Ambiguity on uncertainty and the equity premium

Ambiguity on uncertainty and the equity premium

Finance Research Letters xxx (xxxx) xxxx Contents lists available at ScienceDirect Finance Research Letters journal homepage: www.elsevier.com/locat...

1MB Sizes 0 Downloads 23 Views

Finance Research Letters xxx (xxxx) xxxx

Contents lists available at ScienceDirect

Finance Research Letters journal homepage: www.elsevier.com/locate/frl

Ambiguity on uncertainty and the equity premium☆ Xinfeng Ruan , Jin E. Zhang ⁎

Department of Accountancy and Finance, Otago Business School, University of Otago, Dunedin 9054, New Zealand

ARTICLE INFO

ABSTRACT

Keywords: Ambiguity Multiple priors Equity premium puzzle

This paper considers an asset pricing model with a multiple-priors recursive utility incorporating decision makers’ concern with ambiguity on drift and jumps of driving process. Based on our empirical evidence, given a small relative risk aversion (RRA) coefficient (e.g., RRA = 2), the model can well explain the equity premium puzzle, since the ambiguity aversion, as a complementary aversion of the risk aversion, can increase the equity premium and decrease the riskfree rate. This paper documents that ambiguity on uncertainty is a resolution of the equity premium puzzle.

JEL classification: D53 D81 G12

1. Introduction Mehra and Prescott (1985) and Weil (1989) propose two puzzles in modern asset pricing models, that is the equity premium and risk-free rate puzzles. The literature proposes masses of solutions to explain these two puzzles, for example, crises or jumps in Rietz (1988); Barro (2006) and Branger et al. (2016), habit formation in Constantinides (1990) and Campbell and Cochrane (1999), and long-run risks in Bansal and Yaron (2004). The detailed overview can be found in Mehra (2011). Using a consumption-based asset pricing model with a constant relative risk aversion (CRRA) utility function, higher the risk aversion, higher the equity premium and higher the risk-free rate in the meanwhile. For example, in Mehra and Prescott (1985), they need to use a high relative risk aversion (RRA) (RRA > 40) to explain the high equity premium, but at the same time, the risk-free rate is too high. Even though Weil (1989) incorporates Epstein-Zin recursive preference, he can not explain both puzzles with a small RRA (e.g., RRA < 10). Recently, a large number of research claims that the ambiguity aversion plays a key role to explain the equity premium and riskfree rate puzzles.1 Ju and Miao (2012) propose a generalized recursive smooth ambiguity model in discrete time, in which consumption and dividends follow hidden Markov regime-switching processes. Jahan-Parvar and Liu (2014) develop their consumptionbased equilibrium model into the production-based. Both of them document that the ambiguity model with a low RRA can produce a high equity premium and a low risk-free rate. In continuous-time version, Chen and Epstein (2002) extend the stochastic differential utility in Duffie and Epstein (1992b) (which is a continuous-time case of the recursive utility in Epstein and Zin, 1989) into the recursive multiple-priors utility. In their model, the representative investor have multiple prior beliefs on the state of nature. In others words, the investor has ambiguity on true probability measure. Jeong et al. (2015) empirically test their model and find that the RRA

☆ Jin E. Zhang has been supported by an establishment grant from the University of Otago and the National Natural Science Foundation of China (Project no. 71771199). Please send correspondence to Xinfeng Ruan, We declare that we have no relevant or material financial interests that relate to the research described in this paper. All remaining errors are ours. Previous versions of this paper is circulated under the title “Ambiguity on Uncertainty: A Resolution of the Equity Premium Puzzle”. ⁎ Corresponding author. E-mail addresses: [email protected] (X. Ruan), [email protected] (J.E. Zhang). 1 A comprehensive literature overview can be found in Epstein and Schneider (2010).

https://doi.org/10.1016/j.frl.2020.101429 Received 29 October 2019; Received in revised form 5 December 2019; Accepted 8 January 2020 1544-6123/ © 2020 Elsevier Inc. All rights reserved.

Please cite this article as: Xinfeng Ruan and Jin E. Zhang, Finance Research Letters, https://doi.org/10.1016/j.frl.2020.101429

Finance Research Letters xxx (xxxx) xxxx

X. Ruan and J.E. Zhang

is much lower with ambiguity. This paper is an extension of Chen and Epstein (2002). According to Rietz (1988); Barro (2006) and Branger et al. (2016), the crises or jumps are important to explain the equity premium puzzle. Motivated by them, we extend ambiguity on drift in Chen and Epstein (2002) into ambiguity on both drift and jumps in this paper.2 In reality, the growth rate of the consumption and its jumps are not fully observable. Thus, the investor naturally has ambiguity on both drift and jumps. In addition, this paper provides complimentary evidence for Chen and Epstein (2002) and Jeong et al. (2015). Based on a parsimonious jump-diffusion model, we explicitly show the mechanism about how the ambiguity aversion increases the equity premium and decreases the risk-free rate so that the model with ambiguity can well explain the equity premium puzzle by setting a low RRA. The remainder of this article is organized as follows. Section 2 presents the model and results, and Section 3 provides empirical evidence. Section 4 concludes. 2. Model setup 2.1. Consumption dynamics by3

Motivated by Barro (2006) and Branger et al. (2016), we assume the consumption dynamics on a probability ( ,

dCt = µdt + dBt + LdNt , Ct

, P ) are given

(1)

where Bt is the one-dimension Brownian motion and Nt is a Poisson process with constant jump intensity λ > 0 and constant jump size L < 0 in the probability measure P. μ > 0 and σ > 0 both are constant. 2.2. Ambiguity aversion with jump-diffusion information Following Chen and Epstein (2002), we consider a finite horizon model over [0, T] on a probability ( , generator (ηt, ζt) such that the process mt , is a P-martingale. That is ,

dmt

=

,

mt

t dBt

+

t [dNt

m0 , = 1,

dt ],

, P ) . There is the density

(2)

equivalently,

mt

,

= exp

{

1 2

t

t

| s |2 ds

0

s

0

ds

t 0

s dWs

+

t 0

}

ln(1 + s ) dNs ,

(3)

where t > 1. Given an admissible set Θt, 4 the process (ηt, ζt) ∈ Θt generates a probability measure Qη,ζ on ( , words,

dQ , dP

= mt , , for each t .

) that is equivalent to P. In other

(4)

t

By using the Girsanov Theorem, we have a new Brownian motion in the probability measure Q , η,ζ

dBtQ

,

=

t dt

(5)

+ dBt ,

and the Poisson process Nt in Q Q , t

η,ζ

have a new jump intensity, (6)

= (1 + t ) .

Thus, we can define the corresponding set of priors as

= {Q , : ( t , t ) For each measure Q

t

and Q

,

(7)

given in (4)}.

, there exists a utility VtQ uniquely solving

2

Epstein and Ji (2014) study ambiguity on volatility. In order to isolate the random shocks in the consumption growth rate, volatility and jump intensity, and concentrate on the ambiguity on both drift and jumps, following Barro (2006) and Branger et al. (2016), we assume that the consumption dynamics follow a parsimonious jump-diffusion process. Our model could be extended to more sophisticated models, e.g., Bansal and Yaron (2004); Bansal et al. (2012); Drechsler (2013) and others, which allow us to further explore the ambiguity on the long-run risks or other random shocks. We leave these tasks to future works. 4 For the technical regularity condition, see Chen and Epstein (2002). 3

2

Finance Research Letters xxx (xxxx) xxxx

X. Ruan and J.E. Zhang T

VtQ = E Q

t

f (Cs, V sQ ) ds|

,

t

0

t

T.

(8)

With the multiple priors, the defining utility is the lower envelope,

Vt = min VtQ,

0

Q

t

T.

(9)

To illustrate, we also give the utility in the measure P, T

VtP = E

f (Cs, VsP ) ds|

t

which can be rewritten as

dVtP =

t

,

0

t

VtP ,

as the solution to the integral equation

T,

(10)

5

f (Ct , VtP ) dt +

P t dBt

P t [dNt

+

(11)

dt ],

where and can be endogenously solved. By using the Girsanov Theorem, VtQ follows P t

P t

dVtQ =

f (Ct , VtQ ) dt +

Q t t dt

Q t t

dt +

Q t · dBt

Q t [dNt

+

(12)

dt ].

According to Chen and Epstein (2002), we assume

=

( t , t ) sup | t | < a , sup 0 t T

1<

t


t(

t

)) dt +

0 t T

(13)

and then

dVt =

f (Ct , Vt ) + max ( ( t, t)

t t

+

t · dBt

+

t [dNt

dt ]. (14)

Without ambiguity, following Duffie and Epstein (1992a,1992b), the state-price process (or intertemporal marginal rate of substitution, IMRS) is t

= exp

t

fV (Cs, Vs ) ds fC (Ct , Vt ).

0

(15)

With ambiguity, by using the Girsanov Theorem, the state-price process changes t

=

t mt

*, *

t

= exp

0

fV (Cs, Vs ) ds fC (Ct , Vt ) mt

*, *

,

(16)

where t* and t* are the optimal solutions in (14). 2.3. Representative investor In this paper, we assume that the aggregator of the representative investor is

f (C , J ) =

C1 (1

1

1

)((1

)J )

1

J,

1

(17)

where the subjective time discount factor is denoted by β; ψ is the elasticity of intertemporal substitution (EIS) and γ is the coefficient 1) . Recursive utility allows us to separate the effects of the RRA )/(1 of the relative risk aversion (RRA). In addition, = (1 and the EIS. Without ambiguity, according to Benzoni et al. (2011) and Branger et al. (2016), the representative investor maximizing the recursive utility leads to the following Bellman equation,

0 = f (Ct , Jt ) +

(18)

Jt ,

where the infinitesimal generator

Jt = 1

5

1

µ J

1 2

1

Jt follows via Itô’s Lemma, 1

2

J + [(1 + L)1

1].

It is implied by the Martingale Representation Theorem. 3

(19)

Finance Research Letters xxx (xxxx) xxxx

X. Ruan and J.E. Zhang

We conjecture for the functional form of the value function J as

Ct1 1

J=

e x,

(20) 6

where x is the log wealth-consumption ratio. Plugging Eq. (20) into the Bellman Eq. (18) yields to

e

x

=

1

1

µ+

1 2

1

1

1

2

[(1 + L)1

1].

(21)

Given the value function J in (20), the state-price process as defined in Eq. (15), therefore, can be solved as t

=

t+(

Ct e

1)(e x t + x ).

(22)

Applying Itô’s Lemma,

d

t

=

µ dt

dBt + [(1 + L)

t

1][dNt

dt ],

(23)

where

µ =

1

+

1 1 (1 + ) 2

µ

2

(1 + L)

1+

1

((1 + L)1

1) .

(24)

Following Jeong et al. (2015), the density generator in our paper is chosen as (a, b) so that

dmt = mt

adBt + b [dNt

dt ],

(25)

where a > 0 and b > 1 measure the ambiguity on drift and jumps in the driving process. Then, based on Eq. (16), with ambiguity, the state-price process t = t mt via Itô’s Lemma can be rewritten as

d

t t

=

rf dt

(

+ a) dBt + [(1 + L) (1 + b)

1][dNt

dt ],

(26)

where the negative expected growth rate of the state-price process is the risk free rate (e.g., rf = 2011), i.e.,

rf =

1

+

1 1 (1 + ) 2

µ

r drift >0 f

2

(1 + L)

b [(1 + L)

1

((1 + L)1

1)

r fjump< 0

r diff f <0

a

1+

E [d t / t ]/ dt , see Benzoni et al.,

1] .

r fjump amb< 0

amb r drift <0 f

(27)

From Eq. (27), the risk-free rate can be decomposed into six components. First two positive components are the subjective time preference rate, β, and the expected growth rate of consumption scaled by the inverse of the EIS, r fdiff . The rest of four components captures precautionary savings terms for volatility risk, jump risk, ambiguity risks on drift and jumps. Higher the RRA, higher the precautionary savings term for volatility risk. Similarly, higher the ambiguity aversion on drift and jumps, higher the precautionary savings terms for ambiguity risks on drift and jumps. Thus, the ambiguity aversion has same impacts on the risk-free rate with the RRA. Obviously, higher the jump risks (i.e., more negative jump size L or higher jump intensity λ), higher the precautionary savings terms for jumps. For γ > 1, ψ > 0, L < 0, a > 0, b > 0, we have r fdiff , r fdrift amb, r fjump amb < 0 . In addition, if the investor prefers to late resolution

of uncertainty, that is 1 < γ < 1/ψ or 0 < θ < 1, the precautionary savings term for jumps r fjump is always negative. The negative

r fdiff , r fdrift amb, r fjump amb and r fjump offset the positive contributions from β and r fdiff so that the volatility, jump and ambiguity risks lower the risk-free rate. 2.4. Price-dividend ratio Following Bansal and Yaron, (2004); Wachter (2013) and Branger et al. (2016), we assume that the dividend follows

dDt = µdt + Dt

dBt + [(1 + L)

1] dNt ,

(28)

where ϕ > 1 is the leverage parameter. 6

Please note that given the value function J in (20), the aggregator function can be simplified as f (C , J ) = (e 4

x

) J.

Finance Research Letters xxx (xxxx) xxxx

X. Ruan and J.E. Zhang

As the stock price is a claim to aggregate dividends D(t), it can be obtained by the standard discounted cash-flow formula, T

St = Et

s

t

t

which implies that St

1 dSt Et dt St

Ds ds , t

+

Dt St

+

t 0

(29) s Ds ds

is a martingale. Applying Itô’s Lemma, we get

1 d t dSt Et . dt t St

rf =

(30)

If we guess St = e yDt , where y is the log price-dividend ratio and is constant, then dSt /St = dDt / Dt . Now we substitute St = e yDt , dSt /St = dDt / Dt given in (28) and dπt/πt given in (26) into Eq. (30) and then simplify it as

µ+e

y

rf

dSt D = 1 Et + t St St dt

=

(

+ a)

[(1 + L) (1 + b)

1][(1 + L)

1] .

d t dSt = 1 Et t St dt

rf

(31)

Using the above equation, the log price-dividend ratio y could be solved and it is a constant as we expect. The equity premium (EP) is defined as the excess return on the stock, i.e.,

EP =

1 dSt Et dt St

which also equals

EP = (

1 E dt t

+ a) 2

=

+

Dt St

rf ,

d t dSt t St

(32)

based Eq. (30). So the EP is the same as the right side of Eq. (31), i.e.,

[(1 + L) (1 + b) [(1 + L)

EP diff > 0

1][(1 + L)

log(EP + rf

1] +

1] a

b (1 + L) [(1 + L)

EP diff amb> 0

EP jump> 0

Eq. (31) can be further simplified as µ + e

y=

1][(1 + L)

y

1] .

(33)

EP jump amb> 0

rf = EP, so that the constant log price-dividend ratio can be rewritten as (34)

µ).

As the log price-dividend ratio is constant, the EIS does not affect the EP in Eq. (33). The equity premium can be decomposed into four components, EPdiff, EPjump, EP diff amb and EP jump amb . EPdiff depends on the exposure of the price to diffusive risk and the market price of risk and EPjump depends on the jump risks (i.e., jump size and jump intensity). EP diff amb and EP jump amb are from the ambiguity on drift and jumps. Given L < 0 and a, b > 0, we find all components have positive premiums. Combining with the risk-free rate in (27), we find that higher the risk aversion, higher the EP and lower the risk-free rate. Meanwhile, higher the ambiguity aversion (i.e., a or b), higher the EP and lower the risk-free rate. The ambiguity aversion complements the RRA to explain the equity premium and risk-free rate puzzles. 3. Equity premium and risk-free rate 3.1. No jump In this section, we assume that there is no jump in (1), that is L = 0 and = 0 . In order to calibrate our the parameters of consumption dynamics, we use the consumption and dividend data from 1930–2007, which are used in Drechsler (2013). We borrow their statistics summary in Table 1. As E[ΔC/C] and E[ΔD/D] are very close, we assume µ = 0.0188. The volatility of growth rate of consumption is = 0.0221. Following Drechsler (2013), we set the leverage parameter = 3 and the time preference parameter = ln(0.999). As there is no jump, the ambiguity on jumps is zero, that is b = 0 . Suggested by Mehra and Prescott (1985), the RRA should not be larger than 10. We choose = 2, 5 and 10. Furthermore, we set the EIS in three cases, that is > ,=, < 1/ . In (33), the EIS does not affect the EP, thus, in order to fit the EP = 0.0541, we adjust the RRA and then get = 37 . In order to get small risk-free rate in (27), we choose = 0.6 . Plugging all parameters in (34), we get y = 3.13 which is close to the market data E [ln(S / D)] = 3.15. Based on a geometric Bownian motion for the consumption with Epstein-Zin recursive preference without ambiguity, we have to choose = 37 > 10 to explain the equity premium puzzle. With ambiguity, there are more premiums in the EP in (33), meanwhile, the risk-free rate will be lower in (27). We adjust a as a measure of the ambiguity on drift to fit the EP and risk-free rate. The results are shown in Table 2. As the EIS does not affect the the EP, as long as we choose γ < 37, without ambiguity, the model are not able to explain the equity Table 1 Statistics summary in Drechsler (2013). The time period is from 1930–2007. E[ΔC/C]

σ[ΔC/C]

E[ΔD/D]

E[ln (S/D)]

E[EP]

E[rf]

0.0188

0.0221

0.0154

3.15

0.0541

0.0082

5

Finance Research Letters xxx (xxxx) xxxx

X. Ruan and J.E. Zhang

Table 2 Equity premium and risk-free rate with ambiguity. ψ

=2 0.4 0.4 0.5 0.5 0.6 0.6 =5 0.15 0.15 0.2 0.2 0.25 0.25 = 10 0.08 0.08 0.1 0.1 0.2 0.2

a

rf = 0.0082

EP = 0.0541

y = 3.15

0 0.86 0 0.66 0 0.52

0.0463 0.0083 0.0371 0.0080 0.0310 0.0080

0.0029 0.0599 0.0029 0.0467 0.0029 0.0374

3.4926 3.0072 3.8507 3.3284 4.1890 3.6248

0 0.98 0 0.72 0 0.56

0.1170 0.0087 0.0877 0.0081 0.0701 0.0082

0.0073 0.0723 0.0073 0.0551 0.0073 0.0445

2.2491 2.7777 2.5744 3.1150 2.8367 3.3852

0 0.88 0 0.70 0 0.33

0.2030 0.0086 0.1621 0.0074 0.0803 0.0074

0.0147 0.0730 0.0147 0.0611 0.0147 0.0365

1.6150 2.7686 1.8452 3.0017 2.5744 3.6829

Table 3 Parameters of consumption dynamics from Branger et al. (2016). The calibration is based on yearly consumption data from 1971–2009 that is available on Robert Barro’s webpage, http://scholar. harvard.edu/barro/publications/barro-ursua-macroeconomic-data. All parameters are usually calibrated by matching the unconditional moments of annual consumption growth rates. μ

σ

L

λ

0.033

0.04

−0.05

0.28

premium puzzle. However, we calibrate the ambiguity parameter 0 < a < 1 so that the model can well explain the high EP and the low risk-free rate. Our calibration of the ambiguity parameter a is similar to Jeong et al. (2015). More interestingly, with ambiguity, the CRRA utility preference (i.e., = 1/ ) can well explain the equity premium puzzle. 3.2. With jumps In this subsection, we study the ambiguity on both drift and jumps. In order to calibrate the parameters of consumption dynamics in (1), we use the yearly consumption data from 1971–2009 that is available on Robert Barro’s webpage. Branger et al. (2016) calibrate the parameters by matching the unconditional moments of annual consumption growth rates in Table 3. We borrow their calibration in this subsection and use same = 0.03, = 2 and = 2 in Branger et al. (2016). From the evidence in the above subsection and Jeong et al. (2015), the ambiguity aversion can reduce the estimated RRA. To focus on the ambiguity, we set = = 2 .7 Then the EP and the risk-free rate with/without ambiguity are shown in Table 4. Panel A, Table 4 shows that the model with a low RRA generates a high risk-free rate 3.6% and a low EP 0.93%. With ambiguity on uncertainty, the ambiguity premium increases the EP and the ambiguity aversion offsets the subjective time preference rate, β, and the expected growth rate of consumption scaled by the inverse of the EIS, r fdiff . In our special setting of = = 2,

EP drift amb = r fdrift amb and EP jump amb = r fjump amb in Panel B, Table 4. It is intuitive to show how the ambiguity on uncertainty affects the EP and risk-free rate. For example, in the case a = 0, b = 0.9, due to the ambiguity on jumps, there is ambiguity premium EP jump amb = 2.72% increasing from EP = 0.93% into EP = 3.66% . At the same time, there is a precautionary saving term for ambiguity on jumps, which reduces the risk-free rate into 0.87%. Because of the ambiguity aversion for uncertainty, which is similar to the risk aversion for uncertainty, it adds a ambiguity premium in the EP and a precautionary saving term in the risk-free rate. According to the Table 4, the importance of ambiguity on drift is more important than that of jump component, unless the ambiguity bound for the jump is really big (b = 0.9 ). This suggests that ambiguity on drift (a ≈ 0.2 ~ 0.5) matters more to explain the basic asset pricing moments. This is consistent with the empirical estimates of Jeong et al. (2015), and this result shows a simple comparison with the jump risk component.8 To summarize, corresponding to Jeong et al. (2015), the ambiguity on uncertainty is a complementary 7 8

This setting can be relaxed, but this would not add additional insights concerning our focus. We thank an anonymous referee for advising this insightful suggestion. 6

Finance Research Letters xxx (xxxx) xxxx

X. Ruan and J.E. Zhang

Table 4 Equity premium and risk-free rate with jumps. Panel A a = 0, b = 0

Panel B a = 0.4, b = 0

a = 0, b = 0.9

rf

β

0.0360 EP 0.0093

0.03

rf

rf without ambiguity

0.0040 EP

0.0360 EP without ambiguity

0.0413 rf 0.0087 EP

a = b = 0.25

0.0366 rf 0.0084 EP 0.0369

r drift f

0.0165

0.0093 rf without ambiguity

0.0360 EP without ambiguity 0.0093 rf without ambiguity

0.0360 EP without ambiguity 0.0093

r diff f

r fjump

−0.0024 EPdiff 0.0064

−0.0081 EPjump 0.0029

r drift f

r fjump

amb

amb

−0.0320

0

EP drift amb 0.0320

EP jump 0

r drift f

r fjump

amb

0

amb

−0.0272

EP drift 0

r drift f

amb

amb

amb

EP jump 0.0272

r fjump

amb

amb

−0.0200

−0.0076

EP drift amb 0.0200

EP jump 0.0076

amb

aversion for the risk aversion in terms of explaining the equity premium puzzle. 3.3. Sensitivity checks In this paper, we assume that the price-dividend ratio is a constant. This makes the role of the elasticity of intertemproal substitution smaller. In order to investigate the impact of random shocks in the expected growth rate (μ) and volatility (σ) of consumption, we consider sensitivity checks for our results in this subsection. As an example, we focus on the case with jumps, in which the ambiguity aversion coefficients a = b = 0.25 and then rf = 0.0084 and EP = 0.0369. Fig. 1 shows the equity premium and risk-free rate against percentage changes in the expected growth rate (μ) and volatility (σ) of consumption. The left figure plots EP and rf against percentage changes in μ varying from 10% to 10%. According to Eq. (33), μ does not affect EP, so we find a flat EP in the left figure. rf slightly increases with μ and the slope of rf is 1/ψ given in Eq. (27). The risk-free rate varying from 0.0067 to 0.0100 is still close to the benchmark (rf = 0.0084 ). The right figure shows sensitivity checks on σ. Consistent with Eqs. (27) and (33), higher the volatility of consumption, higher the equity premium and lower the risk-free rate. This leads to EP slightly varying from 0.0337 to 0.0403 and rf varying from 0.0059 to 0.0108. Overall, Fig. 1 suggests that our results are not significantly affected by adding a small random shock to μ or σ. 4. Conclusion This paper extends the ambiguity in Chen and Epstein (2002) and provides a tractable way to show the impacts of the ambiguity

Fig. 1. Sensitivity checks. The figure shows the equity premium and risk-free rate against percentage changes in the expected growth rate (μ) and volatility (σ) of consumption in the case with jumps. 7

Finance Research Letters xxx (xxxx) xxxx

X. Ruan and J.E. Zhang

on the risk-free rate and the EP. Two calibrations suggest that the model with ambiguity on uncertainty can well explain the equity premium and risk-free rate puzzles constricted to a low RRA. It is true for both Epstein-Zin recursive preference and CRRA utility with ambiguity. We support that the ambiguity aversion plays a key role to explain these puzzles. Similarly to Drechsler (2013), our model can be extended with long-run risks assumptions to explain more asset pricing puzzles, e.g., variance risk premium puzzle. CRediT authorship contribution statement Xinfeng Ruan: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - original draft, Writing review & editing, Visualization. Jin E. Zhang: Supervision. References Bansal, R., Kiku, D., Yaron, A., 2012. An empirical evaluation of the long-run risks model for asset prices. Crit. Finance Rev. 1 (1), 183–221. Bansal, R., Yaron, A., 2004. Risks for the long run: a potential resolution of asset pricing puzzles. J. Finance 59 (4), 1481–1509. Barro, R.J., 2006. Rare disasters and asset markets in the twentieth century. Q. J. Econ. 121 (3), 823–866. Benzoni, L., Collin-Dufresne, P., Goldstein, R.S., 2011. Explaining asset pricing puzzles associated with the 1987 market crash. J. Financ. Econ. 101 (3), 552–573. Branger, N., Kraft, H., Meinerding, C., 2016. The dynamics of crises and the equity premium. Rev. Financ. Stud. 29 (1), 232–270. Campbell, J.Y., Cochrane, J.H., 1999. By force of habit: a consumption-based explanation of aggregate stock market behavior. J. Political Econ. 107 (2), 205–251. Chen, Z., Epstein, L., 2002. Ambiguity, risk, and asset returns in continuous time. Econometrica 70 (4), 1403–1443. Constantinides, G.M., 1990. Habit formation: a resolution of the equity premium puzzle. J. Political Econ. 98 (3), 519–543. Drechsler, I., 2013. Uncertainty, time-varying fear, and asset prices. J. Finance 68 (5), 1843–1889. Duffie, D., Epstein, L.G., 1992a. Asset pricing with stochastic differential utility. Rev. Financ. Stud. 5 (3), 411–436. Duffie, D., Epstein, L.G., 1992b. Stochastic differential utility. Econometrica 60 (2), 353–394. Epstein, L.G., Ji, S., 2014. Ambiguous volatility, possibility and utility in continuous time. J. Math. Econ. 50, 269–282. Epstein, L.G., Schneider, M., et al., 2010. Ambiguity and asset markets. Annu. Rev. Financ. Econ. 2 (1), 315–346. Epstein, L.G., Zin, S.E., 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica 57 (4), 937–969. Jahan-Parvar, M.R., Liu, H., 2014. Ambiguity aversion and asset prices in production economies. Rev. Financ. Stud. 27 (10), 3060–3097. Jeong, D., Kim, H., Park, J.Y., 2015. Does ambiguity matter? Estimating asset pricing models with a multiple-priors recursive utility. J. Financ. Econ. 115 (2), 361–382. Ju, N., Miao, J., 2012. Ambiguity, learning, and asset returns. Econometrica 80 (2), 559–591. Mehra, R., 2011. Handbook of the Equity Risk Premium. Elsevier. Mehra, R., Prescott, E.C., 1985. The equity premium: A puzzle. J. Monet. Econ. 15 (2), 145–161. Rietz, T.A., 1988. The equity risk premium a solution. J. Monet. Econ. 22 (1), 117–131. Wachter, J.A., 2013. Can time-varying risk of rare disasters explain aggregate stock market volatility? J. Finance 68 (3), 987–1035. Weil, P., 1989. The equity premium puzzle and the risk-free rate puzzle. J. Monet. Econ. 24 (3), 401–421.

8