce comovements since traders with dierent beliefs tend to take

opposite positions. Consequently, as formalized in Proposition 1, nancial innovation that

2

increases average portfolio risks also decreases average portfolio comovements. Another pre-

diction of the traditional risk sharing view is that trading volume is mainly driven by traders

portfolio rebalancing or liquidity needs. In contrast, trading volume in my model is partly

driven by speculation. Moreover, Proposition 2 illustrates that (under appropriate assump-

tions) nancial innovation generates greater speculative trading volume relative to risk sharing

volume whenever it increases average portfolio risks. A growing literature has invoked belief

disagreements to explain the large trading volume observed in nancial markets (see Hong and

Stein 2007). My analysis suggests that this literature is also making implicit predictions about

portfolio risks. If speculation is the main source of trading volume in some new assets, then

those assets are likely to increase portfolio risks.

The results so far take the new assets as exogenous. In practice, new assets are endogenously

introduced by agents with prot incentives. A sizeable literature emphasizes risk sharing as a

major driving force in endogenous nancial innovation (see Allen and Gale 1994; Du e and

Rahi 1995). A natural question is to what extent the risk sharing motive for nancial innovation

is robust to the presence of belief disagreements. I address this question by introducing a

prot seeking market maker that innovates new assets for which it subsequently serves as the

intermediary. The market makers expected prots are proportional to traderswillingness to

pay to trade the new assets. Thus, tradersspeculative trading motive, as well as their risk

sharing motive, creates innovation incentives. Theorem 2 shows that, when belief disagreements

are su ciently large, the endogenous assets maximize the average variance among all possible

choices. Intuitively, the market maker innovates speculative assets that enable traders to bet

most precisely on their disagreements, completely disregarding the risk sharing motive.

A natural question concerns the normative implications of these results. In the baseline

setting, nancial innovation generates a Pareto improvement even when it increases portfolio

risks. However, this conclusion can be overturned in two natural variants: One in which belief

disagreements emerge from psychological distortions, and another one in which the environment

is associated with externalities. In both settings, a measure of tradersaverage portfolio risks

emerge as the central object in welfare analysis, providing some normative content to Theorems

1 and 2. That said, I do not make any policy recommendations since nancial innovation might

3

also impact welfare through various other channels not captured in this model.3

The rest of the paper is organized as follows. The next subsection discusses the related

literature. Section 2 introduces the basic environment and uses a simple example to illustrate

the main channels. Section 3 characterizes the equilibrium, presents the main result, and

discusses its implications for portfolio comovements and trading volume. Section 4 presents

the results about endogenous nancial innovation. Section 5 discusses the welfare implications

and Section 6 concludes. The appendix contains the omitted derivations and proofs.

1.1 Related Literature

My paper belongs to a sizeable literature on nancial innovation and security design (see, in

addition to the above-mentioned papers, Van Horne 1985; Miller 1986; Ross 1988; Merton

1989; Cuny 1993; Demange and Laroque (1995); Du e and Jackson 1989; Demarzo and Du e

1999; Athanasoulis and Shiller (2000, 2001); Tufano 2003). This literature, with the exception

of a few recent papers (some of which are discussed below), has not explored the implications

of belief disagreements for nancial innovation. For example, in their survey of the literature,

Du e and Rahi (1995) note that one theme of the literature, going back at least to Working

(1953) and evident in the Milgrom and Stokey (1982) no-trade theorem, is that an exchange

would rarely nd it attractive to introduce a security whose sole justication is the opportunity

for speculation.The results of this paper show that this observation does not apply if traders

belief dierences reect their disagreements as opposed to private information.

My paper is most closely related to the work of Brock, Hommes, and Wagener (2009), who

also emphasize the hedge-more/bet-more eect and identify destabilizing aspects of nancial

innovation. The papers are complementary in the sense that they use dierent ingredients,

and they focus on dierent aspects of instability. First, their main ingredient is reinforcement

3Among other things, even pure speculation driven by nancial innovation can provide some social benets bymaking asset prices more informative. On the other hand, there is also a large literature that emphasizes variousadditional channels by which nancial innovation can reduce welfare. Hart (1975) and Elul (1994) show thatnew assets that only partially complete the market may make all agents worse o in view of general equilibriumprice eects. Stein (1987) shows that speculation driven by nancial innovation can reduce welfare throughinformational externalities. I abstract away from these channels by focusing on an economy with single good(hence, no relative price eects) and heterogeneous prior beliefs (hence, no information). More recently, Rajan(2006), Calomiris (2008), and Korinek (2012) argue that nancial innovation might exacerbate agency problems;Gennaioli, Shleifer and Vishny (2012) emphasize neglected risks associated with new assets; and Thakor (2012)emphasizes the unfamiliarity of new assets. The potentially destabilizing role of speculation is also discussed inStiglitz (1989), Summers and Summers (1991), Stout (1995), and Posner and Weyl (2013).

4

learning (that is, traders choose their beliefs according to a tness measure). In contrast, my

analysis applies regardless of how beliefs and disagreements are formed. Second, they focus on

the dynamic instability of prices. I focus on portfolios instead of prices; more specically, my

notion of instability is an increase in portfolio risks.

The hedge-more/bet-more eect also appears in Dow (1998), who analyzes nancial in-

novation in the context of market liquidity with asymmetric information. He considers the

introduction of a new asset that makes arbitrage less risky. In view of the hedge-more/bet-

more eect, this induces arbitrageurs to trade more aggressively. The main result is that more

aggressive arbitrage could then reduce welfare by exacerbating adverse selection. In contrast, I

analyze the eect of the hedge-more/bet-more eect on portfolio risks, and I show that nancial

innovation can increase these risks even in the absence of informational channels.

Other closely related papers include Weyl (2007) and Dieckmann (2009), which emphasize

that increased trading opportunities might increase portfolio risks when traders have distorted

or dierent beliefs. Weyl (2007) notes that cross-market arbitrage might create risks when

investors have mistaken beliefs. Dieckmann (2009) shows that rare-event insurance can increase

portfolio risks when traders disagree about the frequency of these events. The contribution of

my paper is to systematically characterize the eect of nancial innovation on portfolio risks

for a general environment with belief disagreements and mean-variance preferences. I also

analyze endogenous nancial innovation and show that it is partly driven by the speculation

motive for trade. In recent work, Shen, Yan, and Zhang (2013) emphasize that endogenous

nancial innovation is also directed towards mitigating traderscollateral constraints.

Finally, there is a large nance literature which analyzes the implications of belief disagree-

ments for asset prices or trading volume. A very incomplete list includes Lintner (1969), Rubin-

stein (1974), Miller (1977), Harrison and Kreps (1978), Varian (1989), Harris and Raviv (1993),

Kandel and Pearson (1995), Zapatero (1998), Chen, Hong and Stein (2003), Scheinkman and

Xiong (2003), Geanakoplos (2010), Cao (2011), Simsek (2013), Kubler and Schmedders (2012).

The main dierence from this literature is the focus on the eect of belief disagreements on

the riskiness of tradersportfolios, rather than the volatility or the level of prices.

5

2 Basic Environment and Main Channels

Consider an economy with a single period and a single consumption good which will be referred

to as a dollar. The uncertainty is captured by a k1 vector of random variables, v = (v1; ::; vk)0.There are a nite number of traders denoted by i 2 I. Each trader is endowed with ei dollars.She also receives an additional wi (v) = (Wi)

0 v dollars, where Wi is a k 1 vector, whichcaptures her background risks. The presence of background risks generate the risk-sharing

motive for trade in this economy. The following assumption about tradesbeliefs generates an

additional speculation motive for trade.

Assumption (A1). Trader is prior belief for v has a Normal distribution, N (vi ;v), where

vi 2 Rk is the mean vector and v is the k k positive denite covariance matrix. Tradersagree to disagree in the sense that each trader knows all other tradersbeliefs.

The rst part of the assumption says that traders can potentially have dierent beliefs

about the mean of the risks, v. Traders are also assumed to agree on the variance. This feature

ensures closed form solutions but otherwise does not play an important role. The important

ingredient is that traders have dierent beliefs about asset valuations; whether these dierences

come from variances or means is not central.4 The second part of assumption (A1) ensures

that belief dierences correspond to disagreements as opposed to tradersprivate information.

This part circumvents the well known no-trade theorems (e.g., Milgrom and Stokey 1982).

A growing literature suggests that disagreements of this type can explain various aspects of

nancial markets (see Hong and Stein 2007).

Traders in this economy cannot directly take positions on the underlying risks, v. But they

can do so indirectly by trading a nite number of risky nancial assets, j 2 J . Asset j paysaj (v) =

Aj0v dollars, where Aj is a k 1 vector. Let A = Aj

j2J denote the k jJ jmatrix of asset payos, which is assumed to have full rank (so that assets are not redundant).

These assets can be thought of as futures whose payos are linear functions of their underlying

assets or indices. However, the economic insights generalize to non-linear derivatives (such as

options) and more exotic new assets.

4 In a continuous time setting with Brownian motion, Bayesian learning would immediately reveal the ob-jective volatility of the underlying process. In contrast, the mean is much more di cult to learn, lending someadditional credibility to assumption (A1). In view of this observation, the common belief for the variance, v,can also be reasonably assumed to be the same as the objective variance of v.

6

Each asset j is in xed supply normalized to 0, and it is traded in a competitive market at

price pj . Each trader can take unrestricted positive or negative positions on this asset denoted

by xji 2 R. The trader invests the rest of her endowment in cash, which is supplied elasticallyand delivers a gross return normalized to 1. The traders net worth can then be written as:

ni = ei +W0iv + x

0i

A0v p . (1)

Here, the vectors, p =p1; ::; pjJ j

0and xi =

x1i ; ::; x

jJ ji

0, respectively denote the prices and

the traders positions for all risky assets. The trader maximizes subjective expected utility

over net worth. Her utility function takes the CARA form. Since the underlying risks, v, are

Normally distributed, the traders optimization reduces to the usual mean-variance problem:5

maxxi

Ei [ni] i2vari [ni] . (2)

Here, i denotes the traders absolute risk aversion coe cient, while Ei [] and vari [] respec-tively denote the mean and the variance of the traders portfolio according to her beliefs.

The equilibrium in this economy is a collection of risky asset prices, p, and tradersportfo-

lios, fxigi, such that each traders portfolio solves problem (2) and prices clear asset markets,that is,

Pi x

ji = 0 for each j 2 J . I capture nancial innovation as an expansion of the set of

traded assets. Before I turn to the general characterization, I use a simple example to illustrate

the main eects of nancial innovation on portfolio risks.

2.1 An illustrative example

Suppose there are two traders with the same coe cients of risk aversion, i.e., I = f1; 2gand 1 = 2 . The uncertainty is captured by two uncorrelated random variables, v1; v2.Tradersbackground risks depend on a combination of the two random variables. Moreover,

5The results of this paper apply as long as tradersportfolio choice can be reduced to the form in (2) over networth. An important special case is the continuous-time model in which traders have time-separable expectedutility preferences (which are not necessarily CARA), and asset returns and background risks follow diusionprocesses. In this case, the optimization problem of a trader at any date can be reduced to the form in (2) (seeIngersoll 1987). The only caveat is that tradersreduced form coe cients of absolute risk aversion, figi, areendogenous. Hence, in the continuous time setting, the results apply at a trading date conditional on figi.

7

they are perfectly negatively correlated with one another, that is:

w1 = v and w2 = v, where v = v1 + v2 for some 6= 0.

As a benchmark suppose traders have common beliefs about both v1 and v2 given by the

distribution, N (0; 1). In this benchmark, rst consider the case in which there are no assets,

i.e., J = ;. In this case, there is no trade and tradersnet worths are given by:

n1 = e1 + v and n2 = e2 v.

Tradersnet worths are risky because they are unable to hedge their background risks. Next

suppose a new asset is introduced whose payo is perfectly correlated with tradersbackground

risks, a1 = v. In this case, tradersequilibrium portfolios are given by x11 = 1 and x12 = 1(and the equilibrium price is p1 = 0). Tradersnet worths are constant,

n1 = e1 and n2 = e2:

Thus, with common beliefs, nancial innovation enables traders to hedge and diversify their

idiosyncratic risks. This is the textbook view of nancial innovation.

Next suppose traders have belief disagreements about some of the uncertainty in this econ-

omy. In particular, traders have common beliefs for v2 given by the distribution, N (0; 1).

They also know that v1 and v2 are uncorrelated. However, they disagree about the distribu-

tion of v1. Trader 1s prior belief for v1 is given by N ("; 1) while trader 2s belief is given by

N ("; 1). The parameter " captures the level of the disagreement. I next use this specicationto illustrate the two channels by which nancial innovation increases portfolio risks.

Channel 1: New assets generate new bets

Suppose asset 1 is available for trade. Since traders disagree about the mean of v1, they also

disagree about the mean of the asset payo, a1 = v1 + v2. In this case, it is easy to check

that tradersportfolios are:

8

x11 = 1 +"

1

1 + 2and x12 = 1 +

"

1

1 + 2. (3)

In particular, traderspositions deviate from the common belief benchmark in view of their

disagreement. Their net worths can also be calculated as:

n1 = e1 +"

v1 + v21 + 2

and n2 = e2 "

v1 + v21 + 2

.

If " > 1 + 2

, then tradersnet worths are riskier than the case in which no new asset

is introduced. Intuitively, trader 1 is so optimistic about the assets payo that she takes a

positive net position, even though her background risk covaries positively with the asset payo.

Consequently, the new asset increases portfolio risks by generating a new bet.

Channel 2: New assets amplify existing bets

Next suppose a second asset with payo, a2 = v2, is also available for trade. Note that traders

do not disagree about the payo of this asset. Nonetheless, this asset also increases portfo-

lio risks through the hedge-more/bet-more eect. To see this, consider tradersequilibrium

portfolios given by:

x11; x

21

=h1 + "

;"

iand

x12; x

22

=h1 "

; "

i, (4)

Note that traderspositions on asset 1 deviate more from the common belief benchmark relative

to the earlier single asset setting. Tradersnet worths are also riskier and given by:

n1 = e1 +"

v1 and n2 = e2 "

v.

Intuitively, asset 1 by itself provides the traders with only an impure bet on the risk, v1, because

its payo also depends on the risk, v2, on which traders do not disagree. To take speculative

positions, traders must also hold these additional risks, which makes them reluctant to bet

[captured by the 11+2

term in Eq. (3)]. When asset 2 is also available, traders take positions

on both assets to take a purer bet on the risk, v1 [cf. Eq. (4)]. When traders are able to purify

9

their bets, they also amplify them, which in turn leads to greater portfolio risks.

3 Financial Innovation and Portfolio Risks

I next characterize the equilibrium and decompose tradersportfolio risks into the uninsurable

and the speculative variance components. The main result, presented later in this section,

describes the eect of nancial innovation on these two components.

Combining Eqs. (1), (2) and assumption (A1), the traders portfolio choice problem can

be written as:

maxxix0i (i p)

1

2i2x0ii + x

0ixi

.

Here, i A0vi (which is a jJ j 1 vector) denotes the traders beliefs for means of assetpayos; A0vA (which is a jJ j jJ j matrix) denotes the variance of asset payos; andi = A

0vWi (which is a jJ j 1 vector) denotes the covariance of asset payos with thetraders background risk. Solving this problem, the traders demand for risky assets is given

by: xi = 1

1i

(i p)i. Aggregating over all traders and using market clearing, asset

prices are given by:

p =1

jIjXi2I

ii i

, (5)

where Pi2I 1= jIj1 is the harmonic mean of tradersabsolute risk aversion coe cients.Intuitively, an asset commands a higher price if traders are on average optimistic about its

payo, or if it on average covaries negatively with tradersbackground risks.

Using the prices in (5), the traders equilibrium portfolio is also solved in closed form as:

xi = xRi + x

Si , where x

Ri = 1~i and xSi = 1

~ii, (6)

and ~i = i

i

1

jIjX{2I

{, (7)

~i = i 1

jIjX{2I

{{. (8)

Here, (loosely speaking) the expression, ~i, captures the covariance of the traders background

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risk relative to the average trader, and the expression, ~i, captures her relative optimism. Note

that the traders portfolio has two components. The rst component, xRi , is the portfolio that

would obtain if there were no belief disagreements (i.e., if ~i = 0 for each i). Hence, I refer

to xRi as the traders risk sharing portfolio. The optimal risk sharing portfolio is determined

by tradersbackground risks and their risk tolerances. The second component, xSi , captures

tradersdeviations from this benchmark in view of their belief disagreements. Hence, I refer

to xSi as the speculative portfolio.

Eqs. (5) (8) complete the characterization of equilibrium in this economy. The maingoal of this paper is to analyze the eect of nancial innovation on portfolio risks. Given the

mean-variance framework, a natural measure of portfolio risk for a trader i is the variance of

her net worth. I consider an average of this measure across all traders, the average variance,

dened as follows:

=1

jIjXi2I

ivar (ni) =

1

jIjXi2I

i

W0i

vWi + 2x0ii + x

0ixi

. (9)

Note that the portfolio risk of a trader is calculated according to traders (common) belief

for the variance, v. Note also that traders that are relatively more risk averse are given a

greater weight in the average. Intuitively, this weighting captures the risk sharing benets

from transferring risks from traders with high i to those with low i.

I use as my main measure of average portfolio risks for two reasons. First, Section 5 shows

that is a measure of welfare in this economy when traderswelfare is calculated according

to a common belief (as opposed to their own heterogeneous beliefs). The second justication

is provided by the following result.

Lemma 1. The risk sharing portfolios,xRii, minimize the average variance, , among all

feasible portfolios:

minfxi2RjJjgi

s.t.Xi

xi = 0.

Note that, absent belief disagreements, the risk sharing portfolios are the same as equilibrium

portfolios [cf. Eq. (6)]. Thus, Lemma 1 implies that the equilibrium portfolios minimize

the average variance when traders have the same beliefs. Thus, it is natural to take as

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the measure of risks, and to characterize the extent to which it deviates from the minimum

benchmark when traders have belief disagreements.

To this end, I let R denote the minimum value of problem in Lemma 1 and refer to it as

the uninsurable variance. I also dene S = R and refer to it as the speculative variance.This provides a decomposition of the average variance into two components, = R + S .

The appendix characterizes the two components in terms of the exogenous parameters as:

R =1

jIjXi2I

i

W0i

vWi ~0i1~iand S =

1

jIjXi2I

i

~ii

01

~ii

. (10)

Intuitively, the uninsurable variance is lower when the assets provide better risk sharing op-

portunities, captured by larger ~i (in vector norm), whereas the speculative variance is greater

when the assets feature greater belief disagreements, captured by larger ~i.

3.1 Comparative Statics of Portfolio Risks

I next present the main result. Let zJ^denote the equilibrium variable z when the set of

assets is given by J^ .

Theorem 1 (Financial Innovation and Portfolio Risks). Let JO and JN denote respectively

the sets of old and new assets.

(i) Financial innovation always reduces the uninsurable variance, that is:

R (JO [ JN ) R (JO) 0.

(ii) Financial innovation always increases the speculative variance, that is:

S (JO [ JN ) S (JO) 0.

(iii) Suppose tradersbeliefs are given by vi;D = mv0 +Dm

vi for all i, where m

v0 is a vector

that captures the average belief, fmvi gi are vectors that satisfyP

imvi = 0, and D 0 is a

parameter that scales belief disagreements. Suppose also that the inequality in part (ii) is strict

for some D > 0. Then, there exists D 0 such that nancial innovation strictly increases theaverage variance, (JO [ JN ) > (JO), if and only if D > D.

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The rst part, which is a corollary of Lemma 1, formalizes the traditional view of nancial

innovation by establishing that new assets always provide some risk sharing benets. The sec-

ond part establishes that speculation creates a second force that always pushes in the opposite

direction. The third part shows that, when belief disagreements are large, the speculation force

is su ciently strong that nancial innovation increases average portfolio risks.

I next provide a sketch proof for the second part, which is useful to develop the intuition

for the main result. Consider an economy which is identical to the original economy except

that there are no background risks (i.e.,Wi = 0 for all i 2 I), so that the only motive for tradeis speculation. The proof in the appendix shows that the average variance in this economy is

identical to the speculative variance in the original economy. Thus, it su ces to show that

nancial innovation increases average portfolio risks in the hypothetical economy.

Recall that the Sharpe ratio of a portfolio is dened as the expected portfolio return in

excess of the risk-free rate (which is normalized to 0) divided by the standard deviation of the

portfolio return. Traders in the hypothetical economy perceive positive Sharpe ratios because

they think various assets are mispriced. Dene a traders speculative Sharpe ratio as the Sharpe

ratio of her equilibrium portfolio in this economy. Using Eqs. (5) (8), this is given by:

SharpeSi =

xSi0

(i p)qxSi0

xSi

=q~0i1~i.

Next consider the traders portfolio return given by ni=ei (where recall that ei is the traders

initial endowment). The standard deviation of this return can also be calculated as:

Si 1

ei

qxSi0

xSi =1

iei

q~0i1~i.

Note that the ratio, iei, provides a measure of trader is coe cient of relative risk aversion.

Thus, combining the two expressions above gives the familiar result that the standard deviation

of the portfolio return is equal to the Sharpe ratio of the optimal portfolio divided by the

coe cient of relative risk aversion (see Campbell and Viceira 2002). Intuitively, if a trader

nds a risky portfolio with a higher Sharpe ratio, then she exploits this opportunity to such

an extent that she ends up with greater portfolio risks.

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The main result can then be understood from the lenses of this textbook result. Financial

innovation increases each traders speculative Sharpe ratio because it expands tradersbetting

possibilities frontier. That is, when the asset set is JO [ JN , traders are able to make all thespeculative trades they could make when the asset set is JO, and some more. Importantly,

new assets expand the betting frontier through the two channels emphasized before. First,

new assets generate new bets, which creates higher expected returns (thereby increasing the

numerator of the speculative Sharpe ratio). Second, new assets also enable traders to purify

their existing bets (thereby reducing the denominator of the speculative Sharpe ratio). Once a

trader obtains a higher speculative Sharpe ratio, she also undertakes greater speculative risks,

providing a sketch proof for the main result.

3.2 Comparative Statics of Portfolio Comovements

Although Theorem 1 focuses on portfolio risks, it also naturally has implications for portfolio

comovements. One signature of eective risk sharing is that tradersnet worths or consumption

tend to comove (see Townsend 1994). In contrast, speculation naturally reduces comovements

since traders with dierent beliefs tend to take opposite positions. To the extent that nancial

innovation facilitates speculation, it could also decrease comovements. The next result makes

this intuition precise for the case in which traders share the same risk aversion coe cients. To

state the result, let = 1jIj2jIjP

(i12I;i22I j i1 6=i2) cov (ni1 ; ni2) denote the average covariance

between two randomly selected tradersnet worths.6

Proposition 1 (Financial Innovation and Portfolio Comovements). Suppose i for eachi. Then, nancial innovation strictly increases the average variance, (JO [ JN ) > (JO), ifand only if it strictly decreases the average covariance, (JO [ JN ) < (JO).

This result follows from the observation that the variance of aggregate net worth,

varP

i2I ni, is a linear combination of the average variance, , and the average covari-

ance, . Since this is an endowment economy, the variance of aggregate net worth is constant.

6When traders dier in their risk aversion, even pure risk sharing might reduce comovements. To see this,consider an example with two traders with 1 = 0 < 2 who have the same beliefs (so there is no speculation).Suppose traders background risks are the same: w1 = w2 = v1 + v2, where v1; v2 are uncorrelated randomvariables. Consider the introduction of an asset with payo a1 = v1. Then, the equilibrium features therisk-neutral trader holding all of the risk, v1, even though this reduces the covariance of tradersnet worths.

14

Therefore, an increase in is associated with a decrease in .

Theorem 1 and Proposition 1 together predict that, when belief disagreements are large,

nancial innovation increases portfolio risks and decreases portfolio comovements. These re-

sults can be tested directly in some contexts in which tradersnet worths can be measured.

For example, some Scandinavian countries collect data on individualswealth. The changes in

the riskiness and comovement of traderswealth can then be linked to measures of nancial

innovation. In other contexts, it might be possible to obtain close proxies of net worth. For

instance, if traders consumption can be measured, then changes in consumption risks and

comovements can be used to assess the net eect of nancial innovation.

3.3 Comparative Statics of Trading Volume

A notable feature of new nancial assets is the large trading volume they generate. Theorem 1

also has implications for trading volume. This is because tradersportfolios are closely related

to their portfolio risks as follows:

R =1

jIjXi2I

i

W0i

vWi xRi0

xRi

and S =

1

jIjXi2I

i

xSi0

xSi , (11)

where the derivation follows by Eqs. (6) and (10). Intuitively, the two components of portfolios,xRi ;x

Si

i, capture the extent to which the assets are used for the corresponding motive for

trade. Consequently, uninsurable variance is lower when traders risk sharing portfolios are

larger (in vector norm), whereas the speculative variance is greater when tradersspeculative

portfolios are larger. As nancial innovation reduces R and increases S , it tends to increase

both the risk sharing and speculative positions, contributing to trading volume.

More subtly, Eq. (11) also suggests that the net eect of nancial innovation on depends

on the relative size of the risk sharing and the speculative trading volume in new assets. The

next result formalizes this intuition for the case in which new assets are uncorrelated with

existing assets.7 To state the result, dene respectively the risk sharing and the speculative

7When new and old assets are correlated, trading volume in new assets does not fully reect the changes inportfolio risks. Nonetheless, trading volume often provides a useful diagnostic tool even in these more generalcases. To see this, consider Example 2.1 in which a second asset, a2 = v2, that is correlated with the rst asset isintroduced. This asset generates some speculative trading volume, i.e., x2;Si 6= 0 for each i, and also leads to anincrease in speculative variance. In contrast, the asset generates no risk-sharing trading volume, i.e., x2;Ri = 0

15

trading volumes in a new asset, jN , as:

T jN ;S =

1

jIjXi

i

xjN ;Si

2!1=2and T jN ;R =

1

jIjXi

i

xjN ;Ri

2!1=2.

Note that trading volume is dened to be a weighted (and quadratic) average of traders

positions, with a greater weight given to traders that are more risk averse.

Proposition 2 (Financial Innovation and Trading Volume). Suppose that a single new asset,

jN , is introduced whose payo is uncorrelated with old assets (that is,Aj0

vAjN = 0 for all

j 2 JO). Then, nancial innovation increases the average variance, (JO [ JN ) > (JO), ifand only if TS;jN > TR;jN , that is, the new asset leads to a greater speculative trading volume

than risk sharing trading volume.

Proposition 2 and Eq. (11) have two empirical implications. First, these results provide an

alternative method to test Theorem 1 when tradersnet worths are not observable but their

risk sharing and speculative positions can be separately identied. While this is not easy, it

might be possible in some contexts to estimate, or at the very least to bound, traders risk

sharing positions. This requires some knowledge of tradersbackground risks but no knowledge

of their beliefs [cf. Eq. (6)]. Tradersspeculative positions can then be obtained as deviations

of their actual positions from the estimated risk sharing positions. To give one example, risk

sharing typically requires individuals to take short positions in stocks of companies in the

industry in which they work. Thus, given an individual i and stock j in the same industry,

a reasonable upper bound on the risk sharing position might be: xj;Ri 0. If we observethe individual taking a long position, xji > 0, then this also implies a lower bound on the

speculative position, xSi xji . Empirical studies have in fact found that individuals investconsiderably in own company and professionally close stocks (see Doskeland and Hvide 2011).

The second implication of Proposition 2 and Eq. (11) is that belief disagreements generate

joint predictions regarding portfolio risks and trading volume. A growing literature has em-

phasized belief disagreements as a potential explanation for the large trading volume observed

in nancial markets (see Hong and Stein 2007). The above analysis suggests that this literature

for each i, and has no impact on the risk sharing variance.

16

is also making implicit predictions about portfolio risks. If speculation is the main source of

trading volume in some new assets, then those assets are likely to increase portfolio risks.

4 Endogenous Financial Innovation

The analysis so far has taken the set of new assets as exogenous. In practice, many nancial

products are introduced endogenously by economic agents with prot incentives. A large liter-

ature has emphasized risk sharing as a major driving force for endogenous nancial innovation

(see Allen and Gale 1994; Du e and Rahi 1995).8 A natural question is to what extent the risk

sharing motive for nancial innovation is robust to the presence of belief disagreements. To

address this question, this section endogenizes the asset design by introducing a prot seeking

market maker.

Suppose the economy initially starts with no assets. A market maker introduces the assets

for which it then serves as the intermediary. The market maker is constrained to introduce

jJ j m assets, but is otherwise free to choose the assetsloading on the underlying risks (recallthat aj (v) =

Aj0v for each j). The asset design, A =

A1;A2; ::;AjJ j

, aects the traders

relative covariance and relative optimism [cf. Eqs. (7) and (8)] according to ~i (A) = A0v ~Wi

and ~i (A) = A0~vi , where the deviation terms ~Wi and ~

vi are dened in Eq. (A:18) in the

appendix.

The market makers innovation incentives are determined by the surplus from trading, that

is, traderswillingness to pay to trade relative to keeping their endowments. The appendix

derives the total surplus as:

Xi

i2

~i (A)

i ~i (A)

1

~i (A)

i ~i (A)

. (12)

Intuitively, traders are willing to pay to trade assets that facilitate better risk sharing [i.e.,

larger ~i (A)], or those that generate greater belief disagreements [i.e., larger ~i (A)].

In practice, the market maker extracts some surplus by charging commissions or bid-ask

8Risk sharing is one of several drivers of nancial innovation emphasized by the previous literature. Otherfactors include mitigating agency problems, reducing asymmetric information, minimizing transaction costs,and sidestepping taxes and regulation (see Tufano 2003). These other factors, while clearly important, are leftout to focus on the tension between risk sharing and speculation.

17

spreads. I abstract away from institutional detail and assume that the market maker extracts

a constant fraction of the total surplus. With this assumption, the market maker chooses the

asset design, A, to maximize the objective function in (12). The next result characterizes the

optimal asset design. Note that many choices of A represent the same trading opportunities

(and thus, also generate the same prots). Thus, suppose without loss of generality that the

market makers choice is subject to the following normalizations:9

= A0vA = I and

(v)1=2Aj

1 0 for each j 2 J . (13)

Proposition 3 (Optimal Asset Design). Suppose the k k matrix

Xi

i2

(v)1=2

~vii (v)1=2 ~Wi

(v)1=2

~vii (v)1=2 ~Wi

0(14)

is non-singular. Then, the design, A, is optimal if and only if the columns of the jJ j jJ jmatrix, (v)1=2A, are the eigenvectors corresponding to the jJ j largest eigenvalues of thematrix in (14). If the eigenvalues are distinct, then the optimal design is uniquely determined

by this condition along with the normalizations in (13). Otherwise, the design is determined

up to a choice of the jJ j largest eigenvalues.

This result generalizes the results in Demange and Laroque (1995) and Athanasoulis and

Shiller (2000) to the case with belief disagreements. Importantly, Eqs. (12) and (14) show that

nancial innovation is partly driven by the size and the nature of tradersbelief disagreements.

This is because innovation incentives are generated by both speculation and risk sharing mo-

tives for trade. I next present the main result of this section which characterizes the optimal

design in two extreme cases.

Theorem 2 (Endogenous Innnovation and Portfolio Risks). Suppose tradersbeliefs are given

by vi;D = mv0 + Dm

vi for all i, where m

v0 is a vector that captures the average belief, fmvi gi

are vectors that satisfyP

imvi = 0, and D 0 is a parameter that scales belief disagreements.

9Here, (v)1=2 denotes the unique positive denite square root of the matrix, v. The rst condition in (13)normalizes the variance of assets to be the identity matrix, I. This condition determines the column vectors ofthe matrix, (v)1=2A, up to a sign. The second condition resolves the remaining indeterminacy by adopting asign convention for these column vectors.

18

For each D, suppose the matrix in (14) is non-singular with distinct eigenvalues.10 Let D (A)

denote the average variance as a function of the asset design, and AD denote the optimal

design (characterized in Proposition 3).

(i) For D = 0, the market maker innovates assets that minimize the average variance:

A0 2 arg minA

0 (A) subject to the normalizations in (13) .

(ii) Suppose there are at least two traders with dierent beliefs, i.e., vi1 6= vi2 for somei1; i2 2 I. As D !1, the market maker innovates assets that maximize the average variance.That is, the limit of the optimal asset design, A1 limD!1AD, and the limit of the scaledaverage variance, 1 (A) limD!1 1D2D (A), exist and they satisfy:

A1 2 arg maxA

1 (A) subject to the normalizations in (13) .

Without belief disagreements, the market maker innovates assets that minimize average

portfolio risks in this economy, as illustrated by the rst part of the theorem. The second part

provides a sharp contrast to this traditional view. When belief disagreements are su ciently

large, the market maker innovates assets that maximize average portfolio risks, completely

disregarding risk sharing. Among other things, this result might explain why most of the

macro futures markets proposed by Shiller (1993) have not been adopted in practice, despite

the fact that they are in principle very useful for risk sharing purposes.

The intuition for Theorem 2 is that, with large disagreements, speculation becomes the

main motive for trade and the main source of prots for the market maker [cf. Eq. (12)]. As

this happens, the market maker introduces assets that enable the traders to bet most precisely

on their dierent beliefs [cf. Eq. (14)]. As a by-product, the market maker also maximizes

average portfolio risks.

10The assumption of distinct eigenvalues can be relaxed at the expense of additional notation.

19

5 Welfare Implications

While Theorems 1 and 2 establish that nancial innovation may increase portfolio risks, they

do not reach any welfare conclusions. In fact, nancial innovation in the baseline setting results

in a Pareto improvement if traderswelfare is calculated according to their own beliefs. This

is because each trader perceives a large expected return from her speculative positions in new

assets, which justies the additional risks that she is taking.11 I next consider two variants of

the baseline setting in which this welfare conclusion can be overturned.

5.1 Ine ciencies Driven by Belief Distortions

The rst setting concerns an interpretation of disagreements as distortions stemming from

various psychological biases emphasized in behavioral nance (see Barberis and Thaler 2003).

If individualsbeliefs are heterogeneously distorted, then they would naturally come to have

belief disagreements. In this case, traderswelfare should ideally be evaluated according to

a common objective belief, as opposed to their own distorted beliefs. However, there is a

practical di culty because the planner might not know the objective belief. In Brunnermeier,

Simsek and Xiong (2012), we propose a belief-neutral welfare criterion that circumvents this

di culty. In particular, we take any convex combination of agents beliefs as a reasonable

objective belief, and we say that an allocation is belief-neutral ine cient if it is ine cient

according to all reasonable beliefs.

To apply the belief-neutral criterion in this model, consider an arbitrary convex combination

of traders beliefs: N (vh;v), where vh =

Pi hi

vi , hi 0 and

Pi hi = 1. The social

welfare under this belief (denoted by subscript h) is measured by the sum of traderscertainty

equivalent net worths:P

i2IEh [ni] i2 varh (ni)

.12 Using the expressions (1) and (9) along

11Note that the reason for e ciency in this setting is almost the opposite of what has been emphasized inmuch of the previous literature on nancial innovation. In particular, traderswelfare gains do not come from adecrease in their risks, but from an increase in their perceived expected returns. Relatedly, it is not clear whetherthese perceived returns should be viewed as welfare gains because they are driven by belief disagreements. Whileall traders perceive large expected returns, at most one of these expectations can be correct. In fact, a growingtheory literature has argued that Pareto e ciency is not the appropriate welfare criterion for environments withbelief disagreements (see Brunnermeier, Simsek and Xiong 2012 and the references therein).12More specically, any allocation ~x that yields a higher value of this expression than x can also be made to

Pareto dominate x (under belief h) with appropriate ex-ante wealth transfers.

20

with market clearing, the social welfare can be calculated as:

Eh

"Xi2I

ei + wi

#

2

.

The rst component of welfare is agents expected endowment, which is exogenous in the

sense that it does not depend on portfolios. The second and the endogenous component

is proportional to average variance, . Consequently, nancial innovation is belief-neutral

ine cient if and only if it increases .13

Intuitively, trading in this economy does not generate expected aggregate net worth since

it simply redistributes wealth. Hence, trading aects social welfare only through portfolio

risks. When portfolio risks increase, social welfare decreases according to each traders belief

(or their convex combinations). Put dierently, each trader believes her welfare is increasing at

the expense of other traders. Consequently, a planner can conclude that nancial innovation

is ine cient without taking a stand on whose belief is correct.

5.2 Ine ciencies Driven by Externalities

The welfare implications of the baseline setting can also be overturned if traderschoices are

associated with externalities. Such externalities naturally emerge when traders correspond to

nancial intermediaries. These intermediaries might take socially excessive risks either because

of re sale externalities (see, for example, Lorenzoni 2008), or because they enjoy explicit or

implicit government protection (see, for example, Rajan 2010). Perhaps for these reasons,

much existing regulation in the nancial system is concerned with restricting intermediaries

portfolio risks. To the extent that nancial innovation increases these risks, it could lead to

ine ciencies.

I next illustrate these types of ine ciencies in a version of the model in which traders are

under government protection. Suppose each trader, i, corresponds to a nancial intermediary13More precisely, the equilibrium is belief-neutral Pareto ine cient as long as the average variance deviates

from its minimum, > R. However, it might be di cult for the planner to implement the minimum, R, asthis would require monitoring that each trader holds exactly the risk sharing portfolio, xRi . Realistically, theplanner might have to decide whether or not to allow unrestricted trade in new assets. A planner subject tothis restriction would conclude that nancial innovation is belief-neutral ine cient if and only if it increases .Note that assumption (A1) facilitates the belief-neutral welfare analysis by ensuring that traders agree on the

variance. The welfare conclusions extend to the case in which there are relatively small disagreements on thevariance.

21

with limited liability. An intermediary whose net worth falls below zero, ni < 0, is forced

into bankruptcy, and its creditors potentially face losses. However, there is a new agent, the

government, which bails out the creditors of an intermediary that enters bankruptcy (for rea-

sons that are outside this model). For simplicity, suppose the government makes the creditors

whole by paying ni > 0. In addition, the government inicts a non-pecuniary punishment tothe bankrupt intermediary that is equivalent to a loss of ni dollars. This assumption consid-erably simplies the analysis by ensuring that the equilibrium remains unchanged despite the

limited liability feature. It is also a conservative assumption since it eliminates the additional

portfolio risks that would stem from the usual risk-shifting motive, and enables me to focus

on speculative risks.

The welfare analysis is potentially dierent than the baseline setting since the government

is also aected by the intermediariesportfolio choices. Suppose the government is risk-neutral

and its belief about the underlying risks is given by Nvg ;

v. The governments expected

welfare loss, which also corresponds to the social cost of bailouts in this setting, is given by:14

NXi=1

Prg (ni < 0)Eg [ni j ni < 0] .

In particular, the governments loss depends on a measure of intermediariesaverage portfolio

risks that is similar to (although not the same as) the average variance, . Intuitively, portfolio

risks matter because they determine the extent to which nancial intermediaries will need

government assistance. Since nancial innovation can increase these risks, it can also increase

the governments loss, illustrating the negative externalities. Moreover, nancial innovation is

ine cient, even in the usual Pareto sense, if it increases the governments loss more than it

increases intermediariesperceived private benets.

In both settings analyzed in Sections 5.1 and 5.2, a measure of average portfolio risks emerge

as the central object in welfare analysis, providing some normative content to Theorems 1 and

2. These analyses should be viewed as partial exercises, characterizing the welfare eects of

nancial innovation that operate through portfolio risks. In particular, I do not take a strong14Note that bailouts represent negative-sum transfers in this economy (as opposed to zero-sum) because of the

simplifying assumption that the government inicts a non-pecuniary punishment on bankrupt intermediaries.Bailouts are also likely to be negative-sum transfers in practice although possibly for dierent reasons, e.g., taxdistortions.

22

normative stance because the model is missing some important ingredients that could change

the welfare arithmetic. Among other things, speculation driven by nancial innovation could

provide additional social benets by making asset prices more informative. Assessing the net

welfare eect of nancial innovation is an important question which I leave for future research.

6 Conclusion

This paper analyzed the eect of nancial innovation on portfolio risks in a standard mean-

variance setting with belief disagreements. When disagreements are large, nancial innova-

tion increases average portfolio risks, decreases average portfolio comovements, and generates

relatively more speculative trading volume than risk sharing volume. Moreover, nancial in-

novation is endogenously directed towards speculative assets that increase average portfolio

risks.

These results show that belief disagreements can overturn the traditional views regarding

the relationship between nancial innovation and portfolio risks. A natural question is how

large belief disagreements should be for this analysis to be practically relevant. I address

this question in Simsek (2011) by considering a calibration of the model in the context of the

national income markets proposed by Shiller (1993). These assets could in principle facilitate

the sharing of income risks among dierent countries. Athanasoulis and Shiller (2001) analyze

these assets in the context of G7 countries, and argue that they would reduce individuals

consumption risks. Using exactly their data and calibration, I nd that small amounts of

belief disagreements about the GDP growth rates of G7 countries (much smaller than implied

by Philadelphia Feds Survey of Professional Forecasters) imply that these assets would actually

increase average consumption risks. While this calibration exercise is promising, it is far from

conclusive. I leave empirical analysis of the model for future research.

23

Appendix: Omitted Derivations and Proofs

I rst note a couple of identities that will be useful in some of the proofs. Consider vectors,

zi, that satisfyP

i zi = 0. Then:

Xi

z0i1i =

Xi

z0i1~i and

Xi

z0i1ii =

Xi

z0i1i~i, (A.15)

which follow from Eqs. (7) and (8). In words, the belief and the covariance terms, i and ii,

in the sums can be replaced by the deviation terms, ~i and i~i.

Proof of Lemma 1. Using Eq. (9), the rst order conditions are xi + i = ifor each

i 2 I, where is a vector of Lagrange multipliers. Note that xRi = 1~i satises these rstorder conditions for the Lagrange multiplier =

Pi2I i

= jIj. It follows that xRi i is the

unique solution to the problem.

Derivation of Eq. (10). Plugging xRi = 1~i into Eq. (9) implies:

jIjR =Xi

i

W0i

vWi + ~0i1~i

Xi

~0i1 ii

,

=Xi

i

W0i

vWi + ~0i1~i

Xi

~0i1 i~i

,

which in turn yields the expression for R. Here, the second line uses the identity in (A:15)

with zi = ~i.

Next consider the residual, jIjS = jIj R. Using Eqs. (6) and (9), this is given by:Xi

ix0ixi + 2

Xi2I

ix0ii +

Xi2I

i~0i1~i

=Xi2I

i

~ii ~i

01

~ii ~i

+ 2

Xi2I

~ii ~i

01

ii

+Xi2I

i~0i1~i

=Xi2I

i

~ii ~i

01

~ii

+ ~i

+Xi2I

i~0i1~i,

which in turn yields the expression for S . Here, the second line uses the identity in (A:15)

24

with zi =~ii ~i.

Proof of Theorem 1. Part (i). By denition, R is the optimal value of the minimization

problem in Lemma 1. Since nancial innovation expands the constraint set of this problem, it

also decreases the optimal value, proving R (JO [ JN ) R (JO).Part (ii). The proof proceeds in three steps. First, Eq. (6) implies that the speculative

portfolio, xSi , solves the following problem:

maxxi2RJ

(~i)0 xi i

2x0ixi, (A.16)

which corresponds to the mean-variance problem of the trader in the hypothetical economy

described in the main text. Moreover, the speculative variance, S , is found by averaging the

variance costs for each trader at the solution to this problem, that is: S = 1jIjP

ii

xSi0

xSi .

Second, nancial innovation relaxes the constraint set of problem (A:16). Consequently, each

trader i obtains a higher objective value. Third, since the problem is a quadratic optimization,

the linear and the quadratic terms at the optimum have a constant proportion, in particular:

(~i)0 xSi = 2

i2

xSi0

xSi . Consequently, nancial innovation increases the quadratic term for

each trader, proving S (JO [ JN ) R (JO).Part (iii). Note that ~i = DA0 ~mi, where ~mi is dened as in Eq. (8). Thus, the

speculative variance can be written as [cf. Eq. (10)]:

S = D21

jIjXi2I

i

(A0 ~mi)i

01

A0 ~mii

.

Hence, the dierence, S (JO [ JN ) S (JO), is also proportional to D2. Since it is positivefor some D > 0, the dierence is also strictly increasing in D. In contrast, the dierence,

R (JO) R (JO [ JN ), does not depend on D. Consequently, there exists D 0 suchthat S (JO [ JN ) S (JO) > R (JO) R (JO [ JN ) i D > D. This in turn implies

(JO [ JN ) > (JO) iD > D, completing the proof.

Proof of Proposition 2. Since asset jN is uncorrelated with the remaining assets, the

matrices and 1 are both block diagonal. By (6), the introduction of asset jN does not

25

aect the positions on the remaining assets. This observation along with Eq. (11) implies:

RJO R JO [ JN = 1jIjX

i2I

i

xjN ;Ri

0jN jNxjN ;Ri =

jN jNT jN ;R

2,

and similarly RJO [ JN R JO = jN jN T jN ;S2. The proof follows from combining

these expressions.

Derivation of the surplus in Eq. (12). Using Eq. (6), tradersnet worth, ni, can be

written as:

ni = ei x0ip+W0iv +~i (A)

i ~i (A)

01A0v.

The certainty equivalent of this expression is given by:

ei x0ip+W0ivi +~i (A)

i ~i (A)

01i

i2W0i

vWi (A.17)

i2

~i (A)

i ~i (A)

01

~i (A)

i ~i (A)

i

~i (A)

i ~i (A)

1i (A) .

In contrast, the traders certainty equivalent payo in autarky is given by

ei +W0ivi

i2W0i

vWi:

Subtracting this expression from Eq. (A:17), the surplus for each trader can be written as:

x0ip+~i (A)

i ~i (A)

01 (i (A) ii (A))

i2

~i (A)

i ~i (A)

01

~i (A)

i ~i (A)

.

Summing this expression across all traders, using market clearing,P

i x0ip =0, and also using

the identities in (A:15) for zi =~i(A)i ~i (A), the total surplus is given by Eq. (12).

26

Proof of Proposition 3. Note that the market maker aects the tradersrelative covariance

and the relative optimism according to ~i (A) = A0v ~Wi and ~i (A) = A0~vi , where:

~Wi = Wi

i

1

jIjX{2I

W{ and ~vi = vi

1

jIjX{2I

{v{ . (A.18)

It is useful to consider the market makers optimization problem in terms of a linear trans-

formation of assets, B = (v)1=2A, where (v)1=2 is the unique positive denite square root

matrix of v. Note that choosing B is equivalent to choosing A. The normalizations in (13)

can be written in terms of B as B0B = I and Bj1 0 for each j. Using these normalizations,the surplus in Eq. (12) can also be written as:

Xi

i2

(v)1=2

~vii (v)1=2 ~Wi

0BB0

(v)1=2

~vii (v)1=2 ~Wi

= tr

B0B

=Xj

Bj0

Bj , (A.19)

where denotes the matrix dened in (14). Here, the second line uses the matrix identity

tr (XY ) = tr (Y X) and the linearity of the trace operator.

Next consider the alternative problem of choosing B to maximize (A:19) subject to the

constraintBj0Bj = 1 for each j, which is implied by the stronger constraint B0B = I. The

rst order conditions for this problem are given by Bj = jBj for each j where j 2 R+ areLagrange multipliers. It follows that the solution,

Bjj, corresponds to eigenvectors of the

matrix, , with corresponding eigenvalues

jj. Plugging this into Eq. (A:19), the surplus is

given byP

j j . Thus, the objective value is maximized i

jjcorrespond to the jJ j largest

eigenvalues of the matrix, . If the jJ j largest eigenvalues are unique, then the solution, Bjj,

is also uniquely characterized as the corresponding eigenvectors (along with the normalizationsBj0Bj = 1 and ~Aj1 0). If the jJ j largest eigenvalues are not unique, then the solution,

Bjj, is uniquely determined up to a choice of these eigenvalues.

Finally, consider the original problem of maximizing the expression in (A:19) subject to the

stronger constraint, B0B = I. Since is a symmetric matrix, its eigenvectors are orthogonal.

This implies that any solution to the alternative problem,Bjj, also satises the stronger

27

constraint B0B = I. It follows that the solutions to the two problems are the same, completing

the proof.

Proof of Theorem 2. Part (i). Let (A) denote the surplus dened in (12). Note that

~i (A) = DA0 ~mvi , where ~m

vi is dened as in Eq. (8). Thus, D = 0 implies ~i (A) = 0, which

in turn implies (A) =P

ii2~i (A)

01~i (A). This expression is equal to c1 c2R (A) for

some c1 0 and c2 > 0 [cf. Eq. (10)]. Thus, maximizing (A) is equivalent to minimizing

R (A). Moreover, ~i (A) = 0 also implies that

R (A) = (A). It follows that the optimal

design minimizes (A).

Part (ii). Consider the alternative problem of choosing A to maximize the scaled

surplus, 1D2

(A), subject to Eq. (13). The limit of the objective value is given by:

limD!1

1

D2 (A) = lim

D!1

Xi

i2

~i (A)

i~i (A)

D

!01

~i (A)

i~i (A)

D

!

=Xi

i2

(A0 ~mvi )0

i1

A0 ~mvii

. (A.20)

where the rst line uses Eq. (12) and the second line uses ~i = DA0 ~mvi . Since the limit

is nite, the alternative problem has a maximum over the extended space D 2 R+ [ f1g.Moreover, by the Maximum Theorem, the solution is upper hemicontinuous. By Proposition 3

(and the assumption in the problem statement), the solution, AD, is unique and bounded for

any nite D. It follows that the limit, A1 = limD!1AD, exists and maximizes the expression

in (A:20). Next consider the limit of the scaled average variance:

1 (A) = limD!1

SD (A)

D2+

RD (A)

D2

=

1

jIjXi

i

(A0 ~mvi )0

i1

(A0 ~mvi )0

i,

where the second equality uses Eq. (10) and ~i = DA0 ~mvi . Combining this expression with

Eq. (A:20) completes the proof.

28

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31

IntroductionRelated Literature

Basic Environment and Main ChannelsAn illustrative example

Financial Innovation and Portfolio RisksComparative Statics of Portfolio RisksComparative Statics of Portfolio ComovementsComparative Statics of Trading Volume

Endogenous Financial InnovationWelfare ImplicationsInefficiencies Driven by Belief DistortionsInefficiencies Driven by Externalities

Conclusion