Know your (holding) limits: CBDC, financial stability and central bank reliance

Appendix

A Detailed model description

Suppose that a person wants to swap euros from a bank account into digital euros. In theory, she might be able to go to an ATM and withdraw money from her bank account in the form of banknotes and then go to the Eurosystem and deposit those banknotes there to obtain digital euros. The bank’s balance sheet would decrease given that it has lost retail deposit (on the liability side) and the banknotes (on the asset side). The same happens if a bank (digitally) intermediates digital euro demand. Rather than taking out cash, that person would transfer money from her bank account to a digital euro account at the central bank. Again, the bank’s banknotes or reserves at the central bank would be reduced, unless the bank obtains banknotes/reserves from the interbank market or Eurosystem. Our model formalises this intuition in what follows.

Formally, we consider a model economy with many banks. Each bank re-optimises its balance sheet after an amount 𝐷𝐷 of its retail deposits has been converted to digital euros. The re-optimisation is done with the aim of maximising profits, consisting of interest income (𝐼𝐼𝐼𝐼𝐼𝐼) after deduction of interest expense (or cost of funding, 𝐼𝐼𝐶𝐶).

A bank can serve its depositors’ demand for CBDC by reducing its existing reserve holdings at the central bank, denoted by 𝑅𝑅, or by obtaining additional central bank liquidity, denoted by 𝐼𝐼𝐶𝐶. Additionally, a range of interbank funding instruments are considered, including short-term secured (𝑆𝑆𝑆𝑆𝑆𝑆) funding and medium-term secured

(𝑀𝑀𝑆𝑆𝑆𝑆) funding (e.g. repo funding); short-term unsecured (𝑆𝑆𝑆𝑆) funding (e.g. interbank lending); medium-term unsecured (𝑀𝑀𝑆𝑆) funding (e.g. commercial paper); and longterm debt (𝐿𝐿𝑆𝑆) funding (e.g. bonds that are either unsecured or secured by assets not considered to be HQLAs, such as certain types of mortgages). 𝑆𝑆𝑆𝑆 funding is considered to have a maturity less than one-month, 𝑀𝑀𝑆𝑆 funding has a maturity between one and twelve months, and 𝐿𝐿𝑆𝑆 funding has a maturity that exceeds twelve months. These differences in maturity matter for the purpose of the LCR and NSFR. Importantly, each interbank market transaction affects the balance sheet of the borrower and of the lender.

There are three types of central bank funding options included in the model. First, short-term, and long-term central bank funding obtained against HQLA collateral,

𝑆𝑆𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶 and 𝐿𝐿𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶 respectively. These options reflect traditional central bank lending

operations. Second, long-term central bank funding obtained against non-HQLA

central bank eligible collateral (𝐿𝐿𝑆𝑆𝐼𝐼𝐶𝐶). This option reflects unconventional types of central bank lending operations. Finally, as a last resort, we allow for any residual need for liquidity to be satisfied by a special form of central bank lending which is either unsecured or secured against collateral that is currently not eligible, (𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶).

Given the different funding options described above, the change in interest expense of bank i equals

?𝐼𝐼𝐶𝐶_𝑖𝑖 = 𝑟𝑟^𝐷𝐷 * 𝛥𝛥𝐷𝐷_𝑖𝑖 + 𝑟𝑟^{𝑆𝑆𝑆𝑆𝑆𝑆} * ?𝑆𝑆𝑆𝑆𝑆𝑆_𝑖𝑖 + 𝑟𝑟^{𝑀𝑀𝑆𝑆𝑆𝑆} * ?𝑀𝑀𝑆𝑆𝑆𝑆_𝑖𝑖 + 𝑟𝑟^{𝑆𝑆𝑆𝑆} * ?𝑆𝑆𝑆𝑆_𝑖𝑖 + 𝑟𝑟^{𝑀𝑀𝑆𝑆} * ?𝑀𝑀𝑆𝑆_𝑖𝑖 + 𝑟𝑟^{𝐿𝐿𝑆𝑆} * ?𝐿𝐿𝑆𝑆_𝑖𝑖

+ 𝑟𝑟𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 * ?𝑆𝑆𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶𝑖𝑖 + 𝑟𝑟𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 * ?𝐿𝐿𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶𝑖𝑖 + 𝑟𝑟𝐿𝐿 * ?𝐿𝐿𝑆𝑆𝐼𝐼𝐶𝐶𝑖𝑖 + 𝑟𝑟𝑆𝑆𝑆𝑆 * ?𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖

where the interest paid on each source of funding 𝑗𝑗 is denoted by 𝑟𝑟^𝑗𝑗. While we allow for the interest paid to differ between banks, the relative order of funding rates is assumed to be the same for all banks: 𝑟𝑟^{𝑆𝑆𝑆𝑆𝑆𝑆} < 𝑟𝑟^{𝑀𝑀𝑆𝑆𝑆𝑆} < 𝑟𝑟^{𝑆𝑆𝑆𝑆} < 𝑟𝑟^{𝑀𝑀𝑆𝑆} < 𝑟𝑟^{𝐿𝐿𝑆𝑆} < 𝑟𝑟^{𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆} <

𝑟𝑟𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 < 𝑟𝑟𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆. ^[20]

The change in the interest income is equal to the change in the return on reserve holdings and interbank lending instruments

?𝐼𝐼𝐼𝐼𝐼𝐼𝑖𝑖 = 𝑟𝑟𝑅𝑅 * ?𝑅𝑅𝑖𝑖𝑆𝑆𝑇𝑇𝑆𝑆 + 𝑟𝑟𝑆𝑆𝑆𝑆𝑆𝑆 * ?𝑆𝑆𝑆𝑆𝑆𝑆𝐿𝐿𝑖𝑖 + 𝑟𝑟𝑀𝑀𝑆𝑆𝑆𝑆 * ?𝑀𝑀𝑆𝑆𝑆𝑆𝐿𝐿𝑖𝑖 + 𝑟𝑟𝑆𝑆𝑆𝑆 * ?𝑆𝑆𝑆𝑆𝐿𝐿𝑖𝑖 + 𝑟𝑟𝑀𝑀𝑆𝑆 * ?𝑀𝑀𝑆𝑆𝐿𝐿𝑖𝑖

+ 𝑟𝑟^{𝐿𝐿𝑆𝑆} * ?𝐿𝐿𝑆𝑆𝐿𝐿_𝑖𝑖,

where 𝑟𝑟^𝑅𝑅 is the return on central bank reserves, ?𝑅𝑅^{𝑆𝑆𝑇𝑇𝑆𝑆} is the total change in central bank reserves, including reserves exchanged for digital euro (?𝑅𝑅_𝑖𝑖^{𝑇𝑇𝑂𝑂𝑂𝑂}) and those traded on the interbank market (?𝑅𝑅_𝑖𝑖^{𝐼𝐼𝑆𝑆}), equal to the sum of the various types of

interbank loans denoted by 𝑆𝑆𝑆𝑆𝑆𝑆𝐿𝐿, 𝑀𝑀𝑆𝑆𝑆𝑆𝐿𝐿, 𝑆𝑆𝑆𝑆𝐿𝐿, 𝑀𝑀𝑆𝑆𝐿𝐿, 𝐿𝐿𝑆𝑆𝐿𝐿. We assume that the rate on reserves provides a floor to market rates, or 𝑟𝑟^𝑅𝑅 < 𝑟𝑟^{𝑆𝑆𝑆𝑆𝑆𝑆}. Thus, it is profitable to lend excess reserves on the interbank market. We do not consider that in the mediumterm banks might change their loan portfolio in response to a change in funding structure.

We can now specify bank 𝑖𝑖’s re-optimisation problem. For a given outflow of retail deposits, all banks simultaneously choose their balance sheet adjustment, including their role on the interbank market, by solving

^{𝑂𝑂𝑂𝑂𝑂𝑂},𝑆𝑆𝑆𝑆𝑆𝑆,𝑀𝑀𝑆𝑆𝑆𝑆,𝑆𝑆𝑆𝑆,𝑀𝑀𝑆𝑆,𝐿𝐿𝑆𝑆,𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆,𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆max,𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆,𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆,𝑆𝑆𝑆𝑆𝑆𝑆𝐿𝐿,𝑀𝑀𝑆𝑆𝑆𝑆𝐿𝐿,𝑆𝑆𝑆𝑆𝐿𝐿,𝑀𝑀𝑆𝑆𝐿𝐿,𝐿𝐿𝑆𝑆𝐿𝐿 (?𝐼𝐼𝐼𝐼𝐼𝐼_𝑖𝑖 - ?𝐼𝐼𝐶𝐶_𝑖𝑖),

𝑅𝑅_𝑖𝑖

subject to the following constraints

?𝐷𝐷_𝑖𝑖 = ?𝑅𝑅_𝑖𝑖^{𝑇𝑇𝑂𝑂𝑂𝑂} - ( ?𝑆𝑆𝑆𝑆𝑆𝑆_𝑖𝑖 + ?𝑀𝑀𝑆𝑆𝑆𝑆_𝑖𝑖 + ?𝑆𝑆𝑆𝑆_𝑖𝑖 + ?𝑀𝑀𝑆𝑆_𝑖𝑖 + ?𝐿𝐿𝑆𝑆_𝑖𝑖 + ?𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖 + ?𝐿𝐿𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖 + ?𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖)(1)

𝑅𝑅 + ?𝑅𝑅_𝑖𝑖^{𝑇𝑇𝑂𝑂𝑂𝑂} + ?𝑅𝑅_𝑖𝑖^{𝐼𝐼𝑆𝑆} ? 0 (2)

?𝑆𝑆𝑆𝑆𝑆𝑆𝐿𝐿_𝑖𝑖 = ?𝑆𝑆𝑆𝑆𝑆𝑆_𝑖𝑖 (3a)

𝑖𝑖 𝑖𝑖

?𝑀𝑀𝑆𝑆𝑆𝑆𝐿𝐿_𝑖𝑖 = ?𝑀𝑀𝑆𝑆𝑆𝑆_𝑖𝑖 (3b)

𝑖𝑖 𝑖𝑖

?𝑆𝑆𝑆𝑆𝐿𝐿_𝑖𝑖 = ?𝑆𝑆𝑆𝑆_𝑖𝑖 (3c)

𝑖𝑖 𝑖𝑖

?𝑀𝑀𝑆𝑆𝐿𝐿_𝑖𝑖 = ?𝑀𝑀𝑆𝑆_𝑖𝑖 (3d)

𝑖𝑖 𝑖𝑖

?𝐿𝐿𝑆𝑆𝐿𝐿_𝑖𝑖 = ?𝐿𝐿𝑆𝑆_𝑖𝑖 (3e)

𝑖𝑖 𝑖𝑖

𝑐𝑐𝑗𝑗,𝑖𝑖

^{?𝑆𝑆𝑆𝑆𝑆𝑆}𝑖𝑖 ^{+ ?𝑀𝑀𝑆𝑆𝑆𝑆}𝑖𝑖 ^? h𝑎𝑎𝑖𝑖𝑟𝑟𝑐𝑐𝑎𝑎𝑎𝑎 𝑐𝑐_𝑗𝑗 ^, 𝑤𝑤h𝑒𝑒𝑟𝑟𝑒𝑒 𝑐𝑐_{𝑗𝑗,𝑖𝑖} = 𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎𝑒𝑒𝑟𝑟𝑎𝑎𝑐𝑐 𝑗𝑗 𝑎𝑎𝑎𝑎 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏 𝑖𝑖 (4)

𝑗𝑗 𝑐𝑐𝑗𝑗,𝑖𝑖

?𝑆𝑆𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖 + ?𝐿𝐿𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖 ? h𝑎𝑎𝑖𝑖𝑟𝑟𝑐𝑐𝑎𝑎𝑎𝑎 𝑐𝑐_𝑗𝑗 - (?𝑆𝑆𝑆𝑆𝑆𝑆_𝑖𝑖 + ?𝑀𝑀𝑆𝑆𝑆𝑆_𝑖𝑖),

𝑗𝑗

𝑤𝑤h𝑒𝑒𝑟𝑟𝑒𝑒 𝑐𝑐_{𝑗𝑗,𝑖𝑖} = 𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎𝑒𝑒𝑟𝑟𝑎𝑎𝑐𝑐 𝑗𝑗 𝑎𝑎𝑎𝑎 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏 𝑖𝑖 (5a)

𝑐𝑐𝑆𝑆𝑆𝑆𝑗𝑗,𝑖𝑖

?𝐿𝐿𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖 ? - (?𝑆𝑆𝑆𝑆𝑆𝑆_𝑖𝑖 + ?𝑀𝑀𝑆𝑆𝑆𝑆_𝑖𝑖 + ?𝑆𝑆𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖 + ?𝐿𝐿𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖), h𝑎𝑎𝑖𝑖𝑟𝑟𝑐𝑐𝑎𝑎𝑎𝑎 𝑐𝑐_𝑗𝑗

𝑗𝑗

𝑤𝑤h𝑒𝑒𝑟𝑟𝑒𝑒 𝑐𝑐_{𝑆𝑆𝑆𝑆𝑗𝑗,𝑖𝑖} = 𝐼𝐼𝐶𝐶 𝑒𝑒𝑐𝑐𝑖𝑖𝑒𝑒𝑖𝑖𝑏𝑏𝑐𝑐𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎𝑒𝑒𝑟𝑟𝑎𝑎𝑐𝑐 𝑗𝑗 𝑎𝑎𝑎𝑎 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏 𝑖𝑖 (5𝑏𝑏)

𝐿𝐿𝐼𝐼𝑅𝑅_𝑖𝑖 ? 100% + 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏𝑒𝑒𝑟𝑟_𝑖𝑖, (6)

𝐼𝐼𝑆𝑆𝐶𝐶𝑅𝑅_𝑖𝑖 ? 100% + 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏𝑒𝑒𝑟𝑟_𝑖𝑖. (7)

Constraint 1 means that each bank matches its deposit outflow with its own central bank reserves or those obtained on the interbank market.²¹ Constraint 2 ensures that the bank does not draw down or lend more central bank reserves than it owns (𝑅𝑅). Constraints 3a-3e capture the aggregate interbank market liquidity position.

Each interbank lending transaction requires a buying bank and selling bank. Constraint 4 reflects the fact that banks have a limited stock of HQLA collateral suited to obtaining secured market lending. The following unencumbered assets are included in the available stock of HQLAs, at market value and with asset-specific haircuts: government bonds, bonds issued by supra-national institutions, thirdcountry bonds, regional government bonds, corporate bonds, high-quality covered bonds and qualifying asset-backed securities (ABS). Constraints 5(a)-5(b) determine the type of central bank funding, given that each bank only has a limited stock of HQLA and non-HQLA central bank eligible collateral.

Constraints 6 and 7 ensure that the bank continues to meet its LCR and NSFR regulatory requirements, including any possible bank-specific voluntary liquidity buffer. These constraints are not only crucial determinants of a bank’s preferred funding option in case of deposit outflows, but also of a bank’s choice to act as a lender on the interbank market (as long as this does not increase their liquidity risk beyond their preferred levels) and of the choice which collateral to encumber first. The LCR and the NSFR are impacted by the following: 1) if assets are encumbered or sold, they do not qualify as unencumbered HQLAs and require more stable funding; 2) different forms of funding have different liquidity risks depending on their maturity and on whether or not they are secured; 3) the fact that interbank lending lowers reserves and may increase required stable funding and/or expected outflow.

The new LCR, after balance-sheet re-optimisation, is calculated as

𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻 _𝚤𝚤 + ?𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻_𝑖𝑖

𝐿𝐿𝐼𝐼𝑅𝑅^𝑖𝑖 = 𝐸𝐸[𝑐𝑐𝑎𝑎𝑎𝑎𝑏𝑏𝑐𝑐𝑐𝑐𝑤𝑤 ]_𝑖𝑖 + ?𝐸𝐸[𝑐𝑐𝑎𝑎𝑎𝑎𝑏𝑏𝑐𝑐𝑐𝑐𝑤𝑤]_𝑖𝑖

where the variables with a circumflex represent the initial stock of unencumbered

𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻s and the initial expected outflow. Once assets are encumbered, they no

²¹ Note that ?𝐷𝐷_𝑖𝑖,?𝑅𝑅 < 0 while ?𝑆𝑆𝑆𝑆𝑆𝑆_𝑖𝑖, ?𝑀𝑀𝑆𝑆𝑆𝑆_𝑖𝑖, ?𝑆𝑆𝑆𝑆_𝑖𝑖, ?𝑀𝑀𝑆𝑆_𝑖𝑖, ?𝐿𝐿𝑆𝑆_𝑖𝑖, ?𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖, ?𝐿𝐿𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖, ?𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖 > 0.

longer qualify as unencumbered HQLAs. Reserves are considered to be HQLAs, consequently using or lending reserves on the interbank market lowers the stock of unencumbered HQLAs. Thus, the change in 𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻s is given by

?𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻_𝑖𝑖 = 𝛥𝛥𝑅𝑅_𝑖𝑖^{𝑆𝑆𝑇𝑇𝑆𝑆} - 𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻 𝑎𝑎𝑎𝑎𝑗𝑗𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑒𝑒𝑏𝑏𝑎𝑎 𝑎𝑎𝑎𝑎𝑒𝑒 𝑎𝑎𝑐𝑐 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏 𝑖𝑖^'𝑎𝑎 𝑎𝑎𝑒𝑒𝑐𝑐𝑎𝑎𝑟𝑟𝑒𝑒𝑎𝑎 𝑏𝑏𝑐𝑐𝑟𝑟𝑟𝑟𝑐𝑐𝑤𝑤𝑖𝑖𝑏𝑏𝑒𝑒

+ 𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻 𝑎𝑎𝑎𝑎𝑗𝑗𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑒𝑒𝑏𝑏𝑎𝑎 𝑎𝑎𝑎𝑎𝑒𝑒 𝑎𝑎𝑐𝑐 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏 𝑖𝑖^'𝑎𝑎 𝑐𝑐𝑒𝑒𝑏𝑏𝑎𝑎𝑖𝑖𝑏𝑏𝑒𝑒,

𝑤𝑤h𝑒𝑒𝑟𝑟𝑒𝑒 𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻 𝑎𝑎𝑎𝑎𝑗𝑗𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑒𝑒𝑏𝑏𝑎𝑎 𝑎𝑎𝑎𝑎𝑒𝑒 𝑎𝑎𝑐𝑐 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏 𝑖𝑖^'𝑎𝑎 𝑎𝑎𝑒𝑒𝑐𝑐𝑎𝑎𝑟𝑟𝑒𝑒𝑎𝑎 𝑏𝑏𝑐𝑐𝑟𝑟𝑟𝑟𝑐𝑐𝑤𝑤𝑖𝑖𝑏𝑏𝑒𝑒

= ?𝑆𝑆𝑆𝑆𝑆𝑆_𝑖𝑖 + ?𝑀𝑀𝑆𝑆𝑆𝑆_𝑖𝑖 + ?𝑆𝑆𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖 + ?𝐿𝐿𝑆𝑆𝑆𝑆𝐼𝐼𝐶𝐶_𝑖𝑖

𝑎𝑎𝑏𝑏𝑎𝑎 𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻 𝑎𝑎𝑎𝑎𝑗𝑗𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑒𝑒𝑏𝑏𝑎𝑎 𝑎𝑎𝑎𝑎𝑒𝑒 𝑎𝑎𝑐𝑐 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏 𝑖𝑖^'𝑎𝑎 𝑐𝑐𝑒𝑒𝑏𝑏𝑎𝑎𝑖𝑖𝑏𝑏𝑒𝑒 = ?𝑆𝑆𝑆𝑆𝑆𝑆𝐿𝐿_𝑖𝑖 + ?𝑀𝑀𝑆𝑆𝑆𝑆𝐿𝐿_𝑖𝑖 + 𝛼𝛼 ?𝐿𝐿𝑆𝑆𝐿𝐿_𝑖𝑖

The first equality uses the assumption that the haircuts imposed by the market are the same as those imposed by the ECB collateral framework, which are the same as the LCR haircut for the assets under consideration. It should be noted that each collateral asset has a different haircut. The second equality reflects that in repo transactions the HQLA value of the lending bank is unaffected while a long term secured loan to a financial institutions counts as HQLA with a haircut 𝛼𝛼, equal to 15% in the euro area.

Additionally, the expected outflow changes given that i) it decreases by the contribution to the expected outflowof the withdrawn retail deposits, and ii) it increases by the expected outflow of the newly obtained funding instruments:

?𝐸𝐸[𝑐𝑐𝑎𝑎𝑎𝑎𝑏𝑏𝑐𝑐𝑐𝑐𝑤𝑤]_𝑖𝑖 = 𝛥𝛥𝐷𝐷_𝑖𝑖 * 𝑟𝑟𝑎𝑎𝑏𝑏𝑐𝑐𝑏𝑏𝑏𝑏 𝑟𝑟𝑎𝑎𝑎𝑎𝑒𝑒 𝐷𝐷_𝑖𝑖

+ (𝛥𝛥𝑏𝑏_𝑖𝑖 * 𝑟𝑟𝑎𝑎𝑏𝑏𝑐𝑐𝑏𝑏𝑏𝑏 𝑟𝑟𝑎𝑎𝑎𝑎𝑒𝑒 𝑏𝑏_𝑖𝑖)

𝑓𝑓𝑖𝑖=𝑆𝑆𝑆𝑆𝑆𝑆,𝑀𝑀𝑆𝑆𝑆𝑆,𝑆𝑆𝑆𝑆,𝑀𝑀𝑆𝑆,𝐿𝐿𝑆𝑆,𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆,𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆,𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆,𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆

Second, the new NSFR, after balance-sheet re-optimisation, is calculated as

𝐻𝐻𝑆𝑆𝐶𝐶_𝚤𝚤 + ?𝐻𝐻𝑆𝑆𝐶𝐶_𝑖𝑖

𝐼𝐼𝑆𝑆𝐶𝐶𝑅𝑅𝑖𝑖 = 𝑅𝑅𝑆𝑆𝐶𝐶𝚤𝚤 + ?𝑅𝑅𝑆𝑆𝐶𝐶𝑖𝑖

where the variables with a circumflex represent the initial stock of available stable funding (𝐻𝐻𝑆𝑆𝐶𝐶) and the initial stock of required stable funding (𝑅𝑅𝑆𝑆𝐶𝐶).

The available stable funding changes given that i) it decreases by the contribution to the initial 𝐻𝐻𝑆𝑆𝐶𝐶 of the withdrawn retail deposits, and ii) it increases by the contribution of the newly obtained funding sources, or

?𝐻𝐻𝑆𝑆𝐶𝐶_𝑖𝑖 = 𝛥𝛥𝐷𝐷_𝑖𝑖 * 𝐻𝐻𝑆𝑆𝐶𝐶 𝑏𝑏𝑎𝑎𝑐𝑐𝑎𝑎𝑐𝑐𝑟𝑟 𝐷𝐷_𝑖𝑖 + (𝑏𝑏_𝑖𝑖 * 𝐻𝐻𝑆𝑆𝐶𝐶 𝑏𝑏𝑎𝑎𝑐𝑐𝑎𝑎𝑐𝑐𝑟𝑟 𝑏𝑏_𝑖𝑖)

𝑓𝑓𝑖𝑖=𝑆𝑆𝑆𝑆𝑆𝑆,𝑀𝑀𝑆𝑆𝑆𝑆,𝑆𝑆𝑆𝑆,𝑀𝑀𝑆𝑆,𝐿𝐿𝑆𝑆,𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆,𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆,𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆,𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆

The 𝑅𝑅𝑆𝑆𝐶𝐶 increases given that i) encumbered assets obtain a RSF factor of 100%, while unencumbered assets have asset-specific RSF factors, and ii) loans to financial institutions may have a positive RSF factor, while reserves do not

?𝑅𝑅𝑆𝑆𝐶𝐶_𝑖𝑖 = ?𝑅𝑅𝑆𝑆𝐶𝐶 𝑏𝑏𝑎𝑎𝑐𝑐𝑎𝑎𝑐𝑐𝑟𝑟_𝑗𝑗 𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎𝑒𝑒𝑟𝑟𝑎𝑎𝑐𝑐 𝑗𝑗 𝑐𝑐𝑏𝑏 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏 𝑖𝑖 𝑒𝑒𝑏𝑏𝑐𝑐𝑎𝑎𝑎𝑎𝑏𝑏𝑒𝑒𝑟𝑟𝑒𝑒𝑎𝑎 𝑤𝑤h𝑒𝑒𝑏𝑏 𝑏𝑏𝑐𝑐𝑟𝑟𝑟𝑟𝑐𝑐𝑤𝑤𝑖𝑖𝑏𝑏𝑒𝑒

𝑗𝑗

+ (𝑐𝑐_𝑖𝑖 * 𝑅𝑅𝑆𝑆𝐶𝐶 𝑏𝑏𝑎𝑎𝑐𝑐𝑎𝑎𝑐𝑐𝑟𝑟 𝑐𝑐_𝑖𝑖),

𝑙𝑙𝑖𝑖=𝑆𝑆𝑆𝑆𝑆𝑆𝐿𝐿,𝑀𝑀𝑆𝑆𝑆𝑆𝐿𝐿,𝑆𝑆𝑆𝑆𝐿𝐿,𝑀𝑀𝑆𝑆𝐿𝐿,𝐿𝐿𝑆𝑆𝐿𝐿

𝑤𝑤h𝑒𝑒𝑟𝑟𝑒𝑒 ?𝑅𝑅𝑆𝑆𝐶𝐶 𝑏𝑏𝑎𝑎𝑐𝑐𝑎𝑎𝑐𝑐𝑟𝑟_𝑗𝑗 = 𝑅𝑅𝑆𝑆𝐶𝐶 𝑏𝑏𝑎𝑎𝑐𝑐𝑎𝑎𝑐𝑐𝑟𝑟 𝑎𝑎𝑏𝑏𝑒𝑒𝑏𝑏𝑐𝑐𝑎𝑎𝑎𝑎𝑏𝑏𝑒𝑒𝑟𝑟𝑒𝑒𝑎𝑎 𝑐𝑐_{𝑖𝑖,𝑗𝑗} - 𝑅𝑅𝑆𝑆𝐶𝐶 𝑏𝑏𝑎𝑎𝑐𝑐𝑎𝑎𝑐𝑐𝑟𝑟 𝑒𝑒𝑏𝑏𝑐𝑐𝑎𝑎𝑎𝑎𝑏𝑏𝑒𝑒𝑟𝑟𝑒𝑒𝑎𝑎 𝑐𝑐_{𝑖𝑖,𝑗𝑗}

𝑎𝑎𝑏𝑏𝑎𝑎 𝑎𝑎h𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎𝑒𝑒𝑟𝑟𝑎𝑎𝑐𝑐 𝑐𝑐𝑏𝑏 𝑏𝑏𝑎𝑎𝑏𝑏𝑏𝑏 𝑖𝑖 𝑒𝑒𝑏𝑏𝑐𝑐𝑎𝑎𝑎𝑎𝑏𝑏𝑒𝑒𝑟𝑟𝑒𝑒𝑎𝑎 𝑤𝑤h𝑒𝑒𝑏𝑏 𝑏𝑏𝑐𝑐𝑟𝑟𝑟𝑟𝑐𝑐𝑤𝑤𝑖𝑖𝑏𝑏𝑒𝑒

1

= 𝑥𝑥 𝑒𝑒𝑏𝑏𝑐𝑐𝑎𝑎𝑎𝑎𝑏𝑏𝑒𝑒𝑟𝑟𝑒𝑒𝑎𝑎 𝑐𝑐_{𝑖𝑖,𝑗𝑗}

h𝑎𝑎𝑖𝑖𝑟𝑟𝑐𝑐𝑎𝑎𝑎𝑎 𝑐𝑐_𝑗𝑗

Discussion of key assumptions

There are several assumptions required to operationalise our analysis. First, we assume that each bank experiences a range of outflows of retail deposits, set as a percentage of its total retail deposits. Since banks experience the same percentage of outflows, the absolute amounts differ.

Second, we assume an order of relative prices. In normal times, short-term wholesale funding rates tend to be above the deposit facility rate (DFR) and below the main refinancing operations (MRO) rate. Bank bond issuance (or long-term nonHQLA secured market funding) is the most expensive option. It is less clear where the medium-term wholesale funding would sit. As spreads are very compressed currently, we assume that they are more expensive than short-term wholesale funding but less expensive than MROs. We assume that central bank funding could be obtained against HQLA collateral, as well as non-HQLA collateral that is included in the additional credit claim (ACC) framework, but also non-HQLA collateral that is excluded from that framework. The latter is included with the aim of modelling the potential need for unconventional monetary policy operations when the digital euro is introduced. It is only available at a penalty rate and is only used if a bank is unable to obtain liquidity otherwise.

Third, for simplicity we have not explicitly included the option for banks to sell assets to other banks or to the central bank, which could happen, for instance, when there is a quantitative easing programme. Selling an asset has almost the same impact on liquidity risk as using the asset as collateral in secured borrowing with a maturity of more than one year. The only difference between the two options is the haircut that is applied to the asset if it is not sold but encumbered, but we expect this to have a minor impact and therefore not alter our conclusions. Selling assets would, however, have an impact on bank funding structures. Furthermore, we do not take into consideration any changes in bank loan portfolios.

B Additional results

Chart B.1

Required central bank funding by type during a digital euro introduction and with a liquid interbank market for eligible non-HQLA collateral

Notes: The shaded area represents the possible share of deposit outflows in the event of a €3,000 holding limit.

Chart B.2

Changes in central bank reliance, aggregated over the different business models

Notes: The shaded area represents the possible outflows in the event of a €3,000 holding limit.

Chart B.3

Net interbank market funding position

Notes: The shaded area represents the possible outflows in the event of a €3,000 holding limit.

Chart B.4

Significant changes in wholesale funding ratios in less extreme cases

Notes: The historical increases are those since 2016. The shaded area represents the possible outflows in the event of a €3,000 holding limit.

Table B.1

Country-specific ability to accommodate deposit outflows

Acknowledgements

We would like to thank Mitsutoshi Adachi, Ramon Adalid, Daniel Dieckelmann, Anton van der Kraaij, Claudia Lambert, Marco Marrazzo,

Manuel Munoz, Cosimo Pancaro, Antonella Pellicani, Andrea Pinna, Petya Radulova, Costanza Rodriguez D´Acri, Stephan Sauer, Tamarah Shakir, Gabriela Šílová, Jens Tapking, José R. Martinez Resano and Nicola Branzoli for their thought-provoking comments, critical discussions and instructive insights. We are also especially grateful to Petya Radulova also for the data support provided. The views expressed in this paper are those of the authors and do not necessarily reflect the views of the ECB or the Eurosystem. We are fully responsible for any remaining errors or omissions. The preliminary and partial results of our analysis featured in Adalid et al. (2022).

Barbara Meller

European Central Bank, Frankfurt am Main, Germany; email: barbara.meller@ecb.europa.eu

Oscar Soons

De Nederlandsche Bank, Amsterdam, The Netherlands; email: o.c.soons@dnb.nl

Související témata