Time‐varying risk of rare disasters, investment, and asset pricing

• Published date: 01 August 2020
• DOI: http://doi.org/10.1111/fire.12226
• Author: Jinqiang Yang,Bo Liu,Yingjie Niu,Zhentao Zou
• Date: 01 August 2020

DOI: 10.1111/fire.12226

ORIGINAL ARTICLE

Time-varying risk of rare disasters, investment,

and asset pricing

Bo Liu1Yingjie Niu2Jinqiang Yang3Zhentao Zou4

1School of Management and Economics,

University of Electronic Science and Technology

of China, Chengdu, China

2School of Finance, Shanghai University of

Finance and Economics, Shanghai, China

3Shanghai KeyLaboratory of Financial

Information Technology,School of Finance,

Shanghai University of Financeand Economics,

Shanghai Institute of International Finance and

Economics, Shanghai, China

4Economics and Management School, Wuhan

University, Wuhan, China

Correspondence

BoLiu, School of Management and Eco-

nomics,University of Electronic Science

andTechnology of China, Chengdu, China.

Email:Liub@uestc.edu.cn

Fundinginformation

NationalNatural Science Foundation of China,

Grant/AwardNumber: 71573033; Tianfu Ten

ThousandPlan of Sichuan Province; Innova-

tiveResearch Team of Shanghai Universityof

Financeand Economics, Grant/Award Number:

2016110241;National Natural Science Founda-

tionof China, Grant/Award Numbers: 71972122,

71772112;Fundamental Research Funds for

theCentral Universities, Grant/Award Number:

413000172

Abstract

We extend an equilibrium business cycle/assetpricing model of pro-

duction and capital accumulation by introducing a time-varying risk

of rare disasters. It predicts that investment is much more volatile

than output, which provides theoretical support for the empirical

data. Furthermore, the model-generated stationary distribution of

the investment-output ratio fits the data remarkably well. Both of

them exhibit negative skewness, which means that there is a small

probability that this ratio can be very low.Given the observations of

the investment-output ratio, we obtain the values of the jump inten-

sity implicit in the historical data and find those recession periods

coincide with a rapid increase in the probability of a disaster.Finally,

the model shows that the existence of adjustment costs generates

a procyclical price of capital and contributes to resolving the equity

premium puzzle.

KEYWORDS

implied disaster probabilities, investment-output ratio, stationary

distribution, time-varying disaster risk

JEL CLASSIFICATIONS

D81, E22, G30

1INTRODUCTION

Toinvestigate the possible impact of catastrophic events, previous studies (Barro, 2009; Pindyck & Wang, 2013) con-

sider a general equilibrium model of production with rare events. In their models, the mean arrival rate of discrete

downward jumps to the capital stock is constant. However, Berkman, Jacobsen, and Lee (2011) create a crisis index

from real data and find that it shows substantial variation over time. Motivated by this empirical result, our paper pro-

posesa modification by assuming that the probability of a disaster is stochastic and varies over time. Following Wachter

Financial Review.2020;55:503–524. wileyonlinelibrary.com/journal/fire c

2020 The Eastern Finance Association 503

504 LIU ET AL.

(2013), we use the recursive utility function developed by Duffie and Epstein (1992). For simplicity, we assume that

there are no adjustment costs in the baseline model. Then we solve our model in semiclosed form. For the special case

with a unit elasticity of intertemporal substitution (EIS), we obtain the closed-form solutions.

We find that the investment-capital ratio is decreasing in the jump intensity as long as the EIS is higher than one.

Intuitively,as the capital stock faces more severe discrete downward shocks, people prefer to consume output rather

than invest in the capital stock. In our model, due to the time-varying risk of a rare disaster,the optimal investment-

capital ratio is no longer constant. Therefore, a time-varying risk of disaster implies that investment is much more

volatile than output, as is observed in the data. Next, we find that the model-generated stationary distribution fits

the data remarkably well. Both of them exhibit negative skewness, which means that there is a small probability that

the investment-capital ratio can be very low.Furthermore, given observations of the investment-output ratio, we can

invertthis function to find the values of the jump intensity implicit in the historical data. We find that recession periods

coincide with a rapid increase in the probability of a disaster.

In the baseline model, as there are no adjustment costs, Tobin’sqalways equals 1. Togenerate a time-varying Tobin’s

q, we extendthe baseline model and discuss the role of adjustment costs. Since the investment-capital ratiois decreas-

ing in the jump intensity, Tobin’s qis closer to 1 for high jump intensities. In addition, we find that higher adjustment

costs make Tobin’sqmore sensitive to the jump intensity. Therefore, higher adjustment costs increase Tobin’s q,on

average,and make it more volatile.

Finally, we find that the existenceof adjustment costs contributes to resolving the equity premium puzzle. Specif-

ically, the equity risk premium contains three parts. The first term is the equity premium in diffusion models, and the

second term arises from the static jump risk in the capital stock. The last term is the risk premium due to time variation

in disaster risk, which is nonzero when there exist adjustment costs. Quantitatively,we find the last term is significant

in the magnitude.

Our paper is related to the literature that introduces disaster risk into standard equilibrium asset pricing models.

Rietz (1988) represents the first attempt to introduce low-probability economic disasters to resolve the equity risk

premium puzzle (Mehra & Prescott, 1985). Barro (2006) calibrates disaster probabilities based on global twentieth-

century history. Barro (2009) extends Barro (2006) by considering Epstein–Weil–Zin preferences and assesses the

welfare cost of consumption uncertainty. Pindyck and Wang (2013)provide a framework for policy analysis using a

general equilibrium model of production with disaster risk.

Several recent studies extend the assumption of constant jump intensity by considering a time-varying risk of a

rare disaster. Chen, Joslin, and Tran (2012) study an endowment economy with two types of agents who disagree

about the dynamics of a time-varying disaster risk. Gabaix (2012) incorporates a time-varying severity of disasters

into the hypothesis proposed by Barro (2006). Gourio (2012) studies a real business cycle model with time-varying

rare-disaster risk by solving the model numerically in discrete time. Du (2013) presents a consumption-based gen-

eral equilibrium model for valuing foreign exchangecontingent claims with a time-varying risk of rare events. Wachter

(2013) explains aggregate stock marketvolatility based on a time-varying probability of a consumption disaster. Farhi

and Gabaix (2016) propose a new model of exchange rates by assuming that the probability of world disasters and

each country’s exposure to these events are time varying. Tsaiand Wachter (2016) consider a time-varying risk of a

rare disaster in a multisector endowment economy. Kilic and Wachter(2018) offer a rare event–based explanation

for labor marketvolatility. Seo and Wachter (2018) investigate whether a model with a time-varying probability of eco-

nomic disaster can explainthe prices of collateralized debt obligations. Seo and Wachter (2019) reconcile option prices

with macroeconomic data on disaster by incorporating time-varying disaster risk. In contrast, our model incorporates

a time-varying risk of rare disasters into a generalequilibrium asset pricing model of production and capital accumula-

tion.

The remainder of the paper is organized as follows. Section 2 describes the baseline model with no adjustment costs,

which includes preferences, production, and investment. Model solutions are derived in Section 3. In Section 4, we

present the quantitative results and empirical analysis. In Section 5, we extendthe baseline model and discuss the role

of adjustment costs. Finally,Section 6 concludes the paper and proposes directions for future research.