The Dynamic Relations between Attention to News,

Investments, and Economic Conditions ∗

Michael Hasler †

June 12, 2014

Abstract

Investment strategies rely on investors capacity to acquire accurate information.

Given the overwhelming quantity of financial news, accurate information can only be

acquired at a cost. In this context, I show that investors’ optimal attention to news is

a hump-shaped function of expected stock returns and an increasing function of both

stock-return variance and uncertainty. Therefore, investors tend to invest more in risk-

free assets when their attention is high, and more in risky assets when their attention

is low. Moreover, attention is decreasing with both risk aversion and the expensiveness

of information. Our empirical findings lend support to the predictions of the model.

Keywords: Portfolio Choice, Attention to News, Information Acquisition, Learning

∗I would like to thank Daniel Andrei, Tony Berrada, Markus Brunnermeier, Patrick Cheridito, PierreCollin-Dufresne, Peter Christoffersen, Julien Cujean, Giuliano Curatola, Jerome Detemple, Bernard Du-mas, Ruslan Goyenko, Harrison Hong, Julien Hugonnier, Jakub Jurek, Jan Peter Kulak, Semyon Malamud,Roberto Marfe, Remy Praz, Michael Rockinger, Cornelius Schmidt, Ronnie Sircar, Ramona Westermann,Wei Xiong, Qunzi Zhang, and seminar participants at EPFL, HEC Lausanne, INSEAD, NCCR-SFI meet-ing, Princeton University, University of Geneva, University of Toronto, and University of Zurich for theircomments.†University of Toronto, 105 St. George Street, Toronto, ON, M5S 3E6, Canada, Michael.Hasler@

rotman.utoronto.ca, www.rotman.utoronto.ca/Faculty/Hasler.aspx

1

1 Introduction

Recent empirical findings show that investors pay time-varying attention to news (Da, En-

gelberg, and Gao (2011), Lustig and Verdelhan (2012)), and that attention is most likely

counter-cyclical (Vlastakis and Markellos (2012)). In addition, investors’ attention and in-

vestment strategies are linked to each other (Barber and Odean (2008)).

My contribution is to develop a model with optimal attention to news which makes sense

of the joint dynamics of attention and investment flows. We consider a dynamic portfolio

choice and costly information acquisition problem faced by a risk-averse investor. The in-

vestor optimally chooses consumption, a portfolio invested in a stock and a risk-free asset,

and the accuracy of a news signal in order to maximize her expected lifetime utility of con-

sumption. The news signal, whose accuracy can be improved at a cost, provides information

on the stochastic and unobservable expected stock return. In addition to the news signal, the

investor uses two free sources of information to infer the value of the unobservable expected

return, namely the observations of stock returns and their variances.

The optimal accuracy of the news signal acquired by the investor—the investor’s attention

to news1—is principally driven by changes in (estimated) expected returns and uncertainty.2

Changes in stock-return variance drive attention too, but to a lesser extent than do changes

in expected returns and uncertainty.

First, attention is a hump-shaped function of the expected return, which itself is mean-

reverting. Indeed, when the expected return is either high or low, the investor is confident

1See, e.g., Sims (2003), Peng and Xiong (2006), Kacperczyk, Nieuwerburgh, and Veldkamp (2009), Mon-dria (2010), and Mondria and Quintana-Domeque (2013) for a similar interpretation of attention.

2Formally, uncertainty is the variance of the expected stock return conditional on the investor’s informa-tion set. It is commonly called uncertainty, because it measures the inaccuracy of the estimated expectedreturn.

2

that it is going to revert back to its mean. When the expected return is close to its uncon-

ditional mean, noise is its main driver. Therefore, the investor has an incentive to inquire

information regarding its future direction only when it lies around its unconditional mean.

Second, attention increases with uncertainty as long as uncertainty is not too high. This

non-monotone relation holds because two opposite effects are at work. On the one hand, high

uncertainty implies an inaccurately estimated expected return and therefore a willingness to

acquire more accurate information. On the other hand, high uncertainty means high risk,

low risky investments, and therefore less incentive to acquire information. The former effect

dominates the latter for most levels of uncertainty, except the very highest.

Third, attention increases with the stock-return variance as long as the latter is not too

high. The rationale is similar to the case of uncertainty. Indeed, an increase in variance im-

plies a deterioration of risky investment opportunities and therefore an incentive to improve

them through the acquisition of information. But an increase in variance is also associated

to a decrease in the allocation to the risky asset, dissuading the investor to acquire accu-

rate information. As long as the variance of stock return is not too high, the former effect

dominates and implies a positive relation between attention and variance.

In line with existing theoretical findings, risky investments increase with expected returns

(e.g., Merton (1971), Kim and Omberg (1996), Campbell and Viceira (1999)) and decrease

with both variance (Chacko and Viceira (2005), Liu (2007)) and uncertainty (Xia (2001)).

Therefore, after controlling for the hump-shaped relation between attention and expected

returns, the model predicts that attention and risky investments tend to be inversely related.

Furthermore, comparative statics show that attention decreases with both risk aversion

and the expensiveness of information. The reason is that high risk aversion implies low risky

3

investments, while more expensive information decrease the wealth of the investor to a larger

extent. Consequently, an increase in either risk aversion or the expensiveness of information

discourages the investor from gathering information on expected returns.

In my empirical analysis, the fitted value of a predictive regression of future S&P500

returns proxies for expected stock return, a GARCH estimation yields the realized S&P500

return variance, and changes in the S&P500 trading volume provides a measure of investors’

attention (Barber and Odean (2008), Hou, Peng, and Xiong (2009)). The macro-uncertainty

index of Jurado, Ludvigson, and Ng (2013) serves as a proxy for uncertainty, while changes

in the Institutional Money Fund index measures the allocation in risk-free assets.3 I find

that attention is a hump-shaped function of expected returns and an increasing function

uncertainty. Variance, however, does not seem to be a significant driver of attention. In

addition, attention is particularly large when risk-free investments are high, and vice versa.

These empirical findings support the main theoretical predictions of the model.

This paper builds on the literature considering portfolio selection problems with stochas-

tic expected returns (Kim and Omberg (1996), Brandt (1999), Campbell and Viceira (1999),

Ait-Sahalia and Brandt (2001), Wachter (2002)), stochastic volatility (Chacko and Viceira

(2005), Liu (2007)), incomplete information (Detemple (1986), Gennotte (1986), Dothan

and Feldman (1986), Feldman (1989), David (1997), Brennan (1998), Veronesi (1999, 2000),

Brennan and Xia (2001)), and uncertainty about predictability (Kandel and Stambaugh

(1996), Barberis (2000), Xia (2001), Brandt, Goyal, Santa-Clara, and Stroud (2005)).

Furthermore, my study builds on the literature considering costly information acquisition.

3The Institutional Money Fund index is provided by the Federal Reserve Bank of St. Louis. The macro-uncertainty index is provided on Sydney Ludvigson’s website.

4

Detemple and Kihlstrom (1987) analyze the problem of an investor who invests in production

technologies and in an information market. Veldkamp (2006b) and Veldkamp (2006a) show

that costly information acquisition helps explain excess co-movement and the simultaneous

increases in emerging markets’ media coverage and equity prices, respectively. Huang and

Liu (2007) argue that costly information pushes investors to acquire imperfectly accurate

signals at a low frequency.

None of these studies shed light on the dynamic relations between attention, investments,

and economic conditions. This paper provides both theoretical and empirical analysis that

helps understand these fundamental relations.

The remainder of the paper is organized as follows. Section 2 presents and solves the

model, Section 3 provides the theoretical results, Section 4 tests the main predictions of

the model, and Section 5 concludes. Derivations and computational details are provided in

Appendix A.

2 The Model

This section presents and solves the problem of an investor who optimally chooses her port-

folio, consumption, and attention paid to news in order to maximize her expected lifetime

utility of consumption. The key feature of the model is that a news signal, whose accuracy

can be improved at a cost, provides information on expected returns. Therefore, the investor

faces a challenge which consists in optimally trading-off an accurate forecast of future returns

at a high cost with an inaccurate forecast but at a low cost.

The investor is allowed to invest in two assets: one stock and one risk free asset. The

5

prices of the stock and the risk-free asset are denoted by P and M , respectively. The

dynamics of prices and their constituents are

dPtPt

= µtdt+√VtdB1t

dµt = λµ(µ− µt)dt+ σµdB2t

dVt = λV (V − Vt)dt+ σV√VtdB3t

dMt = rMtdt,

where r is the constant risk-free rate, µ is the expected return on the stock4 (Ornstein and

Uhlenbeck (1930)), and V is the stock-return variance5 (Heston (1993)). Importantly, the

investor observes prices and the variance of stock returns,6 but not the expected return.

In order to get an idea of future market moves, the investor estimates the expected return

by observing stock price movements and news releases. The information signal acquired

through the processing of news releases has the following dynamics7

dst = µtdt+ σstdB4t,

where the volatility σs is stochastic and observable, because optimally chosen by the investor.

The processes B1, B2, B3, and B4 are Brownian motions with correlation matrix Λ defined

4See, e.g., Kim and Omberg (1996), Campbell and Viceira (1999), Wachter (2002), and Campbell, Chacko,Rodriguez, and Viceira (2004) for portfolio choice problems with autoregressive expected stock returns.

5See, e.g., Chacko and Viceira (2005), Liu (2007), and Gron, Jorgensen, and Polson (2012) for portfoliochoice problems with stochastic stock-return volatility.

6Since the variance of stock returns is characterized by the quadratic variation of returns, it is necessarilyobservable.

7The structure of the signal is borrowed from Detemple (1986), Veronesi (2000), Scheinkman and Xiong(2003), and Bansal and Shaliastovich (2011) among others.

6

as follows:

Λ =

1 ρP,µ ρP,V 0

ρP,µ 1 ρµ,V 0

ρP,V ρµ,V 1 0

0 0 0 1

.

Let us denote by µ the estimated expected return, henceforth called the filter. The filter

µ and the posterior variance γ are such that

µt ∼ N(µt, γt), (1)

where N(m, v) denotes the Gaussian distribution with mean m and variance v. Equation (1)

shows that the posterior variance, called uncertainty from now on, measures the accuracy

of the filter. Indeed, low uncertainty implies that the filter µ is likely to be close to the

true expected return µ. Conversely, the filter can potentially be relatively far from the true

expected return when uncertainty is large, making it likely to be inaccurate.

Proposition 1 below defines the vector of state variables observed by the investor.

7

Proposition 1. The dynamics of the observed state variables satisfy

dPtPt

=µtdt+

(√Vt 0 0

)dBt (2)

dVt =λV (V − Vt)dt+ σV√Vt

(ρP,V

√1− ρ2

P,V 0

)dBt (3)

dµt =λµ(µ− µt)dt+

(γt√Vt

+ ρP,µσµ√Vt(ρµ,V −ρP,V ρP,µ)σµ−γtρP,V√

Vt√

1−ρ2P,V

γtσst

)dBt (4)

dγtdt

=− γ2t

(1

σ2st

+1

Vt(1− ρ2

P,V

))− 2γt

(λµ +

(ρP,µ − ρP,V ρµ,V )σµ√Vt(1− ρ2

P,V

) )

+

(1− ρ2

µ,V + 2ρP,V ρP,µρµ,V − ρ2P,µ − ρ2

P,V

)σ2µ

1− ρ2P,V

, (5)

where B ≡(B1 B2 B3

)>is a 3-dimensional vector of independent and observable Brow-

nian motions.8

Proof. See Lipster and Shiryaev (2001).

The investor estimates expected returns by using three sources of information: returns,

stock-return variance, and the news signal. When performing his learning exercise, the in-

vestor assigns stochastic weights to the three sources of information (see Equation (4)). Each

weight depends on the relative quality of each information flow and therefore corresponds to

the signal-to-noise ratio of each source of information.

Uncertainty, which measures the inaccuracy of the filter, is locally deterministic (Xia

(2001)) and decreases rapidly when the signal volatility is sufficiently low (see first term in

Equation (5)). When the signal volatility is sufficiently high, however, the last component

of uncertainty outweighs the first two negative components and therefore uncertainty in-

8Note that B1 and B2 are defined in Equations (2) and (3), and B3 is such that dst = µtdt+ σstdB3t.

8

creases. It is worth noting that, because the stock-return variance and the signal volatility

are stochastic, uncertainty never converges to a constant steady-state. This feature is similar

to that uncovered by Xia (2001) and is in contrast to the constant steady-state uncertainty

obtained by Brennan and Xia (2001), Scheinkman and Xiong (2003), Dumas, Kurshev, and

Uppal (2009), and Xiong and Yan (2010).

2.1 Optimization Problem

The investor chooses consumption c, risky investment share q, and the signal volatility σs to

maximizes his expected lifetime utility of consumption. The investor’s problem is written

J(Wt, µt, V, γ) = maxc∈R,q∈R,σs∈R+

Et(∫ ∞

t

e−∆(u−t) c1−αu

1− αdu

)s.t. dWt =

[(r − K(σst)

)Wt + qtWt(µt − r)− ct

]dt+ qtWt

(√Vt 0 0

)dBt, (6)

where ∆ is the subjective discount, α the coefficient of relative risk aversion, and W the

wealth of the investor. The vector of state variables (µ, V, γ) is defined in Proposition 1. Since

the HJB equation (see Appendix A.1) associated to problem (6) involves the information

cost function K(σs) and terms of the order of 1σ2s

only, I choose to define attention to news,

x, and the associated information cost function, K(x), as follows:

x ≡ 1

σ2s

K(x) ≡ kx2 = K(σs). (7)

For tractability purposes, I follow Bansal and Shaliastovich (2011) and assume that the

total information cost is linear in wealth (see Equation (6)). This assumption conserves the

9

homogeneity of the value function and simplifies considerably the resolution of the problem.

In addition, the per-unit of wealth cost function K(x) is assumed to be quadratic9 (increasing

and convex) in attention, which implies that perfect information (x =∞) cannot be achieved.

That is, the investor is never able to observe the true level of the expected return. If the

investor chooses to pay no attention to news (x = 0 and K(x) = 0) and to learn using the

information provided by the price and the variance of stock returns only, then her entire

wealth is invested in the financial market. If, however, the investor decides to pay attention

to news (x > 0 and K(x) > 0), then a positive fraction of her wealth flows to the information

market.10

The investor’s optimal consumption, risky investment share, and attention are charac-

terized in Proposition 2 below.

Proposition 2. The optimal consumption c∗, risky investment share q∗, and attention x∗

satisfy

c∗ = F−1/αW

q∗ =µ− rβV

+

(γ +√V ρP,µσµ

)Fµ

βV F+ρP,V σV FV

βF(8)

x∗ =

x∗ =

γ2(2Fγ−Fµ,µ)4k(α−1)F

, if x∗ ∈ R>0

0 otherwise,

(9)

9Although in different settings, quadratic cost functions are considered by Garleanu and Pedersen (2013)and Collin-Dufresne, Daniel, Moallemi, and Saglam (2014) among others.

10Note that attention to news x can be interpreted as the precision of the news signal s. High attentionimplies that the news signal acquired by the investor is accurate, whereas low attention implies the opposite.

10

where the function F (µ, V, γ) is such that the value function satisfies

J (W, µ, V, γ) =W 1−α

1− αF (µ, V, γ) .

Note that Fy stands for the partial derivative of the function F with respect to the variable

y.

Proof. See Appendix A.1.

The optimal consumption is standard, as it corresponds to the term uncovered by Merton

(1971). The optimal risky share consists in a myopic term (Merton (1971)) and two hedging

terms that depend on the sensitivities of the value function to changes in the filter and the

stock-return variance (Liu (2007)). Since the investor is constrained to choose her optimal

attention on the non-negative real line, the optimal attention x∗ is nil as long as the term

x∗ /∈ R>0.

First, there is a direct11 increasing relation between attention and uncertainty (see Equa-

tion (9)). Intuitively, high uncertainty implies an inaccurate filter and therefore a willingness

to pay more attention to news. Second, there is a direct decreasing relation between atten-

tion and risk aversion. The reason is that high risk aversion implies a low risky investment

share (see Equation (8)). Therefore, the investor’s incentive to gather information on the

expected stock return is weaker under high risk aversion than under low risk aversion.

An approximate solution of the HJB equation is obtained by conjecturing an exponential-

quadratic form for the function F and by solving the associated linearized system of equations

numerically. All derivations and computational details are provided in Appendix A.1.

11Indirect relations apply through the function F and its partial derivatives.

11

3 Theoretical Results

This section presents and discusses the dependence of optimal attention and risky investment

share on expected stock returns, stock-return variance, and uncertainty. Attention is a hump-

shaped function of the expected stock return and an increasing function of both stock-return

variance and uncertainty. The risky investment share is increasing with the expected return

and decreasing with both variance and uncertainty. Therefore, when controlling for the

hump-shaped relation between attention and expected returns, attention is decreasing with

the risky investment share.

If not otherwise specified, parameters and values of the state variables used henceforth

are provided in Table 1. Variance parameters are borrowed from Christoffersen, Jacobs,

and Mimouni (2010), expected return parameters are inspired by Xia (2001), and utility

parameters are standard. The correlation between return and variance is negative (Christof-

fersen et al. (2010)) and reflects the leverage effect (Black (1976) and Christie (1982)). The

correlation between return and expected return is negative as growth stocks feature lower

expected returns than value stocks (Fama and French (1992)). The correlation between

expected return and variance is positive because both are counter-cyclical (Campbell and

Shiller (1988a,b), Schwert (1989), and Mele (2007)). Although the cost parameter k looks

particularly small, I mention further that it implies a non-negligible information cost.

3.1 Attention, Investments, and Economic Conditions

Attention is a hump-shaped function of the filter (Figure 1; Panel (a)). The reason is as

follows. When the filter is either sufficiently large or sufficiently low, the investor knows

12

Parameter Symbol ValueMean-Reversion Speed of Variance λV 6.5Long-Term Mean of Variance V 0.035Volatility of Variance σV 0.45Mean-Reversion Speed of Expected Return λµ 0.2Long-Term Mean of Expected Return µ 0.1Volatility of Expected Return σµ 0.05Correlation Return and Expected Return ρP,µ -0.5Correlation Return and Variance ρP,V -0.8Correlation Expected Return and Variance ρµ,V 0.5Risk Free Rate r 0.03Relative Risk Aversion β 3Subjective Discount Rate ∆ 0.01Cost Parameter k 10−8

Expected Return µ µVariance V VUncertainty γ 1

2γss|{x=0,V=V }

Table 1: Calibration

The “steady-state” uncertainty, γss|{x=0,V=V }, is defined in Equation (17).

that the probability of the expected return reverting back to its mean is relatively high.

Indeed, in these states of the world the drift of the expected return is large in absolute

terms, which provides the investor with no incentive to pay a cost and be attention to news.

In contrast, when the filter is close to its long-term mean, the noise component becomes its

main driver. Therefore, the investor has incentives to be attentive to news and to gather

accurate information in these states of the world. It is worth noting that, even though the

cost parameter k is small, the investor allocates up to 15 bps of her wealth per unit of time

to the information market.

The risky investment share is an increasing function of the filter (Figure 1; Panel (b)),

because the filter measures the quality of risky investment opportunities. Since the relation

between attention and the filter is hump-shaped and that between the risky share and the

filter is monotone increasing, attention is an endogenous hump-shaped function of the risky

13

−0.1 0 0.1 0.2 0.3

0

100

200

300

400

Filter µ

Att

enti

onx∗

(a) Attention vs. Filter

−0.1 0 0.1 0.2 0.3

−1

0

1

2

3

Filter µ

Ris

ky

Shar

eq∗

(b) Risky Share vs. Filter

Figure 1: Attention and Risky Investment Share against Expected Stock Return

share.

The non-monotone relation between attention and stock-return variance (Figure 2; Panel

(a)) applies because two opposite forces are at work. The first force pushes the investor to

pay more attention to news as variance increases, because a large stock-return variance

implies that information provided by the observation of returns is inaccurate. The second

force, however, creates a decreasing relation between attention and variance, because a high

stock-return variance implies a low risky investment share (Figure 2; Panel (b)) and therefore

no incentives to gather information on expected stock returns. These competing forces imply

that attention increases with the variance of stock returns as long as the latter is not too

large. Comparing Figures 1 and 2 shows that changes in expected returns impact attention

significantly more than changes in variance do.

As pointed out by Chacko and Viceira (2005) and Liu (2007), the risky investment share

is decreasing with the variance of stock returns (Figure 2; Panel (b)), because variance is a

14

0 0.05 0.1 0.15 0.2

380

385

390

Variance V

Att

enti

onx∗

(a) Attention vs. Variance

0 0.05 0.1 0.15 0.2

0.5

1

1.5

2

Variance V

Ris

ky

Shar

eq∗

(b) Risky Share vs. Variance

Figure 2: Attention and Risky Investment Share against Stock-Return Variance

measure of risk and the investor is risk averse.

The mechanism that implies the non-monotone relation between attention and uncer-

tainty is similar to that generating the relation between attention and the variance of stock

returns. On the one hand, attention tends to increase with uncertainty because uncertainty

measures the inaccuracy of the filter. On the other hand, attention tends to decrease with

uncertainty, because high uncertainty implies low risky investments (Figure 3; Panel (b)) and

therefore no willingness to acquire accurate information regarding expected stock returns.

Panel (a) of Figure 3 shows that the former effect dominates the latter as long as uncertainty

is not too large.

Intuitively, the risky investment share decreases monotonically with uncertainty (Figure

3; Panel (b)) because uncertainty is, similar to the variance of stock return, a measure of

the risk faced by the investor. Overall, Figures 1, 2, and 3 show that changes in attention

are mainly driven by changes in expected returns and uncertainty, while changes in the

15

0 0.001 0.002 0.003

0

200

400

600

Uncertainty γ

Att

enti

onx∗

(a) Attention vs. Uncertainty

0 0.001 0.002 0.0030.6

0.65

0.7

0.75

Uncertainty γ

Ris

ky

Shar

eq∗

(b) Risky Share vs. Uncertainty

Figure 3: Attention and Risky Investment Share against Uncertainty

risky investment share are mainly driven by changes in expected returns and stock-return

variance. Moreover, when controlling for the hump-shaped relation between attention and

expected returns, attention is generally increasing with both variance and uncertainty.12

Consequently, attention tends to decrease with the risky investment share when the impact

of expected returns is controlled for.

3.2 Comparative Statics: The Impact of Risk Aversion and the

Expensiveness of Information

Surprisingly, attention to news is a decreasing function of relative risk aversion (Figure 4; left

hand side). Although the aforementioned statement seems, a priori, counterintuitive, it is

easily rationalized by the fact that high risk aversion implies low risky investments (Figure

12The term “generally” is used to reflect the fact that attention is decreasing with variance and uncertaintyonly for exceptionally large values of these variables.

16

4; right hand side). Therefore, the incentive to gather accurate information on expected

stock returns is naturally larger for a weakly risk averse investor than for a highly risk averse

investor.

As information becomes more expensive (k increases), an accurate news signal can only

be acquired at a high cost, which decreases the investor’s expected utility. Therefore, the

investor’s incentive to pay attention to news dampens (Figure 5; left hand side). Once the

acquisition of news becomes too expensive, the investor chooses to pay no attention to news

and to learn using the information provided by the observation of returns and their variance

only. Even though the cost parameter k strongly affects the investor’s optimal attention to

news, it has no significant impact on the risky investment share (Figure 5; right hand side).

17

−0.1 0 0.1 0.2 0.3

0

100

200

300

400

Filter µ

Att

enti

onx∗

(a) Attention vs. Filter

α = 3α = 7α = 10

−0.1 0 0.1 0.2 0.3

−1

0

1

2

3

Filter µ

Ris

ky

Shar

eq∗

(b) Risky Share vs. Filter

0 0.05 0.1 0.15 0.2150

200

250

300

350

400

Variance V

Att

enti

onx∗

(c) Attention vs. Variance

0 0.05 0.1 0.15 0.2

0

0.5

1

1.5

2

Variance V

Ris

ky

Shar

eq∗

(d) Risky Share vs. Variance

0 0.001 0.002 0.003

0

200

400

600

Uncertainty γ

Att

enti

onx∗

(e) Attention vs. Uncertainty

0 0.001 0.002 0.003

0.2

0.4

0.6

0.8

Uncertainty γ

Ris

ky

Shar

eq∗

(f) Risky Share vs. Uncertainty

Figure 4: Attention and Risky Investment Share against Risk Aversion

18

−0.1 0 0.1 0.2 0.3

0

100

200

300

400

500

Filter µ

Att

enti

onx∗

(a) Attention vs. Filter

k =∞k = 10−7

k = 10−8

−0.1 0 0.1 0.2 0.3

−1

0

1

2

3

Filter µ

Ris

ky

Shar

eq∗

(b) Risky Share vs. Filter

0 0.05 0.1 0.15 0.2

0

100

200

300

400

Variance V

Att

enti

onx∗

(c) Attention vs. Variance

0 0.05 0.1 0.15 0.2

0.5

1

1.5

2

Variance V

Ris

ky

Shar

eq∗

(d) Risky Share vs. Variance

0 0.001 0.002 0.003

0

200

400

600

Uncertainty γ

Att

enti

onx∗

(e) Attention vs. Uncertainty

0 0.001 0.002 0.0030.6

0.65

0.7

0.75

Uncertainty γ

Ris

ky

Shar

eq∗

(e) Risky Share vs. Uncertainty

Figure 5: Attention and Risky Investment Share against the Expensive-ness of Information

The expensiveness of information is driven by the information cost parameter k.

19

Risk Aversion α Cost Parameter k Cost of Ignoring News3 10−8 0.08447 10−8 0.029910 10−8 0.02033 10−7 0.05567 10−7 0.017710 10−7 0.01083 10−6 0.01497 10−6 0.003610 10−6 0.0019

Table 2: The Cost of Ignoring the News Signal

The cost of ignoring news represents the additional fraction of wealth required by an investorwho ignores the news signal to reach the expected utility of an investor who optimally paysattention to news.

In order to quantify the benefits associated to the acquisition of costly information, I

compute the certainty equivalent measure proposed by Das and Uppal (2004). This measure

is defined as the additional fraction of wealth required by an investor who ignores the in-

formation signal13 to reach the expected utility of an investor who optimally pays attention

to news. Table 2 shows that the cost of ignoring the information signal is non-negligible,

especially when both risk aversion α and the cost parameter k are small. The reason is as

follows. The lower the risk aversion and the cost parameter, the larger the incentive to gather

costly information becomes (Figures 4 and 5), and therefore the larger the cost associated

to the lack of information acquisition is. It is worth noting that irrespective of the value

of the cost parameter k, a highly risk averse investor (e.g., α ≈ 20) enjoys only a marginal

benefit through information acquisition, simply because most of her wealth is invested in the

risk-free asset.

13An equivalent interpretation would be that this investor faces an information cost function with param-eter k =∞.

20

4 Empirical Tests

This section provides evidence that the main predictions of the model are sustained by

the data. First, investors attention is a hump-shaped function of expected returns and

an increasing function of both stock-return variance and uncertainty. Second, attention

increases with risk-free investments and therefore decreases with risky investments.

Our empirical analysis is based on the following dataset. The expected stock return is

the fitted value of a predictive regression that infers the future S&P500 return using the

current dividend yield and S&P500 return. The variance of stock returns is obtained by

fitting a GARCH(1,1) model (Bollerslev (1986)) to the demeaned S&P500 returns. The

1-month ahead macro-uncertainty index constructed by Jurado et al. (2013) is my proxy for

uncertainty. Motivated by Barber and Odean (2008) and Hou et al. (2009)14, the growth

rate of the S&P500 trading volume represents my empirical measure of investors’ attention.15

Finally, the growth rate of the “Institutional Money Fund” index is used to proxy risk-free

investments. The “Institutional Money Fund” index is obtained from the Federal Reserve

Bank of St. Louis. Monthly data from January 1974 to January 2012 are considered.

4.1 Attention and Economic Conditions

In order to test whether attention is a hump-shaped function of expected returns and an

increasing function of both stock-return variance and uncertainty, I perform an Ordinary

Least Squares (OLS) regression analysis. Since both the variance of stock returns and un-

14The empirical findings of Chordia and Swaminathan (2000), Lo and Wang (2000), and Gervais, Kaniel,and Mingelgrin (2001) suggest that trading volume is a reasonable proxy of investors’ attention.

15Note that changes in the S&P500 trading volume is more likely to measure the attention of institutionalinvestors than that of retail investors. As a robustness check, I consider a retail investors measure of attentionin Section 4.3 and show that the predictions of the model also hold in that particular case.

21

certainty are highly auto-correlated by construction, I consider changes in these variables.

We control for the auto-correlation in attention by including the 1-month lagged attention

in the set of predictors. The model-implied hump-shaped relation between attention and

expected returns is captured by including the expected return and its square in the set of

predictors.

The data confirm a clear hump-shaped relation between attention and expected returns

(Table 3). Indeed, the quadratic coefficient is negative, significant, and substantially larger

than the linear coefficient in absolute terms. In addition, the top of the hump is close to

the mean of expected returns,16 lending strong support to the predictions of the model. Ta-

ble 3 shows that uncertainty is another important driver of attention, whereas stock-return

variance is not. Although both uncertainty and variance are positive drivers of attention, sta-

tistical significance holds for the former relation only. It is worth noting that both the model

and the data highlight a strong relation between attention and expected returns/uncertainty,

and a weaker relation between attention and variance.

4.2 Attention and Investments

The model predicts that, once the hump-shaped relation between attention and expected

returns is controlled for, attention tends to increase with the risky investment share. This

prediction is again tested by performing OLS regressions. As mentioned earlier, I use the

growth rate of the “Institutional Money Fund” index provided by the Federal Reserve Bank

of St. Louis to proxy the inverse of the risky investment share.

16Over the sample, the mean of the monthly expected return is 1%, while the top of the hump is reachedat ER ≈ 0.6%.

22

Intercept 0.0171**(2.5583)

ERt 1.9622***(2.9828)

ER2t -95.0355**

(-2.4182)∆Variancet 20.9530

(0.9469)∆Uncertaintyt 0.9307**

(2.0050)Attentiont−1 -0.2424***

(-4.5259)Adj. R2 0.0812

N 455

Table 3: Attention (Trading Volume) against Expected Stock Returns, Stock-Return Variance, and Uncertainty

Monthly data from January 1974 to January 2012 are considered. Standard errors areadjusted using Newey and West (1987)’s procedure with six lags. t-statistics are reported inbrackets and statistical significance at the 10%, 5%, and 1% levels is labeled with */**/***,respectively. The last two rows present the adjusted R2 and the number of observations,respectively.

Column (1) of Table 4 controls for the impact of the expected return and the auto-

correlation of attention, while Column (2) of Table 4 also controls for the impact of the

stock-return variance and uncertainty. Both regressions show that attention and risk-free

investments are strongly interconnected. Indeed, the risk-free investment (RFI) coefficients

are positive and significant at the 99% confidence level, confirming the prediction that insti-

tutional investors are likely to pay high attention when risk-free investments are large, and

therefore when risky investments are low. This prediction is also in line with Vlastakis and

Markellos (2012) and Andrei and Hasler (2014), who show that investors’ attention is most

likely counter-cyclical.

23

(1) (2)Intercept 0.0114* 0.0121*

(1.8730) (1.7871)RFIt 0.2203*** 0.1924***

(3.0719) (2.8616)ERt 1.2769** 1.8966***

(2.1226) (2.9205)ER2

t -68.9174** -88.1649**(-2.5221) (-2.2303)

∆Variancet 12.7083(0.5578)

∆Uncertaintyt 0.8936*(1.8434)

Attentiont−1 -0.2186*** -0.2397***(-4.2740) (-4.4561)

Adj. R2 0.0771 0.0879N 456 455

Table 4: Attention (Trading Volume) against Risk-Free Investments

Monthly data from January 1974 to January 2012 are considered. Standard errors areadjusted using Newey and West (1987)’s procedure with six lags. t-statistics are reported inbrackets and statistical significance at the 10%, 5%, and 1% levels is labeled with */**/***,respectively. The last two rows present the adjusted R2 and the number of observations,respectively.

4.3 Robustness

In order to check that my empirical tests are robust to several measures of attention, I follow

the idea of Da et al. (2011) and construct a measure of investors attention using Google

search volumes17 on the queries “S&P500”, “Dow Jones Industrial Average”, “NYSE”, and

“NASDAQ”.18 The search volume associated to each query is recorded at weekly frequency

from the first week of January 2004 to the last week of January 2012.19 We demean these

time-series by their respective sample average for scaling purposes and sum them to obtain an

17As mentioned by Da et al. (2011), this measure of attention is likely to capture retail investors’ attention.Google search volume data are obtained from the Google Trends website: www.google.com/trends.

18Adding similar queries to the analysis does not alter the qualitative nature of the results presentedthereafter.

19Google Search Volume data are recorded until the end of January 2012 only, because the uncertaintyindex provided by Jurado et al. (2013) ends at that particular time.

24

Intercept 1.3491(1.3972)

ERt 389.5587**(2.2278)

ER2t -13476.9579*

(-1.7836)∆Variancet 198.6561

(1.0678)∆Uncertaintyt 218.0345*

(1.7565)Attentiont−1 1.0620***

(3.7089)Adj. R2 0.4025

N 95

Table 5: Attention (Google Search) against Expected Stock Returns, Sock-Return Variance, and Uncertainty

Monthly data from January 2004 to January 2012 are considered. Standard errors areadjusted using Newey and West (1987)’s procedure with six lags. t-statistics are reported inbrackets and statistical significance at the 10%, 5%, and 1% levels is labeled with */**/***,respectively. The last two rows present the adjusted R2 and the number of observations,respectively.

attention index at weekly frequency. Then, I compute averages over each month to obtain

a proxy for investors’ attention at monthly frequency. The measures of expected return,

variance, uncertainty, and risk-free investments are the same as in the previous section.

Although statistical significance is weaker when considering the Google-based measure of

attention than when considering the trading-based measured of attention, attention is again

a significant hump-shaped function of expected returns, a significant increasing function of

uncertainty, and an insignificant increasing function of stock-return variance (Table 5). This

shows that the relation between attention, expected returns, variance, and uncertainty is

robust to changes in attention measures.

Table 6 further confirms that my empirical results depend only weakly on the attention

proxy considered. Indeed, attention is, as in the previous section, an increasing function of

25

(1) (2)Intercept 0.6450 0.9664

(0.7462) (1.0256)RFIt 116.0485** 71.5277***

(2.1501) (3.0378)ERt 150.3129*** 396.2164**

(5.0907) (2.2792)ER2

t -5463.0919*** -12258.5521*(-1.7200)

∆Variancet 198.0174(1.0954)

∆Uncertaintyt 199.6350(1.6080)

Attentiont−1 0.7161*** 1.0026***(6.3092) (3.6304)

Adj. R2 0.3107 0.4118N 96 95

Table 6: Attention (Google Search) against Risk-Free Investments

Monthly data from January 2004 to January 2012 are considered. Standard errors areadjusted using Newey and West (1987)’s procedure with six lags. t-statistics are reported inbrackets and statistical significance at the 10%, 5%, and 1% levels is labeled with */**/***,respectively. The last two rows present the adjusted R2 and the number of observations,respectively.

risk-free investments. Therefore, consistent with the prediction of the model, investors tend

to pay high attention when risky investments are low.

5 Conclusion

This paper aims to understand the link between attention to news, asset allocation, and

economic conditions. When acquiring information is costly, investors’ attention to news is a

hump-shaped function of expected returns and an increasing function of both stock-return

variance and uncertainty. Therefore, attention to news and risky investments tend to be

inversely related, especially if the hump-shaped relation between attention and expected re-

turns is controlled for. In addition, I show that attention decreases in both risk aversion

26

and the expensiveness of information. Our empirical analysis confirms that attention is a

hump-shaped function of expected returns and an increasing function of uncertainty. The

predicted positive relation between attention and stock-return variance, however, is insignif-

icant. Finally, investors pay high attention to news when investing substantially in risk-free

assets, lending support to the prediction of the model.

27

A Appendix

A.1 HJB Equation

The HJB equation associated to problem (6) is written

maxc∈R+,q∈R,σs∈R+

{−∆J +

c1−α

1− α+ LW,µ,V,γJ

}= 0, (10)

where LW,µ,V,γ is the infinitesimal generator of the vector (W, µ, V, γ). Equations (5) and

(6) show that the signal volatility drives the drifts of uncertainty and wealth, and the third

diffusion component of the filter. Since the third diffusion component is zero for all state

variables but the filter, the signal volatility enters Equation (10) through the cost function

K(σs) and terms of the order of 1σ2s

only.

The change of variable and the cost function defined in Equation (7) imply the following

HJB equation

maxc∈R+,q∈R,x∈R+

{−∆J +

c1−α

1− α+ LW,µ,V,γJ

}= 0. (11)

Solving the first order conditions yields

c∗ = J−1/αW (12)

q∗ =(r − µ)JW − V ρP,V σV JW,V −

(γ +√V ρP,µσµ

)JW,µ

VWJW,W(13)

x∗ =

x∗ =

γ2(−2Jγ+Jµ,µ)4kWJW

, if x∗ ∈ R>0

0 otherwise,

(14)

28

where Jy stands for the partial derivative of the function J with respect to the variable y.

The candidate solution to Equation (11) is written

J (W, µ, V, γ) =W 1−α

1− αF (µ, V, γ) . (15)

Substituting Equation (15) in Equations (12), (13), and (14) yields the expressions provided

in Proposition 2.

Substituting the optimality conditions (12), (13), and (14) and the candidate solution

(15) in the HJB equation (11) yields a 3-dimensional non-linear PDE.

The PDE is solved by approximating the candidate solution as follows

J (W, µ, V, γ) =W 1−α

1− αF (µ, V, γ)

≈ W 1−α

1− αeA0+A1µ+A2µ2+A3V+A4V 2+A5γ+A6γ2+A7µV+A8µγ+A9V γ. (16)

The solution (16) is substituted in the PDE. Then, the resulting expression is approximated

by a second order Taylor expansion around µ = µ, V = V , and γ = 12γss|{x=0,V=V }, where

γss|{x=0,V=V } stands for the “steady-state” uncertainty conditional on x = 0 and V = V .

The “steady-state” uncertainty is obtained by setting x = 0, V = V , and dγtdt

= 0 in Equation

(5). The solution to the aforementioned equation is

γss|{x=0,V=V } =V λµ(ρ2P,V − 1) +

√V (−ρPµ + ρP,V ρµ,V )σµ

+

√V (ρ2

P,V − 1)(V λ2

µ(ρ2P,V − 1)− 2

√V λµ(ρPµ − ρP,V ρµ,V )σµ + (ρ2

µ,V − 1)σ2µ

).

(17)

29

The resulting Taylor expansion yields a system of 10 equations and 10 unknowns (the Ais)

that is solved numerically.

�

30

References

Ait-Sahalia, Y. and M. W. Brandt (2001). Variable selection for portfolio choice. The Journal

of Finance 56 (4), 1297–1351.

Andrei, D. and M. Hasler (2014). Investors attention and stock market volatility. Working

Paper .

Bansal, R. and I. Shaliastovich (2011). Learning and asset-price jumps. Review of Financial

Studies 24 (8), 2738–2780.

Barber, B. M. and T. Odean (2008). All that glitters: The effect of attention and news

on the buying behavior of individual and institutional investors. Review of Financial

Studies 21 (2), 785–818.

Barberis, N. (2000, 02). Investing for the long run when returns are predictable. Journal of

Finance 55 (1), 225–264.

Black, F. (1976). Studies of stock price volatility changes. American Statistical Association,

Proceedings of the 1976 Meetings of the American Statistical Association, Business and

Economic Statistics Section, 177–181.

Bollerslev, T. (1986, April). Generalized autoregressive conditional heteroskedasticity. Jour-

nal of Econometrics 31 (3), 307–327.

Brandt, M. W. (1999, October). Estimating portfolio and consumption choice: A conditional

euler equations approach. Journal of Finance 54 (5), 1609–1645.

31

Brandt, M. W., A. Goyal, P. Santa-Clara, and J. R. Stroud (2005). A simulation approach

to dynamic portfolio choice with an application to learning about return predictability.

Review of Financial Studies 18 (3), 831–873.

Brennan, M. J. (1998). The role of learning in dynamic portfolio decisions. European Finance

Review 1 (3), 295–306.

Brennan, M. J. and Y. Xia (2001). Stock price volatility and equity premium. Journal of

Monetary Economics 47 (2), 249 – 283.

Campbell, J. and R. Shiller (1988a). The dividend-price ratio and expectations of future

dividends and discount factors. Review of Financial Studies 1 (3), 195–228.

Campbell, J. Y., G. Chacko, J. Rodriguez, and L. M. Viceira (2004). Strategic asset allocation

in a continuous-time VAR model. Journal of Economic Dynamics and Control 28 (11),

2195 – 2214.

Campbell, J. Y. and R. J. Shiller (1988b). Stock prices, earnings, and expected dividends.

The Journal of Finance 43 (3), 661–676.

Campbell, J. Y. and L. M. Viceira (1999, May). Consumption and portfolio decisions when

expected returns are time varying. The Quarterly Journal of Economics 114 (2), 433–495.

Chacko, G. and L. M. Viceira (2005). Dynamic consumption and portfolio choice with

stochastic volatility in incomplete markets. Review of Financial Studies 18 (4), 1369–1402.

Chordia, T. and B. Swaminathan (2000). Trading volume and cross-autocorrelations in stock

returns. The Journal of Finance 55 (2), 913–935.

32

Christie, A. A. (1982, December). The stochastic behavior of common stock variances :

Value, leverage and interest rate effects. Journal of Financial Economics 10 (4), 407–432.

Christoffersen, P., K. Jacobs, and K. Mimouni (2010). Volatility dynamics for the s&p500:

Evidence from realized volatility, daily returns, and option prices. Review of Financial

Studies 23 (8), 3141–3189.

Collin-Dufresne, P., K. Daniel, C. Moallemi, and M. Saglam (2014). Strategic asset allocation

with predictable returns and transaction costs. Working Paper .

Da, Z., J. Engelberg, and P. Gao (2011, October). In search of attention. Journal of

Finance 66 (5), 1461–1499.

Das, S. R. and R. Uppal (2004). Systemic risk and international portfolio choice. The Journal

of Finance 59 (6), 2809–2834.

David, A. (1997, December). Fluctuating confidence in stock markets: Implications for

returns and volatility. Journal of Financial and Quantitative Analysis 32 (04), 427–462.

Detemple, J. and R. Kihlstrom (1987). Acquisition dinformation dans un modle intertem-

porel en temps continu. L’Actualit conomique 63, 118–137.

Detemple, J. B. (1986). Asset pricing in a production economy with incomplete information.

The Journal of Finance 41 (2), pp. 383–391.

Dothan, M. U. and D. Feldman (1986, June). Equilibrium interest rates and multiperiod

bonds in a partially observable economy. Journal of Finance 41 (2), 369–82.

33

Dumas, B., A. Kurshev, and R. Uppal (2009). Equilibrium portfolio strategies in the presence

of sentiment risk and excess volatility. The Journal of Finance 64 (2), 579–629.

Fama, E. F. and K. R. French (1992). The cross-section of expected stock returns. The

Journal of Finance 47 (2), pp. 427–465.

Feldman, D. (1989). The term structure of interest rates in a partially observable economy.

The Journal of Finance 44 (3), pp. 789–812.

Garleanu, N. and L. H. Pedersen (2013). Dynamic trading with predictable returns and

transaction costs. The Journal of Finance 68 (6), 2309–2340.

Gennotte, G. (1986). Optimal portfolio choice under incomplete information. The Journal

of Finance 41 (3), pp. 733–746.

Gervais, S., R. Kaniel, and D. H. Mingelgrin (2001). The high-volume return premium. The

Journal of Finance 56 (3), 877–919.

Gron, A., B. N. Jorgensen, and N. G. Polson (2012). Optimal portfolio choice and stochastic

volatility. Applied Stochastic Models in Business and Industry 28 (1), 1–15.

Heston, S. (1993). A closed-form solution for options with stochastic volatility with appli-

cations to bond and currency options. Review of Financial Studies 6 (2), 327–343.

Hou, K., L. Peng, and W. Xiong (2009). A tale of two anomalies: The implications of

investor attention for price and earnings momentum. Working Paper .

Huang, L. and H. Liu (2007, 08). Rational inattention and portfolio selection. Journal of

Finance 62 (4), 1999–2040.

34

Jurado, K., S. C. Ludvigson, and S. Ng (2013, September). Measuring uncertainty. Working

Paper 19456, National Bureau of Economic Research.

Kacperczyk, M., S. V. Nieuwerburgh, and L. Veldkamp (2009, October). Rational attention

allocation over the business cycle. NBER Working Papers 15450, National Bureau of

Economic Research, Inc.

Kandel, S. and R. F. Stambaugh (1996). On the predictability of stock returns: An asset-

allocation perspective. The Journal of Finance 51 (2), pp. 385–424.

Kim, T. S. and E. Omberg (1996). Dynamic nonmyopic portfolio behavior. Review of

Financial Studies 9 (1), 141–61.

Lipster, R. S. and A. N. Shiryaev (2001). Statistics of Random Processes II. Springer Verlag,

New York.

Liu, J. (2007). Portfolio selection in stochastic environments. Review of Financial Stud-

ies 20 (1), 1–39.

Lo, A. and J. Wang (2000). Trading volume: definitions, data analysis, and implications of

portfolio theory. Review of Financial Studies 13 (2), 257–300.

Lustig, H. and A. Verdelhan (2012). Business cycle variation in the risk-return trade-off.

Journal of Monetary Economics 59 (S), S35–S49.

Mele, A. (2007, November). Asymmetric stock market volatility and the cyclical behavior

of expected returns. Journal of Financial Economics 86 (2), 446–478.

35

Merton, R. C. (1971, December). Optimum consumption and portfolio rules in a continuous-

time model. Journal of Economic Theory 3 (4), 373–413.

Mondria, J. (2010). Portfolio choice, attention allocation, and price comovement. Journal

of Economic Theory 145 (5), 1837 – 1864.

Mondria, J. and C. Quintana-Domeque (2013). Financial contagion and attention allocation.

The Economic Journal 123 (568), 429–454.

Newey, W. K. and K. D. West (1987, May). A simple, positive semi-definite, heteroskedas-

ticity and autocorrelation consistent covariance matrix. Econometrica 55 (3), 703–08.

Ornstein, L. S. and G. E. Uhlenbeck (1930, Sep). On the theory of the brownian motion.

Phys. Rev. 36, 823–841.

Peng, L. and W. Xiong (2006). Investor attention, overconfidence and category learning.

Journal of Financial Economics 80 (3), 563 – 602.

Scheinkman, J. A. and W. Xiong (2003). Overconfidence and speculative bubbles. The

Journal of Political Economy 111 (6), 1183–1219.

Schwert, G. W. (1989, December). Why does stock market volatility change over time?

Journal of Finance 44 (5), 1115–53.

Sims, C. A. (2003, April). Implications of rational inattention. Journal of Monetary Eco-

nomics 50 (3), 665–690.

Veldkamp, L. (2006a). Media frenzies in markets for financial information. American Eco-

nomic Review 96(3), 577–601.

36

Veldkamp, L. L. (2006b, 07). Information markets and the comovement of asset prices.

Review of Economic Studies 73 (3), 823–845.

Veronesi, P. (1999). Stock market overreaction to bad news in good times: A rational

expectations equilibrium model. Review of Financial Studies 12 (5), 975–1007.

Veronesi, P. (2000, April). How does information quality affect stock returns? Journal of

Finance 55 (2), 807–837.

Vlastakis, N. and R. N. Markellos (2012). Information demand and stock market volatility.

Journal of Banking and Finance 36, 1808–1821.

Wachter, J. A. (2002). Portfolio and consumption decisions under mean-reverting returns:

An exact solution for complete markets. The Journal of Financial and Quantitative Anal-

ysis 37 (1), pp. 63–91.

Xia, Y. (2001). Learning about predictability: The effects of parameter uncertainty on

dynamic asset allocation. The Journal of Finance 56 (1), pp. 205–246.

Xiong, W. and H. Yan (2010). Heterogeneous expectations and bond markets. Review of

Financial Studies 23 (4), 1433–1466.

37