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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:
[email protected]). 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.
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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.
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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
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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
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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).
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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.
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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.
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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.
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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:
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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
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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.
13
-
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