ACTA VŠFS, 2/2015, vol. 9 115
Information Acquisition and Excessive Risk:
Impact of Policy Rate and Market Volatility
Získávání informací a nadměrné riziko: role
úrokových sazeb a volatility na trzích
VOLHA AUDZEI
Abstract
Excessive risk-taking of financial agents drew a lot of attention in the aftermath of the
financial crisis. Low interest rates and subdued market volatility during the Great Moderation
are sometimes blamed for stimulating risk-taking and leading to the recent financial
crisis. In recent years, with many central banks around the world conducting the policy of
low interest rates and mitigating market risks, it has been debatable whether this policy
contributes to the building up of another credit boom. This paper addresses this issue by
focusing on information acquisition by the financial agents. We build a theoretical model
which captures excessive risk taking in response to changes in policy rate and market volatility.
This excessive risk takes the form of an increased risk appetite of the agents, but also
of decreased incentives to acquire information about risky assets. As a result, with market
risk being reduced, agents tend to acquire more risk in their portfolios then they would
with the higher market risk. The same forces increase portfolio risk when the safe interest
rate is falling. The robustness of the results is considered with different learning rules.
Keywords
Rational Inattention, Interest Rates, Financial Crisis, Risk-taking
Abstrakt
Nadměrné podstupování rizika zástupci finančního trhu získalo po nedávné finanční
krizi mnoho pozornosti. Nízké úrokové sazby a tlumená volatilita na trhu během období
Velkého zklidnění (Great Moderation) jsou někdy obviňovány ze stimulace podstupování
rizika, které vedlo k nedávné finanční krizi. V posledních letech, kdy centrální banky po
celém světě provádí politiku nízkých úrokových sazeb, a zmírňují tržní rizika, je akutní
otázka, zda tato politika nepřispívá k vytvoření další úvěrové konjunktury. Náš článek se
zabývá tímto tématem z pohledu získávání informací zástupců finančního trhu. Sestavíme
teoretický model, který zachycuje nadměrné podstupování rizika v reakci na změny
úrokové sazby a/nebo tržní volatility. Toto nadměrné riziko získává formu zvýšené chuti
zástupců finančního trhu riskovat, ale také snížené motivace získávat informace o rizikových
aktivech. V důsledku sníženého tržního rizika, mají zástupci finančního trhu tendenci
hromadit více rizika ve svých portfoliích než v případě s vysokým tržním rizikem. Stejné
mechanismy zvyšují riziko portfolia, když je úroková sazba snížena. Robustnost získaných
výsledků je posuzována z hlediska různých pravidel učení.
Klíčová slova
Rational Inattention, úrokové sazby, finanční krize, podstupování rizika
ACTA VŠFS, 2/2015, vol. 9116
JEL Codes
E44, E52, G14, D84
Introduction
The paper is motivated by the debate about whether a low policy rate has contributed
to the recent financial crisis and if the ongoing policy of low interest rates is contributing
to the building up of a new financial bubble. There are voices among policy-makers and
academics suggesting that one could observe worrying tendencies of risky asset accumu-
lation1
. There is evidence of an increased risk appetite, which is believed to be attributed
to accommodative monetary policy conditions and subdued market volatility (for the
evidence see, e.g., Bank for International Settlements 2014). At the same time both proponents
and opponents of a low policy rate do not have clear answers as to what tools a
central bank should use in order to maintain price stability and stimulate output growth
on the one side, and financial stability on the other (for a recent debate on this see Stein
2013 and Bernanke 2013).
The question asked in this paper is if endogenous information acquisition can drive overaccumulation
of risk when safe interest rates or market volatility is reduced. It is common
that in portfolio choice models with rational expectations, investment into a risky asset
is linear in excess return. In our model, when the policy rate or market volatility falls, risk
accumulation in the economy increases in a nontrivial way.
We capture the excessive risk accumulation by modeling information decisions. Financial
agents invest in information to reduce the variance of their forecasts. We show that when
market volatility declines, agents invest into information less and acquire more of a risky
asset. This results in an even larger portfolio risk than in the economy with higher market
volatility. With interest rates being lowered, our model not only captures the standard
“search-for-yield” effect, where financial intermediaries invest more into risky assets. We
also show an increase in agents’ ignorance about the asset quality. With low information
investment and large risky asset holdings it implies a larger portfolio risk accumulation.
The main contribution of our model to the current debate is that it mimics excessive
risk-taking of financial agents. We show that average risk monitoring declines with lower
interest rates despite the growth in excess return on a risky asset. Another result is overaccumulation
of risky assets in a low risk environment. That is to say with low variance
of risky asset return, agents take more risk in their portfolio than they would have with a
high risky asset variance. This effect is explained in our model with just one deviation from
rational expectations: agents do not know the future return, but only its distribution, i.e.
there is no assumption of agents’ irrationality. In our model, this result is driven by a decline
in risk monitoring in low risk environment. Combined with an increase in risky asset
acquisition, it results in higher portfolio variance compared to high variance environment.
1	 For the evidence see Stein (2013); the recent examples of uncertainty among policy makers could be found
in articles by Chris Giles "Central Bankers Say They Are Flying Blind " and "IMF warns on risks of excessive
easing" in The Financial Times, April 17, 2013.
ACTA VŠFS, 2/2015, vol. 9 117
To check the robustness of the results, in the spirit of Nieuwerburgh and Veldkamp (2010)
we consider two alternative learning functions, a linear and an entropy based. The rise in
portfolio risk when the safe interest rate falls is robust to a learning rule specification. The
increase in risk with falling market volatility is more pronounced in a linear learning rule.
1	 Related Literature
Our study relates to the several stands of literature. First, there is the literature on the role
of interest rates in mitigating or stimulating asset booms, in particular papers providing
empirical evidence that easier monetary policy is associated with higher risk-taking.
Maddaloni (2011) concludes that, for the euro area and US, low short-term interest rates
cause softening of the banks’lending standards. Additional support for a risk-taking channel
of monetary policy can be found in Gambacorta (2009) and Ongena and Peydro (2011).
Adrian et al. (2010) find empirical support for the notion that monetary policy effects
the supply of credit, operating through the term spreads; and that monetary policy can
influence risk appetite. Ahrend (2010) focuses on a different aspect of the financial imbalances
- on excessive asset prices growth, and finds that low interest rates cause growth
in some asset prices in OECD countries, particularly on the housing market. Detken and
Smets (2004) come to the similar conclusion that low policy rates coincide with asset price
booms. The evidence on the dynamic interaction between stock prices and Federal Reserve
policy rate is provided by Laopodis (2010). White (2012) discusses the "unintended
consequences" of easy monetary policy, among which are misallocation of credit and
structural changes in the financial sector, e.g. movements from traditional banking model
to shadow banking. Statistical evidence that a long period of low interest rate and low
market volatility have contributed to excessive risk-taking is summarized in the Annual
Report of the Bank for International Settlements (2014).
There are theoretical studies focusing on the channels through which monetary policy
affects risk-taking or asset prices. Taylor (2007 and 2010) suggests that the Fed´s low rates
stimulated a house price boom through credit growth. The several mechanisms through
which the risk-taking channel of monetary policy could work are mentioned in Borio and
Zhu (2008). In particular, search-for-yield implies that low interest rates result in a low
return on the safe assets, which pushes investors to accumulate more of the risky ones in
the search for an acceptable portfolio return. Also low interest rates imply a lower discount
factor for evaluation of assets or income flows, causing higher risk tolerance. Our model
incorporates both of these channels within the bank’s portfolio choice problem.
The banks risk monitoring incentives in connection with monetary policy are studied in
the model of Dell Ariccia et al. (2010). Their findings depend on the banks capital structure
and the possibility of adjusting it. They conclude that with a flexible capital structure
monetary policy easing leads to higher leverage and risk-taking. Their approach, however,
is different from that pursued in this paper in several respects. They concentrate on a
partial equilibrium model, where banks choose the probability of loan repayment subject
to costs. Therefore, in their model banks do not learn about the asset quality, but invest
to increase return probability. We build a general equilibrium model where banks are uncertain
about the risky asset return, but might invest in reducing their uncertainty. That
ACTA VŠFS, 2/2015, vol. 9118
is, learning does not influence the return probability, but makes banks more informed.
Therefore, we capture two aspects of risky behavior - investment in an asset known to be
risky and investment into learning about the asset quality.
Another strand of literature our study is related to is dedicated to the learning and expectation
formation and relaxation of the assumption of rational expectations. Among
the papers to support the importance of imperfect expectations and learning are Boz
and Mendoza (2010), Bullard et al. (2010), Kurz and Motolese (2010), Lorenzoni (2009),
Adam and Marcet (2010). Empirical support for the role of imperfect expectations can be
found in Fuhrer (2011) and Beaudry et al. (2011). In this paper we incorporate the idea that
agents do not have perfect foresight and have to form subjective expectations about risky
asset return. We use the approach of Nieuwerburgh and Veldkamp (2010) to model the
banks decisions to invest in learning about the risky asset. In Nieuwerburgh and Veldkamp
(2010), the investor draws an additional signal about asset return, and pays for an increase
in the signal precision before observing it. We modify their formulation for information
acquisition, so that in our model agents select the information budget depending on risk
premia and market volatility.
To conclude, our study is motivated by rich empirical evidence. Our model explores causalities
between monetary policy and agents’ risk-taking. We also show that prolonged
periods of low interest rates or low risk lead to excessive accumulation of risk.
The remainder of the paper begins with analysis of a partial equilibrium model to describe
the intuition for the main results. In section 3 the financial sector is described, and the
intuition for excessive risk-taking is presented in section 4 within a partial equilibrium.
In section 5 we complete the model for general equilibrium and then proceed with the
calibration, simulations and discussion in section 6. The last section concludes.
2	 The Model of Financial Sector
Consider a model with a financial intermediary, bank, a manufacturing firm and a household.
The assets in the economy are manufacturer claims (a risky asset) and reserves (a safe
asset). The risk in manufacturer claims comes from the uncertainty about future productivity.
All the agents in the economy know the productivity distribution. The household puts
savings in the bank (in the form of investment), and the bank transfers all its profit back to
the household. The safe and risky interest rates are set by the market.
The bank is risk-averse, which is motivated by the fact that banks are often subject to regulations
and have reputational concerns for the safety of their deposits. We then expand the
model and grant financial intermediary access to a noisy signal about future productivity.
This signal helps the agents to reduce the variance of their forecast. Yet they have to pay
for it. Banks are Bayesian, they form forecasts of risky returns as a weighted average of
their prior and the signal.
We abstract from any nominal variables in the model. All the prices and returns are real.
In what follows, we present the model set-up. We start with a partial equilibrium model
ACTA VŠFS, 2/2015, vol. 9 B119
to illustrate the mechanism of the excessive risk-taking and information acquisition. Then
we simulate general equilibrium model to study the model dynamic and potential role of
interest rates feedback2
.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next period net return:
	
5
We abstract from any nominal variables in the model. All the prices and returns are real.
In what follows, we present the model set-up. We start with a partial equilibrium model
to illustrate the mechanism of the excessive risk-taking and information acquisition.
Then we simulate general equilibrium model to study the model dynamic and potential
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next period net return:
(1)
where ρ is the risk aversion parameter, ktb is the bank’s risky asset holdings and Πt+1
stands for the next period return. That is, portfolio variance is costly and the bank,
therefore, has incentives to reduce it. The next period return consists of the return on
the bank’s portfolio minus the information budget:
(2)
where dt is household investment, R,r and Rs are respectively gross returns from risky
and safe assets, bt is the information budget selected by the bank. The bank’s future
return depends on the amount of funds it has for investment - dt,and from a composition
of its portfolio - quantity of risky asset, ktb: Note that the return is reduced by the
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger demand fo
a risky asset, it drops, potentially offsetting higher risk appetite.
	(1)
where
5
nal variables in the model. All the prices and returns are real.
the model set-up. We start with a partial equilibrium model
m of the excessive risk-taking and information acquisition.
quilibrium model to study the model dynamic and potential
ck2.
of the financial sector.
se and has mean-variance utility in its next period net return:
(1)
n parameter, ktb is the bank’s risky asset holdings and Πt+1
return. That is, portfolio variance is costly and the bank,
reduce it. The next period return consists of the return on
he information budget:
(2)
stment, R,r and Rs are respectively gross returns from risky
formation budget selected by the bank. The bank’s future
nt of funds it has for investment - dt,and from a composition
f risky asset, ktb: Note that the return is reduced by the
rate could be viewed as a reverse of the asset price. With larger demand for
offsetting higher risk appetite.
is the risk aversion parameter,
5
We abstract from any nominal variables in the model. All the prices and returns are
In what follows, we present the model set-up. We start with a partial equilibrium m
to illustrate the mechanism of the excessive risk-taking and information acquisi
Then we simulate general equilibrium model to study the model dynamic and pote
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next period net r
where ρ is the risk aversion parameter, ktb is the bank’s risky asset holdings and
stands for the next period return. That is, portfolio variance is costly and the b
therefore, has incentives to reduce it. The next period return consists of the retur
the bank’s portfolio minus the information budget:
where dt is household investment, R,r and Rs are respectively gross returns from
and safe assets, bt is the information budget selected by the bank. The bank’s fu
return depends on the amount of funds it has for investment - dt,and from a compos
of its portfolio - quantity of risky asset, ktb: Note that the return is reduced by
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger de
a risky asset, it drops, potentially offsetting higher risk appetite.
is the bank’s risky asset holdings and
5
We abstract from any nominal variables in the model. All
In what follows, we present the model set-up. We start w
to illustrate the mechanism of the excessive risk-taking
Then we simulate general equilibrium model to study the
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance uti
where ρ is the risk aversion parameter, ktb is the bank’s
stands for the next period return. That is, portfolio va
therefore, has incentives to reduce it. The next period re
the bank’s portfolio minus the information budget:
where dt is household investment, R,r and Rs are respec
and safe assets, bt is the information budget selected b
return depends on the amount of funds it has for investme
of its portfolio - quantity of risky asset, ktb: Note that
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse o
a risky asset, it drops, potentially offsetting higher risk appetite.
stands
for the next period return. That is, portfolio variance is costly and the bank, therefore, has
incentives to reduce it. The next period return consists of the return on the bank’s portfolio
minus the information budget:
	
5
In what follows, we present the model set-up. We start with a partial equilibrium model
to illustrate the mechanism of the excessive risk-taking and information acquisition.
Then we simulate general equilibrium model to study the model dynamic and potential
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next period net return:
(1)
where ρ is the risk aversion parameter, ktb is the bank’s risky asset holdings and Πt+1
stands for the next period return. That is, portfolio variance is costly and the bank,
therefore, has incentives to reduce it. The next period return consists of the return on
the bank’s portfolio minus the information budget:
(2)
where dt is household investment, R,r and Rs are respectively gross returns from risky
and safe assets, bt is the information budget selected by the bank. The bank’s future
return depends on the amount of funds it has for investment - dt,and from a composition
of its portfolio - quantity of risky asset, ktb: Note that the return is reduced by the
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger demand fo
a risky asset, it drops, potentially offsetting higher risk appetite.
	 (2)
where
5
m any nominal variables in the model. All the prices and returns are real.
we present the model set-up. We start with a partial equilibrium model
mechanism of the excessive risk-taking and information acquisition.
e general equilibrium model to study the model dynamic and potential
ates feedback2.
description of the financial sector.
is risk-averse and has mean-variance utility in its next period net return:
(1)
sk aversion parameter, ktb is the bank’s risky asset holdings and Πt+1
ext period return. That is, portfolio variance is costly and the bank,
centives to reduce it. The next period return consists of the return on
lio minus the information budget:
(2)
ehold investment, R,r and Rs are respectively gross returns from risky
bt is the information budget selected by the bank. The bank’s future
n the amount of funds it has for investment - dt,and from a composition
quantity of risky asset, ktb: Note that the return is reduced by the
stment, bt:
isky interest rate could be viewed as a reverse of the asset price. With larger demand for
s, potentially offsetting higher risk appetite.
is household investment,
5
We abstract from any nominal variables in the model. All the prices and returns are real.
In what follows, we present the model set-up. We start with a partial equilibrium model
to illustrate the mechanism of the excessive risk-taking and information acquisition.
Then we simulate general equilibrium model to study the model dynamic and potential
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next period net return:
(1)
where ρ is the risk aversion parameter, ktb is the bank’s risky asset holdings and Πt+1
stands for the next period return. That is, portfolio variance is costly and the bank,
therefore, has incentives to reduce it. The next period return consists of the return on
the bank’s portfolio minus the information budget:
(2)
where dt is household investment, R,r and Rs are respectively gross returns from risky
and safe assets, bt is the information budget selected by the bank. The bank’s future
return depends on the amount of funds it has for investment - dt,and from a composition
of its portfolio - quantity of risky asset, ktb: Note that the return is reduced by the
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger demand fo
a risky asset, it drops, potentially offsetting higher risk appetite.
,
5
abstract from any nominal variables in the model. All the prices and returns are real.
what follows, we present the model set-up. We start with a partial equilibrium model
lustrate the mechanism of the excessive risk-taking and information acquisition.
n we simulate general equilibrium model to study the model dynamic and potential
of interest rates feedback2.
start with a description of the financial sector.
ks. The bank is risk-averse and has mean-variance utility in its next period net return:
(1)
ere ρ is the risk aversion parameter, ktb is the bank’s risky asset holdings and Πt+1
nds for the next period return. That is, portfolio variance is costly and the bank,
refore, has incentives to reduce it. The next period return consists of the return on
bank’s portfolio minus the information budget:
(2)
ere dt is household investment, R,r and Rs are respectively gross returns from risky
safe assets, bt is the information budget selected by the bank. The bank’s future
urn depends on the amount of funds it has for investment - dt,and from a composition
ts portfolio - quantity of risky asset, ktb: Note that the return is reduced by the
rmation investment, bt:
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger demand for
ky asset, it drops, potentially offsetting higher risk appetite.
and
5
We abstract from any nominal variables in the model. All the prices and returns are
In what follows, we present the model set-up. We start with a partial equilibrium m
to illustrate the mechanism of the excessive risk-taking and information acquis
Then we simulate general equilibrium model to study the model dynamic and pote
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next period net r
where ρ is the risk aversion parameter, ktb is the bank’s risky asset holdings and
stands for the next period return. That is, portfolio variance is costly and the b
therefore, has incentives to reduce it. The next period return consists of the retu
the bank’s portfolio minus the information budget:
where dt is household investment, R,r and Rs are respectively gross returns from
and safe assets, bt is the information budget selected by the bank. The bank’s fu
return depends on the amount of funds it has for investment - dt,and from a compos
of its portfolio - quantity of risky asset, ktb: Note that the return is reduced b
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger d
a risky asset, it drops, potentially offsetting higher risk appetite.
5
We abstract from any nominal variables in the model. All the prices and returns are
In what follows, we present the model set-up. We start with a partial equilibrium m
to illustrate the mechanism of the excessive risk-taking and information acquis
Then we simulate general equilibrium model to study the model dynamic and pote
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next period net r
where ρ is the risk aversion parameter, ktb is the bank’s risky asset holdings and
stands for the next period return. That is, portfolio variance is costly and the
therefore, has incentives to reduce it. The next period return consists of the retu
the bank’s portfolio minus the information budget:
where dt is household investment, R,r and Rs are respectively gross returns from
and safe assets, bt is the information budget selected by the bank. The bank’s f
return depends on the amount of funds it has for investment - dt,and from a compo
of its portfolio - quantity of risky asset, ktb: Note that the return is reduced b
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger d
a risky asset, it drops, potentially offsetting higher risk appetite.
are respectively gross returns from risky
and safe assets,
5
bles in the model. All the prices and returns are real.
del set-up. We start with a partial equilibrium model
excessive risk-taking and information acquisition.
m model to study the model dynamic and potential
nancial sector.
as mean-variance utility in its next period net return:
(1)
eter, ktb is the bank’s risky asset holdings and Πt+1
That is, portfolio variance is costly and the bank,
it. The next period return consists of the return on
mation budget:
(2)
R,r and Rs are respectively gross returns from risky
on budget selected by the bank. The bank’s future
nds it has for investment - dt,and from a composition
asset, ktb: Note that the return is reduced by the
be viewed as a reverse of the asset price. With larger demand for
higher risk appetite.
is the information budget selected by the bank. The bank’s future return
depends on the amount of funds it has for investment -
5
weighted average of their prior and the signal.
We abstract from any nominal variables in the model. All the prices and
In what follows, we present the model set-up. We start with a partial eq
to illustrate the mechanism of the excessive risk-taking and informa
Then we simulate general equilibrium model to study the model dynam
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next p
where ρ is the risk aversion parameter, ktb is the bank’s risky asset h
stands for the next period return. That is, portfolio variance is costl
therefore, has incentives to reduce it. The next period return consists
the bank’s portfolio minus the information budget:
where dt is household investment, R,r and Rs are respectively gross re
and safe assets, bt is the information budget selected by the bank. Th
return depends on the amount of funds it has for investment - dt,and fro
of its portfolio - quantity of risky asset, ktb: Note that the return is
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price.
a risky asset, it drops, potentially offsetting higher risk appetite.
, and from a composition of its
portfolio - quantity of risky asset,
5
set by the market.
The bank is risk-averse, which is motivated by the fact that banks are often subject t
regulations and have reputational concerns for the safety of their deposits. We the
expand the model and grant financial intermediary access to a noisy signal about futur
productivity. This signal helps the agents to reduce the variance of their forecast. Ye
they have to pay for it. Banks are Bayesian, they form forecasts of risky returns as
weighted average of their prior and the signal.
We abstract from any nominal variables in the model. All the prices and returns are rea
In what follows, we present the model set-up. We start with a partial equilibrium mod
to illustrate the mechanism of the excessive risk-taking and information acquisition
Then we simulate general equilibrium model to study the model dynamic and potenti
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next period net retu
(1
where ρ is the risk aversion parameter, ktb is the bank’s risky asset holdings and Πt+
stands for the next period return. That is, portfolio variance is costly and the ban
therefore, has incentives to reduce it. The next period return consists of the return o
the bank’s portfolio minus the information budget:
(2
where dt is household investment, R,r and Rs are respectively gross returns from risk
and safe assets, bt is the information budget selected by the bank. The bank’s futur
return depends on the amount of funds it has for investment - dt,and from a compositio
of its portfolio - quantity of risky asset, ktb: Note that the return is reduced by th
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger dema
a risky asset, it drops, potentially offsetting higher risk appetite.
: Note that the return is reduced by the information
investment,
5
esian, they form forecasts of risky returns as a
e signal.
s in the model. All the prices and returns are real.
set-up. We start with a partial equilibrium model
cessive risk-taking and information acquisition.
model to study the model dynamic and potential
ncial sector.
mean-variance utility in its next period net return:
(1)
r, ktb is the bank’s risky asset holdings and Πt+1
at is, portfolio variance is costly and the bank,
The next period return consists of the return on
tion budget:
(2)
and Rs are respectively gross returns from risky
budget selected by the bank. The bank’s future
it has for investment - dt,and from a composition
et, ktb: Note that the return is reduced by the
viewed as a reverse of the asset price. With larger demand for
her risk appetite.
:
The bank’s objective is to maximize (1), and the choice variables are information budget,
the safety of their deposits. We then
ry access to a noisy signal about future
uce the variance of their forecast. Yet
y form forecasts of risky returns as a
odel. All the prices and returns are real.
e start with a partial equilibrium model
sk-taking and information acquisition.
tudy the model dynamic and potential
r.
ance utility in its next period net return:
(1)
e bank’s risky asset holdings and Πt+1
folio variance is costly and the bank,
period return consists of the return on
et:
(2)
e respectively gross returns from risky
lected by the bank. The bank’s future
nvestment - dt,and from a composition
ote that the return is reduced by the
reverse of the asset price. With larger demand for
etite.
, and risky asset quantity
5
about future productivity. All the agents in the economy know the productivity
distribution. The household puts savings in the bank (in the form of investment), and the
bank transfers all its profit back to the household. The safe and risky interest rates are
set by the market.
The bank is risk-averse, which is motivated by the fact that banks are often subject to
regulations and have reputational concerns for the safety of their deposits. We then
expand the model and grant financial intermediary access to a noisy signal about future
productivity. This signal helps the agents to reduce the variance of their forecast. Yet
they have to pay for it. Banks are Bayesian, they form forecasts of risky returns as a
weighted average of their prior and the signal.
We abstract from any nominal variables in the model. All the prices and returns are real.
In what follows, we present the model set-up. We start with a partial equilibrium model
to illustrate the mechanism of the excessive risk-taking and information acquisition.
Then we simulate general equilibrium model to study the model dynamic and potential
role of interest rates feedback2.
We start with a description of the financial sector.
Banks. The bank is risk-averse and has mean-variance utility in its next period net return:
(1)
where ρ is the risk aversion parameter, ktb is the bank’s risky asset holdings and Πt+1
stands for the next period return. That is, portfolio variance is costly and the bank,
therefore, has incentives to reduce it. The next period return consists of the return on
the bank’s portfolio minus the information budget:
(2)
where dt is household investment, R,r and Rs are respectively gross returns from risky
and safe assets, bt is the information budget selected by the bank. The bank’s future
return depends on the amount of funds it has for investment - dt,and from a composition
of its portfolio - quantity of risky asset, ktb: Note that the return is reduced by the
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger demand for
a risky asset, it drops, potentially offsetting higher risk appetite.
. Compared to the strand of literature on rational inattention
with exogenous capacity constraint, here we endogenize capacity and formulate it in budget
terms.
Maximizing the bank’s utility, we get its holdings of the risky asset:
	
The bank’s objective is to maximize (1), and the choice variables are information bu
bt , and risky asset quantity . Compared to the strand of literature on rat
inattention with exogenous capacity constraint, here we endogenize capacity
formulate it in budget terms.
Maximizing the bank’s utility, we get its holdings of the risky asset:
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
=
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
𝜌𝜌𝜌𝜌𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is risky asset return variance. Sign ‘^’ stands for posterior variance, upd
after information decisions. As is typical in the literature, the amount of risky a
bought is increasing with excess return, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
, and is decreasing with risk aver
𝜌𝜌𝜌𝜌, and risky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
For simplicity, we make the bank transfer all its profit to the household in return to
savings, dt.
Information Acquisition. The information acquisition is modeled simila
Nieuwerburgh and Veldkamp (2010). In their paper an investor is allocating hi
exogenously limited capacity to learn between different assets depending on hi
portfolio decisions. In our model, we endogenize learning capacity by replacing it
the budget, bt. The bank then chooses the budget to determine how much to
subject to fixed learning costs, a.
	(3)
where
s to maximize (1), and the choice variables are information budget,
quantity . Compared to the strand of literature on rational
ogenous capacity constraint, here we endogenize capacity and
t terms.
’s utility, we get its holdings of the risky asset:
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
=
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
𝜌𝜌𝜌𝜌𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2 (3)
et return variance. Sign ‘^’ stands for posterior variance, updated
cisions. As is typical in the literature, the amount of risky assets
with excess return, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
, and is decreasing with risk aversion,
urn variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
ke the bank transfer all its profit to the household in return to their
ition. The information acquisition is modeled similar to
Veldkamp (2010). In their paper an investor is allocating his/her
capacity to learn between different assets depending on his/her
n our model, we endogenize learning capacity by replacing it with
bank then chooses the budget to determine how much to learn
ing costs, a.
is risky asset return variance. Sign ‘^’ stands for posterior variance, updated after
information decisions. As is typical in the literature, the amount of risky assets bought is
increasing with excess return,
The bank’s objective is to maximize (1), and the choice variables are information budget,
bt , and risky asset quantity . Compared to the strand of literature on rational
inattention with exogenous capacity constraint, here we endogenize capacity and
formulate it in budget terms.
Maximizing the bank’s utility, we get its holdings of the risky asset:
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
=
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
𝜌𝜌𝜌𝜌𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2 (3)
where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is risky asset return variance. Sign ‘^’ stands for posterior variance, updated
after information decisions. As is typical in the literature, the amount of risky assets
bought is increasing with excess return, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
, and is decreasing with risk aversion,
𝜌𝜌𝜌𝜌, and risky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
For simplicity, we make the bank transfer all its profit to the household in return to their
savings, dt.
Information Acquisition. The information acquisition is modeled similar to
Nieuwerburgh and Veldkamp (2010). In their paper an investor is allocating his/her
exogenously limited capacity to learn between different assets depending on his/her
, and is decreasing with risk aversion, ρ, and risky
asset return variance
s objective is to maximize (1), and the choice variables are information budget,
risky asset quantity . Compared to the strand of literature on rational
n with exogenous capacity constraint, here we endogenize capacity and
e it in budget terms.
ng the bank’s utility, we get its holdings of the risky asset:
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
=
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
𝜌𝜌𝜌𝜌𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2 (3)
is risky asset return variance. Sign ‘^’ stands for posterior variance, updated
rmation decisions. As is typical in the literature, the amount of risky assets
increasing with excess return, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
, and is decreasing with risk aversion,
ky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
city, we make the bank transfer all its profit to the household in return to their
t.
on Acquisition. The information acquisition is modeled similar to
urgh and Veldkamp (2010). In their paper an investor is allocating his/her
sly limited capacity to learn between different assets depending on his/her
.
For simplicity, we make the bank transfer all its profit to the household in return to their sav-
ings,
Financial Sector
with a financial intermediary, bank, a manufacturing firm and a
ssets in the economy are manufacturer claims (a risky asset) and
sset). The risk in manufacturer claims comes from the uncertainty
ductivity. All the agents in the economy know the productivity
ousehold puts savings in the bank (in the form of investment), and the
ts profit back to the household. The safe and risky interest rates are
erse, which is motivated by the fact that banks are often subject to
ave reputational concerns for the safety of their deposits. We then
and grant financial intermediary access to a noisy signal about future
signal helps the agents to reduce the variance of their forecast. Yet
for it. Banks are Bayesian, they form forecasts of risky returns as a
of their prior and the signal.
any nominal variables in the model. All the prices and returns are real.
e present the model set-up. We start with a partial equilibrium model
mechanism of the excessive risk-taking and information acquisition.
general equilibrium model to study the model dynamic and potential
es feedback2.
scription of the financial sector.
risk-averse and has mean-variance utility in its next period net return:
(1)
k aversion parameter, ktb is the bank’s risky asset holdings and Πt+1
xt period return. That is, portfolio variance is costly and the bank,
entives to reduce it. The next period return consists of the return on
o minus the information budget:
(2)
hold investment, R,r and Rs are respectively gross returns from risky
t is the information budget selected by the bank. The bank’s future
the amount of funds it has for investment - dt,and from a composition
quantity of risky asset, ktb: Note that the return is reduced by the
ment, bt:
.
Information Acquisition. The information acquisition is modeled similar to Nieuwerburgh
and Veldkamp (2010). In their paper an investor is allocating his/her exogenously limited
capacity to learn between different assets depending on his/her portfolio decisions. In our
2	 In our model a risky interest rate could be viewed as a reverse of the asset price. With larger demand for a
risky asset, it drops, potentially offsetting higher risk appetite.
ACTA VŠFS, 2/2015, vol. 9120
model, we endogenize learning capacity by replacing it with the budget,
5
stands for the next period return. That is, portfolio variance is costly and the bank,
therefore, has incentives to reduce it. The next period return consists of the return on
the bank’s portfolio minus the information budget:
(2)
where dt is household investment, R,r and Rs are respectively gross returns from risky
and safe assets, bt is the information budget selected by the bank. The bank’s future
return depends on the amount of funds it has for investment - dt,and from a composition
of its portfolio - quantity of risky asset, ktb: Note that the return is reduced by the
information investment, bt:
2
In our model a risky interest rate could be viewed as a reverse of the asset price. With larger deman
a risky asset, it drops, potentially offsetting higher risk appetite.
. The bank then
chooses the budget to determine how much to learn subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return forecast by investing into additional
signal and pay costs proportional to the variance reduced.The decision to monitor is
taken ex-ante signal realization. For this purpose, the period is decomposed into sub-periods.
The timing is as in table 1.
Table 1: The Timeline of Information Decisions
subperiod 1 subperiod 2
6
werburgh and Veldkamp (2010). In their paper an investor is allocating his/her
enously limited capacity to learn between different assets depending on his/her
olio decisions. In our model, we endogenize learning capacity by replacing it with
udget, bt. The bank then chooses the budget to determine how much to learn
ct to fixed learning costs, a.
cial intermediaries can reduce the variance of their return forecast by investing into
onal signal and pay costs proportional to the variance reduced. The decision to
tor is taken ex-ante signal realization. For this purpose, the period is decomposed
ub-periods. The timing is as in table 1.
subperiod 1 subperiod 2
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals are realized
pected posterior return is
𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using Bayes rule,
dget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is chosen: 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
Table 1: The Timeline of Information Decisions
ble 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is the posterior the bank
cts to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior variance after observing the
3.
e first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and expected return, 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡, both
iding with true moments of return distribution. The agent decides what budget to
l posterior variables are formed using Bayes rule.
information signals are realized
expected posterior return is
6
Nieuwerburgh and Veldkamp (2010). In their paper an investor is allocating his/her
exogenously limited capacity to learn between different assets depending on his/her
portfolio decisions. In our model, we endogenize learning capacity by replacing it with
the budget, bt. The bank then chooses the budget to determine how much to learn
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return forecast by investing into
additional signal and pay costs proportional to the variance reduced. The decision to
monitor is taken ex-ante signal realization. For this purpose, the period is decomposed
into sub-periods. The timing is as in table 1.
subperiod 1 subperiod 2
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals are realized
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using Bayes rule,
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is chosen: 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
Table 1: The Timeline of Information Decisions
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is the posterior the bank
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior variance after observing the
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and expected return, 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡, both
coinciding with true moments of return distribution. The agent decides what budget to
3
All posterior variables are formed using Bayes rule.
6
Nieuwerburgh and Veldkamp (2010). In their paper
exogenously limited capacity to learn between differ
portfolio decisions. In our model, we endogenize lear
the budget, bt. The bank then chooses the budget t
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of the
additional signal and pay costs proportional to the v
monitor is taken ex-ante signal realization. For this pu
into sub-periods. The timing is as in table 1.
subperiod 1
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) informat
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is form
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portf
Table 1: The Timeline of Informat
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, R
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the post
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎
coinciding with true moments of return distribution. T
3
All posterior variables are formed using Bayes rule.
is formed using Bayes rule,
budget,
6
Nieuwerburgh and Veldkamp (2010). In their paper an investor is allocating his/her
exogenously limited capacity to learn between different assets depending on his/her
portfolio decisions. In our model, we endogenize learning capacity by replacing it with
the budget, bt. The bank then chooses the budget to determine how much to learn
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return forecast by investing into
additional signal and pay costs proportional to the variance reduced. The decision to
monitor is taken ex-ante signal realization. For this purpose, the period is decomposed
into sub-periods. The timing is as in table 1.
subperiod 1 subperiod 2
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals are realized
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using Bayes rule,
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is chosen: 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
Table 1: The Timeline of Information Decisions
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is the posterior the bank
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior variance after observing the
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and expected return, 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡, both
coinciding with true moments of return distribution. The agent decides what budget to
3
All posterior variables are formed using Bayes rule.
and
6
Nieuwerburgh and Veldkamp (2010). In their paper an investor is allocating his/her
exogenously limited capacity to learn between different assets depending on his/her
portfolio decisions. In our model, we endogenize learning capacity by replacing it with
the budget, bt. The bank then chooses the budget to determine how much to learn
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return forecast by investing into
additional signal and pay costs proportional to the variance reduced. The decision to
monitor is taken ex-ante signal realization. For this purpose, the period is decomposed
into sub-periods. The timing is as in table 1.
subperiod 1 subperiod 2
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals are realized
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using Bayes rule,
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is chosen: 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
Table 1: The Timeline of Information Decisions
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is the posterior the bank
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior variance after observing the
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and expected return, 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡, both
coinciding with true moments of return distribution. The agent decides what budget to
3
All posterior variables are formed using Bayes rule.
are chosen and portfolio is chosen:
6
uwerburgh and Veldkamp (2010). In their paper an investor is allocating his/her
genously limited capacity to learn between different assets depending on his/her
tfolio decisions. In our model, we endogenize learning capacity by replacing it with
budget, bt. The bank then chooses the budget to determine how much to learn
ject to fixed learning costs, a.
ancial intermediaries can reduce the variance of their return forecast by investing into
itional signal and pay costs proportional to the variance reduced. The decision to
nitor is taken ex-ante signal realization. For this purpose, the period is decomposed
o sub-periods. The timing is as in table 1.
subperiod 1 subperiod 2
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals are realized
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using Bayes rule,
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is chosen: 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
Table 1: The Timeline of Information Decisions
able 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is the posterior the bank
ects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior variance after observing the
nal3.
he first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and expected return, 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡, both
nciding with true moments of return distribution. The agent decides what budget to
All posterior variables are formed using Bayes rule.
In table 1
6
𝜌𝜌𝜌𝜌, and risky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡 .
For simplicity, we make the bank transfer all its profit to the household in return to their
savings, dt.
Information Acquisition. The information acquisition is modeled similar to
Nieuwerburgh and Veldkamp (2010). In their paper an investor is allocating his/her
exogenously limited capacity to learn between different assets depending on his/her
portfolio decisions. In our model, we endogenize learning capacity by replacing it with
the budget, bt. The bank then chooses the budget to determine how much to learn
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return forecast by investing into
additional signal and pay costs proportional to the variance reduced. The decision to
monitor is taken ex-ante signal realization. For this purpose, the period is decomposed
into sub-periods. The timing is as in table 1.
subperiod 1 subperiod 2
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals are realized
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using Bayes rule,
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is chosen: 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
Table 1: The Timeline of Information Decisions
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is the posterior the bank
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior variance after observing the
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and expected return, 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡, both
coinciding with true moments of return distribution. The agent decides what budget to
3
All posterior variables are formed using Bayes rule.
is the bank’s prior about future return,
6
𝜌𝜌𝜌𝜌, and risky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡 .
For simplicity, we make the bank transfer all its profit to the hou
savings, dt.
Information Acquisition. The information acquisition is
Nieuwerburgh and Veldkamp (2010). In their paper an invest
exogenously limited capacity to learn between different asset
portfolio decisions. In our model, we endogenize learning capa
the budget, bt. The bank then chooses the budget to determ
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return fo
additional signal and pay costs proportional to the variance re
monitor is taken ex-ante signal realization. For this purpose, the
into sub-periods. The timing is as in table 1.
subperiod 1 subper
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using B
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is cho
Table 1: The Timeline of Information Decisio
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior varia
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and ex
coinciding with true moments of return distribution. The agent
3
All posterior variables are formed using Bayes rule.
,
savings, dt.
Information Acquisition. The infor
Nieuwerburgh and Veldkamp (2010).
exogenously limited capacity to learn
portfolio decisions. In our model, we e
the budget, bt. The bank then choose
subject to fixed learning costs, a.
Financial intermediaries can reduce the
additional signal and pay costs propor
monitor is taken ex-ante signal realizat
into sub-periods. The timing is as in tab
subperiod 1
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
)
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen
Table 1: The Timeli
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about f
expects to get after observing the signal
signal3.
In the first subperiod the agent has pr
coinciding with true moments of return
3
All posterior variables are formed using Ba
is the posterior the bank
expects to get after observing the signal.
6
savings, dt.
Information Acquisition. The information acquisition is mod
Nieuwerburgh and Veldkamp (2010). In their paper an investor is
exogenously limited capacity to learn between different assets dep
portfolio decisions. In our model, we endogenize learning capacity b
the budget, bt. The bank then chooses the budget to determine h
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return foreca
additional signal and pay costs proportional to the variance reduce
monitor is taken ex-ante signal realization. For this purpose, the peri
into sub-periods. The timing is as in table 1.
subperiod 1 subperiod 2
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals are r
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using Bayes
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is chosen:
Table 1: The Timeline of Information Decisions
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is the p
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior variance a
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and expecte
coinciding with true moments of return distribution. The agent decid
3
All posterior variables are formed using Bayes rule.
is the posterior variance after observing the
signal3
.
In the first subperiod the agent has prior variance,
6
𝜌𝜌𝜌𝜌, and risky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡 .
For simplicity, we make the bank transfer all its profit to the hou
savings, dt.
Information Acquisition. The information acquisition is
Nieuwerburgh and Veldkamp (2010). In their paper an invest
exogenously limited capacity to learn between different asset
portfolio decisions. In our model, we endogenize learning capa
the budget, bt. The bank then chooses the budget to determ
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return fo
additional signal and pay costs proportional to the variance re
monitor is taken ex-ante signal realization. For this purpose, the
into sub-periods. The timing is as in table 1.
subperiod 1 subper
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using B
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is cho
Table 1: The Timeline of Information Decisio
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior varia
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and ex
coinciding with true moments of return distribution. The agent
3
All posterior variables are formed using Bayes rule.
; and expected return,
where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is risky asset return variance.
after information decisions. As is typica
bought is increasing with excess return, 𝐸𝐸𝐸𝐸
𝜌𝜌𝜌𝜌, and risky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
For simplicity, we make the bank transfer
savings, dt.
Information Acquisition. The inform
Nieuwerburgh and Veldkamp (2010). In
exogenously limited capacity to learn be
portfolio decisions. In our model, we end
the budget, bt. The bank then chooses
subject to fixed learning costs, a.
Financial intermediaries can reduce the va
additional signal and pay costs proporti
monitor is taken ex-ante signal realizatio
into sub-periods. The timing is as in table
subperiod 1
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
)
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen
Table 1: The Timeline
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about fu
expects to get after observing the signal.
signal3.
In the first subperiod the agent has prio
coinciding with true moments of return d
3
All posterior variables are formed using Baye
, both coinciding
with true moments of return distribution. The agent decides what budget to allocate to
information decision. The choice of the budget determines by how much the variance will
be reduced. In the spirit of Nieuwerburgh and Veldkamp (2010) we interpret it as an investment
into purchasing additional market data, when an agent does not have prior knowledge
of what is in the data, but knows that this data will sharpen his/her forecast. We model this
decision as a choice of budget that determines posterior variance,
6
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
=
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝜌𝜌𝜌𝜌𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is risky asset return variance. Sign ‘^’ s
after information decisions. As is typical in the
bought is increasing with excess return, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅
𝜌𝜌𝜌𝜌, and risky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
For simplicity, we make the bank transfer all its pr
savings, dt.
Information Acquisition. The information a
Nieuwerburgh and Veldkamp (2010). In their p
exogenously limited capacity to learn between
portfolio decisions. In our model, we endogenize
the budget, bt. The bank then chooses the bud
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance o
additional signal and pay costs proportional to t
monitor is taken ex-ante signal realization. For th
into sub-periods. The timing is as in table 1.
subperiod 1
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) info
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and
Table 1: The Timeline of Info
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future retu
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the
signal3.
In the first subperiod the agent has prior varian
coinciding with true moments of return distributi
3
All posterior variables are formed using Bayes rule.
. When choosing the
budget and posterior variance, agent takes into account what the return expectations will be
after the signal is observed. In other words, the agent has to form expectations about return
expectations: expected posterior
formulate it in budget terms.
Maximizing the bank’s utility, we get its holdings of the risky a
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
=
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
𝜌𝜌𝜌𝜌𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is risky asset return variance. Sign ‘^’ stands for po
after information decisions. As is typical in the literature, th
bought is increasing with excess return, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
, and is dec
𝜌𝜌𝜌𝜌, and risky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
For simplicity, we make the bank transfer all its profit to the ho
savings, dt.
Information Acquisition. The information acquisition
Nieuwerburgh and Veldkamp (2010). In their paper an inv
exogenously limited capacity to learn between different ass
portfolio decisions. In our model, we endogenize learning ca
the budget, bt. The bank then chooses the budget to deter
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return
additional signal and pay costs proportional to the variance
monitor is taken ex-ante signal realization. For this purpose,
into sub-periods. The timing is as in table 1.
subperiod 1 subp
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information sign
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed usin
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is c
Table 1: The Timeline of Information Dec
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior va
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and
coinciding with true moments of return distribution. The agen
3
All posterior variables are formed using Bayes rule.
. Yet before paying for the signal and observing it, the
expected posterior equals the prior
allocate to information decision. The choice of the budge
variance will be reduced. In the spirit of Nieuwerbur
interpret it as an investment into purchasing additional m
not have prior knowledge of what is in the data, but kno
his/her forecast. We model this decision as a choice of bu
variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
. When choosing the budget and poster
account what the return expectations will be after the sig
the agent has to form expectations about return expect
Yet before paying for the signal and observing it, the expe
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂= 𝜇𝜇𝜇𝜇 .
When taking decisions in subperiod 1, the agent rationa
the risky asset in the subperiod 2 as in (3) where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is po
Thus, with the information investment - budget bt and (3
subject to the learning rule:
and non-forgeting constraint: 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
> 0 . a is cost o
f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) is the learning function. The function is continuo
arguments, it is increasing in initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, and i
Intuitively, the more we reduce the posterior variance rel
.
When taking decisions in subperiod 1, the agent rationally anticipates the demand for the
risky asset in the subperiod 2 as in (3) where
bt , and risky asset quantity . Compared to the strand of lite
inattention with exogenous capacity constraint, here we endoge
formulate it in budget terms.
Maximizing the bank’s utility, we get its holdings of the risky asset:
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
=
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
𝜌𝜌𝜌𝜌𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is risky asset return variance. Sign ‘^’ stands for posterior
after information decisions. As is typical in the literature, the amo
bought is increasing with excess return, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
, and is decreasin
𝜌𝜌𝜌𝜌, and risky asset return variance 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
For simplicity, we make the bank transfer all its profit to the househo
savings, dt.
Information Acquisition. The information acquisition is mo
Nieuwerburgh and Veldkamp (2010). In their paper an investor is
exogenously limited capacity to learn between different assets de
portfolio decisions. In our model, we endogenize learning capacity
the budget, bt. The bank then chooses the budget to determine h
subject to fixed learning costs, a.
Financial intermediaries can reduce the variance of their return forec
additional signal and pay costs proportional to the variance reduce
monitor is taken ex-ante signal realization. For this purpose, the per
into sub-periods. The timing is as in table 1.
subperiod 1 subperiod
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
) information signals are
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
𝜇𝜇𝜇𝜇̂ is formed using Bayes
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen and portfolio is chosen:
Table 1: The Timeline of Information Decisions
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about future return, Rtr+1, 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂ is the
expects to get after observing the signal. 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is the posterior variance
signal3.
In the first subperiod the agent has prior variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
; and expect
coinciding with true moments of return distribution. The agent decid
3
All posterior variables are formed using Bayes rule.
is posterior variance of the return.Thus, with
the information investment - budget bt and (3), the banks utility is rewritten:
	
allocate to information decision. The choice of the budget determines by how much the
variance will be reduced. In the spirit of Nieuwerburgh and Veldkamp (2010) we
interpret it as an investment into purchasing additional market data, when an agent does
not have prior knowledge of what is in the data, but knows that this data will sharpen
his/her forecast. We model this decision as a choice of budget that determines posterior
variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
. When choosing the budget and posterior variance, agent takes into
account what the return expectations will be after the signal is observed. In other words,
the agent has to form expectations about return expectations: expected posterior 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂.
Yet before paying for the signal and observing it, the expected posterior equals the prior
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂= 𝜇𝜇𝜇𝜇 .
When taking decisions in subperiod 1, the agent rationally anticipates the demand for
the risky asset in the subperiod 2 as in (3) where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is posterior variance of the return.
Thus, with the information investment - budget bt and (3), the banks utility is rewritten:
(4)
subject to the learning rule:
(5)
and non-forgeting constraint: 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
> 0 . a is cost of reducing the variance, and
f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) is the learning function. The function is continuous and monotone in both of its
arguments, it is increasing in initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, and is decreasing in posterior, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
Intuitively, the more we reduce the posterior variance relative to the prior, the more we
	 (4)
subject to the learning rule:
	
allocate to information decision. The choice of the budget determines by how much the
variance will be reduced. In the spirit of Nieuwerburgh and Veldkamp (2010) we
interpret it as an investment into purchasing additional market data, when an agent does
not have prior knowledge of what is in the data, but knows that this data will sharpen
his/her forecast. We model this decision as a choice of budget that determines posterior
variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
. When choosing the budget and posterior variance, agent takes into
account what the return expectations will be after the signal is observed. In other words,
the agent has to form expectations about return expectations: expected posterior 𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂.
Yet before paying for the signal and observing it, the expected posterior equals the prior
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂= 𝜇𝜇𝜇𝜇 .
When taking decisions in subperiod 1, the agent rationally anticipates the demand for
the risky asset in the subperiod 2 as in (3) where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is posterior variance of the return.
Thus, with the information investment - budget bt and (3), the banks utility is rewritten:
(4)
subject to the learning rule:
(5)
and non-forgeting constraint: 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
> 0 . a is cost of reducing the variance, and
f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) is the learning function. The function is continuous and monotone in both of its
arguments, it is increasing in initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, and is decreasing in posterior, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
	 (5)
3	 All posterior variables are formed using Bayes rule.
ACTA VŠFS, 2/2015, vol. 9 B121
and non-forgeting constraint:
7
subject to the learning rule:
(5)
and non-forgeting constraint: 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
> 0 . a is cost of reducing the variance, and
f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) is the learning function. The function is continuous and monotone in both of its
arguments, it is increasing in initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, and is decreasing in posterior, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
Intuitively, the more we reduce the posterior variance relative to the prior, the more we
should pay. We assume that the information budget is exhausted so that (5) becomes
equality. Then with the properties of our learning function, the choice of the information
budget, bt, uniquely determines the posterior variance and captures the information
decision of the bank.
In the following section we consider risk-taking decisions of the bank in a partial
equilibrium to identify risk driving forces.
Aggregating Financial Markets. The total investment into the safe asset, res, is given by
the bank’s financial resources not invested into the risky asset:
The investment into the safe asset is determined as deposits, dt, that was not invested
in the risky asset, 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
.
Recall, that the risky asset in the model is the investment in the manufacturing firm,
which uses it to build new capital. The manufacturing firm does not have funds for
investment on its own. To invest it has to sell its claims to the bank. Thus, the total
investment into the capital is then given by the bank’s risky asset holdings:
It = 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
. a is cost of reducing the variance, and
7
subject to the learning rule:
and non-forgeting constraint: 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
> 0 . a is cost of reducing the variance,
f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) is the learning function. The function is continuous and monotone in both
arguments, it is increasing in initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, and is decreasing in posterior
Intuitively, the more we reduce the posterior variance relative to the prior, the mor
should pay. We assume that the information budget is exhausted so that (5) beco
equality. Then with the properties of our learning function, the choice of the inform
budget, bt, uniquely determines the posterior variance and captures the inform
decision of the bank.
In the following section we consider risk-taking decisions of the bank in a p
equilibrium to identify risk driving forces.
Aggregating Financial Markets. The total investment into the safe asset, res, is give
the bank’s financial resources not invested into the risky asset:
The investment into the safe asset is determined as deposits, dt, that was not inve
in the risky asset, 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
.
Recall, that the risky asset in the model is the investment in the manufacturing
which uses it to build new capital. The manufacturing firm does not have fund
investment on its own. To invest it has to sell its claims to the bank. Thus, the
investment into the capital is then given by the bank’s risky asset holdings:
It = 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
is the learning function. The function is continuous and monotone in both of its
arguments, it is increasing in initial variance,
7
subject to the learning rule:
and non-forgeting constraint: 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
> 0 . a is cost of reducing the varian
f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) is the learning function. The function is continuous and monotone in bo
arguments, it is increasing in initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, and is decreasing in poste
Intuitively, the more we reduce the posterior variance relative to the prior, the m
should pay. We assume that the information budget is exhausted so that (5) b
equality. Then with the properties of our learning function, the choice of the info
budget, bt, uniquely determines the posterior variance and captures the info
decision of the bank.
In the following section we consider risk-taking decisions of the bank in a
equilibrium to identify risk driving forces.
Aggregating Financial Markets. The total investment into the safe asset, res, is
the bank’s financial resources not invested into the risky asset:
The investment into the safe asset is determined as deposits, dt, that was not
in the risky asset, 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
.
Recall, that the risky asset in the model is the investment in the manufactur
which uses it to build new capital. The manufacturing firm does not have fu
investment on its own. To invest it has to sell its claims to the bank. Thus, t
investment into the capital is then given by the bank’s risky asset holdings:
It = 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
, and is decreasing in posterior,
subperiod 1
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
)
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chosen
Table 1: The Timel
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prior about
expects to get after observing the signa
signal3.
In the first subperiod the agent has p
coinciding with true moments of retur
3
All posterior variables are formed using Ba
. Intuitively,
the more we reduce the posterior variance relative to the prior, the more we should pay. We
assume that the information budget is exhausted so that (5) becomes equality. Then with
the properties of our learning function, the choice of the information budget,
monitor is taken ex-ante signa
into sub-periods. The timing is
subperiod
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡~𝑁𝑁𝑁𝑁(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
)
expected posterior return is
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂~𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
budget, bt and 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
are chose
Table 1: T
In table 1 𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 is the bank’s prio
expects to get after observing t
signal3.
In the first subperiod the age
coinciding with true moments
3
All posterior variables are forme
, uniquely
determines the posterior variance and captures the information decision of the bank.
In the following section we consider risk-taking decisions of the bank in a partial equilibrium
to identify risk driving forces.
Aggregating Financial Markets. The total investment into the safe asset, res, is given by the
bank’s financial resources not invested into the risky asset:
7
f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) is the learning function. The function is continuous and monotone in both of its
arguments, it is increasing in initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, and is decreasing in posterior, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
Intuitively, the more we reduce the posterior variance relative to the prior, the more we
should pay. We assume that the information budget is exhausted so that (5) becomes
equality. Then with the properties of our learning function, the choice of the information
budget, bt, uniquely determines the posterior variance and captures the information
decision of the bank.
In the following section we consider risk-taking decisions of the bank in a partial
equilibrium to identify risk driving forces.
Aggregating Financial Markets. The total investment into the safe asset, res, is given by
the bank’s financial resources not invested into the risky asset:
The investment into the safe asset is determined as deposits, dt, that was not invested
in the risky asset, 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
.
Recall, that the risky asset in the model is the investment in the manufacturing firm,
which uses it to build new capital. The manufacturing firm does not have funds for
investment on its own. To invest it has to sell its claims to the bank. Thus, the total
investment into the capital is then given by the bank’s risky asset holdings:
It = 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
The investment into the safe asset is determined as deposits, dt, that was not invested in the
risky asset,
7
on we consider risk-taking decisions of the bank in a partial
risk driving forces.
Markets. The total investment into the safe asset, res, is given by
ources not invested into the risky asset:
e safe asset is determined as deposits, dt, that was not invested
sset in the model is the investment in the manufacturing firm,
new capital. The manufacturing firm does not have funds for
. To invest it has to sell its claims to the bank. Thus, the total
pital is then given by the bank’s risky asset holdings:
It = 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
.
Recall, that the risky asset in the model is the investment in the manufacturing firm, which
uses it to build new capital.The manufacturing firm does not have funds for investment on its
own. To invest it has to sell its claims to the bank. Thus, the total investment into the capital
is then given by the bank’s risky asset holdings:
7
Intuitively, the more we reduce the posterior variance relative to the prior, the more we
should pay. We assume that the information budget is exhausted so that (5) becomes
equality. Then with the properties of our learning function, the choice of the information
budget, bt, uniquely determines the posterior variance and captures the information
decision of the bank.
In the following section we consider risk-taking decisions of the bank in a partial
equilibrium to identify risk driving forces.
Aggregating Financial Markets. The total investment into the safe asset, res, is given by
the bank’s financial resources not invested into the risky asset:
The investment into the safe asset is determined as deposits, dt, that was not invested
in the risky asset, 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
.
Recall, that the risky asset in the model is the investment in the manufacturing firm,
which uses it to build new capital. The manufacturing firm does not have funds for
investment on its own. To invest it has to sell its claims to the bank. Thus, the total
investment into the capital is then given by the bank’s risky asset holdings:
It = 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
3	 Excessive Risk-Taking and Information Acquisition
In this section we analyze the two channels through which a bank accumulates risk in the
portfolio when the safe interest rate is reduced or market volatility declines. One of them
is clear from (3): whenever the safe interest rate drops, it increases the risk premium and
makes the risky asset more attractive. Similarly, when asset variance is reduced, the bank
rationally increases holdings of the risky asset. The other channel highlighted in this paper
is a change in information acquisition: reduction in the information budget. Through
this channel, the bank increases the riskiness of the asset per se by choosing to learn less
about it. The portfolio risk then, as a product of risky asset holdings and return variance,
increases with the lower interest rate and, in some cases, lower market volatility.
At first glance, the reduction in information acquisition with increase in risky asset holdings
might seem counter-intuitive. It could be suggested that with larger asset holdings,
agents would like to learn more about them. For example Nieuwerburgh and Veldkamp
(2010) found that when allocating fixed learning capacity between the assets, agents allocate
more to those assets they invest more into. Here, we should remind the reader, that in
our paper we are studying not the allocation of the fixed capacity, but the determination
of this capacity: by how much agents are willing to reduce their expected income in order
to reduce the income variance. Also this capacity, in the form of the information budget,
is itself a function of expected return and initial variance. It describes a trade-off between
ACTA VŠFS, 2/2015, vol. 9122
the return the agent expects to get and variance he/she would like to reduce. Below, we
study the properties of the information budget for specified learning functions.
As learning function choice could influence the results (and we show later that this is the
case), we consider alternative functions. Nieuwerburgh and Veldkamp (2010) show that
the choice of utility function and learning technologies in uences results quantitatively
and, sometimes, qualitatively. They consider mean-variance and exponential utility functions,
and three learning rules: one linear and two entropy based measures. Below, we
study mean-variance utility under linear and entropy learning functions.
Information Budget and Comparative Statics. As in Nieuwerburgh and Veldkamp
(2010) we consider alternative learning functions,
8
would like to reduce. Below, we study the properties of the information budget fo
specified learning functions.
As learning function choice could influence the results (and we show later that this is th
case), we consider alternative functions. Nieuwerburgh and Veldkamp (2010) show tha
the choice of utility function and learning technologies in uences results quantitative
and, sometimes, qualitatively. They consider mean-variance and exponential utilit
functions, and three learning rules: one linear and two entropy based measures. Below
we study mean-variance utility under linear and entropy learning functions.
Information Budget and Comparative Statics. As in Nieuwerburgh and Veldkamp (201
we consider alternative learning functions, f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) in (5): a linear rule and an entropy
based. The linear function implies that the bank pays fixed costs, a, for each unit of the
linear decline in the variance:
bt =𝑎𝑎𝑎𝑎(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) (6
Linear constraint is an intuitive rule and simple to work with. The one caveat is that it
marginally as costly for the agents to reduce the variance by 1% as by 100%. Agent
potentially could choose to learn the whole truth and choose the posterior to be zero
This, of course, is very costly for them in absolute terms of linear costs, a, and this neve
happened in our simulations. But in the general case, one should consider this possibilit
in (5): a linear rule and an
entropy based. The linear function implies that the bank pays fixed costs, a, for each unit
of the linear decline in the variance:
	
8
would like to reduce. Below, we study the properties of the information budget fo
specified learning functions.
As learning function choice could influence the results (and we show later that this is the
case), we consider alternative functions. Nieuwerburgh and Veldkamp (2010) show tha
the choice of utility function and learning technologies in uences results quantitatively
and, sometimes, qualitatively. They consider mean-variance and exponential utility
functions, and three learning rules: one linear and two entropy based measures. Below
we study mean-variance utility under linear and entropy learning functions.
Information Budget and Comparative Statics. As in Nieuwerburgh and Veldkamp (2010
we consider alternative learning functions, f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) in (5): a linear rule and an entropy
based. The linear function implies that the bank pays fixed costs, a, for each unit of the
linear decline in the variance:
bt =𝑎𝑎𝑎𝑎(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) (6
Linear constraint is an intuitive rule and simple to work with. The one caveat is that it i
marginally as costly for the agents to reduce the variance by 1% as by 100%. Agent
potentially could choose to learn the whole truth and choose the posterior to be zero
This, of course, is very costly for them in absolute terms of linear costs, a, and this neve
happened in our simulations. But in the general case, one should consider this possibility
	(6)
Linear constraint is an intuitive rule and simple to work with. The one caveat is that it
is marginally as costly for the agents to reduce the variance by 1% as by 100%. Agents
potentially could choose to learn the whole truth and choose the posterior to be zero.
This, of course, is very costly for them in absolute terms of linear costs, a, and this never
happened in our simulations. But in the general case, one should consider this possibility.
The entropy based constraint implies that the agent pays for each unit of log variance decrease.
One can find some variation in the definition of the entropy based learning rule. For example,
in Nieuwerburgh and Veldkamp (2010) it is the simple ratio of prior to posterior variance.
Mackowiak andWiederholt (2009) use the logarithm of base 2, while there are many papers on
rational inattention using a natural logarithm (e.g. Matejka and McKay (2015) and Cabrales et
al. (2013)). In our definition of entropy we follow Mackowiak and Wiederholt (2009)4
:
	
The entropy based constraint implies that the agent pays for each unit of log variance
decrease. One can find some variation in the definition of the entropy based learning
rule. For example, in Nieuwerburgh and Veldkamp (2010) it is the simple ratio of prior
to posterior variance. Mackowiak and Wiederholt (2009) use the logarithm of base 2,
while there are many papers on rational inattention using a natural logarithm (e.g.
Matejka and McKay (2015) and Cabrales et al. (2013)). In our definition of entropy we
follow Mackowiak and Wiederholt (2009)4:
bt =𝑎𝑎𝑎𝑎 ∙ 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙2 �
𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2� (7)
The advantage of the entropy rule is that when the agent gets closer to learning the true
state of the world (posterior variance goes to zero), the required budget goes to infinity.
The entropy constraint is also well-motivated for analysis of processing the information
subject to limited capacity. In our case, however, the agent’s decision resembles more a
choice of a quality of market report to buy or market expert to pay, than processing
market data him/herself. That is, in our view, both types of constraints are well reasoned
here.
To select the information budget the agent maximizes the utility as in (4), but the
decision is now divided in two subperiods. The information budget is chosen in the first
subperiod:
max
𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡,1 �𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡,2Π𝑡𝑡𝑡𝑡+1 −
1
𝜌𝜌𝜌𝜌
𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡,2(Π𝑡𝑡𝑡𝑡+1)� (8)
2
	 (7)
The advantage of the entropy rule is that when the agent gets closer to learning the true
state of the world (posterior variance goes to zero), the required budget goes to infinity. The
entropy constraint is also well-motivated for analysis of processing the information subject
to limited capacity. In our case, however, the agent’s decision resembles more a choice of a
quality of market report to buy or market expert to pay, than processing market data him/
herself. That is, in our view, both types of constraints are well reasoned here.
To select the information budget the agent maximizes the utility as in (4), but the decision
is now divided in two subperiods. The information budget is chosen in the first subperiod:
	
The entropy based constraint implies that the agent pays for each unit of log variance
decrease. One can find some variation in the definition of the entropy based learning
rule. For example, in Nieuwerburgh and Veldkamp (2010) it is the simple ratio of prior
to posterior variance. Mackowiak and Wiederholt (2009) use the logarithm of base 2,
while there are many papers on rational inattention using a natural logarithm (e.g.
Matejka and McKay (2015) and Cabrales et al. (2013)). In our definition of entropy we
follow Mackowiak and Wiederholt (2009)4:
bt =𝑎𝑎𝑎𝑎 ∙ 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙2 �
𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2� (7)
The advantage of the entropy rule is that when the agent gets closer to learning the true
state of the world (posterior variance goes to zero), the required budget goes to infinity.
The entropy constraint is also well-motivated for analysis of processing the information
subject to limited capacity. In our case, however, the agent’s decision resembles more a
choice of a quality of market report to buy or market expert to pay, than processing
market data him/herself. That is, in our view, both types of constraints are well reasoned
here.
To select the information budget the agent maximizes the utility as in (4), but the
decision is now divided in two subperiods. The information budget is chosen in the first
subperiod:
max
𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡,1 �𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡,2Π𝑡𝑡𝑡𝑡+1 −
1
𝜌𝜌𝜌𝜌
𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡,2(Π𝑡𝑡𝑡𝑡+1)� (8)
subject to (3) and posterior variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
, given by one of the learning rules: (6) or (7).
Note, that in (8) the agent chooses bt in the first subperiod before knowing his expected
return in the second subperiod (before the signal - market report - is realized). Adopting
the formula from Nieuwerburgh and Veldkamp (2010), formula 14, we have:
	
(8)
subject to (3) and posterior variance,
ssive Risk-Taking and Information Acquisition
ction we analyze the two channels through which a bank accumulates risk in
olio when the safe interest rate is reduced or market volatility declines. One of
lear from (3): whenever the safe interest rate drops, it increases the risk
and makes the risky asset more attractive. Similarly, when asset variance is
the bank rationally increases holdings of the risky asset. The other channel
d in this paper is a change in information acquisition: reduction in the
on budget. Through this channel, the bank increases the riskiness of the asset
choosing to learn less about it. The portfolio risk then, as a product of risky
dings and return variance, increases with the lower interest rate and, in some
wer market volatility.
ance, the reduction in information acquisition with increase in risky asset
might seem counter-intuitive. It could be suggested that with larger asset
agents would like to learn more about them. For example Nieuwerburgh and
p (2010) found that when allocating fixed learning capacity between the assets,
ocate more to those assets they invest more into. Here, we should remind the
at in our paper we are studying not the allocation of the fixed capacity, but the
ation of this capacity: by how much agents are willing to reduce their expected
n order to reduce the income variance. Also this capacity, in the form of the
on budget, is itself a function of expected return and initial variance. It
a trade-off between the return the agent expects to get and variance he/she
e to reduce. Below, we study the properties of the information budget for
learning functions.
g function choice could influence the results (and we show later that this is the
consider alternative functions. Nieuwerburgh and Veldkamp (2010) show that
e of utility function and learning technologies in uences results quantitatively
etimes, qualitatively. They consider mean-variance and exponential utility
and three learning rules: one linear and two entropy based measures. Below,
mean-variance utility under linear and entropy learning functions.
on Budget and Comparative Statics. As in Nieuwerburgh and Veldkamp (2010)
der alternative learning functions, f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) in (5): a linear rule and an entropy
e linear function implies that the bank pays fixed costs, a, for each unit of the
line in the variance:
bt =𝑎𝑎𝑎𝑎(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) (6)
, given by one of the learning rules: (6) or (7).
4	 The results with a natural algorithm do not differ qualitatively, and there is a minor quantitative difference.
ACTA VŠFS, 2/2015, vol. 9 B123
Note, that in (8) the agent chooses
9
subject to (3) and posterior variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
, given by one of the learning rules: (6) or (7).
Note, that in (8) the agent chooses bt in the first subperiod before knowing his expected
return in the second subperiod (before the signal - market report - is realized). Adopting
the formula from Nieuwerburgh and Veldkamp (2010), formula 14, we have:
It is instructive to analyze comparative statics of the resulting solutions. In the partial
equilibrium model we take as given both assets returns, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
and its mean, and 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. It
will be convenient then to consider model’s response to change in expected risk premia,
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. In a general equilibrium, both returns will be determined by the market
clearing condition, with a stochastic component influencing risk asset return. In table 2,
the changes in the information budget with respect to variables of interest are described
(for full description of the derivatives, the reader is referred to the appendix).
4
The results with a natural algorithm do not differ qualitatively, and there is a minor quantitat
difference.
in the first subperiod before knowing his expected
return in the second subperiod (before the signal - market report - is realized). Adopting the
formula from Nieuwerburgh and Veldkamp (2010), formula 14, we have:
9
subject to (3) and posterior variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
, given by one of the learning rules: (6) or (7).
Note, that in (8) the agent chooses bt in the first subperiod before knowing his expected
return in the second subperiod (before the signal - market report - is realized). Adopting
the formula from Nieuwerburgh and Veldkamp (2010), formula 14, we have:
It is instructive to analyze comparative statics of the resulting solutions. In the partial
equilibrium model we take as given both assets returns, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
and its mean, and 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. It
will be convenient then to consider model’s response to change in expected risk premia,
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. In a general equilibrium, both returns will be determined by the market
clearing condition, with a stochastic component influencing risk asset return. In table 2,
the changes in the information budget with respect to variables of interest are described
(for full description of the derivatives, the reader is referred to the appendix).
4
The results with a natural algorithm do not differ qualitatively, and there is a minor quantitativ
difference.
It is instructive to analyze comparative statics of the resulting solutions. In the partial equilibrium
model we take as given both assets returns,
9
ct to (3) and posterior variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
, given by one of the learning rules: (6) or (7).
that in (8) the agent chooses bt in the first subperiod before knowing his expected
n in the second subperiod (before the signal - market report - is realized). Adopting
ormula from Nieuwerburgh and Veldkamp (2010), formula 14, we have:
nstructive to analyze comparative statics of the resulting solutions. In the partial
brium model we take as given both assets returns, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
and its mean, and 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. It
e convenient then to consider model’s response to change in expected risk premia,
1 − 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. In a general equilibrium, both returns will be determined by the market
ng condition, with a stochastic component influencing risk asset return. In table 2,
hanges in the information budget with respect to variables of interest are described
ull description of the derivatives, the reader is referred to the appendix).
he results with a natural algorithm do not differ qualitatively, and there is a minor quantitative
nce.
and its mean, and
9
(3) and posterior variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
, given by one of the learning rules: (6) or (7).
in (8) the agent chooses bt in the first subperiod before knowing his expected
he second subperiod (before the signal - market report - is realized). Adopting
a from Nieuwerburgh and Veldkamp (2010), formula 14, we have:
ctive to analyze comparative statics of the resulting solutions. In the partial
m model we take as given both assets returns, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
and its mean, and 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. It
venient then to consider model’s response to change in expected risk premia,
𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. In a general equilibrium, both returns will be determined by the market
ondition, with a stochastic component influencing risk asset return. In table 2,
es in the information budget with respect to variables of interest are described
scription of the derivatives, the reader is referred to the appendix).
sults with a natural algorithm do not differ qualitatively, and there is a minor quantitative
. It will be convenient
then to consider model’s response to change in expected risk premia,
subject to (3) and pos
Note, that in (8) the a
return in the second s
the formula from Nie
It is instructive to an
equilibrium model we
will be convenient the
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. In a gen
clearing condition, w
the changes in the inf
(for full description o
4
The results with a
difference.
.
In a general equilibrium, both returns will be determined by the market clearing condition,
with a stochastic component influencing risk asset return. In table 2, the changes in the information
budget with respect to variables of interest are described (for full description of
the derivatives, the reader is referred to the appendix).
Table 2: Comparative Statics: Information Budget
Information budget
derivatives
Linear rule Entropy rule
Information
budget
derivatives
Linear rule Entropy rule
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
�
negative negative for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
positive positive for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎
positive positive, but negative for relatively small
Table 2: Comparative Statics: Information Budget
Comparing derivatives under both learning rules in table 2, we see the similar
responses. The information budget rises when initial variance rises, so that
volatility in the market, agents are willing to sacrifice a larger budget to reduce u
Also, with a larger expected risk premium agents are willing to invest less in re
uncertainty, as the larger expected return compensates agents for taking a risk.
Table 2 explains the information channel of increase in risk-taking. When the s
rate falls, it decreases the expected risk premium (which is (𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠)), and de
information budget. With a lower information budget, the agent has a large
variance. Similarly, with a lower initial volatility (prior variance), the agent decid
smaller information budget. The initial effect of a reduction in interest rate or init
on the risky asset position is positive. It could be suggested, that a small informa
and larger posterior variance may offset this effect. We show below that this is n
in our model. The bank’s risky position rises, and, together with small i
acquisition, drives up portfolio variance.
negative negative for
Information
budget
derivatives
Linear rule Entropy rule
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
�
negative negative for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
positive positive for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎
positive positive, but negative for relatively small
Table 2: Comparative Statics: Information Budget
Comparing derivatives under both learning rules in table 2, we see the similar
responses. The information budget rises when initial variance rises, so that
volatility in the market, agents are willing to sacrifice a larger budget to reduce
Also, with a larger expected risk premium agents are willing to invest less in r
uncertainty, as the larger expected return compensates agents for taking a risk.
Table 2 explains the information channel of increase in risk-taking. When the s
rate falls, it decreases the expected risk premium (which is (𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠)), and de
information budget. With a lower information budget, the agent has a larg
variance. Similarly, with a lower initial volatility (prior variance), the agent decid
smaller information budget. The initial effect of a reduction in interest rate or ini
on the risky asset position is positive. It could be suggested, that a small informa
and larger posterior variance may offset this effect. We show below that this is
in our model. The bank’s risky position rises, and, together with small
acquisition, drives up portfolio variance.
, otherwise 0
Information
budget
derivatives
Linear rule Entropy rule
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
�
negative negative for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
positive positive for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎
positive positive, but negative for relatively small
Table 2: Comparative Statics: Information Budget
Comparing derivatives under both learning rules in table 2, we see the similar
responses. The information budget rises when initial variance rises, so that
volatility in the market, agents are willing to sacrifice a larger budget to reduce
Also, with a larger expected risk premium agents are willing to invest less in r
uncertainty, as the larger expected return compensates agents for taking a risk
Table 2 explains the information channel of increase in risk-taking. When the
rate falls, it decreases the expected risk premium (which is (𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠)), and de
information budget. With a lower information budget, the agent has a larg
variance. Similarly, with a lower initial volatility (prior variance), the agent decid
smaller information budget. The initial effect of a reduction in interest rate or in
on the risky asset position is positive. It could be suggested, that a small informa
and larger posterior variance may offset this effect. We show below that this is
in our model. The bank’s risky position rises, and, together with small
acquisition, drives up portfolio variance.
positive positive for
Information
budget
derivatives
Linear rule Entropy rule
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
�
negative negative for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
positive positive for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎
positive positive, but negative for relatively small
Table 2: Comparative Statics: Information Budget
Comparing derivatives under both learning rules in table 2, we see the similar s
responses. The information budget rises when initial variance rises, so that w
volatility in the market, agents are willing to sacrifice a larger budget to reduce u
Also, with a larger expected risk premium agents are willing to invest less in re
uncertainty, as the larger expected return compensates agents for taking a risk.
Table 2 explains the information channel of increase in risk-taking. When the sa
rate falls, it decreases the expected risk premium (which is (𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠)), and dec
information budget. With a lower information budget, the agent has a large
variance. Similarly, with a lower initial volatility (prior variance), the agent decide
smaller information budget. The initial effect of a reduction in interest rate or init
on the risky asset position is positive. It could be suggested, that a small informat
, otherwise 0
Information
budget
derivatives
Linear rule Entropy rule
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
�
negative negative for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
positive positive for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎
positive positive, but negative for relatively small
Table 2: Comparative Statics: Information Budget
Comparing derivatives under both learning rules in table 2, we see the sim
responses. The information budget rises when initial variance rises, so t
volatility in the market, agents are willing to sacrifice a larger budget to redu
Also, with a larger expected risk premium agents are willing to invest less
uncertainty, as the larger expected return compensates agents for taking a r
Table 2 explains the information channel of increase in risk-taking. When t
rate falls, it decreases the expected risk premium (which is (𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠)), and
information budget. With a lower information budget, the agent has a l
variance. Similarly, with a lower initial volatility (prior variance), the agent de
smaller information budget. The initial effect of a reduction in interest rate o
on the risky asset position is positive. It could be suggested, that a small info
and larger posterior variance may offset this effect. We show below that thi
in our model. The bank’s risky position rises, and, together with sm
acquisition, drives up portfolio variance.
positive positive, but negative for relatively small
Comparing derivatives under both learning rules in table 2, we see the similar signs of the responses.The
information budget rises when initial variance rises, so that with larger volatility
in the market, agents are willing to sacrifice a larger budget to reduce uncertainty. Also, with
a larger expected risk premium agents are willing to invest less in reducing the uncertainty,
as the larger expected return compensates agents for taking a risk.
Table 2 explains the information channel of increase in risk-taking.When the safe interest rate
falls, it decreases the expected risk premium (which is
ormation
budget
rivatives
Linear rule Entropy rule
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
�
negative negative for , otherwise 0
𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
positive positive for , otherwise 0
𝑡𝑡𝑡𝑡 positive positive, but negative for relatively small
Table 2: Comparative Statics: Information Budget
aring derivatives under both learning rules in table 2, we see the similar signs of the
nses. The information budget rises when initial variance rises, so that with larger
lity in the market, agents are willing to sacrifice a larger budget to reduce uncertainty.
with a larger expected risk premium agents are willing to invest less in reducing the
tainty, as the larger expected return compensates agents for taking a risk.
2 explains the information channel of increase in risk-taking. When the safe interest
alls, it decreases the expected risk premium (which is (𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠)), and decreases the
mation budget. With a lower information budget, the agent has a larger posterior
nce. Similarly, with a lower initial volatility (prior variance), the agent decides to have a
er information budget. The initial effect of a reduction in interest rate or initial variance
e risky asset position is positive. It could be suggested, that a small information budget
arger posterior variance may offset this effect. We show below that this is not the case
r model. The bank’s risky position rises, and, together with small information
sition, drives up portfolio variance.
Accumulation in Partial Equilibrium. Calculating derivatives with respect to risk
ium and prior variance, we find that risky asset holdings decrease in initial variance and
), and decreases the information
budget. With a lower information budget, the agent has a larger posterior variance.
Similarly, with a lower initial volatility (prior variance), the agent decides to have a smaller
information budget. The initial effect of a reduction in interest rate or initial variance on the
risky asset position is positive. It could be suggested, that a small information budget and
larger posterior variance may offset this effect. We show below that this is not the case in
our model. The bank’s risky position rises, and, together with small information acquisition,
drives up portfolio variance.
ACTA VŠFS, 2/2015, vol. 9124
Risk Accumulation in Partial Equilibrium. Calculating derivatives with respect to risk premium
and prior variance, we find that risky asset holdings decrease in initial variance and
increase in risk premium5
. Figure 1 illustrates this point. The graphs were drawn with fixed
interest rates. Later in the paper we analyze a general equilibrium model where interest rates
are set by the market.
Figure 1: Risk Accumulation in a Partial Equilibrium
a b
c d
Linear
Entropy
Note: dotted line corresponds to information budget b, dashed line - to risky asset holdings kb, solid line - to
portfolio variance, bold solid line - steady state portfolio variance
In figure 1 panels a and b correspond to a model with a linear learning rule; and c and d to an
entropy learning rule.The solid black line on all the graphs shows the initial (before reduction
in safe interest rate and variance) portfolio variance. The solid blue line represents portfolio
variance, its rise over the initial level shows the increase in portfolio variance.The channels of
portfolio variance increase are clear from the figure: there is a decline in information acquisi-
tion,
nstraint implies that the agent pays for each unit of log variance
d some variation in the definition of the entropy based learning
Nieuwerburgh and Veldkamp (2010) it is the simple ratio of prior
Mackowiak and Wiederholt (2009) use the logarithm of base 2,
y papers on rational inattention using a natural logarithm (e.g.
2015) and Cabrales et al. (2013)). In our definition of entropy we
d Wiederholt (2009)4:
bt =𝑎𝑎𝑎𝑎 ∙ 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙2 �
𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2� (7)
entropy rule is that when the agent gets closer to learning the true
sterior variance goes to zero), the required budget goes to infinity.
t is also well-motivated for analysis of processing the information
acity. In our case, however, the agent’s decision resembles more a
market report to buy or market expert to pay, than processing
elf. That is, in our view, both types of constraints are well reasoned
ation budget the agent maximizes the utility as in (4), but the
d in two subperiods. The information budget is chosen in the first
max
𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡
𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡,1 �𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡,2Π𝑡𝑡𝑡𝑡+1 −
1
𝜌𝜌𝜌𝜌
𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡,2(Π𝑡𝑡𝑡𝑡+1)� (8)
terior variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
, given by one of the learning rules: (6) or (7).
gent chooses bt in the first subperiod before knowing his expected
ubperiod (before the signal - market report - is realized). Adopting
uwerburgh and Veldkamp (2010), formula 14, we have:
alyze comparative statics of the resulting solutions. In the partial
take as given both assets returns, 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
and its mean, and 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
. It
n to consider model’s response to change in expected risk premia,
eral equilibrium, both returns will be determined by the market
th a stochastic component influencing risk asset return. In table 2,
ormation budget with respect to variables of interest are described
6
, and an increase in risky asset holdings,
7
the risky asset in the subperiod 2 as in (3) where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is posterior variance of the return.
Thus, with the information investment - budget bt and (3), the banks utility is rewritten:
(4)
subject to the learning rule:
(5)
and non-forgeting constraint: 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
> 0 . a is cost of reducing the variance, and
f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) is the learning function. The function is continuous and monotone in both of its
arguments, it is increasing in initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, and is decreasing in posterior, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
.
Intuitively, the more we reduce the posterior variance relative to the prior, the more we
should pay. We assume that the information budget is exhausted so that (5) becomes
equality. Then with the properties of our learning function, the choice of the information
budget, bt, uniquely determines the posterior variance and captures the information
decision of the bank.
In the following section we consider risk-taking decisions of the bank in a partial
equilibrium to identify risk driving forces.
Aggregating Financial Markets. The total investment into the safe asset, res, is given by
the bank’s financial resources not invested into the risky asset:
The investment into the safe asset is determined as deposits, dt, that was not invested
in the risky asset, 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
.
Recall, that the risky asset in the model is the investment in the manufacturing firm,
which uses it to build new capital. The manufacturing firm does not have funds for
investment on its own. To invest it has to sell its claims to the bank. Thus, the total
investment into the capital is then given by the bank’s risky asset holdings:
It = 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
.
Panels a and c in figure 1 show, that when the safe interest rate falls, there is a larger risk
accumulated in the portfolio. The risky asset position increases and the information budget
falls.This resembles the debate that a low interest rate environment stimulated excessive risk-
5	 With the entropy learning, the risky asset position increases in risk premium for large enough
3 Excessive Risk-Taking and Information Acquisition
In this section we analyze the two channels through which a bank accumula
the portfolio when the safe interest rate is reduced or market volatility declin
them is clear from (3): whenever the safe interest rate drops, it increase
premium and makes the risky asset more attractive. Similarly, when asset v
reduced, the bank rationally increases holdings of the risky asset. The othe
highlighted in this paper is a change in information acquisition: reducti
information budget. Through this channel, the bank increases the riskiness of
per se by choosing to learn less about it. The portfolio risk then, as a produ
asset holdings and return variance, increases with the lower interest rate and
cases, lower market volatility.
At first glance, the reduction in information acquisition with increase in r
holdings might seem counter-intuitive. It could be suggested that with la
holdings, agents would like to learn more about them. For example Nieuwer
Veldkamp (2010) found that when allocating fixed learning capacity between t
agents allocate more to those assets they invest more into. Here, we should r
reader, that in our paper we are studying not the allocation of the fixed capaci
determination of this capacity: by how much agents are willing to reduce their
income in order to reduce the income variance. Also this capacity, in the fo
information budget, is itself a function of expected return and initial va
describes a trade-off between the return the agent expects to get and varian
would like to reduce. Below, we study the properties of the information b
specified learning functions.
As learning function choice could influence the results (and we show later that
case), we consider alternative functions. Nieuwerburgh and Veldkamp (2010)
the choice of utility function and learning technologies in uences results qua
and, sometimes, qualitatively. They consider mean-variance and exponen
functions, and three learning rules: one linear and two entropy based measur
we study mean-variance utility under linear and entropy learning functions.
Information Budget and Comparative Statics. As in Nieuwerburgh and Veldka
we consider alternative learning functions, f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) in (5): a linear rule and an
based. The linear function implies that the bank pays fixed costs, a, for each u
linear decline in the variance:
bt =𝑎𝑎𝑎𝑎(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
)
Linear constraint is an intuitive rule and simple to work with. The one caveat
marginally as costly for the agents to reduce the variance by 1% as by 100
potentially could choose to learn the whole truth and choose the posterior t
This, of course, is very costly for them in absolute terms of linear costs, a, and
. All derivatives
are in the appendix.
6	 At some point (panels b-d) the information budget hits zero. At this point, the model behaves the same as
the one without information acquisition. Below this point, a sharper increase in risky asset holdings,
7
allocate to information decision. The choice of the budget determi
variance will be reduced. In the spirit of Nieuwerburgh and V
interpret it as an investment into purchasing additional market data
not have prior knowledge of what is in the data, but knows that t
his/her forecast. We model this decision as a choice of budget that
variance, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
. When choosing the budget and posterior varian
account what the return expectations will be after the signal is obse
the agent has to form expectations about return expectations: ex
Yet before paying for the signal and observing it, the expected post
𝐸𝐸𝐸𝐸𝜇𝜇𝜇𝜇̂= 𝜇𝜇𝜇𝜇 .
When taking decisions in subperiod 1, the agent rationally anticip
the risky asset in the subperiod 2 as in (3) where 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
is posterior v
Thus, with the information investment - budget bt and (3), the ban
subject to the learning rule:
and non-forgeting constraint: 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
− 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
> 0 . a is cost of reducin
f(𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, 𝜎𝜎𝜎𝜎�𝑡𝑡𝑡𝑡
2
) is the learning function. The function is continuous and m
arguments, it is increasing in initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
, and is decreas
Intuitively, the more we reduce the posterior variance relative to t
should pay. We assume that the information budget is exhausted
equality. Then with the properties of our learning function, the cho
budget, bt, uniquely determines the posterior variance and capt
decision of the bank.
In the following section we consider risk-taking decisions of t
equilibrium to identify risk driving forces.
Aggregating Financial Markets. The total investment into the safe
the bank’s financial resources not invested into the risky asset:
The investment into the safe asset is determined as deposits, dt, t
in the risky asset, 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
.
Recall, that the risky asset in the model is the investment in the
which uses it to build new capital. The manufacturing firm doe
investment on its own. To invest it has to sell its claims to the
investment into the capital is then given by the bank’s risky asset h
It = 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
, is
observed.
ACTA VŠFS, 2/2015, vol. 9 B125
taking during the Great Moderation. In our model, we capture also lower incentives to get
information about the risky asset the agent becomes more ignorant about the asset quality.
A similar result is found for reduction in market volatility in panels b and d. Surprisingly, when
the prior variance falls, the agent ends up with a larger portfolio risk than in a higher variance
environment. This result is, again, driven by the information channel: an agent is willing to
pay less for variance reduction when it is already small; and by larger risky asset accumulation
when the risk gets smaller.This finding could be also be applied to the Great Moderation
period, when market volatility was perceived to be low and financial agents demonstrated
a higher risk appetite.
Of course, when trying to explain overaccumulation of risk during the Great Moderation,
other forces besides the low volatility, mentioned, and a low safe interest rate environment
could be considered. We show in this paper, however, that market volatility and low policy
rates could be contributing factors to increase in risk preferences. These are also important
factors to consider when addressing current central banks’ policy of low interest rates and
suppressing market volatility.
Next, we complete the model and consider risk accumulation in a general equilibrium.
4	 General Equilibrium Model
Here we briefly describe the rest of the model and general equilibrium. Then we consider
the equilibrium impact of the interest rate change on risk preferences and information
acquisition, when there is feedback between the agents’asset holdings and market interest
rates.
Household. There is a representative household which maximizes the following utility
function:
	
acquisition, when there is feedback between the agents’ asset holdings and market
interest rates.
Household. There is a representative household which maximizes the following utility
function:
(9)
subject to a budget constraint:
dt + ct =𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
- tt (10)
where dt is household savings, f 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
is realized profit from the financial sector, 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
is
realized profit from manufactures and t is tax. The household decides how much to
consume and to invest in the bank. Its income is generated by the bank s and
manufacturer’s profits net of lump-sum taxes. u(c) is twice differentiable and concave.
Note, that we abstract from any labor decisions.
The consumption Euler equation looks standard and relates gross interest on savings to
the stochastic discount factor:
	 (9)
subject to a budget constraint:
	
acquisition, when there is feedback between the agents’ asset holdings and marke
interest rates.
Household. There is a representative household which maximizes the following utilit
function:
(9
subject to a budget constraint:
dt + ct =𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
- tt (10
where dt is household savings, f 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
is realized profit from the financial sector, 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
realized profit from manufactures and t is tax. The household decides how much t
consume and to invest in the bank. Its income is generated by the bank s an
manufacturer’s profits net of lump-sum taxes. u(c) is twice differentiable and concave
Note, that we abstract from any labor decisions.
The consumption Euler equation looks standard and relates gross interest on savings t
the stochastic discount factor:
𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡) = 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
𝛽𝛽𝛽𝛽𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡+1) (11
	
(10)
where
when there is feedback between the agents’ asset holdings and market
s.
There is a representative household which maximizes the following utility
(9)
budget constraint:
dt + ct =𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
- tt (10)
household savings, f 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
is realized profit from the financial sector, 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
is
fit from manufactures and t is tax. The household decides how much to
d to invest in the bank. Its income is generated by the bank s and
er’s profits net of lump-sum taxes. u(c) is twice differentiable and concave.
e abstract from any labor decisions.
ption Euler equation looks standard and relates gross interest on savings to
ic discount factor:
is household savings,
acquisition, when there is feedback between the agents’ asset holdings and market
interest rates.
Household. There is a representative household which maximizes the following utility
function:
(9)
subject to a budget constraint:
dt + ct =𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
- tt (10)
where dt is household savings, f 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
is realized profit from the financial sector, 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
is
realized profit from manufactures and t is tax. The household decides how much to
consume and to invest in the bank. Its income is generated by the bank s and
manufacturer’s profits net of lump-sum taxes. u(c) is twice differentiable and concave.
Note, that we abstract from any labor decisions.
The consumption Euler equation looks standard and relates gross interest on savings to
the stochastic discount factor:
𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡) = 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
𝛽𝛽𝛽𝛽𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡+1) (11)
is realized profit from the financial sector,
cquisition, when there is feedback between the agents’ asset holdings and market
nterest rates.
ousehold. There is a representative household which maximizes the following utility
unction:
(9)
ubject to a budget constraint:
dt + ct =𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
- tt (10)
where dt is household savings, f 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
is realized profit from the financial sector, 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
is
ealized profit from manufactures and t is tax. The household decides how much to
onsume and to invest in the bank. Its income is generated by the bank s and
manufacturer’s profits net of lump-sum taxes. u(c) is twice differentiable and concave.
ote, that we abstract from any labor decisions.
he consumption Euler equation looks standard and relates gross interest on savings to
he stochastic discount factor:
𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡) = 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
𝛽𝛽𝛽𝛽𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡+1) (11)
is
realized profit from manufactures and t is tax. The household decides how much to consume
and to invest in the bank. Its income is generated by the bank s and manufacturer’s
profits net of lump-sum taxes.
on, when there is feedback between the agents’ asset holdings and market
rates.
old. There is a representative household which maximizes the following utility
:
(9)
o a budget constraint:
dt + ct =𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
- tt (10)
t is household savings, f 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
is realized profit from the financial sector, 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
is
profit from manufactures and t is tax. The household decides how much to
e and to invest in the bank. Its income is generated by the bank s and
turer’s profits net of lump-sum taxes. u(c) is twice differentiable and concave.
at we abstract from any labor decisions.
sumption Euler equation looks standard and relates gross interest on savings to
hastic discount factor:
𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡) = 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
𝛽𝛽𝛽𝛽𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡+1) (11)
𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠
is twice differentiable and concave. Note, that we
abstract from any labor decisions.
The consumption Euler equation looks standard and relates gross interest on savings to
the stochastic discount factor:
	
acquisition, when there is feedback between the agents’ asset holdings and marke
interest rates.
Household. There is a representative household which maximizes the following utilit
function:
(9
subject to a budget constraint:
dt + ct =𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
- tt (10
where dt is household savings, f 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
is realized profit from the financial sector, 𝜋𝜋𝜋𝜋𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
i
realized profit from manufactures and t is tax. The household decides how much t
consume and to invest in the bank. Its income is generated by the bank s an
manufacturer’s profits net of lump-sum taxes. u(c) is twice differentiable and concave
Note, that we abstract from any labor decisions.
The consumption Euler equation looks standard and relates gross interest on savings t
the stochastic discount factor:
𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡) = 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
𝛽𝛽𝛽𝛽𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡+1) (11
𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
= 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
+
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡
(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
) (12
	
(11)
ACTA VŠFS, 2/2015, vol. 9126
	
12
the stochastic discount factor:
𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡) = 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
𝛽𝛽𝛽𝛽𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡+1) (11)
𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
= 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
+
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡
(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
) (12)
Manufacturer. On the production side there is a representative producer with a
production function:
yt+1 = zt+1kt
where z is stochastic productivity.
The producer needs to borrow money to finance investment (make new capital), and the
law of motion for capital is then:
kt+1 = It + (1-δ)kt (13)
The producer maximizes one period profit, which consists of revenues minus payment
on the loan for investment purposes:
(14)
where Rr is the gross interest rate paid to investors in the capital. We define Rr as
	 (12)
Manufacturer. On the production side there is a representative producer with a production
function:
12
𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡) = 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
𝛽𝛽𝛽𝛽𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢′(𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡+1) (11)
𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
= 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
+
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡
(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
) (12)
Manufacturer. On the production side there is a representative producer with a
production function:
yt+1 = zt+1kt
where z is stochastic productivity.
The producer needs to borrow money to finance investment (make new capital), and the
law of motion for capital is then:
kt+1 = It + (1-δ)kt (13)
The producer maximizes one period profit, which consists of revenues minus payment
on the loan for investment purposes:
(14)
where Rr is the gross interest rate paid to investors in the capital. We define Rr as
where z is stochastic productivity.
The producer needs to borrow money to finance investment (make new capital), and the
law of motion for capital is then:
	
12
𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
= 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
+
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡
(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
) (12)
Manufacturer. On the production side there is a representative producer with a
production function:
yt+1 = zt+1kt
where z is stochastic productivity.
The producer needs to borrow money to finance investment (make new capital), and the
law of motion for capital is then:
kt+1 = It + (1-δ)kt (13)
The producer maximizes one period profit, which consists of revenues minus payment
on the loan for investment purposes:
(14)
where Rr is the gross interest rate paid to investors in the capital. We define Rr as
	
(13)
The producer maximizes one period profit, which consists of revenues minus payment on
the loan for investment purposes:	
	
12
𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑑𝑑𝑑𝑑
= 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
+
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡
(𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡+1
𝑟𝑟𝑟𝑟
− 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠
) (1
Manufacturer. On the production side there is a representative producer with
production function:
yt+1 = zt+1kt
where z is stochastic productivity.
The producer needs to borrow money to finance investment (make new capital), and th
law of motion for capital is then:
kt+1 = It + (1-δ)kt (1
The producer maximizes one period profit, which consists of revenues minus payme
on the loan for investment purposes:
(14
where Rr is the gross interest rate paid to investors in the capital. We define Rr as
	 (14)
where Rr
is the gross interest rate paid to investors in the capital. We define Rr
as
	
(15)
That is, Rr depends on future productivity, is decreasing in capital, and is uncertain from
the investors point of view because of the uncertain z. Productivity z is such that the
expected return is as modeled in table 1.
Note, that all variables are expressed in real terms - in the units of final output.
4.1 Central Bank and Government
It is assumed that the government pays gross interest on the safe asset, and finances
expenditures by taxing the household. The government budget is balanced:
(16)
The role of the central bank in this economy is limited. Here we allow for a shock to the
safe interest rate through the household s Euler equation (11) which is supposed to
resemble monetary policy shock.
4.2 Equilibrium
Equilibrium in this model is a set of allocations: such that
given prices and beliefs all agents solve their problems and markets clear.
	 (15)
That is, Rr
depends on future productivity, is decreasing in capital, and is uncertain from
the investors point of view because of the uncertain z. Productivity z is such that the expected
return is as modeled in table 1.
Note, that all variables are expressed in real terms - in the units of final output.
4.1	 Central Bank and Government
It is assumed that the government pays gross interest on the safe asset, and finances expenditures
by taxing the household. The government budget is balanced:
	
(15)
That is, Rr depends on future productivity, is decreasing in capital, and is uncertain from
the investors point of view because of the uncertain z. Productivity z is such that the
expected return is as modeled in table 1.
Note, that all variables are expressed in real terms - in the units of final output.
4.1 Central Bank and Government
It is assumed that the government pays gross interest on the safe asset, and finances
expenditures by taxing the household. The government budget is balanced:
(16)
The role of the central bank in this economy is limited. Here we allow for a shock to the
safe interest rate through the household s Euler equation (11) which is supposed to
resemble monetary policy shock.
4.2 Equilibrium
Equilibrium in this model is a set of allocations: such that
given prices and beliefs all agents solve their problems and markets clear.
	 (16)
The role of the central bank in this economy is limited. Here we allow for a shock to the safe
interest rate through the household s Euler equation (11) which is supposed to resemble
monetary policy shock.
4.2	Equilibrium
Equilibrium in this model is a set of allocations:
(15)
That is, Rr depends on future productivity, is decreasing in capital, and is uncertain from
the investors point of view because of the uncertain z. Productivity z is such that the
expected return is as modeled in table 1.
Note, that all variables are expressed in real terms - in the units of final output.
4.1 Central Bank and Government
It is assumed that the government pays gross interest on the safe asset, and finances
expenditures by taxing the household. The government budget is balanced:
(16)
The role of the central bank in this economy is limited. Here we allow for a shock to the
safe interest rate through the household s Euler equation (11) which is supposed to
resemble monetary policy shock.
4.2 Equilibrium
Equilibrium in this model is a set of allocations: such that
given prices and beliefs all agents solve their problems and markets clear.
5 Simulations
such that given prices and beliefs all agents solve their problems and markets clear.
ACTA VŠFS, 2/2015, vol. 9 B127
5	Simulations
5.1	 Calibration and Parameter Values
In the model, most of the parameters are standard. The only nonstandard parameters
are learning costs, a, moment of productivity distribution -
13
Equilibrium in this model is a set of allocations: such that
given prices and beliefs all agents solve their problems and markets clear.
5 Simulations
5.1 Calibration and Parameter Values
In the model, most of the parameters are standard. The only nonstandard parameters
are learning costs, a, moment of productivity distribution - E (z) and initial variance of
agents’ beliefs, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
.This group of parameters was selected to ensure the existence of
solutions, and non-negative values of information cost, bt
7. Also, for alternative learning
specifications, to ensure the existence of equilibrium, these three parameters have to
be different.
Linear Entropy
ρ risk-aversion 2
α capital share 0.33
δ depreciation 0.02
β discount factor 0.95
7
Condition for the existence of non-negative bt are in appendix.
and initial variance
of agents’ beliefs,
13
given prices and beliefs all agents solve their problems and markets clear.
5 Simulations
5.1 Calibration and Parameter Values
In the model, most of the parameters are standard. The only nonstandard parame
are learning costs, a, moment of productivity distribution - E (z) and initial varianc
agents’ beliefs, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
.This group of parameters was selected to ensure the existenc
solutions, and non-negative values of information cost, bt
7. Also, for alternative lear
specifications, to ensure the existence of equilibrium, these three parameters hav
be different.
Linear Entropy
ρ risk-aversion 2
α capital share 0.33
δ depreciation 0.02
β discount factor 0.95
7
Condition for the existence of non-negative bt are in appendix.
.This group of parameters was selected to ensure the existence of
solutions, and non-negative values of information cost,
13
given prices and beliefs all agents solve their problems and markets clear.
5 Simulations
5.1 Calibration and Parameter Values
In the model, most of the parameters are standard. The only nonstandard parameters
are learning costs, a, moment of productivity distribution - E (z) and initial variance of
agents’ beliefs, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
.This group of parameters was selected to ensure the existence of
solutions, and non-negative values of information cost, bt
7. Also, for alternative learning
specifications, to ensure the existence of equilibrium, these three parameters have to
be different.
Linear Entropy
ρ risk-aversion 2
α capital share 0.33
δ depreciation 0.02
β discount factor 0.95
7
Condition for the existence of non-negative bt are in appendix.
7
. Also, for alternative learning
specifications, to ensure the existence of equilibrium, these three parameters have to be
different.
Table 3: Parameter Values
Linear Entropy
ρ risk-aversion 2
13
ilibrium in this model is a set of allocations: such that
n prices and beliefs all agents solve their problems and markets clear.
Simulations
Calibration and Parameter Values
he model, most of the parameters are standard. The only nonstandard parameters
learning costs, a, moment of productivity distribution - E (z) and initial variance of
nts’ beliefs, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
.This group of parameters was selected to ensure the existence of
tions, and non-negative values of information cost, bt
7. Also, for alternative learning
cifications, to ensure the existence of equilibrium, these three parameters have to
different.
Linear Entropy
ρ risk-aversion 2
α capital share 0.33
δ depreciation 0.02
β discount factor 0.95
ondition for the existence of non-negative bt are in appendix.
capital share 0.33
δ depreciation 0.02
β discount factor 0.95
s interest on the safe asset, and finances
vernment budget is balanced:
(16)
s limited. Here we allow for a shock to the
Euler equation (11) which is supposed to
: such that
problems and markets clear.
andard. The only nonstandard parameters
distribution - E (z) and initial variance of
was selected to ensure the existence of
tion cost, bt
7. Also, for alternative learning
uilibrium, these three parameters have to
Linear Entropy
2
0.33
0.02
0.95
n appendix.
mean productivity 10
a information costs 1 1.5
13
Note, that all variables are expressed in real terms - in the units of final output.
4.1 Central Bank and Government
It is assumed that the government pays gross interest on the safe asset, and finances
expenditures by taxing the household. The government budget is balanced:
(16)
The role of the central bank in this economy is limited. Here we allow for a shock to the
safe interest rate through the household s Euler equation (11) which is supposed to
resemble monetary policy shock.
4.2 Equilibrium
Equilibrium in this model is a set of allocations: such that
given prices and beliefs all agents solve their problems and markets clear.
5 Simulations
5.1 Calibration and Parameter Values
In the model, most of the parameters are standard. The only nonstandard parameters
are learning costs, a, moment of productivity distribution - E (z) and initial variance of
agents’ beliefs, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
.This group of parameters was selected to ensure the existence of
solutions, and non-negative values of information cost, bt
7. Also, for alternative learning
specifications, to ensure the existence of equilibrium, these three parameters have to
be different.
Linear Entropy
ρ risk-aversion 2
α capital share 0.33
δ depreciation 0.02
β discount factor 0.95
7
Condition for the existence of non-negative bt are in appendix.
prior variance 1.1
Table 3 shows the selected parameter values used for the simulation below. In this paper
we are focusing mainly on intuition, how low policy rates and / or subdued market volatility
can influence risk-taking and what the contribution of the information channel could
be. Above, in the section on partial equilibrium, we show that both risk-taking channels
work regardless of parameter values. That is why we consider our procedure for selecting
information costs and prior variance satisfactory for our purpose. If, however, one is
targeting quantitative effects, more rigorous calibration of information costs and market
volatility is necessary. For the mean productivity values, we are targeting that the condi-
tion
E (z) mean productivity 10
a information costs 1 1.5
𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2 prior variance 1.1
Table 3: Parameter Values
s the selected parameter values used for the simulation below. In this paper
sing mainly on intuition, how low policy rates and / or subdued market
influence risk-taking and what the contribution of the information channel
bove, in the section on partial equilibrium, we show that both risk-taking
rk regardless of parameter values. That is why we consider our procedure
g information costs and prior variance satisfactory for our purpose. If,
e is targeting quantitative effects, more rigorous calibration of information
market volatility is necessary. For the mean productivity values, we are
at the condition is satisfied in the steady state. Even
elected number seems to be large, it results in steady state risky asset return
1.0920 for linear and entropy rules respectively.
ubsection we show general equilibrium results for our model of information
ulations
h a linear learning rule model. For the simulations8, we lowered the initial
safe interest rate for 1% and 0.1% respectively for 20 periods. The safe
is satisfied in the steady state. Even though the selected
number seems to be large, it results in steady state risky asset return 1.2369 and 1.0920
for linear and entropy rules respectively.
In the next subsection we show general equilibrium results for our model of information
acquisition.
7	 Condition for the existence of non-negative bt are in appendix.
ACTA VŠFS, 2/2015, vol. 9128
5.2	Simulations
We start with a linear learning rule model. For the simulations8
, we lowered the initial variance
or safe interest rate for 1% and 0.1% respectively for 20 periods. The safe interest rate was
reduced using a deterministic shock to the household’s Euler equation (11). After 20 periods,
both of the variables return to their steady state values, together with other model variables.
Figure 2 reports responses for a linear learning rule model. The vertical dashed lines mark the
start and end of the decline in selected variables. Panel a shows the reaction to a shock to
the Euler equation, which we here call "monetary policy". Recall that there is no money in the
model, and this name is figurative to suggest that the shock to the safe interest rate resembles
monetary authority action in a full-blown New Keynesian model. One also can note from the
panel a that agents are rational and the safe interest change is expected: the slight adjustment
to the change starts ahead of the actual shock realization. Following the decline in the safe
interest rate, the bank’s risky asset holdings increase. The risky asset is investment into capital
in our economy, which is why additional capital is accumulated. Larger capital accumulation
reduces the expected return on capital. This is the force that returns the model to the steady
state after the policy is removed. Before this, there is a drop in the information budget as a
larger risk premium (expected return on risky asset falls less than safe interest rate) makes an
agent tolerate larger risk. Lower information acquisition determines larger posterior variance.
Both larger posterior variance and the risky asset position increase the bank’s portfolio risk.
Figure 2: Linear Learning Rule
a. Accommodative "Monetary Policy" Shock, 0.1%
8 	 The simulations are done using Dynare version 4.2.	
ACTA VŠFS, 2/2015, vol. 9 B129
b. Prior Variance Reduction by 1%
For the change in initial variance, panel b, we also observe some adjustments beforehand.
Anticipating decline in the variance, risky-asset holdings, capital and consumption start
to increase before the actual variance reduction. Accumulation of capital declines the
return on capital, which is the risky asset in our model. At period t =40 when the initial
variance falls, the information budget falls too. Posterior variance, being the difference
of prior variance and the information budget, declines, but two times less than the prior.
Information costs are unity in this model, which is why, without the information channel
the posterior variance from (6) should fall by the same amount as the prior variance. A
decline in the information budget here reduces the effect of initial volatility on the risk
that agents are facing. This and a rise in risky asset portfolio holdings increase portfolio
variance above the steady state level. At period t =50, when the expected return reaches
its minimum value, risky asset holdings and portfolio variance start declining. After the
policy is removed and the level of capital reduced, the increasing expected return returns
the economy back to the steady state.
For the model with the entropy learning rule, figure 3, panel a; a very similar response to
interest rate decline is found. A reduction in safe interest rates simultaneously reduces
information acquisition and increases risky asset holdings. A combination of the two increases
the bank’s portfolio risk.
ACTA VŠFS, 2/2015, vol. 9130
Figure 3: Entropy Learning Rule
a. Accommodative "Monetary Policy" Shock, 0.1%
b. Prior Variance Reduction by 1%
ACTA VŠFS, 2/2015, vol. 9 B131
When considering a reduction in prior variance, figure 3, panel b, a different response
of the information budget and safe interest rate is observed. Risky asset holdings are
increased, raising capital and consumption and decreasing the expected return. At the
same time there is a reduction in the information budget, but unlike in the linear model,
this effect is short-lived, and is reversed in a couple of periods. This leads to short-lived
increase in portfolio variance, which declines afterwards. If in the linear model the information
budget is always below the steady state level for lower prior variance, it is not the
case in entropy. With the entropy constraint, there is a larger effect of falling expected return
on the information budget. With the expected return falling, the information budget
starts to increase, decreasing posterior variance and portfolio risk. Also, the initial fall in
the information budget is less pronounced than in the linear model. The difference is
partially attributed to larger information costs and partially to a different functional form
of learning function.
Conclusions
This paper addresses the debate as to whether periods of low policy rates and low market
volatility could lead to overaccumulation of risky assets. It is motivated by the number of
empirical studies showing that increase in risk appetite is associated with low policy rates.
We contribute to the literature by building a model with rationally inattentive financial
agents, who decide how much to invest in information acquisition subject to information
costs. Information acquisition is modelled as paying for a decline in risky asset variance.
We consider two basic learning functions: entropy and linear learning rule.
It is then shown that with a low safe interest rate there are two channels of increase in
risk-taking: a standard in the literature search-for-yield, and a decline in the information
budget. These two channels result in a high risky asset position and high risk of the asset
per se, as an agent face higher uncertainty about asset returns. As a result, agent accumulates
more risk in his or her portfolio when the safe asset rate falls. These findings are
robust to the learning rule specification.
Another result is larger risk-taking with the decline in risky asset volatility. When the variance
of risky return falls, agents rationally increase their risky asset holdings. At the same
time, they are willing to pay less for further reduction in return variance. Lower incentives
for information acquisition partially offset the drop in initial variance, with posterior variance
falling much less than the prior. In combination with larger risky asset holdings, it
increases agent’s portfolio variance.
Acknowledgements
We are grateful to Sergey Slobodyan, Filip Matejka, Michal Kejak, Mirko Wiederholt, Vincent
Sterk and Jaume Ventura for comments and suggestions. The work was supported by
Charles University Grant Agency grant number 528314 and published as CERGE-EI Working
Paper no 536.
ACTA VŠFS, 2/2015, vol. 9132
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Contact address
Mgr. Volha Audzei
CERGE-EI - a joint workplace of Charles University in Prague and the Economics Institute of
the Academy of Sciences of the Czech Republic/ CERGE-EI – společné pracoviště Univerzity
Karlovy v Praze a Národohospodářského ústavu AV ČR
(e-mail: vaudzei@cerge-ei.cz)
ACTA VŠFS, 2/2015, vol. 9134
Appendix Comparative static
Linear Learning Rule
From table 2 the solution for information budget is positive when information costs are:
Appendix Comparative static
Linear Learning Rule
From table 2 the solution for information budget is positive when information costs are:
That is, larger than one plus the expected return to variance ratio. In this interval, the
derivative with respect to initial variance is positive:
And the derivative with respect to risk premium is non-positive:
The impact of information costs increase is always positive on the interval with
positive bt :
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
Entropy Learning Rule
bt, is positive when
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
The derivative with respect to risk premia is always non-positive:
That is, larger than one plus the expected return to variance ratio. In this interval, the
derivative with respect to initial variance is positive:
Appendix Comparative static
Linear Learning Rule
From table 2 the solution for information budget is positive when information costs are:
That is, larger than one plus the expected return to variance ratio. In this interval, the
derivative with respect to initial variance is positive:
And the derivative with respect to risk premium is non-positive:
The impact of information costs increase is always positive on the interval with
positive bt :
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
Entropy Learning Rule
bt, is positive when
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
And the derivative with respect to risk premium is non-positive:
Appendix Comparative static
Linear Learning Rule
From table 2 the solution for information budget is positive when information costs are:
That is, larger than one plus the expected return to variance ratio. In this interval, the
derivative with respect to initial variance is positive:
And the derivative with respect to risk premium is non-positive:
The impact of information costs increase is always positive on the interval with
positive bt :
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
Entropy Learning Rule
bt, is positive when
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
The impact of information costs increase is always positive on the interval with positive bt
:
Appendix Comparative static
Linear Learning Rule
From table 2 the solution for information budget is positive when information costs are:
That is, larger than one plus the expected return to variance ratio. In this interval, the
derivative with respect to initial variance is positive:
And the derivative with respect to risk premium is non-positive:
The impact of information costs increase is always positive on the interval with
positive bt :
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
Entropy Learning Rule
bt, is positive when
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
The derivative with respect to risk premia is always non-positive:
The effect on risky asset portfolio holdings is characterized by the following derivatives:
Appendix Comparative static
Linear Learning Rule
From table 2 the solution for information budget is positive when information costs are:
That is, larger than one plus the expected return to variance ratio. In this interval, the
derivative with respect to initial variance is positive:
And the derivative with respect to risk premium is non-positive:
The impact of information costs increase is always positive on the interval with
positive bt :
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
Entropy Learning Rule
bt, is positive when
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
The derivative with respect to risk premia is always non-positive:
Entropy Learning Rule
bt
, is positive when
Appendix Comparative static
Linear Learning Rule
From table 2 the solution for information budget is positive when information costs are:
That is, larger than one plus the expected return to variance ratio. In this interval, the
derivative with respect to initial variance is positive:
And the derivative with respect to risk premium is non-positive:
The impact of information costs increase is always positive on the interval with
positive bt :
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
Entropy Learning Rule
bt, is positive when
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
The derivative with respect to risk premia is always non-positive:
ACTA VŠFS, 2/2015, vol. 9 B135
The derivative of budget with respect to initial variance,
19
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
The derivative with respect to risk premia is always non-positive:
is always nonnegative:
19
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
The derivative with respect to risk premia is always non-positive:
The derivative with respect to risk premia
19
t, is positive when
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
The derivative with respect to risk premia is always non-positive:is always non-positive:
19
bt, is positive when
The derivative of budget with respect to initial variance, 𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2
is always nonnegative:
The derivative with respect to risk premia is always non-positive:
The derivative of budget, bt
, with respect to information costs, a, is:
The derivative of budget, bt, with respect to information costs, a, is:
The sign of the derivative is determined by the nominator. The derivative is positive
when:
Since 0.6703 >1.0597, there is a region where the derivative could be negative. The
information budget is decreasing with information cost, when information costs are:
Thus for relatively small information costs, an increase in information cost will reduce the
information budget. For other, feasible values of a, an increase in information costs also
increases the information budget.
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
The sign of the derivative is determined by the nominator. The derivative is positive
when:
The derivative of budget, bt, with respect to information costs, a, is:
The sign of the derivative is determined by the nominator. The derivative is positive
when:
Since 0.6703 >1.0597, there is a region where the derivative could be negative. The
information budget is decreasing with information cost, when information costs are:
Thus for relatively small information costs, an increase in information cost will reduce t
information budget. For other, feasible values of a, an increase in information costs also
increases the information budget.
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
Since 0.6703 >1.0597, there is a region where the derivative could be negative. The information
budget is decreasing with information cost, when information costs are:
The derivative of budget, bt, with respect to information costs, a, is:
The sign of the derivative is determined by the nominator. The derivative is positive
when:
Since 0.6703 >1.0597, there is a region where the derivative could be negative. The
information budget is decreasing with information cost, when information costs are:
Thus for relatively small information costs, an increase in information cost will reduce the
information budget. For other, feasible values of a, an increase in information costs also
increases the information budget.
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
𝜕𝜕𝜕𝜕𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
𝜕𝜕𝜕𝜕�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠�
=
𝑎𝑎𝑎𝑎(−(𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠)2+𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2)
𝜌𝜌𝜌𝜌(�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠�
2
+𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2)2log(4)
>0
Thus for relatively small information costs, an increase in information cost will reduce
the information budget. For other, feasible values of a, an increase in information costs
also increases the information budget.
The effect on risky asset portfolio holdings is characterized by the following derivatives:
The derivative of budget, bt, with respect to information costs, a, is:
The sign of the derivative is determined by the nominator. The derivative is positive
when:
Since 0.6703 >1.0597, there is a region where the derivative could be negative. The
information budget is decreasing with information cost, when information costs are:
Thus for relatively small information costs, an increase in information cost will reduce th
information budget. For other, feasible values of a, an increase in information costs also
increases the information budget.
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
𝜕𝜕𝜕𝜕𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
𝜕𝜕𝜕𝜕�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠�
=
𝑎𝑎𝑎𝑎(−(𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠)2+𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2)
𝜌𝜌𝜌𝜌(�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠�
2
+𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2)2log(4)
>0
if
The derivative of budget, bt, with respect to information costs, a, is:
The sign of the derivative is determined by the nominator. The derivative is positive
when:
Since 0.6703 >1.0597, there is a region where the derivative could be negative. The
information budget is decreasing with information cost, when information costs are:
Thus for relatively small information costs, an increase in information cost will reduce th
information budget. For other, feasible values of a, an increase in information costs also
increases the information budget.
The effect on risky asset portfolio holdings is characterized by the following
derivatives:
𝜕𝜕𝜕𝜕𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡
𝑏𝑏𝑏𝑏
𝜕𝜕𝜕𝜕�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠�
=
𝑎𝑎𝑎𝑎(−(𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠)2+𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2)
𝜌𝜌𝜌𝜌(�𝜇𝜇𝜇𝜇𝑡𝑡𝑡𝑡−𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡
𝑠𝑠𝑠𝑠�
2
+𝜎𝜎𝜎𝜎𝑡𝑡𝑡𝑡
2)2log(4)
>0
if