On the Global Spread of Risk Panics 1

Transcription

1 On the Global Spread of Risk Panics 1 Philippe Bacchetta University of Lausanne CEPR Eric van Wincoop University of Virginia NBER August 17, 21 1 We would like to thank Martina Insam for able research assistance. We gratefully acknowledge financial support from the National Science Foundation (grant SES ), the Hong Kong Institute for Monetary Research, the National Centre of Competence in Research Financial Valuation and Risk Management (NCCR FINRISK), and the Swiss Finance Institute.

2 Abstract The strong co-movement of equity prices across the globe during the 28 financial crisis coincides with a spike in equity price risk (VIX) of similar magnitude across a broad set of developing and developed countries. The literature provides no explanation for the contagion of risk. In this paper we shed light on this comovement of risk by developing a model that allows for the possibility of large self-fulfilling shifts in risk. This builds on Bacchetta, Tille and van Wincoop (21), who develop the concept of risk panics in a closed economy framework. During such a panic a weak macro variable becomes a focal point (coordination device) for a self-fulfilling increase in risk. We show that when macro fundamentals have a limited impact on equity prices during normal times, the risk panic will impact countries very similarly.

3 1 Introduction The financial panic in the Fall of 28 lead to a sharp drop in equity prices and an increase in perceived risk around the world. It has been widely document that the magnitude of equity prices decline was broadly similar across the globe. What is perhaps less known is that the spike in asset price risk, as measured by implied volatility measures (e.g., VIX in the US), was also of similar magnitude across a broad set of countries. This is illustrated in Figure 1 for a group of 12 developed and developing countries. 1 The close co-movement in equity prices during this period was clearly related to the co-movement in risk. Moreover, an interesting feature of recent events is that a country-specific fundamental is related to the increase in global risk perceptions. In the Fall of 28, it was the financial health of leveraged institutions mainly based in the US that lead to nervous financial markets. In the Spring of 21, it was the health of the Greek public sector that was associated with the global increase in risk. In this paper we propose a framework consistent with these features. We develop a model of a global risk panic where countries can experience large selffulfilling shifts in risk and steep declines in equity prices of similar magnitude across countries. The paper builds on Bacchetta, Tille, and van Wincoop (21), from here on BTW, who develop the concept of risk panics in a closed economy context. BTW show that as long as asset demand depends on asset price risk, self-fulfilling shifts in risk are possible. In this case there is a dynamic mapping of risk into itself, because the risk associated with the future asset price depends on uncertainty about future risk. During a risk panic, investors perception of risk may be affected by a weak macro fundamental. This fundamental suddenly takes on an additional role as a coordination device for the self-fulfilling shift in risk. The weaker the fundamental, the larger is the panic. The cross-country spread of risk has not received much attention in the theoretical literature on financial contagion, which we review in Section 2. This literature looks at contagion in asset price levels or returns and does not explain surges in risk. Some recent papers have explained co-movement in equity prices during the recent crisis with co-movement of risk premia. This focus is well placed as it is hard to argue that either of the two other asset pricing determinants, the risk- 1 Data sources are reported in Appendix C. 1

4 free rate and expected dividends, can account for the equity price co-movement. But movements in risk play little or no role in the risk premia considered in the literature. In this paper we also attribute equity price co-movement to co-movement in risk premia. But we depart from the recent literature along two important dimensions. First, in our model fluctuations in risk premia (and especially their spikes during a panic) are explicitly linked to fluctuations in asset price risk itself. Understanding the fluctuations in asset price risk, and their co-movement across countries, then becomes the central aspect of our analysis. Second, in our model a panic generates entirely self-fulfilling shifts in risk and in equity prices while the recent contagion literature relies on shocks to observed macro fundamentals (i.e., technology shocks). The panic in the Fall of 28 came quite unexpectedly and is hard to attribute to a sudden large shock to macro fundamentals. 2 We develop a stylized model which delivers a closed form analytical solution. This makes the mechanisms at work easier to understand for what is otherwise a quite difficult topic. We consider a two-country model where investors trade equity claims with exogenous and stochastic dividends. Two simplifying assumptions are an OLG structure and a constant interest rate on bonds. BTW show that relaxing the latter assumption is not central to risk panics. The only stochastic fundamental is the dividend on equity. The role this fundamental plays during a panic is as a coordination device for a self-fulfilling shift in risk. As emphasized by BTW, the precise nature of the macro variable that becomes the focal point for a risk panic is not so important. They show that results are similar when the key macro fundamental is the net worth of leveraged financial institutions, which fits more closely to the recent crisis. Focusing on stochastic dividends as the only source of macro shocks has the advantage of making the model more standard and analytically tractable. The objective of the paper is to show how a global risk panic can arise and how it spreads across countries. In particular, we analyze the factors that determine how much individual countries are affected by the panic. An important dimension 2 The main fundamental in 28, the deteriorating financial health of leveraged financial institutions due to mortgage market losses, had been weakening for at least a year prior to the crisis. More recently, the Greek debt crisis also did not involve a large sudden change in fundamentals as Greek debt and the deficit had been large and growing for some time. 2

5 is the hedging property of a country asset with respect to the global portfolio. If assets have similar hedging properties, their price collapse in a panic will be of similar magnitude and we can talk about full contagion. We show that a condition for these hedging properties to be similar is that macro fundamentals have limited explanatory power for equity prices during normal times. The remainder of the paper is organized as follows. Section 2 reviews related recent literature on financial contagion. Section 3 describes the model. Section 4 discusses the solution for the world equity price and global risk panics. Section 5 discusses the solution of the equity prices of the two countries. It particularly focuses on how a global risk panic is spread across the two countries. Section 6 concludes. 2 Related Literature The recent financial crisis has spurred renewed interest in the issue of financial contagion. 3 The vast existing literature 4 is being extended, drawing lessons from recent events. This renewed interest in contagion is not surprising as movements in equity prices across the globe were highly synchronized during the Fall of 28. A striking feature however, especially in light of Figure 1, is that the theoretical literature gives little attention to asset price risk. Co-movement of asset prices is not associated in the literature with co-movements of risk. Nonetheless the recent crisis has lead the literature to put more emphasis on common shifts in risk premia to explain asset price co-movements. However, these shifts in risk premia are not associated with asset price risk itself, but rather operate through wealth effects often due to financial frictions. This channel was suggested by Calvo (1999) in the context of the Russian virus and later by Krugman (28) to explain the co-movement of equity prices during the recent crisis. 5 More formally in general equilibrium frameworks, this channel of contagion appears in Gromb 3 There is no precise or agreed-upon definition of contagion. This terminology is generally used (or abused) in the context of increased co-movements in asset prices or returns. 4 See Dornbusch et al. (2) or Karolyi (23) for surveys. 5 In Calvo (1999), price movements are exacerbated by imperfect information by investors. See King and Wadwhani (199) or Calvo and Mendoza (2) for models explaining contagion based on limited information. 3

6 and Vayanos (22), Kyle and Xiong (21) and Pavlova and Rigobon (28). To understand this wealth channel, assume that for whatever reason the Home equity price goes down. This reduces the wealth of Home investors, which reduces their demand for Foreign equity through a wealth effect. Similarly, to the extent that Foreign investors hold Home equity, it also reduces their wealth, which again reduces demand for Foreign equity. This leads to a drop in the Foreign equity price. Risk premia and wealth are related as reduced wealth implies that the holdings of risky assets increase relative to wealth. This leads to higher risk premia. This explanation for contagion during the recent crisis has an important limitation though. The contagion relies critically on the presence of large cross-border asset holdings. Rose and Spiegel (21) find that cross-border asset exposure to the U.S. did not affect the extent to which countries were affected by the crisis. This has lead to a search for additional contagion channels operating through risk premia that do not depend on the extent of cross-border asset holdings. Dedola and Lombardo (21) develop a model where financial intermediaries charge an external finance premium to investors, which is basically a risk premium. As investors wealth decreases (negative technology shock), a higher risk premium on both Home and Foreign equity implies a common drop in their prices. The extent of equity price risk does not affect the finance premium, which is only an exogenous function of wealth. A similar mechanism is present in Devereux and Yetman (21) and Mendoza and Quadrini (21). In Devereux and Yetman (21) investors face a leverage or borrowing constraint. A lower wealth implies a tighter constraint and higher risk premia on all risky assets held by investors. This is because the borrowing constraint makes no distinction between the riskiness of Home versus Foreign equity. In these papers the implied risk premia are the same for Home and Foreign equity independently of the extent of cross-border asset holdings. However, this is essentially by assumption as the finance premium or the borrowing constraint makes no distinction between the riskiness of Home and Foreign assets. In reality borrowing constraints depend on the riskiness of the assets held by investors as well as their exposure to individual assets. This is true both for collateralized and uncollateralized lending. Risk premia will then differ across assets. However, contagion through wealth effects will again depend on the extent of cross-border asset holdings. For example, a drop in Home wealth should raise risk premia on 4

7 Foreign stock more when Home investors allocate more of their wealth to Foreign equity and are therefore more at risk of default with a further drop of Foreign equity prices. A few papers do examine the impact of shocks to uncertainty occurring on one asset. For example, Fostel and Geneakoplos (28) present a model where an increase in uncertainty in a developed country assets leads to a price decline in emerging country asset prices. Schinasi and Smith (2) examine the impact of a volatility increase in one asset under various portfolio rules and determine when contagion occurs. These papers, however, consider exogenous changes in risk and do not lead to co-movement in asset price risk. Asset price volatility plays a much more prominent role in the empirical literature on contagion. Although most of the literature focuses on co-movements in asset prices or returns, it takes time-varying volatility into account (e.g., with Garch models) to correct for heteroskedasticity (e.g., see Bekaert and Harvey, 1995, or King et al., 1994). Moreover, there is a branch of the literature that examines the international transmission of volatility. 6 Although the methodologies and the results vary from study to study, the transmission of volatility seems to be stronger in periods of high volatility (e.g., see Edwards and Susmel, 21, Beirne et al., 29, or Diebold and Yilmaz, 29). Nevertheless, these empirical studies do not provide any theoretical justification for the transmission of volatilities. 3 A Simple Two-Country Portfolio Choice Model In this section we describe a simple two-country portfolio choice model. Each country, Home and Foreign, is inhabited by two-period overlapping generations of consumers-investors with mean-variance preferences. They allocate their portfolio between bonds, Home stocks and Foreign stocks. 7 Financial markets are perfectly integrated. The only uncertainty comes from Home and Foreign shocks that affect dividends. In this section, we derive the optimal portfolios and the equilibrium conditions for equity prices. The equilibria themselves are discussed in the next two sections. 6 See Soriano and Climent (26) for a survey. 7 There is no distinction between Home and Foreign bonds as it is a single good economy. 5

8 3.1 Optimal Portfolios The model complexity is kept to a strict minimum so that it can be solved analytically. We denote the Home and Foreign countries respectively H and F. In both countries the overlapping generations of investors are born with wealth. 8 They invest it in equity and bonds and consume the return on their investment when old. The total number of agents is in the Home country and 1 in the Foreign country. The bond pays an exogenous constant gross return. This implicitly assumes that there is a risk-free technology with a constant real return that is in infinite supply. This short-cut assumption allows us to derive a closed form solution to the model. 9 Equity consists of a claim on trees with stochastic dividends of respectively and in the Home and Foreign countries. The per capita capital stock (number of trees) in both countries is. Equitypricesare and.home and Foreign equity returns from to +1arethen +1 = (1) +1 = (2) The only source of uncertainty in the model is with respect to dividends: = + (3) = + (4) where is a positive constant, is a non-negative parameter, and and are Home and Foreign macro variables. The formulation of (3) and (4) allows us to vary the fundamental role of the macro variables and in affecting asset payoffs. As becomes smaller, both dividends and asset prices become less affected by fundamental shocks. When =,thevariables and no longer 8 As discussed in BTW, the assumption of OLG with initial endowments contributes to simplify the analysis by reducing the number of state variables. However, numerical analysis shows that relaxing this assumption does not affect fundamentally the results. 9 In BTW we relax it in a closed economy setting by introducing a time-varying interest rate solved from bond market equilibrium. While this requires a numerical solution as a result of the non-linearities that it generates, it does not fundamentally alter the findings. 6

9 affect dividends. They then becomes pure sunspots that have no fundamental role in the model. The macro variables follow an AR process: +1 = + +1 (5) =. The innovations +1 and +1 have symmetric distributions with mean zero. Their variance is 2 and their correlation is. Also assume that ( 2 +1) = 2, =. Investors from both countries born at time maximize a mean-variance utility over their portfolio return: +1 5 ( +1) (6) As explained in BTW, the adoption of mean-variance preferences is a simplifying device to make asset demand, and therefore asset prices, depend on future asset price risk. This will also be the case when we introduce financial constraints in an expected utility framework, such as value-at-risk or margin constraints, but the resulting setup will be far more complex. As we will see, the link between asset prices and future asset price risk is key to generating self-fulfilling shifts in risk. The two countries are perfectly integrated, so that they choose the same portfolio allocation and have the same portfolio returns: +1 = (1 ) (7) where denotes the portfolio share invested in equity from country. The equity market clearing conditions are = (8) = (1 ) (9) 3.2 Equilibrium Conditions for Equity Prices Maximization of (6) with respect to and gives = 1 ( +1 ) ( +1 ) ( ) ( +1 ) ( +1 ) ( +1 ) ( ) 2 (1) = 1 ( +1 ) ( +1 ) ( ) ( +1 ) ( +1 ) ( +1 ) ( ) 2 (11) 7

10 Write the excess payoff on stocks as +1 = for =. Substituting the portfolio expressions (1)-(11) into the market clearing conditions (8)-(9) then gives ( +1 ) +1 ( ) +1 = ³ ( +1 ) ( +1 ) ( ) 2 (12) ( +1 ) +1 ( ) +1 = (1 ) ³ ( +1 ) ( +1 ) ( ) 2 (13) Define the world equity price as +1 = +1 +(1 ) +1 and the global dividend payment as +1 = +1 +(1 ) +1. The excess payoff on a world equity claim is then +1 = +1 +(1 ) +1 = Writing (12)-(13) jointly in vector notation and then pre-multiplying them with the matrix ( +1 ) ( ) ( ) ( +1 ) gives +1 = ( ) (14) +1 = ( ) (15) Theequilibrium expected excesspayoff on equity, which is a risk premium, depends on the covariance with the excess payoff on the world equity claim. 1 Taking the weighted sum of (14) and (15), with weights and 1, aswell as their simple difference, gives +1 = ( +1 ) (16) ( )= ( ) (17) 1 These last two equations imply the familiar capital asset pricing model. Using that +1 = +1 +(1 ) +1, they can be written as +1 = ( ) ( +1 ). = +1 for =, where 8

11 These two equations are key to solving for the equilibrium asset prices. (16) tells us that the risk premium on the global equity position depends on the variance of the global excess payoff. Equation (17) simply says that investors demand a higher expected excess payoff on the equity that has a higher covariance with the world. Risk premia also depend on risk aversion, on the asset supply and on wealth, but they are constant in the analysis. Returning to our notation in terms of equity prices, if we define = as the difference in equity price, (16)-(17) become ( ) = ( ) (18) ( ) = ( ) (19) Using (18) and (19) we now solve for and as a function of the state variables and. This also gives the solution for the equity prices of the individual countries. In the next section we focus on the world price, while we examine individual prices and contagion in Section 5. 4 World Equity Price: Multiple Equilibria and Panics The solution for the world equity price can be obtained from (18) alone. This equation is the same as the equilibrium in the closed economy model of BTW, where agents can invest in a single stock and in bonds. Not surprisingly, the nature of the equilibria resulting from (18) is the same as in BTW. There are three types of equilibria: a pure fundamental equilibrium, sunspot-like equilibria, and equilibria that allow for self-fulfilling switches between low and high risk states. We first describe the basic mechanism behind the multiplicity of equilibria and then discuss the three types of equilibria. 4.1 Basic Mechanism A key feature of equation (18) is that the price at time is related to the variance of the price at time +1, ( +1 ). Equation (18) can be rewritten 9

12 as: = ( ) (2) The asset price today depends not only on expectations and on risk associated with future dividends, but also on risk associated with the future world equity price itself. Define this risk as = ( +1 ). It is this dependence of the equity price on risk associated with its future level that creates the possibility of self-fulfilling shifts in risk: depends negatively on.butsince +1 in turn depends on +1, depends on uncertainty associated with +1. Risk does not just depend on uncertainty about future dividends, but also on uncertainty about future risk itself. It is this dynamic mapping of risk into itself that gives rise to the possibility of self-fulfilling shifts in risk. Shifts in beliefs about risk may be determined by a fundamental variable. This is a sunspot-like equilibrium. 11 Thevariablemayplaytheroleofacoordination device for self-fulfilling shifts in risk that is entirely unrelated to its fundamental role. In our model this dual role can be played by or or some linear combination, each generating a different sunspot-like equilibrium. For illustrative purposes, in what follows we will only consider sunspot-like equilibria where plays this dual role. While dividends are the only fundamentals in our simple model, BTW show that other fundamentals, such as the net worth of leveraged institutions, can take on this dual role as well. In principle, any macro variable can become a coordination device for self-fulfilling shifts in risk. During the recent European debt crisis, one can argue that the Greek debt played such a role. 4.2 Fundamental Equilibrium In such a simple model, the fundamental equilibrium is linear in the shocks. Consider a simple linear solution for the world equity price: = + + (21) where, and are constants to be solved. Using (21), we can compute the expectation and variance of Notice that the variance is constant in this case. Substituting the result into (18), and equating on the left and 11 The terminology of sunspot-like equilibria was first introduced by Manuelli and Peck (1992). 1

13 the right hand side the constant term, the term linear in and the term linear in, allows us to solve for the three unknown parameters. The solution is (see Appendix A for details of the algebra and for an expression for the constant term ): = (22) (1 ) = (23) so that = + (24) where +1 = +1 +(1 ) +1. The world equity price depends on the world dividend, whose impact is larger the higher the persistence of dividend shocks. We call this the fundamental equilibrium. The coefficient on goes to zero as we let, in which case no longer plays a fundamental role (does not affect the global dividend). 4.3 Sunspot-like Equilibria The other equilibria, both the sunspot-like equilibria and the switching equilibria, involve self-fulfilling shifts in the perception of risk. These shifts are influenced by fundamental variables. As mentioned, we only consider sunspot-like equilibria where the Home fundamental plays this role. We can find the sunspot-like equilibrium where plays the dual role described above by conjecturing a solution of the type = (25) In comparison to (21) this equilibrium conjecture has an additional term that is quadratic in the Home fundamental. This quadratic term will capture the role of as a variable around which self-fulfilling shifts in beliefs about risk are coordinated. We again use the method of undetermined coefficients to solve for the parameters of the conjectured solution. We first use (25) to compute the expectation and variance of Substituting the result into (18) and equating 11

14 on the left and the right hand side the constant term, the term linear in,the term linear in, and the term quadratic in, allows us to solve for the four unknown parameters. Algebraic details are left to Appendix A. The solution is (leaving again the constant term to the Appendix): = 2 (26) = " + # 2 1 (1 ) (27) (1 ) = (28) The key parameter here is, which multiplies 2 in the equilibrium world equity price. It captures self-fulfilling shifts in risk. To see this, first consider the self-fulfilling part and then the risk part. Changes in the equity price associated with the term 2 are self-fulfilling because they do not capture the fundamental role of. Its fundamental role depends on the parameter. But does not depend on. Even when =,sothat plays no fundamental role in the model at all (does not affect dividends), the coefficient remains the same. Next consider the risk part. The variance of tomorrow s world equity price is ( +1 ) = ( +2 ) ( +2 ) (29) As long as 6=, global asset price risk is time varying in. It therefore follows that the term 2 captures changes in the equity price related to selffulfilling shifts in beliefs about risk. When = this is most clear as is then a pure sunspot and the time-varying risk is clearly unrelated to fundamentals. But thesameisthecasewhen and plays a fundamental role as well. We call this equilibrium a sunspot-like equilibrium, and the variable a sunspot-like variable, because has a role similar to that of a sunspot. clearly is not a pure sunspot as it affects the Home dividend as long as, but its role in generating self-fulfilling shifts in risk is exactly the same as that of a sunspot variable. The term 2 would be the same if were a sunspot. 12

15 Even though plays a fundamental role when, this role is always significantly dominated by its sunspot role. This is reflected not only in the term that is quadratic in, which exclusively reflects the sunspot role of,but also in the linear term in in the world equity price. The coefficient in the linear term in actually has a negative sign, as opposed to a positive sign in the fundamental equilibrium. This change in sign is due to the covariance between the fundamental dividend risk and the self-fulfilling shifts in risk. 12 For illustrative purposes, Figure 2 shows both the fundamental and sunspot-like equilibria for a particular parameterization (at the bottom of Figure 2). It shows how the world equity price and risk depend on the Home fundamental.risk is defined as the standard deviation of +1 divided by. For the purpose ofthefigureweassumethat only takes on the values and +. It is important that the distribution is bounded as itself is then bounded as well (here between plus and minus (1 )), which is needed to assure that the asset price is positive in all states. In the sunspot-like equilibrium the equity price and risk are clearly much more sensitive to,reflecting the self-fulfilling shifts in risk. The additional risk leads to a lower world equity price than under the fundamental equilibrium for all possible values of. At the extreme values of (on either side) there are very high beliefs about risk and correspondingly low equity prices. 4.4 Switching Equilibria and Global Risk Panics There is a third type of equilibria that we refer to as switching equilibria. These are the main focus of this paper. They are closely related to the fundamental and sunspot-like equilibria discussed so far. In a switching equilibrium, there are occasional switches between low and high risk states. The low risk state is similar to the fundamental equilibrium and the high risk state is similar to the sunspot-like equilibrium. We assume that the switch between low and high risk states is driven 12 An increase in raises the expected excess payoff due to the persistent increase in the dividend. In equilibrium there has to be an offsetting drop in the expected excess payoff or increase in risk. In the fundamental equilibrium doesthejobasitraisesthecurrent equity price, which lowers the expected excess payoff. But in the sunspot-like equilibrium, does the job as it raises risk. This is because the variance of has the linear term 4 2, which depends positively on when. 13

16 by a Markov process. With probability 1 we are in the low risk state tomorrow when we are in the low risk state today. Similarly, with probability 2 we are in the high risk state tomorrow if we are in the high risk state today. When 1 = 2 = 1, we always remain in the same state, leading to the fundamental and sunspot-like equilibria discussed above. When we lower the probabilities below one, agents need to take into account that we may switch to another state when computing expectations and risk. The low and high risk states then no longer correspond exactly to the fundamental and sunspot-like equilibria. For example, risk is higher in the low risk state than in the fundamental equilibrium as there is now a possibility of switching to the high risk state. The relationship between the equity price and the state variables remains as in (25), but ones needs to separately solve for the coefficients in the low and high risk states. BTW solve for such switching equilibria. Here we take a somewhat simpler approach, with the benefit ofthefindings in BTW. They show that the solution is a continuous function of the probabilities. As 1 = 2 approaches 1, the low and high risk states approach respectively the fundamental and sunspot-like equilibria. Using this finding, we consider the low risk state to be the fundamental equilibrium and the high-risk state the sunspot-like equilibrium. This is infinitesimally close to the true equilibrium when 1 = 2 are very close to 1. While the probabilities of switching are then very small, we can still consider the impact of a switch. This approach has the advantage that we already know what the low and high risk states are. For larger switching probabilities (lower values of 1 and 2 )asolution can only be obtained numerically. The qualitative results remain similar though. A risk panic involves a switch from the low risk state to the high risk state. At the moment the panic happens there is an immediate increase in perceived risk, coordinated around. In Figure 2 this is the jump in risk from the broken line (fundamental equilibrium) to the solid like (sunspot-like equilibrium). At the time of the panic the fundamental itself does not change. Rather, it suddenly takes on the additional role of a variable around which beliefs about risk are coordinated in a self-fulfilling way. As we can see from Figure 2, the increase in risk can be very large, particularly when the fundamental is either very weak or very strong. We consider the latter case less realistic in practice as sharp increases in risk usually happen at the time that the market is concerned about a bad fundamental. In our model the switch itself is a random event. In practice though, such an event 14

17 is usually associated with a piece of information that draws attention to a weak fundamental. Figure 2 shows that the drop in the world equity price can be very large when the panic happens at a time that the fundamental is weak. This is also illustrated in Figure 3, which shows what happens to the world equity price and risk as a result of the panic that happens at the time that the macro fundamental (Home dividend) is at its weakest, i.e., is equal to We assume that in period 6 the world economy switches to the high risk state, where it stays until period 1. Before and after that we are in the low risk state. The panic leads to a 48% drop in the world equity price. This is caused by an increase in world equity price risk from.4% to 14%. Of course, dependent on the parameters one can get even larger or smaller risk panics. For example, if we lower from.5 to.4, the panic leads to a 61% drop in the world equity price and risk increases from.5% to 23% The Spread of Risk Panics The previous section described how global asset prices could collapse because of a risk panic. An important question is how such a collapse is spread across countries. In this section, we examine the impact of a risk panic on equity prices of individual countries. 5.1 Country Prices and Covariance To look at the different price reactions, we need to determine from (19). We will consider the following solution: = (3) As in the previous section, we assume that only the Home dividend coordinate self-fulfilling shifts in risk. can 13 This value comes from the fact that = = 1 and = While our aim here is certainly not to draw precise quantitative comparisons to the recent crisis, we should point out that the numbers that we reported in Figure 1 for the VIX cannot be directly compared to those reported here for risk. The VIX numbers are risk measures that are multiplied by the square root of 12 in order to annualize them. For example, a VIX of 8 implies that equity price risk over the next month is 23%. 15

18 To determine the parameters in (3) we take the following steps. First, we conjecture (3). Then we compute the expectation of and covariance between and Substituting the result in (19), we can solve for the 4 parameters in the conjecture (3) for.togetherwith this gives the equity price of both countries: the Home and Foreign equity prices are respectively = +(1 ) and =. Details of the algebra are left to Appendix B. We first discuss the fundamental equilibrium and then turn to sunspot-like equilibria and switching equilibria featuring risk panics. In the fundamental equilibrium we have =, = ( ) and = ( ). The expressions for the equity prices of the two countries then become = 1 Ã! = 1 Ã! (31) (32) where = ( ), =, is the covariance between the Home (Foreign) and world dividend innovation. The latter is defined as +1 = +1 + (1 ) +1. It is natural to assume that. Two points are worth making with regards to this fundamental equilibrium. First, the constant terms depend on the covariance of the dividend innovation with the world dividend innovation. The higher is this covariance, the riskier the asset and therefore the lower the price. Second, equity prices only depend on domestic dividend innovations. Thus, in the fundamental equilibrium there is no contagion of shocks across countries. This is because we have shut down the regular channels of contagion through the interest rate and wealth. Both are held constant. Sunspot-like equilibria can be computed following the solution method discussed above. Appendix B computes the values of the four unknown parameters of the conjecture (3) for. The impact of the Foreign fundamental remains the same in the sunspot-like equilibrium as in the fundamental equilibrium. This reflects the fact that the Foreign dividend only plays a pure fundamental role (by assumption). The impact of the Home fundamental is more complex as it coordinates self-fulfilling shifts in risk. In order to understand what drives the solution of asset prices as a function 16

19 , it is useful to consider equation (19). It equates the difference in the excess payoff on Home and Foreign equity to the difference in their respective risk premia. Integrating forward, we have = X = ( ) (33) This equation holds both in the fundamental and sunspot-like equilibrium. A risk panic involves a switch from the fundamental equilibrium to the sunspot-like equilibrium. Since does not change during the panic, the change in during the panic is equal to = X = ( ) (34) When differs from, the panic affects the asset prices of the two countries differently. Equation (34) implies that the panic affects the two countries differently to theextentthatitaffects the relative riskiness of the assets differently. This in turn depends on how it affects the covariance between the equity payoffsofthe individual countries and the global payoff. If this covariance rises more for the Home than the Foreign country, so that the covariance between and increases, then risk increases more in the Home country and its equity price drops more during the panic. Write as the change in the covariance between and from the fundamental to the sunspot-like equilibrium. Using the results from Appendix B, we can write this as = (1 ) 2 ( ) 1 ( )+ " 2 (1 ) 2 # 1 +( ) (35) where a bar stands for the value of the parameter in the fundamental equilibrium. While (35) may appear a complicated expression at first, we will see that it provides insight into the various mechanisms that determine how a global risk panic spreads across the two countries. 17

20 5.2 The Two Factors Determining the Spread of a Panic At a general level, two factors determine how the risk panic spreads across the two countries: (1) an indeterminacy and (2) differences in fundamental hedging properties. We show in Appendix B that is indeterminate in the sunspot-like equilibrium. While we know the coefficient on 2 for the global equity price, how this average coefficient for the global price divides across the two assets is not determinate. This indeterminacy is due to another type of self-fulfilling beliefs. If agents believe that the Home country will be more affected by the panic, then their beliefs will be fulfilled in equilibrium. This can be seen from (35). If the Home equity price depends more negatively on 2, which happens when, then the relative risk of Home equity during the panic depends more positively on 2. This in turn justifies a more negative coefficient on 2 in the equilibrium Home equity price. To put it another way, when the Home equity price depends more negatively on 2 in the sunspot-like equilibrium, the covariance with the world equity price depends more positively on This increases its risk as a function of 2, whichinturnjustifies the more negative coefficient on 2 inthehomeequityprice. The second factor that determines how the panic spreads across the two countries is associated with the fundamental hedging properties of the assets. An asset is a more attractive hedge against global risk when its payoff covaries less with the global payoff. The fundamental payoff on an asset is the payoff in the fundamental equilibrium, when there are no self-fulfilling shifts in risk. As we have seen, the payoff in the fundamental equilibrium only depends on the asset s dividend. A higher covariance between the dividend of an asset and the global payoff in the sunspot-like equilibrium implies a weaker hedge. This leads to a larger impact of the panic. In order to illustrate these points more precisely and obtain a better understanding of the role of the parameters in driving the results, we firstconsiderthe case where the macro variables and have a large fundamental role ( is large) and then the case where they playasmallfundamentalrole( close to ). 15 This is because the world equity price also depends negatively on depends positively on 2. and the variance of 18

21 We will argue that the latter is more realistic. 5.3 Large Fundamental Role of Macro Variables In the case where is much above zero, so that the macro variables have a large fundamental role, we obtain two results. First, the impact of the indeterminacy on the equilibrium prices is limited. Second, the fundamental hedging properties of the assets are such that the panic affects the Home country more than the Foreign country. These two results are illustrated in Figure 4 for the same parameterization as used in Figures 2 and 3. The assumption = 1 implies that the macro variables play an important fundamental role. Figure 4 shows the impact of a risk panic on the equity prices and risk of both countries. As in Figure 3, the panic is assumed to take place when the fundamental is at its weakest ( = 1). Risk is again measured as the standard deviation of the equity price over the next period, divided by its current price. The results are shown as a function of,whichis indeterminate. All values of are considered for which the Home and Foreign equity prices are positive for all possible realizations of the state space ( ). Clearly, the Figure shows that the indeterminacy associated with has very little impact on the outcome and that the panic has a much larger impact on the Home country. The indeterminacy in has little impact because affects not only the coefficient on the quadratic term 2 in the asset prices, but also the coefficient on the linear term in.aswelower below zero, the more negative quadratic term for the Home price by itself increases the magnitude of a risk panic for the Home country. But the coefficient on the linear term in for the Home price will also be lower, which reduces the panic in the Home country when the Home fundamental is weak at the time of the panic ( ). The lower coefficient on follows from (35). As we lower below zero, the relative risk of the Home equity depends more positively on,whichlowers the coefficient on in the Home equity price and more so the larger. Since the world equity price depends negatively on, a more negative quadratic term in for the Home equity price increases the covariance between the Home 19

22 and world equity price as a linear function of. 16 This increase in risk of Home equity as a linear function of reduces the coefficient on in the equilibrium Home equity price. The second factor determining the extent of the panic spread is associated with the fundamental hedging properties of the assets. This leads to a larger impact of the panic in the Home country. The expression for in (35) provides some understanding of these hedging properties. Assume for now that =, so that the relative size of the linear coefficients remains the same as in the fundamental equilibrium (the deviation of from in equilibrium only serves to amplify the results that we are about to describe). Then, since and, it follows from (35) that both the constant term and the coefficient on are negative in the expression for. At =, the risk panic then makes the Home equity a relatively better hedge against global risk, leading to a smaller drop in the Home equity price. The negative linear coefficient on in implies that the weaker the Home fundamental at the time of the panic (the more negative ), the less attractive the Home equity as a hedge against global risk as a result of the panic. This leads to a larger risk panic in the Home country. 17 We find that unless is very close to, the second effect dominates, leading to a larger panic in the Home country. The intuition for these findings is associated with the extent to which the dividend of the equities is a hedge against the global risk faced by the agents in the sunspot-like equilibrium. Since in the high risk equilibrium, this reduces the relative risk of the Home equity as its dividend becomes negatively correlated with, making it a better hedge against global risk and reducing the relative size of the panic in the Home country. The quadratic term in the global equity price is 2. As, the covariance of this term with the Home dividend, 2 2, depends negatively on. This increases the relative risk of the Home asset when and more so the more negative. When the Home fundamental is sufficiently 16 This is because ( )=2 2 depends positively on. 17 These results are reinforced when we take into account that they imply as a negative linear term in implies a more positive coefficient on in the Home equity price than the Foreign equity price. Therefore, which in turn generates even more negative coefficients in both the constant and linear terms in the expression for. 2

23 weak this causes a larger drop in the Home equity price during the panic. Not surprisingly, this difference in hedging properties becomes small when is large. This is illustrated in Figure 5, which is the same as Figure 4 except that is set equal to.99 rather than.5. In that case the hedging properties of the Home and Foreign dividends are virtually identical and the panic affects the covariance with the global payoff virtually the same across the two assets. The size of the country that is the focal point for the panic does not matter much in these results. Even if the Home country is small relative to the global economy, it remains the case that the risk panic is much larger in the Home country. For example, setting = 1 and =, risk increases 55% and 12% in the Home and Foreign country respectively, while their respective equity prices drop by 8% and 46%. 5.4 Weak Fundamental Role of Macro Variables Next consider the case where is close to zero. The macro variables and then play a very limited fundamental role. This should be interpreted more generally as the case where macro variables have limited impact on payoffs ofthe assets in normal (non-panic) times. We believe this is a reasonable assumption for various reasons. First, it is well known that observed macro variables have limited explanatory power for asset prices outside of crisis episodes. 18 Second, even when macro variables affect the payoff directly through dividends, their overall impact on the payoff of equity (dividend plus future equity price) is relatively small. This is because most of the equity payoff volatility is driven by the price. Moreover, it is well-known since the early work by Shiller (1981) that dividend volatility has very limited explanatory power for equity price volatility. When the macro variables have a weak fundamental role ( close to ), the fundamental hedging properties of the assets become very similar. The only fundamental difference between the assets is their dividends, which are now very little affected by the macro variables. This by itself leads the panic to spread equally among the two countries. 18 For equity prices the limited role of public news was first illustrated by Roll (1988). For another asset price, the exchange rate, this disconnect from observed macro fundamentals has received even more attention. 21

24 But the other factor determining the spread of the panic across the two countries, the indeterminacy, now plays a larger role. As can be seen from (35), when is small only impacts relative risk through the quadratic term. There is no longer an offset through the linear term. This is illustrated in Figure 6, which has the same parameterization as for Figure 4 except that =. The panic affects the two countries equally when =. But the indeterminacy associated with now has a big impact on how the panic spreads across the two countries. Nonetheless a good case can be made for = as a plausible outcome. First, any deviation from = is entirely arbitrary. While it is possible for investors to believe that the panic should affect one country more than the other, there is no a priori reason for this to be the case as the assets are not different in any fundamental way. Second, and perhaps more convincing, =isjustifiedbasedonasmall cost of reshuffling portfolios. Assume that investors from both countries can invest in a domestic stock fund and a global stock fund. The latter has constant shares allocated to Home and Foreign equity. When =and =, the equity prices of both countries will drop the same percentagewise in the panic. From the equity market clearing conditions this implies that and remain in the same proportion (both falling). In this case all that investors need to do is sell off the global fund (get out of risky assets). There is no need to also change the allocation to the domestic fund. If the panic affects countries differently, then will change, requiring changes in the allocation to both funds. 5.5 On the Role of Financial Integration How does the extent of financial integration affect the spread of risk panics? So far we have assumed that the two countries are perfectly integrated, leading to identical portfolio shares. We now ask what role financial integration plays in the results. One way to address this issue is to consider the exact opposite assumption of financial autarky. We still assume that only the Home variable can affect risk perceptions. Home investors then invest in Home stocks and bonds, while Foreign investors invest in Foreign stocks and bonds. Consider market equilibrium 22

25 for Home equity. The optimal portfolio share by Home investors is = 1 +1 ( +1 ) (36) The market clearing condition is now Market clearing implies = (37) ( ) = ( ) (38) This has exactly the same form as the market equilibrium condition (18) for the world equity price. The only difference is that the fundamental is now the Home dividend +1 rather than the global dividend +1. The fundamental equilibrium takes the form = + while the sunspot-like equilibrium takes the form = (39) (4) The solution for the Foreign equity price can be derived analogously. fundamental equilibrium takes the form The = + while the sunspot-like equilibrium takes the form = (41) (42) The quadratic terms in the sunspot-like equilibria for Home and Foreign stocks are exactly the same as that for the world equity price under perfect integration in the sunspot-like equilibrium. A weighted averaged of the linear terms in also corresponds exactly to that in the global equilibrium. 23

26 We saw that under full integration two factors determine how much the panic affects individual countries: an indeterminacy and the fundamental hedging properties of the assets. These factors still affect the impact of the panic on the countries under financial autarky, but the nature of both the indeterminacy and hedging properties have changed substantially. Regarding the indeterminacy, there are now only two possibilities. A country is either fully hit by the panic or not at all. The continuum of equilibria associated with that we saw before no longer applies. The reason for this difference is that the market portfolio of risky assets now consists simply of domestic equity rather than the world equity portfolio. If a panic occurs, the magnitude of the panic is well-defined for the market portfolio of risky assets. This used to be the world equity price, but is now the equity price of the individual countries. The fundamental hedging properties of the assets have changed as well. Under full integration what matters is the covariance with the world equity payoff. Under financial autarky, it is the covariance with the domestic equity payoff that matters. The hedging properties now matter much less as agents can no longer reallocate their portfolio between Home and Foreign assets. This is illustrated in Figure 7, which shows the magnitude of the panic in both countries as a function of, assuming both are affected by the panic. We see that the results depend very little on, which under full integration had a large effectasitaffects the fundamental hedging properties of the assets. Figure 7 suggests that it is easy to account for a panic that spreads evenly across countries by assuming financial autarky. But that would be deceptive. There is an implicit assumption that both countries are hit by the panic simultaneously. That is not necessarily the case and one can make a reasonable argument that in fact it is not terribly likely. Theoretically it is possible that investors in the Foreign country panic in response to a macro variable in the Home country that has no fundamental relevance for the return on Foreign equity, the only risky assets held by Foreign investors. The Home macro variable is then a pure sunspot from the perspective of Foreign investors. In practice though, it would seem odd for investors to panic about something that has no relevance to them whatsoever. They would not necessarily even pay any attention. One final point is worth making regarding imperfect financial integration, which regards the findings by Rose and Spiegel (21) that the extent of cross-border 24