Financial Networks and Contagion - Cascades Larger but Less Frequent

4.6 Cascades are Larger but Less Frequent in More Integrated Systems Next, we consider the implications of increased integration in our simple model on the depth of cascades, as illustrated in Figure 5. 100 100 80 80 c = .5 c = .4 c = .3 c = .2 c = .1 60 40 20 c = .5 c = .4 c = .3 c = .2 c = .1 60 40 20 0 0 0.3 2.3 4.3 6.3 8.3 10.3 12.3 14.3 16.3 18.3 (a) Five levels of integration and the percentage of organizations failing as a function of expected degree (θ = 0.93, n = 100). 0.3 2.3 4.3 6.3 8.3 10.3 12.3 14.3 16.3 18.3 (b) Five levels of integration and the percentage of organizations failing as a function of expected degree (θ = 0.96, n = 100). Figure 5: How integration (the fraction c of a typical portfolio held by other organizations) affects the percentage of organizations failing, averaged over 1000 simulations. The horizontal axis corresponds to the diversification level (the expected degree in the random network of cross-holdings). The two figures work with different failure thresholds and depict how the size of cascades varies with the level of integration c ranging from 0.1 to 0.5. Figures 5a and 5b illustrate how the proportion of organizations that fail changes as the level of integration is varied from c = 0.1 to 0.5, for two different values of θ (the fraction of initial value that must be retained for an organization to avoid failure). As integration is increased the curves all shift upward and we see increased cascades. Although the effects in Figures 5a and 5b show unambiguous increases in cascades as integration increases, they work with levels of c ≤ 0.5 for which there is not so much of a tradeoff. In particular, for c ≤ 0.5 the initial firm whose asset price is dropped to 0 always fails (in the range of θ ≥ 0.8 considered in the simulations). As c is increased beyond 0.5, eventually the integration level begins to help avoid first failures, because each organization is less exposed to the failure of own proprietary asset. Then we see the tradeoff between I and II that is present as integration is varied (diversification is held constant, so III – having to do with the connectedness of the network – is not affected). We can see this in Figure 6. Figure 6 shows that as integration increases to very high levels, the percentage of first failures drops: organizations are so integrated that the drop in the value of an organization’s own investments is less consequential to it, and so there is no first failure. 100 80 c = .4 c = .6 c = .7 c = .8 c = .9 60 40 20 0 0.3 2.3 4.3 6.3 8.3 10.3 12.3 14.3 16.3 18.3 Figure 6: How integration affects the percentage of “first failures”: the percentage of simulations with at least one organization failing, for various levels of integration c from 0.4 to 0.9, with the horizontal axis tracking diversification (expected degree) in the network. The failure threshold is constant at θ = 0.8. To summarize, increasing integration (as long as it is not already very high) makes shocks more likely to propagate to neighbors in the financial network and increases contagion via the mechanism of II. For very high levels of integration, each organization begins to carry something close to the market portfolio, and so any first failure caused by the devaluation of a single proprietary asset becomes less likely. 4.7 The Consequences of Diversification and Integration: Analytic Results Fixing any given level of diversification and integration a network can typically be rewired to make it more or much less susceptible to cascades of failures. This makes it difficult to state general analytical comparative static results on integration and diversification. Moreover, even with randomization any network can appear with positive, albeit small, probability. Together these issues present a substantial challenge in proving that the non-monotonicity identified by the simulations holds generally. We can show that the patterns identified in the simulations can be established rigorously using a particular class of networks where degrees are fairly regular. Let G(d, n) be the set of all directed graphs with n nodes indexed by an expected degree d: if d is an integer, then let the network be regular with all nodes having both in- and out-degree equal to d. If d is not an integer then each node’s in-degree (and out-degree) is either bdc or dde; these proportions are chosen such that the overall average in and out degrees are d.30 30 A proportion d − bdc have outdegree (indegree) bdc and a proportion dde − d have outdegree (indegree) dde. Here we assume that d is rational and that n is such that this construction is feasible without violating A regular random network with degree d is a draw from G(d, n) uniformly at random. Each organization has a single asset of value 1 (so D = I and p = (1, . . . , 1)). We set all organizations’ thresholds v i to a common v ∈ (0, 1). For which values of d will a non-vanishing fraction of organizations fail in expectation following the failure of some asset i picked uniformly at random?31 Let Z(n, d) be the (random) number of organizations that fail following the failure of a randomly selected asset in a random regular network with degree d, and define q(d) by: E[Z(n, d)] . n→∞ n q(d) ≡ lim Let vemin (e vmax ) be the random variable equal to the lowest (highest) initial valuation in the realized network (before any failures and with all assets at value 1).32 Proposition 1. If one proprietary asset fails (uniformly at random), a non-vanishing fraction of organizations fail if and only if there are intermediate levels of both integration and diversification: 1. If c(1−c) v emin −v < 1 asymptotically almost surely33 , then q(d) = 0 for all d. 2. If c(1−c) v emax −v ≥ 1 a.a.s. then: (a) q(d) = 0 whenever d < 1, h j ki (b) q(d) > 0 whenever d ∈ 1, vec(1−c) a.a.s., max −v l m (c) q(d) = 0 whenever d ≥ veminc −v a.a.s.. Proposition 1 reaffirms the non-monotonicity of failures in diversification and integration that we saw in the simulations. The condition that vec(1−c) ≥ 1 requires intermediate levels max −v of integration, as the left-hand side expression tends to 0 as c tends to 0 or 1. Conditional on that intermediate integration condition being satisfied, the expected number of failures is non-monotonic as we vary the expected degree d. Only for intermediate d do a positive proportion of organizations fail following the failure of a single organization. The intuition for Proposition 1 is relatively straightforward. A standard result in the random graph literature is that there is a threshold expected degree at which, in the limit for large graphs, the graph structure suddenly changes from many small isolated components of vanishing size to a giant component of non-vanishing size. In the case of regular random graphs considered in this section, that threshold occurs at d = 1. Thus, for d < 1 contagion integer constraints. 31 The exercise here is exactly as in the simulations – see Step 4 in Section 4.4 above. 32 Even though nodes all have the same expected degrees and initial asset holdings, the realized network varies (e.g., with respect to in-degrees) and can lead to variations in the Aij and valuations. 33 We say a statement holds asymptotically almost surely (a.a.s.) if it holds with a probability approaching 1 as n → ∞. This is necessary here as vemin is a random variable. to a positive fraction oflorganizations following the failure of a single proprietary asset is m c impossible. Once d > vemin −v , a single organization’s failure will not cause a sufficient decrease in the value of any other organization to induce a second failure. There can be no contagion. Only for intermediate levels of d can a positive fraction of organizations fail in the cascade. It might seem that the threshold of d = 1 is low for a giant component in the network to form. This is in part due to the uniform randomness in link formation. In practice networks typically have less random structures and organizations have cross-holdings concentrated among a particular subset of others (who are also more likely to cross-hold each other). In such networks the threshold degree at which a giant component emerges will be higher. We examine this in more detail in Section 4.8. Next, we prove a general result about how integration affects the extent of cascades. For this result we relax the assumptions of the parametric model and permit any initial cross-holdings C. The result is also proved for an arbitrary vector β, an arbitrary vector of threshold values v, any direct holdings of assets D, and any underlying asset values p. Before stating the result we introduce the concept of fair trades. Fair trades are exchanges of cross-holdings or underlying assets that leave the (market) values of the organizations unchanged at current asset prices.34 More precisely, the matrices (C, D) and (C0 , D0 ) are said to be related by a fair trade at p if v = v0 , where v = Ap and v0 = A0 p; the matrix A0 is computed as in (5) with C0 and D0 playing the roles of C and D.35 Proposition 2. Consider (C, D) and (C0 , D0 ) that are related by a fair trade at p,36 and such that integration increases: A0ij ≥ Aij whenever i 6= j. Every organization that fails in the cascade at (C, D, p) also fails at (C0 , D0 , p). The reasoning behind Proposition 2 is as follows. As can be seen immediately from equation (5), when organization i fails and incurs failure costs βi , it is the ith column of A which determines who (indirectly) pays these costs. Increasing Aij for all i and j 6= i increases the share of i’s failure costs paid by each other organization. This increases the negative externality i imposes on each organization following its own failure. These other organizations are then more likely to also fail once i fails and so the number of organizations that fail in the cascade weakly increases. To tie Proposition 2 back to the effects of increasing integration in our simple model, the following proposition shows how increased integration weakly increases Aij for all i and all j 6= i and strictly increases at least one off-diagonal entry of A in each column. Proposition 3. Suppose that Cij = cGij /dj for some adjacency matrix G, with 0 < c ≤ 21 and each di ≥ 1. (In other words, a fraction c of each organization is shared out equally 34 So, absent failure, the values of organizations are the same before and after fair trades. We show in Section 6.1, that there are circumstances under which organizations may have incentives to undertake “unfair” trades because of the failure costs. 36 The definition of a fair trade ignores any failure costs – i.e., the values before and after a trade are calculated as if failures do not occur. This offers a clear benchmark. 35 among those who hold it according to G.) Then Aii is decreasing in c and Aij is increasing in c: 1. ∂Aii ∂c < 0 for each i; 2. ∂Aij ∂c ≥ 0 for all i 6= j; 3. ∂Aij ∂c > 0 for all i 6= j so that there is a path37 from j to i in G. Note that Proposition 3 does impose any assumptions on the underlying graph G other than each organization being cross-held by at least one other.38 By Proposition 3 increasing c weakly increases Aij for all i and all j 6= i and so corresponds to an increase in integration. As long as these trades are fair, which could be ensured by transfers of a numeraire, Proposition 2 can then be applied to show that more organizations will fail in a cascade as the system becomes more integrated.39 4.8 Alternative Network Structures Additional insights emerge from examining some other random graph models of financial interdependencies. 4.8.1 A Core-Periphery Model As a stylized representation of the interbank lending market, we examine a core-periphery model where 10 large organizations are completely connected among themselves, and each of 90 smaller organizations has one connection to a random core organization.40 Each of the ten large core organizations has proprietary assets with an initial value of 8. Each of the 90 peripheral organizations has proprietary assets with an initial value of 1. We then vary different facets of integration:41 the level CCC of cross-holdings of each core organization by other core organizations, the level CP C of cross-holdings of each core organization by peripheral organizations, and the level CCP of cross-holdings of each core bii , are as follows: organization by other core organizations. The remaining private holdings, C bii = 1 − CCC − CP C for a core organization, and C bii = 1 − CCP for a peripheral one. C 37 Recall footnote 8. Interestingly, the monotonicity identified in Proposition 3 does not always hold for c > 1/2. For such c, there are graph structures where further increases in c result in the immediate neighbors of i depending less on i. The increase in Aij for non-neighbors of i can come at the expense of both Aii and Aij for j such that Cij > 0. 39 As long as initial holdings of this numeraire are sufficiently large, there will always exist transfers of the numeraire that will make any given trade fair. 40 Soromaki et al. (2007) map the US interbank network based on the Fedpayments system. They identify a clique of 25 completely connected banks (including the very largest ones), and thousands of less connected peripheral regional and local banks. 41 Note that in this model the diversification (degree) structure is essentially fixed given the structure of ten completely inter-connected organizations and the peripheral ones each having one connection; the only randomness comes from the random attachment of each peripheral organization to a single core organization. 38 We first explore what happens when a core organization fails. As we see in the lefthand part of Figure 7a, the fraction of peripheral organizations that fail along with the core organization is increasing in CP C . Once the core organizations become sufficiently integrated among themselves, starting around CCC = .29, the core organization’s failure begins to cascade to other core organizations, and then wider contagion occurs. How far this ultimately spreads is governed by the combination of integration levels. 100 100 90 CPC = .06 80 .04 Percent of Orgs that Fail 70 60 .03 50 .5 80 .55 .45 70 60 50 40 40 30 .4 30 .02 20 .01 10 0.35 20 0 (a) One core organization’s asset initially fails 0.38 0.36 0.34 0.3 0.32 0.28 0.26 0.24 0.2 0.22 0.18 0.1 0.38 0.37 0.36 0.34 0.33 0.32 0.3 0.31 0.29 0.28 0.27 0.26 0.25 Core dependence on core (0.2 ≤ CCC ≤ 0.38) 0.16 .35 0 0.14 10 0.12 Percent of Orgs that Fail CCP = .6 90 .05 Core dependence on core (0.1 ≤ CCC ≤ 0.38) (b) One peripheral organization’s asset initially fails Figure 7: The consequences of failure in the core-periphery model. The horizontal axis is the fraction of each core organization cross-held by other core organizations (integration of core to core). In Figure 7a, curves correspond to different levels of cross-holdings of each core organization by peripheral organizations. In Figure 7b, they correspond to different levels of cross-holdings of peripheral organizations by core ones. The failure threshold is θ = .98. The more subtle effects are seen in in Figure 7b. The curves are layered in terms of integration between the core and periphery CP C , with increased integration leading to higher failure rates due to an initial failure of a peripheral organization. However, the magnitude of the failure rates is initially increasing in core integration (CCC < .25) and then decreasing in core integration (CCC > .25). Initial increases in core-integration enable contagion from one core organization to another, which leads to widespread cascades. Once core integration becomes high enough, however, core organizations become less exposed to their own peripheral organizations, and so then are less prone to fail because of the failure of a peripheral organization. 4.8.2 A Model with Segregation among Sectors Second, we considered a model that admits segregation (homophily) among different segments of an economy: for instance among different countries, industries, or sectors. In this model, there are ten different groups of ten nodes each. The key feature being varied is the relative intensity of nodes’ connections with others in their own group compared to other groups. This captures the difference between integration across industries and integration within industries. Varying this difference leads to the results captured in Figure 8. An obvious effect is that increasing homophily can eventually sever connections between groups of organizations and lead to lower contagion. However, as we see in Figure 8, the curves associated with different levels of diversification (expected degrees d) cross each other. With medium diversification (e.g., d = 3 or d = 5) there is initially a higher level of contagion than with higher diversification (e.g., d = 7 or d = 9). This is because organizations are more susceptible to each other with medium degrees than with high degrees and the network is still connected enough to permit widespread contagion. However, lower-degree networks fragment at lower levels of homophily than high degree networks. So at high levels of homophily, lower-degree networks are actually more robust. For example, once at least 95 percent of relationships are within own group (in expectation), then we see lower contagion rates with diversifications d = 3, 5 than with d = 7, 9. 100 d=5 Percent of Orgs that Fail d=3 80 d=7 60 d=9 40 20 d=1 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Homophily Index Figure 8: Ten groups of ten organizations each. The vertical axis is the fraction of organizations that fail as a function of the homophily. The horizontal axis is the fraction of expected cross-holdings in same-type organizations. Curves correspond to different diversification levels (expected degrees d). The failure threshold is θ = .96. 4.8.3 Power Law Distributions We also examined networks with more extreme degree distributions, such as a power-law distribution. Those results are described in detail in Section 10.2.1 in the Supplementary Appendix and are in line with the original regular networks. More extreme exponents in the power law actually lead to smaller contagions on average, but larger contagions conditional on some high-degree organization’s failure. 4.8.4 Correlated and Common Assets An important concern that emerged from the recent financial crisis is that many organizations may have investments with correlated payoffs, which could potentially exacerbate contagions, as many organizations’ values may be low at the same time. In Sections 10.2.2 and 10.2.3 we examine two variations with correlated values. As one might expect, increasing correlation increases the failure rate. The more interesting part is that the increase occurs abruptly at a particular level of correlation. We also examine a model in which organizations have some holdings of both an idiosyncratic and a common asset, with the possibility of leverage in holdings of the common asset. Some organizations are long the asset and others can be short. This results in some interesting patterns in cascades: even low leverage levels can lead to increased cascades by increasing organizations’ exposures. However, organizations that are short the common asset might escape a cascade triggered by a shock to that asset. 5 Avoiding a First Failure In this section we analyze the changes in the structure of interdependencies among organizations that can help prevent a first failure. It is conceivable that if an organization is at risk of eventual failure but away from its failure frontier, there could exist some fair trades (as defined in Section 4.7) that would unambiguously make that organization safer: prone to failure at a smaller set of prices. The proposition of this section shows that, at least when it comes to saving the most vulnerable organization, there are always tradeoffs: new holdings that avoid failure at one set of prices make it more likely at another set of nearby prices. To state this result, it is helpful to introduce some notation. We write organization i’s value assuming no failures at asset prices p, cross-holdings C and direct holdings D as vi (p, C, D). An organization i is closest to failing at positive asset prices p, cross-holdings C, and direct holdings D if there exists a (necessarily unique) λ > 0 such that at asset prices λp, organization i is in on its failure frontier, vi (λp, C, D) = v i , while all other organizations are solvent, vj (λp, C, D) > v j for j 6= i. Define q(p, C, D) := λp. Proposition 4. Suppose an organization i is closest to failing at asset prices p, crossholdings C, and direct holdings D. Consider new cross-holdings and direct holdings C0 and D0 resulting from a fair trade at p so that row i of A0 is different from that of A. Then, for any ε > 0, there is a p0 within an ε-neighborhood of q(p, C, D),42 such that i fails at prices p0 after the fair trade but not before: vi (p0 , C0 , D0 ) < v i < vi (p0 , C, D). 42 I.e. p0 such that ||p0 − q(p, C, D)||1 < ε. 6 Endogenously High Failure Costs and Thresholds due to Moral Hazard Whether an organization fails depends on its failure threshold. The impact that its failure has on other organizations depends on its failure costs. If organizations have some control over their failure thresholds and costs, then we might hope that they would choose to limit these. We show in this section that organizations can actually have incentives to increase both their failure costs and thresholds. 6.1 Organization Values Can Be Endogenous Our previous analysis has assumed that exchanges of cross-holdings or assets between organizations occur through fair trades at the current asset prices (recall Section 4.7). That was useful for illustrating the workings of the model and identifying the general effects of diversification and integration. However, the value to an organization of a trade depends not only on the value of the bundle of assets being received, but also on the implications of the trade for ensuing failures. Solvent or liquid organizations may have incentives to bail out insolvent or illiquid ones in order to avert a contagion (as pointed out, e.g., by Leitner 2005). For instance, it can be that by relinquishing some holdings (in either assets or another organization) an organization’s value actually increases! This means that we cannot value organizations solely based on their implied underlying asset holdings, but need also to consider the solvency of all other organizations. Trades can be “incentive compatible” when they are not “fair” (as evaluated by pricing the traded assets at the prices p and neglecting failure costs). We first illustrate the endogeneity of values through a simple example, and then explore the associated moral hazard issues. 6.2 An example Consider a world with two assets and two organizations. We begin with a case where asset holdings are D1 = (1, 0), D2 = (0, 1). Initial cross-holdings are C1 (0) = (0, 1/2) and C2 (0) = (1/2, 0), such that each organization has a one half stake in the other (so bii = 1/2). C From equation (5) it is easily verified that the organizations’ indirect holdings of the underlying assets are given by ! A= 2 3 1 3 1 3 2 3 . With the initial cross-holdings organization 1 receives 2/3 of asset 1’s value while organization 2 receives 1/3. The opposite is true for the asset 2. Let both asset 1 and asset 2 have price p1 = p2 = 10. Thus, without any failure costs, the values of the organizations would be v1 = v2 = 10. We let v 1 = 0 and v 2 = 11; let organization 2’s failure costs be β2 = 6. This means that if there are no changes in cross-holdings, from (5) the values of the two organizations are 8 and 6.43 Suppose now that organization 1 can make a transfer to organization 2. If organization 1 were to make a transfer of 1 to organization 2, organization 2 would not fail and the values of the two organizations would be 9 and 11. Thus by making a transfer to organization 2, organization 1 is able to increase its value from 8 to 9! Such a payment might be a direct transfer of cash or implemented through a trade in underlying assets or cross-holdings. For example, organization 1 might simply give organization 2 an increased stake in itself.44 Organization 1 is incentivized to “save” organization 2.45 Suppose we now extend the above example to permit organization 2 to have some control over its failure costs β2 and failure threshold v 2 . For simplicity we suppose that organization 2 can choose from β2 ∈ {0, 5, 10} and from v 2 ∈ {10, 11, 12, 13, 14}. Note that organization 2 can avoid failure without any intervention from organization 1 by choosing v 2 = 10. However, such a choice is not in the best interest of organization 2. We assume organization 1 will ‘save’ organization 2 if doing so weakly increases its value. If organization 2 needs saving (v 2 > 10), 1’s value after just saving 2 will be v10 = 10−(v 2 −10) while its value will be 10 − (β2 /3) if it does not save organization 2. Organization 1 will therefore save organization 2 if and only if v 2 > 10 and: β2 > (v 2 − 10). 3 The left hand side is the increase in value 1 receives from 2 remaining solvent and the right hand side is the cost of saving 2 – the transfer 1 must make to 2 for 2 to remain solvent. Table 1 below shows the transfers that organization 1 will make to organization 2 for the different values of v 2 and β2 that organization 2 can choose. These choices of v 2 and β2 then result in different values for organization 2 as shown in Table 2: As can be seen in Tables 1 and 2, for a fixed failure threshold, organization 2 is only saved when its failure costs are sufficiently large. Conditional on being saved 2’s value is increasing in his failure threshold and conditional on not being saved, organization 2’s value is weakly decreasing in his failure threshold. For sufficiently high failure thresholds organization 2 is never saved and for sufficiently low failure threshold organization 2 doesn’t fail. To maximize its utility after a bailout, organization 2 must set the highest failure costs it can and then carefully choose its failure threshold so that organization 1 is just incentivized to save it. In 43 Values before failure costs are 10 for both organizations. Organization 2 therefore fails and its failure cost of 6 reduces the effective value of its proprietary asset from 10 to 4. Organization 2 ultimately incurs 2/3 of this loss while organization 1 incurs 1/3. 44 One of the lower cost ways in which organization 1 might “save” organization 2 is to simply take over organization 2. 45 All the parameter values in the example can be varied slightly without generating a discontinuous change in the equilibrium. In this sense the example presented is not a knife-edge case. Failure Threshold v 2 10 11 12 13 14 Failure 0 0 0 0 0 0 Costs 5 0 1 0 0 0 β2 10 0 1 2 3 0 Table 1: Transfer made from 1 to 2. Failure Threshold v 2 10 11 12 13 14 Failure 0 10 10 10 10 10 Costs 5 10 11 6 2/3 6 2/3 6 2/3 β2 10 10 11 12 13 3 1/3 Table 2: Value of 2 after the transfer. this example, this requires organization 2 choosing a failure threshold of 13 and failure costs of 10. Of course, if organizations can commit not to bail each other out, then these moral hazard problems can be avoided. However, firms have a fiduciary obligation to maximize shareholder value, even if this involves bailing out a failing organization they have a stake in. This can make it difficult for organization to commit not to bail out another, and absent such a commitment device, organizations can have strong incentives to increase their failure costs and manipulate their failure thresholds. The moral hazard problem in this example occurs absent any intervention by the government. Failure costs alone are sufficient for moral hazard problems to arise.46 It arises because organizations do not fully bear their failure costs. As other organizations pay (indirectly through the devaluation of holdings in i) some of organization i’s failure costs (βi ), these other organizations will be prepared to expend resources bailing out i. As the proportion of i’s failure costs that i pays is given by Aii , a natural measure of the severity of the moral hazard problem is 1 − Aii . When 1 − Aii = 0 there is no moral hazard problem and the extent of the moral hazard problem is monotonic in 1 − Aii = 0 in the following sense: If 1 − Aii is increased by redistributing shares of i from outside shareholders to other organizations, such that all other organizations’ claims on i weakly increase, any organization that previously would have bailed out i faces weakly stronger incentives to bail out i while organizations who previously would not have found it profitable to bail out i may now find it profitable to dos o. We saw in Section 3.1 that cascades of failure can occur, amplifying and propagating shocks if failure costs are sufficiently large and failure thresholds are sufficiently high. The analysis in this section has identified an endogenous mechanism through which organizations are willing to invest in increasing their failure costs and possibly their failure thresholds. Although such investments are valuable to an organization only in the event that it is bailed out, and in an uncertain world such bailouts may or may not be forthcoming, the misalignment of incentives due to the moral hazard problem can nevertheless result in systems endogenously conducive to cascades of failure. 46 This moral hazard problem also distorts organizations’ investment decisions, both in terms of their investments in risky projects and their investments in cross-holdings. 7 Illustration with European Debt Cross-Holdings We close the paper with an illustration of the model with data on the cross-holdings of debt among six European countries (France, Germany, Greece, Italy, Portugal and Spain). We include this as a proof of concept, and emphasize that the crude estimates that we use for cross-holdings make this noisy enough that we do not see the conclusions as robust, but merely as illustrative of the methodology. Moreover, in the simulations, when a country “fails”, it defaults on 50% of its obligations to foreign countries. Such losses may corresponds more closely to a sequence of disorderly bankruptcies than the more orderly writing down of Greek debt that has occurred over time. For the purposes of this illustrative exercise, we treat these countries as a closed system with no holdings by other countries outside of these six. 7.1 The Data Data on the cross-holdings are for the end of December 2011 from the BIS (Bank for International Settlements) Quarterly Review (Table 9B). The data used for this exercise are the consolidated foreign claims of banks from one country on debt obligations of another country. The data looks at the immediate borrower rather than the final borrower47 when a bank from a country different from the final borrower serves as an intermediary.48 This gives following raw cross-holdings matrix, where the column represents the country whose debt is being held and the row is the country which holds that debt. So, for example, through their banking sectors Italy owes France $329,550M, while France only owes Italy $40,311M.   (France) (Germany) (Greece) (Italy) (Portugal) (Spain)  (France) 0 174862 1960 40311 6679 27015     (Germany) 198304  0 2663 227813 2271 54178     (Greece) 39458 32977 0 2302 8077 1001  .    (Italy) 329550 133954 444 0 2108 29938     (Portugal) 21817 30208 51 3188 0 78005  (Spain) 115162 146096 292 26939 21620 0 To convert the above matrix into our fractional cross-holdings matrix, C, we then estimate the total amount of debt issued by each country. To do this, we estimate the ratio of 47 Which basis is appropriate is discussed in section 10.8 of the Supplementary Appendix. For illustrative purposes, we examine holdings at a country level, so that all holdings of Italian debt by banks or other investors in France are treated as being held by the entity “France,” and we suppose that substantial losses by banks and investors in France would lead to a French default on National debt. It would be more accurate to disaggregate and build a network of all organizations and investors, if such data were available. 48 Greece .18 France Spain .11 Germany .05 .12 .11 .13 Italy Portugal Figure 9: Interdependencies in Europe: The matrix A, describing how much each country ultimately depends on the value of others’ debt. The widths of the arrows are proportional to the sizes of the dependencies; the area of the oval for each country is proportional to its underlying asset values. foreign to domestic holdings by 1/3, in line with estimates of by Reinhart and Rogoff (2011). b − C)−1 implies: Then, the formula A = C(I   (France) (Germany) (Greece) (Italy) (Portugal) (Spain)  (France) 0.71 0.13 0.13 0.17 0.07 0.11     (Germany) 0.18 0.72 0.12 0.11 0.09 0.14      A =  (Greece) 0.00 0.00 0.67 0.00 0.00 0.00  .    (Italy) 0.07 0.12 0.03 0.70 0.03 0.05     (Portugal) 0.01 0.00 0.02 0.00 0.67 0.02  (Spain) 0.03 0.03 0.02 0.02 0.14 0.68 The matrix A can be pictured as a weighted directed graph, as in Figure 9. The arrows show the way in which decreases in value flow from country to country. For example, the arrow from Greece to France represents the value of France’s claims on Greek assets, and thus how much France is harmed when Greek debt loses value. The areas of the ovals represent the value of each country’s direct holdings of primitive assets. All dependencies of less than 5% have been excluded from Figure 9 (but appear in the table above). We treat the investments in primitive assets as if each country holds its own fiscal stream, which is used to pay for the debt, and presume that the values of these fiscal streams are proportional to GDP. Thus, D = I and p is proportional to the vector of countries’ GDPs.49 Normalizing Portugal’s GDP to 1, the initial values in 2011 are:      v0 = Ap =     7.2 0.71 0.18 0.00 0.07 0.01 0.03 0.13 0.72 0.00 0.12 0.00 0.03 0.13 0.12 0.67 0.03 0.02 0.02 0.17 0.11 0.00 0.70 0.00 0.02 0.07 0.09 0.00 0.03 0.67 0.14 0.11 0.14 0.00 0.05 0.02 0.68           ·       11.6 14.9 1.3 9.2 1.0 6.3            =       12.7 14.9 0.8 9.4 0.9 7.1 (France) (Germany) (Greece) (Italy) (Portugal) (Spain)      .    Cascades To illustrate the methodology, we consider a simple scenario. The failure thresholds are set to θ multiplied by 2008 values.50 If a country fails, then the loss in value is v i /2, so that half the value of its debt is lost. We examine the best equilibrium values for various levels of θ. Greece’s value has already fallen by well more than ten percent, and so it has hit its failure point for all of the values of θ. We then raise θ to various values and see which cascades occur. Value of θ .9 .93 .935 First Failure Greece Greece Greece Second Failure Portugal Third Failure Spain Fourth Failure France, Germany Fifth Failure Italy .94 Greece, Portugal Spain France Germany, Italy Table 3: Hierarchies of Cascades in the Best Equilibrium Algorithm, as a Function of the Failure Threshold θ. We see that Portugal is the first failure to be triggered by a contagion. Although it is not particularly exposed to Greek debt directly, the fact that its GDP has dropped substantially means that it is triggered once we get to θ = .935. Once Portugal fails, then Spain fails due to its poor initial value and its exposure to Portugal. Then the large size of Spain, and the exposure of France and Germany to Spain cause them to fail. Pushing θ up to .94 causes Portugal to fail directly, and then leads to a similar sequence. (Increasing θ further would not change the ordering; it would just cause some countries to fail at earlier waves.) Interestingly, Italy is the last in each case: this is due to its low exposure to others’ debts. Its GDP is not particularly strong, but it does not hold much of the other countries. 49 We work in the scale of GDPs – that is, we do not carry around an explicit constant of proportionality relating the value of the fiscal streams p to the value of GDP; we merely take the entries of the vector p to be the GDP values. 50 Those values are calculated in the same way as the values above, being proportional to 2008 GDP values instead of 2011 and again normalized by setting Portugal’s 2011 to 1. Clearly the above exercise is based on rough numbers, ad hoc estimates for the default thresholds, and a closed (six country) world. Nonetheless, it illustrates the simplicity of the approach and makes it clear that much more accurate simulations could be run with access to precise cross-holdings data, default costs and thresholds.51 We re-emphasize that the cascades are (hopefully!) off the equilibrium path, but that understanding the dependency matrix and the hierarchical structure of potential cascades can improve policy interventions and also help predict the structure of bailouts. 8 Concluding Remarks Based on a simple model of cross-holdings among organizations that allows discontinuities in values, we have examined cascades in financial networks. We have highlighted several important features. First, diversification and integration are usefully distinguished as they have different effects on financial contagions. Second, both diversification and integration entail tradeoffs in how they affect contagion. These tradeoffs result in nonmonotonic effects where middle ranges are the most dangerous with respect to cascades of failures. The tradeoffs can also be related to important realistic aspects of a network, such as its coreperiphery and segregation structure. Finally, potential cascades introduce interesting moral hazard problems, where organizations can have incentives to increase their failure thresholds or costs in order to receive larger bailouts at an interim stage when they are close to failure. A fully endogenous study of the network of cross-holdings and of asset holdings is a natural next step.52 The moral hazard issues that we have demonstrated suggest that modeling the endogenous structures will be delicate and that a simple general equilibrium approach will not suffice. This presents interesting challenges for future research. 9 Appendix: Proofs Proof of Lemma 1: One representation of A is as the following infinite sum, known as the Neumann series: b A=C ∞ X p=0 b +C b Cp = C ∞ X Cp (6) p=1 bii and that there is equality if and only if there are It follows immediately that Aii ≥ C b and C such that C bii =  for no cycles involving i. Part (2) can be proved by considering C bii → 0 but A all i and Cij = (1 − )/(n − 1) for all i and all j. Taking  → 0, we have C tends to the matrix with all entries 1/n. 51 Of course, a linear cross-holdings structure is also an important simplification. A further refinement would involve modeling the holdings in greater detail, and solving for the ultimate dependencies of organizations on assets (analogous to computing the A matrix) in that more complicated world. 52 For some analyses of network formation in other financial settings, see Babus (2009), Cohen-Cole, Patacchini and Zenou (2012), and Baral (2012). Proof of Proposition 1: A contagion of failures flows through organizations’ indegrees. When organization i fails organization j might also fail if Cji > 0. We therefore wish to know when there is a giant component in indegree such that following only organizations’ indegrees a non-vanishing fraction of organizations can be reached in expectation from an organization selected uniformly at randomly. From results from Newman, Strogatz and Watts (2001), for a regular random graph with indegree and outdegrees in either bdc, dde, a giant component (in indegree) appears for: X X (2kk 0 − k − k 0 )θkk0 ≥ 0, k∈{bdc,dde} k0 ∈{bdc,dde} where θkk0 is the proportion of nodes with indegree k and outdegree k 0 . Simplifying the above equation there will be a giant component for d ≥ 1 and so for all d < 1 it follows that q(d) = 0. Suppose that organization j has holdings in organization i and recall that if organization i fails, organization j’s value will decrease by Aji . A lower bound on Aji for a regular random graph can be found by considering a tree network. If contagion would occur within a tree then any feedback effects can only increase contagion. We therefore have that Aji ≥ c(1−c) . dde Organization j will therefore fail, following the failure of organization i if: vemax − c(1 − c) < v dde We therefore get contagion for sure within a component for any dde < vec(1−c) . Combining  max −v  c . the above results we therefore have that q(d) > 0 for d ∈ 1, b vec(1−c) max −v c Finally, we derive an upper bound on possible contagion. From Lemma 2, Aji ≤ bdc for each i 6= j. It follows that there will be no contagion if: c vemin − > v bdc c bdc > vemin − v There is thus no contagion for d > d veminc −v e. This completes the proof of Proposition 1. Proof of Proposition 2. Following the failures of organizations Zk−1 , the value of organization i is: n n n X X X vi (Zk−1 ) = Aij Djk pk + Aij (Djk pk − βj ) = vi (∅) − Aij βj . j ∈Z / k−1 j∈Zk−1 j∈Zk−1 As fair trades hold constant vi (∅), this equation shows that the value of organization i given failures Zk−1 is weakly decreasing in Aij for all j 6= i. Holding fixed the hierarchies in which all other organizations fail, after a weak increase in Aij for all i and all j 6= i, if organization i failed in hierarchy k it will now fail (weakly) sooner in hierarchy k 0 ≤ k and if organization i did not fail in any hierarchy it might now fail in some hierarchy. Moreover, as failures are complementary, if organization i fails strictly sooner in hierarchy 0 k weakly more organizations will be included in all subsequent failure sets Zk00 , for all k 00 > k 0 . This is because more failure costs are summed over in the above equation when calculating a firm’s value in each failure hierarchy. Proof of Proposition 3: Let C = Gd−1 and note that by the Neumann series we may write ∞ X t A = (1 − c) ct C t=0 ∂A = (1 − c) ∂c ∞ X t=1 t−1 tc t C − ∞ X t t c C = −I + t=0 ∞ X t (t(1 − c) − c)ct−1 C . t=1 t Since c ≤ 12 , every term in the summation over t is nonnegative. Moreover, ct−1 C has a strictly positive entry whenever there is a path of length t from i to j in C, or equivalently in G. This shows claims 2 and 3 in the proposition. To verify claim 1, note that every column of A sums to 1. Claim 3 along with the assumption that every node in G has at least one neighbor shows that every column has an off-diagonal entry that strictly increases in c; and no off-diagonal entry decreases by claim 2. So the diagonal entry strictly decrease in c. Proof of Proposition 4. As any trade involving organization i must change composition of i’s dependency on underlying assets, after any trade there must exists a price vector p00 within an  neighborhood of λp, such that vi (p00 , C0 , D0 |Z = ∅) 6= vi (p00 , C, D|Z = ∅) = v i . For the Proposition to be false, it must then be that vi (p00 , C0 , D0 |Z = ∅) > vi (p00 , C, D|Z = ∅). Define price p0 such that 12 p00 + 21 p0 = λp. As ||p0 − λp||1 = ||p00 − λp||1 and p00 is within an  neighborhood of λp, p0 is also within an  neighborhood of λp. By the linearity of firms’ values, absent any failure, and as the trade was fair 1 1 vi (p00 , C0 , D0 |Z = ∅) + vi (p0 , C0 , D0 |Z = ∅) = vi (λp, C0 , D0 |Z = ∅) 2 2 = vi = vi (λp, C, D|Z = ∅) 1 1 = vi (p00 , C, D|Z = ∅) + vi (p0 , C, D|Z = ∅) 2 2 Thus as vi (p00 , C0 , D0 |Z = ∅) > vi (p00 , C, D|Z = ∅), vi (p0 , C0 , D0 |Z = ∅) < v i < vi (p0 , C, D|Z = ∅).