SSuffocating Fire Sales
Nils Detering ∗ , Thilo Meyer-Brandis † , Konstantinos Panagiotou † , Daniel Ritter † June 16, 2020
Abstract
Fire sales are among the major drivers of market instability in modern financial systems.Due to iterated distressed selling and the associated price impact, initial shocks to someinstitutions can be amplified dramatically through the network induced by portfolio over-laps. In this paper we develop models that allow us to investigate central characteristicsthat drive or hinder the propagation of distress. We investigate single systems as well asensembles of systems that are alike, where similarity is measured in terms of the empiricaldistribution of all defining properties of a system. This asymptotic approach ensures a greatdeal of robustness to statistical uncertainty and temporal fluctuations, and we give variousapplications. A natural characterization of systems resilient to fire sales emerges, and weprovide explicit criteria that regulators may exploit in order to assess the stability of anysystem. Moreover, we propose risk management guidelines in form of minimal capital re-quirements, and we investigate the effect of portfolio diversification and portfolio overlap.We test our results by Monte Carlo simulations for exemplary configurations.
The inevitably complex structure of dependencies among institutions is one of the most definingcharacteristics of the modern financial environment. This complexity is visible on multiplelevels, including for example the intricate distribution of corporate obligations and also the –sometimes significant – overlap in the asset holdings of the participating institutions. While thistremendous blend of dependencies and interactions provides the members with a great numberof opportunities, it is also the source of an enormous threat: the interlocking and pervasivestructures make the system in large parts or even as a whole fragile to possible initial localshock events. The problem of modeling, understanding, and managing this notion of systemicrisk has been an important research topic, and it has become particularly prominent after thestrike of the global financial crisis in the years 2007/08.One of the first papers in the context of financial mathematics to address systematically theeffect of – what they termed – cyclical dependencies among the market participants was theseminal work by [18]. They considered the channel of default contagion, that is, the successivepropagation of balance-sheet insolvency across financial institutions, and showed, among otherresults, that under mild assumptions a unique clearing vector exists that clears the obligationsof all members. From today’s viewpoint, however, it is well understood that there are variousother important channels of contagion. For instance, in his book [25] lists
Asset Correlation , ∗ Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106, USA.Email: [email protected] † Department of Mathematics, University of Munich, Theresienstraße 39, 80333 Munich, Germany. Emails:[email protected], [email protected] and [email protected] a r X i v : . [ q -f i n . R M ] J un
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B C A x A1 x C6 x A2 x A3 x C2 x C5 x B3 x B4 x B5 c c c c c c Figure 1: An illustration of a system with n = 6 financial institutions (circles) and M = 3 assets (squares). Edges represent investments of the institutions in the assets and their thicknessindicates the investment volume x mi . Furthermore, capitals c i are attached to the institutions. Default Contagion , Liquidity Contagion and
Asset Fire Sales as the four main channels of directand indirect default propagation.Here we focus on the fire sales channel, which is widely accepted as one of the main driversof market instability, see below for many related papers. The underlying dynamics that weconsider are defined abstractly as follows. Consider a system of institutions that are invested incertain assets, see also Figure 1, where each institution i is equipped with some initial capital(equity) c i and holds some number x Ai of shares of an asset A . The portfolios of the institutionsmay therefore overlap, and the dependencies are quantified by the collection of the x Ai ’s. Then,as a reaction to some initial shock event, one or more of the institutions sell according to anindividual strategy a non-negligible number of shares of their assets. These sales may causea decline of the assets’ share prices and all investors in the sold assets incur losses in theirportfolio values. This may start another round of asset sales, possibly on an extended set ofassets, again reducing share prices and so on. By this iterative process the initial stress ispropagated through the system and can be amplified considerably. In particular, institutionswho were spared from the initial shock can get into trouble, if their portfolios overlap with thoseof distressed investors. Related Work
Various approaches have been developed to model and understand the impact offire sales. The works [11, 31, 33] extend the classical setting from [18] for given financial networksso that various aspects of fire sales are incorporated. [9] use a branching process approximationto model fire sales, and they find that the system is stable when certain market parameters arebelow a critical value that they specify. [26] study the benefits and disadvantages of overlappingportfolios not only from the perspective of single institutions, but also for the market as a whole.An important consequence of their model is that there may arise a divergence between privateand social welfare depending on the various statistical features that they consider. Alike andrelated considerations are also made in [5], where a small number of assets is considered, and in[32], where using a microfounded model it is shown that the risk of joint liquidation motivatesinstitutions to create heterogeneous portfolios. A related setting for reinsurance markets withoverlapping insured objects is analyzed by [28]. [14] analyze the impact of fire sales on assetprice dynamics and correlation in continuous time. In [15], extending and building upon workof [27], they further describe the impact of the liquidation of large portfolios on the covariancestructure of asset returns and provide a quantitative explanation for spikes in volatility andcorrelations observed.Apart from the theoretical work there is a significant amount of empirical studies thatconsider the effect of fire sales, develop viable models and propose measures to capture quanti-2atively their effects. In [23] the topology of the network of common asset holdings is analyzedand [7] propose a network representation to quantify the interrelations induced by common assetholdings. [12] develop a stress testing framework for fire sales and propose indices of centralityfor institutions participating in a fire sales process in [13]. There are several works that developexposure- and market-based measures to depict the effect of fire sales. For example, [22] use thescalar product of two portfolios’ weights, and [29] use the so-called absorption ratio, the degreeof variation induced by the first principal components of asset returns, to measure dependencebetween different sources of risk in a portfolio. Additional measures are constructed from astatistical analysis of the equity returns of institutions, as in [1, 8, 2].
Our Results & Perspective
In this paper we study various aspects of fire sales, in particularregarding its modeling and the associated risk management.
The Deterministic Model.
As already mentioned, the underlying parameters of our modelassociate to each institution i a capital c i and to each pair ( i, A ) of an institution i and an asset A a number x Ai of shares of A that i holds. Apart from that, we equip each institution i withan abstract strategy ρ that dictates how many of the held shares are sold in case of losses, andwe assume that there is a function h that determines the impact of the ongoing sales to theprices of the shares. In contrast to previous works we only impose minimal assumptions at thispoint: the “strategy function” ρ is right-continuous and non-decreasing – the more the pricedrops the more shares are pushed off – and the “price impact function” h is continuous andnon-decreasing – the more shares are sold the more the price drops. Thus, the basic setting thatwe consider is quite flexible and it enables us to model various different scenarios and strategiesfor the institutions. Given all these parameters, one of our main results, given in Section 3 (inan auxiliary model) and 4, describes the eventual loss caused by the initial stress (initial losses (cid:96) i for institution i ) in the system. Let us remark that we only consider dynamics provoked byprice impact of sold shares, assuming that exogenous price changes are negligible over the shortperiod of time in which the fire sales take place (unlike for example as in [14]).Let us also remark at this point that in this paper we work with the actual losses incurredin each round of the fire sales process, which makes the model complex and its dynamics quiteinvolved. To the best of our knowledge and in direct comparison with previous works, this is thefirst model that considers the actual losses instead of using upper bounds and approximationsthat are easier to handle. The Stochastic Model.
The results described so far enable us for any given set of institutionsand assets equipped with all relevant parameters to determine for each institution the final lossincurred by fire sales. Thus, they are readily applicable to given systems. However, we areinterested in a more fundamental question: what are the important underlying structures thatpropel the process of fire sales? In other words, which system characteristics favor the emergenceof large fire sales cascades, and which ones prohibit them? Going even one step further, wewant to understand how large cascades can be prevented, for example by determining minimalcapital requirements for all institutions, so that the system as a whole is resilient to small initialshocks.The main contribution of this paper is a qualitative and quantitative answer to these ques-tions, where we proceed as follows. Instead of restricting our attention only to a single system,we consider an ensemble of systems that share common characteristics. This “structural re-semblance” is measured in terms of the joint empirical distribution of all parameters that weconsider: the capital of each institution, the number of shares of each asset that it holds andthe initial losses that it suffers. Equivalently, it is the distribution of a vector ( X , C, L ), where C is the capital, L is the initial loss, and X the vector of the number of shares (there is onecomponent for each asset) of an institution drawn uniformly at random among all institutions3n the system. The route that we take is thus to depart from any particular system and todescribe it solely by the distribution of the vector ( X , C, L ); hence we view any given system asa typical realization of a random experiment where the asset holdings, the capital and the initialloss of each institution are independent samples from ( X , C, L ). Similar to the deterministicmodel, we describe in Section 2 for this stochastic model the final state of the system after thefire sales cascade is completed.This approach constitutes a (further) key difference between our work and the previousliterature on fire sales. By reducing any particular system to its bare bone characteristics, weobtain robustness and flexibility. For example, moderate uncertainties of the precise systemconfiguration are absorbed. Moreover, our model is applicable to various configurations underthe premise that these are typical samples of some given distribution – this puts us in a positionto study future configurations or to develop system architectures that are resilient to fire sales.Finally, the model can readily be calibrated based on the empirical distribution of the assetholdings of the institutions in any given system.The setting considered here is related to a prominent approach in a different community,namely the study of random graphs/networks . There, sequences of networks with alike statisticalproperties are studied, and the long term behavior is investigated. This approach was also usedrecently to study default contagion in the financial mathematics community, see the papers [3,16, 17]. Some of our results are similar in nature, but the process that we consider has entirelydifferent dynamics. Risk Management.
More than a mere description of the fire sales process, it is a central objectiveto identify unstable system configurations and to devise measures to prevent disastrous firesales cascades. A common approach taken in the literature and in the works already cited is tospecify certain parameters, for example shock sizes and quantiles of the final damage, that areadmissible for a system to pass a stress test.The stochastic model studied here enables us to develop an attractive supplement andalternative to those approaches, as it puts us in the position to define in a natural and parameter-free way a notion of resilience (stability) of a given system. More specifically, in Section 2.2 wecall a system non-resilient if an arbitrary shock ( no matter how small it is) leads to a cascadewith the consequence that eventually a positive fraction of assets are sold, i.e. there exists a positive lower bound (which is given explicitly) for the total fraction of assets sold, independentof the size and nature of the initial shock. Note that this definition makes no sense for any givenfinite system – the outcome of the process may depend on the precise specification of the c i and x Ai – but here the effect of the stochastic model is greatly beneficial: we do not consider a systemwith a given number of institutions, but rather all systems where the empirical distribution ofthe asset holdings, capital and initial loss is (close to) the distribution of ( X , C, L ). In thissetting, (non-)resilience is a property solely of ( X , C, L ) that may become apparent only for largesystems; however, we demonstrate in Section 5.2 that the behavior converges rather quickly andthe results are a good approximation already for systems of moderate size.Finally, as an application we are able to derive in Section 5.1 explicit capital requirementsthat ensure stability of the system, which are of fundamental interest to regulators. It turns outthat for each institution this capital buffer is solely based on its own asset holdings, thus ensuringfull transparency and fairness, which is an essential question to address when determining thesystemic riskiness of the involved institutions. We want to stress again at this point thatthe particular feature of our model is that the requirements are independent of any chosenadditional parameters. As a further application, in Section 5.2 our tools allow us to quantifythe effects of portfolio diversification and similarity, thus contributing to the latest discussionsin the literature, see [5, 20, 26, 32] for example. Outline
In Section 2 we present our model of fire sales, we introduce the aforementioned notion4f resilience and outline our main results. Section 3 deals with the analysis of an auxiliary model,which is rather pessimistic and overestimates the losses for all institutions. The advantage of thismodel is that it is much easier to study, in particular we can derive bounds for several relevantparameters (like the total nubmer of sold assets) in terms of a solution of a fixed-point equation.Then, in Section 4 we build the bridge between the two models, and in particular we deriveabstract and explicit criteria that guarantee resilience for the fire sales model. Section 5 dealswith applications of the developed theory and provides simulations, as outlined in the previousparagraph. The papers closes with a conclusion that summarizes future research directions, inparticular the problem of applying our results to systems of moderate size. All proofs that areomitted in the main part and some generalized results can be found in the Section 7.
In this section we define our model for fire sales, we introduce the parameters that we studyand then we outline the results that we obtain.
Model Parameters
We consider a financial system consisting of n ∈ N institutions that caninvest in M ∈ N different (not perfectly liquid) assets or asset classes. That is, to each institution i ∈ [ n ] := { , . . . , n } we assign a number x mi ∈ R + , of held shares of asset m ∈ [ M ] (or any otherindex set of size M ). See Figure 1 for an illustration. Further, we denote by c i ∈ R + the initialcapital of institution i (for example the equity for leveraged institutions or the portfolio valuefor institutions that exclusively invest in assets) and we assume that it incurs exogenous losses (cid:96) i ∈ R + , due to some shock event. In the case of a market crash for instance, it could be that (cid:96) i = (cid:80) ≤ m ≤ M x mi δ m p m , where p m denotes the initial price of one share of asset m ∈ [ M ] and δ m ∈ (0 ,
1] is the relative price shock on the asset. Furthermore, let the empirical distribution F n : R M + , × R + , ∞ × R + , → [0 ,
1] of the institutions’ parameters be denoted by F n ( x , c, (cid:96) ) = n − (cid:88) i ∈ [ n ] { x i ≤ x , . . . , x Mi ≤ x M , c i ≤ c, (cid:96) i ≤ (cid:96) } (2.1)and let in the following ( X n , C n , L n ) be a random vector with distribution F n . Asset Sales
We assume that due to the exogenous losses some of the institutions are forcedto liquidate parts of their asset holdings in order to comply with regulatory or market-imposedconstraints (e. g. leverage constraints), self-imposed risk preferences and policies to adjust theportfolio size, or to react to investor redemption. These sales are described by a non-decreasingfunction ρ : R + , → [0 ,
1] such that each institution i ∈ [ n ] incurring a loss of Λ sells x mi ρ (Λ /c i )of its shares of asset m . The fraction Λ /c i describes the relative loss of institution i measuredagainst its initial equity. It is hence sensible to assume that ρ (0) = 0 , ρ ( u ) ≤ ρ ( u ) = ρ (1) for all u ≥ . If at default of an institution the whole portfolio is to be liquidated, then ρ (1) = 1. In general,however, the remaining assets at default may be frozen by the insolvency administrator andonly be sold to the market on a longer time scale. In this case, ρ (1) = lim u → − ρ ( u ) ∈ [0 , ρ are rather mild and allow for a flexible description of various scenarios. Aconcrete and simple but useful example for a sales function is given by ρ ( u ) = { u ≥ } , whichdescribes complete liquidation of the portfolio at default (if the institution is leveraged) resp. dis-solution. Other more involved examples, derived from various types of leverage constraints or5onsidering different parameters, can be found in Section 7.3.Some remarks are in place here. First, to simplify the exposition and the notation in themain part we restrict our analysis to continuous ρ , and we refer to Section 7, where we provideproofs of all our results in full generality. Second, it is possible to consider an even more generalmodel in which we choose different sale functions ρ m for the assets m ∈ [ M ]. It is then possible toreplace the scalar function ρ ( u ) by the diagonal matrix diag( ρ ( u ) , . . . , ρ M ( u )) in all forthcomingconsiderations, so as to highlight different aspects, like liquidity constraints. Further, we maypartition the set of institutions into different types (banks, insurance companies, hedge funds,. . . ) and choose different ρ or ρ m for each type. Finally, our proofs in this article also workfor arguments (of ρ ) other than Λ /c i (where Λ are the losses), but for the sake of simplicity westick to this particular form. Price Impact
Since the assets are not perfectly liquid (the limit order book has finite depth),the sales of shares triggered by the exogenous shock cause prices to descent. This on the otherhand causes further losses for all the institutions invested in the assets due to mark-to-marketaccounting. We model the price loss of asset m ∈ [ M ] by a continuous function h m : R M + , → [0 , y = ( y , . . . , y M ) ∈ R M + , and ny m sharesof asset m have been sold in total, then we assume that the price of m drops by h m ( y ).Note the relative parametrization with the number of institutions n , where we assumed that ny m (instead of y m ) shares of asset m are sold. For fixed n this is arbitrary; however, in duecourse we will consider the stochastic model (see Assumption 2.1 below), and this parametriza-tion will turn out to be rather convenient to state our results. In particular, it reflects the factthat the market depth scales with the number of institutions involved. Fire Sales
The fire sales process that we consider combines all previous ingredients. Triggeredby some exogenous event the institutions start selling a portion of their assets and drive downthe prices. Due to mark-to-market effects, this means that institutions experience further lossesand are forced into further sales. This iterative process continues until the system stabilizesand no further sales, losses and price changes occur. More specifically, let us denote by τ ( k ) =( τ j ( k ) ) ≤ j ≤ M the vector of cumulatively sold shares at the beginning of round k ∈ N , that is, thenumber of actually sold shares in round k is τ ( k +1) − τ ( k ) and, obviously, τ (1) = 0. Moreover,for each bank i ∈ [ n ] and any round k let • x i,k = ( x ji,k ) ≤ j ≤ M be the number of shares that i holds at the beginning of round k ; • (cid:96) i,k be the total loss of i accumulated until the beginning of round k .From these definitions we readily obtain that the number of shares held at the beginning ofround k x i, = x i , x i,k = x i (cid:0) − ρ ( (cid:96) i,k − /c i ) (cid:1) i ∈ [ n ] , k ≥ , (2.2)and the total number of sold shares τ (1) = , τ ( k ) = (cid:88) i ∈ [ n ] x i ρ (cid:0) (cid:96) i,k − /c i (cid:1) , k ≥ . (2.3)Moreover, from the specification of the fires sales process we obtain (see also for a justificationafter the display) that (cid:96) i,k = (cid:96) i + x i,k h (cid:0) τ ( k ) /n (cid:1) + k − (cid:88) (cid:96) =1 ( x i,(cid:96) − x i,(cid:96) +1 ) h (cid:0) τ ( (cid:96) ) /n (cid:1) , i ∈ [ n ] , k ∈ N , (2.4)as in the beginning of round k there remain x i,k shares which are affected with a price impactof h ( τ ( k ) /n ), and in any preceding round 1 ≤ (cid:96) ≤ k − x i,(cid:96) − x i,(cid:96) +1 h ( τ ( (cid:96) ) /n ). Thus, by induction ( τ ( k ) ) k ∈ N is non-decreasing componentwise andbounded by (cid:80) i ∈ [ n ] x i . The limit n ψ n := lim k →∞ τ ( k ) , (2.5)that is, the vector of finally sold shares, exists. In addition to ψ n we will also be interested inthe number of defaulted institution given by D n := { i ∈ [ n ] : lim k →∞ (cid:96) i,k ≥ c i } and hence n − |D n | = n − (cid:88) i ∈ [ n ] { lim k →∞ (cid:96) i,k ≥ c i } . (2.6)The set D n provides additional information about the vulnerability of the financial system andthe impact of fire sales on leveraged institutions such as banks or hedge funds. This completesthe description of the model for fire sales that we study here. Up to now we have described the fire sales process in any specific (finite) system. Our aimis, however, to understand qualitatively how and which characteristics of a system promote orhinder the spread of fire sales. As portrayed in detail in the introduction, in the following wethus consider an ensemble of systems that are similar in the sense that they all share some(observed) statistical characteristics. This similarity is measured in terms of the most naturalparameters, namely the joint empirical distribution function (2.1) of the asset holdings, thecapital/equity, and the initial losses. In particular, we assume that we have a collection ofsystems with a varying number n of institutions with the property that the sequence ( F n ) n ∈ N stabilizes, i.e., has a limit. Additionally, we assume convergence of the average asset holdingsto a finite value; this is a standard assumption avoiding condensation of the distribution of theasset holdings. Assumption 2.1.
Let M ∈ N . For each n ∈ N consider a system with n institutions and M assets specified by sequences x ( n ) = ( x i ( n )) ≤ i ≤ n of asset holdings, c ( n ) = ( c i ( n )) ≤ i ≤ n ofcapitals and (cid:96) ( n ) = ( (cid:96) i ( n )) ≤ i ≤ n of exogenous losses. Let F n be the empirical distributionfunction of these parameters for n ∈ N (as in (2.1) ) and let ( X n , C n , L n ) = (cid:0) ( X n , . . . , X Mn ) , C n , L n ) (cid:1) ∼ F n . Then assume the following.(a)
Convergence in distribution:
There is a distribution function F such that as n → ∞ , F n ( x , y, z ) → F ( x , y, z ) at all continuity points of F .(b) Convergence of means:
Let ( X , C, L ) = (( X , . . . , X M ) , C, L ) ∼ F . Then as n → ∞ , E [ X mn ] → E [ X m ] < ∞ , m ∈ [ M ] . An ensemble of systems satisfying Assumption 2.1 will be called an ( X , C ) -system withinitial shock L in the sequel. A particular and probably the most relevant scenario is as follows.Suppose that the distribution F is specified, for example by considering a real system. Then,for each n ∈ N we construct a system by assigning to each institution i ∈ [ n ] independentlyasset holdings, capital and losses sampled from F . Then, by the strong law of large numbers,with probability 1, the sequence of systems we obtain satisfies Assumption 2.1. As in the(deterministic) model our aim is to describe in this broader setting the final state of the system.As institutions without any asset holdings play no role in the process and can be removed, weshall assume that P ( X = 0 , . . . , X m = 0) = 0. 7n order to simplify the presentation in the main part of this article we assume that thesales function is strictly increasing around 0. It can be waived and the results in full generalityare presented the Section 7.1 and proved in 7.2. Assumption 2.2.
The sales function ρ : R + , → [0 , is continuous and strictly increasing ina neighbourhood of . (Non-)Resilience In this part of the section we approach the heart of the matter and developa notion of how resilient or non-resilient a given ( X , C )-system is with respect to fire salestriggered by some initial shock L . Note that all information about an initial shock comes fromthe random variable L , whereas the system itself is specified by ( X , C ). One thing that wecan certainly then do is to consider shocks of different magnitudes on the same, crucially, apriori unshocked system. From a regulator’s perspective, for example, a desirable property ofan ( X , C )-system is the ability to absorb small local shocks L without larger parts of the systembeing harmed. In our model we can even vary the statistical properties of L arbitrarily. Thefollowing way of defining resilience thus emerges naturally: we let the shock L ‘become small’in the sense that E [ L/C ] →
0, and we call the system resilient if the asymptotic number of soldshares ψ mn (that now – explicitly – depends on L ) also tends to 0. Definition 2.3 (Resilience) . An ( X , C )-system is said to be resilient , if for any (cid:15) > δ > L with E [ L/C ] < δ it holds lim sup n →∞ ψ mn ≤ (cid:15) for all m ∈ [ M ].Note that our definition of resilience is in terms of the final number of sold shares. There arealso other options, for example we could have based it instead on the final fraction of institutionsthat defaulted, which is the ‘standard’ choice in the default contagion literature, see i.e. [16, 17].However, we selected the final number of sold shares, as the economic cost of fire sales can belarge even when the number of defaults is rather small. In any case, all our results can easilybe adapted to a resilience condition based on the default fraction.We now move on to define non-resilient systems. In complete analogy to the previousconsiderations, we call a financial system non-resilient if the fraction of eventually sold sharesis lower bounded by some positive constant that is – crucially – independent of the shock L , aslong as this is positive. In order to ensure that the process starts at all, the shock has to affectsome banks, so we require L to be such that P ( L > > Definition 2.4 (Non-Resilience) . A ( X , C )-system is non-resilient , if there exists ∆ ∈ R + andan asset m ∈ [ M ] such that lim inf n →∞ ψ mn ≥ ∆ for any initial shock L with P ( L > > P ( L > >
0, and apartfrom that, it is not really the complement of the notion of resilience in Definition 2.3. Rathersurprisingly, as we will show later, under Assumption 2.2 there is essentially no gap betweenDefinition 2.3 and Definition 2.4, that is, a system is either resilient or non-resilient. However,when ρ is not strictly increasing, the situation may be different. This is a particularly interestingand important setting, as it describes a situation where shock driven asset sales only occur ifsome institutions have been affected to some larger extend. It thus ensures that minor losses(e.g. resulting from mark-to-market accounting of naturally volatile assets) and major shocksas the default of Lehman Brothers in 2008 are treated differently. As in Section 7 we treat theimportant example of a sales function ρ that is not strictly increasing close to 0 we provide therequired notion of weak non-resilience here, which is the complement of Definition 2.3. Definition 2.5 (Weak Non-Resilience) . A ( X , C )-system is weakly non-resilient , if it is notresilient, i.e. there exists ∆ ∈ R + and an asset m ∈ [ M ] such that for every ε > L such that 0 < E [ L/C ] ≤ ε but lim inf n →∞ ψ mn ≥ ∆.8 verview of the Results Having defined the notions of (non-)resilience we can give an in-formal overview of our main findings. Given an ( X , C )-system with initial shock L we firstderive explicit bounds for the final number of sold shares ψ mn , m ∈ [ M ] (Section 3.2), and forthe default fraction n − |D n | in Section 7.2. To this end, we resort to a different process, theso-called auxiliary model described in the next section, which makes an important simplificationto the rather involved fire-sales process given by (2.2)–(2.6): instead of considering the actuallosses incurred in some round k , we overestimate them and use the simple upper bound of x i · h ( τ ( k ) ), where τ ( k ) is the vector of the total number of sold shares in the first k rounds. Theadvantage is, although also challenging, that this auxiliary process can be handled analytically;the disadvantage, however, is that it is not clear how much this simplification affects the actualbehavior. Here our contribution is to provide an explicit connection: we manage to ‘sandwich’the fire sales process between two carefully crafted auxiliary processes, so that (non-)resilienceproperties are not or minimally affected (see Section 4.1).With this coupling at hand we manage to derive explicit (non-)resilience criteria for ( X , C )-systems. As it turns out (see Section 4.1 and 4.2), the behavior of such a system can be capturedby a collection of functions ( f m ) m ∈ [ M ] , which essentially describe the evolution of the price ofthe corresponding asset during the execution of the (auxiliary) process. The joint zero of thesefunctions coincides with the final number of sold shares and thus allows us to study resilientand non-resilient cases. With these tools at hand, we are able to derive qualitative criteriathat depend on the parameters of system – X , C – and of the process – the sales function ρ and the price impact h – only and characterize resilience. For example, we show that if E [ X /C ] ρ (cid:48) (0) >
1, then the system is non-resilient: the first factor says informally that thecapital is too small in direct comparison to the asset holdings, and the second that sellingbegins immediately as a response to a price drop. On the other hand, if E [ X /C ] ρ (cid:48) (0) < ρ (cid:48) (0) >
0, then the system is resilient. We also consider cases in which ρ (cid:48) (0) = 0; there we showthat the actual interplay between ρ and h (price drop) determines if the system is resilient ornot. Several results of this kind are presented in Section 4.2. As already described previously, the aim of this section is to formulate an alternative to the firesales process defined by (2.2)–(2.6), which is more amenable to an analytic treatment. Notethat the essence of the fire sales process lies in Equation (2.4): the behavior is rather complex,as we need to keep track of the asset sales in all rounds. In order to make the analysis tractable,we will substitute (2.4) with a conservative assumption that overestimates the loss as follows:we will forget the actual prices at which assets were sold in previous rounds, and we will boundthem from below by using the price impact in the current round – which is of course at least aslarge as the impact in previous rounds. More specifically, note that (cid:96) i,k ≤ (cid:96) i + x i h ( τ ( k ) /n ) , i ∈ [ n ] , k ∈ N . (3.1)In the following we describe an auxiliary fire sales process, in which we replace the actual loss (cid:96) i,k by the simple upper bound. (As we already said, this process will serve as a tool to gainunderstanding of the real process.) We now provide a description of the final state of the systemafter this auxiliary fire sales process is completed; in particular, we are interested in the vector χ n of the number of finally sold shares divided by n and the final price impact h m ( χ n ) on anyasset m ∈ [ M ]. Since we will be using it throughout the rest of the paper, Table 2 contrastsour notation for the fire sales process and the auxiliary process defined in the current section.9ire Sales Process Auxiliary Processvector of finally sold shares (divided by n ) ψ n = ( ψ n , . . . , ψ mn ) χ n = ( χ n , . . . , χ mn )vector of sold shares in rounds 1 , . . . , k τ ( k ) = ( τ k ) , . . . , τ M ( k ) ) σ ( k ) = ( σ k ) , . . . , σ M ( k ) )Figure 2: Notation for the fire sales and the auxiliary process.In order to describe χ n we again consider the process in rounds, where in each roundinstitutions react to the price changes from the previous round. Denote by σ ( k ) = ( σ k ) , . . . , σ M ( k ) )the vector of cumulatively sold shares in round k . We readily obtain σ (1) = (cid:88) i ∈ [ n ] x i ρ (cid:18) (cid:96) i c i (cid:19) = n E (cid:20) X n ρ (cid:18) L n C n (cid:19)(cid:21) . Similarly, in round k ≥ σ ( k ) = (cid:88) i ∈ [ n ] x i ρ (cid:32) (cid:96) i + x i · h ( n − σ ( k − ) c i (cid:33) = n E (cid:34) X n ρ (cid:32) L n + X n · h ( n − σ ( k − ) C n (cid:33)(cid:35) . (3.2)Thus, by induction ( σ ( k ) ) k ∈ N is non-decreasing componentwise and bounded by n E [ X n ]. Thelimit n χ n := lim k →∞ σ ( k ) – the vector of finally sold shares in the auxiliary process – exists.In contrast to the fire sales process, it turns out that for the auxiliary process, the numberof sold shares χ n can be derived by solving a rather simple equation. Proposition 3.1.
Consider the auxiliary fire sales process as described above. Then χ n = n − lim k →∞ σ ( k ) , the number of sold shares divided by n at the end of the auxiliary process, isthe smallest (componentwise) solution of E (cid:20) X n ρ (cid:18) L n + X n · h ( χ ) C n (cid:19)(cid:21) − χ = 0 . (3.3) Proof.
Proof. By continuity of ρ and the dominated convergence theorem χ n = n − lim k →∞ σ ( k ) = lim k →∞ E (cid:34) X n ρ (cid:32) L n + X n · h ( n − σ ( k − ) C n (cid:33)(cid:35) = E (cid:34) X n ρ (cid:32) L n + X n · h ( n − lim k →∞ σ ( k − ) C n (cid:33)(cid:35) = E (cid:20) X n ρ (cid:18) L n + X n · h ( n − χ n ) C n (cid:19)(cid:21) and χ n is thus a solution of (3.3).By the Knaster-Tarski theorem there must exist a least fixed point ˆ χ n . Clearly, σ (0) := ≤ n ˆ χ n . Hence assume inductively that σ ( k ) ≤ n ˆ χ n for k ≥
1. Then σ ( k +1) = (cid:88) i ∈ [ n ] x i ρ (cid:32) (cid:96) i + x i · h ( n − σ ( k ) ) c i (cid:33) ≤ (cid:88) i ∈ [ n ] x i ρ (cid:18) (cid:96) i + x i · h ( ˆ χ n ) c i (cid:19) = n ˆ χ n (3.4)by monotonicity of ρ , and hence χ n = n − lim k →∞ σ ( k ) ≤ ˆ χ n . By definition of ˆ χ n it thus holdsthat χ n = ˆ χ n .Further, given χ n , we readily obtain that under the auxiliary process the set of finally10efaulted institutions D n := { i ∈ [ n ] : (cid:96) i + x i · h ( χ n ) ≥ c i } and hence n − |D n | = n − (cid:88) i ∈ [ n ] { (cid:96) i + x i · h ( χ n ) ≥ c i } = P ( L n + X n · h ( χ n ) ≥ C n ) . (3.5)Note that we slightly ‘overload’ the notation, as we use D n for both the fires sales and theauxiliary process. Since it will be always clear from the context which processes is studied, thisshould cause no confusion.Let us close this section with a remark about the case of non-continuous sales function ρ .The following example shows that also in this case it may be possible to determine the finalstate of the system by the smallest solution of (3.3). Consider ρ ( u ) = { u ≥ } , that is,institutions sell their portfolio as they go bankrupt. Then σ ( k ) (cid:54) = σ ( k − only if in round k atleast one institution defaults that was solvent in round k −
1. Since there are only n institutions,the fire sales process stops after at most n − χ n of finally sold sharesdivided by n solves (3.3). Again by (3.4) we then obtain χ n = ˆ χ n is the smallest solution of(3.3). By a similar reasoning we derive more generally for any sale function ρ with finitely manydiscontinuities that χ n = ˆ χ n . Proposition 3.1 describes the final state of any (finite) system when we consider the auxiliaryprocess. Here we extend our considerations to stochastic systems, as we did in the case of thefire sales process. We are given an ( X , C )-system with initial shock L , that is, an ensembleof systems, where for each n ∈ N a system with n institutions and M assets is specified bysequences of asset holdings, capitals and exogenous losses with joint distribution ( X n , C n , L n ).We also assume that the system satisfies Assumption 2.1, that is, the distribution of ( X n , C n , L n )converges to that of ( X , C, L ) and the asset holdings are in L .Applying Proposition 3.1 to each one of the systems in the ensemble, we obtain that foreach n ∈ N , the number of sold shares divided by n at the end of the auxiliary process is thesmallest solution of (3.3). Since ( X n , C n , L n ) → ( X , C, L ) it is thus natural to believe that forlarge n , the number of finally sold shares χ n is close to the smallest (componentwise) solutionof the ‘limiting equation’ E (cid:20) X ρ (cid:18) L + X · h ( χ ) C (cid:19)(cid:21) − χ = 0 . (3.6)As it will turn out, except for few pathological situations, this intuition is indeed correct andmotivates the following definitions. Let f m ( χ ) := E (cid:20) X m ρ (cid:18) L + X · h ( χ ) C (cid:19)(cid:21) − χ m , m ∈ [ M ] . (3.7)Further, let S := (cid:92) m ∈ [ M ] (cid:8) χ ∈ R M + , : f m ( χ ) ≥ (cid:9) (3.8)the subset where all of the functions f m , m ∈ [ M ], are non-negative. Let ˆ χ be the smallestjoint root (which always exists by the Knaster-Tarski theorem) of the functions f m , m ∈ [ M ].While the necessity of Assumption 2.1 is obvious it is surprising that in most cases it is actuallyalso sufficient to ensure that ˆ χ = lim n →∞ χ n . (3.9)Finally, in analogy to (3.5) we can therefore expect for the stochastic model that the final11 .2 0.4 0.6 0.8 - - - S χ (a) - - - S χ χ * (b) - - - S χ (c) Figure 3: Figure corresponding to Example 3.2. Plot of function f ( χ ) (blue) for three differentexample systems. The set S is depicted in orange.fraction of defaulted institutions in the auxiliary model is given bylim n →∞ n − |D n | = g ( ˆ χ ) , where g ( χ ) := P ( L + X · h ( χ ) ≥ C ) . (3.10)To illustrate under which conditions the previous statements are true and the intuition is right,let us look at an instructive example with one asset only ( M = 1). This is a carefully crafted ‘toyexample’ that is unlikely to model a real system, but it will reveal the important characteristicsthat determine the underlying behavior. Example 3.2.
Consider a system defined by h ( χ ) = χ, ρ ( y ) = { y ≥ } , X ∼ Exp(1) , C ≡ c, P ( L = c ) = 0 . and P ( L = 0) = 0 . , where L is independent of X and c = 0 . . See Figure 3(a) for an illustration of f , S and ˆ χ . There ˆ χ is the only root of f and the set S is the interval [0 , ˆ χ ] . The situation changes,however, as we vary the capital. See Figure 3(c) for the case of c = 0 . . Then S consists oftwo disjoint intervals and f has three roots. In fact, one can observe the splitting of S and thusa discontinuity of ˆ χ at c ≈ . , see Figure 3(b). The situation in Figure 3(b) presents a ratherspecial and pathological case. In fact, only in this very special case above heuristics fail and thesituation becomes more complex than suggested in (3.9) and (3.10) . In order to formalize the special situation encountered in the previous example, let us denoteby S the largest connected subset of S containing (clearly ∈ S ). Further, let χ ∗ ∈ R M + , with ( χ ∗ ) m := sup χ ∈ S χ m . (3.11) Lemma 3.3.
There exists a smallest joint root ˆ χ of all functions f m ( χ ) , m ∈ [ M ] , with ˆ χ ∈ S .Further, χ ∗ as defined above is a joint root of the functions f m , m ∈ [ M ] , and χ ∗ ∈ S . The special case from Figure 3(b) is then described by ˆ χ (cid:54) = χ ∗ , and in all other cases ˆ χ = χ ∗ .We shall henceforth term the former case as ‘pathological’, since we can only obtain it if wemaliciously fine-tune the parameters of the system to provoke such a behavior. We do notexpect such situations to occur or to be relevant, and in any case, in all settings that we studyhere they don’t.Our first result provides in non-pathological cases (that is, when ˆ χ = χ ∗ ) the asymptoticnumber of finally sold shares for the auxiliary process and in all other cases upper and lowerbounds. Theorem 3.4.
Consider a ( X , C ) -system with initial shock L satisfying Assumptions 2.1, 2.2.Then the number χ mn of finally sold shares of asset m ∈ [ M ] divided by n in the auxiliary process atisfies ˆ χ m + o (1) ≤ χ mn ≤ ( χ ∗ ) m + o (1) . In particular, for the final price impact h m ( χ n ) on asset m ∈ [ M ] h m ( ˆ χ ) + o (1) ≤ h m ( χ n ) ≤ h m ( χ ∗ ) + o (1) . In the previous section we derived results that allow us to determine the final default fraction in( X , C )-systems caused by the auxiliary process and sparked by some exogenous shock L . In thissection we go one step further and investigate whether a given system in an initially unshockedstate is likely to be resilient to small shocks or susceptible to fire sales.Note that all information about an initial shock comes from the random variable L , whereasthe system itself is specified by ( X , C ). So we can easily consider shocks of different magnitude L on the same a priori unshocked system. In the following, whenever we use the notation g , f m and χ ∗ , we always mean the quantities (3.7)-(3.10) from the previous section for the ( X , C )-systemwith initial shock L ≡
0. We shall first provide resilience criteria for the auxiliary system.
Theorem 3.5.
For each (cid:15) > there exists δ > such that for all L with E [ L/C ] < δ thenumber nχ mn,L of finally sold shares of each asset m ∈ [ M ] for the auxiliary process satisfy lim sup n →∞ χ mn,L ≤ ( χ ∗ ) m + (cid:15), m ∈ [ M ] . We immediately obtain the following handy resilience criterion.
Corollary 3.6 (Resilience Criterion) . If χ ∗ = 0 , then the ( X , C ) -system is resilient under theauxiliary process. Note that g ( ) = 0 and hence there are only few defaults if the system is resilient (i.e. χ ∗ = , S = { } ). It is possible, however, that g ( χ ∗ ) = 0 while χ ∗ (cid:54) = in which case a shock triggersa large fraction of asset sales but only very few defaults.We now turn to non-resilience criteria and study the case χ ∗ >
0. Since ρ (0) = 0 and L ≡ f m (0) = 0 , m ∈ [ M ], that is, ˆ χ = 0 and thus Theorem 3.4 does not give anyuseful lower bound for the number of sold shares. However, if we consider a positive shock L onthe system, then 0 is not anymore a joint root of the functions (3.7), and Theorem 3.4 guaranteesthat χ mn ≥ ∆ m − o (1) as n → ∞ , where ∆ is the smallest such joint root. In particular, if weneglect pathological situations as described in Section 3.2, then ∆ should be (close to) χ ∗ ; asthe next result shows, this is for example the case when there is no joint root of the ( f m ) m ∈ [ M ] between 0 and χ ∗ . Theorem 3.7.
Consider a ( X , C ) -system such that χ ∗ > and such that every z ∈ R M \{ , χ ∗ } with ≤ z ≤ χ ∗ (coordinate-wise) is not a joint root of the functions ( f m ) m ∈ [ M ] . Then thesystem is non-resilient under the auxiliary process, in particular χ mn,L ≥ ( χ ∗ ) m − o (1) , m ∈ [ M ] , where nχ mn,L is the number of finally sold shares of asset m . As already remarked previously, we do not expect that situations other than the ones coveredby the previous result will ever be relevant. In any case, the conclusion of non-resilience is forexample also true if we request that there are only finitely many roots, or that the roots arebounded away from 0, an observation we shall regularly use in proofs. Constructing such highly13 a) (b)
Figure 4: Figure corresponding to Example 3.8. Plot of the zero lines for f (blue) and f (yellow) of the unshocked system (solid) and the shocked system (dashed). Left (a) depicts aresilient system and right (b) a non-resilient system.artificial examples, where the infimum of the set of roots is 0, would require an enormousfine-tuning of the system parameters (and enough malicious energy).Let us finally remark here that Theorem 3.7 is not true without the assumption that ρ isstrictly increasing. This becomes clear by looking at the particular example ρ ( u ) = { u ≥ } ofsales at default. Asset sales will only happen if some banks default initially, that is, P ( L > C ) >
0; a shock which is such that P ( L > C ) = 0 can not trigger any sales. In Theorem 7.5 we show,however, that for χ ∗ > Example 3.8.
Let f ( x ) = 1 . x − / [ x ≥ and let X be a random variable with density f .Consider a system with two assets ( M = 2 ) defined by h ( χ ) = χ, ρ ( u ) = min { , u } , X = X = X, C = 10 X γ Moreover, we consider a shock L independent of X with P ( L = C ) = 0 . . See Figure 4(a)for an illustration of f , f when γ = 0 . and 4(b) for the case γ = 0 . . The solid yellow andblue lines mark the zero sets of the functions f and f for the unshocked system. For γ = 0 . (a) the system is resilient ( χ ∗ = 0 , S = { } ) as the solid lines cross in , for γ = 0 . (b)the system is non-resilient and S is the area bounded by the solid lines, which cross first in χ ∗ (cid:54) = 0 . The dashed lines are the zero lines for the functions for the shocked system and thecoordinates of their intersection mark the number of sold shares. The smaller the shock, thecloser the dashed lines of the shocked system will approach the lines of the unshocked system.While in the resilient case (a) this pushes their intersection towards and only few shares aresold, in the non-resilient system, as the shock gets smaller, the intersection of the dashed linesare approaching χ ∗ (cid:54) = 0 and thus lower bounded, no matter how small the shock is. The Fire Sales Process
In this section we establish an explicit connection between the fire sales process (2.2)–(2.6) andthe auxiliary process described and studied in the previous section. In particular, we will usethe final number of sold assets χ n = n − lim k →∞ σ ( k ) in a specific auxiliary processes to boundthe asset sales ψ n = n − lim k →∞ τ ( k ) in the real process.Let us start with a simple observation. For a given finite financial system ( X n , C n ) withshock L n and specified functions ρ and h , it easily follows from (2.4) and (3.1) that the auxiliaryprocess leads to more asset sales, to a higher price impact and as a result also to more defaults.Thus, the auxiliary process always serves as an upper bound for the fire sales process (regardingthe actual asset sales and also the final default fraction). This is summarized in the followingtheorem. In the rest of this section we (silently) consider a ( X , C )-system with initial shock L that satifies Assumptions 2.1 and 2.2. Theorem 4.1.
Let ψ mn be the number of finally sold shares of asset m ∈ [ M ] divided by n inthe fire sales process. Let χ ∗ be as in (3.11) . Then lim sup n →∞ ψ mn ≤ ( χ ∗ ) m . Proof.
Proof According to our assumptions, the functions ρ : R + , → [0 ,
1] and h m : R M + , → [0 ,
1] are non-decreasing and τ (1) = σ (1) = 0. From (2.4) and (3.1) we obtain that (cid:96) i,k = (cid:96) i + x i,k h ( τ ( k ) /n ) + k − (cid:88) (cid:96) =1 ( x i,(cid:96) − x i,(cid:96) +1 ) h ( τ ( (cid:96) ) /n ) ≤ (cid:96) i + x i h ( τ ( k ) /n )it follows that ρ ( (cid:96) i,k ) ≤ ρ ( (cid:96) i + x i h ( τ ( k ) /n )) and as a result τ ( k ) ≤ σ ( k ) . Thus ψ n = lim k →∞ τ ( k ) n − ≤ lim k →∞ σ ( k ) n − = χ n . By Theorem 3.4, we know that for the auxiliary process, lim sup n →∞ χ mn ≤ ( χ ∗ ) m and thus it follows that lim sup n →∞ ψ mn ≤ ( χ ∗ ) m .More surprisingly, as we shall see below, we can adapt carefully the parameters of the givensystem so as to also obtain lower bounds from the auxiliary process. The central idea is todefine the auxiliary process on another system having smaller initial asset holdings and a salesfunction that is dominated point-wise by the original one. More specifically, let us fix ε > δ ε := ρ − ( ε ), where ρ − denotes the generalized inverse of ρ . In addition to ρ , define thesales function ρ ε : R + , → [0 ,
1] by ρ ε ( u ) = ρ ( u ) { u ≤ δ ε } + ε { δ ε < u } . Further, we consider a system that is derived from the ( X , C )-system with initial shock L butwith asset holdings (1 − ε ) X instead. In analogy to (3.7) we consider for this system f mε ( χ ) := E (cid:20) (1 − ε ) X m ρ ε (cid:18) L + (1 − ε ) X · h ( χ ) C (cid:19)(cid:21) − χ m , m ∈ [ M ] . (4.1)Further, also in analogy to (3.8), let S ε := (cid:92) m ∈ [ M ] (cid:8) χ ∈ R M + , : f mε ( χ ) ≥ (cid:9) S ε, be the largest connected subset of S ε containing and further, let ˆ χ ε ∈ S ε, thesmallest joint root of the functions f mε , m ∈ [ M ]. We obtain the following crucial lower bound: Theorem 4.2.
Let ψ mn be the number of finally sold shares of asset m ∈ [ M ] divided by n inthe fire sales process. For every ε > n →∞ ψ mn ≥ ˆ χ mε . Proof.
Proof For every n ∈ N we consider a system ( X n , C n , L n ) satisfying Assumptions 1 and2. Then it is immediate that the sequence ((1 − ε ) X n , C n , L n ) n ∈ N satisfies Assumption 2.1 withlimiting distribution ((1 − ε ) X , C, L ). We now fix n ∈ N . As usual, let τ ( k ) be the total numberof assets sold in round k in ( X n , C n , L n ) with sales function ρ under the fire sales process.Further, let σ ( k ) be the total number of assets sold in round k in ((1 − ε ) X n , C n , L n ) with salesfunction ρ ε in the auxiliary process. We shall show that σ ( k ) ≤ τ ( k ) , which implies that χ n =lim k →∞ σ ( k ) n − ≤ lim k →∞ τ ( k ) n − = ψ n and since by Theorem 3.4, lim inf n →∞ χ n ≥ ˆ χ mε , theclaim follows. In order to show that σ ( k ) ≤ τ ( k ) we use induction. By definition σ (1) = τ (1) = 0.So, lets assume that σ (1) ≤ τ (1) , . . . , σ ( k ) ≤ τ ( k ) . We determine the number of assets bank i sellsin round k . First consider the case that (cid:96) i,k /c i ≤ δ , that is, the loss of bank i at the beginningof round k is bounded by δ and thus the fraction of shares sold so far is less than ε . As a result x i,k ≥ (1 − ε ) x i and thus x i (1 − ε ) ρ ε (cid:32) (cid:96) i + (1 − ε ) x i · h ( σ ( k ) n − ) c i (cid:33) ≤ x i ρ (cid:32) (cid:96) i + (1 − ε ) x i · h ( σ ( k ) n − ) c i (cid:33) (4.2) ≤ x i ρ (cid:32) (cid:96) i + x i,k · h ( σ ( k ) n − ) c i (cid:33) ≤ x i ρ (cid:32) (cid:96) i + x i,k · h ( τ ( k ) n − ) c i (cid:33) ≤ x i ρ (cid:18) (cid:96) i,k c i (cid:19) . (4.3)If, on the contrary, (cid:96) i,k /c i > δ , then x i (1 − ε ) ρ ε (cid:32) (cid:96) i + (1 − ε ) x i · h ( σ ( k ) n − ) c i (cid:33) ≤ x i (1 − ε ) ε ≤ x i ρ (cid:18) (cid:96) i,k c i (cid:19) and again bank i has sold more shares in step k of the fire sales process in ( X n , C n , L n ) withsales function ρ than in the auxiliary process in ((1 − ε ) X n , C n , L n ) with sales function ρ ε . Weinfer σ ( k +1) = (cid:88) i ∈ [ n ] x i (1 − ε ) ρ ε (cid:32) (cid:96) i + (1 − ε ) x i · h ( σ ( k ) n − ) c i (cid:33) ≤ (cid:88) i ∈ [ n ] x i ρ (cid:18) (cid:96) i,k c i (cid:19) = τ ( k +1) and the induction step is completed. As considered before for the auxiliary process, we now investigate resilience properties of thefire sales process. We denote by χ ∗ ,ε the smallest joint root of the functions f mε , m ∈ [ M ] asdefined in Section 4.1 with L = 0. Due to the conservative character of the auxiliary process,it is immediate that resilience of the auxiliary process for system ( X , C ) implies resilience ofthe fire sales process for the system. Contrary, non-resilience under the fire sales process of in( X , C ) implies non-resilience also under the auxiliary process for ( X , C ).16 orollary 4.3. If χ ∗ = , then the ( X , C ) -system is resilient under the fire sales process. Ifthere exists ε > such that χ ∗ ,ε > and such that every z ∈ R M \ { , χ ∗ ,ε } with ≤ z ≤ χ ∗ ,ε (coordinate-wise) is not a joint root of the functions ( f mε ) m ∈ [ M ] , then the ( X , C ) -system isnon-resilient under the fire sales process.Proof. Proof The first claim about resilience is a straightforward consequence of Theorem 4.1.For the second part note that if there exist ε > χ ∗ ,ε > > , then the system((1 − ε ) X , C ) with sales function ρ ε is non-resilient for the auxiliary process by Theorem 3.7(and its extension in 7.1.3) and for every shock L , the number of sold assets over n , χ n itholds that lim inf n →∞ χ mn ≥ χ ∗ ,ε > . By Theorem 4.2 it follows that under the real processin the system ( X , C ) with sales function ρ the number nψ mn of assets m ∈ [ M ] sold fulfillslim inf n →∞ ψ mn ≥ ( χ ∗ ,ε ) m > for every shock L , which implies non-resilience.We will now present several more explicit (non-)resilience criteria for the fire sales process;in particular, we will show that in many natural situations the fire sales and the auxiliaryprocesses behave the same with respect to their resilience properties. A starting point is thefollowing observation, which states that in many cases an analysis of the derivatives at 0 of thefunctionals f m , m ∈ [ M ] describing the auxiliary system can be used to study (non-)resilience. Theorem 4.4.
Assume that ρ is differentiable in , and that there exists m ∈ [ M ] such thatall partial right-derivatives ∂∂χ m h (cid:96) ( ) , (cid:96) ∈ [ M ] exist (possibly with value ∞ ) and E X m ρ (cid:48) (0) (cid:88) (cid:96) ∈ [ M ] X (cid:96) ∂∂χ m h (cid:96) ( ) /C > . (4.4) Then the ( X , C ) -system with sales function ρ is non-resilient under the fire sales process.Proof. Proof Let χ m be the M dimensional vector which has value χ in the m -th entryand zero in all other entries. It is sufficient to show that there exists ε > χ → f mε ( χ m ) /χ > f mε is monotone increasing in the arguments [ M ] \ { m } . Usingthe lemma of Fatou and that ρ ( u ) = ρ ε ( u ) for small u , we obtainlim inf χ → f mε ( χ m ) /χ = lim inf χ → E (cid:20) (1 − ε ) X m ρ ε (cid:18) (1 − ε ) X · h ( χ m ) C (cid:19) /χ (cid:21) − ≥ E (cid:20) lim inf χ → (1 − ε ) X m ρ ε (cid:18) (1 − ε ) X · h ( χ m ) C (cid:19) /χ (cid:21) − E (cid:20) (1 − ε ) X m lim inf χ → ρ (cid:18) (1 − ε ) X · h ( χ m ) C (cid:19) /χ (cid:21) −
1= (1 − ε ) E X m ρ (cid:48) (0) (cid:88) (cid:96) ∈ [ M ] X (cid:96) ∂∂χ m h (cid:96) ( ) /C − , and by (4.4) this expression is larger than zero for small ε >
0. Thus non-resilience follows forthe fire sales process.The previous theorem allows us to show that in many relevant situations actually the auxil-iary and the real process are equivalent in terms of resilience. Let us consider the one dimensionalsystem (i.e., m = 1) in the important case of linear price impact (i.e., h ( χ ) = χ ). The followingcorollary shows that in most cases the resilience properties of the processes coincide. Corollary 4.5.
Let h ( χ ) = χ and let ρ be differentiable in . Then the following holds. . E [ X /C ] = ∞ and ρ (cid:48) (0) > : the system is non-resilient under both (fire sales andauxiliary) process.2. E [ X /C ] < ∞ and ρ (cid:48) (0) > : the systems are resilient under both processes if E [( X /C ) ρ (cid:48) (0)] < and non-resilient if E [( X /C ) ρ (cid:48) (0)] > .3. E [ X /C ] < ∞ and ρ (cid:48) (0) = 0 : the systems are resilient under both processes.Proof. Proof For 1. non-resilience follows from Theorem 4.4 as the expectation in (4.4) isinfinite. For 2., if E [( X /C ) ρ (cid:48) (0)] > f = f . To show resilience for E [( X /C ) ρ (cid:48) (0)] <
1, note that by the Dominated ConvergenceTheorem f (cid:48) (0) = E (cid:20) Xρ (cid:48) (0) (cid:18) X ∂∂χ h (0) (cid:19) /C (cid:21) − E [ X ρ (cid:48) (0) /C ] − < S = { } and χ ∗ = 0, which implies with Corollary 3.6 that the system is resilientunder the auxiliary process, and so also under the fire sales process. Case 3. follows similarly.Corollary 4.5 misses the case E [ X /C ] = ∞ und ρ (cid:48) (0) = 0. This case represents a scenarioof a heterogeneous system with a rather low market reaction to stress, and is thus of particularinterest. Note that in this case the expectation in (4.4) equals 0 and based on Theorem 4.4 wecannot prove non-resilience of the real system. Moreover, we cannot use dominated convergenceto derive the derivative of f = f and to conclude (non-)resilience.In order to address this case we make more refined assumptions. In particular, we assumethat h ( χ ) = χ ν for some ν and ρ ( u ) = u q ∧ q ∈ (0 , ∞ ). In addition we allow for q = ∞ which describes sales at default (i.e. ρ ( u ) = { u ≥ } .) The next result, whose proof is presentedin Section 7.2.2, shows that (non-)resilience is a property that depends on the interplay of ρ and h . Theorem 4.6.
Let m = 1 and P ( X, C >
0) = 1 . Then the following holds.1. − νq > : the system is non-resilient.2. − νq = 0 : the system is resilient if E [ X ( X/C ) /ν ] < and non-resilient if E [ X ( X/C ) /ν ] > .3. − νq < or q = ∞ : Let α ∗ = sup { α ∈ R : E [ X α /C α ] < ∞} . If α ∗ > /ν , then thesystem is resilient. If α ∗ < /ν and f (cid:48) (0) exists, then the system is non-resilient. The previous results are restricted to systems with one asset only. They can be generalizedto systems with multiple assets in various forms. For example, we can assume that for eachasset m ∈ [ M ] with M ≥ h m ( χ ) = ( χ m ) ν m . Thenan argument as before shows that if 1 − q min m ∈ [ M ] ν m >
0, then the system is non-resilient.Moreover, if 1 − q min m ∈ [ M ] ν m = 0 and max m ∈ [ M ] E X m (cid:16) (cid:88) (cid:96) ∈ [ M ] X (cid:96) /C (cid:17) /ν > , then the system is non-resilient (corresponding to a generalization of 2. in Theorem 4.6). Re-silience is more difficult to characterize, as it may depend on the interplay among differentassets. In these cases, Theorem 4.4 and Corollary 4.3 may be used to study resilience propertiesfor the system at hand. We leave it as a research problem to extract further and more explicit(non-)resilience conditions from our main results.18 Applications: Minimal Capital Requirements & The Effect ofthe Portofolio Structure
In this section we apply the theory developed in previous sections to investigate which structuresor properties of systems hinder or promote the emergence and spread of fire sales. In the nextsubsection we address the important question of formulating adequate capital requirements: howmuch capital is necessary and sufficient for each bank, so that the system at hand is resilient?We give an answer to this question in the specific setting where the asset holdings follow a power-law distribution, which is the most typical case to consider. Then, in Section 5.2 we considersystems parametrized by two orthogonal characteristic quantities: portfolio diversification and portfolio similarity . We first study their effect analytically based on our previous results, andwe also verify our findings with simulations for finite systems of reasonable size. Our examplesdemonstrate the effects of portfolio diversification and similarity on resilience in various sensiblesettings.
While the results in the previous section allow regulators to test the resilience of financial systemswith respect to fire sales, a natural question from a regulatory perspective is to determine capitalrequirements that actually ensure resilience.Historically, the first approach to systemic risk was to apply a monetary risk measure tosome aggregated, system-wide risk factor, see [10, 30, 24] for an axiomatic characterization ofthis family of risk measures. Another approach is to determine the total risk by specifyingcapital requirements on an institutional level before aggregating to a system-wide risk factor,see [4, 6, 19]. An important question is then whether from an institutional viewpoint therequirements correspond to fair shares of the systemic risk, see e.g. [6]. In particular, a majorproblem in the implementation of the methodologies proposed in the aforementioned papers isthat a given individual capital requirement in general depends on the configuration of the wholesystem. As a consequence, one institution’s capital requirement may be manipulated by otherinstitutions’ behaviour, or the entrance of a new institution into the system would potentiallyalter the capital requirements of all other institutions, for example. An important contributionof the methodology proposed here is that, besides certain global parameters that need to bedetermined by a regulating institution, the implied requirement for a given institution i ∈ [ n ]depend on its local parameters, i.e., its capital and asset holdings, only.Abstractly, a capital requirement is a function b : R m + → R + that assigns to any bank i ∈ [ n ] with asset holdings x i , . . . , x ni a (minimal) required capital c i = b ( x , . . . , x n ). In ourstochastic setting this implies that C = b ( X , . . . , X m ) is a random variable and we would liketo give conditions on b that ensure resilience. We provide here only stylized examples for whichanalytic derivations are possible to illustrate the power of the stochastic model. For generalsystem configurations one can always resort to numerical calculations.Let us now assume that the distribution F X of asset holdings has power law tail in the sensethat there exist constants B , B ∈ (0 , ∞ ) such that for x large enough B x − β ≤ − F X ( x ) ≤ B x − β , (5.1)for some β >
2. There is empirical evidence for power laws in investment volumes, see e.g. [21].Moreover, we assume the setting of Section 4.2, that is, we assume that h ( χ ) = χ ν for some ν and ρ ( u ) = u q ∧ q ∈ (0 , ∞ ].One natural choice for the capitals c i as a function of the asset holdings is of power form.We immediately get the following corollary, whose proof is a direct consequence of Theorem 4.6,19hat determines sharply the magnitude of the minimal capital requirements to ensure resilience. Corollary 5.1.
Let − νq < or q = ∞ and c i = αx γi for α ∈ R + and γ ∈ R + , . Then,1. if γ > − ν ( β − , then the system is resilient.2. if γ < − ν ( β − , then the system is non-resilient.Proof. Proof We verify the conditions of Theorem 4.6, i.e., we show that α ∗ > /ν (resp. < /ν )where α ∗ = sup { α ∈ R : E [ X α /C α ] < ∞} . By the choice of capital C = X γ we get that E [ X α /C α ] = E [ X α (1 − γ ) ] which is < ∞ (resp. ∞ ) for α < ( β − / (1 − γ ) (resp. α > ( β − / (1 − γ )). Under Assumption 1. then α ∗ > /ν while under Assumption 2. α ∗ < /ν .Additionally we can derive sufficient capital requirement in the multidimensional case. Inthis setting we assume there exist constants B ,m , B ,m ∈ (0 , ∞ ) and β m for m ∈ [ M ] such thatfor x large enough B ,m x − β m ≤ − F X m ( x ) ≤ B ,m x − β m . (5.2) Corollary 5.2.
Let − νq < or q = ∞ and c i = ( (cid:80) m ∈ [ M ] αx mi ) γ for some α ∈ R + and γ ∈ R + , . Let β min := min { β , . . . , β m } . Then,1. if γ > − ν ( β min − , then the system is resilient.2. if γ < − ν ( β min − , then the system is non-resilient.Proof. Proof The proof is immediate from Lemma 5.3 below, which allows to reduce to the onedimensional situation.The following result might be known, but we did not find reference to it in the literature.We provide a proof for completeness.
Lemma 5.3.
Let X m , m ∈ [ M ] be random variables with power law tails as above and let β min := min { β , . . . , β m } . Then there exists B , B > such that B x − β min ≤ − F X + ...X M ( x ) ≤ B x − β min . (5.3) Proof.
Proof We assume that M = 2; the result for general M follows by induction. Let usfurther assume without loss of generality that F X ( x ) = F X ( x ) = 0 for x < β min = β ≤ β . Clearly,1 − F X + X ( x ) = P ( X + X > x ) ≥ P ( X > x ) = 1 − F X ( x ) ≥ B , x − β , which provides the lower bound. By the union bound1 − F X + X ( x ) = P ( X + X > x ) ≤ P ( X > x/
2) + P ( X > x/ ≤ − F X ( x/ − F X ( x/ ≤ B , ( x/ − β + B , ( x/ − β ≤ ( B , β − + B , β − ) x − β and we obtain the upper bound. 20 .2 The Effect of Portfolio Diversification and Similarity For simplicity, throughout this section we assume that the limiting total asset holdings X tot = X + . . . + X M are Pareto distributed with density f X tot ( x ) = ( β − x − β { x ≥ } for someexponent β >
2. One can generalize the example also to more general distributions. Further,we make the assumptions that ρ ( u ) = { u ≥ } and h m ( χ ) = 1 − e − χ m to simplify calculations,but also for other sensible choices our observations below are applicable.In a first setting we consider a system of institutions whose investment in each asset m ∈ [ M ]makes up a fraction λ m ∈ R + of their total asset holdings, where (cid:80) m ∈ [ M ] λ m = 1. Example 5.4.
For a system as described above the functions f m ( χ ) are given by f m ( χ ) = λ m E (cid:34) X tot (cid:40) X tot M (cid:88) (cid:96) =1 λ (cid:96) (cid:16) − e − χ (cid:96) (cid:17) ≥ C (cid:41)(cid:35) − χ m , m ∈ [ M ] . Let us write t = (cid:80) ≤ (cid:96) ≤ M λ (cid:96) (1 − e − χ (cid:96) ) for short. Now assume similar to Corollary 5.1 that C = α ( X tot ) γ for some constants α, γ ∈ R + , . Then f m ( χ ) = λ m E (cid:104) X tot (cid:110) X tot ≥ ( α/t ) − γ (cid:111)(cid:105) − χ m = λ m (cid:90) ∞ max { , ( α/t ) / (1 − γ ) } ( β − x − β d x − χ m = λ m β − β − (cid:110) , ( tα − ) β − − γ (cid:111) − χ m . Motivated by the symmetry of the functions, we consider f m ( χ ) along direction v ∈ R M + , with v m = ( λ m ) − . Then f m ( χ v ) λ m = β − β − (cid:32) α − M (cid:88) (cid:96) =1 λ (cid:96) (cid:16) − e − χ/λ (cid:96) (cid:17)(cid:33) β − − γ − χ ( λ m ) . Let γ c := 3 − β and α c := (cid:80) m ∈ [ M ] ( λ m ) ( β − / ( β − . We infer that if χ ∈ R + , is smallenough and γ > γ c or γ = γ c and α > α c , then dd χ f m ( χ v ) < for all m ∈ [ M ] . That is, χ ∗ = and the auxiliary system is resilient by Corollary 3.6. On the other hand, if either γ < γ c or γ = γ c and α < α c , then dd χ f m ( χ v ) > for all m ∈ [ M ] and the system is non-resilient byTheorem 3.7. Since γ c does not depend on the choice of { λ m } m ∈ [ M ] , it makes sense to consider α c as a measure for stability of the system (the smaller α c , the more stable the system). Clearly, α c becomes minimized for λ m = M − for all m ∈ [ M ] and hence a perfectly diversified systemrequires least capital and can be seen as the most stable (see also discussion after Example 5.5). Next, we consider a financial system that comprises of S ∈ N subsystems of equal size n/S . Foreach subsystem s ∈ [ S ] there is a set of D s = D ∈ N specialized assets that can only be investedin by institutions from s . In addition to these S · D specialized assets, there is a set of J ∈ N joint assets that can be invested in by any institution. Thus, each institution can choose from∆ := D + J different assets to invest in. We call ∆ the diversification of the system. Further,let Σ := J/ ∆ be the (portfolio) similarity in the system. Then, as in Example 5.4, one possibleroute to take is it determine the optimal investments for each institution (that is shifted towardsinvesting in the specialized assets to avoid overlap with other subsystems). Instead we assumein the following example that each institution still perfectly diversifies its investment over the∆ = D + J assets available to it. Example 5.5.
Consider a system as described above consisting of S subsystems and allowingeach institution to invest in D specialized assets and in J joint assets in equal shares. Then the
10 20 30 400.00.20.40.60.81.0 Δ f i na l de f au l tf r a c t i on (a) Σ f i na l de f au l tf r a c t i on (b) Figure 5: Figure corresponding to the simulation study. (a) The effect of varying portfoliodiversification ∆ as Σ = 0 . is fixed. (b) The effect of varying portfolio similarity as ∆ = 20 is fixed. In blue: the theoretical final default fraction. In orange: exemplary simulations. Ingreen: the median over simulations. system is described by the following functions: f j ( χ ) := S − S (cid:88) s =1 E (cid:34) X tot D + J (cid:40) X tot D + J (cid:32) J (cid:88) k =1 (cid:16) − e − χ k (cid:17) + D (cid:88) d =1 (cid:16) − e − χ s,d (cid:17)(cid:33) ≥ C (cid:41)(cid:35) − χ j ,f s,d ( χ ) := S − E X tot D + J X tot D + J J (cid:88) j =1 (cid:16) − e − χ j (cid:17) + D (cid:88) e =1 (cid:0) − e − χ s,e (cid:1) ≥ C − χ s,d , where j ∈ [ J ] , s ∈ [ S ] , d ∈ [ D ] and χ = ( χ , . . . , χ J , χ , , . . . , χ S,D ) ∈ R J + SD + , with small misuseof notation. Similar as in Example 5.4 we derive that γ c = 3 − β and α c = J + DS ( D + J ) β − β − S − S β − β − . From the formula it is obvious that α c decreases (i. e. capital required to ensure resilience) as ∆ increases or Σ decreases. The examples show that diversification reduces the capital necessary to ensure resiliencewhereas stronger similarity between the institutions’ portfolios increases it. The fact that di-versification is favourable might seem at odds with other studies in the literature (see for example[26, 5, 32]), which find that diversification can increase externalities and is less favourable from asystemic risk perspective. In the second example this discrepancy is not surprising, as increaseddiversity arises due to an increased number of assets. In the first example it is due the factthat capital requirements ensure resilience and that amplification effects are small. The salespressure arising from the shock to some banks is then distributed among more assets, each ofthem being less impacted. Spillover effects to other institutions are contained due to the natureof the capital requirements based on X tot . Simulation Study
Note that all previous conclusions build on the (asymptotic) theory de-veloped in the previous sections. To verify and back up the result for finite systems, however,we also give a simulation based verification for a series of moderate size ( n = 10 ) systems. Inthe setting of the last example our choices of the parameters are as follows. We set β = 3 and S = 2 and D = J = 10. Then we obtain ∆ = 20, Σ = 0 . γ c = 0 and α c = 0 . c i = α c . Further, we draw the total asset holdings x tot i for eachinstitution i ∈ [ n ] as random numbers according to the above described Pareto distribution.Finally, we randomly choose a set of initially defaulted institutions of size 0 . n and equallydistribute it across the S subsystems.To see the effect of diversification, we first fix Σ = 0 . D = J vary from 1 to 20(i.e. ∆ ∈ [40]). The results are plotted in Figure 5(a). Since we calibrated the capitals c i = α c to the values ∆ = 20 and Σ = 0 .
5, the theoretical (asymptotic) final default fraction is 1 for∆ ≤
20 and 0 otherwise. This curve is shown in blue. In orange we illustrate 10 of the 10 simulations. One can see that in each simulation the final default fraction rapidly decreases ata certain value for ∆ close to the theoretical value of 20. In green finally, we plot the medianover all 10 simulations which is very close to the theoretical curve despite the finite systemsize and hence verifies that systems become more resilient as ∆ increases. Deviations from thetheoretical curve become smaller as n increases.Furthermore, in the same setting we conducted simulations for systems of fixed diversifica-tion ∆ = 20 and varying similarity Σ between 0 and 1 ( J ∈ [0 ,
20] and D = 20 − J ). The resultsare shown in Figure 5(b). Again, in blue we plot the theoretically predicted curve which is 0for Σ < . . simulations for each Σ can be seen in green and it verifies that the system becomes less resilientas the similarity Σ increases. Again deviations from the theoretical curve are due to finite sizeeffects and become smaller as n increases. In this paper we introduced and studied a model for fire sales. Our model describes the state ofa financial system by specifying for each participant the capital and the amount of m distinctassets that she posseses. All this information is concisely represented by the emprical distribu-tion function of the previously mentioned parameters. Then this system gets initially shocked– the participants suffer losses – and as consequence assets gets sold, which causes the prices togo down, and so an iterated process of selling and price decline starts rolling. Here we studiedthe final state of the system, in particular the number of sold assets and the final default frac-tion. Our main driving question was to understand which structural characteristics promoteor hinder the outbreak of fire sales, and thus our approach is fundamentally asymptotical innature: we ‘gathered’ all systems that are similar – in terms of the empirical distribution – andstudied, as the system size gets large, the effect of various statistical parameters of the (limiting)distribution function. Our results provide in many settings a characterization of resilient andnon-resilient systems, and allow us to formulate explicit capital requirements as well as to studythe effect of portofolio correlations.There is a whole bunch of questions that may be investigated within the framework developedhere. • Finite Systems
Our results are completely explicit in the stochastic model and thus, apriori , apply to large systems only. Our simulation studies show that, however, they arealready good approximations for moderate-sized systems. A challenging direction for fu-ture research is to quantify precisely the finite-size effect : how do our results qualitativelycarry over to systems of a given size n ? How can we adapt the notions of (non-)resilience,and what effect does it have, for example on the capital requirements that we develop?23 Refined Models
Our model describes the process of (distressed) selling of the assets.However, there is no explicit modelling of other parties, like for example the buyers ofassets, or effects, like the buying of assets when the price drops. The effect of other partiesis captured here in an abstract way by the price impact function h . More refined modelsrepresenting the statistical properties of further market participants and actions wouldconstitute significant extensions of our model. • Capital Allocation Strategies
The results presented here allow us, among otherthings, to formulate minimal capital requirements that make the system resilient to firesales. Thus, they constitute a building block of a more global capital allocation strategythat should be capable of making complex systems with also other types of interactions(like default/ liquidity contagion) resilient to initial stress. For example, our capitalrequirements can be combined with classical value-at-risk requirements used in the BaselIII framework. Developing such global strategies and studying the interplay of variousinteractions is a significant research topic. • Portofolio Optimization
Our results are not only applicable from the view of regula-tors; they may also be utilized from market participants to optimize their portfolio withrespect to systemic risk considerations. Developing adequate strategies by taking intoconsideration the effect of fire sales is a challenging open problem. ρ To focus on the underlying phenomena of fire sales rather than be restrained by technical details,in the main part of this article we considered the special case of continuous sale functions ρ .As already the simple example of sales at default ρ ( u ) = { u ≥ } shows, however, it is notuntypical for thresholds to exist at which institutions sell a positive fraction of their assets andthus ρ to be discontinuous. By the explanation via thresholds, right-continuous sale functionsare more natural than sale functions with discontinuities from the right, and we thus assume ρ to be right-continuous in the sequel and denote ◦ ρ ( u ) := lim (cid:15) → ρ ((1 − (cid:15) ) u ) its left-continuousmodification. Replacing ρ by its right-continuous modification ρ ( u ) := lim (cid:15) → ρ ((1 + (cid:15) ) u ) inthe following, however, really makes our results applicable to any non-decreasing sale function ρ . In addition to the results presented in the main body of the paper, the results presented herequantify, besides the final number of sold shares, also the final number of defaulted banks. As for the continuous case in the main part of this article, we start by considering the auxiliaryprocess in the deterministic model. For a general non-decreasing sales function function ρ theequation E (cid:20) X n ρ (cid:18) L n + X n · h ( χ ) C n (cid:19)(cid:21) − χ = 0 (7.1)(see (3.3)) has a smallest solution χ n by the Knaster-Tarski theorem, and as in (3.4) it holds n − lim k →∞ σ ( k ) ≤ χ n . For left-continuous ρ in fact we would derive equality but for right-continuous ρ it is in general possible that n − lim k →∞ σ ( k ) < χ n . This is the case if lim k →∞ σ ( k ) sold shares would be enough to start a new round of fire sales but this quantity is actually never24eached in finitely many rounds. Then the following holds and extends Proposition 3.1; theproof is straight-forward by bounding ρ from below with its left-continuous modification ◦ ρ . Proposition 7.1.
Consider the auxiliary process with a right-continuous sales function ρ andthe corresponding left-continuous modification ◦ ρ . Let χ n ∈ R M + , denote the smallest solution of (7.1) . Moreover, let ˆ χ n ∈ R M + , be the smallest solution of E (cid:20) X n ◦ ρ (cid:18) L n + X n · h ( χ ) C n (cid:19)(cid:21) − χ = 0 . Then the number of sold shares divided by n at the end of the fire sales process satisfies ˆ χ n ≤ n − lim k →∞ σ ( k ) ≤ χ n . (7.2)The equilibrium vector n χ n (in the sense of (7.1)) is thus a conservative bound on the finalnumber of sold shares n χ n = lim k →∞ σ ( k ) . However, as discussed above the convergence of thefire sales process to a non-equilibrium heavily relies on the assumption of arbitrarily small salesizes towards the end of the process. For real systems this is obviously not realistic since theleast possible number of shares sold by an institution is lower bounded by 1. For all practicalpurposes it will therefore hold that χ n = χ n and fire sales stop at an equilibrium state. We now consider a sequence of financial systems of increasing size n ∈ N as before satisfyingAssumption 2.1. As in Section 3.2 let f m , g : R M + , → R , m ∈ [ M ] be given by f m ( χ ) := E (cid:20) X m ρ (cid:18) L + X · h ( χ ) C (cid:19)(cid:21) − χ m , m ∈ [ M ] , g ( χ ) := P ( L + X · h ( χ ) ≥ C ) . Moreover, motivated by the lower bound in Proposition 7.1 define the lower semi-continuousfunctions ◦ f m ( χ ) := E (cid:20) X m ◦ ρ (cid:18) L + X · h ( χ ) C (cid:19)(cid:21) − χ m , m ∈ [ M ] , (7.3)and let ◦ g ( χ ) := P ( L + X · h ( χ ) > C ) . In addition to the sets S and S from Section 3.2, now also define ◦ S := (cid:92) m ∈ [ M ] (cid:110) χ ∈ R M + , : ◦ f m ( χ ) ≥ (cid:111) and denote by ◦ S the largest connected subset of ◦ S containing (clearly ◦ f m ( ) ≥ m ∈ [ M ] and thus ∈ S ). Note that still S and S are closed sets as f m , m ∈ [ M ], are uppersemi-continuous for the choice of right-continuous sale function ρ . Finally, as in Section 3.2 set χ ∗ ∈ R M + , with ( χ ∗ ) m := sup χ ∈ S χ m . We can then state the following generalization of Lemma 3.3.
Lemma 7.2.
There exists a smallest joint root ˆ χ of all functions ◦ f m ( χ ) , m ∈ [ M ] , with ˆ χ ∈ S .Further, χ ∗ as defined above is a joint root of the functions f m , m ∈ [ M ] , and χ ∗ ∈ S .
25n complete analogy to Theorem 3.4, but now without neccesitating Assumption 2.2, we candescribe the extent and impact of the auxiliary process together with bounds for the number ofdefaulted institutions.
Theorem 7.3.
Consider a ( X , C ) -system with initial shock L satisfying Assumption 2.1. As-sume that ρ is right-continuous. Then for χ mn , the number of finally sold shares of asset m ∈ [ M ] divided by n in the auxiliary process, and for the final default fraction n − |D n | , ˆ χ m + o (1) ≤ χ mn ≤ ( χ ∗ ) m + o (1) , ◦ g ( ˆ χ ) + o (1) ≤ n − |D n | ≤ g ( χ ∗ ) + o (1) . In particular, for the final price impact h m ( χ n ) on asset m ∈ [ M ] h m ( ˆ χ ) + o (1) ≤ h m ( χ n ) ≤ h m ( χ ∗ ) + o (1) . In addition to instability of the joint root ˆ χ (as is explained in Subsection 3.2) anotherexplanation for ˆ χ (cid:54) = χ ∗ could be that ˆ χ is a point of discontinuity for some f m but this againis a rather pathological case. It then usually holds ◦ g ( ˆ χ ) = g ( χ ∗ ) and Theorem 7.3 determinesthe limits of χ n and n − |D n | as n → ∞ . In this section we study (non-)resilience properties of ( X , C )-systems in the general setting; notethat these notions remains unaffected by our previous Assumption 2.2. The following statementthus generalizes Theorem 3.5 and also includes a statement about the default fraction. Theorem 7.4.
Consider a ( X , C ) -system satisfying Assumption 1. Assume that ρ is right-continuous. Then for each (cid:15) > there exists δ > such that for all initial shocks L with E [ L/C ] < δ , the number nχ mn,L of finally sold shares of asset m ∈ [ M ] and the final set ofdefaulted institutions D n,L in the shocked system satisfy lim sup n →∞ χ mn,L ≤ ( χ ∗ ) m + (cid:15), m ∈ [ M ] and lim sup n →∞ n − |D n,L | ≤ g ( χ ∗ ) + (cid:15). Using the notations of Section 7.1.2 also the wording of Corollary 3.6 remains completelyunchanged; that is, if χ ∗ = 0, then the ( X , C )-system satisfying Assumption 1 with right-continuous ρ is resilient.We now turn to the case χ ∗ >
0. Theorem 3.7 remains unchanged in case ρ is strictlyincreasing in a neighbourhood of 0. If ρ is not strictly increasing, we distinguish betweennon-resilience and weak non-resilience. Theorem 7.5.
Consider a ( X , C ) -system satisfying Assumption 1. Assume that ρ is right-continuous. Let (cid:15) > . For every δ > there exists a shock L with E [ L/C ] < δ and lim inf n →∞ χ mn,L > ( χ ∗ ) m − (cid:15) and lim inf n →∞ n − |D n,L | > ◦ g ( χ ∗ ) − (cid:15). In particular from Theorem 7.5 we immediately get the following result.
Corollary 7.6 (Non-resilience Criterion) . If χ ∗ > , then the ( X , C ) -system is weakly non-resilient under the auxiliary process. .2 Proofs of Statements Proof.
Proof of Lemma 7.2. Existence of ˆ χ follows from the Knaster-Tarski theorem. We nowconstruct a joint root ◦ S (cid:51) ¯ χ ≤ ˆ χ such that we can conclude ˆ χ = ¯ χ ∈ ◦ S . It holds ◦ f m ( ˆ χ ) = 0for all m ∈ [ M ] and thus (for any fixed m ∈ [ M ]) ◦ f m ( χ ) ≤ χ ≥ χ ∈ R M + , such that χ m = ˆ χ m by monotonicity of ◦ f m . Consider then the following sequence ( χ ( k ) ) k ∈ N ⊂ R M + , : • χ (0) = ∈ ◦ S • χ (1) = ( χ , , . . . , ≤ χ ≤ ˆ χ is the smallest possible value such that ◦ f ( χ (1) ) = 0. It is possible to find such χ since ◦ f ( χ ) + χ is monotonically increasingin χ , ◦ f ( ) ≥ ◦ f ( ˆ χ , , . . . , ≤
0. By monotonicity of ◦ f m with respect to χ for all m ∈ [ M ] \{ } , it then holds ◦ f m ( χ (1) ) ≥ ◦ f m ( ) ≥ (cid:54) = m ∈ [ M ] and in particular χ (1) ∈ ◦ S . • χ (2) = χ (1) + (0 , χ , , . . . , ≤ χ ≤ ˆ χ is the smallest value such that ◦ f ( χ (2) ) = 0. Again it is possible to find such χ since ◦ f ( χ ) + χ is monotonicallyincreasing in χ , ◦ f ( χ (1) ) ≥ ◦ f ( χ (1) + (0 , ˆ χ , , . . . , ≤
0. By monotonicity of ◦ f m with respect to χ for all m ∈ [ M ] \{ } , it then holds ◦ f m ( χ (2) ) ≥ ◦ f m ( χ (1) ) ≥ (cid:54) = m ∈ [ M ] and in particular χ (2) ∈ ◦ S . • χ ( i ) , i ∈ { , . . . , M } , are found analogously, changing only the corresponding coordinate. • χ ( M +1) = χ ( M ) + ( χ M +1) − χ M ) , , . . . , χ M ) ≤ χ M +1) ≤ ˆ χ is the smallestvalue such that ◦ f ( χ ( M +1) ) = 0, which is again possible by monotonicity of ◦ f ( χ ) + χ , ◦ f ( χ ( M ) ) ≥ ◦ f ( χ ( M ) +( ˆ χ − χ M ) , , . . . , ≤
0. Further, it still holds χ ( M +1) ∈ ◦ S . • Continue for χ i , i ≥ M + 2.The sequence ( χ ( k ) ) k ∈ N constructed this way has the following properties: it is non-decreasingin each coordinate and bounded inside [ , ˆ χ ]. Hence by monotone convergence, each coordinateof χ ( k ) converges and so ¯ χ = lim k →∞ χ ( k ) exists. Since the convergence is from below, ◦ f m ( ¯ χ ) = E (cid:20) X m ◦ ρ (cid:18) L + X · h (lim k →∞ χ ( k ) ) C (cid:19)(cid:21) − lim k →∞ χ m ( k ) = lim k →∞ E (cid:20) X m ◦ ρ (cid:18) L + X · h ( χ ( k ) ) C (cid:19)(cid:21) − χ m ( k ) = lim k →∞ ◦ f m ( χ ( k ) ) ≥ χ ∈ ◦ S . Now suppose there is m ∈ [ M ] such that ◦ f m ( ¯ χ ) >
0. By lower semi-continuity of ◦ f m then also ◦ f m ( χ ( k ) ) > (cid:15) for some (cid:15) > k large enough. This, however, isa contradiction to the construction of the sequence ( χ ( k ) ) k ∈ N since ◦ f m ( χ ( k ) ) = 0 in every M -thstep. Hence ◦ f m ( ¯ χ ) = 0 for all m ∈ [ M ] and ¯ χ is a joint root of all functions ◦ f m , m ∈ [ M ].Now turn to the proof that χ ∗ ∈ S . We first consider the case that ρ is continuous. Weapproximate χ ∗ ∈ S by the sequence ( ˆ χ ( (cid:15) )) (cid:15)> of smallest fixpoints for the functions f m ( χ )+ (cid:15) .This allows us to apply the Knaster-Tarski Theorem and the monotonicity properties of f m + (cid:15)
27s above. Simple topological arguments will then allow us to conclude that χ ∗ ∈ S . Let for (cid:15) > S ( (cid:15) ) := (cid:92) m ∈ [ M ] (cid:8) χ ∈ R M + , : f m ( χ ) ≥ − (cid:15) (cid:9) and denote by S ( (cid:15) ) the connected component of in S ( (cid:15) ). By the same procedure as for ˆ χ above, we now derive that there exists a smallest (componentwise) point ˆ χ ( (cid:15) ) ∈ S ( (cid:15) ) such that f m ( ˆ χ ( (cid:15) )) = − (cid:15) for all m ∈ [ M ]. Clearly, ˆ χ ( (cid:15) ) is non-decreasing (componentwise) in (cid:15) and hence˜ χ := lim (cid:15) → ˆ χ ( (cid:15) ) exists (we will show that ˜ χ = χ ∗ in fact).Now by monotonicity of S ( (cid:15) ), we derive that ˆ χ ( δ ) ∈ S ( δ ) ⊆ S ( (cid:15) ) for all δ ≤ (cid:15) . Since S ( (cid:15) ) is a closed set, it must thus hold that also ˜ χ = lim δ → ˆ χ ( δ ) ∈ S ( (cid:15) ) for all (cid:15) > χ ∈ (cid:84) (cid:15)> S ( (cid:15) ). Further, we derive that (cid:84) (cid:15)> S ( (cid:15) ) ⊆ (cid:84) (cid:15)> S ( (cid:15) ) ⊆ S . Moreover, (cid:84) (cid:15)> S ( (cid:15) ) is the intersection of a chain of connected, compact sets in the Hausdorff space R M and it is hence a connected, compact set itself. Since it further contains , we can then concludethat (cid:84) (cid:15)> S ( (cid:15) ) ⊆ S and thus ˜ χ ∈ S .Consider now an arbitrary χ ∈ S . We want to show that χ ≤ ˜ χ componentwise and thus˜ χ = χ ∗ . It suffices to show that S ⊂ [ , ˆ χ ( (cid:15) )] for all (cid:15) . Then χ ≤ ˆ χ ( (cid:15) ) and χ ≤ lim (cid:15) → ˆ χ ( (cid:15) ) =˜ χ . Hence assume that S (cid:54)⊂ [ , ˆ χ ( (cid:15) )]. By connectedness of S we find ¯ χ ∈ S with ¯ χ m ≤ ˆ χ m ( (cid:15) )for all m ∈ [ M ] and equality for at least one coordinate (otherwise S ∩ ∂ [ , ˆ χ ( (cid:15) )] = ∅ and S = (cid:0) S ∩ (cid:0) R M + , \ [ , ˆ χ ( (cid:15) )] (cid:1)(cid:1) ∪ ( S ∩ [ , ˆ χ ( (cid:15) ))) is the union of two open non-empty sets andhence not connected). W. l. o. g. let this coordinate be ¯ χ . By monotonicity of f with respectto χ m for every 1 (cid:54) = m ∈ [ M ], we thus derive that f ( ¯ χ ) ≤ f ( ˆ χ ( (cid:15) )) = − (cid:15) < χ ∈ S .Now consider the general case that ρ is right-continuous and let ( ρ r ( u )) r ∈ N a sequence ofcontinuous sale functions approximating ρ from above. Denoting by S r the analogue of S forthe sale function ρ r , we derive that S = (cid:84) r ∈ N S r since clearly S r ⊇ S for all r ∈ N and furtherby dominated convergence for every χ ∈ (cid:84) r ∈ N S r , χ m ≤ E (cid:20) X m ρ r (cid:18) L + X · h ( χ ) C (cid:19)(cid:21) → E (cid:20) X m ρ (cid:18) L + X · h ( χ ) C (cid:19)(cid:21) , as r → ∞ , so that (cid:84) r ∈ N S r ⊆ S . If we further let S r denote the largest connected subset of S r containing , then S r is compact and connected for every r ∈ N and hence so is (cid:84) r ∈ N S r . Since further ∈ (cid:84) r ∈ N S r , we derive that (cid:84) r ∈ N S r = S . Let now χ ∗ r denote the analogue of χ ∗ for the salefunction ρ r . Then lim r →∞ χ ∗ r ∈ S R for all R ∈ N and hence lim r →∞ χ ∗ r ∈ (cid:84) R ∈ N S R = S . Nowsuppose there existed a vector χ ∈ S and m ∈ [ M ] such that χ m > lim r →∞ ( χ ∗ r ) m . Then alsofor R large enough, χ m > ( χ ∗ R ) m and hence χ (cid:54)∈ S R . This, however, contradicts the assumptionthat χ ∈ S = (cid:84) R ∈ N S R . Hence there exists no such χ ∈ S and χ ∗ = lim r →∞ χ ∗ r ∈ S .Finally, we show that χ ∗ is a joint root of f m , m ∈ [ M ]: Since χ ∗ ∈ S , it holds that f m ( χ ∗ ) ≥ m ∈ [ M ]. Assume now that f m ( χ ∗ ) > m ∈ [ M ]. We can thengradually increase the m -coordinate of χ ∗ (until f m ( χ ∗ ) = 0). By monotonicity of f k ( χ ) withrespect to χ m for every m (cid:54) = k ∈ [ M ], however, we can be sure that we do not leave the set S by this procedure which is a contradiction to the definition of χ ∗ . Hence χ ∗ is a joint root of f m , m ∈ [ M ]. Remark 7.7.
In the proof of Lemma 3.3, for the case that ρ is continuous, we constructed χ ∗ as the limit of a sequence ( ˆ χ ( (cid:15) )) (cid:15)> such that f m ( ˆ χ ( (cid:15) )) = − (cid:15) for all m ∈ [ M ]. For non-continuous ρ by the Knaster-Tarski theorem we still know that there exists a smallest vectorˆ χ ( (cid:15) ) such that f m ( ˆ χ ( (cid:15) )) = − (cid:15) , but the construction of ˆ χ ( (cid:15) ) as for ˆ χ in the proof of Lemma3.3 fails and we can hence not be sure a priori that ˆ χ ( (cid:15) ) ∈ S ( (cid:15) ). Hence let further ˜ χ ( (cid:15) ) be28efined as the smallest vector in S ( (cid:15) ) such that f m ( ˜ χ ( (cid:15) )) = − (cid:15) . This vector exists again by theKnaster-Tarski theorem noting that analogue to Lemma 3.3 S ( (cid:15) ) contains its componentwisesupremum χ ∗ ( (cid:15) ). Then by the same means as above, we derive that χ ∗ = lim (cid:15) → ˜ χ ( (cid:15) ).The rest of the section is devoted to the proof of Theorem 7.3 (and Theorem 3.4). Therewe are considering a sequence of financial systems. The following technical lemma about theconvergence of the smallest joint roots will be quite handy. Lemma 7.8.
Let a sequence (for r ∈ N ) of financial systems be described by functions ◦ f mr , m ∈ [ M ] , with smallest joint root ˆ χ r . If lim inf r →∞ ◦ f mr ( χ ) ≥ ◦ f m ( χ ) pointwise for every m ∈ [ M ] ,then lim inf r →∞ ˆ χ r ≥ ˆ χ , where ˆ χ denotes the smallest joint root of the functions ◦ f m , m ∈ [ M ] .Proof. Proof. The main difficulty in showing the clraim is that lim inf r →∞ ◦ f mr ( χ ) ≥ ◦ f m ( χ ) onlypointwise but not uniformly in χ . A further difficulty is the multidimensionality. The mainidea is to construct a path in analogy to the construction in Lemma 7.2 that leads to a point˜ χ ( (cid:15) ) smaller but close to ˆ χ . On this path the functions ◦ f mr , m ∈ [ M ] are all positive for r large.It can then be compared componentwise with a path leading to ˆ χ r .For this consider the construction of ˆ χ in Lemma 7.2 and change it in such a way thatin each step k = LM + m (where L ∈ N and m ∈ [ M ]) a point χ ( k ) ( (cid:15) ) is chosen such that ◦ f m ( χ ( k ) ( (cid:15) )) ≤ (cid:15) for some fixed (cid:15) > χ m ( k ) ( (cid:15) ) ≥ χ m ( k − ( (cid:15) ) as the smallest possible valuesuch that this inequality holds; it will then either be ◦ f m ( χ ( k ) ( (cid:15) )) = (cid:15) or χ ( k ) ( (cid:15) ) = χ ( k − ( (cid:15) )).Note that ◦ f m ( χ ( k ) ( (cid:15) )) < (cid:15) can only happen if ◦ f m ( ) < (cid:15) in which case there exists k ∈ N ∞ suchthat χ m ( k ) = 0 and ◦ f m ( χ ( k ) ) < (cid:15) for all k ≤ k but ◦ f m ( χ ( k ) ) ≥ (cid:15) and χ m ( k ) > k > k . Then( χ ( k ) ( (cid:15) )) k ∈ N is a non-decreasing (componentwise) sequence bounded by ˆ χ and hence ˜ χ ( (cid:15) ) =lim k →∞ χ ( k ) ( (cid:15) ) exists. Further, it holds that ◦ f m ( ˜ χ ( (cid:15) )) ≤ lim inf k →∞ ◦ f m ( χ ( k ) ( (cid:15) )) ≤ (cid:15) . Finally,˜ χ ( (cid:15) ) is non-increasing componentwise in (cid:15) and bounded inside [ , ˆ χ ] and thus ˜ χ = lim (cid:15) → ˜ χ ( (cid:15) )exists. Moreover, ◦ f m ( ˜ χ ) ≤ lim inf (cid:15) → ◦ f m ( ˜ χ ( (cid:15) )) ≤ lim inf (cid:15) → (cid:15) = 0 and hence in particular˜ χ = ˆ χ .Fix now δ > (cid:15) > χ m ( (cid:15) ) > ˆ χ m (1 − δ ) / for all m ∈ [ M ]. Further, choose K = K ( (cid:15) ) ∈ N large enough such that χ m ( K ) ( (cid:15) ) > ˜ χ m ( (cid:15) )(1 − δ ) / forall m ∈ [ M ]. In particular, χ m ( K ) ( (cid:15) ) > ˆ χ m (1 − δ ). Now note that ˆ χ r can be constructed by asequence ( χ ( k,r ) ) k ∈ N analogue to ˆ χ in the proof of Lemma 3.3 as well. We can then in each step k ∈ N cap the element of the constructing sequence χ ( k,r ) at χ ( k ) ( (cid:15) ), which clearly does notincrease the limit of the sequence. We want to make sure that in fact the cap is used in every step k ≤ K if we only choose r large enough. Then we can conclude that ˆ χ r ≥ χ ( K ) ( (cid:15) ) ≥ ˆ χ (1 − δ )and hence letting δ →
0, lim inf r →∞ ˆ χ r ≥ ˆ χ .We now show that the cap is applied in every step k ≤ K for r large enough by an inductionargument. For k = 0, clearly χ ( k,r ) = χ ( k ) ( (cid:15) ) = and the cap is applied. Now lets assume itholds for k ≤ k < K . If χ ( k +1) ( (cid:15) ) = χ ( k ) ( (cid:15) ), then of course the cap is also applied in step k +1as the sequence χ ( k,r ) is increasing. Otherwise, note that by definition of χ ( k +1) ( (cid:15) ), it holds ◦ f m ( χ ) ≥ (cid:15) for all χ ∈ R M + , such that χ m ∈ [ χ m ( k ) ( (cid:15) ) , χ m ( k +1) ( (cid:15) )] and χ (cid:96) = χ (cid:96) ( k ) ( (cid:15) ) = χ (cid:96) ( k +1) ( (cid:15) )for all (cid:96) ∈ [ M ] \{ m } . Now choose a discretization { χ j } ≤ j ≤ J of [ χ m ( k ) ( (cid:15) ) , χ m ( k +1) ( (cid:15) )] for J < ∞ such that χ = χ m ( k ) ( (cid:15) ), χ J = χ m ( k +1) ( (cid:15) ) and χ j − < χ j < χ j − + (cid:15)/ j ∈ [ J ]. We nowuse the assumption that lim inf r →∞ ◦ f mr ( χ j ) ≥ ◦ f m ( χ j ) for every 0 ≤ j ≤ J , where χ mj = χ j and χ (cid:96)j = χ (cid:96) ( k ) ( (cid:15) ) for (cid:96) ∈ [ M ] \{ m } . Then for r large enough, ◦ f mr ( χ j ) ≥ ◦ f m ( χ j ) − (cid:15)/ ≥ (cid:15)/ χ = α χ j − + (1 − α ) χ j between χ j − and χ j ( α ∈ [0 , ◦ f mr ( χ ) ≥ ◦ f mr ( χ j − ) + χ mj − − χ mj ≥ (cid:15)/ − (cid:15)/ (cid:15)/ . Hence the cap is applied in step k + 1. As there are only finitely many steps k ≤ K , thisfinishes the proof. Proof.
Proof of Theorem 7.3. We start by proving the lower bound. Recall from Proposition 7.1that χ n ≥ ˆ χ n . Using weak convergence of the random vector ( X n , C n , L n ) and approximating ◦ ρ from below by a sequence of continuous sale functions ( ρ r ) r ∈ N , we derive for U ∈ R + thatpointwiselim inf n →∞ E (cid:20) X mn ◦ ρ (cid:18) L n + X n · h ( χ )) C n (cid:19)(cid:21) ≥ lim n →∞ E (cid:20) ( X mn ∧ U ) ρ r (cid:18) L n + X n · h ( χ )) C n (cid:19)(cid:21) = E (cid:20) ( X m ∧ U ) ρ r (cid:18) L + X · h ( χ )) C (cid:19)(cid:21) . Hence as U → ∞ and r → ∞ by monotone convergence,lim inf n →∞ E (cid:20) X mn ◦ ρ (cid:18) L n + X n · h ( χ ) C n (cid:19)(cid:21) − χ m ≥ ◦ f m ( χ )) (7.4)and we can use Lemma 7.8 to derive that lim inf n →∞ χ n ≥ lim inf n →∞ ˆ χ n ≥ ˆ χ .We now want to show the lower bound on the final default fraction. Fix some δ > n large enough such that χ n ≥ ˆ χ n ≥ (1 − δ ) ˆ χ componentwise. Then n − |D n | = P ( L n + X n · h ( χ n ) ≥ C n ) ≥ P ( L n + X n · h ((1 − δ ) ˆ χ ) > C n ) . However, using weak convergence of ( X n , C n , L n ) and approximating the indicator function { y > } from below by continuous functions ( φ t ) t ∈ N , we derive thatlim inf n →∞ n − |D n | ≥ lim n →∞ E (cid:20) φ t (cid:18) L n + X n · h ((1 − δ ) ˆ χ ) C n (cid:19)(cid:21) = E (cid:20) φ t (cid:18) L + X · h ((1 − δ ) ˆ χ ) C (cid:19)(cid:21) and as t → ∞ , lim inf n →∞ n − |D n | ≥ P ( L + X · h ((1 − δ ) ˆ χ ) > C ) = ◦ g ((1 − δ ) ˆ χ ) . This quantity now tends to ◦ g ( ˆ χ ) as δ → ◦ g .Now we approach the second part of the theorem. Recall from Proposition 7.1 that χ n ≤ χ n .By the construction of χ ∗ in the proof of Lemma 3.3, we have a non-increasing (in (cid:15) ) sequence( ˆ χ ( (cid:15) )) (cid:15)> such that lim (cid:15) → ˆ χ ( (cid:15) ) = χ ∗ . (See Remark 7.7 for non-continuous ρ .) In particular, χ ∗ ≤ ˆ χ ( (cid:15) ) for every (cid:15) > f m ( ˆ χ ( (cid:15) )) = − (cid:15) . Using weak convergence of ( X n , C n , L n ) wederive for U ∈ R + and ( ρ s ) s ∈ N an approximation of ρ from above by continuous sale functionsthatlim sup n →∞ E (cid:20) X mn ρ (cid:18) L n + X n · h ( ˆ χ ( (cid:15) )) C n (cid:19)(cid:21) = E [ X m ] − lim inf n →∞ E (cid:20) X mn (cid:18) − ρ (cid:18) L n + X n · h ( ˆ χ ( (cid:15) )) C n (cid:19)(cid:19)(cid:21) ≤ E [ X m ] − lim inf n →∞ E (cid:20) ( X mn ∧ U ) (cid:18) − ρ s (cid:18) L n + X n · h ( ˆ χ ( (cid:15) )) C n (cid:19)(cid:19)(cid:21) = E [ X m ] − E (cid:20) ( X m ∧ U ) (cid:18) − ρ s (cid:18) L + X · h ( ˆ χ ( (cid:15) )) C (cid:19)(cid:19)(cid:21) U → ∞ , s → ∞ , by monotone convergencelim sup n →∞ E (cid:20) X mn ρ (cid:18) L n + X n · h ( ˆ χ ( (cid:15) )) C n (cid:19)(cid:21) ≤ f m ( ˆ χ ( (cid:15) )) + ˆ χ m ( (cid:15) ) = ˆ χ m ( (cid:15) ) − (cid:15). Hence for n large enough it holds E (cid:20) X mn ρ (cid:18) L n + X n · h ( ˆ χ ( (cid:15) )) C n (cid:19)(cid:21) − ˆ χ m ( (cid:15) ) ≤ − (cid:15)/ < m ∈ [ M ]. In particular, we know that χ n ≤ ˆ χ ( (cid:15) ). Letting (cid:15) →
0, this shows thatlim sup n →∞ χ mn ≤ lim sup n →∞ χ mn ≤ ( χ ∗ ) m for all m ∈ [ M ] and hence completes the proof ofthe upper bound on finally sold assets.For the upper bound on the final default fraction n − |D n | = P ( L n + X n · h ( χ n ) ≥ C n ),approximate the indicator function { y ≥ } from above by continuous functions ( ψ t ) t ∈ N anduse weak convergence of ( X n , C n , L n ) to derivelim sup n →∞ n − |D n | ≤ lim n →∞ E (cid:20) ψ t (cid:18) L n + X n · h ( ˆ χ ( (cid:15) )) C n (cid:19)(cid:21) = E (cid:20) ψ t (cid:18) L + X · h ( ˆ χ ( (cid:15) )) C (cid:19)(cid:21) and as t → ∞ , lim sup n →∞ n − |D n | ≤ g ( ˆ χ ( (cid:15) )). Letting (cid:15) →
0, thus shows the second part ofthe theorem by upper semi-continuity of g .In the following two proofs we use the notations g , f m , ◦ g , ◦ f m , ˆ χ and χ ∗ as introduced in7.1.2 but for an unshocked ( X , C )-system. If instead we index these quantities by · L , we meanthe system shocked by L . Proof.
Proof of Theorem 3.5. By Remark 7.7, there exists a sequence of vectors ˜ χ ( γ ) ∈ R M + , such that f m ( ˜ χ ( γ )) = − γ for all m ∈ [ M ] and arbitrary γ ∈ R + . Now for arbitrary α ∈ R + itholds that f mL ( χ ) = E (cid:20) X m ρ (cid:18) L + X · h ( χ ) C (cid:19)(cid:21) − χ m ≤ E (cid:20) X m (cid:26) LC ≥ α (cid:27)(cid:21) + E (cid:20) X m ρ (cid:18) αC + X · h ( χ ) C (cid:19)(cid:21) − χ m . Since E [ L/C ] < δ , by Markov’s inequality it holds that P ( L/C ≥ α ) ≤ δ/α and hence for δ > E [ X m { L/C ≥ α } ] ≤ γ/ E [ X m ] < ∞ ). By dominatedconvergence and right-continuity of ρ , it thus holds that f mL ( χ ) ≤ f m ( χ ) + 2 γ/ α > E (cid:20) X m ρ (cid:18) αC + X · h ( χ ) C (cid:19)(cid:21) ≤ E (cid:20) X m ρ (cid:18) X · h ( χ ) C (cid:19)(cid:21) + γ/ . In particular, f mL ( ˜ χ ( γ )) ≤ − γ/ < χ ∗ L < ˜ χ ( γ ) for δ small enough. By similarmeans, we further derive that for δ small enough it holds g L ( ˜ χ ( γ )) ≤ g ( ˜ χ ( γ )) + (cid:15)/
3. Togetherwith Theorem 3.4, we thus derive thatlim sup n →∞ n − |D n,L | ≤ g L ( χ ∗ L ) + (cid:15)/ ≤ g L ( ˜ χ ( γ )) + (cid:15)/ ≤ g ( ˜ χ ( γ )) + 2 (cid:15)/ . Now since ˜ χ ( γ ) → χ ∗ and by upper semi-continuity of g , we can choose γ > g ( ˜ χ ( γ )) ≤ g ( χ ∗ ) + (cid:15)/ n →∞ n − |D n,L | ≤ g ( χ ∗ ) + (cid:15) .For the bound on ( χ mn,L ) choose γ and δ small enough such that ( χ ∗ L ) m ≤ ˜ χ m ( γ ) + (cid:15)/ ≤ ( χ ∗ ) m + 2 (cid:15)/ n →∞ ( χ mn,L ) ≤ ( χ ∗ L ) m + (cid:15)/ ≤ ( χ ∗ ) m + (cid:15). Proof.
Proof of Theorem 3.7. For some l ∈ [ M ] it must hold that P ( L, X l > > δ > A ⊂
Ω such that
L, X l ≥ δ on A . Then we can find ˜ A ⊂ A and x large such that C ≤ x on ˜ A and therefore L/C ≥ δ/x on ˜ A . Since ρ is assumed to be strictlyincreasing we can conclude that ε := ρ ( δ/x ) >
0. We obtain that E (cid:20) X l ρ (cid:18) L + X · h (0) C (cid:19)(cid:21) ≥ εδ P ( ˜ A ) > ◦ f lL ( ) >
0. Therefore is not a joint root of the functions ◦ f mL ( ) , m ∈ [ M ] and infact the auxilary process starts. Because of ◦ f mL ( χ ) ≥ ◦ f m ( χ ) it follows that ˆ χ L ≥ χ ∗ and thusTheorem 7.4 completes the proof. Proof.
Proof of Theorem 7.5. For a given δ > P ( L =2 C ) = δ/ P ( L = 0) = 1 − δ/
3. Such shock can be chosen to be independent of X . Thenclearly E [ L/C ] = (2 / δ < δ . Moreover since ρ (1) = 1 it follows that ◦ f mL ( ) = E [ X m ](2 / δ for m ∈ [ M ]. Since E [ X l ] > l ∈ [ M ], it follows that ◦ f lL ( ) > l ∈ [ M ]. Nowthe same arguments as in the proof of Theorem 3.7 can be applied. Proof.
Proof of Theorem 4.6 We start with the case 1 − νq >
0. Using Theorem 4.2 it is enoughto show for some fixed ε ∈ (0 ,
1) that lim inf n →∞ E [ Xρ ε (cid:16) Xh ( a n ) C (cid:17) ] /a n > a n ) n ≥ with lim n →∞ a n = 0. Using Fatou’s lemma we get thatlim inf n →∞ E (cid:20) Xρ ε (cid:18) Xh ( a n ) C (cid:19) /a n (cid:21) ≥ E (cid:20) lim inf n →∞ Xρ ε (cid:18) Xh ( a n ) C (cid:19) /a n (cid:21) = E (cid:20) lim inf n →∞ X (cid:18) Xa νn C (cid:19) q /a n (cid:21) = E (cid:20) lim inf n →∞ a νq − n X (cid:18) XC (cid:19) q (cid:21) = E (cid:2) ∞ · { X ( X/C ) q (cid:54) =0 } (cid:3) = ∞ , and thus the real system is non-resilient.We next look at the case 1 − νq = 0. Let E [ X ( X/C ) q ] <
1. We show resilience of the auxiliaryprocess from which resilience for the real process follows by Theorem 4.1. As before, let ( a n ) n ≥ be a sequence with lim n →∞ a n = 0. We need to show that lim sup n →∞ E [ Xρ ( Xh ( a n ) /C )] /a n <
32. Define the event A n := { ( X/C ) q ≤ /a n } . First observe that since q = 1 /νXa − n ρ (cid:18) Xh ( a n ) C (cid:19) = Xa − n (cid:18) Xh ( a n ) C (cid:19) q A n + Xa − n A cn = X ( X/C ) q A n + Xa − n A cn ≤ X ( X/C ) q A n + X ( X/C ) q A cn = X ( X/C ) q . It follows thatlim n →∞ E (cid:20) Xa − n ρ (cid:18) Xh ( a n ) C (cid:19)(cid:21) ≤ lim n →∞ E [ X ( X/C ) q ] = E [ X ( X/C ) q ] < , and thus the system is resilient. On the contrary if E [ X ( X/C ) q ] >
1, then for fixed ε > A n := { ( X/C ) q ≤ (1 /a n ) ε } and observe that A n (cid:37) Ω. It follows for ε > n →∞ E (cid:20) (1 − ε ) Xa − n ρ ε (cid:18) Xh ( a n ) C (cid:19)(cid:21) ≥ lim n →∞ E [(1 − ε ) X ( X/C ) q A n ] > , and thus the real system is non-resilient.Now consider the remaining case 1 − νq <
0. First assume that α ∗ > /ν , where weclaim resilience of the auxiliary process. Chose 1 < δ such that δ (1 /ν ) < min { α ∗ , q } and set A n := { ( X/C ) /ν ≤ /a n } . Then E (cid:20) Xa − n ρ (cid:18) Xh ( a n ) C (cid:19)(cid:21) = E (cid:20) Xa − n (cid:18) Xh ( a n ) C (cid:19) q A n (cid:21) + E (cid:2) Xa − n A cn (cid:3) = a νq − n E (cid:20) X (cid:18) XC (cid:19) q A n (cid:21) + E (cid:2) Xa − n A cn (cid:3) . (7.5)For the first term observe that a νq − n E (cid:20) X (cid:18) XC (cid:19) q A n (cid:21) = a νq − n E (cid:34) X (cid:18) XC (cid:19) δ/ν (cid:18) XC (cid:19) q − δ/ν A n (cid:35) = a νq − n E (cid:34) X (cid:18) XC (cid:19) δ/ν (cid:18) XC (cid:19) q − δ/ν { ( X/C ) q − δ/ν ≤ (1 /a n ) νq − δ } (cid:35) ≤ a νq − n E (cid:34) X (cid:18) XC (cid:19) δ/ν a − νq + δn { ( X/C ) q − δ/ν ≤ a − νq + δn } (cid:35) ≤ a δ − n E (cid:34) X (cid:18) XC (cid:19) δ/ν (cid:35) , and lim n →∞ a δ − n E (cid:2) X ( X/C ) δ/ν (cid:3) = 0 by the choice of δ and our assumption E (cid:2) X ( X/C ) δ/ν (cid:3) < ∞ . For the second term in (7.5) we obtain by Markov’s inequality that E (cid:2) Xa − n A cn (cid:3) = a − n E (cid:104) X { ( X/C ) δ/ν > (1 /a n ) δ } (cid:105) ≤ (1 /a n ) − δ E (cid:104) X ( X/C ) δ/ν (cid:105) and lim n →∞ (1 /a n ) − δ E (cid:2) X ( X/C ) δ/ν (cid:3) = 0 by the choice of δ .To show non-resilience for α ∗ < /ν , note that we assumed that the right derivative of f ( · )in 0 exists (possibly with value ∞ ). It is thus sufficient to find some ε > a n ) n ≥ with lim n →∞ a n = 0 such thatlim n →∞ E (cid:20) X (1 − ε ) a − n ρ ε (cid:18) Xh ( a n ) C (cid:19)(cid:21) > . A n := { ( X/C ) /ν ε − /qν ≥ /a n } . Since E (cid:20) X (1 − ε ) a − n ρ ε (cid:18) Xh ( a n ) C (cid:19)(cid:21) ≥ E (cid:20) X (1 − ε ) a − n ρ ε (cid:18) Xh ( a n ) C (cid:19) A n (cid:21) ≥ a − n E [ X (1 − ε ) ε A n ] , it is sufficient to find ( a n ) n ≥ such that E [ X A n ] (cid:29) a n . For this, choose δ < δ/ν > α ∗ . Using Fubini we obtain that E (cid:34) X (cid:18) XC (cid:19) δ/ν (cid:35) = E (cid:34) X (cid:90) ( X/C ) δ/ν t (cid:35) = E (cid:20) X (cid:90) ∞ { t ≤ ( X/C ) δ/ν } d t (cid:21) = E (cid:20) X (cid:90) ∞ { t ≤ ( X/C ) δ/ν } d t (cid:21) = E (cid:20) X (cid:90) ∞ { t /δ ε − /qν ≤ ( X/C ) /ν ε − /qν } d t (cid:21) = (cid:90) ∞ E (cid:104) X { t /δ ε − /qν ≤ ( X/C ) /ν ε − /qν } (cid:105) d t = (cid:90) ∞ E (cid:104) X { u ≤ ( X/C ) /ν ε − /qν } u δ − ε − /qν (cid:105) d u. Since E [ X ( X/C ) δ/ν ] = ∞ it follows that E (cid:104) X { u ≤ ( X/C ) /ν ε − /qν } u δ − ε − /qν (cid:105) can not be domi-nated by the function Cu − α for any α > C >
0. Chose now α such that δ + α < C = 1and a sequence ( u n ) n ≥ such that E (cid:104) X { u ≤ ( X/C ) /ν ε − /qν } u δ − ε − /qν (cid:105) ≥ u − αn . Set now a n = 1 /u n . It follows by the choice of α and assuming that (1 − ε ) ε ≥ / n →∞ a − n E [ X (1 − ε ) ε A n ] ≥ lim n →∞ u n E (cid:104) X { u n ≤ ( X/C ) /ν ε − /qν } (cid:105) / n →∞ u − α − δn / ∞ . Some concrete examples for sales functions ρ are as follows: • The perhaps simplest non-trivial example is ρ ( u ) = { u ≥ } . It describes completeliquidation of the portfolio at default (if the institution is leveraged) resp. dissolution. • A more involved example can be derived from a leverage constraint that prohibits aninstitution from investing more money into risky assets than a certain multiple λ max ≥ xp/c =: λ ≤ λ max , where x denotes the number of shares held, p is the price per share and c denotes the institution’scapital. Assume now that while the asset price p stays constant, the institution suffers anexogenous shock (cid:96) and c is reduced to ˜ c = c − (cid:96) . – If (cid:96) ≤ (1 − λ/λ max ) c , then the leverage constraint xp/ ˜ c ≤ λ max is satisfied and noreaction is required by the institution. – However, if (cid:96) > (1 − λ/λ max ) c , then the institution must get rid of some shares;suppose that it sells δx of them, for some 0 < δ ≤
1. In order for the leverageconstraint (1 − δ ) xp/ ˜ c ≤ λ max to hold, it is easy to verify that δ ≥ − (1 − (cid:96)c ) λ max λ .The relative asset sales are hence given by ρ ( (cid:96)/c ), where ρ ( u ) := (1 − (1 − u ) λ max /λ ) + for u ∈ [0 , − λλ − is reached. 34 Taking an alternative route in the previous example, suppose that the loss of the institutionstems only from a price change p → ˜ p < p , which reduces the capital to ˜ c = c − x ( p − ˜ p ). If˜ p ≥ p (1 − λ ) / (1 − λ max ), then no action is required to comply with the leverage constraint.In the remaining cases the institution must sell a fraction of 1 − λ max + λ max p ˜ p (1 − λ ) oftheir assets, and we obtain ρ ( u ) = (1 − λ max (1 − u ) / ( λ − u )) + for u ∈ [0 , • Finally, it can be shown that for price changes combined with exogenous losses the salefunction is bounded from above and below by the two previous cases. Leverage constraintshence imply a sale function which is 0 below a certain threshold and then grows linearly.
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