International Asset Pricing with Risk-Sensitive Agents

Transcription

1 International Asset Pricing with Risk-Sensitive Agents Riccardo Colacito Mariano M. Croce Abstract We propose a frictionless general equilibrium model in which two international consumers with recursive preferences trade two consumption goods and a complete set of date- and state-contingent securities. Consumption home bias and concern for the temporal distribution of risk generate rich dynamics for international prices and quantities. In our model, exchange rate movements are as volatile as they are in the data. Furthermore, both the volatility of the exchange rate movements and risk premia are endogenously time varying and history dependent. JEL classification: C62; F31; G12. This draft: June 30, Both authors are affiliated with the University of North Carolina at Chapel Hill, Kenan Flagler School of Business. We thank Jonathan Berk, Hui Chen, Diego Garcia, Lorenzo Garlappi, Lars Hansen, Anh Le, Karen Lewis, Deborah Lucas, Hanno Lustig, Martin Schneider, Chris Telmer, Harald Uhlig, Adrien Verderlhan, Pietro Veronesi, and Stan Zin, along with seminar participants at Boston University, Carnegie-Mellon University, the Federal Reserve Bank of Kansas City, MIT (Sloan), New York University, the University of Minnesota (Carlson), UNC-Chapel Hill, University of Pennsylvania (Wharton), the 2010 Western Finance Association Meeting in Victoria, the 2010 Society for Economic Dynamics Meeting in Montreal, the 2010 Econometric Society World Congress in Shanghai, and the 2011 Annual Meeting of the American Economic Association in Denver. Lynn Hand provided excellent copyediting support. All errors remain our own. This research was funded in part by The Jefferson- Pilot Faculty Development Fund.

2 1 Introduction One of the best-established facts in empirical finance is the tendency of volatilities and correlations to change over time. An extensive literature has documented that the volatility of exchange rate movements varies over time (see, for example, Engle (1982), Gagnon (1993), Thursby and Thursby (1987), and McKenzie (1999) for studies considering different frequencies). Numerous studies have also shown that the correlations of international asset returns are far from being constant (Longin and Solnik (1995) and Cappiello, Engle and Sheppard (2006)); that consumption volatility changes over time (Kim and Nelson (1999)); and that equity risk premia are countercyclical (Campbell and Cochrane (1999)). This paper studies an international asset pricing model in which time variation in volatilities and correlations arises endogenously from an optimal risk-sensitive trading arrangement. The economy is populated by two agents with recursive preferences that we denote as home and foreign. Consumers are each endowed with the stochastic supply of one country-specific good. Endowments are i.i.d. and less than perfectly correlated. Preferences display a bias for the consumption of the domestic good. Trade occurs in frictionless goods markets and in financial markets featuring a complete set of state- and date-contingent securities. Since the consumers in the two countries enjoy their respective goods with different intensities and their endowments are imperfectly correlated, a risk-sharing opportunity arises. Our economy is standard except for the fact that the agents have recursive utilities. Our choice to employ recursive preferences is motivated by our interest in the joint dynamics of international quantities and prices. Brandt, Cochrane and Santa- Clara (2006) have shown the inability of time-additive preferences to generate both highly volatile and highly cross-country-correlated stochastic discount factors. The high volatility is required to replicate the equity premia that we observe in a large cross-section of countries (see Hansen and Jagannathan (1991)). The high correla- 1

3 tion is necessary to account for the degree of volatility of real exchange rate movements that we typically observe between major industrialized countries (see Backus, Foresi and Telmer (1996)). Colacito and Croce (2011) show that a framework featuring Epstein and Zin (1989) preferences and exogenously specified homoscedastic consumption processes, which are highly correlated in the long-run, can resolve this joint puzzle. Our paper differs from previous work in that we allow consumption to be an endogenous variable determined in a general equilibrium model in which agents optimally choose how much to save and endowments have a low cross-country correlation at all horizons. Furthermore, Lustig and Verdelhan (2007) and Lustig, Roussanov and Verdelhan (2011a), point out that large common components and time-varying risk in the dynamics of international stochastic discount factors are needed to account for key features of currency risk premia. Our model provides a micro foundation to these findings, as it endogenously produces time-varying volatility and shows that such common components can be related to the distribution of wealth across countries. In this respect, we differ from Stathopoulos (2011) and Verdelhan (2010), whose analyses focus on the international distribution of the processes of habit formation. To study the international asset pricing implications of our model, we first compute efficient allocations by solving a social planner s problem. We document that agents preferences play a crucial role in the implementation of the optimal risk-sharing scheme. With time additive preferences, allocations are history-independent functions of the endowments. With recursive preferences à la Epstein and Zin (1989) and Hansen and Sargent (1995), allocations are history-dependent functions of the endowments. This history dependence can conveniently be summarized by endogenously time-varying Pareto weights, which alter the conditional distribution of consumption across dates and states of the world. In our model, in fact, consumption growth becomes predictable and its volatility is no longer constant over time, even if endowments are i.i.d. and markets complete. 2

4 Since time-varying Pareto weights are a telltale sign of time-varying distribution of international wealth, our model provides a link between the dynamics of a country s savings and the dynamics of the perceived risk of prices and quantities. First of all, we document that in each country the conditional volatility of the stochastic discount factor varies over time with the domestic level of wealth. This relationship is explained by the very specific attitude toward risk associated with the type of agents considered in this paper. Our agents are willing to give up a conspicuous share of consumption when the supply of the good that they care the most about is abundant, in order to obtain a marginal increment of consumption during periods of scarce supply. Given a history of low realizations of the most-preferred good, the affected consumer receives positive transfers from the insurance assets previously accumulated and can afford to demand an increasing amount of insurance for the future, despite a lower level of endowment. At the same time, the other country s wealth decreases, since a conspicuous share of previously purchased insurance assets expires worthless. This second country finds it increasingly costly to provide more insurance on such a history. At the equilibrium, it is willing to provide insurance only at a price that is increasingly higher than the price that it is willing to pay to be insured. As securities prices reflect state- and date-contingent marginal utilities of consumption, a time-varying distribution of prices is equivalent to time-varying volatility of stochastic discount factors and consumption. Moreover, we document that such variation of the volatility is countercyclical in the spirit of Lustig, Roussanov and Verdelhan (2011b) and can be easily explained as mean-variance trade-off. With recursive preferences, indeed, agents care not only about expected future utility, but also about the conditional variance of continuation utilities. When affected by positive shocks to the relative supply of the good that is enjoyed the most, our agents are willing to give up on average more future resources in exchange for insurance assets reducing the conditional volatility of their future utilities. The conditional expected returns of such assets are ultimately countercyclical 3

5 as well. We also show that the cross-country correlation of the continuation utilities is timevarying. This result can be explained as follows. For each realization of the endowments there are two effects on utilities. The first effect (an income effect) is related to the size of the exogenous supply of the two goods: utilities are monotonically increasing in the endowment level. The second effect (a substitution effect), in contrast, is related to the relative supply of the two goods: the risk-sharing mechanism requires that a wealthy country redistribute part of its resources abroad when experiencing a positive shock to the relative supply of its own most-preferred good. Focusing on utilities, the second effect mitigates the first and becomes more intense as the crosscountry distribution of savings spreads out. As a consequence, the international correlation of utilities depends on the time-varying relative intensity of the income and substitution effects. At the equilibrium, the cross-country correlation of utilities is indeed an increasing function of international wealth inequality. Since risk-sensitive agents are risk averse in future utility in addition to future consumption, their continuation utilities are a crucial component of their marginal rates of substitution. This has two consequences in our economy. First of all, the desire to smooth future utilities leads to stochastic discount factors that are more correlated than one-period-ahead consumption growth rates. Equivalently, international risk sharing, measured as the degree of comovement of the intertemporal marginal rates of substitution, can be as high as needed to replicate the volatility of exchange rates fluctuations, despite the modest correlation of consumption across countries. Second, dynamic wealth redistribution allows marginal utilities to be more or less correlated depending on the degree of wealth inequality across countries. By no-arbitrage, the time-varying cross-country correlation of the marginal rates of substitution induces time variation in both exchange rate s volatility and returns correlations. To make the model closer to what we observe in data, we borrow an insight from the 4

6 rare-events literature and postulate a small probability of a large endowment drop in either of the two countries. As shown by Rietz (1988), Barro (2006), Gabaix (2009), and Fahri, Fraiberger, Gabaix, Ranciere and Verdelhan (2009) this element generates the necessary volatility of intertemporal marginal rates of substitution. We show that rare events can amplify the time variation in the market price of risk and other second moments related to asset prices. The benefit of introducing rare events is twofold. First, rare events improve the quantitative performance of our model without affecting the qualitative implications of the optimal risk-sharing scheme. Second, rare disasters allow us to better understand the optimal trading scheme implemented by countries exposed to nonsynchronized depressions. We also show that all our results are confirmed and quantitatively enhanced when we introduce serial correlation in the level of the endowments. Intuitively, rare but severe and persistent drops in output promote even further risk-sensitive risk-sharing. The model delivers a number of qualitative predictions that appear to be consistent with international financial data. Countries with higher international debt tend to have lower consumption volatility. Countries with modest international debt, in contrast, tend to have higher exchange rate volatility. Finally, countries whose stock markets returns are more correlated show lower currency volatility. We check these predictions, focusing on both US data and a wider cross-section of countries, and show that our model is broadly consistent with the data. This paper is not the first one to feature endogenously time-varying Pareto weights. Indeed, it is well known that they naturally arise when markets are not complete. A number of papers have investigated the impact of frictions in international financial markets on international business cycles (see, for example, Baxter and Crucini (1995), Heathcote and Perri (2002), Kehoe and Perri (2002), and Kollmann (1996)). In Pavlova and Rigobon (2007), Pareto weights vary over time thanks to exogenous country-specific demand shocks. Chen, Joslin and Tran (2010a) and Chen, Joslin and Tran (2010b) show that disagreement about the probability of disasters can generate 5

7 a time-varying distribution of wealth in an economy populated by agents with timeadditive preferences. Our work differs from those just mentioned in at least three major dimensions. First, we obtain rich endogenous dynamics in the international distribution of wealth in an environment with no frictions. Second, we explicitly address the relevance of the time-variation of wealth for both quantities and international asset prices. Third, we document that the equilibrium of the economy can be interpreted as one in which agents have a preference for robustness. Following this interpretation, we show that disagreement about the transition probabilities across states arises endogenously as a function of the degree of home bias. Characterizing the dynamics of risk-sensitive allocations is challenging even when preferences are defined over only one consumption good. Lucas and Stokey (1984), Ma (1993), and Kan (1995) provide sufficient conditions for the existence of a recursive representation. Anderson (2005) shows that when preferences are heterogenous it is in general very hard to ensure a stationarity equilibrium. Garleanu and Panageas (2010) use an OLG approach to get stationarity. Backus, Routledge and Zin (2009) study the dynamics of a two-agent economy, in which one agent is infinitely riskaverse and the other is risk-neutral with respect to static gambles. In this paper, we document that consumption home bias is both a natural and convenient way to introduce heterogeneity in agents preferences in an international setting. This produces rich allocation dynamics in a stationary environment in which no agent eventually receives a negligible amount of wealth. The remainder of this paper is organized as follows. The next section describes the setup of the economy and outlines the solution to the Pareto problem. Section 3 provides a simple two-state example that is used to build on the intuitions of the dynamic economy. Section 4 discusses the generalized setup in which rare events are also part of the state space and the endowment processes are allowed to have some persistence. Section 5 looks at the data, by checking the empirical predictions of the model. Section 6 provides an alternative interpretation of the model in terms of disagreement 6

8 about the transition probabilities of the endowments. Section 7 concludes the paper. 2 The economy Endowments, preferences, and markets. In our economy there exist two goods, X and Y, whose endowments are stochastic. In each period, there is a realization of a random event s t. Let s t = (s 0,...,s t ) denote the history of events up to time t. Endowments in period t are time-invariant measurable functions of s t : (X t (s t ),Y t (s t )) = (X t (s t ),Y t (s t )). Let π(s t s t 1 ) be a Markov chain, with given initial realization s 0, such that π(s 0 ) = 1. We assume that endowments are i.i.d.: π(s t s t 1 ) = π(s t ). The economy consists of two countries, home (h) and foreign (f), each populated by a representative consumer. The countries have risk-sensitive preferences defined over domestic and foreign consumption bundles: U i,t (s t ) = (1 δ)logc i,t (s t )+δθlog { } Ui,t+1 (s t+1 s t ) exp π(s t+1 s t ), i {h,f} θ s t+1 where θ = 1 1 γ and γ is the coefficient of atemporal risk-aversion. This specification was initially proposed by Hansen and Sargent (1995) and is used by Tallarini (2000) and Anderson (2005), among others. Our economy departs from the constant relative risk-aversion case often analyzed in the literature primarily in the fact that these preferences are non-time-additive and they allow agents to be risk-averse in future utility in addition to future consumption. 1 Alternatively these preferences can be interpreted as the special case of Epstein and Zin (1989) preferences in which the intertemporal elasticity of substitution equals 1. The period utility functions are defined over consumption aggregates of good X and 1 Precisely the CRRA case is nested within this specification as the limiting case in which γ 1. 7

9 good Y : C h,t = x α h,ty 1 α h,t and C f,t = x 1 α f,t y α f,t (1) where x i,t and y i,t denote the consumption of good X and good Y in country i {h,f} at date t. We let the home country be endowed with good X and the foreign country be endowed with good Y. The parameter α captures the degree of bias of the consumption of each representative agent. Specifically, by letting α be larger than 1/2, we model the assumption of consumption home bias toward good X and good Y for the home and foreign country, respectively. At each date, trade occurs in a set of claims to one-period-ahead state-contingent consumption. There is a complete set of these claims. At each date t > 0, the budget constraints of the home and foreign agents are as follows: x h,t (s t )+p t (s t )y h,t (s t )+ q t (s t+1 s t )a h,t+1 (s t+1,s t ) X t (s t )+a h,t (s t ) s t+1 x f,t (s t )+p t (s t )y f,t (s t ) q t (s t+1 s t )a h,t+1 (s t+1,s t ) p t (s t )Y t (s t ) a h,t (s t ), s t+1 where p t (s t ) denotes the relative price of good Y and good X (the terms of trade), a i,t (s t ) denotes country i s claims to time t consumption of good X, and q t (s t+1 s t ) gives the price of one unit of timet+1 consumption of goodx, contingent on the realization s t+1 at t+1, when the history at t is s t. Pareto problem. Efficient allocations can be computed as the solution to the planner s problem. We decentralize this equilibrium in a later section with the complete set of sequential state- and date- contingent securities discussed above. The planner attaches nonnegative Pareto weights µ h = µ and µ f = 1 µ on the 8

10 consumers and chooses allocations {(x i,t (s t ),y i,t (s t ))} + t=0, i {h,f} to maximize Q(s 0 ) = µu h,0 (s 0 )+(1 µ)u f,0 (s 0 ), subject to a sequence of two economy-wide feasibility constraints: x h,t (s t )+x f,t (s t ) X t (s t ) y h,t (s t )+y f,t (s t ) Y t (s t ). Anderson (2005) suggests a recursive way to characterize this problem in a onegood economy, and Colacito and Croce (2009) extend this technique to multiple-goods economies. The insight is that the solution can be conveniently cast in terms of a time-varying vector of Pareto weights, even though technically there is only a time 0 vector of Pareto weights. LetM t (s t ) = µ h,t (s t )/µ f,t (s t ) denote the timethistorys t ratio of Pareto weights. The optimal consumption allocation rule is a sequence of functions that maps a historys t into a choice of timetconsumption of each good in each country: x h,t (s t ) = αm t (s t s t 1 ) (1 α)+αm t (s t s t 1 ) X t(s t ), x f,t (s t ) = y h,t (s t ) = (1 α)m t(s t s t 1 ) α+(1 α)m t (s t s t 1 ) Y t(s t ), y f,t (s t ) = (1 α) (1 α)+αm t (s t s t 1 ) X t(s t ) α α+(1 α)m t (s t s t 1 ) Y t(s t ), where M t (s t s t 1 ) = M t 1 (s t 1 ) exp s t exp { Uh,t (s t s t 1 ) θ { Uh,t (s t s t 1 ) θ } } π(s t s t 1 ) / exp s t exp { Uf,t (s t s t 1 ) θ { Uf,t (s t s t 1 ) θ } }, π(s t s t 1 ) t 1 and M 0 (s 0 ) = µ/(1 µ). The ratio of the Pareto weights, M t (s t ), indexes inequality at time t and summarizes the effects of the history of the exogenous state on consumption allocations. By interpreting the Pareto weights as time varying, consumption allocation rules have the 9

11 above straightforward representation, and the Pareto problem can be written recursively, as described in the Appendix. 3 A two-states example The environment. To develop the intuitions of the dynamic economy, we focus first on a simplified framework. For each history s t 1, let there be only two equally likely states for time t: s HL t = Xt(st ) Y t(s t ) = k and slh t = Xt(st ) Y t(s t ) = 1/k, for some k > 1. Accordingly, we index all equilibrium prices and quantities with a superscript HL or LH, depending on which of the two i.i.d. states materializes. We calibrate the endowments to be equal to 100 in the low-supply state and k = The coefficient of risk aversion γ is set to 25 and the subjective discount factor δ to 0.95 to reflect a yearly decision problem. Home bias is embedded in the parameter α = Countercyclical Pareto weights. Figure 1 highlights the most important feature of the dynamics of Pareto weights: they are countercyclical. In periods of large supply of the most-preferred good, the domestic Pareto weight drops. In periods of scarce supply of the most preferred good, the domestic Pareto weight rises. This result can be formalized by the following proposition. Proposition 1. Let θ < 0 and α > 1/2. Then: M t (s HL t s t 1 ) < M t 1 (s t 1 ) and M t (s LH t s t 1 ) > M t 1 (s t 1 ) Proof. See Appendix. 2 The properties of the risk-sharing scheme do not depend on the level of the endowments, as far as they are bounded away from zero. We choose a level of 100 simply for numerical reasons. The parameter κ = 1.03 is chosen to have a moderate amount of volatility in the growth rate of the two goods. 10

12 µ HL t+1 (st ) x 10 3 µ LH t+1 (st ) x µ t (s t ) Figure 1: Countercyclical Pareto weights. The two panels report the change in the home Pareto weight contingent on the two states as a function of the current Pareto weight. According to our notation, µ HL t+1 (st ) := µ t+1 (s HL t+1 st ) µ t (s t ) and µ LH t+1 (st ) := µ t+1 (s LH t+1 st ) µ t (s t ). In periods of large supply of the most-preferred good, the domestic Pareto weight drops. In periods of scarce supply of the most preferred good, the domestic Pareto weight rises. µ t (s t ) Trade-off between expected utility and utility variance. The analysis of the two-states example is instructive, in that it highlights at least one key feature of the specific attitude toward risk of the agents considered in this paper. Upon the realization of a good (relative) endowment draw, our agents are willing to give up current shares of consumption, and consequently to decrease the expected level of their utilities, in exchange for the decreased volatility of their future utility profiles. This trade-off between the first and second moments of the conditional distribution of the utility can be readily appreciated by taking a second-order Taylor expansion of the utility function: 3 where U i,t (s t ) (1 δ)logc i,t (s t )+δe[u i,t+1 (s t+1 s t )]+ δ 2θ V [U i,t+1(s t+1 s t )] (2) E[U i,t+1 (s t+1 s t )] = s t+1u i,t+1 (s t+1 s t )π(s t+1 s t ) 3 The expansion is taken about E[U i,t+1 (s t+1 s t )]. Additionally, the term log ( θ V [U 2 i,t+1 (s t+1 s t )] ) is approximated as 1 2θ V [U 2 i,t+1 (s t+1 s t )]. 11

13 and V [U i,t+1 (s t+1 s t )] = s t+1 ( Ui,t+1 (s t+1 s t ) E[U i,t+1 (s t+1 s t )] ) 2 π(st+1 s t ), i {h,f} denote, respectively, the expectation and the variance of the utility conditional on the current state s being s t. Ignoring the very last term in (2), the standard time-additive utility case obtains. With risk-sensitive preferences, however, agents care not only about the expected future utility level, but also about the conditional distribution of utility. Equivalently, the conditional variance of the utility matters, and the preference parameter θ = 1/(1 γ) measures how much agents care about the temporal distribution of risk. Since for any value of risk aversion in excess of 1 the risk-sensitivity parameter θ is negative, agents are willing to trade off expected utility levels for a lower expected utility volatility. To further illustrate this point, in figure 2 we plot the conditional utility volatility against the expected utility obtained in equilibrium over the domain of the home country s Pareto weight. Different curves are associated with different degrees of risk sensitivity. When γ = 1, agents have standard time-additive preferences and in equilibrium they will bear the same amount of utility risk, regardless of the expected utility level. For larger values of γ the situation changes in at least two relevant ways. First, agents are willing to hold a smaller amount of utility risk. This explains the vertical shift of the lines for higher values of γ. Second, agents are willing to substitute a lower expected level for a lower volatility of their utilities. This explains the positive slope of the lines. Risk-sensitive agents are interested in smoothing both future utility and future consumption. For this reason, time-varying conditional volatilities are a natural equilibrium outcome in a risk-sensitive exchange economy. 12

14 8 x γ=1 σ[u h,t+1 (s t+1 s t )] γ=10 γ= E[U h,t+1 (s t+1 s t )] Figure 2: The expected utility-volatility trade-off. The horizontal axis reports the conditional expectation of the utility of the home agent. The vertical axis reports the conditional volatility of the utility of the home agent. The three lines are plotted for increasing values of the riskaversion coefficient, γ, and are obtained using equilibrium utilities for different values of the home country s Pareto weight. For comparison, the lines are obtained using utility values over a symmetric interval that includes 95% of the probability distribution for the case of γ = 10. Each curve shows the incentive to trade-off future resources (measured by E[U h,t+1 (s t+1 s t )]) for future utility risk (measured by σ[u h,t+1 (s t+1 s t )]). Countercyclical risk. In this economy, the conditional variance of the utility is high during periods of low supply of the most-preferred good, and it is low during times of abundant supply of the most-preferred good. That is, risk is high during bad times and low during good times. The countercyclicality of risk can be better appreciated by means of the example depicted in figure 3. Assume that the home country is exposed to a sequence of 50 periods of positive endowment realizations (top panel). The optimal international risk-sharing arrangement requires that the home country redistribute part of its consumption share abroad (middle panel). In exchange, the same country is promised 13

15 X/Y HL LH µ h x 10 8 V t (U h,t+1 ) Figure 3: Counter-cyclical utility variance. The three panels report the relative supply of the two goods (top), the current Pareto weight for the home consumer (middle), and the conditional variance of the utility for the home consumer (bottom) over an history of 100 periods. In the first 50 periods, the home country is assumed to be luckier than the foreign country (X/Y > 1) and for this reason it endogenously experiences a reduction in conditional utility risk, V t (U h,t+1 ). In the remaining 50 periods the opposite occurs. to receive a larger future consumption share during times of adverse realizations of its most-preferred good. This results in a decreased conditional variance of its future utility profile (bottom panel). Conversely, as the home country is exposed to a sequence of bad endowment realizations (the subsequent 50 periods in figure 3), it will receive an increasing share of consumption back at the cost of having to give up part of the relative smoothness of its continuation utility. Risk is therefore countercyclical in the relative supply of the two goods. We show next how this countercyclicality of risk translates into time-varying equity risk premia. 14

16 a HL h,t+1 (st ) 10 a LH h,t+1 (st ) µ t (s t ) µ t (s t ) 0 x 10 3 x ã HL h,t+1 (st ) ã LH h,t+1 (st ) ã h,t (s t ) ã h,t (s t ) Figure 4: Asset holdings. The top two panels report the home holdings of securities contingent on the two states as a function of the current Pareto weight. According to our notation, ã HL h,t+1 (st ) := ã h,t+1 (s HL t+1 st ) ã h,t (s t ) and ã LH h,t+1 (st ) := ã h,t+1 (s LH t+1 st ) ã h,t (s t ).The bottom two panels report the change in the holdings of the two state contingent securities as a function of the current asset holdings. These four panels show the monotonic positive correspondence between international wealth (i.e., net foreign assets) and Pareto weight. Asset holdings and state prices. A competitive equilibrium is characterized by means of an initial distribution of wealth {a i,0 (s 0 )}, an allocation {x i,t (s t ),y i,t (s t )} i,s t, asset portfolios {a i,t+1 (s t+1 s t )} i,st+1, and pricing kernels {q t (s t+1 s t )} st+1. By virtue of the welfare theorems of economics, we can associate a level of claims to date t + 1 history s t+1 consumption, a i,t+1 (s t+1 s t ), to each current Pareto weight, µ t (s t ). In the top two panels of figure 4 we document that asset holdings in the home country are monotonically increasing in the Pareto weight, regardless of the realization of the endowment shock. The bottom two panels of figure 4 suggest that households will reduce their amount of savings when there is a large supply of the most-preferred good, and increase their savings otherwise. 4 4 For scaling reasons, the bottom two panels report the monotonic transformation ã h,t (s t ) = exp{a h,t (s t )} 1. 15

17 For each history s t, one of the following two values for the pricing kernel q t (s t+1 s t ) obtains: q HL t (s t ) = δ q LH t (s t ) = δ ( x HL h,t+1 (st ) x h,t (s t ) ( x LH h,t+1 (s t ) x h,t (s t ) ) 1 exp { U HL h,t+1 (st )/θ } s t+1 exp{u h,t+1 (s t+1 s t )/θ}π(s t+1 ) ) 1 exp { Uh,t+1 LH (st )/θ } s t+1 exp{u h,t+1 (s t+1 s t )/θ}π(s t+1 ) Taking the ratio of the two prices yields the following relationship between asset prices, future allocations and continuation utilities: q HL q LH t (s t ) t (s t ) { } = xlh h,t+1 (st ) U HL x HL h,t+1 (st ) exp h,t+1 (st ) Uh,t+1 LH (st ) θ (3) Figure 5(a) documents that the pricesq HL t (s t ) andq LH t (s t ) are, respectively, decreasing and increasing in the current Pareto weight attached to the home country. This has an intuitive explanation in light of the results reported in the previous subsections. As the home country is hit by a negative endowment shock, its Pareto weight increases (see figure 1) and its savings get larger and larger (see figure 4). In a competitive equilibrium, this translates into an increasing price of purchased insurance, q LH t (s t ), and a decreasing price of provided insurance, q HL t (s t ). Figure 5(b) decomposes the ratio of pricing kernels into the share associated to the ratio of current allocations and the share being contributed by relative continuation utilities in the two states of the world. This confirms that when agents care about the temporal distribution of risk, the dynamics of prices are mainly determined by future utilities. This has an important implication in our setup. As the gap between security prices widens (see figure 5(a)), so does the gap between the future utilities of the home agent in the two states of the world. Hence, as long as the international distribution of wealth is dynamic, the conditional volatility of utility will vary over time. In the next section we show that this stochastic volatility property carries over to all quantities and prices in the economy. 16

18 q LH t (s t+1 ) { } U exp HL h,t+1 (st ) U LH h,t+1 (st ) θ q HL t (s t+1 ) 0.96 x LH h,t+1 (st ) x HL h,t+1 (st ) q HL t (s t ) qt LH (s t ) µ t(s t ) µ t(s t ) (a) Figure 5: Pricing kernels in the two-state model. The left panel shows the prices of the statecontingent securities upon realization of either of the two states. The right panel decomposes the ratio of the prices into the ratio of future consumption and the term related to continuation utilities. The depicted time-varying volatility of asset prices reflects mainly endogenous timevarying utility risk. (b) 4 Quantitative implications So far we have focused on a two-state i.i.d. economy in order to highlight the key features of our recursive risk-sharing scheme. In this section we look at applications that allow us to better quantify the asset pricing implications of our model. We proceed in two steps. First, we retain the i.i.d. endowment assumption and introduce a small probability of a large drop in the endowment of either one of the two goods. This allows us to work with a simple economy with a finite number of states and only one state variable (i.e. the ratio of the Pareto weights), consistent with the previous section. It also enables us to study endogenous variations of the conditional second moments around average levels consistent with the data. In a second step, we relax our i.i.d. assumption by introducing persistence in the level of the endowments. We show that after the introduction of a degree of persistence consistent with the data, the main features of the model are preserved and our quantitative implications are in fact enhanced. 17

19 Table 1: Calibrated Economy Endowments X Y Probability σ( c) σ( x) Results σ(m) E(M) σ( e) ρ( c h, c f ) ρ(m h,m f ) E(rh c rf h ) E(r c h rf h ) σ(rh c rf h ) Model Data Notes - All the statistics are annual and multiplied by 100 (except for correlations and Sharpe ratios). All data sources are described in the Appendix. All statistics are computed as an average of major industrialized countries. Preference parameters are calibrated as γ = 25, δ = 0.95, and α = An economy with rare events In this section, we assume that there are nine possible states for the joint distribution of the endowments of the two goods. Four of them are equally likely no-disaster states in which the endowment of each good is either 100 or 103. Five states are equally likely disaster states in which the supply of at least one of the two goods is 60. We assume that there is a combined 5% probability of a disaster state. These numbers are in line with the closed economy model of Barro (2006). We retain the same preference parameters used in the previous section. Table 1 summarizes the calibration. In what follows we first outline the differences in the optimal dynamics of the Pareto weights that we obtain in this environment as compared to the previous two-state case. We then examine the properties of the international stochastic discount factors and their implications for asset prices and exchange rates. 18

20 µ 2 x X=103, Y= x X=100, Y= x 10 3 X=103, Y= x 10 4 X=100, Y= X=103, Y= X=100, Y=60 9 x 106 Without Rare Events µ With Rare Events µ X=60, Y= µ X=60, Y= µ (a) x X=60, Y= µ µ h Figure 6: The left panels report the phase diagrams in the model with rare events. Each panel shows the one period ahead change in the home Pareto weight as a function of the current Pareto weight, conditional on each of the nine endowments realizations. The right panel shows the invariant distribution of Pareto weights in the model with and without rare events. A small probability of a large endowment decline is able to generate large movements in the cross-country distribution of wealth. The results are obtained by simulating the model over 100 million of periods. (b) Pareto weights in the rare event model Figure 6(a) documents the dynamics of Pareto weights in this environment. The explanation of the cases involving and unequal supply of the two goods is the same as in the discussion reported in the previous section. Notice that in these cases the magnitude of the change in the one-period-ahead Pareto weight is directly related to the difference in the supply of the two goods. During times of equal endowment realizations, Pareto weights can be either increasing or decreasing in the current distribution of wealth. This reflects the way in which the risk-sharing scheme is implemented in this economy. One country is willing to lend resources to the other country which is facing a poor realization of its endowment in the expectation of receiving back part of those resources as soon as the second country fares comparatively better. Even a small probability of a large endowment decline is able to generate large movements in the cross-country distribution of wealth. Figure 6(b) documents this find- 19

21 ing. The solid line shows the ergodic distribution of Pareto weights with rare events, while the dashed line is obtained in an economy without disaster events. 5 In both cases, because of our symmetric calibration, the distribution is symmetric around 0.5. The dispersion of the Pareto weights around their mean, however, is significantly greater once rare events are introduced into the economy. When disasters are not perfectly synchronized across countries, risk-sensitive agents promote a more intense risk-sharing scheme in which Pareto weights or, equivalently, international assets are exposed to relevant adjustments over time Stochastic discount factors in the rare-events model The logarithm of the intertemporal marginal rates of substitution through which future uncertain payoffs are being discounted in the two countries is defined as m i,t+1 = log U i,t/ C i,t+1 (4) U i,t / C i,t = logδ logc i,t+1 + U i,t+1 log { } Ui,t+1 exp π(s t+1 ), i {h,f}, θ θ s t+1 where, to simplify the notation, we denoted history s t variables without (s t ). 6 We establish the following four properties of the stochastic discount factors related to their volatilities and cross-country correlations. Property 1: The volatility of stochastic discount factors is high. In figure 7(a) we show the conditional volatility of the home intertemporal marginal rate of substitution divided by its conditional mean as a function of the Pareto weight in the same country. Our stochastic discount factors inherit the high volatility property that has 5 We simulate the model over 100 million periods and compute in both cases an histogram with 25 equally distant bins. Since the pictures are constructed on a discrete grid, the sum of the areas underneath each curve does not necessarily equal one. 6 We express the stochastic discount factors in units of the respective consumption bundles, because this is the relevant measure for calculating exchange rates in the subsequent sections. 20

22 been documented in the rare-events literature. This implies that assets Sharpe ratios, our metric to asses the asset pricing performance of the model, are relatively large in this economy. Table 1 reports the equity risk premium and its ratio to the volatility for an asset that pays the consumption bundle as its dividend. The average excess return is about 1.7%, consistent with Lustig, Van Nieuwerburh and Verdelhan (2011c), implying a sizeable Sharpe-ratio of roughly 20%. 7 Property 2: The volatility of stochastic discount factors varies over time. Figure 7(a) shows that the volatility of stochastic discount factors is not only high, but also time varying, ranging from 25% to 34%. The dynamics are driven by the optimal risk-sharing scheme. A country that experiences a sequence of positive endowment realizations becomes relatively safer: its market price of risk declines as its Pareto weight decreases. A country that is hit by a sequence of negative shocks is able to smooth out its utility over time at the cost of having to accept an increasing volatility of consumption: its market price of risk rises as its Pareto weight increases. Time-varying market prices of risk translate into time-varying risk premia. Figure 7(b) shows that expected excess returns vary from a little over 1% to a little over 2.1% in this economy. Note that the one-country version of our model is associated with µ t = 1. In the one-country economy, the expected excess return would be constant at 2.1%. In our economy, in contrast, the excess return is time varying around an unconditional mean of 1.7%. The 400-basis-point difference is a measure of the overall reduction in risk obtained through international trade of goods and securities. In relative terms this difference is very relevant, as it suggests that international trade can reduce equity premia by 20%. 7 The moderate equity premium produced in this economy is due to an excessively high risk-free rate. Barro (2006) shows that by bringing into the model a default probability on government bonds in the case of a disaster, the model is able to generate a higher equity risk premium through a reduction in the average risk-free rate. 21

23 f σ t (M h,t+1 )/E t (M h,t+1 ) µ h (a) ) f r h,t f c E t (r h,t µ h (b) 22 ), c t+1 corr t ( c t+1 h µ h ),m t+1 corr t (m t+1 h σ t ( e t+1 ) µ h (c) (d) Figure 7: Variables in the rare-events model as a function of the current Pareto weight in the home country. Panel (a): Conditional market price of risk in the home country. Panel (b): Conditional equity risk premium of the home country. Panel (c): Correlations of growth rates of consumption and marginal utilities across countries. The left scale reports correlations of consumption growth (dashed line). The right scale reports correlations of marginal rates of substitution (solid line). Panel (d): Conditional volatility of the real exchange rate movement. Dashed line: The correlation of stochastic discount factors is set to its unconditional value. Solid line: The conditional correlation of stochastic discount factors varies as in panel (c).

24 Property 3: The correlation of stochastic discount factors is high. Brandt et al. (2006) pointed out that with time-additive preferences, stochastic discount factors are as correlated as consumption growth rates. This is not necessarily the case with the type of preferences studied in this paper. When agents care about the temporal distribution of risk, they will attempt to equalize their continuation utilities as much as possible, even though consumption profiles are not perfectly aligned. Figure 7(c) documents that intertemporal marginal rates of substitution are at least twice as correlated as consumption. This bears an important consequence for real exchange rates. Since markets are complete, the growth rate of the real exchange rate, e t+1, is simply the difference of the two log-stochastic discount factors, m f,t+1 and m h,t+1. Given our symmetric calibration, the unconditional volatility of the exchange rate growth is as follows: σ( e t+1 ) = σ(m h,t+1 ) 2(1 ρ(m h,t+1,m f,t+1 )), As shown in Table 1, thanks to the high correlation of the stochastic discount factors, ρ(m h,t+1,m f,t+1 ), we are able to reproduce the observed volatility of real exchange rate fluctuations, despite the modest correlation of consumption growth rates across countries. Moreover, we are not exposed to the Brandt et al. (2006) puzzle: in our model, the volatility of the exchange rate is consistent with the data even though stochastic discount factors are as volatile as required by the Hansen and Jagannathan (1991) bound. Property 4: The correlation of stochastic discount factors varies over time. The reallocation of consumption shares that takes place through the optimal risksharing scheme makes continuation utilities more or less correlated as a function of the relative wealth of the two countries. Specifically, as the wealth spread increases, the concavity of the utility functions is such that a positive endowment shock in the 23

25 high-pareto-weight country leads to increased utility in the other country also, even if the second country experiences a negative endowment draw. Hence, risk sharing is the highest exactly when it is needed the most that is, when either of the two countries is close to dying. By no arbitrage, the conditional volatility of exchange rate growth is equal to σ t ( e t+1 ) = σ 2 t (m f,t+1 )+σ 2 t (m h,t+1 ) 2ρ t (m h,t+1,m f,t+1 )σ t (m h,t+1 )σ t (m f,t+1 ). The combination of time-varying volatilities and the correlation of stochastic discount factors results in time-varying volatility of exchange rate movements. Figure 7(d)) shows that the dynamics of the exchange rate volatility are mainly driven by timevarying conditional correlations. Indeed, by shutting down this channel and by allowing only the conditional volatilities of stochastic discount factors to change, the resulting exchange rate volatility is pretty much flat on the Pareto weight domain (dashed line, figure 7(d)). As the conditional correlation varies over time (solid line, figure 7(d)), so does the volatility of the exchange rate, which can range from as low as 10.5% to as high as 14.5%. Hence, the dynamic risk-sharing arrangement drives the time-varying riskiness of the exchange rate Sensitivity analysis In table 2 we report the results of a sensitivity analysis with respect to some of the preference parameters and to the probability distribution across states of the world. Three aspects seem to emerge. First, without rare events, the model is not able to produce sufficiently volatile stochastic discount factors. This results in equity risk premia that are on average smaller than 0.03% per year and in exchange rates that are too smooth compared to the actual data. Second, the introduction of a small probability of a joint large endowment decline al- 24

26 lows our model to deliver highly volatile stochastic discount factors and equity premia as large as 3% per year. However, the model s implied exchange rate volatility is still too low due to the extremely high cross-country correlation of marginal rates of substitution. This correlation is mainly driven by the high correlation of consumption growth, which in this calibration arises from the simultaneous realization of large endowment declines. Third, spreading the disaster probability to allow for the case of a large endowment drop in one country together with a no-disaster state in the other country, delivers both large equity premia and sufficiently volatile fluctuations of exchange rates. Thus, to summarize, rare events can improve the performance of the model if they are not perfectly correlated across countries. 4.2 Introduction of serial correlation We now introduce serial correlation into the endowments process in order to determine the change in our results when endowment shocks are long-lasting. In this setup, the current values of the endowments become additional state variables as they alter the conditional distribution of future endowment level. We work with continuous log-normal shocks because they enable us to use fast high-order perturbation methods as in Colacito and Croce (2009). Our endowment processes, therefore, are specified as follows: logx t = C +ρlogx t 1 +σǫ X,t ǫ Xre,t (5) logy t = C +ρlogy t 1 +σǫ Y,t ǫ Yre,t where ǫ X,t and ǫ Y,t are i.i.d.n(0,1) shocks and are assumed to be uncorrelated, while ǫ Xre,t and ǫ Yre,t capture rare events modeled as in Gourio (2011) with a total assigned 25

27 26 Table 2: Sensitivity Analysis σ( c) σ( x) σ(m)/e(m) σ( e) corr( c h, c f ) corr(m h,m f ) E(rh c rf h ) E(r c h rf h ) σ(rh c rf h ) (103, 103) (103, 100) (100, 103) (100, 100) (103,60) (103,80) (100,60) (100,80) (60,103) (80,103) (60,100) (80,100) (60, 60) (80,80) γ Notes - The table reports the variables of interest (top panel) when changing the states and their associated probabilities (bottom panel). All other parameters are calibrated as in Table 1.

28 probability of 5%, as in the previous section. 8 As before, the constant C > 0 plays no role in the risk-sharing scheme and is normalized so that the average endowment level is 100. As reported in the first column of table 3, our new endowment specification preserves all the results shown in table 1 in the case of no serial correlation. In table 3 we also report the correlation of the implied endowment growth rates. Even though rare events are not correlated, the growth rates of the endowment in the two countries have a correlation of 30%. This is related to the lack of persistence: when a joint disaster occurs, both countries experience a substantial rebound in the subsequent period, making the unconditional correlation across countries positive. In an economy with persistent shocks and no disaster (second column), the correlation of the endowment growth rates is now perfectly zero, while the correlation of the consumption growth rates is higher thanks to risk-sharing. The introduction of persistent shocks makes the utility-smoothing motive even stronger than before, as documented by the much higher correlation of the stochastic discount factors and the implied moderate volatility of the exchange rate. Without rare events the consumption claim risk premium is six times lower than in column (1), and the implied Sharpe ratio is half of the previous level. This shows that shocks to the endowment level alone are not enough to explain the observed risk premia, even with positive serial correlation. Column (3) shows that when disasters have long-lasting effects, we can replicate our benchmark asset pricing results reported in table 1 with a rare event of smaller magnitude, although this results in an excessive cross-country correlation of the consumption growth rates. As shown in column (4), we can solve this problem by reducing the cross-country correlation of the disaster shocks to keep the correlation of the endowment growth rates below 20%. In our final specification, we work with serially cor- 8 To keep our set of equations continuous and differentiable, we use a logistic transformation of a Gaussian random variable to approximate Poisson shocks. 27