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Working Paper Series
Asymmetric Information and the Lack of
International Portfolio Diversification
WP 05-07
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Juan Carlos Hatchondo
Federal Reserve Bank of Richmond
Asymmetric Information and the Lack of International Portfolio
Diversification∗
Working Paper No. 05-07
Juan Carlos Hatchondo
†
Federal Reserve Bank of Richmond
September 8, 2005
Abstract
There is pervasive evidence that individuals invest primarily in domestic assets and thus
hold poorly diversified portfolios. Empirical studies suggest that informational asymmetries
may play a role in explaining the bias towards domestic assets. In contrast, theoretical studies based on asymmetric information fail to produce significant quantitative effects. The
present paper develops a theoretical model in which the presence of informational asymmetries explains a significant fraction of the home equity bias observed in the data. The main
departure from previous theoretical work is the assumption that local investors outperform
foreign investors in identifying the correct ranking of local investment opportunities instead
of possessing superior information about the aggregate performance of the domestic stock
market. The other key assumption is based on the evidence that short-selling is a costly
activity. This paper studies the case of a two-country world. There are two assets in each
country. Only local investors receive informative signals about local assets. Thus, domestic
agents have an incentive to concentrate their investments in the local asset favored by the
signal realization, and reduce the position held in the other local asset. When the signal is
sufficiently informative and short-sales are costly, local investors decide not to finance purchases of the perceived “good” local asset by selling short the perceived “bad” local asset.
Instead they invest a lower fraction of their portfolio in foreign securities. This liberates
resources that can be allocated in the local asset perceived to pay higher expected returns.
Keywords: International portfolio diversification, home bias, asymmetric information.
JEL Classification: D82, F30, G11, G15
∗
This paper is part of my dissertation at the University of Rochester. I am greatly indebted to Per Krusell for
encouragement and advice. This work has also benefited from comments and discussions with Rui Albuquerque, Irasema
Alonso, Eva De Francisco, Margarida Duarte, Fernando Leiva, Leonardo Martinez, Joseph Perktold, Martin Schneider
and Alan Stockman. The project started while visiting the Institute for International Economic Studies at Stockholm
University. I am grateful for their hospitality. The usual disclaimer applies. Any opinions expressed in this paper are
those of the author and do not necessarily reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve
System.
†
Address: Research Department, Federal Reserve Bank of Richmond, 701 E. Byrd St., Richmond, VA 23261, USA.
E-mail: JuanCarlos.Hatchondo@rich.frb.org.
1
Introduction
The last decades have witnessed a remarkable increase in international capital flows. Gross cross-border
transactions in bond and equity of US residents represented 4 percent of the GDP in 1975, but increased
to 320 percent by 2003. Yet the fraction of the US equity portfolio invested overseas has remained quite
low. Figure 1 provides a measure of the bias towards domestic stocks. The dashed line describes the
fraction of foreign stock holdings in the equity portfolio of US residents. In order to assess whether those
values are high or low, we need a theory. The solid line describes an index that interprets the data from
the perspective of the International version of the Capital Asset Pricing Model (ICAPM). The latter
predicts that in a frictionless world, the composition of every investor’s portfolio coincides with the
world market portfolio composition.1 By construction, the index takes a value of one when the fraction
of foreign stocks is zero, and takes a value of zero when the fraction of foreign equity holdings coincides
with the prediction of the ICAPM. The graph shows that even though the bias has been receding over
time, it is still significant. This discrepancy between the theory and the data is known in the literature
as the home equity bias puzzle. It was initially documented by French and Poterba (1991), Cooper and
Kaplanis (1994) and Tesar and Werner (1995).
This finding has motivated a vast body of literature. There may be various reasons why domestic
investors are reluctant to invest abroad. For example, there are domestic regulations that limit the
foreign exposure of institutional investors. Some foreign countries impose limits on the fraction of a
firm that can be owned by non-nationals. Transaction costs may be higher for cross-border transactions.
Exchange rate fluctuations increase the risk of investing in foreign assets if domestic investors care only
about returns nominated in domestic currency. However, none of these factors have offered a satisfactory
explanation. Lewis (1999) and Karolyi and Stulz (2003) offer a detailed survey of this literature.
An alternative explanation rests on the conjecture that local investors can collect more precise
information about domestic assets.2,
3
The typical setup in this literature features a two-country world
with a single asset per country. Domestic investors receive informative signals about domestic and
foreign assets, but the signal about the local asset is more precise. The reason why this generates
home equity bias is that the informational disadvantage about the foreign asset turns it into a more
1
Baxter and Jermann (1997) argue that the amount invested overseas should be even larger than that, given that the
returns on human capital are correlated with the returns on domestic financial assets.
2
See Gehrig (1993), Zhou (1998) and Coval (2000).
3
Several papers find empirical evidence in favor of the presence of informational asymmetries in financial markets. A
survey is provided in the next section.
1
1.0
0.8
Fraction of US Equity holdings invested abroad
0.6
Fraction of Foreign Equities in World Portfolio
0.4
0.2
0.0
1990
1992
1994
1996
1998
2000
2002
2004
Figure 1: Measures of foreign investment positions. Sources: Department of Commerce and International Federation of Stock Exchanges
risky investment option from the perspective of local investors. However, this explanation faces two
limitations. First, it relies on effects that are of second order importance. Once the parameters in the
model are replaced with realistic values, the fraction of the bias that can be explained with the model
is quite low. Second, this approach produces high volatility in the fraction invested overseas, which is
not observed in the data.
The distinctive feature of this paper is that the information asymmetry between domestic and foreign
investors relates to the performance of individual stocks rather than to the market portfolio. This paper
assumes that domestic investors do not outperform foreign investors in predicting the performance
of aggregate variables, but they do enjoy an advantage in identifying the best individual investment
opportunities. The latter is motivated on the grounds that there is more scope for the existence of
disparities in information about individual firms than about aggregates like the stock market. The
second assumption that separates this work from previous theoretical papers is based on the extensive
2
evidence that shows that short-selling is a costly activity.4
This paper maintains the two-country world setup but with two technologies in each country. All
technologies are publicly listed. Each technology receives either a high productivity shock (high returns) or a low productivity shock (low returns). The probability distribution over the next period’s
productivity shock is not publicly observed. Instead, domestic investors receive an informative signal
about local assets. A majority of domestic investors receives a signal favorable to the local asset that it
is more likely to pay high returns. The remaining local investors receive an incorrect signal, encouraging
investment in the local asset that pays high returns with the lowest probability. Each investor does not
know whether the signal received is correct or not.5
Agents diversify their portfolios in two dimensions: across countries and across assets within each
country. The information structure described above has two implications regarding the portfolio composition. First, agents want to concentrate the local component of their portfolios in the local technology
favored by the signal realization (the “good” asset). Second, since the information received is uncorrelated with any index of aggregate performance, local investors do not have an incentive to invest more
heavily in the domestic or foreign country. This implies that when the degree of information is such
that local agents still demand a positive amount of both local assets, the home equity bias is nil. The
result changes when the degree of information conveyed by the signal is such that agents are willing
to finance the purchases of the perceived “good” local stock by selling short the perceived “bad” local
stock. When short-selling is costly, local agents find it optimal to finance the purchases of the “good”
local asset by investing less in foreign stocks, instead of short-selling the perceived “bad” local stock.
This mechanism is what generates home bias in the model. The result is driven by an effect of first
order importance, i.e., differences in expected returns. This allows us to explain a high fraction of the
bias observed in the data for realistic parameters values. In addition, and in line with what we observe
in the data, the fraction of foreign investments generated by the model is stable over time.
The paper is also consistent with other dimensions of the portfolio behavior of US investors. The
paper predicts that the foreign component of the equity portfolio is more diversified than the local
component. In addition, if the precision of the signal was allowed to vary across agents, the model would
state that the fraction invested in foreign stocks is inversely proportional to the degree of concentration
4
See section 2.1 on page 10.
For simplicity, it is assumed that local agents do not receive information about foreign assets. The results will not be
affected as long as the foreign signal is sufficiently less informative than the local signal.
5
3
of the portfolio of local stocks. This coincides with the findings of Albuquerque et al. (2005a). The
present paper predicts that investors not only invest more heavily in local stocks but they also enjoy
higher returns on their trades of these stocks. This is consistent with Coval and Moskowitz (1999),
Coval and Moskowitz (2001), and Ivkovich and Weisbenner (2005). In other studies, Ivkovick et al.
(2004), and Kacperczyk et al. (2004) find that agents with more concentrated portfolios earn abnormal
returns (after controlling for risk). The present paper offers a theoretical explanation for this.
The remainder of the paper is organized as follows. The next two subsections review related empirical
and theoretical literature. Section 2 introduces the model and discusses the assumptions. Section 3
describes the testable implications of our model. Section 4 extends the model to a case with positive
cross-asset return correlation. Section 5 allows for endogenously determined stock prices in a context
with partially revealing prices. Section 6 computes the shadow price of the short-sales constraint. Section
7 provides some analytical characterization of our main results using the CARA-Gaussian setup. Finally,
Section 8 concludes.
1.1
Related empirical literature
This paper relies on the assumption that domestic investors have better information about domestic
stocks than foreign investors. This can be justified on many grounds. Equity investment in foreign
companies requires understanding different accounting practices and legal environments. Domestic
investors are exposed to a wide array of sources of local news that can convey useful information about
the performance of domestic companies. In addition, the geographic proximity allows for face-to-face
contacts with local corporate executives, employees and other individuals that may have valuable private
information.
Coval and Moskowitz (1999, 2001) study the portfolio composition of more than 2,000 mutual funds
in the US. They find evidence of home equity bias within US boundaries. Fund managers invest more in
companies with headquarters located near the fund’s offices. Moreover, they earn substantial abnormal
returns in nearby investments, while at the same time stocks held predominantly by local investors tend
to show higher expected returns. The evidence strongly suggests that fund managers are exploiting an
informational advantage in their selection of nearby stocks. Ivkovich and Weisbenner (2005) study a
sample of 78,000 households and also find evidence of a strong preference for local stocks. In addition,
the excess return on local investments is larger among companies not listed in the Standard and Poor
4
500 index. The latter are presumably firms with wider informational asymmetries between local and
non-local investors.6
In a similar vein, Ivkovick et al. (2004) show that the stocks purchased by individuals with concentrated portfolios display higher returns than the stocks purchased by individuals with more diversified
portfolios. A similar finding is reported by Kacperczyk et al. (2004) for a sample of US equity mutual
funds. Both papers argue that this is consistent with the presence of informational asymmetries across
market participants.
It should be stressed that these studies analyze the behavior of agents who are trading in the most
developed and transparent financial market in the world. If informational differences can persist in
this environment, the case for asymmetric information across country boundaries is even stronger. The
barriers that can account for the information asymmetries in the US must be lower than the barriers
that prevent information from flowing across countries. Surprisingly, the empirical evidence in this area
is not conclusive.
On the one hand, several papers find evidence supporting the hypothesis that local investors are better informed than foreign investors. Kang and Stulz (1997) study foreign stock ownership in Japanese
firms. They find that foreign investors concentrate their portfolios in large firms, firms with good
accounting performance, and firms with high exports. These are the types of companies where the
information asymmetries are presumed to be the lowest. Dahlquist and Robertsson (2001) find qualitatively similar results for Swedish firms. Choe et al. (2004) analyze foreign trades in Korea and find
that foreign investors buy at higher prices than domestic investors, and sell at lower prices. A similar result is found by Dvor̆ák (2001) for Indonesia. Shukla and van Inwegen (1995) provide evidence
that UK mutual funds obtain lower returns from their investments in the US compared to US funds.
Ahearne et al. (2004) study the home bias of US investors against specific countries. They find that
the home bias decreases with the fraction of the foreign country market value that is cross-listed in the
US stock market. Frankel and Schmukler (1996) show that domestic investors were “front-runners” in
the Mexican crisis of 1994: they tried to sell their local investments before foreign investors did. Portes
and Rey (1999) analyze the determinants of cross-border transaction flows. They find that distance
6
Hubermann (2001) also documents that investors tilt their portfolio composition toward local companies. But he
argues that this behavior is due to the fact that investors prefer to invest in firms that are familiar to them, independently
of their prospects. If that were the case, we should not expect to observe abnormal returns on local investments. However,
the evidence provided by Coval and Moskowitz (1999, 2001) and Ivkovich and Weisbenner (2005) suggests that a significant
fraction of investors indeed behave rationally.
5
has a significant negative impact, which they argue is a proxy for informational asymmetries. Hau
(2001) finds that traders located in Frankfurt and in German speaking cities in Europe show higher
proprietary trading profits on German stocks. There is also evidence of higher profits for traders located
near corporate headquarters.
On the other hand, other papers find evidence that foreign investors outperform local investors,
which implies that either foreign agents do not have less information than local residents, or that the
informational disadvantage does not play a significant role. Karolyi (2002) shows that foreign investors
obtain higher returns in Japan. Grinblatt and Keloharju (2000) reach the same conclusion for Finland.
Seasholes (2004) provides evidence that foreign investors in Taiwan buy before price increases and sell
before price decreases. Froot and Ramadorai (2003) use evidence of changes in close-end country fund
prices, and the net value of the underlying assets. They find that cross-border flows positively forecast
changes in both prices. This is interpreted as evidence favoring the assumption that foreigners are
better informed about fundamentals than domestic investors are.
One reason behind the mixed results is that it is not easy to isolate the role played by differences
in information. Even when the comparison is made across similar classes of agents, there may be other
factors affecting the different behavior of domestic and foreign agents. In this sense, one advantage of
the samples used by Coval and Moskowitz (1999, 2001), Ivkovick et al. (2004), and Kacperczyk et al.
(2004) is that they consist of a homogenous set of actors who are subject to the same legal environment.
1.2
Related theoretical literature
The above evidence suggests that the assumption of a local information advantage has an intuitive
appeal, and is consistent with most empirical studies. From a theoretical point of view, it has already
been said that the present paper is not the first attempt to explain the home bias assuming informational
asymmetries between domestic and foreign investors. Gehrig (1993) and Brennan and Cao (1997) use
the workhorse model of rational expectations equilibrium developed by Grossman (1976), Grossman and
Stiglitz (1980) and Admati (1985). They assume that every agent receives informative signals about
the future performance of domestic and foreign assets. The domestic signal conveys more information
than the foreign one. This leads to more imprecise assessments about future performance of the foreign
asset. Domestic agents thus perceive the foreign stock as more risky than the local stock, and reduce
their holdings of the foreign asset. However, Glassman and Riddick (2001) and Jeske (2001) argue that
6
the implied risk aversion needed to generate quantitatively significant results is unreasonably high.
Zhou (1998) considers a two asset model with a more sophisticated learning process. Agents face
the so-called “infinite regress” problem: forecasting the forecasts of forecasts ... of others. But that
feature does not help him to obtain any sizeable effect. Coval (2000) extends the framework in Zhou
(1998) by introducing direct investment decisions and simplifies the learning process. He also obtains a
small impact on the home bias.
In addition to the poor quantitative performance, Jeske (2001) argues that the previous modelling
strategies do not seem suitable to address the home bias puzzle. Since domestic agents hold better
information about domestic assets, sufficiently low expected local dividends induce residents to liquidate
their local positions in favor of foreign assets. On the other hand, foreign investors unaware of the poor
expected performance of local assets may find it convenient to purchase local stocks at a discount. As a
consequence, these models predict unrealistic fluctuations of the fraction of foreign investments (which
can turn into foreign bias for certain shock realizations). These limitations lead him to conclude that
asymmetric information does not stand up as a compelling theoretical explanation for the home equity
bias.
The reason behind the lack of success of previous attempts is that the burden of the explanation
relies on effects that are of second order importance. In the setups considered, agents can be neither
systematically pessimistic nor optimistic with respect to any asset. The explanation for why agents show
a preference for domestic assets is that they are perceived to be less risky or that they provide better
hedge against consumption risk (see Coval (2000)). But this plays a secondary role in the standard
expected utility framework with the HARA utility functions commonly assumed for macroeconomic
analysis.
Van Nieuwerburgh and Veldkamp (2005) consider a CARA-Gaussian structure but allow for a more
complex learning process. Agents decide how much to reduce the variance of the signal they learn from.
They face a limited “learning capacity”. Thus, the decision of how much to learn from each asset is
non-trivial. They show that when local investors enjoy an information advantage in local assets, the
optimal strategy is to concentrate the learning capacity in those assets. Thus, allowing agents to learn
from local and foreign assets not only does not reduce the information asymmetry between domestic and
foreign investors, but also magnifies it. In contrast to the previous literature, they obtain quantitatively
significant results, but their model is ill-suited to deal with the second critique –the excess volatility in
7
the fraction invested abroad.
Epstein and Miao (2003) and Alonso (2005) assume preferences that allow for ambiguity aversion
and are able to explain a significant fraction of the bias. The reason is that they introduce an effect of
first order importance: domestic agents are systematically pessimistic about foreign stocks. The present
paper relies also on effects of first order importance, yet shows that similar quantitative results can be
obtained without a major departure from the mainstream model.
2
The model
We consider a two-country world. Each country is inhabited by a large number of infinitely-lived,
identical agents. Agents have preferences defined over a stream of tradable consumption goods:
E
with
"
∞
X
#
β t u (ct ) | I0 ,
t=0
u (c) =
c1−σ − 1
.
1−σ
The perceived probability distribution of future consumption flows depends on each agent’s initial
information set, denoted by I0 .
Each country hosts two risky technologies. Each technology produces the same consumption and
investment good. The output provided by each technology depends on the capital allocated in the
previous period and the current productivity shock. The productivity shocks vary across technologies.
For simplicity, it is assumed that productivity shocks may take either a high value (A h ) or a low value
(Al ). Technologies do not require labor as an input and display constant returns to scale, i.e., they are
of the AK type. The probability that technology i is hit with a high productivity shock in period t is
denoted by νit , where i = 1, 2, 1∗ , 2∗ . The superscript “*” is used to denote foreign variables. The
probability values are drawn from a joint distribution with density f (ν1t , ν2t , ν1∗ t , ν2∗ t ). The density
function f is time invariant.
Investors are not able to observe the probabilities that govern the distribution of productivity shocks,
i.e., they do not observe the realizations of νit+1 . Instead, they receive informative signals about the
relative expected performance of local technologies. A fraction φ of domestic investors receive a signal
favorable to the local technology that it is more likely to receive a high productivity shock, while a
8
fraction 1 − φ receive an incorrect signal, suggesting to invest in the technology that is less likely to
receive a high productivity shock. Each investor does not know whether the signal received is correct or
not. The value of φ is assumed to be larger than 0.5 (the signals are informative). Finally, it is assumed
that agents are not allowed to hold short positions.
?
Beginning of period t. Productivity shocks Ait are
realized.
?
?
Production
takes place and
capital income
is distributed.
?
Nature
draws ν1t+1 ,
ν2t+1 , ν1∗ t+1 ,
ν2∗ t+1 .
Agents
receive
private
signals.
?
Agents
invest and
consume
ct .
?
Beginning of
period t+1.
Shocks Ait+1
are realized.
?
···
Figure 2: Order of events in period t.
The timing of the model is described in Figure 2. Productivity shocks are realized at the beginning of
P
each period. Then, the production process takes place and agents receive income y = i kit Ait . Recall
that capital is the only factor input. After that, Nature draws the vector (ν 1t+1 , ν2t+1 , ν1∗ t+1 , ν2∗ t+1 ),
that determines the probability distribution over next period’s productivity shocks. This vector also
conditions the distribution of signals received by domestic and foreign agents. Each agent uses the
signal realization to update the probability distribution over next period’s productivity shocks. At the
end of the period, agents decide how much to invest in each of the four technologies, and how much to
consume.
The investor’s problem has a recursive structure. This is due to the fact that the density function
f is time invariant, and does not depend on previous shock realizations.7 From the perspective of an
individual investor, the relevant state variables are his wealth level (ω)and the signal received: (s). The
last piece of information is helpful to forecast the probability distribution for next period’s shocks. The
agent’s optimization problem can then be summarized by the following Bellman equation.
V (ω, s) =
7
M ax
k10 ,k20 ,k10 ∗ ,k20 ∗
u (c) + βE V ω 0 , s0
(1)
If the density depended on past realizations of νi and νi∗ , agents would typically need to keep track of the entire
history of shocks in order to update their beliefs.
9
subject to
c + k10 + k20 + k10 ∗ + k20 ∗ = ω,
X
ki0 A0i ,
ω0 =
i=1,2,1∗ ,2∗
and
ki0
≥ 0 for i = 1, 2, 1∗ , 2∗ .
The last inequality rules out short-selling. An alternative modelling strategy would be to impose
a “fee” when agents decide to short-sell one of the assets. The actual volume of short-sales in the US
market is very low. This means that the short-selling constraint is not a very restrictive assumption.
2.1
Discussion of assumptions
The signal structure captures the hypothesis that local investors do not possess significantly better
information about the future performance of the local stock market compared to what foreign investors
know, but they do outperform foreign investors in spotting the best domestic investment opportunities.
For instance, it is not clear why American investors should do systematically worse at predicting the
performance of the German stock market when compared to German investors. In this case, the relevant
information set consists mostly of public news and past performance of aggregate variables, which are
readily available to every investor at the same time. It may be possible that some local investors get
privileged access to information about policy decisions (like a proximate declaration of default), but
this relates to rare events and should not play a significant role in developed countries, with good
institutions. On the contrary, domestic investors are exposed to a wide array of sources of local news
that can convey useful information about the performance of domestic companies. In addition, the
geographic proximity allows for face-to-face contacts with local corporate executives, employees and
other individuals that may have valuable private information.
One shortcoming of the present paper is that the allocation of information is exogenously given. This
contrasts with the fact that individuals can actually acquire information from various sources: Investing
in a portfolio managed by a Mutual Fund is probably the easiest one. But individuals can also pay
for expert investment advice.8 In fact, the last two decades have witnessed a sensible decrease in the
home bias, and a simultaneous increase in the fraction of equity investments managed by institutional
investors. One possible interpretation is that a higher participation of institutional investors has helped
8
The expansion of the world wide web has also made possible access to an enormous amount of data. On the other hand,
we could reasonably conjecture that this type of information cannot be used to design profitable investment strategies.
10
to reduce the information asymmetry between investors in different countries, inducing a more diversified
portfolio.9 In spite of this, several studies surveyed in the previous section find evidence consistent with
the fact that information is not evenly distributed.
Several reasons prevent information from flowing freely across agents. It is in the best interest of
any investor to maintain an information advantage if he can profit from it. But even financial firms
may find it convenient to hide some of the information they possess. For example, a Mutual Fund that
has invested heavily in certain stocks may not be interested in spreading bad news about those stocks.
In addition, in most cases it is not easy to verify the quality of the information received. This favors
agents who do not want to truthfully disclose all the information they have.
This paper does not explore the causes that explain the presence of informational asymmetries. 10
Instead, we take as given the fact that some investors know more about certain assets than other
investors.
The second key assumption in the model is that agents face a short-sale constraint. There is plenty
of evidence that short-selling is more costly than buying a stock. The common method of shorting an
equity is to borrow the security and sell it. Later, the short-seller needs to buy it back to return it to the
lender. The explicit cost of the transaction is the fee that the short-seller pays for the loan. But there
are other costs. The standard practice is that the equity lender can ask the loan to be repaid (“recall”
the shares) at any time, which exposes the short-seller to risk. There are also various regulations that
increase the cost of short-selling: the proceeds from short-selling are taxed at the short-term capital
gain tax rate independently of how long the short position is open; sell orders that are short sales can be
executed only after the stock price has increased (on an “uptick”), etc. 11 Some institutional investors
are even prohibited from taking short positions. Almazan et al. (2004) find that by 2000, 69 percent
of US equity funds were not permitted to engage in short-selling practices. Among the ones that were
not constrained, only 10 percent held short positions.12 All of the above may help to explain why the
9
Another interpretation is that the more active participation of Mutual Funds, and other institutional investors, has
decreased the cost of foreign equity investments. Nonetheless, previous empirical studies have not found convincing evidence
in favor of the transaction cost explanation of the home bias.
10
Van Nieuwerburgh and Veldkamp (2005) provide an explanation for why local agents may find it optimal to learn
mostly about local stocks.
11
See D’Avolio (2002), Dechow et al. (2001) and Duffie et al. (2002) for a description of the institutional details of equity
lending markets and regulations applied to short-selling. An additional implicit cost of short-selling is described in Lamont
(2004). He argues that firms have incentives to impede short sales of their stock. He analyzes a sample of 266 firms who
threatened, took action against, or accused short sellers of illegal activities. His findings suggest that those firms succeeded
in raising the costs of short selling.
12
Koski and Pontiff (1999) find that 79 percent of equity mutual funds make no use of derivatives suggesting that funds
are also not finding synthetic ways to take short positions.
11
market for equity loans is so thin. D’Avolio (2002) reports that in June 2001 the total amount of stocks
shorted represented 1.7 percent of the market capitalization. Dechow et al. (2001) document that short
positions represented only 0.2 percent of the market value in 1976 but increased to 1.4 percent in 1993.
The previous evidence shows that short-selling activities are quite limited in the US, the most developed
financial market. This suggests that although the assumption of no-short-sales constraints made is this
paper may appear extreme, it is in line with the trading limitations observed in actual markets.
Other features of the model require further explanation. Notice that agents can only invest in four
assets while there are sixteen possible state realizations, depending on the realization of the productivity
shocks. This means that markets are incomplete. Not only is this a realistic assumption, but it is also
necessary in order to have a well defined measure of the home bias. Otherwise, it would not be clear
how to classify assets that pay contingent on joint domestic and foreign state realizations. Besides,
under complete markets there would be fifteen endogenous prices. That is more than enough to reveal
all the information agents need to know.
The model laid down above does not feature endogenous factor prices. The return on capital is
exogenously given by productivity shocks, and the share prices of the four technologies are equal to
one.13 The sectors’ sizes fully adjust in every period in response to the aggregate demand of each stock.
Assuming a more standard production process with decreasing marginal productivity on capital and
labor would allow for endogenous factor prices. But the presence of incomplete markets implies that
the model cannot be solved as if each economy was inhabited by a representative agent. In order to
forecast future factor prices, individuals will typically need to keep track of the wealth distribution in
each country. Even though this is an interesting extension, it involves a high level of complexity and is
not necessary to illustrate the main point of the paper.
The constant-share-price result can be relaxed by imposing a sluggish adjustment in the supply
of stocks. The standard model with asymmetric information used in the finance literature assumes a
constant asset supply over time, except for shocks due to liquidity trading. We choose the opposite
extreme in order to prevent investors from extracting valuable information from prices. If relative stock
prices are allowed to differ, it would be necessary to allow for additional sources of uncertainty in order
to mask the expected relative performance of each asset. This is not a trivial extension. To the best of
our knowledge, the only model structure with multiple assets and partially revealing prices corresponds
13
This is because there is only one good in the economy, and there are no adjustments costs.
12
to the one developed by Admati (1985). Her results relies on the Gaussian-CARA framework. However,
it is not possible to accommodate that modelling strategy in the present work. The assumption of noshort-sales breaks down the Bayesian updating scheme over normally distributed variables, making the
problem analytically intractable. Section 5 develops a simple environment that allows for endogenously
determined stock prices. It shows that the main conclusion of the paper is not affected as long as prices
do not reveal too much information.
2.2
Implications for the home bias
This section makes an extra simplification to the model introduced above. The probability distribution
from which the values of νi are drawn is assumed to be independent across technologies and across time,
and follow a uniform distribution with support [0, 1]. The previous simplification implies that there is
no persistence in returns.14 The second implication induced by this simplified setup is that signals
about domestic technologies do not convey information about foreign assets. It is straightforward to
generalize the model and allow domestic agents to receive some information about foreign assets. The
main results will not be affected as long as foreign signals are sufficiently less informative than local
ones.
The conditional expectation of the value function can then be expressed as
0
0
E V ω 0 , s0 | s, = E [P r (h, h, h, h, 1) | s] V ωh,h,h,h
, 1 + · · · + E [P r (l, l, l, l, 1) | s] V ωl,l,l,l
,1 +
0
0
E [P r (h, h, h, h, 2) | s] V ωh,h,h,h
, 2 + · · · + E [P r (l, l, l, l, 2) | s] V ωl,l,l,l
, 2 (, 2)
where
ωi0 ,j 0 ,i∗ 0 ,j ∗ 0 = Ai0 k1 + Aj 0 k2 + Ai∗ 0 k1∗ + Aj ∗ 0 k2∗ .
The term E [P r (i, j, i∗ , j ∗ , s0 ) | s] denotes the conditional joint probability of receiving a signal s0 in
the following period and observing a future vector of productivity shocks equal to (i, j, i ∗ , j ∗ ). The first
two components in this vector refer to productivity shocks to domestic technologies 1 and 2 respectively.
The last two components denote shock realizations of foreign technologies 1 and 2 respectively. The
conditional expectation is computed using Bayes’ rule.
14
There is mixed evidence in this respect: some authors find that past prices do not convey useful information about
future returns while other papers find some effect. But even the latter do not find a large effect. This suggests that
eliminating serial correlation in returns is not a very restrictive assumption. See Malkiel (2003) for a discussion on the
topic.
13
ν2
6
1
A
0.5
u
u
B
-
0.5
1
ν1
Figure 3: Change in perceived distribution of returns upon the arrival of a signal favoring local
asset 1.
Figure 3 illustrates how a domestic investor updates his beliefs. Before receiving any signal, the only
information the agent possesses is that a set of values of νi0 has been drawn from a uniform distribution.
Thus, the expected probability of receiving a high return when investing in domestic technology i
coincides with the unconditional mean of νi . In this case, the latter is equal to 0.5. Thus, the beliefs of
both domestic technologies are represented by point A. The perceived return process of foreign stocks is
not affected by the signal realization, so point A also illustrates the beliefs of a local agent with respect
to foreign stocks. The perceived distribution of returns of local stocks changes upon the receipt of a
signal. For example, if an investor receives a signal favoring technology 1, he believes that it is more
likely that the actual realization of (ν10 , ν20 ) is below the diagonal, than above the diagonal. This explains
why the conditional expectation of (ν10 , ν20 ) lies on a point like B. The agent becomes optimistic about
asset 1 and pessimistic about asset 2. This explains the incentive to concentrate the local component
of his portfolio in one local stock.
14
Formally,
0
0
∗0
∗0
0
E Pr i , j , i , j , s | s
Pr (s0 ) E Pr i0 , j 0 , i∗ 0 , j ∗ 0 , s
Pr (s)
0
Pr (s ) E [Pr (A01∗ = Ai∗ 0 )] E Pr A02∗ = Aj ∗ 0 E Pr A01 = Ai0 , A02 = Aj 0 | s
,
Pr (s)
=
=
where the second equation makes use of the assumptions stated at the beginning of this section. More
explicitly, the previous expectation is computed as follows:
E Pr i0 , j 0 , i∗ 0 , j ∗ 0 , s | s =
0
E Pr A02∗
Pr (ν10 > ν20 ) Pr (s0 | ν10 > ν20 ) +
E Pr A01∗ = Ai∗ 0 ×
Pr (ν10 < ν20 ) Pr (s0 | ν10 < ν20 )
R1Rν
0
0
Pr (s | ν1 > ν2 ) 0 0 1 Pri (ν1 ) Prj (ν2 ) f (ν1 ) f (ν2 ) dν2 dν1 +
= Aj ∗ 0
Pr (s | ν < ν ) R 1 R 1 Pri0 (ν ) Prj 0 (ν ) f (ν ) f (ν ) dν dν
1
2
1
2
2 1
1
2 0 ν1
Pr (ν1 > ν2 ) Pr (s | ν1 > ν2 ) + Pr (ν1 < ν2 ) Pr (s | ν1 < ν2 )
,
where νi0 denotes the probability with which technology i will pay high returns on the following period.
0
Pri (ν) denotes the probability that technology i receives productivity shock A i0 if the actual probability
of being hit with a high productivity shock is ν. Formally,
ν
if i0 = h,
0
Pri (ν) =
1 − ν if i0 = l.
The dynamic optimization problem is solved by finding policy functions that satisfy the Euler
Equations. It is easy to check that the assumptions of a constant relative risk aversion utility function
and linear technology lead to individual policy functions that are linear in wealth. However, it is not
possible to find an explicit expression for the slope coefficients, so the problem is solved using numerical
techniques.15 Appendix 7 considers a model with CARA utility function and a Gaussian process for
the productivity shocks. That framework allows us to fully characterize the optimal investment policy.
We find qualitatively similar results to the ones described in this section.
The parameter values are chosen in such a way that the domestic return process resembles some key
statistics of the US stock market. The assumption that foreign assets share the same characteristics
15
Equation 2 shows that each individual needs to allocate future consumption across 32 states. The assumption that
technologies are identical means that the signal received does not affect the perceived discounted utility of future consumption streams. This reduces the future state space by half. In addition, investors do not receive any information that
can help them differentiate between the two foreign assets so the latter are perceived to be equivalent. In practice, then,
it is only necessary to solve for three policy functions: the investments in the two domestic technologies and the foreign
one. Also, the state space can be reduced to 12 possible realizations. With logarithmic utility function, the solution for
the policy functions consists of the root of a system of three polynomials of 12th order!
15
with domestic assets is made for simplicity. A period in the model corresponds to one year. The value
for the high productivity shock (Ah ) is set equal to 1.27. The value for the low productivity shock (Al )
is set equal to 0.85. This yields an average return on each stock market of 6 percent, with a standard
deviation of 14.8 percent.16 Finally, we choose standard preference parameters: a logarithmic utility
function and a subjective discount factor (β) of 0.96.
1.0
0.8
Fraction invested in local stocks
0.6
Fraction invested in the domestic stock
with good signal
0.4
Fraction invested in each foreign stock
0.2
0.0
0.50
0.55
0.60
0.65
0.70
φ (degree of information)
0.75
0.80
Figure 4: Fractions invested in each local and foreign asset, and home bias as functions of the
precision of the signal
Figure 4 describes the main result of the paper. The graph shows how investors’ policy functions
depend on the precision of the signal. A completely uninformative signal corresponds to the case where
φ = 0.5. Half of domestic agents receive a signal favorable to technology 1 and the other half receive a
signal favorable to technology 2, regardless of the actual probability distribution of domestic returns. In
this case the signal is pure noise and agents hold a perfectly diversified portfolio: they invest a quarter
of their savings in each asset. As the signal becomes more informative, the signal realization has an
effect on the investment decisions. Every local investor increases his position in the local technology
favored by the signal realization and decreases his position in the other local technology. The overall
proportion of local assets in his portfolio is barely affected: He still invests roughly half of his savings
in foreign assets.
16
The statistics correspond to the case where both technologies receive the same weight.
16
When the signal is sufficiently informative, local investors want to finance purchases of the perceived
high-returns asset by selling short the perceived low-returns asset. When short-selling is a costly activity,
the optimal strategy is to lower the fraction invested in foreign stocks. This liberates resources that can
be invested in the local asset with higher expected returns. It is in this range of values of φ where the
model can generate a significant level of home bias. For instance, when the proportion of local investors
that receive the “correct” signal is around 65 percent, domestic investors always hold 70 percent of their
portfolios in local assets. More precisely, 65 percent of domestic investors invest 70 percent of their
savings in the local asset that is actually paying higher returns with higher probability. The remaining
30 percent of their portfolio is split between the two foreign stocks. The remaining local investors are
investing 70 percent of their savings in the local asset with worse prospects (though they do not know
this), and split the remaining 30 percent of their savings between the two foreign stocks.
100%
3.0%
2.0%
80%
Excess expected return
on local stocks
70%
Fraction invested
in local stocks
Excess return
Fraction of portfolio
90%
1.0%
60%
50%
0.0%
0.50
0.55
0.60
0.65
0.70
0.75
0.80
φ
Figure 5: Home bias and risk adjusted excess return on local portfolio.
The reason why local investors tilt their portfolio toward domestic assets is because they know that,
on average, they earn abnormal returns on the local portfolio. Figure 5 illustrates the risk adjusted
difference between the expected return of the local portfolio, versus the expected return of the foreign
portfolio.17 The figure shows that the range of values of excess returns required to explain a significant
17
We do not model explicitly the market for risk-free bonds. However, all agents in the model are identical except for the
signal realization. This implies that no agent, either local or foreign, is willing to borrow or lend at the equilibrium risk-free
17
fraction of the bias is not unreasonable. For example, Coval and Moskowitz (2001) find that mutual
funds earn an excess return of 1.18 percent on local stocks compared to what they earn on non-local
stocks. Ivkovich and Weisbenner (2005) find that households earn an additional 3.2 percent on their
local stocks.
The model does a good job in explaining a significant fraction of the bias we observe in the data.
It has also a richer structure compare to previous papers in the literature. Thus, if we wanted to run
additional tests, we should check whether the portfolio implications are aligned with what we observe
in the data. This is discussed in the next section.
Some comments on the robustness of the results are in order. This section builds on several restrictive assumptions. Some of them are abandoned in later sections. Section 4 considers the case where
there is positive cross-asset return correlation between local stocks. 18 Not only this is a more realistic
assumption, but it also reduces the role played by differences in information about the relative performance of local assets. For instance, when the returns of local assets are perfectly correlated, the model
collapses to the case of one asset per country. Section 5 considers a different setup in which local and
foreign stocks are in fixed supply –except for unobserved liquidity shocks. This introduces a nontrivial
signal extraction problem: agents learn from the signals and market prices. The result described in this
section is robust to the previous generalizations.
3
Testable implications
This section lays down the main empirical implications of the model described before. The fact of
considering a framework with multiple assets distinguishes this paper from previous work and allows
for a richer set of implications. We find that not only is the present model able to explain a significant
fraction of the home equity bias observed in the data, but it is also consistent with several patterns
e
rate –denoted by Rf . This simplifies the computation of the equilibrium risk-free rate. Denote by RL
the expected return
e
e
on the portfolio of local stocks, σL
the standard deviation of the return on the portfolio of local stocks, RF
the expected
e
return on the portfolio of foreign stocks, σF the standard deviation of the returns on the portfolio of foreign stocks. The
risk adjusted excess return on the local portfolio (ERL )is computed as follows:
ERL =
e
RL
− Rf e
e
σF − (RF
− Rf ) .
e
σL
The previous equation computes the abnormal returns earned on a portfolio that has the same risk as the foreign portfolio
and the same Sharpe ratio as the local portfolio.
18
Another extension would be to consider the scenario where there is positive cross-country return correlation. This will
reduce the incentives to hold a diversified portfolio and therefore, it will help the model to explain a higher fraction of the
bias.
18
of foreign investment behavior. Albuquerque et al. (2005b) analyze the equity portfolios of a set of
individuals who traded through a large investment broker between 1991 and 1996. They find evidence
suggesting that the portfolio of foreign stocks is more diversified than the portfolio of local stocks. Local
agents hold a larger fraction of their foreign equity investments through mutual funds, compared to the
fraction of US stocks held through institutional investors. They also find that the share of foreign
investments is negatively correlated with the degree of concentration of the portfolio of local stocks.
US investors who invest heavily in a few domestic firms tend to allocate a lower fraction of their equity
investments in foreign stocks.
For simplicity, the present paper restricts attention to the case where domestic agents do not receive
information about foreign assets. In this scenario, local investors specialize in one domestic stock but
hold a perfectly diversified foreign portfolio: they invest the same amount in each foreign stock. 19 This
is in line with the first finding described above. With respect to the second finding, our model predicts
that agents who receive precise information about the relative performance of local stocks display a
bias towards local assets. In contrast, local investors who receive less informative signals hold a more
diversified local portfolio and display no bias: they invest half of their wealth in foreign assets.
A general prediction of the current setup is that more concentrated portfolios show better performance. When there is heterogeneity in the quality of the information received, the model predicts that
individuals with more precise information show a higher specialization and on average, enjoy higher returns. This is consistent with the evidence found by Ivkovick et al. (2004) for a sample of US households.
They show that the stocks purchased by individuals with concentrated portfolios display higher returns
than the stocks purchased by individuals with diversified portfolios. A similar finding is reported by
Kacperczyk et al. (2004) for a sample of US equity mutual funds.
Tesar and Werner (1995) report that the turnover rate on foreign equity portfolios is significantly
higher than the turnover rate on domestic equity portfolios. This finding has been used as evidence
against theoretical explanations that rely on informational asymmetries. If domestic agents receive
more precise information about domestic firms than foreign firms, they should trade more intensively
on local stocks.20 However, Warnock (2002) argues that these findings are based on data published
19
Allowing domestic agents to receive signals about foreign stocks induces a less diversified foreign portfolio. However,
it does not affect the proportion invested in foreign equity as long as the foreign signal is not so informative that local
investors fully specialize in one foreign asset.
20
The exception is Brennan and Cao (1997). They develop a model with asymmetric information that generates higher
turnover rates on foreign equity portfolios.
19
before reliable cross-border holdings data were available. He uses more accurate data and finds foreign
turnover rates significantly lower than the ones reported in Tesar and Werner (1995), and roughly
comparable to domestic turnover rates. Our model features lower foreign turnover rates but given the
previous evidence, we do not interpret this fact as a severe limitation of the model.
4
The case with positive return correlation between domestic assets
Given that the signals observed by local agents relate only to future relative performance of domestic assets, the role played by differences in information decreases with the correlation of local returns.
For example, if the returns of both assets were perfectly correlated, there would be no scope for differences in performance. That case would resemble the framework analyzed in previous studies (one
asset per country), which we already know does not help to explain the lack of international portfolio
diversification.
In order to allow for cross-asset return correlation, it is assumed that the values of ν 1 and ν2 are
drawn in two steps. The steps are summarized in Figure 6. First, Nature draws a value ν, which can
be interpreted as an aggregate shock. The variable ν satisfies ν =
ν1 +ν2
2 .
A realization of ν corresponds
to a point on the main diagonal in Figure 6. For instance, suppose that a point like A has been drawn.
In a second stage, Nature draws a value η that determines the relative performance of domestic assets.
In this case, η corresponds to a point on the line that goes through A, has a slope equal to -1, and is
contained in the unit square.
This approach introduces two mechanisms that help to generate positive cross-asset return correlation. One is to allow for a higher probability mass on the extreme values of ν. The other is to allow
for a distribution of η such that it increases the probability mass of realizations of (ν 1 , ν2 ) close to the
diagonal. When this happens, local assets tend to share a similar probability distribution and therefore
display unconditional return correlation.21 We maintain the assumption that local and foreign stocks
are ex ante identical. A formal description proceeds in the following paragraphs.
For a given realization of ν and η, ν1 is given by the following equations:
2νη
if ν < 0.5,
ν1 =
(2ν − 1) (1 − η) + η if ν > 0.5.
The value taken by ν2 is just 2ν − ν1 . This means that when the realization of η equals 0.5, the
21
It should be noticed that the return of both assets are uncorrelated once we condition on the realizations of ν 1 and ν2 .
20
ν2 6
1
@
A@s
@s
@
B
-
1
ν1
Figure 6: How positive cross-asset return correlation is generated.
returns of both domestic technologies share the same probability distribution. For values of η larger
than 0.5, technology 1 yields higher expected returns. For values of η below 0.5, technology 2 yields
higher expected returns. The random variable η follows a Beta distribution. This section maintains the
assumption that local assets have the same ex ante distribution of returns, which implies that η has a
mean of 0.5. This pins down one of the parameters of the Beta distribution. The remaining parameter
is used to control for the volatility of η.
The random variable ν has support [0, 1] and density f (ν; α), with
2αν+1−α
if ν < 0.5,
1− α
2
f (ν; α) =
2α(1−ν)+1−α
if ν > 0.5.
1− α
2
.
The above distribution collapses to the Uniform distribution when α = 0, and displays a probability
distribution shifted towards the corners when α < 0.
The random variables that determine aggregate and relative performance are assumed to be independent. Thus, the density function of ν and η can be written as
g β (η; ση ) 2αν+1−α
if ν < 0.5,
1− α
2
.
h (ν, η) =
g β (η; σ ) 2α(1−ν)+1−α
if
ν
>
0.5.
α
η
1−
2
The function g β (η; ση ) denotes the density function of a random variable with Beta distribution and
h
i
parameters 81 σ12 − 4 . The second term allows us to determine how much probability mass is allocated
η
21
near the extremes, i.e., when both assets pay high or low returns with certainty. The density function
over ν1 and ν2 is obtained after a change of variables.
2α(ν1 +ν2 )+1−α
ν1
1
gβ
;
σ
η
ν1 +ν2
1− α
2(ν1 +ν2 )
2
f (ν1 , ν2 ) =
α(2−ν1 −ν2 )+1−α
1−ν2
gβ
1
;
σ
η
2−ν1 −ν2
1− α
2(2−ν1 −ν2 )
2
if ν1 + ν2 < 1,
if ν1 + ν2 > 1.
This section maintains the assumption that both countries are identical. This means that foreign
assets are subject to the same return process as local assets. It is also assumed that returns from
domestic and foreign assets are uncorrelated. Before choosing the values of σ η and α, it is necessary
to determine what degree of correlation is consistent with the data. The capital asset pricing model
(CAPM) provides a simple framework in order to retrieve a sensible value. The CAPM states that:
Ri = Rf + βi (Rm − Rf ) + i ,
(3)
where Ri denotes the return of asset i, Rf denotes the risk free interest rate, Rm denotes market return
and i denotes the idiosyncratic shock to asset i. This setup captures a simple mechanism that generates
cross-asset return correlation: The return of every asset in the economy depends on a single aggregate
variable, i.e., the excess return of the market portfolio. The next step is to provide an interpretation
for the assets in our model. If the model were followed literally, each asset would correspond to a
portfolio of local firms. This approach implies a high level of aggregation, so we would expect to obtain
a strong cross-asset correlation. However, the same reason we utilized to abstract from informational
asymmetries about aggregate variables could be applied to those portfolios. That is why we would like
to interpret local and foreign assets as firms or specific industries. The fact that the paper considers
only two stocks per country is just a simplification necessary for tractability purposes.
If each asset stands for a firm, the results in Fama and French (1992) imply a zero cross-asset
correlation.22 They study a sample of 2,267 stocks and conclude that the average estimation of β is
not significantly different than zero. Fama and French (1993) sort individual stocks into 25 portfolios
according to firm sizes and book-to-market ratio. Their estimation of the single factor model produces a
mean return correlation across portfolios of 78 percent.23 Fama and French (1997) sort stocks according
to the industry they belong to. They construct 48 industry-portfolios and obtain a mean cross-asset
correlation of 63 percent.
22
23
The implicit assumption is that idiosyncratic shocks (denoted by i ) are independent across assets.
The mean R2 is taken as the estimated cross-asset correlation.
22
An alternative procedure to estimate the cross-asset return correlation is to use Equation (3) and
assume that both assets have a β of one, i.e., they are two representative assets. Then,
Corr (Ri , Rj ) =
V ar (Rm − Rf )
Cov (Ri , Rj )
=
=
σ (Ri ) σ (Rj )
V ar (Rm − Rf ) + V ar ()
1+
1
V ar()
V ar (Rm −Rf )
,
which shows that cross-asset correlation depends on the ratio of idiosyncratic risk to aggregate risk.
Campbell et al. (2001) use the CAPM structure to estimate return volatility at the market, industry,
and firm levels. Using their estimates, a correlation of 0.6 is obtained at the industry level, similar to the
value obtained in Fama and French (1997). As expected, the correlation at the firm level is significantly
lower, ranging from 0.19 to 0.25 depending on whether returns are computed on a daily or weekly basis.
The previous evidence illustrates that the correlation can take almost any positive value between 0
and 0.8, depending on how assets are defined. We follow Campbell et al. (2001) and choose a correlation
of 0.25 as the benchmark value. The baseline value of α is set to 0 and the baseline value of σ η is set
to 0.25. We also consider the case α = −6 and ση = 0.1, which generates a correlation of 0.45. Figure
7 reports the results. As expected, the fraction invested overseas decreases as the correlation increases.
But the bias can still be significant for reasonable levels of cross-asset return correlation. However, when
the cross-asset correlation is relatively high, there is less room for disagreement about asset returns and
agents tend to hold a more diversified portfolio.
5
Endogenous asset prices
The objective of this section is to illustrate that the main result of the paper does not depend on the
assumption that the supply of stocks is infinitely elastic, which implies constant stock prices. When
the last assumption is abandoned, equilibrium prices typically reveal valuable information. In order to
prevent prices from being fully revealing, it is necessary to allow for additional sources of uncertainty.
This is not an easy task once we depart from the standard environment with a CARA utility function and
Gaussian returns. In accordance with most of the previous finance literature, the model below assumes
that prices are only partially revealing because of the existence of supply shocks (noise traders). The
difference with respect to the previous literature is that we restrict attention to an ad hoc structure
of shocks that has the advantage of reducing the dimensionality of the price realizations that can be
observed in equilibrium. Instead of extracting information from a multidimensional continuum space,
23
Fraction invested in domestic assets
1.0
ρ=0
0.9
ρ = 0.25
0.8
0.7
ρ = 0.45
0.6
0.5
0.5
0.6
0.7
0.8
0.9
φ
Figure 7: Sensitivity of home equity bias to cross-asset return correlation
agents learn from a finite set of prices.24 We conclude the section by showing that a significant home
bias can still be observed as long as prices do not convey too much information.
The main features of the model are the following ones: As before, the world is composed of two
countries. There are two trees in each country. The trees display the same unconditional distribution of
dividends, i.e., they are ex ante identical. For simplicity, it is assumed that agents live for two periods. 25
Agents are initially endowed with exogenous income and shares of trees. It is assumed that every agent
is entitled to an equal amount of shares of local and foreign trees. There is a measure 1 of agents in
each economy. The last two assumptions imply that every agent is endowed with 0.5 shares of each
tree. Consumption goods are perishable. Agents can only allocate consumption across time and states
by trading shares of trees. As before, short-sales are not allowed.
24
The approach taken in this section is similar to Wallace (1992).
It easy to verify that the modeling strategy followed in the previous section implies that the fraction of savings invested
in local stocks is invariant to the time horizon. This allows us to compare the results of the present section with the findings
in the case of constant stock prices.
25
24
Trees pay dividends in the second period and then die. Tree i pays high dividends dh with probability
νi , and low dividends dl with probability 1 − νi . Dividend payoffs are independent across assets. Each νi
is drawn from a uniform distribution with support [0,1]. Agents do not observe the actual realizations
of ν’s but receive informative signals. The signal structure is the same as the one defined in Section 2.
The consumer’s optimization problem can be stated as follows:
X X X X
u (c0 ) + β
Pr (i, j, i∗ , j ∗ | I) u (ci,j,i∗ ,j ∗ )
M ax
a1 ,a2 ,a1∗ ,a2∗
∗
∗
(4)
i=l,h j=l,h i =l,h j =l,h
subject to
c0 = y + p1 (0.5 − a1 ) + p2 (0.5 − a2 ) + p1∗ (0.5 − a1∗ ) + p2∗ (0.5 − a2∗ ),
ci,j,i∗ ,j ∗
= d i a 1 + d j a 2 + d i∗ a 1 ∗ + d j ∗ a 2 ∗
f or i, j, i∗ , j ∗ ∈ {h, l} ,
am ≥ 0 f or m ∈ {1, 2, 1∗ , 2∗ } ,
where am denotes holdings of asset m, pm denotes the market price of asset m, y denotes the exogenous
income received in the first period, and I denotes the agent’s information set.
It is necessary to differentiate between two types of state realizations. There is a current state and a
future state. The current state is determined by the realization of ν1 , ν2 , ν1∗ , and ν2∗ . Given the signal
structure, if agents could pool all the information available, they would only be able to differentiate
between four possible state configurations, depending on which is the best asset in each country. 26 This
means that there are four possible observable current states. They are described in Table 1. On the
other hand, the future state realization is determined by the actual dividend shocks experienced by each
of the four trees. As before, there are 16 possible future states.
Current state
Description
I
ν 1 > ν 2 ; ν1 ∗ > ν 2 ∗
II
ν 1 > ν 2 ; ν1 ∗ < ν 2 ∗
III
ν 1 < ν 2 ; ν1 ∗ > ν 2 ∗
IV
ν 1 < ν 2 ; ν1 ∗ < ν 2 ∗
Table 1: Partitions of pooled information
26
This is due to the fact that the fraction φ does not depend on the absolute values of (ν 1 , ν2 ) or (ν1∗ , ν2∗ ). In other
words, the intensity of the signal is independent from the actual gap in expected relative performances.
25
5.1
A heuristic description of the model with partially revealing prices
Without additional sources of uncertainty, the current setup features fully revealing prices. For instance,
suppose that current state I has taken place. Thus, a majority of domestic agents has received a signal
favoring asset 1. The resulting higher demand of that asset translates into a higher relative price of
local stock 1. Thus, domestic and foreign agents would be able to infer which one is the best domestic
asset just by looking at market prices. A similar result would apply to foreign assets. This implies that
there would be no heterogeneity across agents. On top of the egalitarian distribution of endowments
assumed in this section, agents would share the same information set. Prices would adjust in such a
way that agents decide not to trade and keep half of their wealth in foreign assets. Figure 8 illustrates
the mapping from states to price vectors in this economy. Depending on the signal realization, each
asset can have either a high, or a low price, The former is denoted by p̄ F R , and the latter is denoted by
pF R .
¯
I = (ν1 > ν2 , ν1∗ > ν2∗ )
q
II = (ν1 > ν2 , ν1∗ < ν2∗ )
q
III = (ν1 < ν2 , ν1∗ > ν2∗ )
q
IV = (ν1 < ν2 , ν1∗ < ν2∗ )
q
R
p
~F
= p̄F R , pF R , p̄F R , pF R
I
¯
¯
-q
q
FR
R
, pF R , pF R , p̄F R
p
~F
II = p̄
¯
¯
-q
q
R
FR
p
~F
, p̄F R , p̄F R , pF R
III = p
¯
¯
-q
q
R
FR
p
~F
, p̄F R , pF R , p̄F R
IV = p
¯
¯
-q
q
I
II
III
IV
p~PI R
-q
p~PIIR
-q
p~PIIIR
-q
p~PIVR
-q
Figure 9: Partially revealing prices.
Figure 8: Fully revealing prices.
In order to make the problem more interesting, the current section features a non-trivial information
structure. It assumes that the supply of trees is subject to shocks. These shocks can be thought of
as asset demand that arises from unmodelled agents, or due to non-informational reasons. The typical
interpretation in the literature is that they reflect trades of investors faced with liquidity shocks. In
this scenario, a high price of one of the assets does not necessarily signal high expected dividends. It
is also possible that the asset has become valuable because its demand was hit with a large inelastic
component that left few shares available to the remaining agents.
Figure 9 illustrates how the model works in the case where the current state is characterized by
26
higher expected returns of local and foreign tree 1. This section studies a case in which there are four
possible combinations of supply shocks. One is that there is no noise, i.e., no shocks have taken place.
The remaining three alternatives are such that the equilibrium prices observed in those cases mimic
the equilibrium prices observed in the remaining states with no shocks. A solid line is used in Figure
9 to represent the mapping from states to equilibrium prices when the shocks have taken zero value.
A dashed line represents the mapping from states to prices when the shocks are non-zero. Agents face
a non-trivial signal extraction problem. They observe a vector of equilibrium prices but cannot realize
whether the prices reveal the actual ranking of local and foreign stocks, or they are just the result of
noise.
5.2
A formal description of the model and the equilibrium concept
Formally, each supply shock consists of a four dimensional vector. Component i represents the shock
to stock i. There are four possible supply shocks for each current state realization. This means that,
unconditionally, there are sixteen possible supply shocks. One of the four possible shock vectors is the
null vector, This is independent of the initial state that has been realized. When the null vector is
realized, the asset supplies are unaffected. However, the remaining shocks are such that the resulting
equilibrium prices can mimic prices observed in other states with zero shocks. Formally, denote by
{~
µij }4j=1 the set of possible supply shocks in current state i. Notice that µ
~ ij ∈ R4 ∀i, j, and µ
~ ii = ~0 ∀i.
Let us denote by p~i ∈ R4 the equilibrium price vector in state i with zero shocks. When the vector of
supply shocks takes a value µ
~ ij , the equilibrium price vector equals p~j , independently of the state i that
has been realized.
There is a probability q +
occur with probability
1−q
4 .
1−q
4
that the supply shock takes null values. All other shock realizations
The degree of informativeness of market prices is summarized in the value
taken by q. If q = 0 prices are fully uninformative. If q = 1, prices are fully informative. Prices are
partially revealing in all other cases.
Denote by ~λi ∈ R4 the vector of measures of agents in current state i. The first two components of
vector ~λi correspond to the fractions of domestic agents receiving signals 1 and 2, respectively. The last
two components correspond to the fractions of foreign agents receiving signals 1 ∗ and 2∗ respectively.
These measures are summarized in Table 2.
Let ~a (~
p, s) denote the vector of asset demands that solves optimization problem (4). Namely,
27
Local agents
Current state
Foreign agents
Signal 1
Signal 2
Signal 1∗
Signal 2∗
I
φ
1−φ
φ
1−φ
II
φ
1−φ
1−φ
φ
III
1−φ
φ
φ
1−φ
IV
1−φ
φ
1−φ
φ
Table 2: Measure of agents depending on the current state realization
k (~
~a (~
p, s) = [a1 (~
p, s) , a2 (~
p, s) , a1∗ (~
p, s) , a2∗ (~
p, s)]. Let Zij
p) denote the excess demand of asset k in
state i with supply shock ij and price vector p~. Formally,
k
Zij
(~
p) =
4
X
~λi (l) ~ak (~
p, ~s (l)) − (1 + µ
~ ij (l)) for i, j ∈ {I, II, III, IV } ,
k ∈ {1, 2, 1∗ , 2∗ } ,
l=1
where ~x (l) denotes the component l of vector ~x. The term ~s denotes the vector of possible signal
realizations. The current state, indexed by i, determines the measure of agents receiving a signal favoring
local tree 1, as well as the possible values that the supply shocks may take. The shocks, indexed by j,
determine the available net supply of assets once the inelastic component has been incorporated.
Definition 1 A rational expectations equilibrium (REE) consists of a set of price vectors p~ I , p~II , p~III ,
p~IV , and individual demands ~a (~
p, s) such that:
(i) ~a (~
p, s) solves each consumer’s optimization problem for market prices p~ and individual signal
s.
k (~
(ii) Markets Clear: Zij
p) = 0 ∀ i, j ∈ {I, II, III, IV } where k ∈ {1, 2, 1∗ , 2∗ }.
(iii) Agents update their beliefs using their private signal and market prices according to Bayes’
rule.27
Given the particular uncertainty structure assumed in this section, the equilibrium satisfies an extra
condition:
• Prices are partially revealing: p~ = p~j whenever µ
~ =µ
~ ij
27
∀i, j ∈ {I, II, III, IV }.
Appendix contains a formal description of the beliefs’ updating scheme.
28
Unfortunately, the above problem does not allow for a closed-form solution. This implies that the
equilibrium must be found using numerical techniques. Given that the trees are ex ante identical and
that the optimization problems of domestic and foreign investors are entirely symmetric, it is sufficient
to solve for equilibrium prices in two cases: first, when the price of domestic asset 1 is higher than the
price of domestic asset 2 and ν1 > ν2 ; second, when the same ranking of local prices is combined with
ν1 < ν2 . Local agents do not receive signals about foreign assets, so the results are invariant to the
ranking of prices in foreign markets. We show the results for the first case. The second one features
higher levels of home bias in the range of values we are interested in, i.e., when prices do not reveal too
much information.28
The model is solved assuming that agents share a logarithmic utility function. The low dividend
value is set to 0.8 and the high dividend value is set to 1.2. Finally, q and φ are left as free parameters.
The first one controls for the degree of informativeness of market prices. The second one determines
how informative individual signals are. The results are shown in Figure 10.
If prices are very informative or individual signals are not sufficiently informative, local agents invest
roughly half of their portfolios in domestic assets. This corresponds to values of q close to 0, or values
of φ close to 0.5. As prices become less informative or individual signals become more informative,
investors start to bias their portfolio toward domestic securities. The graph shows that the home equity
bias can be quite significant when prices are not very informative.
The graph also shows that the bias may decrease with the precision of the signal if the latter
is already sufficiently informative. The reason is the following. Agents that display the strongest
preference toward domestic assets are the ones receiving an incorrect signal. Their beliefs mirror the
beliefs of agents receiving the correct signal, but the price of the asset for which they expect higher
dividends is lower. As the signal becomes more precise (φ increases), the fraction of individuals with this
strong preference for local assets decreases, driving down the overall fraction invested in local assets.
Table 3 illustrates the magnitude of supply shocks for a case where 65 percent of domestic portfolios
are composed of local assets. It shows that it is not necessary to consider extreme shocks in order
to observe a significant level of home bias. Even though the model presented in this section relies on
ad-hoc assumptions, we conjecture that extending the model to less arbitrary distributions of shocks
or allowing for other sources of uncertainty that mask the current state realization, will not lead to
28
The reason is that a majority of domestic agents receive a signal favoring the cheapest local asset, which reinforces
the desire to invest locally
29
Figure 10: Fraction invested in local assets as a function of the information conveyed by prices and
individual signals
different qualitative conclusions. The difference is that in a more general case, agents need to learn
over a fourth dimensional space. This is due to the fact that there are four prices that convey valuable
information. The mechanism leading to partially revealing prices would not be qualitatively different
from the one assumed in this section, but the level of complexity would be significantly larger.
6
The shadow price of the short-sales constraint
The restriction on short-sales captures in a simple way the fact that short-selling is a costly activity. A
more general formulation can be developed assuming, for example, that agents are required to pay a fee
in order to hold short positions. The fee should include not only the direct cost derived from the equity
loan, but also the implicit cost due to legal restrictions and the extra risk incurred by short-selling (like
an early recall). This section finds the implicit fee that prevents agents from selling short.
It is assumed that agents pay a fee τ whenever they short-sell. The fee is proportional to the amount
30
Stated mimicked
Asset 1
Asset 2
Asset 1∗
Asset 2∗
I
0.000000
0.000000
0.000000
0.000000
II
-0.080300
0.010677
-0.358955
0.437354
III
-0.358955
0.437354
-0.080300
0.010677
IV
-0.439255
0.448031
-0.439255
0.448031
Table 3: Supply shocks to domestic and foreign assets when q = 0.1 and φ = 0.65 Expressed as a
fraction of the average supply of each asset.
sold short. The investor’s optimization problem is set out below. The only difference with respect to
the optimization problem in the baseline model is the individual’s budget constraint. Without loss of
generality, we consider the problem of an agent with a signal realization that favors asset 1.
V (ω) =
M
ax
0 0
k10 ,k2 ,k1∗ ,k20 ∗
subject to
where
u (c) + βE V ω 0 | s = 1
(5)
c + k10 1 + I k10 τ + k20 1 + I k20 τ + k10 ∗ 1 + I k10 ∗ τ + k20 ∗ 1 + I k20 ∗ τ = ω,
−1 if x < 0,
I (x) =
0
if x > 0.
The value of τ consistent with the observation that agents do not hold short positions can be retrieved
from the first order conditions in the problem with short-sales constraints. 29 The results shown in Figure
11 are based on the benchmark parameterization used in Section 4, where a cross-asset return correlation
of 0.25 is assumed. The graphs illustrate that the model is capable of generating significant levels of
home bias without imposing high costs on short sales. For instance, when the fraction invested in local
stocks is around 75 percent, the implicit fee is 2.5 percent.
29
As it was said before, policy functions are linear in wealth. Thus, the policy function of asset i can be written as
(ω) = αi ω with i ∈ {1, 2, 1∗ , 2∗ }. There are 16 future state realizations depending on the productivity shock faced by
each local and foreign technology. Let Ahi denote the productivity shock received by technology i in state h. Since we
consider the case where the local signal favors technology 1, the short-sales constraint binds when the investor wants to
short sell stocks of technology 2. The implicit value of τ consistent with no-short-sales is therefore obtained as follows:
ki0
τ =1−β
16
X
Pr (j | s = 1)
h=1
Ah
P 2
i=1,2,1∗ ,2∗
31
αi Ahi
.
0.9
3
0.8
2
0.7
1
0.6
0
Fraction investested in domestic country
Shadow price of short-sales constraint (%)
4
0.5
0.5
0.6
0.7
0.8
0.9
φ
Figure 11: Home bias and shadow price of short-sales constraint in the baseline model with crossasset return correlation of 0.25.
7
Analytical characterization in the CARA-Gaussian framework
This section considers a model that shares the blueprints of the framework laid down in Section 2 and
has the advantage of being analytically tractable. However, it uses a more complex and nonstandard
structure for macroeconomic analysis. In addition, it generates a volatile fraction of foreign investments.
For these reasons it is not taken as our benchmark model.
The setup analyzed in this section assumes that agents live for two periods. They consume at the
end of the second period. Only investment activities take place in the first period. As before, there
are two technologies available in each country. Production technology is of the “AK” type, but the
productivity shock follows a different process. The shock to technology i (A i ) consists of two parts,
Ai = µ i + i ,
32
where both µi and i are normally distributed.30 More precisely,
µi ∼ N θ, σµ2 ,
i ∼ N 0, σ2
∀i = 1, 2, 1∗ , 2∗ .
.
This section maintains the assumption that the signal received by local agents reveals information
about relative performance of local technologies, but not about the aggregate performance of the home
country. Formally, each local agent observes a private signal s that satisfies the following:
s = µ1 − µ2 + ξ,
ξ ∼ N 0, σξ2 .
The realization of ξ is idiosyncratic. It is assumed that agents cannot pool the signals. They only
observe their own signal. Each foreign investor receives a private signal s ∗ , with
s∗ = µ∗1 − µ∗2 + ξ.
For simplicity, it is assumed that all normal variables introduced before are uncorrelated. The
final modification with respect to our benchmark framework is that investors display preferences with
constant coefficient of absolute risk aversion. The utility function of local and foreign agents has the
following form:
u (c) = −e−λc .
Consider now the problem of a local investor endowed with ω units of the good and signal s. His
objective is to maximize his expected utility of consumption, i.e.
h
i
E −e−λ[k1 A1 +k2 A2 +k1∗ A1∗ +(ω−k1 −k2 −k1∗ )A2∗ ] | s ,
subject to
ki ≥ 0 ∀i = 1, 2, 1∗ .
The demand of the fourth asset (foreign technology 2) is obtained as a residual. The preferences,
dividend distribution, and signal structure assumed in this section allow us to reduce the optimization
30
Since the only information agents receive is about µi , the component i determines how useful that information is.
33
problem to maximizing the certainty equivalent of consumption. After rearranging terms, the objective
function simplifies to the following expression:
E [A1 − A2∗ | s]
h
i
E [A2 − A2∗ | s] λ h
−
k1 k2 k1∗ ω
k1 k2 k1∗
E [A1∗ − A2∗ | s] 2
E [A2∗ | s]
k1
k2
ω Σ
.
k1∗
ω
i
The matrix Σ denotes the covariance matrix of [A1 − A2∗ , A2 − A2∗ , A1∗ − A2∗ , A2∗ ] conditional on
the information conveyed by the signal. The conditional expectations and covariance matrix can be
found using the projection theorem.31
and
4
σµ
2
2σµ +σξ2
4
σµ
2 +σ 2
2σµ
ξ
2 σµ2 + σ2 −
σ2 + σ2 +
µ
Σ=
σµ2 + σ2
− σµ2 + σ2
µ
σµ2
+
E [A1 − A2∗ | s]
σµ2 + σ2
ξ
σ2
− σµ2 + σ2
2
2
2
2
σµ + σ
− σµ + σ
− σµ2 + σ2
2 σµ2 + σ2
− σµ2 + σ2
σµ2 + σ2
σ4
µ
σµ2 + σ2 + 2σ2 +σ
2
µ
ξ
4
σµ
2 σµ2 + σ2 − 2σ2 +σ
2
− σµ2 + σ2
2s
σµ
2 +σ 2
2σ
µ2
−σµ s
2σ2 +σ2
µ
E [A2 − A2∗ | s]
=
E [A1∗ − A2∗ | s]
∗
E [A2 | s]
0
θ
.
The expressions for the conditional expectations show that the expected return of asset 1 is corrected
upward upon the arrival of a positive signal, while the expected return of asset 2 is corrected downward.
Also, the main diagonal of Σ shows that the availability of some information about domestic assets
drives their perceived variance down compared to that of foreign assets. The domestic signal does not
carry any information about foreign assets, so its perceived probability distribution coincides with the
unconditional distribution.
31
Consider two normally distributed random vectors, say X and S.
ΣX,X ΣX,S
µX
X
,
∼N
ΣS,X ΣS,S
µS
S
The distribution of X given S = s is also normal.
−1
(X | S = s) ∼ N µX + ΣX,S Σ−1
S,S (s − µS ) , ΣX,X − ΣX,S ΣS,S ΣS,X
34
Equation (6) reports the optimal investment behavior in the unconstrained problem, i.e., when
agents do not face short-sales constraints.
2
σµ
k
1
(σµ2 +σ2 )(2σµ2 +σξ2 )−2σµ4
1 ω s
2
−σµ
.
k2 = 1 +
2
4
2
2
2
4 λ (σµ +σ )(2σµ +σξ )−2σµ
k1∗
1
0
(6)
If the signal is not very informative (σξ2 is high), asset holdings resemble the perfectly diversified
portfolio, where an equal amount is invested in each asset. A similar result holds if investors are highly
risk averse (high λ) or returns are volatile (high σµ2 + σ2 ). However, it is easy to verify that agents
always allocate half of their portfolios in domestic assets, regardless of the signal realization.
In the constrained problem, the solution coincides with (6) whenever the signal does not take extreme
values. Equation (7) describes the solution in the case where the short-sales constraint is binding for
one of the local assets.
2 + σ2
2
σ
2
k
σ
1
µ
,
µ |s| + ω 2σµ2 + σξ2
i=
4
σµ
2
2
λ
2
2
2
2
4
σ
+
σ
−
∗
3 σµ + σ 2σµ + σξ − 2σµ
−1
k1
2 +σ 2
µ
2σµ
(7)
ξ
where
i
i
h
2
2σµ + σξ2 − 2σµ4
ωλ σµ2 + σ2 2σµ2 + σξ2 − σµ4
,
i = 1 if s ∈
,
4
σµ2
σµ2
h
h
i
i
2
2
2 + σ2
2 + σ2
2 − 2σ 4
2 − σ4
λ
σ
ωλ
σ
2σ
+
σ
2σ
+
σ
µ
µ
µ
µ
µ
µ
ξ
ξ
ω
.
i = 2 if s ∈ −
, −
4
σµ2
σµ2
ωλ
h
σµ2 + σ2
It is easy to check from the previous equation that the fraction invested in local assets grows as the
signal increases in absolute value. Agents fully specialize in one of the domestic assets when the signal
received is sufficiently large in absolute value.
The previous solution shows a result also observed in the benchmark model: The bias decreases with
the degree of risk aversion (λ). Similarly, we may also expect to observe more diversified portfolios as
asset returns become more volatile. The following proposition shows that this is not always the case.
Proposition 2 The home equity bias increases with σµ2 if 2
3σ 2 σξ 2 − 4σµ 4 + 2σµ 2 σµ 2 σξ 2 + 2 σµ 2 + σξ 2 σ 2 > 0.
35
σµ 2 + σ 2
2σµ 2 + σξ 2 − σµ 4 ×
Lemma 3 When σµ2 is sufficiently small, the home equity bias increases with σµ2 when σµ2 <
q
and decreases with σµ2 when σµ2 > 34 σ σξ .
q
3
4 σ σξ
The explanation is that changes in σµ2 induce a horse race between two effects: On the one hand,
as the component of the return from which agents learn become more volatile, there is more room for
a wider dispersion of beliefs at the individual level. This increases the bias. On the other hand, more
volatile returns induce a stronger desire to hold diversified portfolios. This discourages full specialization
in one of the local stocks and, henceforth, it reduces the bias.
36
8
Conclusions
There is pervasive evidence that individuals and institutional investors favor stocks of their own country. In addition, empirical studies show that there exists a home equity bias within US boundaries.
Households and mutual funds prefer stocks of proximate companies. These studies also show that the
returns agents enjoy on local stocks exceed the returns on non-local stocks. This suggests that the
lack of portfolio diversification is based on rational behavior. It also points towards the presence of
informational asymmetries in financial markets.
This paper develops a theoretical model that can explain a significant fraction of the bias observed
in the data. The model differs from the standard theory in three aspects. First, it considers the case
of multiple stocks per country. Second, it assumes that local investors are able to collect more precise
information about the ranking of local stocks than that of foreign stocks. Third, it assumes that shortsales are costly. In this environment, each domestic investor displays a strong preference for certain local
stocks. When the information collected is sufficiently precise, local investors find it convenient to finance
purchases of the perceived good local stocks by selling short the perceived bad local stocks. However, if
the cost of short-selling is high, domestic investors decide to sacrifice the diversification services provided
by their foreign investments in order to concentrate their equity portfolio in the (domestic) stocks that
are thought to offer higher expected returns. Unlike previous papers in the literature, the underlying
mechanism that explains the bias for local stocks is based on first order effects, i.e., differences in
expected returns. This explains the ability of the model to generate significant quantitative results. In
addition, the model has several testable implications regarding portfolio behavior that are in line with
previous empirical studies.
We show that the strong home equity bias implied by the model is robust to several changes in
the baseline specification. However, there was one extension that was not pursued in the paper: the
case of persistence in stock dividends. The introduction of permanent shocks has a double effect. On
the one hand, it increases the power of private information. The latter can now be used to forecast
the future stream of returns. Without persistence, it only helps to predict next period returns. This
effect strengthens the mechanism that generates home bias. On the other hand, there is more public
information available. If dividends have a persistent component, agents can learn from past realizations
of dividends. This reduces the role of private information and undermines the incentives to invest
37
heavily in local stocks.32 However, this extension poses two challenges from a technical point of view.
First, it can only be solved under a recursive structure. The problem becomes intractable if agents need
to keep track of all past dividend realizations and signals received in order to compute their beliefs.
Second, it requires dealing with multiple state variables with continuous domain.
32
This conclusion depends on the fact that stock prices adjust to the information contained in past dividend realizations.
If prices are constant over time, as in the “AK” model, the adjustment is made through quantities. In this case, the
fraction invested overseas is low on average, but can display a high volatility.
38
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42
A
Beliefs updating scheme when agents learn from signals and prices
Agents need to infer the probability distribution over future states before solving their optimization
problem. They receive two pieces of information: prices and individual signals. Both of them reveal
information regarding the relative values of ν1 , ν2 , ν1∗ and ν2∗ . Agents’ beliefs consist of the expected
probability distribution over future states conditional on the information received. 33 In order to simplify
the exposition, it is assumed that the market price vector corresponds to the equilibrium prices agents
would observe in current state I without supply shocks, i.e., p~ = p~I . It is straightforward to generalize
the formulas to other cases. The formal expression for the expected probability of future state i, j, i ∗ , j ∗
given prices p~ and private signal s is illustrated below.
E [P r (i, j, i∗ , j ∗ ) | p~, s] =
Pr (ν1 > ν2 ; ν1∗ > ν2∗ )
1−q
4
E [Pr (i, j, i∗ , j ∗ , p~, s)]
=
Pr (~
p, s)
+ q Pr (s | ν1 > ν2 ) E [Pr (i, j, i∗ , j ∗ ) | ν1 > ν2 ; ν1∗ > ν2∗ ] +
∗ ∗
∗
∗
Pr (ν1 > ν2 ; ν1∗ < ν2∗ ) 1−q
4 Pr (s | ν1 > ν2 ) E [Pr (i, j, i , j ) | ν1 > ν2 ; ν1 < ν2 ] +
∗ ∗
∗
∗
Pr (ν1 < ν2 ; ν1∗ > ν2∗ ) 1−q
4 Pr (s | ν1 < ν2 ) E [Pr (i, j, i , j ) | ν1 < ν2 ; ν1 > ν2 ] +
∗ ∗
∗
∗
Pr (ν1 < ν2 ; ν1∗ < ν2∗ ) 1−q
4 Pr (s | ν1 < ν2 ) E [Pr (i, j, i , j ) | ν1 < ν2 ; ν1 < ν2 ]
Pr (ν1 > ν2 ; ν1∗ > ν2∗ ) 1−q
4 + q Pr (s | ν1 > ν2 ) +
Pr (ν1 > ν2 ; ν1∗ < ν2∗ ) 1−q
4 Pr (s | ν1 > ν2 ) +
Pr (ν1 < ν2 ; ν1∗ > ν2∗ ) 1−q
4 Pr (s | ν1 < ν2 ) +
Pr (ν1 < ν2 ; ν1∗ < ν2∗ ) 1−q
4 Pr (s | ν1 < ν2 )
The second equation above uses the law of conditional probabilities and the third one uses Bayes’
rule. Every current state realization could lead to the observed market price p~. Thus, when agents
compute their beliefs, they span over the four possible current states. The first element in each term
on the numerator denotes the a priori probability of being in each current state. The second and third
components capture the probability of observing prices p~ and signal s for each current state. Finally,
the fourth component computes the expected probability that the future dividend shocks take values
i, j, i∗ , j ∗ for each current state realization.
33
It is sufficient to compute the expectation of these probabilities because the latter enter linearly in the individual’s
first order conditions.
43
B
Proof of proposition 2
Denote by Φ the fraction invested in local assets. There is a home bias when Φ > 0.5. The aggregate
fraction invested in local assets depends on the actual realizations of µ1 and µ2 . The latter conditions
the distribution of information across agents. For instance, if the difference between these two variables
is large, a high fraction of local investors will receive extreme signals. In order to allow for a general
statement, we consider the ex ante expectation of Φ. That is, the unconditional expected fraction
invested in local assets. The latter is computed as follows:
#
"
Z ∞ Z ∞
1
(µ1 − θ)2 (µ2 − θ)2
E (Φ) =
×
exp −
−
2
2σµ 2
2σµ 2
−∞ −∞ σµ 2π
R
−s̄ f (s | µ1 , µ2 ) dsdµ1 dµ2 + R −¯s g (s) f (s | µ1 , µ2 ) dsdµ1 dµ2 + R s̄ f (s|µ1 ,µ2 ) dsdµ1 dµ2 +
2
−∞
−s̄
−s
,
¯
R
R
s̄
∞
g
(s)
f
(s
|
µ
,
µ
)
dsdµ
dµ
+
f
(s
|
µ
,
µ
)
dsdµ
dµ
1
2
1
2
1
2
1
2
s
s̄
¯
where
g (s) =
"
1
#
σµ2
2|s| + 2σµ2 + σξ2 σµ2 + σ2
ωλ
2σµ2 + σξ2 − 2σµ4
h
i
2
2 − 2σ 4
2 + σ2
2σ
+
σ
λ
σ
µ
µ
µ
ξ
ω
s=
¯ 4
σµ2
h
i
ωλ σµ2 + σ2 2σµ2 + σξ2 − σµ4
s̄ =
.
σµ2
3 σµ2 + σ2
We are interested in the derivative
∂E (Φ)
.
∂σµ 2
(B.1)
It is easy to verify that the derivatives of the threshold values (s and s̄) with respect to σµ2 cancel out
¯
and therefore, do not play a role. This implies that the sign of Equation (B.1) depends on the sign of
the derivatives of the values taken by g(s). Without loss of generality, we consider now the case where
s > 0.
∂g (s)
=
∂σµ2
2s
λω
h
i
3σ2 σξ2 − 4σµ4 + 2σµ2 σµ2 σξ2 + 2 σµ2 + σξ2 σ2
i2
h
3 σµ2 + σ2 2σµ2 + σξ2 − 2σµ4
The sign of the expression above is ambiguous. However, it is possible to find a sufficient condition
that can help us to identify cases where the derivative takes positive values. If the sign is positive for
44
s = s̄, then it must be positive for all possible signals belonging to the range [s , s̄]. This substitution
¯
yields the sufficient condition stated in the text. From the previous equation, it is easy to see that when
σµ2 is sufficiently small, the first term in the numerator dominates the entire expression and, therefore
it determines the sign of the derivative.
45
@FedFRASER