DP2020-33

Time-Varying Risk Attitude andBehavioral Asset Pricing
Shoka HAYAKI

December 15, 2020

Time-Varying Risk Attitude and Behavioral Asset Pricing
Shoka Hayaki1
Abstract
We propose a new behavioral asset pricing model that can flexibly express the time-variability or statedependence of aggregate risk attitude. We suppose that individual preferences are heterogeneous, and eachpreference changes depending on the state of the economy. While most previous studies, including Chanand Kogan (2002) which is the basis of our model, imply that aggregate risk aversion is counter-cyclicalby assuming either heterogeneity or changeability of preferences, considering both of these can result inaggregate risk aversion being procyclical in a particular economic situation. The status quo hypothesis we
propose suggests that the aggregate risk attitude is procyclical during recessions and counter-cyclical during
booms and depressions. This stems from the fact that in our setting only risk-tolerant individuals face losses
large enough to be loss-averse in order to maintain the status quo. We analyze the weekly returns of 10
major stock markets, including Canada, China, Eurozone, France, Germany, Hong Kong, India, Japan, theUK and the US, and provide some evidences to support our status quo hypothesis in Western stock markets,
where stock markets are relatively mature. We conclude that it is essential at least in such matured stock
markets to allow both heterogeneity and changeability of preferences in order to obtain an accurate
aggregate risk attitude.
Keywords: risk attitude, heterogeneous preferences, status quo, loss-aversion, prospect theory
JEL classification: G12, G15, G40

1

Junior Research Fellow, Research Institute for Economics and Business Administration, Kobe
University / Graduate School of Business Administration, Kobe University /shoka.hayaki0817@gmail.com

11. Introduction
An aggregate risk attitude is time-varying. A well-known study by Campbell and Cochrane (1999)
showed that the relative risk aversion of a representative individual fluctuates with changes in wealth.
There are three major factors that cause the aggregate risk attitude to fluctuate over time. First, thepreference parameters of individuals in the economy change such as Guiso et al. (2018) and SchildbergHörisch (2018). These studies focus on changes in individual risk attitudes rather than changes in an
aggregate risk attitude. Second, the risk aversion and other preference parameters of a representative
individual change such as Campbell and Cochrane (1999) and Barberis et al. (2001).2 This means that thepreferences of individuals in the economy are homogeneous. Third, the preference parameters ofindividuals in the economy does not change, but those of aggregate individual change such as Wang (1996),
Chan and Kogan (2002), and Verdelhan (2010). The individual preferences are heterogeneous. Althoughthese three factors rely on essentially different models and theories, they draw common conclusions. Thatis, the aggregate risk attitude is counter-cyclical.
The counter-cyclicality of aggregate risk attitude is extremely intuitive. Put briefly, people becomerisk averse when the economy goes bad. This corresponds to the claim of the first group. If this change is
generally homogeneous for all individuals, it corresponds to the claim of the second group. On the other
hand, Chan and Kogan (2002), in the third group, argue that the economic downturn increases the relative
wealth of relatively risk-averse individuals, resulting in an increase in wealth-weighted aggregate risk
aversion. This is because relatively risk-averse individuals are less willing to hold risk assets than relatively
risk-tolerant individuals and therefore do not significantly impair wealth in the event of an economic
downturn.
Few models of heterogeneous individuals that change their preferences have been considered. The
reason is probably that the consequences of these explanations are the same. If the economic downturn
makes individuals risk-averse, and the wealth of relatively risk-averse individuals increases, aggregate risk
aversion should be even more risk-averse. It remains important to take into account both changes inindividual preferences and their heterogeneity, nevertheless. This is because there are three possible
hypotheses in which the aggregate risk attitude becomes more risk-tolerant as the economy deteriorates.
The first hypothesis is that in the event of an economic downturn, the preference of a sufficient numberof individuals to affect the economy as a whole becomes more risk-tolerant. Some studies have shown that
individuals who experience catastrophic shocks such as the Great East Japan Earthquake and the Coronadisaster (Hanaoka et al., 2018, and Tsutsui and Tsutsui-Kimura, 2020, respectively) are rather risk tolerant.
The reason for this phenomenon may be interpreted as “habituation” to repeated stress according to Tsutsui
and Tsutsui-Kimura. Despite these extremely interesting results, most studies, including Guiso et al. (2018)
and Schildberg-Hörisch (2018), suggest that catastrophic shocks such as the financial crisis make people
more risk-averse. At least in the analysis of financial markets, this hypothesis seems to be somewhat
2Studies of similar settings are Menzly et al. (2004) and Santos and Veronesi (2010).

2

inadequate. In any case, when this hypothesis is correct at the aggregate level, the worse the economy, thelower the aggregate risk attitude.
The second hypothesis is that when the economy deteriorates, individuals become risk-loving.
Prospect theory provided by Kahneman and Tversky (1979) and Tversky and Kahneman (1992) suggeststhat individuals facing loss become risk-loving and loss-averse. When considering a CRRA (constantrelative risk aversion) type utility function, if the relative risk aversion is negative, the utility function meansrisk lover. Their weighted aggregate risk attitude may decrease in a recession where many individuals facelosses. A general equilibrium involving individuals with convex utility functions, however, generally does
not hold. Araujo et al. (2017) and Araujo et al. (2018) have shown the existence of such a generalequilibrium, but some conditions are required; for example, the condition that there must be enough riskaverse individuals compared to risk-loving individuals.
The third is the status quo hypothesis, and we pay the most attention to this hypothesis. This meansthat when the economy goes bad, individuals become more loss-averse. As in the second hypothesis, arecession means that many individuals face losses. People should be more loss-averse according to prospect
theory. Loss avoidance is expressed by the slope of the value function (utility function) when facing loss
becomes steeper than when facing gain. Interpreted in the framework of the expected utility theory, thismeans that the marginal utility of consumption increases, or the marginal negative utility of reduction in
consumption increases. This implies that individuals who are forced to reduce their consumption in the faceof loss cannot reduce their consumption as the original marginal utility suggests in order to maintain theirstatus quo. In other words, it can be said that it expresses the status quo propensity.3 This may represent
the cost stickiness in households and businesses.
This status quo hypothesis can be used to link the economic downturn with a reduction in aggregate
risk aversion, while using the general equilibrium theory without breaking it. Before explaining thismechanism, we explain the Chan and Kogan (2002) model with heterogeneous individuals who do notchange their preference parameters. During a recession, optimal consumption generally decreases
regardless of individual risk attitudes. Relatively risk-tolerant individuals hold more risk assets thanrelatively risk-averse individuals. They are likely to suffer significant losses first when the economy entersa recession. Consequently, relatively risk-tolerant individuals have a greater reduction in optimal
consumption than relatively risk-averse individuals, and their aggregate consumption share (the ratio of
their consumption to aggregate consumption) is lower. The influence of relatively risk-tolerant individuals
is reduced, and the aggregate risk attitude become larger, or more risk-averse.
On the other hand, when relatively risk-tolerant individuals facing losses switch to loss-averse
preferences, their optimal consumption does not decrease as much as the traditional model. In a recession,
3

The expression status quo bias may be more common. Our status quo hypothesis represents a situationin which consumption cannot be reduced even if it is desired. This is not just due to psychological bias.
Structural problems that lead to cost stickiness in households and businesses are also relevant. Therefore,
it may be inappropriate to call it bias. We should collectively call these the status quo propensity.

3

relatively risk-averse individuals also reduce consumption. The aggregate consumption share of relatively
risk-tolerant individuals may increase compared to that of relatively risk-averse individuals. The influenceof relatively risk-tolerant individuals becomes rather large, and the deterioration of the economy leads to adecrease in the aggregate risk aversion.
The status quo hypothesis is thought to show interesting properties during further economic downturns
such as the global financial crisis. In a minor recession, only relatively risk-tolerant individuals with large
risk assets face significant losses. That is, individual loss-aversions are heterogeneous. As the economic
downturn becomes more severe, relatively risk-averse individuals, however, also face significant losses. In
other words, individual loss-aversions become homogeneous. The aggregate consumption share of
relatively risk-averse individuals will rise, and the aggregate risk aversion will rise again.
From the above viewpoints, the aggregate risk aversion is expected to be procyclical in recessions,counter-cyclical in depressions, and counter-cyclical in booms. In order to capture more flexible movements,
the aggregate risk aversion expressed as a function of the state variable of the economy is approximated bya polynomial. The GARCH-M (generalized autoregressive conditional heteroskedasticity in mean) model
provided by Engle, Lilien and Robins (1987) is used to analyze the aggregate risk aversion. Since theconventional GARCH-M model does not allow time-variability of aggregate risk aversion, the constantaggregate risk aversion is replaced by the time-varying aggregate risk aversion as a function of the statevariable.
We use the weighted average of historical returns in the stock market as the proxy for state variable.
Chan and Kogan (2002) used “relative log-consumption” as a variable to capture the state of the economy.
This is defined as the logarithm of the ratio of current consumption to the consumption benchmark, whichis the weighted sum of past consumptions. This can be rewritten as a weighted average of past consumption
or endowment growth rates. Data representing consumption or endowment, such as household consumption,
CPI, or GDP, are less frequent than stock return, however. Furthermore, these data are lagging indicesrelated to the business cycle, and it must be said that consumption as data is a little slow to show the businesssentiment that investors feel. In fact, the stock market index will be the most important factor for investors
to judge the future growth potential of consumption, productivity and endowment in the economy.
Therefore, in this paper, as a variable to capture the state of the economy, we use a variable in which theaggregate consumption or endowment growth rate in the “relative log-consumption” is replaced by thestock market return.
We analyze weekly returns from July 2005 to June 2020 in the 10 major stock markets, including
Canada, China, Eurozone, France, Germany, Hong Kong, India, Japan, the UK and the US. The timevarying (or state-dependent) model is supported in all Western markets, but not in Asian markets. As
expected, in the Western markets, the aggregate risk attitude tends to decrease during the economic
downturn. It is suggested that loss aversion, or status quo hypothesis, regarding consumption plays animportant role during the recession. This does not necessarily deny the possibility that people will become4relatively risk-averse during the recession. At the market level as a whole, the effect of the shift to more
loss averse preference is greater than that. In addition, the aggregate risk attitude tends to rise again during
extreme economic downturns such as the 2008 global financial crisis and the 2020 COVID-19 crisis. This
result strongly supports the status quo hypothesis because the other two hypotheses cannot explain thisphenomenon. In the boom period, as shown by the conventional models, a counter-cyclical relationship isconfirmed between the state of the economy and the aggregate risk aversion.
We have shown that loss aversion, or status quo hypothesis, affects the market as a whole byconsidering both heterogeneity and changeability of individual preferences. Moreover, our empirical model
uses only market returns and interest rates and can minimize the problems of variable selection arbitrarinessand overfitting. Loss-aversion or status quo propensity is universal properties for humankind, and it isthought that these properties will be maintained in the mature stock market.
Section 2 introduces a Chan and Kogan (2002) model with heterogeneous individuals who do notchange their preferences and provide some useful interpretations for the following sections. In Section 3,
the Chan and Kogan model is extended so that the preference changes depending on the state of the economy.
Section 4 provides some evidences in support of our loss avoidance hypothesis and Section 5 concludes.
2. Heterogeneous individuals and time-varying risk attitude
This section explains the time-variability of the (aggregate) risk attitude according to Chan and Kogan(2002). Their model is a continuous-time consumption-based asset pricing model with catching up with theJoneses preferences, or external habit formation. In their setting, while all individuals have constant relativerisk aversions (hereafter, CRRA(s)), each CRRA coefficient is allowed to be different. Such heterogeneousrisk attitude model implies that the aggregate risk aversion in such an economy is counter-cyclical. Next,
we discuss the characteristics of the relative log consumption, or state variable, which plays an importantrole in their model.
2.1. Heterogeneous constant risk aversions
Chan and Kogan (2002) constructed a new continuous-time consumption-based asset pricing modelthat relaxes the assumptions of representative individuals in the conventional model. When an individual
has the same CRRA-type utility function, the representative individual also has a CRRA-type utility
function. This is a typical setting. This is not always the case if individuals can have different CRRA-typeutility function. Wang (1996) is considering an exchange economy in which there are two individuals withdifferent risk aversions. In their model, as wealth increases, a single type of individual dominates the market.
Chan and Kogan used catching up with Joneses preference (Abel, 1990; Abel, 1999) to prevent theasymptotic cross-sectional distribution of wealth from degenerating. In the Chan and Kogan model, an5individual with a constant relative risk aversion 𝛾 𝑖𝑛𝑑 ∈ (1, ∞) has the following utility function.4𝑢(𝐶𝑡 , 𝑋𝑡 ; 𝛾

𝑖𝑛𝑑1
𝐶𝑡 1−𝛾)=()
1 − 𝛾 𝑖𝑛𝑑 𝑋𝑡

𝑖𝑛𝑑

,

(1)

where 𝐶𝑡 is an individual consumption and 𝑋𝑡 is an external benchmark which represents the standardof living in the economy. This reflects that the marginal utility of consumption is increasing in the
consumption level of other individuals. In other words, individuals consume more during boom times when
the standard of living rises. The dynamics of 𝑥𝑡 = ln 𝑋𝑡 are defined as follows using the past values oflogarithmic aggregate endowment 𝑦𝑡 = ln 𝑌𝑡 :𝑡

𝑥𝑡 = 𝑥0 𝑒−𝜅𝑡 + 𝜅 ∫ 𝑒−𝜅(𝑡−𝑠) 𝑦𝑠 d𝑠 ,𝜅 > 0.

(2)0

The parameter 𝜅 governs the degree of history dependence in 𝑋𝑡 .
Considering the budget constraint up to the aggregate endowment 𝑌𝑡 , the consumption of anindividual with a risk aversion 𝛾 𝑖𝑛𝑑 is a function of 𝑌𝑡 and 𝑋𝑡 , that is, 𝐶𝑡 = 𝐶𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾 𝑖𝑛𝑑 ). The∞

budget constraint at time 𝑡 is expressed as ∫1 𝐶𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 ≤ 𝑌𝑡 . Let 𝑓(𝛾 𝑖𝑛𝑑 ) be the social∞

weight attached by the central planner, or the distribution of individuals, that is ∫1 𝑓(𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 = 1. In
an exchange economy, no intertemporal transfer of wealth is possible at the aggregate level, and the
optimization problem can be separated at each time 𝑡 ∈ [0, ∞). The utility function of representative (oraggregate) individual at time 𝑡 is∞𝑈(𝑌𝑡 , 𝑋𝑡 ) =

sup

∫ 𝑓(𝛾

𝑖𝑛𝑑

{𝑐𝑡 (𝑌𝑡 ,𝑋𝑡 ;𝛾 𝑖𝑛𝑑 );𝛾 𝑖𝑛𝑑 >1} 1
∞

s. t.

1
𝐶 (𝑌 , 𝑋 ; 𝛾 𝑖𝑛𝑑 ))( 𝑡 𝑡 𝑡
)
𝑖𝑛𝑑
1−𝛾
𝑋𝑡

∫ 𝐶𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾

𝑖𝑛𝑑

)d𝛾

𝑖𝑛𝑑

1−𝛾 𝑖𝑛𝑑d𝛾 𝑖𝑛𝑑(3)≤ 𝑌𝑡 .

1

We summarize the main implications of their model.5 First, the aggregate relative risk aversion isexpressed as follows.
𝛾(𝑌𝑡 , 𝑋𝑡 ) ≡−𝑌𝑡 𝑈𝑌𝑌 (𝑌𝑡 , 𝑋𝑡 )= −ℎ′ (𝑧𝑡 ) > 1,
𝑈𝑌 (𝑌𝑡 , 𝑋𝑡 )(4)

where 𝑧𝑡 ≡ 𝑦𝑡 − 𝑥𝑡 = ln(𝑌𝑡 /𝑋𝑡 ) is relative log-consumption according to Chan and Kogan (2002)
because it measures aggregate consumption relative to the standard of living, and the function ℎ(𝑧𝑡 ) is∞logarithmic Lagrange multiplier and implicitly defined as ∫ 𝑓(𝛾 𝑖𝑛𝑑 )1/𝛾
1𝑖𝑛𝑑

exp{ℎ(𝑧𝑡 ) − 𝑧𝑡 } d𝛾 𝑖𝑛𝑑 = 1.

Since 𝑧𝑡 is convenient for describing the state of the economy, we call it simply state variable. Eq. (4)implies the aggregate risk aversion is the function of the state variable:𝛾(𝑌𝑡 , 𝑋𝑡 ) = 𝛾(𝑧𝑡 ).

(5)

Therefore, the aggregate risk aversion becomes time-varying although the risk aversion of each individual

The constraint 𝛾 𝑖𝑛𝑑 > 1 is necessary for the complementarity between individual consumption andthe consumption benchmark described below.
5
See Chan and Kogan (2002) and Munk (2013) for detail.
46

is constant.
Second, the first derivative of the aggregate risk aversion satisfies the following properties.
𝛾 ′ (𝑧𝑡 ) < 0.

(6)This is derived from the property that ℎ(𝑧𝑡 ) is a decreasing and convex function. This property implies
that the aggregate risk aversion is counter-cyclical.
Third, the optimal aggregate consumption share 𝑐𝑡∗ ≡ 𝐶𝑡∗ /𝑌𝑡 is
𝑐𝑡∗ (𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 ) = 𝑓(𝛾 𝑖𝑛𝑑 )1/𝛾

𝑖𝑛𝑑

exp {−∞1𝛾ℎ(𝑧𝑡 ) − 𝑧𝑡 }𝑖𝑛𝑑

where ∫ 𝑐𝑡∗ (𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 = 1.

(7)

1

Differentiating with respect to 𝑧𝑡 ,d𝑐𝑡∗ (𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 )1= 𝑖𝑛𝑑 (𝛾(𝑧𝑡 ) − 𝛾 𝑖𝑛𝑑 )𝑐𝑡∗ (𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 ).
d𝑧𝑡
𝛾

(8)

It is implied that when the state variable increases, individuals with risk aversion 𝛾 𝑖𝑛𝑑 lower (higher)
than aggregate risk aversion 𝛾(𝑧𝑡 ) increase (decrease) their optimal aggregate consumption share.
Relatively risk-tolerant individuals have a greater share of risky asset in their wealth than relatively riskaverse individuals regardless of the state of the economy. As the state of the economy improves, theirwealth share may increase by growth of the risky asset, and they can consume more. In contrast, when thestate of the economy deteriorates, they face great losses. Therefore, they are forced to reduce theiraggregate consumption share.
2.2. Useful interpretations
We provide three useful interpretations of the Chan and Kogan (2002) model in this subsection.
First, we provide the property that the state variable is (approximately) a function of the history of theaggregate endowment growth. Using d𝑥𝑡 = 𝜅(𝑦𝑡 − 𝑥𝑡 )d𝑡 according to Chan and Kogan, the current
state variable, 𝑧𝑡 , can be expressed as a function of the state variable Δ𝑡 period ago, 𝑧𝑡−∆𝑡 , and the(log) aggregate endowment growth from Δ𝑡 period ago to the present, 𝑦𝑡 − 𝑦𝑡−∆𝑡 :𝑧𝑡 = 𝑦𝑡 − 𝑦𝑡−∆𝑡 + (1 − 𝜅Δ𝑡)𝑧𝑡−∆𝑡 .

(9)

If 𝑡 is large enough, the effect of 𝑧0 can be ignored. Eq. (9) implies that 𝑧𝑡 is a function of the historyof the (log) aggregate endowment growth 𝑦𝑡 − 𝑦𝑡−∆𝑡 for 𝑡 ∈ (Δ𝑡, 𝑡]. For sufficiently large 𝑡, the statevariable can be approximated as follows.
𝑧𝑡 ≈ 𝑧(𝜓𝑡 ; 𝜅),(10)where 𝜓𝑡 is the history of the (log) aggregate endowment growth.
Second, we provide the property about reference period. Using eq. (2) and the fact that𝑡𝜅 ∫ 𝑒−𝜅(𝑡−𝑠) d𝑠 = 1 − 𝑒−𝜅𝑡 , the state variable,0𝑧𝑡 = 𝑦𝑡 − 𝑥𝑡 , can be rewritten as follows.

𝑡𝑧𝑡 = ∫ 𝑒−𝜅(𝑡−𝑠) (𝑦𝑡 − 𝑦𝑠 )d𝑠 + 𝑒−𝜅𝑡 (𝑦𝑡 − 𝑥0 ).
0

7

(11)

If 𝑡 is sufficiently large compared to a unit of period, the second term can be ignored. In this case, 𝑧𝑡
can be interpreted as a weighted average of 𝑦𝑡 − 𝑦𝑠 , which means the (log) endowment growth from 𝑡 −𝑠 periods ago. The coefficient 𝑒−𝜅(𝑡−𝑠) implies the influence of individuals who refer to 𝑡 − 𝑠 periodsago. Assuming that the period referred to by the individual is not changed, while the individual who refers
to 𝑡 − 𝑠 periods ago appears once in the market, the individual who refers to one period ago will appear
in the market 𝑡 − 𝑠 times. Thus, the distribution of such individuals is expressed as follows.
𝑔(𝑠; 𝜅) = (𝑡 − 𝑠)𝜅2 𝑒−𝜅(𝑡−𝑠) .

(12)

−1

The peak of the distribution is equal to 𝜅 . For example, given 𝜅 =0.02, it means that 50 periods ago is
the most referenced.
Third, we provide an intuitive interpretation of the negative state variable. Let 𝒯+𝑡 be the set of time−𝑠 satisfied that 𝑦𝑡 − 𝑦𝑠 ≥ 0 for all 𝑠 ∈ 𝒯+
𝑡 . Similarly, let 𝒯𝑡 be the set of time 𝑠 satisfied that 𝑦𝑡 −𝑦𝑠 < 0 for all 𝑠 ∈ 𝒯+
𝑠 . Then, the state variable can be rewritten as follows.
𝑧𝑡 = ∫ 𝑒−𝜅(𝑡−𝑠) |𝑦𝑡 − 𝑦𝑠 |d𝑠 − ∫ 𝑒−𝜅(𝑡−𝑠) |𝑦𝑡 − 𝑦𝑠 |d𝑠
𝒯+
𝑡

(13)

𝒯−
𝑡

The first term is the weighted average of the influence of the individual facing the gain, 𝑒−𝜅(𝑡−𝑠) for 𝑠 ∈+𝒯+𝑡 , by the magnitude of each positive growth, |𝑦𝑡 − 𝑦𝑠 | for 𝑠 ∈ 𝒯𝑡 . Similarly, the second term is theweighted average of the influence of the individual facing the loss by the magnitude of each negative
growth. Consequently, the negative state variable means that the growth-weighted aggregate influence ofan individual facing a gain outweighs that of an individual facing a loss.
3. State-dependent preferences of heterogeneous individuals, and status quo hypothesis
This section extends the Chan and Kogan (2002) model to allow individual preferences to change
depending on the state of the economy. As shown in eq. (13), it is expected that the smaller the state variable
𝑧𝑡 , the greater the influence of individuals facing losses.6 Individuals facing loss are considered to be lossaverse as shown by Kahneman and Tversky (1979) and Tversky and Kahneman (1992). Alternatively,
feared individuals may be risk-averse as Guiso et al. (2018) suggest, and individuals facing significant
losses may be risk-averse. First, we consider the case where only the individual risk aversions change. Thisis expressed by switching the distribution of social weights depending on the state of the economy. Next,
we examine the situation in which the individual loss aversions change. It can be shown that an increase inloss aversion for individuals with a certain risk aversion is virtually equivalent to an increase in their social
weight. We call it status quo hypothesis.
3.1. Switching risk aversions
Switching risk aversions means that a given distribution of social weights changes. For example, we
consider the case where relatively risk-tolerant individuals change their risk aversion to be more risk-averse.
6

Barberis et al. (2001) use similar variables to describe the situation in which an individual faces loss.

8

At this time, the social weight assigned to relatively risk-tolerant individuals will decrease, and the social
weight assigned to relatively risk-averse individuals will increase.
To investigate this effect, we make the distribution of social weights in the Chan and Kogan model
dependent on the latent state 𝑘:
1−𝛾 𝑖𝑛𝑑∞

𝑈 (𝑌𝑡 , 𝑋𝑡 ; 𝑘) =sup∫ 𝑓𝑘 (𝛾𝑖𝑛𝑑{𝑐𝑡 (𝑌𝑡 ,𝑋𝑡 ;𝛾 𝑖𝑛𝑑 );𝛾 𝑖𝑛𝑑 >1} 1∞

s. t. ∫ 𝐶𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾

1
𝐶 (𝑌 , 𝑋 ; 𝛾 𝑖𝑛𝑑 ))( 𝑡 𝑡 𝑡
)
𝑖𝑛𝑑
1−𝛾
𝑋𝑡

𝑖𝑛𝑑

)d𝛾

𝑖𝑛𝑑

d𝛾 𝑖𝑛𝑑 ,(14)≤ 𝑌𝑡 .

1

Given various 𝑓𝑘 (𝛾 𝑖𝑛𝑑 ), the optimal allocation of consumption in a sufficiently large number of Pareto
equilibria is potentially derived. Let 𝑆𝑡 be the latent variable that captures the latent state. 𝑆𝑡 = 𝑘determines the structure of 𝑓𝑘 (𝛾 𝑖𝑛𝑑 ), and the Pareto equilibrium under its Pareto weight is adopted. It isthought that the latent state changes when the number of individuals facing loss increases. Therefore, we
can naturally assume that the latent variable 𝑆𝑡 depends on the state variable 𝑧𝑡 , that is, 𝑆𝑡 ≡ 𝑆(𝑧𝑡 ).
This procedure provides the distribution of social weights pseudo time-variability. The aggregate risk
aversion 𝛾 is still a function of 𝑧𝑡 because the pseudo time-variability is captured by 𝑧𝑡 .
The optimal aggregate consumption shares of individuals with 𝛾 𝑖𝑛𝑑 in latent state 𝑘 are given below.
1/𝛾 𝑖𝑛𝑑1∗
𝑐𝑘𝑡(𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 ) = (𝑓𝑘 (𝛾 𝑖𝑛𝑑 ))exp {− 𝑖𝑛𝑑 ℎ𝑘 (𝑧𝑡 ) − 𝑧𝑡 } ,
𝛾
(15)
∞
∗
where ∫ 𝑐𝑘𝑡(𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 = 1.
1

Eq. (15) holds for any 𝛾 𝑖𝑛𝑑 ∈ (1, ∞). Let 𝛾𝑖𝑖𝑛𝑑 and 𝛾𝑗𝑖𝑛𝑑 in (1, ∞) be individual risk attitudes where
𝛾𝑖𝑖𝑛𝑑 > 𝛾𝑗𝑖𝑛𝑑 , that is, individuals with 𝛾𝑖𝑖𝑛𝑑 are more risk-averse than individuals with 𝛾𝑗𝑖𝑛𝑑 . Eq. (15) canbe rewritten as follows.
𝑖𝑛𝑑∗𝑓𝑘 (𝛾𝑖𝑖𝑛𝑑 ) 𝑐𝑘𝑡(𝑧𝑡 ; 𝛾𝑖𝑖𝑛𝑑 )𝛾𝑖=
exp{(𝛾𝑖𝑖𝑛𝑑 − 𝛾𝑗𝑖𝑛𝑑 )𝑧𝑡 } .
𝑖𝑛𝑑𝑓𝑘 (𝛾𝑗 ) 𝑐∗ (𝑧 ; 𝛾 𝑖𝑛𝑑 )𝛾𝑗𝑖𝑛𝑑
𝑘𝑡

𝑡

(16)

𝑗

∗
An increase in the social weight ratio 𝑓𝑘 (𝛾𝑖𝑖𝑛𝑑 )/𝑓𝑘 (𝛾𝑗𝑖𝑛𝑑 ) means an increase in 𝑐𝑘𝑡(𝑧𝑡 ; 𝛾𝑖𝑖𝑛𝑑 ), a decrease∗in 𝑐𝑘𝑡(𝑧𝑡 ; 𝛾𝑗𝑖𝑛𝑑 ) , or both. This means an increase in the influence of 𝛾𝑖𝑖𝑛𝑑 (relatively risk-averse

individuals), a decrease in the influence of 𝛾𝑗𝑖𝑛𝑑 (relatively risk-tolerant individuals), or both. This isbecause an increase in the optimal aggregate consumption share of individuals with 𝛾 𝑖𝑛𝑑 means anincrease in the influence of 𝛾 𝑖𝑛𝑑 on the aggregate risk aversion (see Appendix for detail). In any case, as
the social weight of relatively risk-averse individuals increases, the aggregate risk attitude increase.7

7

As shown by Schildberg-Hörisch (2018) and Guiso et al. (2018), if the economic downturn makes eachindividual’s risk aversion more risk-averse, then the aggregate risk attitude becomes counter-cyclical.
This makes the counter-cyclicality suggested by the conventional Chan and Kogan model more
prominent.

93.2. Status quo hypothesis: switching loss aversions
Next, we consider the case where individuals with a certain risk aversion shifts to be loss averse. We
provide the status quo hypothesis by considering this case. This case holds almost the same argument as
the case of switching a given distribution of social weights substantially. This is because an increase in theparameter representing loss aversion in individuals with a certain risk aversion 𝛾 𝑖𝑛𝑑 has the same effectas an increase in its social weight.
To investigate this effect, eq. (14) is rewritten as follows.
𝑈̂ (𝑌𝑡 , 𝑋𝑡 ; 𝑘) =1−𝛾 𝑖𝑛𝑑∞sup∫ 𝑓𝑘 (𝛾𝑖𝑛𝑑{𝑐𝑡 (𝑌𝑡 ,𝑋𝑡 ;𝛾 𝑖𝑛𝑑 );𝛾 𝑖𝑛𝑑 >1} 1∞

s. t. ∫ 𝐶𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾𝜆 (𝛾 𝑖𝑛𝑑 ) 𝐶 (𝑌 , 𝑋 ; 𝛾 𝑖𝑛𝑑 )) 𝑘 𝑖𝑛𝑑 ( 𝑡 𝑡 𝑡
)
1−𝛾
𝑋𝑡

𝑖𝑛𝑑

)d𝛾

𝑖𝑛𝑑

d𝛾 𝑖𝑛𝑑 ,(17)≤ 𝑌𝑡 .

1

where 𝜆𝑘 (𝛾 𝑖𝑛𝑑 ) is the parameter of loss-aversion for individuals with 𝛾 𝑖𝑛𝑑 in the latent state 𝑘.8 An
increase in loss aversion implies that individuals facing loss and wanting to reduce their consumption
cannot reduce their consumption as they wish. This slowdown in consumption decline may be due to thestatus quo propensity.9 This may represent the cost stickiness in households and businesses.
The optimal aggregate consumption shares of individuals with 𝛾 𝑖𝑛𝑑 in latent state 𝑘 are given below.
1/𝛾 𝑖𝑛𝑑1∗𝑐𝑘𝑡
̂ (𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 ) = (𝜆𝑘 (𝛾 𝑖𝑛𝑑 )𝑓𝑘 (𝛾 𝑖𝑛𝑑 ))exp {− 𝑖𝑛𝑑 ℎ̂𝑘 (𝑧𝑡 ) − 𝑧𝑡 } ,
𝛾
(18)
∞
∗
where ∫ 𝑐𝑘𝑡̂ (𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 = 1.
1

In the same way as eq. (16), we have𝑖𝑛𝑑∗
𝜆𝑘 (𝛾𝑖𝑖𝑛𝑑 )𝑓𝑘 (𝛾𝑖𝑖𝑛𝑑 ) 𝑐𝑘𝑡(𝑧𝑡 ; 𝛾𝑖𝑖𝑛𝑑 )𝛾𝑖=
exp{(𝛾𝑖𝑖𝑛𝑑 − 𝛾𝑗𝑖𝑛𝑑 )𝑧𝑡 } .
𝑖𝑛𝑑𝑖𝑛𝑑
𝜆𝑘 (𝛾𝑗 )𝑓𝑘 (𝛾𝑗 ) 𝑐∗ (𝑧 ; 𝛾 𝑖𝑛𝑑 )𝛾𝑗𝑖𝑛𝑑
𝑘𝑡

When

𝑓𝑘 (𝛾𝑖𝑖𝑛𝑑 )/ (𝜆𝑘 (𝛾𝑗𝑖𝑛𝑑 )𝑓𝑘 (𝛾𝑗𝑖𝑛𝑑 ))

𝑡

(19)

𝑗

is fixed, an increase in the loss aversion 𝜆𝑘 (𝛾𝑖𝑖𝑛𝑑 ) means an∗∗increase in 𝑐𝑘𝑡(𝑧𝑡 ; 𝛾𝑖𝑖𝑛𝑑 ), a decrease in 𝑐𝑘𝑡(𝑧𝑡 ; 𝛾𝑗𝑖𝑛𝑑 ), or both. As with social weights, in any case, as theloss-aversion of relatively risk-averse individuals increases, the aggregate risk attitude increase. Inversely,as the loss-aversion of relatively risk-tolerant individuals increases, the aggregate risk attitude decrease. If

8The loss avoidance parameter 𝜆 does not necessarily have to be the same for individuals with the same𝛾 𝑖𝑛𝑑 . For example, if individuals with 𝛾 𝑖𝑛𝑑 = 2 and 𝜆 = 3 and individuals with 𝛾 𝑖𝑛𝑑 = 2 and 𝜆 = 1
divide the social weight belonging to individuals with 𝛾 𝑖𝑛𝑑 = 2 in half, these can be regarded as a single
group with 𝛾 𝑖𝑛𝑑 = 2 and 𝜆 = 1.5.
9
At a glance, the slowdown in consumption decline may also be explained by switching to be more riskaverse. This is because 𝐶/𝑋 is generally a very small value and eq. (1) implies 𝑢𝐶 =𝑖𝑛𝑑
𝑋 −1 (𝐶/𝑋)−𝛾 , and consequently, the more risk averse, the greater the marginal utility of consumption.
The large 𝛾 𝑖𝑛𝑑 , however, makes 𝑢𝐶 extremely volatile because exponential terms are included in𝑖𝑛𝑑𝑖𝑛𝑑
𝑢𝐶𝐶 = −𝛾 𝑖𝑛𝑑 𝑋 −2 (𝐶/𝑋)−𝛾 −1 and 𝑢𝐶𝑋 = (𝛾 𝑖𝑛𝑑 − 1)𝑋 −2 (𝐶/𝑋)−𝛾 . This property is not suitablefor expressing the rigidity of the consumption level. On the other hand, the switch to loss-averse
preference changes 𝑢𝐶𝐶 and 𝑢𝐶𝑋 only in a multiplier, which prevents 𝑢𝐶 from becoming extremely
volatile. Studies such as Guiso et al. (2018) that suggest that people become more risk-averse during an
economic downturn may not be able to distinguish between switching to be loss-averse and risk-averse.

10relatively risk-tolerant individuals become loss-averse due to the economic downturn, the influence ofrelatively risk-tolerant individuals on the aggregate risk aversion will increase. That is, the aggregate risk
aversion decreases in the economic downturn, or is procyclical. Note that when all individuals become
homogeneously loss-averse, the optimal aggregate consumption share of each individual does not change.
The relationship between the aggregate risk attitude and the state of the economy is summarized below
as the status quo hypothesis. Relatively risk-tolerant individuals are more likely to face losses than relatively
risk-averse individuals during recession because the former allocate more to risk assets than the latter. The
aggregate risk aversion will be procyclical when the effect of such heterogeneous loss-aversion outweighs
the effect that lead to counter-cyclicality such as increasing the social weight of relatively risk-averse
individuals (e.g. Guiso et al., 2018) and increasing the influence of relatively risk-averse individuals on theaggregate risk aversion (e.g. Chan and Kogan, 2002). A serious economic downturn such as 2008 global
financial crisis and 2020 COVID-19 crisis is expected to make aggregate risk aversion counter-cyclicalagain because it homogenizes the loss-aversion of each individual.
4. Empirical evidences
This section uses data to analyze how the aggregate risk attitude fluctuate in response to changes inthe state of the economy. First, we construct empirical models with time-varying (state-dependent)
aggregate risk attitude. Second, we explain our data for market returns in some main market and provide
simple analysis. Third, we provide evidence that the aggregate risk attitudes in stock markets of Westerncountries are time-varying, and the status quo hypothesis is supported for the time-variability.
4.1. Empirical model
The aggregate risk attitude can be expressed as a function of the state of the economy, which represents
the business-cycle of the economy. Most conventional models imply that the aggregate risk aversion ismonotonic decreasing function of state variable. In contrast, our status quo hypothesis implies that the effectof state variable on aggregate risk attitudes depends on the level of state variable (or business sentiment).
In other word, the function 𝛾(𝑧) may have one or more inflection points. For this reason, we make somepolynomial approximations to capture the various relationships between the state variable and the aggregate
risk aversion:⎧𝛿0 + 𝛿1 (1 + 𝑧){{
𝛿0 + 𝛿1 (1 + 𝑧) + 𝛿2 (1 + 𝑧)2
𝛾(𝑧) =⎨𝛿0 + 𝛿1 (1 + 𝑧) + 𝛿2 (1 + 𝑧)2 + 𝛿3 (1 + 𝑧)3{{⎩𝛿0 + 𝛿1 (1 + 𝑧) + 𝛿2 (1 + 𝑧)2 + 𝛿3 (1 + 𝑧)3 + 𝛿4 (1 + 𝑧)4(20)

The first line in eq. (20), linear model, corresponds to the Campbell and Cochrane (1999) model, Chan and
Kogan (2002) model, and other main conventional models if 𝛿1 < 0. The second to fourth lines, quadratic
to quartic models, approximate the situation where the inflection point exists. Our status quo hypothesis

11suggests procyclicality during the recession and counter-cyclicality during the boom and depression.
Therefore, the status quo hypothesis is supported by the cubic function model. Note that these parameters,
𝛿s, are not constrained to flexibly capture market movements.
Moreover, an appropriate 𝜅 is required to specify the state variable since 𝑧𝑡 ≈ 𝑧(𝜓𝑡 ; 𝜅) from eq.
(10). We consider 9 patterns of 𝜅; 0.02, 0.025, 0.0325, 0.0375, 0.045, 0.055, 0.075, 0.115 and 0.25. Fromeq. (12), these mean that the peaks in the distribution of past time referred to in the weekly data are 1 year,
9 months, 7 months, 6 months, 5 months, 4 months, 3 months, 2 months, and 1 month, respectively (see fordetails in Section 2.2). As a result, the number of models is 4 × 9 = 36. Let the model with the smallest AIC(Akaike’s information criterion) be the time-varying or state-dependent model. We also consider a constantmodel, which has a constant aggregate risk aversion, as a benchmark model. That is, 𝛾(𝑧) = 𝛿0 .
We use the weighted average of historical returns in the stock market as the proxy for state variable.
Chan and Kogan (2002) used “relative log-consumption” as a variable to capture the state of the economy.
Data representing consumption or endowment, such as household consumption, CPI, or GDP, are lessfrequent than stock return, however. Furthermore, these data are lagging indices related to the businesscycle, and it must be said that consumption as data is a little slow to show the business sentiment thatinvestors feel. In fact, the stock market index will be the most important factor for investors to judge thefuture growth potential of consumption, productivity and endowment in the economy. Therefore, as avariable to capture the state of the economy, we use a variable in which the aggregate consumption or
endowment growth rate in the relative log-consumption is replaced by the stock market return.
We use GARCH-in-mean (generalized autoregressive conditional heteroskedasticity in mean) model
proposed by Engle, Lilien and Robins (1987) in order to get the aggregate risk attitude of the economy:2𝑟𝑀𝑡 − 𝑟𝑓𝑡 = 𝛾(𝑧𝑡−1 )𝜎𝑡|𝑡−1+ 𝜖𝑡 ,

2𝜖𝑡 ∼ 𝒩(0, 𝜎𝑡|𝑡−1
).

(21)

where 𝑟𝑀𝑡 and 𝑟𝑓𝑡 are the stock market return and risk-free rate at time 𝑡, respectively. We assume thatthe dynamics of the conditional variance follows the GARCH (1,1) process:22𝜎𝑡|𝑡−1= 𝛼0 + 𝛼1 𝑒2𝑡−1 + 𝛽𝜎𝑡−1|𝑡−2,(22)

where 𝑒𝑡−1 is the residual at time 𝑡 − 1, and 𝛼0 , 𝛼1 and 𝛽 are constant. The specification corresponds
to the discrete-time capital asset pricing model with the time-varying state-dependent aggregate risk
aversion. 10
Ultimately, the only variables needed in our model are market returns and risk-free rates. Suchsimplicity can suppress the problems of variable selection arbitrariness and overfitting.

10

Chan and Kogan (2002) imply that the Sharpe ratio of the single risky asset is expressed as the productof the state-dependent aggregate risk attitude and the volatility of the aggregate endowment growth rate.
Since the volatility of the aggregate endowment growth rate in the Chan and Kogan model is treated as aconstant, it is necessary to verify the state dependence of the Sharpe ratio rather than the state dependenceof the aggregate risk attitude in order to accurately verify the Chan and Kogan model. As mentioned
above, Statistical data on consumption and endowment do not necessarily represent the intrinsic
aggregate endowment growth rate, and there is no guarantee that its volatility is constant. We use market
returns as a proxy for aggregate endowment growth, and finally, we obtain eq. (21).

124.2. Data and summary
The data required in this paper are (log) market returns and (log) risk-free rates. We analyze the 10
major stock markets to investigate whether the mechanism for time-variability in the aggregate risk attitudevaries from country to country; Canada, China, Eurozone, France, Germany, Hong Kong, India, Japan, theUnited Kingdom, and the United States. The index of each market is used for the weekly market return;
TSX, SSE, STOXX 600, CAC 40, DAX, HSI, SENSEX, TOPIX, FTSE 100, and S&P 500, respectively.
The risk-free rate of each market is used for the 3 months interbank rate of each market or currency except
for Canada and China. The risk-free rate of Canada is used for Canadian 3 months government bond yield,
and that of China is used for Chinese 3 months interbank repo rate. The length of the sample period is 783
weeks from July 1, 2005 to June 26, 2020 in order to have the same period in each market.11 We need,however, longer sample to eliminate the arbitrariness of the initial value for the state variable 𝑧 because
the state variable constructed by historical stock market returns in discrete-time framework can be rewritten
as 𝑧𝑡 = 𝑟𝑀𝑡 + (1 − 𝜅)𝑧𝑡−1 = (1 − 𝜅)𝑡 𝑧0 + ∑𝑡−1
(1 − 𝜅)𝑠 𝑟𝑀,𝑡−𝑠 from eq. (9). We set the initial value of𝑠=0
𝑧0 = 𝑦0 − 𝑥0 to zero and burn-in-period to 100 weeks (about 2 years). Therefore, the precise length ofmarket returns is 883 weeks.12
Table 1 shows the summary statistics of logarithmic returns on 10 market indices. The skewness ofthe market index is negative in all markets. The kurtosis of the market index is well above 3 in all markets.
All distributions of market indices are distorted in the negative direction and are fat-tail. This is aspreviously pointed out, and these characteristic does not change in emerging markets such as China andIndia. The rightmost column shows the correlation with the US market. It can be seen that the stock markets
in Western countries including Canada, Eurozone, France, Germany, and the UK are highly correlated with
the US. On the other hand, the stock markets of Asian countries including China, Hong Kong, India andJapan have extremely low correlation with the US.
[Table 1]
4.3. Results
We analyze the structure of state-dependent aggregate risk attitude specified by equations (20) and(21) with conditional volatility specified by eq. (22). Table 2 shows the results of constant and statedependent aggregate risk attitude model with conditional volatility. The constant aggregate risk attitude
model in the Canadian stock market provides the estimate of constant term 𝛿0 is 2.589. Its standard error
is 1.312, and the parameter is statistically significant at 5% level. The AIC (Akaike’s information criterion)of the constant model is 3,225.055. On the other hand, the state-dependent aggregate risk attitude modelwith cubic function and 𝜅 = 0.0375 in the Canadian stock market has the smallest AIC. 𝜅 = 0.0375

11

It also aims to eliminate the period when the exchange rates of each country are unstable due to theintroduction of the euro.
12
All these data are taken from Bloomberg Professional.

13

implies the peak of individual reference time point is 6 months ago (see also 2.2 in this paper for details).
All estimates of coefficients, 𝛿0 , 𝛿1 , 𝛿2 and 𝛿3 , are statistically significant at 5% level. The AIC of statedependent model is 3,222.594 and smaller than that of constant model. In addition, likelihood ratio test (LRtest in Table 2) shows that the null hypothesis that the constant model is correct is rejected at the 5% level.
To sum up, the aggregate risk attitude in the Canadian market is state-dependent.
In the same way, it can be seen that stock markets in Western countries, including Canada, Eurozone,
France, Germany, the UK and the US have a representative individual with state-dependent aggregate risk
attitude. Inversely, representative individuals in Asian stock markets, including China, Hong Kong, India
and Japan have constant aggregate risk attitudes. In the Chinese market, the state-dependent model hassome insignificant parameters in spite of being selected by the likelihood ratio test and AIC. In the Indian
market, the state-dependent model is not selected by the likelihood ratio test and AIC although allparameters are significant. The constant model is generally supported in the Asian stock markets. At least,
polynomial approximations from linear to quartic functions cannot capture their state-dependence or timevariability.
[Table 2]
Figure 1 shows the structure of aggregate risk attitude with conditional volatility. The horizontal andvertical axes represent the state variable and the aggregate risk attitude, respectively. Western stock markets
have similar cubic structure. On the other hand, the structures of aggregate risk attitudes in Asian stock
markets are very different from those in Western stock markets and are different from each other. We willconsider only the Western market from now on since the state-dependent model with polynomial
approximations is not supported in the Asian stock market.13
Cubic curves in these markets commonly have the first inflection point or local minimum point around-0.25, and the second inflection point or local maximum point near 0. The decreasing structure in the range
over 0, the movement between the upper left and the lower right, corresponds to the counter-cyclicality ofaggregate risk attitude at times when most individuals face gain, that is, booms. It can be seen that theconventional model such as Campbell and Cochrane (1999) and Chan and Kogan (2002) holds in such astable economic environment.
The aggregate risk attitude, however, become increasing function of state variable in the range -0.25
to 0. This structure corresponds to the procyclicality of aggregate risk attitude at times when risk-taking
individuals face loss, that is, recessions. This implies two hypotheses as shown in Section 3. The first
hypothesis is that the social weight of relatively risk-tolerant individuals increases due to the economic
13

We also analyze the Sharpe ratio by a similar way, in order to match the formation of Chan and Kogan(2002) model (not listed in table). That is, we use following model instead of eq. (21).
2𝜇𝑡 − 𝑟𝑓𝑡 = 𝛾(𝑧𝑡−1 )𝜎𝜎𝑡|𝑡−1 + 𝜖𝑡 ,𝜖𝑡 ∼ 𝒩(0, 𝜎𝑡|𝑡−1(21′ ))
where 𝜎 is the constant volatility of consumption (or endowment) growth. Results in the US stock
market and German stock market also support our status quo hypothesis while results in the other markets
reject our status quo hypothesis. It may be necessary to consider the time-variability of the consumption
volatility, the dependence of the market on the US market, and the effects of exchange rates.

14

downturn. The second hypothesis, or the status quo hypothesis, is that the relatively risk-tolerant individuals
will switch to more loss-averse preferences in order to maintain their status quo as the economy
deteriorates.14
The decreasing structure outside the range below -0.25 corresponds to the counter-cyclicality ofaggregate risk attitude at times when most individuals face loss, that is, depressions. The first hypothesis
cannot explain this counter-cyclicality. This is because if an individual facing a loss becomes risk-tolerant,
the aggregate risk attitude should still be procyclical during depressions. On the other hand, the second
status quo hypothesis can explain this counter-cyclicality because the depression provides almost
homogeneous switching to loss aversions. In other words, the depression causes loss not only to individuals
who take risks but also to individuals who do not take risks, and eliminates distortions based on differencesin preferences. Consequently, the relationship between aggregate risk attitude and state variable becomes
the same negative relationship as during the boom period.
[Figure 1]
Finally, the relationship between inflection points and the state variable 𝑧 is described. Figure 2 show
the time-series of 𝑧 and inflection points. Areas filled in purple (blue) mean that the state of the economy
is below the first (second) inflection point. The first inflection point is around -0.25 and the second
inflection point is around 0 as can be seen from Figure 1. The purple areas generally correspond to the 2008
global financial crisis and the 2020 COVID-19 crisis. During such a serious crisis, almost all individuals
may switch to loss-averse preferences in order to maintain their status quo. The blue areas are observed in2011 and 2016-2017, except around those two crises. The former may respond to the Greek and Spanish
crises, and the latter may respond to the Brazilian economic crisis. It can be said that shocks that are notdue to their own country cannot reduce the wealth of relatively risk-averse individuals who do not takerisks, except for two global crises.
[Figure 2]
5. Conclusion
We improved Chan and Kogan (2002) model and constructed a new behavioral asset pricing modelwith heterogeneous individuals whose preferences change depending on the state of the economy. Basedon this model, we proposed a status quo hypothesis. In our status quo hypothesis, individuals facing loss
switch to loss-averse preferences to maintain status quo. In the boom period, most individuals face gains,so the aggregate risk attitude is countercyclical as in the conventional model. In a minor recession, the effectof switching to loss averse preference on aggregate risk attitudes is heterogeneous, as only risk-taking
individuals face significant losses. As a result, the aggregate risk attitude becomes procyclical. On the other14

A hypothesis in which the proportion of risk-loving individuals increases can also be considered. In
this case, the general equilibrium theory may not hold. This hypothesis is beyond the scope of this paper.
Recent studies have also developed a general equilibrium model of the economy, including risk-loving
individuals (Araujo et al., 2017; Araujo et al., 2018).

15

hand, in a major recession, or depression, almost all individuals face large losses. The effect of loss aversion
on aggregate risk attitudes is nearly homogeneous. Consequently, the aggregate risk attitude becomescountercyclical again.
Next, the structure of aggregate risk attitude was approximated by a polynomial to take into accountthe possibility of multiple inflection points. As a result, the structures of aggregate risk attitudes in the stock
markets of Western countries including Canada, Eurozone, France, Germany, the UK and the US could bewell approximated by a cubic function. This cubic function implies the counter-cyclicality of theseaggregate risk attitudes during booms and depressions and the procyclicality during recessions. This result
supports our status quo hypothesis. We conclude that it is essential to allow both heterogeneity andchangeability of preferences in order to determine an accurate aggregate risk attitude.

16

Appendix
We provide a formal proof that an increase in the optimal aggregate consumption share of individuals
with 𝛾 𝑖𝑛𝑑 means an increase in the influence of 𝛾 𝑖𝑛𝑑 on the aggregate risk aversion, while this is
intuitively clear.
Chan and Kogan (2002) proved that the aggregate risk aversion is expressed as follows.
∞𝛾(𝑧𝑡 ) = (∫
1

−1

1

𝛾

𝑐∗ (𝑧 ; 𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 )𝑖𝑛𝑑 𝑡 𝑡∞s. t. ∫ 𝑐𝑡∗ (𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 = 1.

(A1)

1

We focus on a specific range (𝛾 𝑖𝑛𝑑 , 𝛾̅ 𝑖𝑛𝑑 ) and define 𝑐𝑡∗̂ , 𝛾1𝑖𝑛𝑑 and 𝛾2𝑖𝑛𝑑 as follows.
̅̅ ̅ 𝑖𝑛𝑑𝛾∗𝑐𝑡̂ = ∫𝑐𝑡∗ (𝑧𝑡 ; 𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑
𝛾 𝑖𝑛𝑑

𝛾1𝑖𝑛𝑑 =
𝛾2𝑖𝑛𝑑 =𝑐𝑡∗̂̅̅ ̅ 𝑖𝑛𝑑𝛾∫𝛾𝑖𝑛𝑑

1 𝑐∗ (𝑧 ; 𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑
𝛾 𝑖𝑛𝑑 𝑡 𝑡

(A2)
(A3)

,

1 − 𝑐𝑡∗̂
1 𝑐∗ (𝑧 ; 𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑∫𝛾𝑖𝑛𝑑 ∈(1,∞)∖(𝛾𝑖𝑛𝑑 ,𝛾̅̅̅𝑖𝑛𝑑 ) 𝑖𝑛𝑑
𝑡 𝑡
𝛾

(A4)

𝑐𝑡∗̂ means the aggregate consumption share of the individuals with 𝛾 𝑖𝑛𝑑 ∈ (𝛾 𝑖𝑛𝑑 , 𝛾̅ 𝑖𝑛𝑑 ). 𝛾1𝑖𝑛𝑑 means theaggregate risk aversion of only these individuals. 𝛾2𝑖𝑛𝑑 means the aggregate risk aversion except for these
individuals. For example, if 𝛾1𝑖𝑛𝑑 < 𝛾2𝑖𝑛𝑑 , the individuals with 𝛾 𝑖𝑛𝑑 ∈ (𝛾 𝑖𝑛𝑑 , 𝛾̅ 𝑖𝑛𝑑 ), or simply 𝛾1𝑖𝑛𝑑 , mean
relatively risk-tolerant individual than representative (aggregate) individual since 𝛾2𝑖𝑛𝑑 ≈ 𝛾 for small 𝑐𝑡∗̂ .
Eq. (A1) can be rewritten as follows.
−1

1
1𝛾(𝑧𝑡 , 𝑐𝑡∗̂ ) = ( 𝑖𝑛𝑑 𝑐𝑡∗̂ + 𝑖𝑛𝑑 (1 − 𝑐𝑡∗̂ ))
𝛾1
𝛾2

.

(A5)

Consequently,𝜕𝛾(𝑧𝑡 , 𝑐𝑡∗̂ )11𝛾22(𝛾 𝑖𝑛𝑑 − 𝛾1𝑖𝑛𝑑 ).
=
−𝛾
(
−
)
=
−
𝜕𝑐𝑡∗̂
𝛾1𝑖𝑛𝑑 𝛾2𝑖𝑛𝑑
𝛾1𝑖𝑛𝑑 𝛾2𝑖𝑛𝑑 2(A6)

If 𝛾1𝑖𝑛𝑑 < 𝛾2𝑖𝑛𝑑 , an increase in the optimal aggregate consumption share of individuals with 𝛾 𝑖𝑛𝑑 ∈(𝛾 𝑖𝑛𝑑 , 𝛾̅ 𝑖𝑛𝑑 ), who are relatively risk-tolerant individuals, decrease the aggregate risk aversion. This implies
that the representative (aggregate) individual become more risk-tolerant. Similarly, increasing the optimal
aggregate consumption share of relatively risk-averse individuals makes representative individual morerisk-averse.

17Reference
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18

Table 1

Summary statistics
Mean

Std. Dev
2
2.425

3-1.514414.405Min

5-17.542

Med

7
0.293

Max

Canada

1
0.053

Skew

Kurt

9
12.817

Corr
10
0.822

China

0.127

3.411

-0.3595.512-14.898

0.237

13.9450.191

Eurozone

0.033

2.717

-1.79016.755-24.2520.314

12.437

0.834

France

0.020

3.069

-1.43512.989-25.050

0.268

12.432

0.812

Germany

0.124

3.150

-1.24912.513-24.3470.441

14.942

0.813

Hong Kong

0.070

2.967

-0.3336.075-17.8150.327

11.719

0.604

India

0.203

2.990

-0.475

6.948-17.3810.378

13.171

0.557

Japan

0.038

2.926

-1.0449.202-22.0190.338

12.871

0.620

UK

0.025

2.546

-1.63918.139-23.632

0.200

12.584

0.829

US

0.118

2.534

-1.13112.747-20.084

0.253

11.4241.000

Note: Results are based on market index returns [%, weekly] from 1 Jul 2003 to 26 Jun 2020 (783 weeks).
The statistics of “Corr” means the correlation with the US stock returns.

8.4604 **5.2637 *3,960.718

3,558.762

6.9589 *3,557.803

3,794.387

2.441 **
(1.098)0.778 ***
(0.043)0.173 ***
(0.035)

7.1914 *3,793.196

0.0450(5 months)-1,153 **(458)

3,026 **
(1,208)-2,605 **(1,048)736 **(298)

0.791 ***
(0.041)0.163 ***
(0.034)

France4
4
constant time-varying
100.528 ***
0.489 ***
(0.164)(0.153)

Note: Values in parentheses denote standard errors. ***,** and * denote statistical significance at 1, 5 and 10% levels, respectively.

LR test3,961.982

3,222.594

AIC

3,225.055

0.0450(5 months)

0.0375(6 months)

𝜅0.1150(2 months)-858 **(384)-1,449 **
(564)

𝛿3

2,200 **
(993)

-1,842 **(843)

505 **
(235)

88 *
(50)

3.079 ***
(1.118)0.721 ***
(0.045)

3,776 **
(1,484)73 (49)0.712 ***
(0.046)0.232 ***
(0.042)

𝛿2-0.368 (0.979)

0.847 ***
(0.030)0.237 ***
(0.042)-163 (100)899 **(364)

2.589 **
(1.312)𝛿00.845 ***
(0.030)0.135 ***
(0.029)Eurozone
3
3
constant time-varying
100.490 ***
0.462 ***
(0.130)(0.124)-3,221 **(1,284)0.727 ***
(0.035)0.726 ***
(0.035)𝛽0.136 ***
(0.029)China
22constant time-varying
100.245 ***
0.248 ***
(0.094)(0.095)

𝛿1

0.229 ***
(0.035)0.228 ***
(0.035)𝛼1𝛼0Canada1
1
constant time-varying
100.277 ***
0.267 ***
(0.066)(0.063)GARCH estimation (1/2)

Table 2

3,824.198

3.411 ***
(0.999)0.634 ***
(0.065)0.267 ***
(0.051)

9.4026 **3,820.795

0.0750(3 months)-1,485 ***(453)

3,944 ***
(1,211)-3,431 ***(1,064)979 ***(307)

0.656 ***
(0.058)0.255 ***
(0.049)Germany
5
5constant time-varying
101.122 ***
1.000 ***
(0.313)(0.269)

3,460.558-2,091 *(1,149)

578 *
(323)

0.749 ***
(0.047)0.200 ***
(0.041)5.4234 -3,694.102

0.0325(7 months)

0.3139 -3,782.059

Note: Values in parentheses denote standard errors. ***,** and * denote statistical significance at 1, 5 and 10% levels, respectively.

0.5491 -3,798.647

AIC

LR test0.0200(1 year)𝜅

7.2458 *3,459.313

0.0200(1 year)-965 *(508)
0.0200(1 year)3.227 **
(1.265)0.731 ***
(0.046)0.215 ***
(0.041)-280 **(120)

3.185 (5.735)-0.439 (5.639)

0.576 ***
(0.095)0.251 ***
(0.052)9time-varying
00.394 ***
(0.114)UK

𝛿33,780.373

2.618 **
(1.145)0.573 ***
(0.098)0.251 ***
(0.053)9
constant
10.429 ***
(0.117)2,484 *
(1,336)-667 **(299)

186 **(86)0.802 ***
(0.038)0.162 ***
(0.032)Japan
8
8
constant time-varying
10
1.544 ***
1.520 ***
(0.542)(0.525)764 **(334)

3,693.526

1.965 *
(1.133)0.804 ***
(0.037)0.160 ***
(0.031)India
7
7
constant time-varying
100.311 ***
0.305 ***
(0.101)(0.102)

𝛿2

3,797.196

6.403 (6.262)

1.769 (1.197)

𝛿0-4.704 (6.267)

0.892 ***
(0.027)0.891 ***
(0.028)𝛽𝛿10.080 ***
(0.021)0.081 ***
(0.021)𝛼1𝛼0Hong Kong
6
6
constant time-varying
100.227 **
0.227 **
(0.097)(0.097)GARCH estimation (2/2)

Table 2

3,338.053

4.130 ***
(1.151)0.648 ***
(0.045)0.285 ***
(0.042)10
constant
10.445 ***
(0.107)8.5357 **3,335.517

0.0375(6 months)-1,097 **(439)

2,867 **
(1,141)-2,443 **(973)

680 **(271)

0.661 ***
(0.043)0.273 ***
(0.041)10time-varying
00.425 ***
(0.101)USFigure 1

Structure of aggregate risk attitude

4
2
0
−2
−6

−4

aggregate risk attitude

6

8

Western markets

US
Canada−0.3

Eurozone
France−0.2

Germany
UK
−0.1

0.0

0.1

0.0

0.1

state variable

4
2
0
−2
−4
−6

aggregate risk attitude

6

8

Asian markets

US
China
−0.3

Hong Kong
India
−0.2

Japan

−0.1
state variableFigure 2

Time-series of state variable and inflection points

Under first inflection point

Under second inflection point

2010

2015

0.0
−0.2

2005

Under second inflection point

2010

2015

Germany

2020

Under second inflection point

2010

2015

0.0
−0.2
−0.4
−0.6

−0.2

state of the economy

0.0

0.2

date (weekly)

France

−0.4
−0.6

2020

Under first inflection point
2005

Under second inflection point

2010

2015

US

2020

−0.6

Under first inflection point
2005

2010

Under second inflection point
2015

date (weekly)

2020

0.0
−0.2
−0.4

−0.4

−0.2

state of the economy

0.0

0.2

date (weekly)UK
0.2

date (weekly)−0.6

state of the economy

Under first inflection point

date (weekly)

Under first inflection point
2005

state of the economy

−0.4

2020

0.2

2005

−0.6

state of the economy

0.0
−0.2
−0.4
−0.6

state of the economy

0.2

Eurozone

0.2

CanadaUnder first inflection point
2005

2010

Under second inflection point
2015

date (weekly)2020