University:
Boston University
Course:
QST FE 200 | Principles of Finance
Academic year:

2024
Views:

185

Pages:

13
Author:

Carleigh Vargas

10 Supplementary Appendix: For Online Publication 10.1 Debt and other Liabilities Throughout the paper we suppose that organizations have linear cross-dependencies. Although such dependencies behave like equity when organizations are away from their failure frontiers, the focus of this paper in on cascades of failures and so the region we are interested in is around organizations failure thresholds. In this region, we view our linear cross dependencies as approximations for debt cross-holdings. This is shown in Figure 10 below: 𝐴𝑖𝑗 𝑫𝒑 − 𝒃 𝒗 𝑗 Value of debt holding in j 0 Discontinuous drop Value of debt holding in j 0 Value of j’s asset (𝑝𝑗 ) (a) Linear Model Value of debt holding in j 0 Value of j’s asset (𝑝𝑗 ) (b) Debt – no uncertainty Value of debt holding in j Value of j’s asset (𝑝𝑗 ) (c) Debt – uncertainty Value of debt holding in j 0 0 Value of j’s asset (𝑝𝑗 ) (d) Region of interest Value of j’s asset (𝑝𝑗 ) (e) Approximation Figure 10: Approximation of a debt model by a linear model. The model is easily adapted to general sorts of cross liabilities beyond the linear crossholdings. These could reflect any sort of debt or other contractual agreement, which could be contingent on the market value of the organizations (for instance, the debt cannot exceed the organization’s market value if there is limited liability). If we let Lji (V) be the amount owed to j by i as a function of book value, and L the corresponding matrix (with 0’s on its diagonal, as an organization cannot have debt to itself) book values become: Vi = X j6=i Cij Vj + X (Lij (V) − Lji (V)) + j6=i X k 41 Dik pk − βi Ivi 1/d negative holdings of the common asset are possible. We then adjust the value of each organization’s proprietary asset so that their total initial asset value is 1 – as before. Next, we group the organizations into 10 groups of 10, as in our homophily simulations. However, unlike before, this grouping was not entirely random. A parameter ρ governs the extent to which the grouping is random versus based on organizations’ positions in the common asset. When ρ = 0 it is entirely random. When ρ = 1 it is based entirely on holdings of the common asset. As before a homophily parameter h governs the relatively likelihood of links within groups versus across groups. Thus, when h > 0 and ρ > 0 organizations with similar exposures to the common asset are more likely to be linked to each other. We can now look at the effect of correlating risks in a system with homophily/segregation by holding h constant and comparing ρ = 0 to ρ > 0. And in a system with no homophily/segregation, we can see the effect of correlated risks by reducing the leverage parameter – exposure to the common asset becomes more correlated as the parameter ` decreases, with perfect correlation for ` = 0. Interestingly, in this model, correlating risks by adjusting the ρ parameter has a minimal impact regardless of homophily. The key parameter that had a very substantial impact is the leverage parameter. For even small shocks to the common asset of 5 percent, large cascades occur (across the range of other parameters) for ` > 1.5. Note that for these higher levels of leverage, the correlation in exposure to the common asset is actually lower. The threshold value of the parameter ` for which a large cascade occurs decreases in the size of the shock. However, for large shocks to the common asset of 20 percent, increasing the parameter ` reduces the extent of the cascade. Intuitively, a large parameter ` means 44 that some organizations have negative holdings of the common asset (short positions - e.g., Goldman Sachs in the 2008 crisis) and their value can increase sufficiently for them to survive the failure of many other organizations. 10.3 Using the Dependency Matrix This section validates the direct use and manipulation of the dependency matrix A. Proposition 5 shows that absent any discontinuities (i.e. with failure costs of zero for all organizations), any change in C or A can be represented as changes in D alone. Proposition 6 then identifies a simple necessary and sufficient condition for the A to be valid – that is for there to exist direct cross-holdings C it can be derived from. This second result allows one to directly manipulate A. Proposition 5. Assuming there are no failures, for any D, C there is a D0 , C0 with C being b being the identity that results in the same organization values the matrix of zeros and C for any underlying asset prices p. Similarly, for any A, D there exists D0 with C being the b being the identity that results in the same organization values for any matrix of zeros and C underlying asset prices p. Proposition 5 follows directly from letting b − C)−1 )D = AD. D0 = (C(I Thus, in the absence of failure, it is simply the indirect holdings of underlying assets that matter, and so one can equivalently work with them in understanding organizations’ values. The proposition implies that instead of considering trades in cross-holdings, when we are working to understand what might trigger a first failure (so that none have yet occurred) there is always some trade in underlying assets that replicates the trade in cross-holdings. However, in practice, at least some of the underlying assets are non-tradeable and so can only be held through cross-holdings.55 To work in the underlying asset space we therefore want to know when trades of underlying assets can be replicated through an exchange of cross-holdings, keeping the organizations’ asset holdings (D) constant. Proposition 6 provides necessary and sufficient conditions on A for it to be a valid representation of some C. b + C (i.e. one that is column Proposition 6. There exists a valid cross-holdings matrix C stochastic, contains non-negative entries and has strictly positive entries on the lead diagonal) −1 that generates A if and only if A−1 ii > 0 for all i and Aij ≤ 0 for all i and all j 6= i. 55 If all underlying assets were freely tradeable then there would be no reason for any cross-holdings. Any portfolio of claims to underlying assets held through cross-holdings could be replicated as direct holdings and without any risk of devaluation through failure. 45 Proof of Proposition 6: Recall from (5) that b − C)−1 . A = C(I If A is invertible, manipulating this equations yields that: b − C)−1 )−1 A−1 = (C(I b −1 A−1 = (I − C)C b = I−C A−1 C b C = I − A−1 C (9) If we can represent the right hand side of this equation just in terms of the A matrix, we will have found a way to map an A matrix into a C matrix. We will then just need to find conditions under which the C matrix we are deriving is column stochastic and has all non-negative elements (and strictly positive elements on the lead diagonal) when added to b When these conditions are met, the A matrix will have an associated valid C matrix it C. can be derived from and we can work directly with it. b is a diagonal matrix: Considering entry (i, i) of this matrix equation, and recalling that C bii . Cii = 1 − (A−1 )ii C bii = 1/(A−1 )ii . This puts the left hand side of (9) Since Cii = 0 by assumption, we find C b be the matrix thus defined, set in terms of just A. Letting C b S = I − A−1 C. (10) Thus the matrix A can be derived from a valid C (equal to the S matrix in equation 10) b is column stochastic such that column j of S sums to 1 − C bjj and if and only if (i) S + C b are non-negative and the lead diagonal is strictly positive. (ii) all entries of S + C b is column stochastic. All valid A matrices are column stochastic First we prove that S+ C and so A−1 is also column stochastic. To see this let 1 be the vector of ones such that 1A = 1. This is the definition of A being column stochastic. Now post multiply by A−1 . We then find that 1 = 1A−1 and so A−1 is also column stochastic. P b to bjj = C bjj Pn (A−1 )ij = C bjj . Adding C As A−1 is column stochastic, ni=1 (A−1 )ij C i=1 both sides of equation 10 we then have that: n X i=1 bij = Sij + C n X bjj + C bij = 1 − C bjj + C bjj = 1 Iij − (A−1 )ij C i=1 b is always column stochastic, there exists a valid C representation of A if and As S + C 46 b are non-negative and all entries of C b are strictly positive. only if all entries of S + C From equation 10 the elements of S are: −1 −1 1 1 bii = 1 − (A )ii + Sii + C = −1 −1 −1 (A )ii (A )ii (A )ii bii = − (A )ij , Sij + C (A−1 )jj and for all i and all j 6= i. Thus all elements of S are well-defined and weakly positive if and only if (A−1 )ii > 0 and (A−1 )ij ≤ 0 for all i and all j 6= i. 10.4 Bounds on the Dependency Matrix We provide some useful upper bounds on the possible values of the dependency matrix A. bkk , and Let c = maxk 1 − C bii c max Cik Aij = C bkk 1 − c k6=i 1 − C and bii Aii = C c Cik 1+ max bkk 1 − c k6=i 1 − C . bii = 1 − c for all i, so Lemma 2. Aij is an upper bound on Aij for all i and j. Therefore, if C that each organization holds c of its holdings in other organizations and 1 − c in itself, then Aij ≤ maxk6=i Cik for each i and j 6= i, and Aii ≤ (1 − c) + maxk6=i Cik . Proof. Recall that b − C)−1 , A = C(I or alternatively that b A=C ∞ X Ct . t=0 Cij bjj . 1−C Let C be the matrix for which we set C ij = Then, b A≤C ∞ X t ct C . t=0 t−1 Note that C is a column stochastic matrix. It follows that C is also a column stochastic t t−1 for any t ≥ 1 (because it is a column-stochastic matrix raised to a power). Write C = CC . t−1 From this, given the fact that C is column stochastic for each t, it follows that the ij-th t entry of C is no more than maxk6=i maxk6=i 1−CCik b . Also, note that for t = 0, the ij-th entry kk 47 t of C when j 6= i is 0. Thus, for i 6= j, bii Aij ≤ C ∞ X ct max C ik . t=1 Then since 1/ P∞ t=1 k6=i ct = c/(1 − c) it follows that bii c max C ik , Aij ≤ C 1 − c k6=i 0 This is the claimed expression for j 6= i. For j = i we also have the ii-then entry of C being bii = 1 − c for all i follow directly. 1. The simplifications for C 10.5 Multiple Equilibria and Discontinuities in Organizations’ Values In the absence of any failure issues, equation (5) is a standard pricing equation describing how the values of organizations depend on the primitive asset values v = A [Dp]. The novel and interesting part of equation (5) comes from the failure costs b(v). These terms generate several complexities that equation (5) illuminates. In particular, the presence of failure introduces several forms of discontinuity which result in multiple equilibria. Discontinuities in the value of a given organization i can come from two sources. The basic form is that the failure costs of organization i can be triggered when the values of other organizations or underlying assets fall which then lead i to hit its failure threshold. The other form is due to another organization, in which i has cross-holdings, hitting its failure threshold, which then leads to a discontinuous drop in the value of i’s holdings and consequently its value. In terms of multiplicities of equilibria, there are also different ways in which these can occur. The first is that taking other organizations’ values and the value of underlying assets as fixed and given, there can be multiple possible consistent values of organization i that solve equation (5). There may be a value of vi satisfying equation (5) such that 1vi ≤vi = 0 and another value of vi satisfying equation (5) such that 1vi ≤vi = 1; even when all other prices and values are held fixed. This generates the a first source of multiple equilibria corresponding to the standard story of self-fulfilling bank runs (such as those in classic models such as Diamond and Dybvig (1983)). The second is the interdependence of the values of the organizations: the value of i depends on the value of organization j, while the value of organization j depends on the value of organization i, and given the discontinuities possible in prices due to failure costs, there can be multiple solutions. There might then be two consistent joint values of i and j: one consistent value in which both i and j fail and another consistent value in which both i and j remain solvent. This second source of multiple equilibria is different from the individual 48 bank run concept, as here organizations fail because people expect other organizations to fail, which then becomes self-fulfilling. Although governments may be able to give assurances such as insuring deposits that manipulate expectations regarding the self-fulfilling value of a single organizations, it seems more difficult to control expectations when an organization’s value depend on the expected values of many other organizations. For example, an organization’s value can depend on the expected value of an organization that falls under the regulatory oversight of another government. Suppose organizations A and B have cross-holdings in each other and organization B also has cross-holdings in organization C. Investors in organization A may then become less confident investors will keep their money in organization B, or less confident the investors in B have confidence in them or in the investors in C, and so on. 10.6 Including Multiple Equilibria Due to Bank Runs This section extends the example in section 3.2.2. The same parameter values are used in Figure 13 as were used in section 3.2.2 and Figure 2, although the scale of the axis has been adjusted. As can be seen the scope for multiple equilibria increases a great deal once bank runs are permitted. Note i’s failure threshold conditional on i failing is shift out twice as far as i’s failure threshold conditional on j failing because i effectively pays 2/3 of his failure costs but only 1/3 of j’s. As shown in Figure 13d there is a large set of prices for which it is consistent for both 1 and 2 to both fail such that total failure costs of 100 are incurred and failure costs of 50 are paid by each organization. 49 p2 p2 FF1|1 FF1|2 FF1 FF1 Multiple Equilibria due only to interdependencies 100 100 50 50 FF2 0 Multiple Equilibria of 1 due only to bank runs 50 FF2|1 0 p1 100 (a) Multiple Equilibria due only to interdependencies and without bankruns 50 (b) Multiple Equilibria of 1 due to bank runs p2 p2 FF1|1 FF1|1,2 All Regions of Multiple Equilibria FF1|2 Multiple Equilibria of 2 due only to bank runs 100 FF1 100 50 FF2|1,2 50 FF2 0 p1 100 50 100 FF2|2 FF2 0 p1 (c) Multiple Equilibria of 2 due to bank runs 50 FF2|1 100 FF2|2 p1 (d) All Multiple Equilibria Figure 13: The total set of multiple equilibria is much larger once bank runs are permitted. Nevertheless, the interdependencies provide an additional source of multiplicity even when bank runs are permitted. 10.7 Best case and worst case trade-offs We now return to considering multiple equilibria due only to the interdependencies between organizations, and turn off the multiplicity due to standard self-fulfilling bank runs by assuming that assurances such as insuring deposits can control expectations. We identify a the tension between limiting failures in the best case equilibrium and worst case equilibrium. Trades that prevent any organizations failing in the best case outcome can also make more organizations fail in the worst case outcome. We say that cross-holdings are best-case safest when they maximize the percentage decrease in asset prices that would be necessary for a first organization to fail. More formally, cross-holdings are best-case safest at D, p if in the best equilibrium all organizations survive and the cross-holdings solve the following maximization problem: 50 max min C i vi (C, p) − v i vi (C, p) It is possible for all organizations to fail if the total value of primitive assets less all failure costs can be allocated in a way that leaves all organizations XX X below Xtheir failure thresholds. Such an allocation exists if and only if: Dik pik − βi < vi. i k i i Proposition 7. Suppose organizations’ failure costs are X a constant proportion γ of the value of their direct asset holdings such that βi = γ Dik pk and it is possible for all k organizations to fail. Then all asset holdings that are best-case safest at prices p also result in all organizations failing in worst-case equilibrium. Proof. If no organization fails, then their market values are: v = ADp. In order to be best case safest, we need to maximize the percentage loss that any organization can suffer without failing. As all assets have positive value, this requires equalizing the proportional loss in value each organization must suffer to fail. If this was not equalized, reallocating assets at the margin from the set of organizations furthest from their failure constraints to those organizations closest to them would increase the percentage loss in value that any organization can suffer without failing. Thus, in a best case safest asset allocation: v = ADp = θv for some scalar θ. As by assumption it is possible for all organizations to fail at the same time and so: XX i Dik pk − X βi < i k X vi i As failure costs are a constant proportion of the value of organizations’ direct asset holdings and as A is column stochastic: XXX X (1 − γ)Aij Dik pk < vi j i i k Using equation 10.7: (1 − γ)θ X vi < i and so (1 − γ)θ < 1. Suppose now all organizations fail. In this case: 51 X i vi v = A(Dp − β) = ADp(1 − γ) = (1 − γ)θv < v Thus, in the worst case equilibrium, all organizations fail. Proposition 7 illustrates that if trades are undertaken with the sole purpose of achieving the best case safest outcome, these same trades can also result in the worst possible outcome occurring in the worst-case equilibrium – all organizations failing. 10.8 Details: Cascades of Default in Europe We first discuss the data used and then provide the calculations of the v i s. There is data available from the Bank of International Settlements on aggregated cross-liabilities between countries on both an immediate borrower basis (which reports all contracts) and a final borrower basis (which nets out contracts with intermediaries replacing them with contracts between the final parties). If two parties trade through an intermediary we assume that intermediary writes separate contracts with the two parties (or acts as some kind of guarantor). In this case default by the intermediary would affect both parties and it is appropriate to use the intermediate borrower basis data.56 The calculations of the v i s are based on the peak GDPs from 2008. The normalized GDPs (relative to Portugal’s GDP in 2011) are:          12.0 15.3 1.5 9.7 1.1 6.7      .    This leads to values based on the A matrix of:      v0 = Ap =     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 56           ·       12.0 15.3 1.5 9.7 1.1 6.7            =       13.1 15.4 1.0 9.8 1.0 7.5 (France) (Germany) (Greece) (Italy) (Portugal) (Spain)      .    Note that calculating the A matrix is far more involved than just looking at the final borrower basis data. 52 Thus      v = θ    13.1 15.4 1.0 9.8 1.0 7.5 (France) (Germany) (Greece) (Italy) (Portugal) (Spain)      ,     and 53   θ  β=  2   13.1 15.4 1.0 9.8 1.0 7.5 (France) (Germany) (Greece) (Italy) (Portugal) (Spain)      .   

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