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Disaster Risk and Business Cycles

Francois Gourio

2010
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Social Science Research Network
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This paper proposes a tractable business cycle model with large, volatile, and countercyclical risk premia. Risk premia are driven by a small, exogenously time-varying risk of economic disaster, and macroeconomic aggregates respond to this time-varying risk. The model is consistent with the second moments of quantities, of asset returns, and matches well the relations between quantities and asset prices. An increase in the probability of disaster leads to a collapse of investment and a
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... ment and a recession, with no current or future change in productivity. Demand for precautionary savings increases, leading yields on safe assets to fall, while spreads on risky securities increase. To assess the empirical validity of the model, I infer the probability of disaster from observed asset prices and feed it into the model. The variation over time in this probability appears to account for a signi...cant fraction of business cycle dynamics, especially sharp downturns in investment and output such as the last quarter of 2008. This is consistent with the then-widespread fear of a repeat of the Great Depression. for helpful discussions. This paper was ...rst circulated in February 2009 under the title "Time-varying risk premia, time-varying risk of disaster, and macroeconomic dynamics." 1 The empirical ...nance literature has provided substantial evidence that risk premia are time-varying (see for instance Campbell and Shiller (1988) , Fama and French (1989) , Ferson and Harvey (1991), Cochrane (2005)). Yet, standard business cycle models such as the real business cycle model, or the DSGE models used for monetary policy analysis, largely fail to replicate the level, the volatility, and the cyclicality of risk premia. This seems an important neglect, since empirical work suggests a tight connection between risk premia and economic activity. For instance, Philippon (2008) and Gilchrist and Zakrajsek (2007) show that corporate bonds spreads are highly correlated with real physical investment, both in the time series and in the cross-section. A large research, summarized in Backus, Routledge and Zin (2008) , shows that the stock market, the term premium, and (negatively) the short rate all lead the business cycle. I introduce time-varying risk premia in a standard real business cycle model, through a small, stochastically time-varying risk of economic "disaster", following the work of Rietz (1988) , Barro (2006), and Gabaix (2007) . Existing work has so far been con...ned to endowment economies, and hence does not consider the feedback from time-varying risk premia to macroeconomic activity. I prove two theoretical results, which hold under the assumption that a disaster reduces total factor productivity (TFP) and the capital stock by the same amount. First, when the risk of disaster is constant, the path for macroeconomic quantities implied by the model is the same as that implied by a model with no disasters, but a di¤erent discount factor . This "observational equivalence" (in a sample without disasters) is similar to Tallarini (2000) : macroeconomic dynamics are essentially una¤ected by the amount of risk or the degree of risk aversion. Second, when the risk of disaster is time-varying, an increase in probability of disaster is observationally equivalent to a preference shock. This is interesting since these shocks appear to be important in accounting for the data, according to estimation of DSGE models with multiple shocks such as Smets and Wouters (2003) . An increase in the perceived probability of disaster can create a collapse of investment and a recession, as risk premia rise, increasing the cost of capital. Demand for precautionary savings increase, leading the yield on less risky assets to fall, while spreads on risky securities increase. These business cycle dynamics occur with no change in current or future total factor productivity. Quantitatively, I ...nd that this parsimonious model can match many asset pricing facts -the mean, volatility, and predictability of returns -while maintaining the basic success of the RBC model in accounting for quantities. This is important since many asset pricing models which are successful in endowment economies do not generalize well to production economies (as explained in Jermann (1998), Lettau and Uhlig (2001), Kaltenbrunner and Lochstoer (2008)). This second shock also substantially increases the correlation between asset prices, or risk premia, and economic activity, making it closer to the data. One obvious limitation of the paper is that the probability of disaster is hard to observe. As an empirical exercise, I infer the probability of disaster from asset prices. I then feed into the model this estimated probability of disaster. The variation over time in this probability appears to account for a signi...cant fraction of business cycle dynamics, and it especially helps matching sharp downturns. 2 This risk of an economic disaster could be a strictly rational expectation, or more generally it could re ‡ect a time-varying belief, which may di¤er from the objective probability -i.e., waves of optimism or pessimism (see e.g. Jouini and Napp (2008) ). For instance, during the recent ...nancial crisis, many commentators, including well-known macroeconomists 1 , have highlighted the possibility that the U.S. economy could fall into another Great Depression. My model studies the macroeconomic e¤ect of such time-varying beliefs. 2 This simple modeling device captures the idea that aggregate uncertainty is sometimes high, i.e. people sometimes worry about the possibility of a deep recession. It also captures the idea that there are some asset price changes which are not obviously related to current or future TFP, i.e. "bubbles", "animal spirits", and which in turn a¤ect the macroeconomy. Introducing time-varying risk premia requires solving a model using nonlinear methods, i.e. going beyond the ...rst-order approximation and considering "higher order terms". Researchers disagree on the importance of these higher order terms, and a fairly common view is that they are irrelevant for macroeconomic quantities. Lucas (2003) summarizes: "Tallarini uses preferences of the Epstein-Zin type, with an intertemporal substitution elasticity of one, to construct a real business cycle model of the U.S. economy. He ...nds an astonishing separation of quantity and asset price determination: The behavior of aggregate quantities depends hardly at all on attitudes toward risk, so the coe¢ cient of risk aversion is left free to account for the equity premium perfectly." 3 My results show, however, that these higher-order terms can have a signi...cant e¤ect on macroeconomic dynamics, when we consider shocks to the probability of disaster. 4 Substantively, the results of this paper suggests that the channel through which variation in asset prices a¤ect investment and output is the cost-of-capital: the model matches the data well, provided that variation in risk premia are fed into the model. Beside this substantive contribution, the paper provides a tractable framework that incorporates volatile, countercyclical risk premia in a standard macroeconomic model. This framework lends itself naturally to useful extensions, such as incorporating 1 Greg Mankiw (NYT, Oct 25, 2008): "Looking back at [the great Depression], it's hard to avoid seeing parallels to the current situation. (...) Like Mr. Blanchard at the I.M.F., I am not predicting another Great Depression. But you should take that economic forecast, like all others, with more than a single grain of salt." Robert Barro (WSJ, March 4, 2009): "... there is ample reason to worry about slipping into a depression. There is a roughly one-in-...ve chance that U.S. GDP and consumption will fall by 10% or more, something not seen since the early 1930s." Paul Krugman (NYT, Jan 4, 2009): "This looks an awful lot like the beginning of a second Great Depression." 2 Of course in reality this change in probability of disaster may be an endogenous variable and not an exogenous shock. But it is useful to understand the e¤ect of an increase in aggregate risk (premia) on the macroeconomy. 3 Note that Tallarini (2000) actually picks the risk aversion coe¢ cient to match the Sharpe ratio of equity. Since return volatility is very low in his model -there are no capital adjustment costs -he misses the equity premium by several order of magnitudes. 4 Cochrane (2005, p. 296-297) also discusses in detail the Tallarini (2000) result: "Tallarini explores a di¤erent possibility, one that I think we should keep in mind; that maybe the divorce between real business cycle macroeconomics and ...nance isn't that short-sighted after all (at least leaving out welfare questions, in which case models with identical dynamics can make wildly di¤erent predictions). (...) The Epstein-Zin preferences allow him to raise risk aversion while keeping intertemporal substitution constant. As he does so, he is better able to account for the market price of risk (...) but the quantity dynamics remain almost unchanged. In Tallarini's world, macroeconomists might well not have noticed the need for large risk aversion." where x t+1 is a binomial variable which is 1 with probability p t and 0 with probability 1 p t : A disaster does not a¤ect productivity A. I will relax this assumption in section 3. 8 Finally, I assume that the 6 It is easy to extend this example to the case where A is stochastic; this does not a¤ect the results, so in the interest of simplicity I omit this extension in the example and consider it in the full model of section 3. 7 In a large downturn, the demand for some luxury goods such as boats, private airplanes, etc. would likely fall sharply. If this situation were to last, the boats-producing factories would never operate at capacity. 8 In an AK model, a permanent reduction in productivity would lead to a permanent reduction in the growth rate of the economy, since permanent shocks to A a¤ect the growth rate of output permanently. > 1: Hence, i is increasing in p if > 1; it is decreasing in p if < 1; and it is independent of p if = 1: 9 Note, however, that the return on capital will not be volatile -in this example, it is constant in a sample without disasters. Adding leverage can create substantial volatility. 6 The intuition for this result is as follows: if p goes up, investment in physical capital becomes more risky and hence less attractive, i.e. the risk-adjusted return goes down. 10 The e¤ect of a change in the return on the consumption-savings choice depends on the value of the IES, because of o¤setting wealth and substitution e¤ects. If the IES is unity (i.e. utility is log), savings are unchanged and thus the savings or investment rate does not respond to a change in the probability of disaster. But if the IES is larger than unity, i.e. < 1, the substitution e¤ect dominates, and i is decreasing in p. Hence, an increase in the probability of disaster leads initially, in this model, to a decrease in investment, and an increase in consumption, since output is unchanged on impact. Next period, the decrease in investment leads to a decrease in the capital stock and hence in output. This simple analytical example thus shows that a change in the perceived probability of disaster can lead to a decline in investment and output.

doi:10.2139/ssrn.1573097
fatcat:du7eko4xsjegffxktgzsmenjce