Long-Term Securities and Banking Crises

Layer 1: Overview

This paper asks how bank holdings of long-term government securities interact with interest-rate-driven monetary tightening to amplify macroeconomic downturns and generate banking crises. The motivating empirical fact is the sharp rise in US commercial banks’ long-term security holdings: the portfolio share of all long-term securities (Treasury bonds, MBS, and agency debt with maturity above one year) reached 25.8% of bank assets in 2021Q2, and long-term Treasuries alone reached 12.2%, based on bank-level call report data from 1997Q2 to 2021Q2. SVB’s failure in March 2023 illustrates the mechanism the paper studies: interest-rate hikes reduce long-term bond prices, impair bank net worth, and can trigger depositor runs.

The paper builds a dynamic New Keynesian (DNK) DSGE model that incorporates a banking sector following Gertler-Karadi (2011, 2013) and Gertler-Kiyotaki (2010, 2015). Banks take deposits, lend to nonfinancial firms, and hold long-term government bonds with geometrically declining coupon structure (decay parameter ρ = 0.96, calibrated to a five-year weighted average maturity). An agency problem between banks and depositors generates an endogenous leverage constraint. Households face asset-management costs for directly holding bonds and equity, which produces firesale prices when banks are forced to liquidate. Cost-push shocks are introduced via a tax-subsidy on retailer revenues following Adam and Woodford (2012), generating an ARMA(1,1) disturbance to the New Keynesian Phillips curve. The model is calibrated to quarterly US data: bank leverage of 6, annualized excess equity return of 4%, excess long-term bond return of 2%, dividend payout ratio of 24%, long-term bonds at 22% of bank assets, and public debt-to-GDP of 100%. Nonlinear perfect-foresight solutions are computed using Dynare for both normal (no-run) and bank-run equilibria.

The central quantitative findings are as follows. First, in the no-run baseline, long-term bond holdings amplify contractionary shocks more than short-term bonds because prices of longer-maturity bonds decline more sharply when interest rates rise — a standard duration effect augmented by a feedback loop through impaired bank net worth. Second, and more strikingly, the model generates self-fulfilling bank runs. When a 10-standard-deviation cost-push shock raises annualized inflation to 7%, the Taylor-rule response raises the annual interest rate passively to 4.2%, and the recovery rate xt (the ratio of liquidation value to deposit claims) falls below 1 from periods 1 through 10, meaning a bank run is feasible across that window. A representative bank run in period 4 causes the capital price to fall by 20% and the long-term bond price by 14%, with severe and prolonged effects on investment and output. If instead the central bank actively tightens — adding two consecutive 25-basis-point surprise hikes on top of the Taylor rule — the nominal rate rises to 5.8%, the capital price falls by an additional 3 percentage points (−8% versus −5%), the bond price by an additional 1 percentage point (−7% versus −6%), and bank net worth falls by 45% rather than 25%, extending the window of bank-run vulnerability from period 10 out to period 16. Crucially, when banks hold only short-term bonds (ρ = 0), the recovery rate never falls below 1 under the same shock sequence, so no run equilibrium exists. The model’s calibrated additional output loss from a banking panic (2.19% averaged over 12 quarters after the run) closely matches the cross-country estimate from Baron, Verner, and Xiong (2021) of 2.3% over a three-year window across 46 countries from 1870–2016.

On the policy side, the paper studies two macroprudential instruments targeting bank long-term bond holdings. A permanent tax τl = 0.07 on those holdings is optimal: it shifts the household share of long-term bonds from 70% to 90% in steady state, reduces the liquidation price drop to 5% (from 14%), shortens the run-vulnerability window from period 16 to period 13, yields a conditional welfare gain of 0.009% (no-run case) or 0.068% (when the tax actually prevents a run), and reduces the bank-run probability by 4.9%. A cyclical subsidy-when-rates-rise (ϕl = −1.5) shortens the vulnerability window from period 16 to period 9. The optimal cyclical policy is at a corner (ϕl = −2) in the searched range. The paper also documents complementarity between the two instruments: more dovish monetary policy (smaller ϕπ) reduces run probabilities for any given macroprudential stance, and more aggressive cyclical macroprudential policy (larger |ϕl|) reduces run probabilities for any given monetary stance.

Layer 2: Deep Dive

What is the identification strategy and what are the main threats to it?

There is no empirical identification exercise in the conventional sense. The paper is a calibrated DSGE model evaluated by impulse response and welfare analysis. The calibration targets observable steady-state moments (bank leverage = 6, excess equity return = 4% p.a., excess long-term bond return = 2% p.a., dividend payout ratio = 24%, long-term bonds = 22% of bank assets, debt-to-GDP = 100%, average bond maturity = 5 years) and shock process parameters borrowed from Gelain and Ilbas (2017). The main ‘identification’ challenge is the choice of ξ (the fraction of pre-run net worth restored to the banking system one period after a run), which is calibrated so the model’s additional output loss (2.19% over 12 quarters) matches the Baron-Verner-Xiong (2021) cross-country estimate of 2.3% over three years. Threats to quantitative conclusions include: (i) the assumption that bank runs are unanticipated (zero perceived probability); (ii) the single aggregate bank (no cross-sectional heterogeneity across institutions); (iii) perfect foresight after shock realization; (iv) no credit policy or unconventional monetary policy; and (v) the cost-push shock being the only inflation driver (no demand or supply shock interaction).

What is the core mechanism by which long-term bond holdings amplify shocks?

Two related channels operate. First, a standard duration channel: the price of a bond portfolio with geometric maturity structure equals the discounted sum of future coupons weighted by the bank’s stochastic discount factor (SDF). A longer maturity (higher ρ) means that a given reduction in the bank’s SDF (caused by deteriorating net worth) is applied to more future coupon payments, so the bond price falls more. The paper formalises this via the bank’s bond pricing equation: Ql_t = sum_{j=1}^∞ ρ^{j-1} Ω̃_{t,t+j}, so a higher ρ maps each deterioration in future SDFs into a larger price decline. Second, a feedback loop: a lower bond price further reduces bank net worth, further lowering the bank’s SDF, further reducing the bond price. This amplification is absent when ρ = 0 (short-term bonds) because the one-period bond price is simply 1/(R_t^n z_t) and is only directly exposed to one period’s interest rate change.

How is the bank-run equilibrium structured, and what determines whether a run is possible?

The paper follows Gertler-Kiyotaki (2015): bank runs are modelled as rollover panics rather than Diamond-Dybvig sequential service. Depositors who rolled over deposits in period t−1 decide in period t whether to roll over again or withdraw. A bank-run equilibrium exists if the recovery rate xt — the ratio of the liquidation value of bank assets at firesale prices to the face value of outstanding deposits — is strictly less than 1. When xt < 1, depositors who believe others will run are individually rational to run (the bank cannot fully repay them in liquidation), making the run self-fulfilling. Liquidation prices are below normal prices because households face asset-management costs for directly holding bonds and equity, so when banks dump all assets on households, prices drop. A sunspot variable shifts the economy from the no-run to the run equilibrium whenever xt < 1. The paper only models unanticipated runs (depositors assign zero probability to a run when making their deposit decision).

What is the role of maturity structure in run likelihood, and how is this demonstrated?

Figure 7 is the key comparison. Under the same shock sequence (10-standard-deviation cost-push shock plus two 25-bp monetary policy shocks), the model is solved for both ρ = 0 (three-month bonds) and ρ = 0.96 (five-year bonds). When ρ = 0, the recovery rate xt stays above 1 at every period — no bank run is possible. When ρ = 0.96, xt falls below 1 from period 1 through period 16, and a bank run is possible in any of those 16 quarters. The output path in the no-run equilibrium is similar across the two maturities, which isolates the run risk channel as the distinctive effect of long-term holdings rather than a simple level effect on investment. This provides the paper’s core result: long-term bonds are not worse per se in normal times, but they create an existential fragility when interest rates rise sharply.

How does active monetary tightening compare to passive Taylor-rule tightening in the bank-run model?

The paper compares two scenarios in Figures 5 and 6. In Figure 5, the central bank responds passively by following the Taylor rule with the baseline ϕπ = 1.98. The cost-push shock raises inflation to 7% and the annual interest rate passively reaches 4.2%. Bank net worth falls 25%, capital price falls 5%, bond price falls 6%, and xt < 1 for periods 1–10. In Figure 6, two consecutive surprise 25-bp hikes are added. Inflation on impact is lower (4.8% rather than 7%), but the interest rate rises further (to 5.8% by period 4). Bank net worth falls 45%, capital price falls 8%, bond price falls 7%, and xt < 1 through period 16. Recession severity (investment, output, consumption) is similar across the two cases, but bank fragility is substantially worse under active tightening. Inflation control comes at the cost of extended bank-run vulnerability.

How does the permanent bond tax work and what are its trade-offs?

The permanent tax τl raises the after-tax cost of holding long-term bonds for banks, inducing a shift from banks to households in the steady state. At τl = 0.07, the household share of long-term bonds rises from 70% to 90%. The tax has two effects: (i) a steady-state effect that reduces bank net worth and capital intermediated by banks — a welfare cost; and (ii) a dynamic effect that reduces the bank’s exposure to bond-price declines when rates rise — a welfare benefit. The optimal rate τl = 0.07 balances these two effects and yields a welfare gain of 0.009% in consumption-equivalent units in the no-run equilibrium, and 0.068% if the tax actually prevents a run that would otherwise occur. The liquidation drop in bond prices falls from 14% to 5% at this tax rate. Bank-run vulnerability (xt < 1) shortens from periods 1–16 to periods 1–13.

How does the cyclical tax/subsidy work and why might the permanent tax be preferred in some dimensions?

The cyclical policy sets τl_t = ϕl (R^n_t − R^n): the tax rate falls when interest rates rise, which amounts to a subsidy on bank long-term bond holdings during rate hikes. This directly offsets the adverse balance sheet effect of bond price declines. Unlike the permanent tax, it does not change the steady state and therefore avoids the steady-state contraction in bank balance sheets and capital. The subsidy with ϕl = −1.5 shortens the run window from period 16 to period 9. The unconstrained optimum is at the corner ϕl = −2 of the searched range, suggesting that the marginal benefit of stabilisation still exceeds marginal cost at the boundary; an interior optimum would require introducing distortionary financing costs for the subsidy, which the paper leaves for future work. Both policies reduce run probability, and the two complement each other and monetary policy in the interaction analysis (Table 2).

What does Table 2 show about the interaction between monetary policy and macroprudential policy?

Table 2 reports the percentage reduction in bank-run probability (relative to the baseline case ϕπ = 1.98, ϕl = 0) under nine combinations of three monetary policy aggressiveness levels (ϕπ = 1.5, 1.98, 2.2) and three cyclical macroprudential parameters (ϕl = −1, −1.5, −2). Key findings: (i) for any given macroprudential rule, more dovish monetary policy (lower ϕπ) reduces run probabilities more — for ϕl = −1.5, the reduction is 10.64% for ϕπ = 1.5 but only 6.12% for ϕπ = 1.98 and 5.21% for ϕπ = 2.2; (ii) for any given monetary rule, a more aggressive macroprudential subsidy (more negative ϕl) further reduces run probability — for ϕπ = 1.98, the reduction goes from 6.12% (ϕl = −1) to 8.39% (ϕl = −1.5) to 10.08% (ϕl = −2). This documents substitutability between looser monetary policy and macroprudential policy in preventing bank runs, and complementarity between their stabilisation effects.

How does this paper relate to and differ from Gertler-Karadi (2013)?

Gertler-Karadi (2013) is the closest predecessor. Both study banks holding government bonds in a DSGE model. Three principal differences: (i) Bond maturity — Gertler-Karadi (2013) uses infinite-maturity console bonds; this paper uses finite-maturity bonds with a geometric coupon structure that can be calibrated to the empirical five-year average maturity, which is quantitatively important for the run conditions. (ii) Bank runs — Gertler-Karadi (2013) features no bank-run equilibrium; this paper explicitly models the possibility and conditions for runs. (iii) Policy focus — Gertler-Karadi (2013) studies unconventional monetary policy (large-scale asset purchases) during crises triggered by capital quality shocks; this paper studies macroprudential taxes on long-term bond holdings during crises triggered by inflation and interest rate hikes.

How does this paper relate to and differ from Gertler-Kiyotaki (2015) and Gertler-Kiyotaki-Prestipino (2020a)?

Gertler-Kiyotaki (2015) and Gertler-Kiyotaki-Prestipino (2020a) introduce rollover-panic bank runs (following Cole-Kehoe 2000 and Calvo 1988) into DSGE models requiring global nonlinear solution methods. This paper follows the same run modelling approach. The key differences: this paper focuses on cost-push shocks and the resulting inflation-interest rate dynamics as the trigger, whereas Gertler-Kiyotaki-Prestipino (2020a) focus on capital quality shocks (‘financial panics’). The paper also introduces variable capital as in Gertler-Kiyotaki-Prestipino (2020a), but the shock environment and policy instruments are distinct — this paper studies two novel macroprudential policies targeting long-term bond holdings, which are absent from those papers.

What is the pecuniary externality underlying the macroprudential policy rationale?

Individual banks, when choosing their long-term bond holdings, fail to internalise two aggregate effects: (i) their leverage decisions affect asset prices through the incentive constraint and the bank SDF, and (ii) their bond holding choices affect the probability of a systemic run, because a deterioration of any individual bank’s balance sheet is identical to all others in the representative-bank model and thus raises the system-wide recovery rate below 1. The externality follows the Lorenzoni (2008) pecuniary externality framework: private agents do not account for the impact of their portfolio choices on equilibrium asset prices. The macroprudential tax corrects this by internalising the effect of long-term bond holdings on the fragility of the overall banking system.

What are the scope conditions and caveats on the paper’s results?

Several scope conditions are important: (i) The paper models only unanticipated bank runs (zero probability assigned by depositors ex ante). Anticipated run risk would alter the ex ante deposit decision and calibration. (ii) The model has a representative bank, so runs are on the entire banking system, not idiosyncratic institution-level runs as at SVB specifically. (iii) The paper does not model the recent bank failures directly and does not claim to replicate SVB or the March 2023 events. (iv) The welfare gains from both macroprudential policies are small in the no-run equilibrium (0.009% for the permanent tax) because the exercises are conditional on specific small-shock sequences; they would be larger for more severe or more persistent shocks. (v) The interior optimum for the cyclical policy is not characterised because the marginal cost of the subsidy (distortionary taxes needed to finance it) is not modelled. (vi) Credit policy and unconventional monetary policy (e.g., QE) are explicitly excluded.

What robustness checks does the paper conduct?

The paper checks that nonlinear perfect-foresight solutions are close to the log-linearised solutions. It compares the two monetary policy regimes (passive Taylor rule versus active surprise hikes) and documents that run conditions differ substantially. It varies ρ across 0 and 0.96 to confirm the maturity-structure mechanism. It explores the permanent tax rate across the full range (Figure 9) to confirm a unique interior optimum at τl = 0.07. It examines the cyclical policy for ϕl in {−1.5, 0, 1.5} and confirms that positive ϕl amplifies shocks (Table 2 range is ϕl in {−1, −1.5, −2} crossed with three ϕπ values). The paper does not conduct formal Bayesian or simulated method of moments estimation, so there is no sensitivity analysis over the full parameter vector.

What are the policy implications and their scope conditions?

The paper supports two macroprudential policy recommendations. First, a permanent tax on bank holdings of long-term bonds reduces run vulnerability and has an optimal rate around 7% in the calibration, but the welfare gain is quantitatively small unless a run is actually prevented (in which case it is about seven times larger, 0.068%). Second, a cyclical subsidy on bank long-term bond holdings during rate hikes acts as an automatic stabiliser and can be more effective at reducing run vulnerability without distorting the steady state; the optimal level exceeds what is studied in the paper. These results apply in the context of cost-push inflation shocks that generate interest rate hikes, which is the environment most relevant for the 2021–2023 episode. The paper’s policy design does not address the role of existing deposit insurance, resolution mechanisms, or capital adequacy requirements, so complementarity or substitutability with those tools is unexplored.

Key Concepts

Recovery rate (x_t): In the paper’s bank-run model, the ratio of the liquidation value of bank assets (valued at firesale prices) to the total nominal claims of depositors. A run equilibrium is possible if and only if x_t < 1; when x_t ≥ 1, a run cannot be self-fulfilling because depositors would be fully repaid even in liquidation.

Rollover panic / sunspot run: A bank-run mechanism (following Cole-Kehoe 2000 and Calvo 1988) in which each depositor’s decision not to roll over deposits is individually rational if and only if they believe other depositors will also not roll over. The run is triggered by a sunspot (a coordination device) rather than a fundamental shock, but its feasibility depends on the fundamental condition x_t < 1.

Geometric maturity structure: A bond portfolio specification (following Cochrane 2001 and Woodford 2001) in which one unit of the portfolio purchased at t pays ρ^{j−1} dollars at t+j for each j ≥ 1. The parameter ρ ∈ (0,1) controls effective maturity: ρ = 0 is a one-period bond and ρ = 0.96 corresponds to a five-year weighted average maturity. This device allows a tractable, single-state-variable representation of long-term debt in a DSGE model.

Incentive (leverage) constraint: In the paper’s agency problem, the constraint that prevents a banker from diverting a fraction θ of assets: the bank’s franchise value V_t must be at least θ times total assets. When binding, this constraint endogenously limits leverage and ties the total credit available to the economy to bank net worth, generating procyclical bank balance sheets.

Firesale price: The equilibrium asset price that obtains when the banking system is fully liquidated and households must absorb all assets directly. Firesale prices are below normal levels because households face asset-management costs (quadratic in their holdings relative to steady-state levels), so they require higher expected returns to absorb the assets, depressing current prices. Firesale prices are the key link between bank illiquidity and real losses.

Cyclical macroprudential tax: A tax (or subsidy when negative) on bank holdings of long-term bonds where the rate responds linearly to the deviation of the nominal interest rate from its steady state: τl_t = ϕl(R^n_t − R^n). When ϕl < 0, the policy subsidises bank long-term bond holdings when rates rise, acting as an automatic stabiliser against interest-rate-driven impairment of bank balance sheets.

Cost-push shock: A disturbance to the New Keynesian Phillips curve that shifts the inflation-output gap trade-off, modelled here (following Adam and Woodford 2012) as a random tax/subsidy on retailer revenues. The paper models it as an ARMA(1,1) process. It raises inflation without a corresponding increase in output, forcing the central bank to tighten and setting off the adverse bank balance-sheet dynamics studied in the paper.

Procyclical bank balance sheet: The property that bank net worth, total assets, and credit intermediated by banks all shrink when contractionary shocks hit, amplifying the original shock. In the paper, the amplification runs through the incentive constraint: when bond or equity prices fall, bank net worth falls, tightening the constraint, raising the marginal cost of funds, reducing investment and output further.