we-solve (beta)

Econ-155 · economics · Economics

Liquidity backstops for nonbanks

LLM status Open unverified

Statement

Original subject group: Banking, credit and financial stability

Audit classification: Explicit published open question. The BIS primary open-question section explicitly addresses central-bank intervention design; the Bank agenda supplies related scope. Collateral haircuts, fiscal accounting and endogenous buffers are expanded editorially.

Definitions and precise research target

Fix nonbank institutions with assets, short-term funding, private liquidity buffers and potential access to central-bank lending. A facility $a$ specifies eligible institutions, collateral, haircuts, interest charges, borrowing caps and activation conditions. Institution $i$ chooses a pre-crisis buffer $b_i(a)$ anticipating the facility. Let $L(a,\omega;b(a))$ be crisis resource losses after this endogenous response, $F(a,\omega)$ expected public losses, and $C(a)$ routine operating costs. The task is to characterize \[J(a)=E[L(a,\omega;b(a))+F(a,\omega)]+C(a),\qquad J_*=\inf_{a\in A}J(a).\] Here $A$ is the feasible facility set. Define shock state $\omega$, its distribution, the institution perimeter and fiscal-risk accounting explicitly. Impose incentive and eligibility constraints ensuring institutions cannot profit from manufacturing collateral or misreporting solvency. Quantify the reduction in liquidity externalities and the offsetting loss of private self-insurance. Compare standing facilities with crisis-only lending while accounting for anticipated discretionary rescues. Determine when access must be coupled to supervision or prudential requirements. State which objects are identified from existing interventions and which require structural assumptions. A liquidity facility that prevents one historical fire sale does not establish an optimal access perimeter or demonstrate that moral hazard is negligible.

Additional formal definitions and scope (editorial). For collateral $j$ of market value $P_j q_j$, haircut $h_j\in[0,1]$ permits at most $(1-h_j)P_j q_j$ of secured lending, subject to any separate borrowing cap. Buffer $b_i(a)$ is an optimal private holding given anticipated access, with explicit private funding cost and losses in stress. Public fiscal cost is the unrecovered resource or distortionary-financing loss after collateral recovery; a loan disbursement alone is not a welfare loss. Specify repayment priority, maturity, central-bank balance-sheet limits and whether monetary-policy costs are included. The expectation in the objective uses a common shock law across facilities. If private choices or crisis prices are multiple, state a selector or minimize worst-case losses over the equilibrium correspondence. Standing-access and discretionary-rescue problems have different commitment assumptions. For a nonempty feasible set and a finite optimal value, the design target is an $\varepsilon$-optimal feasible rule for each specified $\varepsilon>0$: its cost is at most the infimum plus $\varepsilon$, or its welfare is at least the supremum minus $\varepsilon$. Report infeasibility or an unbounded value separately. An exact optimizer is requested only under additional attainment assumptions; it need not exist for the general rule class.

Variants and assumption changes

  1. Committed standing facility (source-motivated). Announce access and haircuts before buffer choice and jointly choose prudential constraints; quantify private self-insurance changes.
  2. Discretionary stress lending (editorial). Let authorities reoptimize after shocks with fiscal limits; distinguish the optimal crisis action from the ex ante effects of anticipating rescue.

References and source audit

  1. BIS WP 972 cover verifies authors; October 2021 original, January 27, 2022 revision inspected; no DOI assigned. Locator: Section 5, printed pp.38--39 (PDF pp.41--42). Support: Nonbank regulation/backstop agenda. Full January 2022 revision accessible. URL: https://www.bis.org/publications/working-paper-972-non-bank-financial-intermediaries-and-financial-stability.pdf.
  2. Official January 9, 2026 web version; locator: Central Bank Balance Sheet / Central banking toolkit, public-backstop paragraphs. Broad agenda; exact model editorial. URL: https://www.bankofengland.co.uk/research/bank-of-england-agenda-for-research.
  1. Sirio Aramonte; Andreas Schrimpf; Hyun Song Shin. 2021. Non-bank financial intermediaries and financial stability. Section 5, printed pp.38--39 (PDF pp.41--42). https://www.bis.org/publications/working-paper-972-non-bank-financial-intermediaries-and-financial-stability.pdf.
  2. Bank of England. 2026. Bank of England Agenda for Research, 2025–2028. Central Bank Balance Sheet / Central banking toolkit, public-backstop paragraphs. https://www.bankofengland.co.uk/research/bank-of-england-agenda-for-research.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-155-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-155

Liquidity backstops for nonbanks Buffer crowd-out, credible rescue, and a supervision wedge

Problem ID: E58-152. Status: Unresolved; restricted results and explicit gaps.

Question and deliberately restricted institution

The target is facility design with endogenous liquidity buffers, fiscal accounting, eligibility, and credible crisis intervention. The calculation below solves an explicit one-institution design problem and identifies a strict local benefit of buffer requirements. It does not determine an optimal nonbank perimeter. Aramonte, Schrimpf, and Shin [bis], Section 5, printed pp. 38--39, explicitly discuss the design of central-bank interventions; their preceding discussion connects anticipated support with self-insurance and oversight. Our formulas are independent model derivations, not theorems attributed to that source.

There are dates 0, 1, and 2, one institution, and a continuum of identical replicas if a population interpretation is desired. Its date-1 cash demand $X$ is uniform on $[0,1]$, under every policy. At date 0 it chooses cash $b\in[0,1]$ at real opportunity cost $cb$, where $0<c<d$. An announced facility supplies at most $a\in[0,1]$ at date 1. Lending and uncovered demand are \[ Q=\min\{a,(X-b)_+\},\qquad Y=(X-b-a)_+. \] Uncovered demand destroys real resources with social cost $DY$, $D\geq d$; the institution internalizes $dY$. These are linear state losses, although their expectations will be quadratic. There is no private borrowing fee. The facility incurs real administration or balance-sheet opportunity cost $\mu Q$, with $0\leq\mu<D$, plus setup cost $\eta a^2/2$, $\eta>0$.

The institution has an independently registered, nonmanufacturable date-2 collateral claim with certain recovery at least one. It cannot sell or repledge that claim, reports no private solvency information, and the facility has first priority. Loans mature at date 2 and are fully recovered. A cap $a$ can be implemented by a haircut $1-a$ on this unit eligible claim. Thus loan principal is not counted as a fiscal loss; $\mu Q$ is an explicitly separate resource cost. Balance-sheet capacity is one, and monetary-policy costs beyond this declared cost are excluded. These restrictions eliminate collateral fabrication and insolvency misreporting by assumption. They do not solve screening for a heterogeneous real-world perimeter.

The private and social problems, with equal welfare weight on domestic resources, are \[ \min_{0\leq b\leq1}\{cb+d{\mathbb{E}} Y\},\qquad J(a,b)=cb+D{\mathbb{E}} Y+\mu{\mathbb{E}} Q+\eta a^2/2. \] The buffer cost is included once. External liquidity losses are represented by $(D-d)Y$; market prices are not endogenized in this benchmark.

Exact design with anticipated support

Put $s=c/d$ and $A=1-s$, so $0<A<1$. For an interval $I$, write ${\operatorname{clip}}_I(x)$ for its Euclidean projection.
Theorem (Buffer response and global facility choice): The unique private response is $b(a)=(A-a)_+$. Consequently, for $a\leq A$, \[ {\mathbb{E}} Y=s^2/2,\quad {\mathbb{E}} Q=sa+a^2/2,\quad J(a,b(a))=c(A-a)+Ds^2/2+\mu(sa+a^2/2)+\eta a^2/2. \tag{1} \] For $A\leq a\leq1$ the objective is \[ J(a,0)=D(1-a)^2/2+\mu(a-a^2/2)+\eta a^2/2. \tag{2} \] Define \[ a_L={\operatorname{clip}}_{[0,A]}\frac{c-\mu s}{\mu+\eta},\qquad a_R={\operatorname{clip}}_{[A,1]}\frac{D-\mu}{D-\mu+\eta}. \] The complete set of optimal capacities consists of the members of $\{a_L,a_R\}$ with the smaller value of $J$. In particular, an optimal capacity exists, but need not be unique.
Proof: For $u\geq0$, integration gives ${\mathbb{E}}(X-u)_+=(1-u)_+^2/2$. The private objective has derivative $c-d(1-b-a)_+$ in $b$. Where the shortfall is positive this derivative increases strictly, and after it vanishes the derivative is the positive constant $c$. The unique constrained minimizer is therefore $(1-a-c/d)_+$. The identity $Q=(X-b)_+-(X-b-a)_+$ proves (1) and (2) by substitution.

On $[0,A]$ the derivative of the objective is $-c+\mu s+(\mu+\eta)a$, with positive second derivative $\mu+\eta$. On $[A,1]$ its derivative is $-(D-\mu)+(D-\mu+\eta)a$, again strictly increasing. Each interval has the stated unique clipped minimizer. The formulas agree at $A$, so these intervals cover the entire continuous optimization problem. Comparing their minimum values proves the characterization, including possible ties. ∎

The left and right derivatives at $A$ differ by $-(D-d)s-\mu A$. For $D>d$ or $\mu>0$ the derivative jumps downward. Consequently imposing global convexity would discard legitimate competing local optima. Formula (1) also exhibits complete crowd-out: expanding access below $A$ reduces private buffers one for one and does not change expected uncovered demand. Even there, a facility can reduce total cost by replacing expensive private cash; crowd-out alone is not a welfare ranking.

Proposition (Discretionary rescue is anticipated): Fix a legally enforceable capacity $\bar a$. At date 1 suppose the authority minimizes $D(X-b-q)_++\mu q$ over $0\leq q\leq\min\{\bar a,(X-b)_+\}$, taking $b$ as sunk. Its unique choice is $q=\min\{\bar a,(X-b)_+\}$ unless the interval is a singleton, in which case that same formula applies. The subgame-perfect buffer is $b(\bar a)$, identical to committed standing access with this capacity. An announcement of no lending is therefore not credible when $\bar a>0$ and a positive shortfall occurs.
Proof: On the feasible interval the uncovered demand is $X-b-q$ whenever that interval has positive length. The derivative of the crisis objective is $\mu-D<0$. Hence the largest feasible loan is uniquely optimal. Anticipating this action gives exactly the private date-0 objective in Theorem 1, whose unique minimizer has already been established. ∎
This result is conditional on a fixed enforceable cap. If crisis authorities can also revise the cap or collateral eligibility, their action space and the date-0 anticipation problem change; naming a facility "crisis-only" is not a commitment technology.

A quantified role for prudential requirements

A supervisor can verify cash and impose $b\geq m$, $m\in[0,1]$, without additional administrative cost. The resulting private response is $b(a,m)=\max\{m,b(a)\}$, by the derivative argument above.
Proposition (A strict supervision wedge): Fix $0\leq a<A$. At $b=b(a)$ the derivative of social cost with respect to an independently imposed buffer is \[ J_b(a,b(a))=c-Ds-\mu a. \tag{3} \] If $D>d$ or $\mu a>0$, there is $\varepsilon_0>0$ such that every requirement $m=b(a)+\varepsilon$, $0<\varepsilon<\varepsilon_0$, strictly lowers social cost. More generally, choosing $a,m\in[0,1]$ has an attained global minimum of $J(a,b(a,m))$.
Proof: At $a<A$, $b(a)>0$ and $b(a)+a=A<1$, so both expected-shortfall formulas are differentiable in a neighborhood. Their derivatives are $\partial_b{\mathbb{E}} Y=-(1-b-a)$ and $\partial_b{\mathbb{E}} Q=-(1-b)+(1-b-a)=-a$. Substitution gives (3), which is negative under the stated condition because $c=ds$. The derivative remains negative on a sufficiently small right neighborhood; integration proves the strict decrease. The requirement binds there because the unconstrained minimizer remains $b(a)$. Finally $b(a,m)$ and the objective are continuous on the compact square, so a convergent subsequence of a minimizing sequence attains the infimum. ∎

With supervision cost $H(m)$ differentiable from the right, the relevant local derivative is (3) plus $H'(b(a))$. Thus the proposition is a marginal benefit calculation, not an assertion that supervision is free in practice. It also distinguishes a genuine liquidity externality, $D-d$, from the social cost of substituting public balance-sheet resources for cash.

What this identifies, and what remains

Under the stipulated uniform law, observing an interior buffer at known capacity identifies $c/d=1-a-b(a)$, not $c$ and $d$ separately. For example, $(c,d)$ and $(tc,td)$ for every $t>0$ produce the same buffer schedule. The statement follows by direct substitution in Theorem 1. Borrowing and shortfalls identify neither the external loss coefficient $D$ nor the resource cost $\mu$. A historical reduction in $Y$ therefore does not identify the minimizing capacity or the supervision wedge.

The exact solution is a benchmark for testing claims about rescue commitment and buffer crowd-out. Endogenous collateral acquisition, solvency screening, equilibrium fire-sale prices, heterogeneous institutions, loss-sharing, and causal estimation of external damage remain outside it. Those omitted objects are essential to the full canonical target, which remains unresolved. No empirical calibration or numerical verification is claimed; all results above are analytic.

References

Completion Estimate

30%

Go to contribution →