Banking Crises Under a Central Bank Digital Currency (CBDC)

2.1 General Central Bank Balance Sheet Dynamics of CBDC Issuance

The idea of opening the central bank balance sheet to deposits by the public is not new. Quite contrary, it was a common central bank practice until mid-20th century. As e.g. elaborated by Fernández-Villaverde et al. (2021), historically many major central banks not only allowed deposits but also granted loans to private firms and individuals. The idea of enabling deposits at the central bank for the public was prominently brought back on stage and into the electronic sphere by Tobin (1985, 1987) from which the current discussion mostly picks up.

The introduction of a CBDC would create a new type of liability that can trigger different adjustments on the central bank balance sheet, contributing in different ways to determine the equilibrium impact of a CBDC. For instance, Barker, Bholat, and Thomas (2017) argue that “although there has been a lot of discussion about how central bank digital currency could radically change payment systems – and even the financial sector as a whole – the implication for the assets on central bank balance sheets could be just as critical”. However, in most CBDC model analyses, the modelling choice that complements CBDC issuance on the central bank balance sheet remains in the background and its implications are not explicitly discussed.^[7] In practice, these balance sheet adjustments will depend on monetary policy, the economic and institutional context and will be difficult to link directly to CBDC. Nonetheless, for analytical purposes, evaluating the impact of different measures in isolation can provide valuable insights.

This paper focuses on the most relevant CBDC scenarios which would lead to an increase in the central bank balance sheet. CBDC issuance will either lead to rearrangements on the liability side of the central bank balance sheet or to an increase in both, liabilities and assets. In most cases, either cash or deposits will be substituted into CBDC.^[8] The substitution of cash into CBDC would simply constitute a change in the type of central bank money held by the public and would be neutral from a financial stability perspective.^[9] Therefore, the following analysis abstracts from cash and focuses only on deposit substitution into CBDC. The substitution of deposits into CBDC would either lead to a reduction in excess reserves (no impact on the size of the central bank balance sheet) or require a counterposition on the asset side (increasing the size of the central bank balance sheet).^[10] Situations in which deposit outflows cannot be absorbed by a reduction in excess reserves pose more serious financial stability concerns. Therefore, for simplicity the following analysis abstracts from modelling reserves and focuses on scenarios in which CBDC issuance leads to an increase in the central bank balance sheet.

2.2 Deposit Substitution into CBDC via Central Bank Credit (‘Credit Policy’)

As a first and most prominently discussed option, the central bank could increase its credit operations to banks in order to complement CBDC issuance. Such adjustments would channel lost deposit funding back to the banks in the form of central bank credit. Lending to financial institutions is already a substantial component of central bank assets in normal times but becomes particularly important in times of crisis. The role of the central bank as ‘lender of last resort’, dating back to Bagehot (1873), is also documented by Pattipeilohy (2016) who shows a clear shift towards private sector lending (including banks) by the Federal Reserve, Bank of England and the Eurosystem in response to the financial crisis from 2007 and onward.

Given the particular relevance of bank lending in times of crisis, the following model framework will explicitly review CBDC issuance with ‘credit policy’ as illustrated in Table 1.^[11] This scenario also provides the basis for the ‘equivalence result of public and private money’ of Brunnermeier and Niepelt (2019) who show that equilibrium allocations stay unaltered under the assumption that CBDC is fully redistributed as credit to private banks (under the same conditions). Moreover, CBDC issuance with ‘credit policy’ is employed for instance also in the model analyses of Skeie (2021) and Kim and Kwon (2023).

2.3 Deposit Substitution into CBDC via the Sale of Assets (Asset Policy)

As a second option, the central bank could accept non-bank assets as a counterposition to CBDC issuance. Within the model framework, such a policy scenario would imply that deposits substituted into CBDC will not be replaced with central bank funding, leading to an immediate reduction in the private bank’s asset-side operations by the same amount. The assets exchanged in return for CBDC issuance would depend on what asset types would be accepted by the central bank and the assets available to banks. Within the stylised model framework of this paper, the ‘asset policy’ scenario would translate into purchases of capital securities by the central bank, visualised in Table 2, as banks only hold capital securities as asset. In practice, this scenario would be equivalent to central bank purchases of corporate bonds. Although still an unconventional monetary policy tool, corporate bond purchases have gained importance since the financial crisis and for instance have been applied to mitigate the impact of the coronavirus pandemic (e.g. see Cavallino et al. 2020). In the literature, CBDC issuance against corporate bonds has been modelled indirectly by Fernández-Villaverde et al. (2021) and directly by Schilling, Fernández-Villaverde, and Uhlig (2024), building on the seminal paper of Diamond and Dybvig (1983) in which the central bank invests into long-term projects (through an investment bank).

In practice, government bonds will also be a prominent assets class to be accepted for CBDC issuance under the ‘asset policy’ scenario. Government bonds represent the largest asset share in the balance sheet of many central banks. This feature is vividly illustrated by Pattipeilohy (2016) who developed a classification scheme of central banks by their balance sheet composition, in which ‘Treasury holder’ is one important type. According to their analysis, most of the major central banks fell into this category in 2015. In the literature, CBDC issuance against government bonds has been modelled by Kim and Kwon (2023), Barrdear and Kumhof (2022), and Kumhof and Noone (2021) who even argue that CBDC should only be issued against government bonds.

Both types of asset policy options, CBDC issuance against government bonds and corporate bonds would lead to a reduction in private banks’ balance sheets. However, the second round and equilibrium effects will likely differ and depend on the presence of friction in the (model) economy. Within the model framework, central bank purchases of capital securities affect the valuation of capital and thus the financial accelerator. The implications of government bond purchases would depend on monetary-fiscal interactions (such as government debt dynamics), the modelling of the government sector (such as the fiscal regime) and the incentives for banks to hold government debt.^[12] The model framework abstracts from these considerations on monetary-fiscal interactions and abstains from explicitly modelling the government sector to remain focused on financial stability considerations.^[13]

3.1 General Model Framework

There are two types of goods in the economy: a non-durable consumption good C _t and a durable capital good K _t . The durable capital good acts as a stylised proxy for the firm sector, is of fixed supply, normalised to unity, and does not depreciate. A unit of capital at time t, K _t , yields Z_t+1 units of consumption in the next period and can be sold at the price Q _t . The productivity of capital Z _t is subject to a multiplicative aggregate shock and described by

where Z is aggregate productivity in steady state and ϵ _z with E (ϵ _z ) = 0 is a shock to productivity.

As in GK15, there are households and bankers in the economy, each with a unit mass of one. Households consume the nondurable consumption good and save. In addition to (i) investing directly in capital and (ii) holding bank deposits, I include as third option the possibility that households can (iii) hold CBDC. Banks are more efficient in managing capital but are subject to a leverage constraint. They maximise profits by investing funds from households and the central bank (under CBDC with ‘credit policy’) into capital. Furthermore, I introduce a central bank into the model that issues CBDC to households and complements funds on the asset side either via ‘credit policy’ by granting loans to banks or via ‘asset policy’ by investing into capital securities.

3.2 Bank Failures in the Model Environment

The equilibrium outlined above is based on the mutual beliefs of households about the deposit decisions of other households, in particular that they will not participate in a bank run. Under certain conditions, a second equilibrium among the above outlined ‘banking equilibrium’ exists, in which the belief of a bank run triggers a bank run in equilibrium (‘bank run equilibrium’). The following section examines the conditions under which a bank run equilibrium exists and how it unfolds under the different CBDC policy scenarios. The analysis considers only systemic bank runs, where banks are identical and subject only to aggregate shocks. In addition, the productivity shock to the economy that opens up the possibility of a bank run is unanticipated and therefore bank runs are also unanticipated.^[21]

4.1 Calibration and Simulation

Table 4 lists all the parameter choices. Most of the parameters present in the ‘no CBDC’ scenario are taken from GK15, with the exception of the share of divertable assets θ = 0.22, which is updated to the value of Gertler, Kiyotaki, and Prestipino (2020). The introduction of a CBDC introduces additional parameters. Under CBDC issuance with ‘credit policy’, central bank credit to banks is, similar to deposit funding, also subject to a moral hazard constraint but only to half the extent and thus ω is set to 0.5 (e.g. as in Gertler and Karadi 2013). Similarly, under CBDC issuance with ‘asset policy’, the management cost of capital purchases α _cb that the central bank faces is set to half the efficiency costs that households face.

There is no empirical counterpart to guide the choice of values of the CBDC interest rate rule. The central bank supply of CBDC in steady state M (‘base issuance’) in set to equal 10 % of total household savings.^[26] As most central banks aim to design CBDC as a means of exchange rather than an attractive store of value (Group of Central Banks 2020), this is a relatively high and conservative value and in line with the cash-to-GDP ratio for 2021 for the US (9.2 %) and euro area (12.8 %) (Bank for International Settlements 2020).^[27]

The CBDC interest rate is determined by the joint impact of the CBDC demand and output coefficient. The output coefficient of the CBDC rule δ _Y set to 0.26, adopting the value of Minesso, Mehl, and Stracca (2022) who analyse the international implications of CBDC with a Taylor-type rule. It is important to differentiate that, in contrast to conventional monetary policy via a Taylor rule, in this model context an increase CBDC interest rate has a stabilising effect. By increasing the opportunity costs, it makes it less attractive for households to invest in less efficient capital securities which stabilises capital prices and hence output. The CBDC rule demand coefficient δ _M is set to 0.1 such that an increase in the demand for CBDC has a distinct but moderate effect on the CBDC interest rate. Finally, periods are calibrated to be equal to one quarter.

The simulation largely follows the computational procedure of Gertler and Kiyotaki (2015), outlined in more detail in the Online Appendix. The response to a shock without a bank run is computed as a shock to the nonlinear perfect foresight model. To simulate a shock followed by a bank run, the equilibrium path is split into three components and calculated starting backwards. First, the path of the economy after the bank run is calculated, starting from one period after the bank run. This is possible because the terminal values (i.e. steady state values) and the initial values of the backward-looking variables are known. Based on the post-bank run paths of the variables, the forward-looking bank run equilibrium can be calculated. Finally, until the period in which the bank run occurs, the equilibrium path is the same as under the shock without a bank run. While in principle a bank run can occur in any period, the period of the shock is the period most susceptible. Therefore, the following analysis focuses on bank runs emerging in the period of the shock and only last one period. Furthermore, the analysis restricts deposits, bank net worth, and bank consumption to be non-negative.

4.2 Steady State Implications of a CBDC

In steady state, the issuance of a CBDC does not affect aggregate quantities, only allocations. Table 5 compares the steady state values of the model under ‘credit policy’, ‘asset policy’, and the baseline scenario with ‘no CBDC’.^[28] Steady state output and most prices are the same across all three scenarios. Accordingly, steady state welfare, defined as aggregate consumption, is the same across all scenarios.^[29]

While total welfare is unchanged by CBDC, there is redistribution from banks to households. Household consumption slightly benefits from CBDC issuance and the banking sector – especially banks’ net worth and consumption – is strained by CBDC in steady state. However, the increase in household consumption by less than 1 % is marginal and originates from the lump-sum transfers of the central bank to households. While the portfolio allocation of household savings changes more substantially (Figure 1), this does not affect household wealth as equilibrium returns are equal across assets and prices remain unchanged. Per calibration target, CBDC holdings account for 10 % of total household savings. The share of capital investment remains relatively constant. Therefore, flows into CBDC originate entirely from deposits and reduce the percentage of deposit holdings by about 10 %-points.

Figure 1:

Households savings in steady state.

The consumption of bankers is lower due to lesser net worth and profitability of banks. Under CBDC with ‘asset policy’, the issuance of CBDC leads to a reduction in the size of the balance sheet, while under CBDC with ‘credit policy’ the lost deposit funding is replaced by central bank credit, leading to a balance sheet size that remains similar (Figure 2). Both policy scenarios strain profitability: while the issuance of CBDC under ‘credit policy’ reduces profitability through higher funding costs of central bank credit, the issuance of CBDC under ‘asset policy’ reduces net worth through a smaller balance sheet size. This smaller balance sheet, and hence the capital holdings of banks under CBDC with ‘asset policy’ is compensated by central bank capital holdings and the central bank takes over some of the capital investments which were previously held by banks (Figure 3). To summarise, both CBDC policy options do not change steady state aggregate output and welfare but only affect the distributions between households and banks.

Figure 2:

Bank funding in steady state.

Figure 3:

Capital allocation in steady state.

4.3 Financial Shocks Under a CBDC

The following section investigates the impact of a CBDC in response to an unanticipated 5 % shock to capital productivity that does not trigger a bank run. Figure 4 shows the response of key variables to the shock. Overall, the issuance of CBDC tends to reduce output losses, albeit only marginally. The slightly dampened impact on output can be explained by the stabilising effect of CBDC issuance on capital prices under both policy scenarios. Under CBDC with ‘credit policy’, capital prices are stabilised by refinancing bank funding with increased loans from the central bank. Under CBDC with asset policy capital prices are stabilised through increased capital holdings by the central bank that substitute for household capital holdings.

Figure 4:

No-bank-run response to a 5 % shock to capital productivity.

The simulated shock to capital quality tightens the bank’s leverage constraint, forcing a reduction in bank capital. Under CBDC issuance with ‘asset policy’, bank capital and deposits fall similarly strong but more persistently than in the ‘no CBDC’ scenario. In contrast to the ‘no CBDC’ scenario, households do not have to fully absorb the reduction in bank capital but can also hold CBDC which increases in response to the shock. To complement the inflow in CBDC holdings on the balance sheet, the central bank increases its investments in bank capital. Those activities alleviate household managements costs and stabilise the price of capital which leads to a smaller drop in bank net worth than in the alternative scenarios.

Under CBDC issuance with ‘credit policy’, the decline in bank capital is smaller than under the alternative scenarios. The decrease in deposits that is replaced by CBDC is channelled back to the bank in the form of central bank credit, mitigating the drop in bank capital and stabilising the price of capital. However, central bank credit is provided at higher funding costs and leads to a similar decline in bank net worth as in the ‘no CBDC’ scenario. Nevertheless, the optimal funding structure of banks still shifts towards central bank credit because the increase in the bank’s interest margin is higher for loans than for deposits. This also explains why the decline in deposits is steeper than in the alternative scenarios.

In all scenarios, the decline in household consumption is similar, being least severe under ‘credit policy’, while the decline in bankers’ consumption is least severe under ‘asset policy’. Considering household and bank consumption jointly, the aggregated and discounted welfare across periods in response to the shock is higher under the CBDC scenarios (‘asset policy’: 4.9566, ‘credit policy’: 4.9573) than in the ‘no CBDC’ scenario (4.9546). Taking stock, CBDC issuance tends to mitigate the response to capital quality shocks by stabilising asset prices but which comes at the cost of larger deposit outflows. Appendix A presents variations in the calibration of the less established parameters δ _Y and δ _M in the CBDC interest rate rule, the capital management costs of the central bank α _cb , as well as the leverage constraint for central bank credit to banks ω.

4.4 Bank Failures

Following on from Section 3.2 which distinguished between the different types of bank failures, this section investigates bank failures and the different equilibrium regions in the numerical example. Specifically, it addresses the questions: how large does a shock to the steady state have to be (i) for a self-fulfilling bank run equilibrium to emerge and enabling a bank failure due to illiquidity, and (ii) to cause a bank failure due to insolvency?

Figure 5 provides the answer to these questions by showing the equilibrium regions for the different policies in the context of the numerical example.^[30] In the baseline case without CBDC, self-fulfilling bank runs become possible when capital productivity shocks exceed 2.5 %. When capital shocks exceed 8.04 %, all deposits are withdrawn regardless of the beliefs of other households and banks become insolvent. Including CBDC issuance in the model framework improves financial stability by postponing both types of bank failure to larger shocks. Under CBDC issuance with ‘asset policy’, self-fulfilling runs become possible when capital productivity shocks exceed 8.57 % and banks become insolvent at shock sizes above 11.3 %. Under CBDC issuance with ‘credit policy’, the lender of last resort measures prevent self-fulfilling runs in the baseline calibration, but not in general. At the same time, under CBDC issuance with ‘credit policy’, banks already become insolvent at shock sizes above 8.26 %, a region similar to the insolvency threshold of no CBDC issuance. The following subsections will further dissect the mechanisms at play for the equilibrium regions in the different policy scenarios.

Figure 5:

Shock sizes to steady state levels that are needed i) for the emergence of a self-fulfilling run, leading to default due to illiquidity and ii) for triggering a fundamental bank run, leading to insolvency in the period of the shock.

4.4.1 Bank Failures Due to Illiquidity

4.4.1.1 Bank Runs Without a CBDC

Figure 6 compares the bank run and no bank run scenarios after a 5 % shock to capital productivity in the case without CBDC issuance. As long as households believe that other households will not lose faith and run on the bank, the banking equilibrium prevails and the shock has the dynamics as displayed in Figure 4 (black solid lines in Figure 6 and black dotted lines in Figure 4). However, if households believe that the shock prompts others to withdraw their deposits, it triggers a bank run in the period of the shock (grey dashed lines in Figure 6).

Figure 6:

‘No CBDC scenario’ – 5 % shock to productivity with and without a bank run in the period of the shock.

The bank run leads to a complete withdrawal of deposits, forcing the banks to liquidate all their capital. All banks fail as their net worth is wiped out during the run. Only households are left to purchase capital and as they are less efficient than banks, they face increasing marginal costs for managing capital. Therefore, capital is sold at a fire-sale price, a price much lower than it would have been in the absence of a run. Overall, the bank run leads to a sharp drop in output and consumption. In the period after the bank run, the banking sector starts to recapitalise. Due to the reduced capital productivity and price, banks are initially able to earn high profit margins, leading to high interest rates and a rapid recovery.

4.4.1.2 Bank Runs Under CBDC with Asset Policy

CBDC issuance with ‘asset policy’ requires a much larger shock to enable the possibility of self-fulfilling runs. Figure 7 compares the bank run (light red dashed lines) and no bank run (red solid lines) scenarios after a 10 % shock to capital productivity. In response to the bank run, the price of capital falls sharply, although in relative terms less than in the ‘no CBDC’ scenario. The dampened decline in capital prices translates into relatively less severe output losses and consumption losses compared to the ‘no CBDC’ case, but still more severe than the no bank run scenario.

Figure 7:

CBDC with asset policy scenario – 10 % shock to productivity with and without a bank run in the period of the shock.

The central bank’s asset purchases stabilise the fire-sale price of assets, thereby mitigating output losses in the bank run scenario. When withdrawing deposits from the bank, households can now hold both capital and CBDC. The additional option to hold CBDC makes the run less costly as households do not have to hold all their funds in the form of capital. Instead, some of the capital is also held by the central bank. Jointly holding the capital stock during the bank run stabilises the fire-sale price for two reasons: first, although the central bank is less efficient in managing capital than the bank, it is more efficient and faces lower capital management costs than households. Second, capital is held jointly by households and the central bank and, as capital management costs are convex, this further mitigates the overall efficiency losses of capital.

The stabilising influence of central bank asset purchases on the fire-sale price of capital is sufficiently large to fully cover deposit claims even in response to larger shocks and thereby postpones the occurrence of self-fulfilling bank runs. While a bank run under CBDC issuance with asset policy still leads to a welfare deterioration, the additional welfare loss is smaller which leads to a more resilient economy towards self-fulfilling runs.

4.4.1.3 Bank Runs Under CBDC with Credit Policy

The issuance of CBDC with ‘credit policy’ increases the threshold for self-fulfilling runs further. In the baseline calibration, there is no region in which the banking and the bank run equilibrium coexist. This result rests on the assumption that the central bank does not participate in self-fulfilling bank runs but only withdraws funding once the bank is insolvent.^[31]

There are two forces that jointly impede the occurrence of bank runs under CBDC with ‘credit policy’. First, in the event of a bank run, only a fraction of the bank’s liabilities are withdrawn, as both deposits and central bank credit are on the bank’s balance sheet. Therefore, the fire-sale price of assets needs to cover only the share of liabilities that are deposits as the central bank does not engage in the bank run. In this way, central bank credit provides the bank with a stable funding source which creates an additional buffer on the bank’s balance sheet. This additional central bank credit buffer results in a lower share of assets that would be withdrawn in a bank run scenario and makes the funding outflow easier to absorb. The larger the pre-run central bank credit holdings are, the higher the hurdle for the emergence of a self-fulfilling bank run. Second, instead of withdrawing funds, the central bank simultaneously provides the bank with additional liquidity in the amount of inflows into CBDC. The continued credit provision enables the bank to invest in capital even in case of a bank run. With the bank being more efficient in managing capital, this greatly stabilises the capital price. Acting as lender of last resort, the central bank provision of credit further impedes the emergence of bank runs. Those two forces break the ‘doom-loop’ of self-fulfilling runs in the baseline calibration.

How can banks under CBDC with ‘credit policy’ become insolvent without opening the possibility for self-fulfilling runs? As described above, the price threshold Q ¯ t * that triggers self-fulfilling banks is only conditional on the fraction of deposits Q ¯ t * = R t d D t − 1 K t − 1 b − Z t . In contrast, the price threshold Q ¯ t that triggers insolvency is easier to be reached because the central bank also stops funding in the event of insolvency: Q ¯ t = R t d D t − 1 K t − 1 b + R t l L t − 1 K t − 1 b − Z t . While Q _t and Q t * are the two equilibrium prices in the banking and bank run equilibrium, R t d D t − 1 K t − 1 b − Z t are the same in both scenarios. For this reason, under the baseline calibration, even on the brink of insolvency, the fire-sale value of assets would be sufficient to cover deposit claims and would prevent a self-fulfilling run.^[32] At the same time, the non-fire-sale value of assets will not be sufficient to cover all liabilities of the bank (i.e. deposits and central bank credit). Therefore, under the baseline calibration, the bank can fail due to insolvency without opening up the possibility of a self-fulfilling run.

Overall, by interceding as lender of last resort with the funds from CBDC, the central bank altogether eliminates the possibility of a run in the context of the numerical example. However, this does not preclude self-fulfilling bank runs under credit policy in the general setting. Self-fulfilling bank runs under CBDC with credit policy are possible under alternative calibrations.

Which of the two channels preventing the bank run under the baseline calibration is stronger and what are the implications of reducing both forces, individually and jointly? The first force works via the share of credit on the bank’s liabilities in the period before the bank run. Reducing δ₀ in the central bank’s CBDC rule to δ₀ = 1.01015 diminishes central bank credit in steady state to almost zero, with L_st.st. = 0.0005 (instead of L_st.st. = 0.0941 under the baseline calibration with δ₀ = 1.0194). Muting the first channel is not sufficient to enable self-fulfilling bank runs before the insolvency threshold. The second channel of providing central bank credit via CBDC inflows is sufficiently strong to still prevent the bank run from the outset. The second channel can be relaxed by increasing the reaction to CBDC demand in the CBDC rule to δ _M to disincentivise CBDC uptake. Setting δ _M  = 0.5 (instead of δ _M  = 0.1 under the baseline calibration), self-fulfilling bank runs become possible from shock sizes above 6.3 %. Relaxing both forces jointly, with δ₀ = 1.01015 and δ _M  = 0.5, enables self-fulfilling bank runs already from shock sizes above 2.9 %.

4.4.1.4 The Role of Friction for the Emergence of Self-Fulfilling Runs

Both types of CBDC issuance – with ‘asset policy’ and ‘credit policy’ – delay the emergence of the bank run equilibrium to larger shocks by reducing frictions during a bank run. This stabilises the fire-sale price of assets, mitigates welfare losses and makes banks runs more difficult to emerge. The contributions of the different model frictions to the emergence of bank runs are examined in more detail below.

The main friction that makes bank runs costly is the capital management cost of households α. Reducing the household capital management costs to almost zero with α = 0.001 (α = 0.008 in the baseline calibration) delays the emergence of self-fulfilling runs in the ‘no CBDC’ scenario to the brink of insolvency at shocks above 21.9 % (vs. 2.5 % under the baseline calibration). Conversely, increasing capital management costs to α = 0.015 in the scenario without CBDC enables a bank run equilibrium already in steady state in the absence of any shocks.

While additional frictions in the CBDC scenarios also play a role for the emergence of bank runs, they are not as central as household capital management costs α and the parameters of the CBDC reaction rule. In the scenario of CBDC issuance with ‘asset policy’, capital management costs of the central bank are introduced as additional friction. Reducing the central bank’s capital management costs to α _cb  = 0.001 (instead of α _cb  = 0.004 in the baseline calibration) delays the emergence of self-fulfilling runs from shocks above 8.6 % under the baseline calibration to shocks above 9.7 %. Conversely, setting central bank capital management costs to the capital management costs of the households with α _cb  = 0.008 reduces the shock threshold for the emergence of bank runs to shocks above 7.7 %. In the CBDC scenario with ‘credit policy’, the leverage constraint for central bank credit ω is introduced as an additional friction. Setting ω = 1 removes the leverage constraint but increases the costs of central bank credit which leads to the possibility of self-fulfilling bank runs at shocks above 2.1 %. Conversely, setting ω = 0 leads to the same leverage constraint but also to the same funding costs as on deposits. Similarly to the baseline calibration, this does not open up the possibility of self-fulfilling runs before the insolvency threshold.

Moreover, bank runs are not anticipated. How would the results change if bank runs were anticipated, for example as in Gertler and Kiyotaki (2015) or Gertler, Kiyotaki, and Prestipino (2016a)? Accounting for the possibility of bank failures would lead to a risk premium on deposits and increase bank funding costs. As the introduction of CBDC in the model framework makes bank runs less likely by delaying their occurrence to larger shocks, the risk premium on deposits would decrease in the presence of CBDC. Therefore, introducing anticipation of bank runs into the analysis would not materially change the insights of this analysis and results would be strengthened further. In addition, allowing for anticipation of bank failures, CBDC would be more favourable for the profitability of banks as the introduction of CBDC would reduce the risk premium and thus the deposit funding costs and therefore further mitigate financial fragility.

4.4.2 Bank Failures Due to Insolvency

Figure 8 displays the evolution of deposits to increasing shock sizes under the different policy scenarios. Banks become insolvent in response to a shock when the non-fire-sale value of assets becomes smaller than the liabilities. Thus, the conditions for a default due to illiquidity and default due to insolvency only differ in the price of capital for the ‘no CBDC’ and CBDC with ‘asset policy’ scenarios. For CBDC with ‘credit policy’, the conditions for bank failure due to illiquidity and insolvency also differ in the assets that are taken into account. While the central bank does not engage in self-fulfilling runs and the fire-sale value of assets only needs to cover deposits, the central bank credit claims are taken into account in the calculation of the insolvency conditions.

Figure 8:

Evolution of deposits in the period of the shock with increasing capital shock sizes under the different policy scenarios and without emergence of self-fulfilling runs.

The size of shocks that thrust banks into insolvency under CBDC with ‘credit policy’ is similar but a bit later than the scenario without CBDC. While in both scenarios the overall balance sheet size is similar, the stabilisation of capital prices under CBDC with ‘credit policy’ leads to a deferral of insolvency to slightly larger shock. In contrast, the shock size that triggers insolvency under CBDC with ‘asset policy’ is substantially larger. The higher insolvency threshold can be explained by the smaller bank balance sheet under CBDC with ‘asset policy’ and also by the stabilisation of the capital price.

To summarise, under both policy scenarios CBDC affects the equilibrium regions in a way that supports financial stability. Under CBDC with ‘asset policy’, the emergence of both bank run types is postponed to substantially larger shock. Under CBDC with ‘credit policy’, the possibility of self-fulfilling runs does not arise in the baseline calibration, while insolvency is triggered at a similar shock size as in the ‘no CBDC’ scenario.