Capital Regulation under Price Impacts and Dynamic Financial Contagion
aa r X i v : . [ q -f i n . M F ] A ug Capital Regulation under Price Impacts and Dynamic FinancialContagion
Zachary Feinstein ∗ Stevens Institute of Technology
August 23, 2019
Abstract
We construct a continuous time model for price-mediated contagion precipitated by a com-mon exogenous stress to the banking book of all firms in the financial system. In this setting,firms are constrained so as to satisfy a risk-weight based capital ratio requirement. We usethis model to find analytical bounds on the risk-weights for assets as a function of the marketliquidity. Under these appropriate risk-weights, we find existence and uniqueness for the jointsystem of firm behavior and the asset prices. We further consider an analytical bound on thefirm liquidations, which allows us to construct exact formulas for stress testing the financial sys-tem with deterministic or random stresses. Numerical case studies are provided to demonstratevarious implications of this model and analytical bounds.
Key words:
Finance; financial contagion; fire sales; risk-weighted assets; stress testing
Financial contagion occurs when the negative actions of one bank or firm causes the distress of aseparate bank or firm. Such events are of critical importance due to their relation to systemic risk.In this work we consider price-mediated contagion that occurs through impacts to mark-to-marketwealth as firms hold overlapping portfolios. Price-mediated contagion can occur due to the priceimpacts of liquidations in a crisis and can be exacerbated by pro-cyclical regulations. Importantly,this kind of contagion can be self-reinforcing, causing extreme events and ultimately a systemiccrisis as witnessed in, e.g., the 2007-2009 financial crisis.Systemic risk and financial contagion has been studied in a network of interbank payments by[12]. We refer to [20] for a review of this payment network model and extensions thereof to include,e.g., bankruptcy costs. The focus of this paper is on price-mediated contagion and fire sales.This single contagion channel causes impacts globally to all other firms due to mark-to-marketaccounting. As prices drop due to the liquidations of one bank, the value of the assets of all other ∗ Stevens Institute of Technology, School of Business, Hoboken, NJ 07030, USA, [email protected] . n bank financial system. The modeling is completed in Section 3.3 to present a market with a n bank financial system and m tradable illiquid assets. As this model has no closed-form solution ingeneral, we propose an analytical approximation that bounds the system response for stress testingpurposes in Section 4. These analytical results allow for a bound on, e.g., the probability thatthe terminal asset price is above some threshold in a probabilistic setting. Numerical case studiesare provided in Section 5 to demonstrate simple insights from this model and provide numericalaccuracy of the analytical bounds from Section 4. The proofs are presented in the appendix. Consider a firm with stylized banking book depicted in Figure 1, but with m ≥ x ≥
0, tradable illiquid investments (e.g., tradablecredit positions) denoted by s ∈ R m + , and nontradable illiquid investments (e.g., residential loans)denoted by ℓ ≥
0. For simplicity and without loss of generality, we will assume that the initialprice of all assets is 1, thus the mark-to-market assets for the bank at time 0 is equal to x + s + ℓ .The firm has liabilities in the total amount of ¯ p ≥
0. For simplicity in this work, we will assumethat all liabilities are not held by any other firms in this system; additionally, we will assume thatno liabilities come due during the (short time horizon) of the fire sale cascade under study, but areliquid enough that they can frictionlessly be paid off early with liquid assets. The capital of thefirm, at time 0, is thus provided by x + s + ℓ − ¯ p . For further simplicity, in Sections 3.1 and 3.2we will assume a single, representative, tradable illiquid asset only. This is along the lines of themodeling undertaken in, e.g., [8, 1, 4]. 3 nitial Banking BookAssets Liabilities Liquid x Illiquid(Tradable) s Illiquid(Nontradable) ℓ Total¯ p Capital x + s + ℓ − ¯ p Updated Banking BookAssets Liabilities
Liquid x Ψ( t )Ψ( t )Ψ( t )Illiquid(Tradable)( s − Γ( t )) q ( t )Illiquid(Nontradable) ℓ Total¯ p Capital x + Ψ( t )+( s − Γ( t )) q ( t )+ ℓ − ¯ p Figure 1: Stylized banking book for a firm before and after price and liquidation updates with 1tradable illiquid asset.Capital ratios are used for regulatory purposes to bound the risk of financial institutions. We willassume that the tradable illiquid assets have risk-weight α ∈ R m + and price process q : [0 , T ] → R m ++ (with q k (0) = 1 for every asset k ). The bank may liquidate assets over time. We will assume that,at time t , they liquidate the tradable illiquid assets at a rate of γ ( t ) ∈ R m . The total amount ofcash gained from liquidations up to time t is provided by Ψ( t ) = R t γ ( u ) ⊤ q ( u ) du ∈ R and the totalnumber of units liquidated up to time t is provided by Γ( t ) = R t γ ( u ) du ∈ R m . Thus, as depictedin Figure 1, at time t , the liquid assets for the firm are provided by x + Ψ( t ) and the tradableilliquid assets by ( s − Γ( t )) ⊤ q ( t ). Throughout this work we will assume that prices drop over timeand as a function of the liquidations, so the total assets and therefore also capital will drop overtime as shown by the crossed out portions of the banking book in Figure 1. More discussion onthe price changes will be provided in Section 3.1 below. Additionally, the liquid assets have 0 riskweight ( α x = 0) and nontradable illiquid assets have risk-weight α ℓ ≥
0. In settings with morethan one bank, we allow for the risk-weights of the nontradable assets to be heterogeneous betweeninstitutions.The capital ratio for a firm at time t is given by total capital divided by the risk-weighted assets.Mathematically, this is formulated as θ ( t ) = ( x + Ψ( t ) + P mk =1 [ s k − Γ k ( t )] q k ( t ) + ℓ − ¯ p ) + P mk =1 α k [ s k − Γ k ( t )] q k ( t ) + α ℓ ℓ . (1)The capital ratio requirement specifies that all institutions must satisfy the condition that θ ( t ) ≥ θ min for all times t for some minimal threshold θ min >
0. We wish to note that the capital ratiois related to the leverage ratio (assets over equity) by choosing α k = 1 for every tradable asset k , α ℓ = 1, and θ min = 1 /λ max for leverage requirement λ max >
0. This relationship is utilized inExample 5.3. 4 ssumption 2.1.
Throughout this work, we assume α k , θ min > with α k θ min < for all assets k and α ℓ ≥ . Additionally, any firm in the financial system will be assumed to satisfy the capitalratio at the initial time , i.e., θ (0) ≥ θ min . Remark 2.2. If α k ∗ θ min ≥ k ∗ , the assumption that the capital ratio at time 0 isabove the regulatory threshold guarantees that the risk-weighted capital ratio is nonincreasing inthe price of that asset. To see this we note that, at time t = 0 (i.e., before any intervention fromthe bank):( x + P mk =1 s k q k (0) + ℓ − ¯ p ) + P mk =1 α k s k q k (0) + α ℓ ℓ ≥ θ min ⇔ x + (1 − α ℓ θ min ) ℓ − ¯ p ≥ m X k =1 ( α k θ min − s k q k (0) . However, the capital ratio being nonincreasing in the price of asset k ∗ is contrary to the under-standing of how a regulatory threshold usually works. In particular, for the considerations of thispaper, this monotonicity implies that, as the price drops in that asset (without the intervention ofthe firm), the bank will always satisfy the capital regulation, and thus no rebalancing of assets willever need to occur.The assumption that α k > k is slightly stronger than exhibited in reality, asthere are liquid assets that are not cash-like instruments. However, as found in the main resultsbelow (see, e.g., Theorem 3.6), if an asset has risk-weight of 0 then it must have no market impactsfrom liquidation. Thus, from the modeling perspective of this work, liquid assets exhibit manybehaviors of cash and are merged for the purposes of this work. In this section we consider a single firm attempting to satisfy its risk-weighted capital ratio whensubject to price impacts. We will consider this in continuous time and determine conditions thatprovide unique liquidations for the bank to satisfy the capital requirement. In particular, wedetermine a condition relating the risk-weight and the price impacts.Consider a single bank with a single tradable illiquid asset. As the crisis we wish to modelis generically on a short time horizon, we will consider all price impacts to be permanent for theduration of the considered time [0 , T ] ⊆ R + . Further, we will assume the price of the illiquid asset issubject to market impacts given by a nonincreasing inverse demand function F : R + × R → R ++ suchthat F (0 ,
0) = 1. That is, F ( t, Γ) is a function of time and units sold; the inclusion of time allowsfor exogenous shocks, e.g., F ( t, Γ) = exp( − at { t Throughout this paper we assume Assumption 3.1, i.e., the clearing prices follow thepath f t ( t ) f Γ (Γ( t )). However, realistically the price effects from time occur due to asset liquidationsoutside of the firm of interest, i.e., F ( t, Γ) = f Γ ( η ( t ) + Γ) for some (nondecreasing) exogenousliquidation function η . Letting the full inverse demand function be defined by the exponentialinverse demand function with strictly positive price impact, i.e., f Γ (Γ) := exp( − b Γ) with b > F ( t, Γ) = exp( − bη ( t )) exp( − b Γ). In particular, we can provide the one-to-one correspon-dence: f t ( t ) = exp( − bη ( t )) and η ( t ) = − b log( f t ( t )). If f Γ were chosen otherwise, the exogenousliquidations would need to be defined as a function of bank liquidations Γ as well.Recall the setting described in Section 2. That is, consider the firm with initial banking bookmade up of liquid assets of x ≥ 0, liabilities of ¯ p ≥ 0, and illiquid holdings of s, ℓ ≥ θ (0) ≥ θ min > θ over time when θ ( t ) > θ ( t ) = ˙ q ( t )[ s − Γ( t )][ α (¯ p − x − Ψ( t ) − ℓ ) + α ℓ ℓ ] + α ˙Γ( t ) q ( t )([ s − Γ( t )] q ( t ) − [¯ p − x − Ψ( t ) − ℓ ])( α [ s − Γ( t )] q ( t ) + α ℓ ℓ ) (2)where the change in prices and recovered cash from liquidations are provided by˙ q ( t ) = f ′ t ( t ) f Γ (Γ( t )) + ˙Γ( t ) f t ( t ) f ′ Γ (Γ( t )) , (3)˙Ψ( t ) = ˙Γ( t ) q ( t ) . (4)As a simplifying assumption, no liquidations will occur except if θ ( t ) ≤ θ min . Therefore the first timethat the firm takes actions is at time τ such that f t ( τ ) = ¯ q := ¯ p − x − (1 − α ℓ θ min ) ℓ (1 − αθ min ) s . If inf t ∈ [0 ,T ] f t ( t ) > ¯ q then no fire sale will occurs. Once the firm starts acting, we assume that it does so only to theextent that it remains at the capital ratio requirement. Assuming it is possible (as proven later inthis section) that a firm is capable of remaining at the regulatory requirement for all times throughliquidations alone, i.e., θ ( t ) ≥ θ min for all times t , we can drop the indicator function on the firm’scapital being positive in ˙ θ ( t ) as it is always satisfied for θ ( t ) ≥ θ min . Thus by solving for ˙ θ ( t ) = 0(with the indicator function in (2) set equal to 1), we can conclude that:˙Γ( t ) = − ˙ q ( t )[ s − Γ( t )][ α (¯ p − x − Ψ( t ) − ℓ ) + α ℓ ℓ ] αq ( t )([ s − Γ( t )] q ( t ) − [¯ p − x − Ψ( t ) − ℓ ]) { θ ( t ) ≤ θ min } . (5)For notational simplicity, we will construct the mapping: Z ( t, Γ( t ) , q ( t ) , Ψ( t )) = [ s − Γ( t )][ α (¯ p − x − Ψ( t ) − ℓ ) + α ℓ ℓ ] αq ( t )([ s − Γ( t )] q ( t ) − [¯ p − x − Ψ( t ) − ℓ ]) { θ ( t ) ≤ θ min } . 6n fact, by the monotonicity of the inverse demand function, we further have that a firm will remainat the θ ( t ) = θ min boundary for any time t ≥ τ = inf { t ∈ [0 , T ] | θ ( t ) ≤ θ min } provided it does notrun out of illiquid assets to sell. Therefore, by solving for the price as a function of liquidations forthe equation θ ( t ) = θ min , we find that q ( t ) = ¯ p − x − Ψ( t ) − ℓ (1 − αθ min )( s − Γ( t )) + α ℓ θ min − αθ min ℓ ∈ R ++ (6)for t ≥ τ . This provides the price directly as a function of the bank’s book.With the representation of the price q ( t ) from (6), we rearrange terms to find that[ s − Γ( t )] q ( t ) − [¯ p − x − Ψ( t ) − ℓ ] = αθ min [¯ p − x − Ψ( t ) − ℓ ] + α ℓ θ min ℓ − αθ min for any time t ≥ τ (equivalently if θ ( t ) ≤ θ min ). Therefore we can rewrite Z to only depend on timeand the liquidations via Z ( t, Γ) = (1 − αθ min )[ s − Γ] αθ min f t ( t ) f Γ (Γ) { t ≥ τ } . In fact, we can decouple ˙Γ( t ) from ˙ q ( t ) and thus consider q ( t ) = f t ( t ) f Γ (Γ( t )) directly and ˙Γ( t ) tosolve the differential equation: ˙Γ( t ) = − Z ( t, Γ( t )) f ′ t ( t ) f Γ (Γ( t ))1 + Z ( t, Γ( t )) f t ( t ) f ′ Γ (Γ( t )) (7) Remark 3.3. Of particular interest is that ˙Γ( t ) f ′ t ( t ) , f ′ Γ (Γ) ≤ t and liquidations Γ. Using the prior computations, as pre-viously discussed for any time t ≥ τ , we can conclude that Z ( t, Γ( t )) ≥ 0. Therefore ˙Γ( t ) = − Z ( t, Γ( t )) f ′ t ( t ) f Γ (Γ( t ))1+ Z ( t, Γ( t )) f t ( t ) f ′ Γ (Γ( t )) ≥ f t ( t ) f ′ Γ (Γ( t )) ≥ − Z ( t, Γ( t )) , otherwise ˙Γ( t ) < purchase assets at the given price q ( t ). As both financial theory and practice indicate suchpurchasing does not occur in times of a crisis, we utilize the following results in order to calibratethe risk-weights of our model so as to appropriately consider fire sales.Formally, as above, let τ := inf { t | θ ( t ) ≤ θ min } = inf { t | f t ( t ) ≤ ¯ q } be the first time the firmhits the regulatory boundary. Lemma 3.4. Let the inverse demand function f Γ be such that ( s − Γ) f ′ Γ (Γ) /f Γ (Γ) ≤ is nonde-creasing for all Γ ∈ [0 , s ) . If α ∈ ( − sf ′ Γ (0)(1 − sf ′ Γ (0)) θ min , θ min ) then any solution Γ : [ τ, T ] → R of (7) issuch that Γ( t ) ∈ [0 , s ) and ˙Γ( t ) ≥ for all times t . Remark 3.5. In the prior lemma we require a monotonicity condition on ( s − Γ) f ′ Γ (Γ) f Γ (Γ) . This termis the “equivalent” marginal change in units held to the price change when Γ units are liquidated(with the next marginal unit is liquidated externally). That is, the firm’s wealth drops by the sameamount under the marginal change in price as if the firm held (cid:12)(cid:12)(cid:12) ( s − Γ) f ′ Γ (Γ) f Γ (Γ) (cid:12)(cid:12)(cid:12) fewer illiquid assets in7heir book. In this sense, this term provides the number of units needed to be sold at the currentprice in order to counteract the price movement. Therefore the assumed monotonicity propertyimplies that the firm need not increase the speed it is selling the illiquid assets solely to counteractits own market impacts. Theorem 3.6. Consider the setting of Lemma 3.4 with α ∈ ( − sf ′ Γ (0)(1 − sf ′ Γ (0)) θ min , θ min ) . There exists aunique solution (Γ , q, Ψ) : [0 , T ] → [0 , s ) × R ++ × [0 , ¯ p − x ) to the differential system (7) , (3) , and (4) (and thus for θ as well for (2) ). Remark 3.7. Noting that − sf ′ Γ (0)1 − sf ′ Γ (0) ∈ [0 , 1) because f ′ Γ (0) ≤ α ∈ ( − sf ′ Γ (0)(1 − sf ′ Γ (0)) θ min , θ min ). Ifthe risk-weight were set too low, i.e., α ∈ [0 , − sf ′ Γ (0)(1 − sf ′ Γ (0)) θ min ), then the bank would instead purchaseassets to remain at the regulatory threshold rather than liquidating as is expected and observedin practice. The existence and uniqueness results follow for α < − sf ′ Γ (0)(1 − sf ′ Γ (0)) θ min as well, thoughwe will only focus on the risk-weights that match with reality. In fact, this lower threshold on therisk-weight α can be viewed as a function to map the illiquidity of the asset (measured by f ′ Γ (0))to an acceptable risk-weight, rather than choosing based on heuristics. Remark 3.8. The existence and uniqueness results above state that the firm will never liquidatetheir entire (tradable) portfolio. This is not be the case if the liquid or untradable assets weredecreasing in value over time as well; in that scenario, the firm can run out of assets to liquidate.As the liquidation dynamics (up until the time that the firm becomes completely illiquid), includingthe existence and uniqueness results, appear similar to the setting stated herein, we focus on thesimpler setting in which untradable assets have fixed value over the (short) time horizon [0 , T ]. Thevalue of studying this simpler setting is that it provides ready access to determining the appropriaterisk-weight α by capturing the dynamics of a fire sale process before the bank fails without needingto model the failure event as well.We will conclude this section by considering two example inverse demand functions f Γ : linearand exponential price impacts. Markets without price impacts is a special case of either inversedemand function by setting b = 0. Example 3.9. Consider the case in which the firm’s actions impact the price linearly, i.e., F ( t, Γ) = f t ( t )(1 − b Γ) for b ∈ [0 , s ). The condition on the inverse demand function for Lemma 3.4 is satisfiedfor any choice b ∈ [0 , s ). Further, the risk-weight condition, α > − sf ′ Γ (0)(1 − sf ′ Γ (0)) θ min , is satisfied if andonly if α > sb (1+ sb ) θ min . In particular, if α ≥ θ min then the fire sale situation is always actualizedwithout dependence on the price impact parameter b . Example 3.10. Consider the case in which the firm’s actions impact the price exponentially, i.e., F ( t, Γ) = f t ( t ) exp( − b Γ) for b ≥ 0. The condition on the inverse demand function for Lemma 3.4is satisfied for any choice b ≥ 0. Further, the risk-weight condition, α > − sf ′ Γ (0)(1 − sf ′ Γ (0)) θ min , is satisfiedif and only if α > sb (1+ sb ) θ min . 8 .2 Capital Ratio Requirements in an n Bank System with Single Representa-tive Tradable Illiquid Asset Consider the same setting as in Section 3.1 but with n ≥ i have initial banking book defined by x i units of liquid asset, s i units of (tradable)illiquid asset, ℓ i units of untradable illiquid asset, and ¯ p i in obligations. Further, we will considerthe (pre-fire sale) market cap for the tradable illiquid asset to be given by M ≥ P ni =1 s i . Theinverse demand function will still be assumed to follow Assumption 3.1.With the assumption that θ i (0) ≥ θ min , we know that firm i will not take any actions unless θ i ( t ) ≤ θ min . As in the 1 bank case, this first occurs at ¯ q i = ¯ p i − x i − (1 − α ℓ,i θ min ) ℓ i (1 − αθ min ) s i . If inf t ∈ [0 ,T ] f t ( t ) > max i ¯ q i then no fire sale occurs. When a firm does need to take action, we will make the assumptionthat it is only enough so that the firm remains at the capital ratio requirement. Thus by solvingfor ˙ θ i ( t ) = 0 when θ i ( t ) ≤ θ min (constructed as in the n = 1 bank setting of Section 3.1), we canconclude: ˙Γ i ( t ) = − ˙ q ( t )[ s i − Γ i ( t )][ α (¯ p i − x i − Ψ i ( t ) − ℓ i ) + α ℓ,i ℓ i ] αq ( t )([ s i − Γ i ( t )] q ( t ) − [¯ p i − x i − Ψ i ( t ) − ℓ i ]) { θ i ( t ) ≤ θ min } with ˙ q ( t ) = f ′ t ( t ) f Γ ( P ni =1 Γ i ( t )) + hP ni =1 ˙Γ i ( t ) i f t ( t ) f ′ Γ ( P ni =1 Γ i ( t )) and˙Ψ i ( t ) = ˙Γ i ( t ) q ( t ) . (8)As in the prior section (after consideration of how the prices must evolve so that the firmsremain at the required capital ratio), let us consider the mapping Z i ( t, Γ) = (1 − αθ min )[ s i − Γ i ( t )] αθ min f t ( t ) f Γ ( P nj =1 Γ j ) { θ i ( t ) ≤ θ min } . With this mapping, we can consider the joint differential equation of Γ and q :˙Γ( t ) = − I + (cid:16) Z ( t, Γ( t )) ~ ⊤ (cid:17) f t ( t ) f ′ Γ ( n X j =1 Γ j ( t )) − Z ( t, Γ( t )) f ′ t ( t ) f Γ ( n X j =1 Γ j ( t )) (9)˙ q ( t ) = f ′ t ( t ) f Γ ( P ni =1 Γ i ( t ))1 + [ P ni =1 Z i ( t, Γ( t ))] f t ( t ) f ′ Γ ( P ni =1 Γ i ( t )) (10)where ~ , , . . . , ⊤ ∈ R n .Let τ = 0, τ k +1 := inf { t ∈ [ τ k , T ] | ∃ i : θ i ( t ) ≤ θ min , θ i ( τ k ) > θ min } , and τ n +1 = T . For theremainder, we will order the banks so that ¯ q i ≥ ¯ q i +1 for every i . Due to the monotonicity propertiesthis implies that bank k hits the regulatory threshold only after the first k − Lemma 3.11. Let the inverse demand function f Γ be such that ( M − Γ) f ′ Γ (Γ) /f Γ (Γ) ≤ is nonde-creasing for any Γ ∈ [0 , M ) . If α ∈ ( − Mf ′ Γ (0)(1 − Mf ′ Γ (0)) θ min , θ min ) then any solution Γ : [0 , T ] → R n of (9) is such that Γ( t ) ∈ [0 , s ) , ˙Γ( t ) ∈ R n + , and ˙ q ( t ) ≤ for all times t . Corollary 3.12. Consider the setting of Lemma 3.11 with α ∈ ( − Mf ′ Γ (0)(1 − Mf ′ Γ (0)) θ min , θ min ) . Thereexists a unique solution (Γ , q, Ψ) : [0 , T ] → [0 , s ) × R ++ × [0 , ¯ p − x ) to the differential system (9) , (10) , and (8) (and thus for θ as well). Remark 3.13. As in the single bank n = 1 setting, we can consider a situation in which therisk-weight was set too low. Under such parameters eventually one bank may begin purchasingassets rather than liquidating in order to satisfy the capital requirements. Existence of a solutionwould still exist in this setting for the n bank case, but uniqueness will no longer hold. Remark 3.14. We wish to extend on the comment of Remark 3.8 to the setting with n banks.As in the single firm setting, if the nontradable assets have decreasing value over time due to thefinancial shock then firms may run out of liquid assets to sell and thus can fail. Therefore the strongresult that all banks survive for all time presented in, e.g., Corollary 3.12 only holds in the specialcase that the nontradable assets have constant value over time. As with the single asset settingconsidered in Remark 3.8, at times between bank failures the system dynamics behave as describedin this work; after a bank failure the system parameters would update appropriately (from possibledefault contagion), then the continuous fire sale model presented herein would begin again untileither the terminal time T was reached or another bank failed. n Bank System with m Tradable IlliquidAssets Consider the same setting as in Section 3.1 but with n ≥ m ≥ i liquidates its tradable assets in proportion toits holdings s i ∈ R m + . Notationally, we denote the proportion of assets liquidated by bank i attime t is given by Π i ( t ). In this way we can define the vector of total liquidations is given byΓ i ( t ) = s i Π i ( t ). As in the prior section, we will consider the (pre-fire sale) market cap for the k th tradable illiquid asset to be given by M k ≥ P ni =1 s ik . The inverse demand function for eachasset will still be assumed to follow Assumption 3.1, i.e., asset k has inverse demand function F k ( t, Γ k ) := f t,k ( t ) f Γ ,k (Γ k ) for any time t and asset liquidations Γ k . We will often consider thevector of inverse demand functions f t ( t ) , f Γ (Γ) ∈ R m ++ to simplify notation.As in the prior sections, we can construct the derivative of θ i over time in order to determinethe necessary liquidations so that all firms satisfy the capital ratio requirement. Using the samelogic as above, we can consider the joint differential equation for the fractional liquidations Π, thevector of prices q , and the cash obtained from liquidating tradable assets Ψ:˙Π( t ) = − (cid:16) I + Z ( t, diag[Π( t )] s ) diag[ f t ( t )] diag[ f ′ Γ ( s ⊤ Π( t ))] s ⊤ (cid:17) − × Z ( t, diag[Π( t )] s ) diag[ f ′ t ( t )] f Γ ( s ⊤ Π( t )) (11)10 q ( t ) = (cid:16) I + diag[ f t ( t )] diag[ f ′ Γ ( s ⊤ Π( t ))] s ⊤ Z ( t, diag[Π( t )] s ) (cid:17) − diag[ f ′ t ( t )] f Γ ( s ⊤ Π( t )) (12)˙Ψ( t ) = diag[ ˙Π( t )] sq ( t ) (13) Z ( t, Γ) = diag (cid:2) { θ ( t ) ≤ θ min } (cid:3) diag h s diag[ αθ min ] diag[ f t ( t )] f Γ (Γ ⊤ ~ i − ( s − Γ)( I − diag[ αθ min ]) . (14) Lemma 3.15. Let the inverse demand function f Γ be such that ( M k − Γ k ) f ′ Γ ,k (Γ k ) /f Γ ,k (Γ k ) ≤ is nondecreasing for any Γ k ∈ [0 , M k ) for every asset k . If α k ∈ ( − M k f ′ Γ ,k (0)(1 − M k f ′ Γ ,k (0)) θ min , θ min ) for everyasset k then any solution Π : [0 , T ] → R n of (11) is such that Π( t ) ∈ [0 , n , ˙Π( t ) ∈ R n + , and ˙ q ( t ) ∈ − R m + for all times t . Using this result on monotonicity of the processes, we are able to determine a result on theexistence and uniqueness of the system under financial contagion. Corollary 3.16. Consider the setting of Lemma 3.15 with α k ∈ ( − M k f ′ Γ ,k (0)(1 − M k f ′ Γ ,k (0)) θ min , θ min ) for everyasset k . There exists a unique solution (Π , q, Ψ) : [0 , T ] → [0 , n × R m ++ × [0 , ¯ p − x ) to the differentialsystem (11) , (12) , and (13) (and thus for θ as well). As described in the proofs of Lemmas 3.4, 3.11, and 3.15, we are able to determine upper boundsfor the number of assets being sold for each firm in the system. In the following results we willrefine these estimates and use this to determine simple analytical worst-case results for the healthof the financial system. As such, given the initial banking book for each firm, a heuristic for thehealth of the system can be determined with ease. Mathematically this is provided by Theorem 4.1.Following this result, we will present a quick example to demonstrate the value of these bounds toconsider a stochastic stress test. Throughout, we will be recalling that, in the single asset setting,firm i hits the regulatory threshold θ min when q ( t ) = ¯ q i .For the remainder of this section we will consider decomposition of the capital ratio as under-taken in the proof of Lemma 3.15. With this notion we will define the individual price bounds forliquidations as ¯ q i = ¯ p i − x i − (1 − α ℓ,i θ min ) ℓ i P mk =1 (1 − α k θ min ) s ik for any bank i . Note that these thresholds do not depend on the asset being considered. Withoutloss of generality, and as previously discussed, we will assume that firms are ordered so that ¯ q i is anonincreasing sequence. We wish to note that the following analytical bounds, while tight for thesingle asset m = 1 setting (see the numerical examples in Section 5 below), are typically very weakin the m ≥ q as a measure of the risk of eachfirm is one that requires further study in the m ≥ heorem 4.1. Consider the setting of Corollary 3.16 with n ≥ banks (ordered by decreasing ¯ q )and m ≥ assets. Define approximate hitting times ˜ τ k and bounds on the firm behavior t ˜Π( t ) for k = 1 , ..., n : ˜Π i ( t ) = max l =1 ,...,m ( ˜Γ nil ( t ) s il | s il > ) ˜Γ kil ( t ) = { t< ˜ τ kl } ˜Γ k − il ( t ) + { t ≥ ˜ τ kl } " s il − ( s il − ˜Γ k − il (˜ τ kl )) (cid:16) f t,l ( t ) f t,l (˜ τ kl ) (cid:17) − αlθ min αlθ min ˜Λ kl if i ≤ k else ˜ τ kl = inf ( t ∈ [˜ τ k − ,l , T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f t,l ( t ) f Γ ,l k − X i =1 ˜Γ k − il ( t ) ! ≤ ¯ q k ) ˜Λ kl = 1 + 1 − α l θ min α l θ min k X j =1 ( s jl − ˜Γ k − jl (˜ τ kl )) f ′ Γ ,l (cid:16)P k − j =1 ˜Γ k − jl (˜ τ kl ) (cid:17) f Γ ,l (cid:16)P k − j =1 ˜Γ k − jl (˜ τ kl ) (cid:17) where ˜ τ l = 0 , ˜ τ n +1 ,l = T , and ˜Γ il ( t ) ≡ . Then Π i ( t ) ≤ ˜Π ni ( t ) for all times t ∈ [0 , T ] and all firms i = 1 , ..., n . With this general analytical construction, we now wish to turn our attention to a specificchoice of inverse demand function to provide some additional results. In particular, as noted inRemark 3.2, we will choose the exponential inverse demand function considered in Example 3.10to deduce exact analytical formulations. For the remainder of this section we will make use of theLambert W function W : [ − exp( − , ∞ ) → [ − , ∞ ], i.e., the inverse mapping of x x exp( x ). Corollary 4.2. Consider the setting of Theorem 4.1. Fix asset k = 1 , ..., m . Further, consider anexponential inverse demand function f Γ ,k (Γ k ) := exp( − b k Γ k ) as in Example 3.10 with b k ≥ . Theanalytical stress test bounds can be explicitly provided for any i = 1 , ..., n : ˜Γ nik ( t ) = s ik − n Y j = i (cid:18) f t,k ( t ∧ ˜ τ j +1 ,k ) f t,k ( t ∧ ˜ τ jk ) (cid:19) − αkθ min αkθ min ˜Λ jk ˜ τ ik = f − t,k (¯ q k ) if i = 1 f − t,k ˜Λ i − ,k W (cid:18) νi − ,k ˜Λ i − ,k exp (cid:18) − αkθ min αkθ min ˜Λ i − ,k [ log(¯ q ik )+ b k P i − j =1 s jk ] (cid:19)(cid:19) ν i − ,k αkθ min ˜Λ i − ,k − αkθ min if i ∈ { , ..., n } ˜Λ ik = 1 − b k − αθ min αθ min i X j =1 s jk i − Y h = j (cid:18) f t,k (˜ τ h +1 ,k ) f t,k (˜ τ hk ) (cid:19) − αkθ min αkθ min ˜Λ hk ν ik = 1 − ˜Λ ik f tk (˜ τ ik ) − αkθ min αkθ min ˜Λ ik here ∧ denotes the minimum operator. Remark 4.3. The expanded form ˜Γ nik provided in Corollary 4.2 holds for any inverse demandfunction f Γ ,k and is not dependent on the choice of the exponential form. However, the forms of˜ τ ik and ˜Λ ik are specific to the exponential inverse demand function considered in Corollary 4.2.This analytical stress test bound has significant value in considering probability distributions.All results in this paper, up until now, would require Monte Carlo simulations in order to approxi-mate the distribution of the health of the financial system if there is uncertainty in the parameters.However, with this analytical bound, we are able to determine analytical worst-case distributionsthat would be almost surely worse than the actualized results due to the results of Theorem 4.1.Thus if the system is deemed healthy enough under this analytical results, it would pass the stresstest under the true dynamics as well. Corollary 4.4. Consider the setting of Corollary 4.2 with exponential price response in time f t,l ( t ) := exp( − a l t { t We can generalize the bound for any random price response in time f t from Corol-lary 4.4 by considering P ( q ( t ) ≥ q ∗ ) ≥ P f t,l ( t ) ≥ ˜Λ k l l W (cid:18) ν kll ˜Λ kll exp (cid:18) − α l θ min α l θ min ˜Λ kll h log ( q ∗ ) + b l P k l j =1 s jl i(cid:19)(cid:19) ν k l l αθ min ˜Λ kll − αlθ min ∀ l = 1 , ..., m where q ∗ l ∈ [¯ q k l +1 ,l , ¯ q k l ) for some k l = 1 , ..., n for every asset l .This result allows us to consider the case for jointly random price response f t and price impactparameters b l ∈ [0 , α l θ min (1 − α l θ min ) M l ) with marginal density g b through an integral representation. Theupper bound on the price impact parameters b is so as to guarantee the selected risk-weight satisfiesthe sufficient conditions considered within this work.13 Case Studies In this section we will consider four numerical case studies to consider implications of the proposedmodel. For simplicity, each of these case studies is undertaken with an exponential inverse demandfunction. Further, as the untradable assets do not impact the liquidation dynamics, we will considerexamples with ℓ = 0. The first three of these numerical case studies is limited to the m = 1 assetsystem with a single, representative, asset as in [3, 4] with α = 1 / (2 θ min ) throughout. As such, ineach example, we limit the price impact parameters so that b < /M as discussed in Remark 4.5.The case studies are as follows. First, we will consider a 20 bank system and determine theeffects of the market impacts on the health of the financial system. Second, we will consider asystem with random parameters to study a probabilistic stress test. Third, we will consider theeffects of changing the regulatory capital ratio threshold. Finally, we will consider the implicationsof diversification for a 2 bank, 2 asset system. In these numerical examples we will consider boththe numerical solutions to the differential system introduced in Section 3 and the stress test boundsconsidered in Section 4. Example 5.1. Consider a financial system with n = 20 banks, a single tradable illiquid m = 1asset, and a crisis that lasts until the terminal time T = 1. Assume that each bank has liabilities¯ p i = 1 and liquid assets x i = i − for i = 1 , ..., 20. Additionally, each bank is given s i = 2 unitsof the illiquid asset; accordingly we set the market capitalization M = P i =1 s i = 40. We willconsider the regulatory environment with threshold θ min = 0 . 10 and risk-weight α = θ min = 5.Finally, we will take the inverse demand function to have an exponential form, i.e., F ( t, Γ) =exp( − at { t< } − a { t ≥ } − b Γ) with a = − log(0 . ≈ . b ∈ [0 , M ) which satisfies the conditions of Corollary 3.12. In this example we will demonstrate thenonlinear response that market impacts b introduce to the health of the firms and clearing prices.First we wish to consider the impact over time that the market impacts can cause. To do sowe compare the asset prices without market impacts ( b = 0) to those with high market impacts( b ≈ M ). As depicted in Figure 2a we see that the prices with and without price impacts arecomparable for (approximately) t ∈ [0 , . t ≈ . 80. At that point 18 of the 20 firms (90%) have hit the regulatory threshold and thefeedback effects of their actions are quite evident. We wish to note the distinction between thissteep drop in the prices to the subtle price drop for t ∈ [0 , . 29] when only the first 3 banks have hittheir regulatory threshold. The times at which the firms hit the regulatory threshold at differentliquidity situations (i.e. no, medium, and high market impacts) are summarized in Table 1.With the notion of how high market impacts effect the prices over time, and how the feedbackeffects can cause virtual jumps in the price, we now wish to consider these effects in more detailby studying only the final state of the system. In Figure 2b we see that, as more banks hitthe threshold capital ratio, the range of price impact thresholds that match that state shrink.That is, the system becomes more sensitive to the price impact parameter as more banks are at theregulatory threshold. This is due to the same feedback effects seen in the high price impact scenario14 = = . b = + − Firm Numerical Bounds Numerical Bounds Numerical Bounds θ min in the full simulation and in the analytical stress test bounds over no price impacts ( b = 0),mid-level price impacts ( b = 0 . /M ), and high price impacts ( b ≈ /M ).of Figure 2a. Further, we see that until about 90% of the banks (18 out of the 20 firms) hit theregulatory threshold (at about b . . /M ), the terminal price is principally affected by the pricechange in time ( f t (1) = 0 . b & . /M ) the feedback effectsof firm liquidations on each other causes the terminal price to drop drastically. Thus, providingonly a small amount of liquidity to the market can have outsized effects on the health of the systemby decreasing the price impacts, though this type of response to a financial crisis would have quicklydecreasing marginal returns as evidenced by Figure 2b.Finally, we wish to consider the analytical stress test bounds. We see the response of the stresstest bounds in the high market impact scenario ( b ≈ M ) in Figure 2a. This is not depicted in thesetting without market impacts as there is no distinction between the exact price process and thebounded price process in this case. In the high market impact scenario, we see that the exact priceprocess and the stress test bounds provide virtually indistinguishable results for the t ∈ [0 , . b ∈ [0 , . M ].15 Time t P r i c e q Price as a Function of Time b = 0Numerical: b = 1/(M+10 -8 )Bounds: b = 1/(M+10 -8 ) (a) A comparison of the asset price over time ina low b = 0 and high b = M +10 − market impactenvironment. Price Impacts b T e r m i na l P r i c e q ( ) B an ks a t R egu l a t o r y R equ i r e m en t ( % ) Impact of Price Impacts on Financial Crisis Prices: NumericalPrices: Bounds% of Banks (b) The final price q (1) and percentage of firms thathave hit their regulatory threshold as a function ofthe price impact parameter b . Figure 2: Example 5.1: The effects of price impacts on market response in a 20 bank system underthe exact differential equation and the analytical stress test bounds. Example 5.2. Consider the setting of Example 5.1 with exponential inverse demand function F ( t, Γ) = exp( − at { t< } − a { t ≥ } − b Γ) for a ∼ Exp( µ ), µ = log(20)log(20) − log(19) ≈ . 4, and b = . M . The choice of the exponential distribution for a with parameter µ is so that P ( F (1 , ≤ . 95) = 0 . 05. We wish to compare the true distribution for q (1) to the analytical stress test boundgiven in Corollary 4.4. In comparison to the analytical cumulative distribution function givenin Corollary 4.4, the true distribution was found numerically through repeated computation ona (log scaled) regular interval. Figure 3 displays the cumulative distribution functions P ( q ( t ) ≤ q ∗ ) without market impacts (black dashed line), with market impacts (black solid line), and theanalytical stress test bound (blue solid line). Notably, the analytical bound, as seen in Figure 3a, isa very accurate estimate of the true distribution while the market without price impacts distinctlyunderestimates large price drops. This is more pronounced in Figure 3b, which is the same figure butfocused on the region for q ∗ ∈ [0 . , . P ( f t (1) ≤ . ≈ . 002 whereas P ( q (1) ≤ . ≈ P ( f t ( t ) f Γ ( P ni =1 ˜Γ ni (1)) ≤ . ≈ . P ( f t (1) ≤ . ≈ P ( q (1) ≤ . ≈ . 014 and P ( f t (1) f Γ ( P ni =1 ˜Γ ni (1)) ≤ . ≈ . 02. Thus the analytical stress test is a bound for the true distribution, but an accurateone (as seen in Figure 3a).While considering the probabilistic setting, we can also consider and plot the response to varyingthe stress scenario given by a . This is depicted in Figure 4 by plotting the terminal price q (1) asa function of the price without market impacts f t (1). The setting without market impacts isthe diagonal line by definition. Market impacts cause feedback effects that drive the price below f t (1). All settings coincide for low stress scenarios ( f t (1) & . 98) as few banks are driven to the16 .7 0.75 0.8 0.85 0.9 0.95 1 q * P ( q ( ) ≤ q * ) Comparison of the CDF for the Terminal Price CDF of f t (1)True CDF of q(1)Analytical Stress Test CDF of q(1) (a) The distribution of terminal prices q (1) withand without price impacts. q * P ( q ( ) ≤ q * ) Comparison of the CDF for the Terminal Price CDF of f t (1)True CDF of q(1)Analytical Stress Test CDF of q(1) (b) Zoomed in view of the distribution of terminalprices q (1). Figure 3: Example 5.2: True and analytical stress test distributions for the terminal price q (1)under a randomly stressed financial system of 20 banks.regulatory threshold. Further, the analytical stress test bound is demonstrably worse than thenumerical terminal value for most stresses; however, these occur at typically unrealistic stresses. Terminal Stress f t (1) T e r m i na l P r i c e q ( ) Impact of Exogenous Stress on Financial Crisis b = 0Numerical: b = 0.9/MBounds: b = 0.9/M Figure 4: Example 5.2: The impacts of the stress scenario on market response in a 20 bank systemunder the exact differential equation and the analytical stress test bounds. Example 5.3. Consider a single bank ( n = 1) and single asset ( m = 1) system with crisis thatlasts until terminal time T = 1. For simplicity, assume that this bank holds no liquid assets, i.e., x = 0. Further, we will directly consider the setting of a leverage constrained firm with varyingmaximal leverage λ max > 1. As we change this leverage requirement, we will assume that the initialbanking book for the firm is such that they begin (at time 0) exactly at the leverage constraintand have a single unit of capital, i.e., s = λ max and ¯ p = λ max − 1. For comparison we will fix theinverse demand function to have an exponential form, i.e., F ( t, Γ) = exp( − at { t< } − a { t ≥ } − b Γ)with a = − log(0 . ≈ . b = − log(0 . − / log(0 . ≈ . λ max ∈ (1 , − . ) ≈ (1 , . λ max ≈ . 15 the firm has adecreasing number of terminal assets as the leverage requirement increases; this is despite the firmhaving a greater number of initial assets. Thus the combination of increasing percentage of assetsliquidated and increasing number of initial assets as the leverage requirement λ max increases, theterminal prices decrease as the leverage requirement increases (as depicted in Figure 5b). Finally,we notice that the analytical stress test bounds are accurate for λ max . . 5. However, for leveragerequirements above that threshold the analytical stress test bounds stop performing well, thoughclearly are a worst-case bound for the health of the financial system. Leverage Requirement λ max T e r m i na l A ss e t s ( ) T e r m i na l A ss e t s ( % o f I n i t i a l H o l d i ng s ) Impact of Leverage Requirement on Asset Holdings Terminal Asset Holdings: NumericalWorst Case Terminal Asset Holdings: Bounds% of Original Asset Holdings: Numerical% of Original Asset Holdings: Bounds (a) The asset holdings and percentage of assets thefirm has remaining in its banking book at time 1as a function of the leverage constraint λ max . Leverage Requirement λ max T e r m i na l P r i c e q ( ) Impact of Leverage Requirement on Prices Numerical SolutionStress Test Bounds (b) The terminal price of the illiquid asset as afunction of the leverage constraint λ max . Figure 5: Example 5.3: The impacts of the leverage requirement on asset holdings and prices underthe exact differential equation and the analytical stress test bounds. Example 5.4. Consider a two bank ( n = 2) and two asset ( m = 2) system with crisis that lastsuntil terminal time T = 1. For simplicity, assume that both banks hold no liquid assets, i.e., x = 0. Further, assume that both banks have liabilities ¯ p i = 0 . 98 and total initial mark-to-marketilliquid assets s i + s i = 2. Additionally, we set the market capitalization of each asset to be M k = s k + s k = 2. In this example we will study the implications of diversification by alteringthe individual portfolios; parameterizing by ζ ∈ [0 , s = (1 − ζ/ M , s = ( ζ/ M , s = ( ζ/ M , and s = (1 − ζ/ M . We will capture the regulatory environment with threshold θ min = 0 . 10 and risk weights α = α = θ min = 5. Finally, we will take exponential inverse demand18unctions F ( t, Γ) = (cid:0) exp( − a t { t< } − a { t ≥ } − b Γ ) , exp( − b Γ ) (cid:1) ⊤ with a = − log(0 . ≈ . a = 0) and b = . × ~ ζ ∈ [0 , 1] for the remainderof this example. For completeness, the aggregate system (assets x = 0, s = M = (2 , ℓ = 0 and liabilities ¯ p = ¯ p + ¯ p = 1 . 96) is considered as well to show the implications of systemheterogeneity.In Figure 6a, we clearly see that diversification of assets does not uniformly improve the marketcapitalization. Though the price of asset 1 rises from approximately 0.76 up to nearly 0.87 asthe firms become more diversified until perfect diversification ( ζ = 1), the price of asset 2 fallsfrom a price of 1 down to approximately 0.92. The optimal total market capitalization is foundat ζ = 0 . 15, i.e., at a 7.5%-92.5% split of assets. Such a portfolio has very little overlap, thusdemonstrating that the contagion effects from holding similar portfolios can easily outweigh thebenefits of diversification. On the other extreme, the completely diverse investment decision ( ζ = 0)has the lowest total market capitalization. Figure 6b, demonstrates that the second bank does notstart liquidating assets until we reach the optimal ζ for the market capitalization, i.e., Π (1) = 0 if ζ < . 15. In fact, this demonstrates that the contagion effects are exactly those that cause increaseddiversification to harm the system. Finally, we note that the aggregated system of this symmetric2 bank system behaves exactly like the perfectly diversified setting ζ = 1, but can differ greatly inoutcome from even a small heterogeneous system. Diversification T e r m i na l T o t a l M a r k e t C ap i t a li z a t i on k M k q k ( ) T e r m i na l P r i c e q k ( ) Impact of Diversification on Market Capitalization and Prices Total Market Capitalization: 2 BanksTotal Market Capitalization: Aggregate SystemPrice of Asset 1: 2 BanksPrice of Asset 1: Aggregate SystemPrice of Asset 2: 2 BanksPrice of Asset 2: Aggregate System (a) The terminal market capitalization and assetprices as a function of portfolio diversity and di-versification ζ . Diversification F r a c t i ona l L i qu i da t i on s T i m e o f F i r s t L i qu i da t i on Impact of Diversification on Liquidations Liquidations: Bank 1Liquidations: Bank 2Liquidations: AggregateTime: Bank 1Time: Bank 2Time: Aggregate (b) The terminal fractional liquidations Π i (1) andfirst liquidation time τ i as a function of portfoliodiversity and diversification ζ . Figure 6: Example 5.4: The impacts of portfolio diversity and diversification on asset prices andholdings. 19 Conclusion In this paper we considered a dynamic model of price-mediated contagion that extends the workof [8, 15, 3, 4]. The focus of this work was on capital ratio requirements and risk-weighted assets.In analyzing this model, we determine bounds for appropriate risk-weights for an asset that isdependent on the liquidity of the asset itself, as modeled through the price impacts of liquidating theasset. Under the appropriate risk-weights, we find existence and uniqueness for the firm behaviorand system health. However, though the output of the model can be computed with standardmethods, an analytical solution cannot be found; an analytical bound on the health of the systemin a stressed scenario was provided. This analytical stress test bound can be used to analyzerandom stresses and find the probability for the system health. We wish to note that an importantextension of this model is to more fully consider the setting in which the nontradable assets haveprices that drop over time, thus allowing for bank failures. By determining bank failures over time,properly modeling the failure event and default contagion, and determining the updated systemparameters after default (to be run on the current proposed model) would be an interesting exerciseto model the reality of a contagious event more closely. A Proofs from Section 3.1 Define the mapping Λ( t ) := 1 + Z ( t, Γ( t )) f t ( t ) f ′ Γ (Γ( t )), which will be utilized throughout many ofthe following proofs. A.1 Proof of Lemma 3.4 Proof. We will demonstrate that if a solution exists then it must satisfy the monotonicity property.To do so, first, we note that the condition on the risk-weight α is equivalent to αθ min < τ ) > 0. Therefore we find that ˙Γ( τ ) > 0. Now we wish to show that ˙Λ( t ) has the same sign as˙Γ( t ), i.e., ˙Γ( t ) ˙Λ( t ) ≥ t ) = 1 + Z ( t, Γ( t )) f t ( t ) f ′ Γ (Γ( t )) = 1 + 1 − αθ min αθ min s − Γ( t ) f Γ (Γ( t )) f ′ Γ (Γ( t ))˙Λ( t ) = 1 − αθ min αθ min ddt ( s − Γ( t )) f ′ Γ (Γ( t )) f Γ (Γ( t )) = (1 − αθ min ) ˙Γ( t ) αθ min dd Γ (cid:20) ( s − Γ) f ′ Γ (Γ) f Γ (Γ) (cid:21) Γ=Γ( t ) Therefore, by αθ min ∈ (0 , t ) ˙Λ( t ) ≥ dd Γ (cid:20) ( s − Γ) f ′ Γ (Γ) f Γ (Γ) (cid:21) Γ=Γ( t ) ≥ t ) > t ∈ [ τ, T ]: • At time τ we have (by assumption) that Λ( τ ) > For any time t ∈ [ τ, T ) such that Λ( t ) > u ) > u ∈ [ t, t + ǫ ]for some ǫ > • For any time t ∈ ( τ, T ] such that Λ( u ) > u ∈ [ τ, t ) then ˙Γ( u ) > u ) ≥ u ∈ [ τ, t ). This implies Λ( t ) > τ ) > t ) ≥ Λ( τ ) for all times t ≥ τ (which implies ˙Γ( t ) ≥ t ≥ τ ).Finally, we will demonstrate that, if a solution Γ : [0 , T ] → R exists, then Γ( t ) < s for alltimes t ∈ [0 , T ]. By definition Γ( t ) = 0 for all times t ∈ [0 , τ ], so we begin with Γ( τ ) = 0. Take T ∗ = inf { t ∈ [ τ, T ] | Γ( t ) ≥ s } and assume this infimum is taken over a nonempty set. On u ∈ [ τ, T ∗ ) we have that:˙Γ( u ) = − Z ( u, Γ( u )) f ′ t ( u ) f Γ (Γ( u ))Λ( u ) ≤ − (1 − αθ min ) inf t ∈ [ τ,T ∗ ] f ′ t ( t ) αθ min f t ( T ∗ )Λ( τ ) ( s − Γ( u ))and inf t ∈ [ τ,T ∗ ] f ′ t ( t ) is attained as we are infimizing a continuous function over a compact space.This differential equation implies Γ( u ) ≤ s h − exp (cid:16) (1 − αθ min ) inf t ∈ [ τ,T ∗ ] f ′ t ( t ) αθ min f t ( T ∗ )Λ( τ ) ( u − τ ) (cid:17)i < s for anytime u ∈ [ τ, T ∗ ). In particular, by continuity, this implies that Γ( T ∗ ) < s . A.2 Proof of Theorem 3.6 Proof. We will use Lemma 3.4 to prove the existence and uniqueness of a solution (Γ , q, Ψ). First,for all times t ∈ [0 , τ ] there exists a unique solution given by Γ( t ) = 0, q ( t ) = f t ( t ), and Ψ( t ) = 0.Now consider the initial value problem with initial condition at t = τ . We will consider thedifferential equation for Γ given in (7). As this equation is no longer dependent on either q or Ψ wecan consider the existence and uniqueness of the liquidations Γ separately. Indeed, from Γ, we candefine q ( t ) = f t ( t ) f Γ (Γ( t )) for all times t , thus the existence and uniqueness of Γ provides the sameresults for q . The results for Ψ follow from the same logic as Γ and thus will be omitted herein. Inour consideration of (7) we will consider a modification of the function Λ( t ) to be given by ¯Λ(Γ)so that its dependence on the liquidations is made explicit:˙Γ( t ) = − (1 − αθ min )( s − Γ( t )) f ′ t ( t ) αθ min f t ( t ) ¯Λ(Γ( t )) =: g ( t, Γ( t )) and ¯Λ(Γ) = 1 + (1 − αθ min )( s − Γ) αθ min f Γ (Γ) f ′ Γ (Γ) . Now we wish to consider the initial value problem for Γ with dynamics given by g and initial valueΓ( τ ) = 0. Before continuing we wish to note that the function ¯Λ is constant in time, i.e., onlydepends on the total number of units liquidated Γ and not on the time.Define the domain U = n Γ ∈ [0 , s ) | ¯Λ(Γ) > Λ( τ ) = h (1 − αθ min ) sαθ min f ′ Γ (0) io . We wish to notefrom the previous proof that Λ is nondecreasing in time, thus any solution must lie in U , i.e., itmust satisfy Γ( t ) ∈ U for all times t ∈ [ τ, T ]. From the definition of U as well as the property that ¯Λis constant in time, we can conclude αθ min f t ( t ) ¯Λ(Γ) > αθ min f t ( t )Λ( τ ) > ∈ U and anytime t ∈ [ τ, T ], and thus the denominator in g is always strictly greater than 0. From this we can21onclude that g and ∂∂ Γ g are continuous mappings over [ τ, T ] × U and thus Γ ∈ U g ( t, Γ) is locallyLipschitz for any time t ∈ [ τ, T ]. This implies there exists some δ > τ, τ + δ ] → U is the unique solution satisfying ˙Γ( t ) = g ( t, Γ( t )) for all times t ∈ [ τ, τ + δ ]. From a sequentialapplication of this approach (i.e., consider now an initial value problem starting at time τ + δ ) wecan either conclude that there exists a unique solution over the entire time range Γ : [ τ, T ] → U orthere exists some maximal domain [ τ, T ∗ ) ( [ τ, T ] over which we can conclude the existence anduniqueness. We will finish by focusing on this second case to prove a contradiction. To do this we willfirst show that g is bounded on [ τ, T ] × U . By definition, we have that g ( t, Γ) ≥ t ∈ [ τ, T ]and Γ ∈ U . In fact, we find that 0 ≤ g ( t, Γ) ≤ − − αθ min ) s inf u ∈ [ τ,T ] f ′ t ( u ) αθ min f t ( T )Λ( τ ) where inf u ∈ [ τ,T ] f ′ t ( u ) isattained as this is optimizing a continuous function over a compact space. With the boundednessof g we find that the limit Γ( T ∗ ) := lim t ր T ∗ Γ( t ) exists. Furthermore, ¯Λ(Γ( T ∗ )) ≥ Λ( τ ) > Λ( τ )and Γ( T ∗ ) < s (by the result of Lemma 3.4). Thus we can continue our solution to Γ : [ τ, T ∗ ] → U and find a contradiction to [ τ, T ∗ ) being the maximal domain. B Proofs from Section 3.2 Throughout this section, without loss of generality, assume that banks are ordered with decreasing¯ q . B.1 Proof of Lemma 3.11 Proof. We will consider this argument by induction. In the n bank case, defineΛ( t, Γ) := 1 + n X j =1 Z j ( t, Γ) f t ( t ) f ′ Γ ( n X j =1 Γ j ) = det I + (cid:16) Z ( t, Γ) ~ ⊤ (cid:17) f t ( t ) f ′ Γ ( n X j =1 Γ j ) . By the ordering of banks and the assumption that no firm will modify its portfolio until it hits theregulatory threshold we know that Γ i ( t ) = 0 if t < τ i . We will consider this proof by induction.Note first that τ = 0 by construction. By Γ i ( t ) = 0 if t < τ i , the results are trivial for t ∈ [0 , τ ).Now take k ∈ { , , ..., n } . For any time t ∈ [ τ k , τ k +1 ) (or t ∈ [ τ k , T ] if τ k +1 ≥ T )Λ( t, Γ( t )) = 1 + 1 − αθ min αθ min hP kj =1 ( s j − Γ j ( t )) i f ′ Γ ( P kj =1 Γ j ( t )) f Γ ( P kj =1 Γ j ( t ))˙Λ( t, Γ( t )) = 1 − αθ min αθ min dd Γ h(cid:16)P kj =1 s j (cid:17) − Γ i f ′ Γ (Γ) f Γ (Γ) Γ= P ki =1 Γ i ( t ) k X i =1 ˙Γ i ( t )Using the same logic as in Lemma 3.4, we recover that ˙Λ( t, Γ( t )) ≥ P ki =1 ˙Γ i ( t ) ≥ dd Γ h(cid:16)h(cid:16)P kj =1 s j (cid:17) − Γ i f ′ Γ (Γ) (cid:17) /f Γ (Γ) i ≥ P ki =1 Γ i ( t ). To prove this sufficient con-dition, consider the assumptions on the inverse demand function f Γ and assume Γ = P ki =1 Γ i ( t ) ∈ , P ki =1 s i ) ⊆ [0 , M ): If Γ is such that f ′ Γ (Γ) ≥ f Γ (Γ) f ′′ Γ (Γ) then dd Γ h(cid:16)P kj =1 s j (cid:17) − Γ i f ′ Γ (Γ) f Γ (Γ) Γ= P ki =1 Γ i ( t ) = dd Γ [ M − Γ] f ′ Γ (Γ) f Γ (Γ) − h M − (cid:16)P kj =1 s j (cid:17)i f ′ Γ (Γ) f Γ (Γ) Γ= P ki =1 Γ i ( t ) = dd Γ [ M − Γ] f ′ Γ (Γ) f Γ (Γ) − M − k X j =1 s j f Γ (Γ) f ′′ Γ (Γ) − f ′ Γ (Γ) f Γ (Γ) Γ= P ki =1 Γ i ( t ) ≥ . Otherwise f ′ Γ (Γ) < f Γ (Γ) f ′′ Γ (Γ) and the result follows directly from the construction of the deriva-tive.Further, by construction, if Λ( τ k , Γ( τ k )) > q ( τ k ) ≤ τ k ) exists. We nowwant to demonstrate that Λ( τ k , Γ( τ k )) > 0. By construction this is true if and only if α > − ( P k − i =1 [ s i − Γ i ( τ k )]+ s k ) f ′ Γ ( P k − i =1 Γ i ( τ k )) /f Γ ( P k − i =1 Γ i ( τ k )) ( − ( P k − i =1 [ s i − Γ i ( τ k )]+ s k ) f ′ Γ ( P k − i =1 Γ i ( τ k )) /f Γ ( P k − i =1 Γ i ( τ k )) ) θ min . With Γ k ( τ k ) = 0 by definition and by theassumption on the inverse demand function 0 ≥ ( P ki =1 s i − P ki =1 Γ i ( τ k ) ) f ′ Γ ( P ki =1 Γ i ( τ k )) f Γ ( P ki =1 Γ i ( τ k )) ≥ ( P ki =1 s i ) f ′ Γ (0) f Γ (0) ≥ Mf ′ Γ (0) f Γ (0) = M f ′ Γ (0). Therefore, if α > − Mf ′ Γ (0)(1 − Mf ′ Γ (0)) θ min then Λ( τ k , Γ( τ k )) > t ) ∈ R n + for all times t . As originally constructed we have thatfor any i ≤ k : ˙Γ i ( t ) = ˙ q ( t )[ s i − Γ i ( t )][¯ p i − x i − Ψ i ( t )] q ( t )[( s i − Γ i ( t )) q ( t ) − (¯ p i − x i − Ψ i ( t ))] { θ i ( t ) ≤ θ min } . If a bank is brought above theregulatory threshold they will not perform any transactions, i.e., ˙Γ i ( t ) = 0, but this can only occurif ˙ q ( t ) > 0. Otherwise (1 − αθ min )( s i − Γ i ( t )) q ( t ) = ¯ p i − x i − Ψ i ( t ) as the firm will need to remain atthe regulatory threshold. As such we can simplify ˙Γ i ( t ) as ˙Γ i ( t ) = − ˙ q ( t )(1 − αθ min )[ s i − Γ i ( t )] αθ min q ( t ) { θ i ( t ) ≤ θ min } .This allows us to conclude that ˙Γ i ( t ) has the opposite sign of ˙ q ( t ), i.e., ˙Γ( t ) ∈ R n + .Finally, we will now demonstrate that Γ( t ) ∈ [0 , s ) for all times t ∈ [ τ k , τ k +1 ) (or t ∈ [ τ k , T ] if τ k +1 ≥ T ) by induction for any k ∈ { , , ..., n } . As noted above, we find that ˙ q ( t ) ≤ t ∈ [ τ k , τ k +1 ). By assumption Γ( t ) ∈ [0 , s ) for all times t ∈ [0 , τ k ], so we begin with Γ( τ k ) ∈ [0 , s ).Take T ∗ = inf { t ∈ [ τ k , τ k +1 ) | ∃ i : Γ i ( t ) ≥ s i } and assume this infimum is taken over a nonemptyset. On u ∈ [ τ k , T ∗ ) we have that:˙ q ( u ) = f ′ t ( u ) f Γ ( P nj =1 Γ j ( u ))Λ( u ) ≥ inf t ∈ [ τ k ,T ∗ ] f ′ t ( t ) f Γ ( P nj =1 Γ j ( u ))Λ( τ k , Γ( τ k ))˙Γ i ( u ) = − (1 − αθ min ) ˙ q ( u )[ s i − Γ i ( u )] αθ min f t ( u ) f Γ ( P nj =1 Γ j ( u )) ≤ − (1 − αθ min ) inf t ∈ [ τ k ,T ∗ ] f ′ t ( t ) αθ min f t ( T ∗ )Λ( τ k , Γ( τ k )) ( s i − Γ i ( u )) . Thus Γ i ( u ) ≤ s i − ( s i − Γ i ( τ k )) exp (cid:16) (1 − αθ min ) inf t ∈ [ τk,T ∗ ] f ′ t ( t ) αθ min f t ( T ∗ )Λ( τ k , Γ( τ k )) ( u − τ k ) (cid:17) < s i for any time u ∈ [ τ k , T ∗ ).As in the proof of Lemma 3.4 we note that inf t ∈ [ τ,T ∗ ] f ′ t ( t ) is attained as we are infimizing acontinuous function over a compact space. Thus, by continuity, this implies that Γ i ( T ∗ ) < s i forall banks i . 23 .2 Proof of Corollary 3.12 Proof. We will use Lemma 3.11 to prove the existence and uniqueness of a solution. First, for alltimes t ∈ [0 , τ ] there exists a unique solution given by Γ( t ) = 0, q ( t ) = f t ( t ), and Ψ( t ) = 0. Asin the Proof of Theorem 3.6, we will consider the differential equation for Γ given by (9). Wenote that, though we considered the joint differential equation for Γ and q previously, (9) onlydepends on q through the collection of indicator functions on θ i ( t ) ≤ θ min ; for the purposes of thisproof we will replace the i th condition with f t ( t ) f Γ ( P nj =1 Γ j ( t )) ≤ ¯ q i . From the solution Γ we canimmediately define q ( t ) = f t ( t ) f Γ ( P ni =1 Γ i ( t )) for all times t , thus the existence and uniqueness ofΓ provides the same results for q . The results for Ψ follow from the same logic as Γ and thus willbe omitted herein. We will consider an inductive argument to prove the existence and uniqueness.Assume that we have the existence and uniqueness of the solution Γ( t ) up to time τ k for some k ∈ { , , ..., n } , then we wish to show we can continue this solution until τ k +1 ∈ [ τ k , T ].By Lemma 3.11, ˙Γ i ( τ k ) ≥ i . Define the process Γ ∗ ( t ) = P ni =1 Γ i ( t ) = P ki =1 Γ i ( t )with initial condition Γ ∗ ( τ k ) = P k − i =1 Γ i ( τ k ). Following the initial formulation for ˙Γ i ( t ) we find˙Γ ∗ ( t ) = − Z ∗ k ( t, Γ ∗ ( t )) f ′ t ( t ) f Γ (Γ ∗ ( t ))1 + Z ∗ k ( t, Γ ∗ ( t )) f t ( t ) f ′ Γ (Γ ∗ ( t )) with Z ∗ k ( t, Γ ∗ ) = (1 − αθ min )[ P ki =1 s i − Γ ∗ ] αθ min f t ( t ) f Γ (Γ ∗ ) { t ≥ τ k } . We note that this follows the differential equation of the 1 bank setting (with possibly non-zeroinitial value). Therefore we can conclude that Γ ∗ ( t ) exists and is unique for t ∈ [ τ k , τ k +1 ] (where τ k +1 is a stopping time determined solely by Γ ∗ ) via an application of Theorem 3.6. Utilizing thisunique process Γ ∗ we find that for any bank i = 1 , ..., k :˙Γ i ( t ) = g i ( t, Γ) = (1 − αθ min )[ f ′ t ( t ) f Γ (Γ ∗ ( t )) + ˙Γ ∗ ( t ) f t ( t ) f ′ Γ (Γ ∗ ( t ))] αθ min f t ( t ) f Γ (Γ ∗ ( t )) [ s i − Γ i ( t )] . As Γ ∗ ( t ) and ˙Γ ∗ ( t ) are bounded in finite time we are able to deduce that g i is uniformly Lipschitzin Γ and thus the existence and uniqueness of Γ i is guaranteed on the domain [ τ k , τ k +1 ]. C Proofs from Section 3.3 C.1 Proof of Lemma 3.15 Proof. We will consider this argument by induction. First, using Sylvester’s determinant identity,we note that det h I + Z ( t, Γ) diag[ f t ( t )] diag[ f ′ Γ (Γ ⊤ ~ s ⊤ i = det h I + diag[ f t ( t )] diag[ f ′ Γ (Γ ⊤ ~ s ⊤ Z ( t, Γ) i for any Γ ∈ R n × m . Therefore, ˙Π is well defined if and only if ˙ q is well defined.Define Y ( t, Γ) := − Z ( t, Γ) diag[ f t ( t )] diag[ f ′ Γ (Γ ⊤ ~ s ⊤ and r ( t, Γ) = s diag[ αθ min ] diag[ f t ( t )] f Γ (Γ).Note that Y ij ( t, Γ) ≥ r i > i, j = 1 , ..., n and all times t and liquidation matricesΓ. Therefore, utilizing results on the Leontief inverse, I − Y ( t, Γ) is invertible if r ( t, Γ) ⊤ Y ( t, Γ) < ( t, Γ) ⊤ . The i th element of r ( t, Γ) ⊤ [ I − Y ( t, Γ)] can be expanded as: r i ( t, Γ) − h r ( t, Γ) ⊤ Y ( t, Γ) i i = m X k =1 α k θ min f Γ ,k ( n X j =1 Γ jk ) + (1 − α k θ min ) f ′ Γ ,k ( n X j =1 Γ jk ) n X j =1 ( s jk − Γ jk ) { θ j ( t, Γ) ≤ θ min } s ik f t,k ( t )where θ j ( t, Γ) is the capital adequacy ratio for firm j at time t given the liquidation matrix Γ. This isstrictly greater than 0 for every firm i if α k θ min f Γ ,k ( P nj =1 Γ jk )+(1 − α k θ min ) f ′ Γ ,k ( P nj =1 Γ jk ) P nj =1 ( s jk − Γ jk ) { θ j ( t, Γ) ≤ θ min } > k . In particular, along the path of a solution Π( t ), the in-equality in asset k is equivalent to0 < Λ k ( t ) := 1 + 1 − α k θ min α k θ min hP nj =1 s jk (1 − Π j ( t )) { θ j ( t ) ≤ θ min } i f ′ Γ ( P nj =1 s jk Π j ( t )) f Γ ( P nj =1 s jk Π j ( t ))For simplicity of notation, given a solution Π( t ), we will assume that the banks are orderedwith increasing regulatory hitting times, i.e., τ i ≤ τ i +1 for every firm i (with the construction that τ = 0 and τ n +1 = T ) where τ j := inf { t ∈ [0 , T ] | θ j ( t ) ≤ θ min } . We will consider this proof byinduction. By Π i ( t ) = 0 if t < τ i , the invertibility of I − Y ( t, diag[Π( t )] s ) is trivial for t ∈ [0 , τ ).Now take i ∈ { , , ..., n } and k ∈ { , , ..., m } . For any time t ∈ [ τ i , τ i +1 ) (or t ∈ [ τ i , T ] if τ i +1 ≥ T )˙Λ k ( t ) = 1 − αθ min αθ min ddt hP ij =1 s jk (1 − Π j ( t )) i f ′ Γ ,k ( P ij =1 s jk Π j ( t )) f Γ ,k ( P ij =1 s jk Π j ( t ))= 1 − αθ min αθ min dd Γ k h(cid:16)P ij =1 s jk (cid:17) − Γ k i f ′ Γ ,k (Γ k ) f Γ ,k (Γ k ) Γ k = P ij =1 s jk Π j ( t ) i X j =1 s jk ˙Π j ( t ) . Using the same logic as in Lemma 3.4, we recover that ˙Λ k ( t ) ≥ P ij =1 s jk ˙Π j ( t ) ≥ dd Γ k h(cid:16)h(cid:16)P ij =1 s jk (cid:17) − Γ k i f ′ Γ ,k (Γ k ) (cid:17) /f Γ ,k (Γ k ) i ≥ k = P ij =1 s jk Π j ( t ). This sufficientcondition follows identically to the proof of Lemma 3.11.Further, by construction, if Λ k ( τ i ) > k then ˙Π( τ i ) ∈ R n + by the non-negativityof the Leontief inverse and ˙ q ( τ i ) ∈ − R m + by construction from ˙Π( τ i ). We now want to demonstratethat Λ k ( τ i ) > k . Using the same construction as in the proof of Lemma 3.11, if α k > − M k f ′ Γ ,k (0)(1 − M k f ′ Γ ,k (0)) θ min then Λ k ( τ i ) > k ( t ) > k and all times t ∈ [ τ i , τ i +1 ) (or t ∈ [ τ i , T ] if τ i +1 ≥ T ). Therefore by the abovearguments and the non-negativity of the Leontief inverse, we can conclude that ˙Π( t ) ∈ R n + (andthus ˙ q ( t ) ∈ − R m + ) for all times t ∈ [ τ i , τ i +1 ).Finally, we will now demonstrate that Π( t ) ∈ [0 , n for all times t ∈ [0 , T ]. We wish toconsider two cases for this proof. If ¯ p i ≤ x i + (1 − α ℓ,i θ min ) ℓ i then bank i will never be required toliquidate any assets (see Remark 2.2). For the remainder of this proof we will consider the setting25hat ¯ p i ≥ x i + (1 − α ℓ,i θ min ) ℓ i . We wish to consider a decomposition of the capital ratio θ i by θ i ( t ) ≥ min k : s ik > ˜ θ ik ( t ) where:˜ θ ik ( t ) := c ik x i + R t s ik ˙Π i ( u ) q k ( u ) du + s ik (1 − Π i ( t )) q k ( t ) + c ik ℓ i − c ik ¯ p i α k s ik (1 − Π i ( t )) q k ( t ) + c ik α ℓ,i ℓ i for all banks i and assets k (with s ik > 0) where P mk =1 c ik = 1. In particular, we will choosethe levels c ik ≥ c ik = (1 − α k θ min ) s ik P ml =1 (1 − α l θ min ) s il for every bank i and asset k . This choice is madesince ˜ θ ik (0) ≥ θ min if and only if c ik ≤ (1 − α k θ min ) s ik ¯ p i − x i − (1 − α ℓ,i θ min ) ℓ i =: ¯ c ik . It can trivially be shown that P mk =1 ¯ c ik ≥ θ i (0) ≥ θ min which holds by assumption. Therefore c ik = ¯ c ik / P ml =1 ¯ c il ≤ ¯ c ik constructs a single asset problem with capital ratio ˜ θ ik satisfying all the conditions of Lemma 3.11and Corollary 3.12 that can be solved independently. Denote ˜Γ ik to be the liquidation function forbank i in asset k so that the capital ratio ˜ θ ik ( t ) ≥ θ min for all times t . By Lemma 3.11 it followsthat ˜Γ ik ( t ) < s ik . By construction if ˜ θ ik ( t ) ≥ θ min for every asset k then θ i ( t ) ≥ θ min ; thus setting˜Π i ( t ) := max k =1 ,...,m n ˜Γ ik ( t ) /s ik | s ik > o < i ( t ) ≤ ˜Π i ( t ) for all times t sinceselling more assets than necessary will drive the capital ratio above the regulatory threshold. C.2 Proof of Corollary 3.16 Proof. We will use Lemma 3.15 to prove the existence and uniqueness of a solution. First, forall times t ∈ [0 , τ ] there exists a unique solution given by Π( t ) = 0, q ( t ) = f t ( t ), and Ψ( t ) = 0.Following the same argument as in the proof of Corollary 3.12, we will follow an inductive argumentto prove the existence and uniqueness. For the purposes of this proof we will replace the i th condition θ i ( t ) ≥ θ min with P mk =1 (1 − α k θ min ) s ik f t ( t ) f Γ ( P nj =1 s jk Π j ( t )) ≤ ¯ p i − x i − (1 − α ℓ,i θ min ) ℓ i as, byconstruction, once a bank has hit the regulatory threshold it will remain there until the terminaltime T . Assume that we have the existence and uniqueness of the solution Π( t ) up to time τ k forsome k ∈ { , , ..., n } , then we wish to show we can continue this solution until τ k +1 ∈ [ τ k , T ]. Asin the proof of Lemma 3.15, we will reorder the banks so that between times τ k and τ k +1 , only thefirst k banks have begun liquidating assets. That is, θ i ( τ k ) ≤ θ min if and only if i ≤ k .By Lemma 3.15, ˙Π i ( τ k ) ≥ i . Let Λ be as in the proof of Lemma 3.15, definethe domain U k = T ml =1 (cid:8) Π ∈ [0 , n | ¯Λ kl ( s ⊤ Π) ≥ Λ l ( τ k ) (cid:9) where ¯Λ k : [0 , M ] → R m is defined by¯Λ kl (Γ) := 1 + − α l θ min α l θ min [( P kj =1 s jl ) − Γ l ] f ′ Γ ,l (Γ l ) f Γ ,l (Γ l ) for all l = 1 , ..., m . We wish to note that from the abovethat by Λ kl nondecreasing in time over [ τ k , τ k +1 ], any solution must lie in U k , i.e., it must satisfyΠ( t ) ∈ U k for any time t ∈ [ τ k , τ k +1 ]. Thus by the logic of Theorem 3.6 we are able to conclude thatthere exists a unique solution on either [ τ k , τ k +1 ] or [ τ k , T ∗ ) with T ∗ ≤ τ k +1 . In the former case, theresult is proven. The latter is contradicted using the same bounding argument as in Theorem 3.6with upper bound to ˙Π provided in Lemma 3.15.26 Proofs from Section 4 For the following proofs, we will focus solely on the m = 1 asset framework with banks ordered bydecreasing ¯ q . The general case is a result of the bounding argument in the proof of Lemma 3.15. D.1 Proof of Theorem 4.1 Proof. We will prove this inductively for the single asset case. Recall the definition of Λ from theproof of Lemma 3.11, i.e., Λ( t, Γ) = 1 + P nj =1 Z j ( t, Γ) f t ( t ) f ′ Γ ( P nj =1 Γ j ).1. First, by definition it is clear that ˜ τ = τ and ˜Γ( t ) = Γ( t ) = 0 for all times t ∈ [0 , ˜ τ ]. Thus˜Λ = Λ( τ , 0) as well. By the proof of Lemma 3.4, we know ˙Γ ( t ) ≤ − (1 − αθ min ) f ′ t ( t ) αθ min f t ( t )˜Λ ( s − Γ ( t ))for t ∈ [ τ , τ ]. As expressed in the proof of Lemma 3.4 we can conclude ˜Γ k ( t ) ≥ Γ ( t ) forall times t ∈ [˜ τ , τ ] and for any iteration k = 1 , ..., n by construction as ˜Γ is the maximalsolution to this differential inequality.2. Fix k ∈ { , , ..., n − } and assume ˜Γ ki ( t ) ≥ Γ i ( t ) for all times t ∈ [0 , τ k +1 ] and any firm i = 1 , ..., k . This implies, for any ˆ k ≥ k , ˜Γ ˆ ki ( t ) ≥ Γ i ( t ) for all times t ∈ [0 , τ k +1 ] as well. Assume τ k +1 < T or else the proof is complete. By monotonicity of the inverse demand function,˜ τ k +1 ≤ τ k +1 with ˜Γ ki (˜ τ k +1 ) ≥ Γ i (˜ τ k +1 ) for any i = 1 , ..., k . In particular, this implies ˜Λ k +1 ≥ Λ(˜ τ k +1 , Γ(˜ τ k +1 )). By the proof of Lemma 3.11 we can show ˙Γ i ( t ) ≤ − (1 − αθ min ) f ′ t ( t ) αθ min f t ( t )˜Λ k +1 ( s i − Γ i ( t ))for t ∈ [˜ τ k +1 , τ k +2 ) and firm i = 1 , ..., k + 1. We note that this is a stricter bound than thatgiven in Lemma 3.11, but exists using the same logic. Solving for the maximal solution tothis differential inequality provides the solution ˜Γ k +1 which must satisfy ˜Γ k +1 i ( t ) ≥ Γ i ( t ) forall times t ∈ [0 , τ k +2 , T ]. D.2 Proof of Corollary 4.2 Proof. First, we will demonstrate that ˜Γ ni ( t ) has the expanded form provided.˜Γ ni ( t ) = { t< ˜ τ n } ˜Γ n − i ( t ) + { t ≥ ˜ τ n } s i − (cid:18) f t ( t ) f t (˜ τ n ) (cid:19) − αθ min αθ min ˜Λ n + ˜Γ n − i (˜ τ n ) (cid:18) f t ( t ) f t (˜ τ n ) (cid:19) − αθ min αθ min ˜Λ n = { t< ˜ τ n − } ˜Γ n − i ( t ) + { t ≥ ˜ τ n − } ˜Γ n − i (˜ τ n − ) n Y k = n − (cid:18) f t ( t ∧ ˜ τ k +1 ) f t ( t ∧ ˜ τ k ) (cid:19) − αθ min αθ min ˜Λ k + { t ≥ ˜ τ n − } s i n X j = n − − (cid:18) f t ( t ∧ ˜ τ j +1 ) f t ( t ∧ ˜ τ j ) (cid:19) − αθ min αθ min ˜Λ j n Y k = j +1 (cid:18) f t ( t ∧ ˜ τ k +1 ) f t ( t ∧ ˜ τ k ) (cid:19) − αθ min αθ min ˜Λ k = { t< ˜ τ i } ˜Γ i − i ( t ) + { t ≥ ˜ τ i } s i n X j = i − (cid:18) f t ( t ∧ ˜ τ j +1 ) f t ( t ∧ ˜ τ j ) (cid:19) − αθ min αθ min ˜Λ j n Y k = j +1 (cid:18) f t ( t ∧ ˜ τ k +1 ) f t ( t ∧ ˜ τ k ) (cid:19) − αθ min αθ min ˜Λ k s i n X j = i − (cid:18) f t ( t ∧ ˜ τ j +1 ) f t ( t ∧ ˜ τ j ) (cid:19) − αθ min αθ min ˜Λ j n Y k = j +1 (cid:18) f t ( t ∧ ˜ τ k +1 ) f t ( t ∧ ˜ τ k ) (cid:19) − αθ min αθ min ˜Λ k = s i − n Y j = i (cid:18) f t ( t ∧ ˜ τ j +1 ) f t ( t ∧ ˜ τ j ) (cid:19) − αθ min αθ min ˜Λ j . The penultimate line uses the fact that ˜Γ i − i ( t ) = 0 for all times t by construction and f t ( t ∧ ˜ τ j +1 ) /f t ( t ∧ ˜ τ j ) = 1 for every j ≥ i if t < ˜ τ i .Now, let us consider the form of ˜Λ i taking advantage of the exponential form for f Γ :˜Λ i = 1 + 1 − αθ min αθ min i X j =1 ( s j − ˜Γ i − j (˜ τ i )) f ′ Γ (cid:16)P i − j =1 ˜Γ i − j (˜ τ i ) (cid:17) f Γ (cid:16)P i − j =1 ˜Γ i − j (˜ τ i ) (cid:17) = 1 − b − αθ min αθ min i X j =1 s j (cid:18) f t (˜ τ i ∧ ˜ τ j +1 ) f t (˜ τ i ∧ ˜ τ j ) (cid:19) − αθ min αθ min ˜Λ j n Y k = j +1 (cid:18) f t (˜ τ i ∧ ˜ τ k +1 ) f t (˜ τ i ∧ ˜ τ k ) (cid:19) − αθ min αθ min ˜Λ k = 1 − b − αθ min αθ min i X j =1 s j i − Y k = j (cid:18) f t (˜ τ k +1 ) f t (˜ τ k ) (cid:19) − αθ min αθ min ˜Λ k Finally, let us consider the time at which the analytical worst-case pricing process hits ¯ q i , i.e.,the time when firm i reaches the regulatory threshold θ min provided all firms follow the worst-casepath. As no firms act before ˜ τ = τ , this can easily be computed as ˜ τ = f − t (¯ q ). Consider now i = 2 , ..., n , recall that ¯ q ≥ ¯ q ≥ ... ≥ ¯ q n , and assume t ≥ ˜ τ i − :¯ q i = f t ( t ) f Γ i − X j =1 ˜Γ nj ( t ) ⇔ ¯ q i = f t ( t ) f Γ i − X j =1 s j − i − X j =1 s j i − Y k = j (cid:18) f t ( t ∧ ˜ τ k +1 ) f t (˜ τ k ) (cid:19) − αθ min αθ min ˜Λ k ⇔ log(¯ q i ) − i − X j =1 s j = log( f t ( t )) + b i − X j =1 s j i − Y k = j (cid:18) f t ( t ∧ ˜ τ k +1 ) f t (˜ τ k ) (cid:19) − αθ min αθ min ˜Λ k ⇔ log(¯ q i ) − i − X j =1 s j = log( f t ( t )) + bf t ( t ) − αθ min αθ min ˜Λ i − f t (˜ τ i − ) − αθ min αθ min ˜Λ i − i − X j =1 s j i − Y k = j (cid:18) f t (˜ τ k +1 ) f t (˜ τ k ) (cid:19) − αθ min αθ min ˜Λ k ⇔ log(¯ q i ) − i − X j =1 s j = log( f t ( t )) + (cid:18) αθ min − αθ min (cid:19) ν i − f t ( t ) − αθ min αθ min ˜Λ i − ⇔ f t ( t ) = ˜Λ i − W (cid:16) ν i − ˜Λ i − exp (cid:16) − αθ min αθ min ˜Λ i − h log (¯ q i ) + b P i − j =1 s j i(cid:17)(cid:17) ν i − αθ min ˜Λ i − − αθ min .3 Proof of Corollary 4.4 Proof. First, before we prove the bound provided in Corollary 4.4 we need to demonstrate that ˜Λ k and ν k do not depend on the parameter a of the inverse demand function f t , i.e., they are constantsin this problem. We will do this by induction jointly on ˜Λ k , ν k , and f (˜ τ k ) for k = 1 , ..., n (triviallythis is the case for the assumed values ˜Λ = 1, ν = 0, and f (˜ τ ) = 1).1. Fix k = 1, then ˜Λ = 1 − b − αθ min αθ min s , f (˜ τ ) = ¯ q , and ν = − ˜Λ f t (˜ τ ) − αθ min αθ min ˜Λ1 . Since ˜Λ and f (˜ τ )do not depend on the parameter a then neither does ν .2. Fix k ∈ { , ..., n } and assume ( ˜Λ i , ν i , f (˜ τ i )) k − i =1 do not depend on the parameter a . By Corol-lary 4.2, f ( τ k ) only depends on ˜Λ k − and ν k − , and thus does not depend on the parameter a . Additionally, ˜Λ k only depends on ( f t (˜ τ i )) ki =1 , which (from the prior statement) does notdepend on a . Finally, ν k only depends on ( ˜Λ i , f t (˜ τ i )) ki =1 , thus it does not depend on a either.We will prove the bound on the probability by induction:1. Let q ∗ ∈ [¯ q , 1] (i.e., k = 0). For such an event to occur, no firms will have hit the regulatorythreshold and thus it must be the case that q ( t ) = f t ( t ). Therefore, P ( q ( t ) ≥ q ∗ ) = P ( f t ( t ) ≥ q ∗ ) = P (cid:18) a ≤ − t log( q ∗ ) (cid:19) = P (cid:18) a ≤ t Φ − (log( q ∗ )) (cid:19) . 2. Assume the provided bound is true for any q ∗ ∈ [¯ q k , q ∗ ∈ [¯ q k +1 , ¯ q k ). P ( q ( t ) ≥ q ∗ ) = P ( q ( t ) ≥ ¯ q k ) + P ( q ( t ) ∈ [ q ∗ , ¯ q k )) ≥ P a ≤ Φ − k − (log( q ∗ ) + b k − X i =1 s i ) ! + P f t ( t ) f Γ k X i =1 ˜Γ ni ( t ) ! ∈ [ q ∗ , ¯ q k ) ! Now we wish to show the form for the last term in our bound. P f t ( t ) f Γ k X i =1 ˜Γ ni ( t ) ! ∈ [ q ∗ , ¯ q k ) ! = P − at − b k X i =1 ˜Γ ni ( t ) ∈ [log( q ∗ ) , log(¯ q k )) ! = P − at + (cid:18) αθ min − αθ min (cid:19) ν k f t ( t ) − αθ min αθ min ˜Λ k ∈ [log( q ∗ ) + b k X i =1 s i , log(¯ q k ) + b k X i =1 s i ) ! = P − at + (cid:18) αθ min − αθ min (cid:19) ν k exp (cid:18) − at − αθ min αθ min ˜Λ k (cid:19) ∈ [log( q ∗ ) + b k X i =1 s i , log(¯ q k ) + b k X i =1 s i ) ! = P a ∈ (Φ − k log(¯ q k ) + b k X i =1 s i ! , Φ − k log( q ∗ ) + b k X i =1 s i ! ] ! . − k − (log(¯ q k ) + b P k − i =1 s i ) = Φ − k (log(¯ q k ) + P ki =1 s i ) as shown below:Φ − k − log(¯ q k ) + b k − X i =1 s i ! = (cid:18) αθ min − αθ min (cid:19) ν k − f t (˜ τ k ) − αθ min αθ min ˜Λ k − − " log(¯ q k ) + b k − X i =1 s i = (cid:18) αθ min − αθ min (cid:19) ν k f t (˜ τ k ) − αθ min αθ min ˜Λ k − " log(¯ q k ) + b k X i =1 s i = αθ min ˜Λ k − αθ min ! − ˜Λ k ˜Λ k ! − " log(¯ q k ) + b k X i =1 s i = αθ min ˜Λ k − αθ min ! 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