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Journal of Applied Mathematics
Volume 2012, Article ID 743656, 27 pages
Research Article

Bank Liquidity and the Global Financial Crisis

1Department of Mathematics, Faculty of Science, University of Namibia, Private Bag 13301, Windhoek 9000, Namibia
2Research Division, Faculty of Commerce and Administration, North-West University, Private Bag x2046, Mmabatho 2735, South Africa
3Economics Division, Faculty of Commerce and Administration, North-West University, Private Bag x2046, Mmabatho 2735, South Africa

Received 2 November 2011; Revised 22 January 2012; Accepted 5 February 2012

Academic Editor: Chuanhou Gao

Copyright © 2012 Frednard Gideon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate the stochastic dynamics of bank liquidity parameters such as liquid assets and nett cash outflow in relation to the global financial crisis. These parameters enable us to determine the liquidity coverage ratio that is one of the metrics used in ratio analysis to measure bank liquidity. In this regard, numerical results show that bank behavior related to liquidity was highly procyclical during the financial crisis. We also consider a theoretical-quantitative approach to bank liquidity provisioning. In this case, we provide an explicit expression for the aggregate liquidity risk when a locally risk-minimizing strategy is utilized.

1. Introduction

During the global financial crisis (GFC), banks were under severe pressure to maintain adequate liquidity. In general, empirical evidence shows that banks with sufficient liquidity can meet their payment obligations while banks with low liquidity cannot. The GFC highlighted the fact that liquidity risk can proliferate quickly with funding sources dissipating and concerns about asset valuation and capital adequacy realizing. This situation underscores the important relationship between funding risk (involving raising funds to bankroll asset holdings) and market liquidity (involving the efficient conversion of assets into liquid funds at a given price). In response to this, the Basel Committee on Banking Supervision (BCBS) is attempting to develop an international framework for liquidity risk measurement, standards, and monitoring (see, e.g., [1]). Although pre-Basel III regulation establishes procedures for assessing credit, market, and operational risk, it does not provide effective protocols for managing liquidity and systemic risks. The drafting of Basel III represents an effort to address the latter (see, e.g., [24]).

Current liquidity risk management procedures can be classified as micro- or macroprudential. In the case of the former, simple liquidity ratios such as credit-to-deposit ratios (nett stable funding ratios), liquidity coverage ratios and the assessment of the gap between short-term liabilities and assets are appropriate to cover the objectives of bank balance sheet analysis. The ratio approach for liquidity risk management is a quantitative international accepted standard for alerting banks about any possible adverse economic downturns. For instance, the credit-to-deposit ratio assesses the relationships between sources and uses of funds held in the bank’s portfolio but has limitations which ultimately do not reflect information on market financing with short-term maturity. By contrast, the liquidity coverage ratio (LCR) performs better by ensuring the coverage of some of the immediate liabilities. Since the LCR depends only on bank balance sheet data, it does not take into account the residual maturities on various uses and sources of funds. Also, in a global context, a quantitative approach may not take financial market conditions into account. In this case, a more comprehensive characterization of the bank system’s liquidity risk through designed stress testing and constructed contingency plans is considered. The Basel Committee on Banking Supervision suggested best practices related to international liquidity standards. In this case, a well-designed policy monitoring instrument to measure and regulate the dynamics of foreign currency is considered to best take financial market conditions into account. Also, central banks (CBs) have a pivotal role to play in managing liquidity inflows via macroeconomic management of exchange rate and interest rate responses. The modeling of capital markets as well as stock and bond behavior also contribute to the liquidity response for possible stress conditions observed. The above approaches for liquidity analysis take into account the macroprudential liquidity management of banks.

In this paper, in Section 2, we discuss balance sheet items related to liquid assets and nett cash outflow in order to build a stochastic LCR model. Before the GFC, banks were prosperous with high LCRs, high cash inflows, low interest rates, and low nett cash outflows. This was followed by the collapse of liquidity, exploding default rates, and the effects thereof during the GFC. Next, in Section 3, we apply a dynamic provisioning strategy to liquidity risk management. In this case, we address the problem of dynamic liquidity provisioning for a mortgage, Λ, which is an underlying (illiquid) nonmarketable asset, by substituting (liquid) marketable securities, 𝑆. In the light of the above, banks prefer to trade in a Treasury bond market because of liquidity reasons. Since the loan process (Λ𝑡)0𝑡𝑇 is not completely correlated with the substitute, it creates the market incompleteness. In other words, we will employ non-self-financing strategy to replicate the trading process. Therefore the banks would require that the uncertainty involved over the remaining of the trading period be minimized. In this case, we specifically minimize at each date, the uncertainty over the next infinitesimal period. In the dynamics trading there is always a residual risk emanating from the imperfection of the correlation between the Brownian motions. Due to the no-arbitrage opportunities there are infinitely many equivalent martingale measures so that pricing is directly linked to risk. Therefore, we choose a pricing candidate (equivalent martingale measure) under which the discounted stock price follows a martingale. This equivalent measure is chosen according to a provisioning strategy which ensures that the value of Λ is the value of the replicating portfolio. We also provide a framework for assessing residual aggregate liquidity risk stemming from the application of the above strategy.

1.1. Literature Review

The documents formulated in response to the proposed Basel III regulatory framework are among the most topical literature on bank liquidity (see, e.g., [1]). During the GFC, unprecedented levels of liquidity support were required from CBs in order to sustain the financial system and even with such extensive support a number of banks failed, were forced into mergers, or required resolution. The crisis illustrated how quickly and severely liquidity risks can crystallize and certain sources of funding can evaporate, compounding concerns related to the valuation of assets and capital adequacy (see, e.g., [24]). A key characteristic of the GFC was the inaccurate and ineffective management of liquidity risk. In recognition of the need for banks to improve their liquidity risk management and control their exposures to such risk, the BCBS issued Principles for Sound Liquidity Risk Management and Supervision in September 2008 (see, e.g., [1]). Supervisors are expected to assess both the adequacy of a bank’s liquidity risk management framework and its liquidity risk exposure. In addition, they are required to take prompt action to address the banks risk management deficiencies or excess exposure in order to protect depositors and enhance the overall stability of the financial system. To reinforce these supervisory objectives and efforts, the BCBS has recently focussed on further elevating the resilience of internationally active banks to liquidity stresses across the globe, as well as increasing international harmonization of liquidity risk supervision (see, e.g., [1]). The BCBS hopes to develop internationally consistent regulatory standards for liquidity risk supervision as a cornerstone of a global framework to strengthen liquidity risk management and supervision (see, e.g., [24]).

In [5] it is asserted that bank liquidity behavior can be described by straightforward indicators constructed from firm-specific balance sheet data (see, also, [6, 7]). Also, their analysis underscores the relevance of using several indicators of liquidity risk at the same time, given the different leads and lags of the measures with systemic risk. Our study is related to theirs in that we make use of balance sheet items to determine bank behavior. Another similarity is that we make use of data from [6] to formulate conclusions in a numerical quantitative framework (compare with the analysis in Section 3 below).

The contribution [8] studies the role of securitization in bank management. A new index of “bank loan portfolio liquidity” which can be thought of as a weighted average of the potential to securitize loans of a given type, where the weights reflect the composition of a bank loan portfolio. The paper uses this new index to show that by allowing banks to convert illiquid loans into liquid funds, securitization reduces banks holdings of liquid securities and increases their lending ability. Furthermore, securitization provides banks with an additional source of funding and makes bank lending less sensitive to cost of funds shocks. By extension, the securitization weakens the ability of regulators to affect banks lending activity but makes banks more susceptible to liquidity and funding crisis when the securitization market is shutdown. We conduct a similar analysis in Section 4 of this paper where illiquid underlying loans are substituted by liquid marketable securities.

In [9], we use actuarial methods to solve a nonlinear stochastic optimal liquidity risk management problem for subprime originators with deposit inflow rates and marketable securities allocation as controls (see [10]). The main objective is to minimize liquidity risk in the form of funding and credit crunch risk in an incomplete market. In order to accomplish this, we construct a stochastic model that incorporates originator mortgage and deposit reference processes. Finally, numerical examples that illustrate the main modeling and optimization features of the paper are provided. Our work in this paper also has a connection with [9] in that the nexus between funding risk and market liquidity is explored. However, this paper is an improvement on the aforementioned in that bank balance sheet features play a more prominent role (see, Sections 2, 3, and 4).

1.2. Main Questions and Article Outline

In this subsection, we pose the main questions and provide an outline of the paper.

1.2.1. Main Questions

In this paper on bank liquidity, we answer the following questions.

Question 1 (banking model). Can we model banks’ liquid assets and nett cash outflows as well as LCRs in a stochastic framework? (compare with Section 2).

Question 2 (bank liquidity in a numerical quantitative framework). Can we explain and provide numerical examples of bank liquidity dynamics? (refer to Section 3).

Question 3 (bank liquidity in a theoretical quantitative framework). Can we devise a liquidity provisioning strategy in a theoretical quantitative framework? (compare with Section 4).

1.2.2. Paper Outline

The rest of the paper is organized as follows. Section 1 introduces the concept of liquidity risk while providing an appropriate literature review. A stochastic LCR model for bank liquidity is constructed in Section 2. Issues pertaining to bank liquidity in a numerical quantitative framework are discussed in Section 3. Section 4 treats liquidity in a theoretical quantitative manner. Finally, we provide concluding remarks in Section 5.

2. Bank Liquidity Model

In the sequel, we use the notational convention “subscript 𝑡 or 𝑠” to represent (possibly) random processes, while “bracket 𝑡 or 𝑠” is used to denote deterministic processes. The assessment of a bank’s relative composition of the stock of high-quality liquid assets (liquid assets) and nett cash outflows, is one of the primary ways of analyzing its liquidity position. In this regard, we consider a measure of liquidity offered by the LCR. Before the GFC, banks were prosperous with high LCRs, high cash inflows, low interest rates, and low nett cash outflows. This was followed by the collapse of liquidity, exploding default rates, and the effects thereof. We make the following assumption to set the space and time index that we consider in our LCR model.

Assumption 2.1 (filtered probability space and time index). Throughout, we assume that we are working with a filtered probability space  (Ω,,)  with filtration  {𝑡}𝑡0  on a time index set  [0,𝑇].  We assume that the aforementioned space satisfies the usual conditions. Under  ,  {𝑊𝑡;0𝑡𝑇,𝑊0=0}  is an  𝑡-Brownian motion.

Furthermore, we are able to produce a system of stochastic differential equations that provide information about the stock of high-quality liquid assets (liquid assets) at time 𝑡 with 𝑥1Ω×[0,𝑇]+ denoted by 𝑥1𝑡 and nett cash outflows at time 𝑡 with 𝑥2Ω×[0,𝑇]+ denoted by 𝑥2𝑡 and their relationship. The dynamics of liquid assets, 𝑥1𝑡, is stochastic in nature because it depends in part on the stochastic rates of return on assets and cash inflow and outflow (see [9] for more details) and the securitization market. Also, the dynamics of the nett cash outflow, 𝑥2𝑡, is stochastic because its value has a reliance on cash inflows as well as liquidity and market risk that have randomness associated with them. Furthermore, for 𝑥Ω×[0,𝑇]2 we use the notation 𝑥𝑡 to denote 𝑥𝑡=𝑥1𝑡𝑥2𝑡,(2.1) and represent the LCR with 𝑙Ω×[0,𝑇]+ by 𝑙𝑡=𝑥1𝑡𝑥2𝑡.(2.2) It is important for banks that 𝑙𝑡 in (2.2) has to be sufficiently high to ensure high bank liquidity.

2.1. Liquid Assets

In this section, we discuss the stock of high-quality liquid assets constituted by cash, CB reserves, marketable securities, and government/CB bank debt issued.

2.1.1. Description of Liquid Assets

The first component of stock of high-quality liquid assets is cash that is made up of banknotes and coins. According to [1], a CB reserve should be able to be drawn down in times of stress. In this regard, local supervisors should discuss and agree with the relevant CB the extent to which CB reserves should count toward the stock of liquid assets.

Marketable securities represent claims on or claims guaranteed by sovereigns, CBs, noncentral government public sector entities (PSEs), the Bank for International Settlements (BIS), the International Monetary Fund (IMF), the European Commission (EC), or multilateral development banks. This is conditional on all the following criteria being met. These claims are assigned a 0% risk weight under the Basel II standardized approach. Also, deep repo-markets should exist for these securities and that they are not issued by banks or other financial service entities.

Another category of stock of high-quality liquid assets refers to government/CB bank debt issued in domestic currencies by the country in which the liquidity risk is being taken by the bank’s home country (see, e.g., [1, 4]).

2.1.2. Dynamics of Liquid Assets

In this section, we consider 𝑑𝑡=𝑟𝑡𝑑𝑡+𝜎𝑡𝑑𝑊𝑡𝑡,0=0,(2.3) where the stochastic processes Ω×[0,𝑇]+ are the return per unit of liquid assets, 𝑟+ is the rate of return per liquid asset unit, the scalar 𝜎𝑇 is the volatility in the rate of returns, and 𝑊Ω×[0,𝑇] is standard Brownian motion. Before the GFC, risky asset returns were much higher than those of riskless assets, making the former a more attractive but much riskier investment. It is inefficient for banks to invest all in risky or riskless securities with asset allocation being important. In this regard, it is necessary to make the following assumption to distinguish between risky (e.g., marketable securities and government/CB bank debt) and riskless assets (cash) for future computations.

Assumption 2.2 (liquid assets). Suppose from the outset that liquid assets are held in the financial market with  𝑛+1  asset classes. One of these assets is riskless (cash) while the assets  1,2,,𝑛  are risky.

The risky liquid assets evolve continuously in time and are modelled using an 𝑛-dimensional Brownian motion. In this multidimensional context, the asset returns in the 𝑘th liquid asset class per unit of the 𝑘th class is denoted by 𝑦𝑘𝑡,𝑘𝑛={0,1,2,,𝑛} where 𝑦Ω×[0,𝑇]𝑛+1. Thus, the return per liquid asset unit is 𝑦=𝐶(𝑡),𝑦1𝑡,,𝑦𝑛𝑡,(2.4) where 𝐶(𝑡) represents the return on cash and 𝑦1𝑡,,𝑦𝑛𝑡 represents the risky return. Furthermore, we can model 𝑦 as 𝑑𝑦𝑡=𝑟𝑦𝑡𝑑𝑡+Σ𝑦𝑡𝑑𝑊𝑦𝑡𝑡,𝑦0=𝑦0,(2.5) where 𝑟𝑦𝑇𝑛+1 denotes the rate of liquid asset returns, Σ𝑦𝑡(𝑛+1)×𝑛 is a matrix of liquid asset returns, and 𝑊𝑦Ω×[0,𝑇]𝑛 is standard Brownian motion. Notice that there are only 𝑛 scalar Brownian motions due to one of the liquid assets being riskless.

We assume that the investment strategy 𝜋𝑇𝑛+1 is outside the simplex 𝑆=𝜋𝚁𝑛+1𝜋𝜋=0,,𝜋𝑛𝑇,𝜋0++𝜋𝑛=1,𝜋00,𝜋𝑛0.(2.6) In this case, short selling is possible. The liquid asset return is then Ω×𝑅+, where the dynamics of can be written as 𝑑𝑡=𝜋𝑇𝑡𝑑𝑦𝑡=𝜋𝑇𝑡𝑟𝑦𝑡𝑑𝑡+𝜋𝑇𝑡Σ𝑦𝑡𝑑𝑊𝑦𝑡.(2.7) This notation can be simplified as follows. We denote 𝑟𝐶(𝑡)=𝑟𝑦0(𝑡),𝑟𝐶𝑇+𝑟,therateofreturnoncash,𝑦𝑡=𝑟𝐶𝑟(𝑡),𝑦𝑡𝑇+𝑟𝐶(𝑡)1𝑛𝑇,̃𝑟𝑦𝑇𝑛,𝜋𝑡=𝜋0𝑡,𝜋𝑇𝑡𝑇=𝜋0𝑡,𝜋1𝑡,,𝜋𝑘𝑡𝑇,𝜋𝑇𝑘,Σ𝑦𝑡=Σ00𝑦𝑡,Σ𝑦𝑡𝑛×𝑛,𝐶𝑡=Σ𝑦𝑡Σ𝑦𝑡𝑇𝜋.Then,wehavethat𝑇𝑡𝑟𝑦𝑡=𝜋0𝑡𝑟𝐶(𝑡)+𝜋𝑡𝑗𝑇̃𝑟𝑦𝑡+𝜋𝑡𝑗𝑇𝑟𝐶(𝑡)1𝑛=𝑟𝐶(𝑡)+𝜋𝑇𝑡̃𝑟𝑦𝑡,𝜋𝑇𝑡Σ𝑦𝑡𝑑𝑊𝑦𝑡=𝜋𝑇𝑡Σ𝑦𝑡𝑑𝑊𝑦𝑡,𝑑𝑡=𝑟𝐶(𝑡)+𝜋𝑇𝑡̃𝑟𝑦𝑡𝑑𝑡+𝜋𝑇𝑡Σ𝑦𝑡𝑑𝑊𝑦𝑡𝑡,0=0.(2.8)

2.2. Nett Cash Outflows

In this section, we discuss nett cash outflows arising from cash outflows and inflows.

2.2.1. Description of Nett Cash Outflows

Cash outflows are constituted by retail deposits, unsecured wholesale funding secured funding and additional liabilities (see, e.g., [1]). The latter category includes requirements about liabilities involving derivative collateral calls related to a downgrade of up to 3 notches, market valuation changes on derivatives transactions, valuation changes on posted noncash or non-high-quality sovereign debt collateral securing derivative transactions, asset backed commercial paper (ABCP), special investment vehicles (SIVs), conduits, special purpose vehicles (SPVs), and the currently undrawn portion of committed credit and liquidity facilities.

Cash inflows are made up of amounts receivable from retail counterparties, amounts receivable from wholesale counterparties, receivables in respect of repo and reverse repo transactions backed by illiquid assets, and securities lending/borrowing transactions where illiquid assets are borrowed as well as other cash inflows.

According to [1], nett cash inflows is defined as cumulative expected cash outflows minus cumulative expected cash inflows arising in the specified stress scenario in the time period under consideration. This is the nett cumulative liquidity mismatch position under the stress scenario measured at the test horizon. Cumulative expected cash outflows are calculated by multiplying outstanding balances of various categories or types of liabilities by assumed percentages that are expected to roll off and by multiplying specified draw-down amounts to various off-balance sheet commitments. Cumulative expected cash inflows are calculated by multiplying amounts receivable by a percentage that reflects expected inflow under the stress scenario.

2.2.2. Dynamics of Nett Cash Outflows

Essentially, mortgagors are free to vary their cash inflow rates. Roughly speaking, this rate should be reduced for high LCRs and increased beyond the normal rate when LCRs are too low. In the sequel, the stochastic process 𝑢1Ω×[0,𝑇]+ is the normal cash inflow rate per nett cash inflow unit whose value at time 𝑡 is denoted by 𝑢1𝑡. In this case, 𝑢1𝑡𝑑𝑡 turns out to be the cash inflow rate per unit of the nett cash inflow over the time period (𝑡,𝑡+𝑑𝑡). A notion related to this is the adjustment to the cash inflow rate per unit of the nett cash inflow rate for a higher or lower LCR, 𝑢2Ω×[0,𝑇]+, that will in closed loop be made dependent on the LCR. We denote the sum of 𝑢1 and 𝑢2 by the cash inflow rate 𝑢3Ω×[0,𝑇]+, that is, 𝑢3𝑡=𝑢1𝑡+𝑢2𝑡,𝑡.(2.9) Before the GFC, the cash inflow rate increased significantly as a consequence of rising liquidity. The following assumption is made in order to model the LCR in a stochastic framework.

Assumption 2.3 (cash inflow rate). The cash inflow, 𝑢3, is predictable with respect to {𝑡}𝑡0.
The cash inflow provides us with a means of controlling LCR dynamics. The dynamics of the cash outflow per unit of the nett cash outflow, 𝑒Ω×[0,𝑇], is given by 𝑑𝑒𝑡=𝑟𝑒𝑡𝑑𝑡+𝜎𝑒𝑡𝑑𝑊𝑒𝑡𝑡,𝑒0=𝑒0,(2.10) where 𝑒𝑡 is the cash outflow per unit of the nett cash outflow, 𝑟𝑒𝑇 is the rate of outflow per unit of the nett cash outflow, the scalar 𝜎𝑒𝑇 is the volatility in the outflow per nett cash outflow unit, and 𝑊𝑒Ω×[0,𝑇] is standard Brownian motion.
Next, we take 𝑖Ω×[0,𝑇]+ as the nett cash outflow increase before cash outflow per monetary unit of the nett cash outflow, 𝑟𝑖𝑇+ is the rate of increase of nett cash outflows before cash outflow per nett cash outflow unit, the scalar 𝜎𝑖 is the volatility in the increase of nett cash outflows before outflow, and 𝑊𝑖Ω×[0,𝑇] represents standard Brownian motion. Then, we set 𝑑𝑖𝑡=𝑟𝑖𝑡𝑑𝑡+𝜎𝑖𝑑𝑊𝑖𝑡𝑡,𝑖0=𝑖0.(2.11) The stochastic process 𝑖𝑡 in (2.11) may typically originate from nett cash flow volatility that may result from changes in market activity, cash supply, and inflation.

2.3. The Liquidity Coverage Ratio

This section discusses ratio analysis and liquidity coverage ratio dynamics.

2.3.1. Ratio Analysis

Ratio analysis is conducted on the bank’s balance sheet composition. In this case, the LCR measures a bank’s ability to access funding for a 30-day period of acute market stress. In this paper, as in Basel III, we are interested in the LCR that is defined as the sum of interbank assets and securities issued by public entities as a percentage of interbank liabilities. The LCR formula is given by LiquidityCoverageRatio=StockofHighQualityLiquidAssets.NettCashOutowsovera30-dayPeriod(2.12) This ratio measures the bank system’s liquidity position that allows the assessment of a bank’s capacity to ensure the coverage of some of its more immediate liabilities with highly available assets. It also identifies the amount of unencumbered, high-quality liquid assets a bank holds that can be used to offset the nett cash outflows it would encounter under a short-term stress scenario specified by supervisors, including both specific and systemic shocks.

2.3.2. Liquidity Coverage Ratio Dynamics

Using the equations for liquid assets 𝑥1 and nett cash outflow 𝑥2, we have that 𝑑𝑥1𝑡=𝑥1𝑡𝑑𝑡+𝑥2𝑡𝑢3𝑡𝑑𝑡𝑥2𝑡𝑑𝑒𝑡=𝑟𝐶(𝑡)𝑥1𝑡+𝑥1𝑡𝜋𝑇𝑡̃𝑟𝑦𝑡+𝑥2𝑡𝑢1𝑡+𝑥2𝑡𝑢2𝑡𝑥2𝑡𝑟𝑒𝑡+𝑥𝑑𝑡1𝑡𝜋𝑇𝑡Σ𝑦𝑡𝑑𝑊𝑦𝑡𝑥2𝑡𝜎𝑒𝑑𝑊𝑒𝑡,𝑑𝑥2𝑡=𝑥2𝑡𝑑𝑖𝑡𝑥2𝑡𝑑𝑒𝑡=𝑥2𝑡𝑟𝑖𝑡𝑑𝑡+𝜎𝑖𝑑𝑊𝑖𝑡𝑥2𝑡𝑟𝑒𝑡𝑑𝑡+𝜎𝑒𝑑𝑊𝑒𝑡=𝑥2𝑡𝑟𝑖𝑡𝑟𝑒𝑡𝑑𝑡+𝑥2𝑡𝜎𝑖𝑑𝑊𝑖𝑡𝜎𝑒𝑑𝑊𝑒𝑡.(2.13) The SDEs (2.13) may be rewritten into matrix-vector form in the following way.

Definition 2.4 (stochastic system for the LCR model). Define the stochastic systemfor the LCR model as 𝑑𝑥𝑡=𝐴𝑡𝑥𝑡𝑥𝑑𝑡+𝑁𝑡𝑢𝑡𝑑𝑡+𝑎𝑡𝑥𝑑𝑡+𝑆𝑡,𝑢𝑡𝑑𝑊𝑡,(2.14) with the various terms in this stochastic differential equation being 𝑢𝑡=𝑢2𝑡𝜋𝑡[],𝑢Ω×0,𝑇𝑛+1,𝐴𝑡=𝑟𝐶(𝑡)𝑟𝑒𝑡0𝑟𝑖𝑡𝑟𝑒𝑡,𝑁𝑥𝑡=𝑥2𝑡𝑥1𝑡𝑟𝑦𝑡𝑇00,𝑎𝑡=𝑥2𝑡𝑢1𝑡0,𝑆𝑥𝑡,𝑢𝑡=𝑥1𝑡𝜋𝑇𝑡Σ𝑦𝑡𝑥2𝑡𝜎𝑒00𝑥2𝑡𝜎𝑒𝑥2𝑡𝜎𝑖,𝑊𝑡=𝑊𝑦𝑡𝑊𝑒𝑡𝑊𝑖𝑡,(2.15) where 𝑊𝑦𝑡,𝑊𝑒𝑡, and 𝑊𝑖𝑡 are mutually (stochastically) independent standard Brownian motions. It is assumed that for all 𝑡𝑇, 𝜎𝑒𝑡>0, 𝜎𝑖𝑡>0 and 𝐶𝑡>0. Often the time argument of the functions 𝜎𝑒 and 𝜎𝑖 is omitted.

We can rewrite (2.14) as follows: 𝑁𝑥𝑡𝑢𝑡𝑥=2𝑡0𝑢2𝑡+𝑥1𝑡𝑟𝑦𝑡𝑇0𝜋𝑡𝑥=0100𝑡𝑢3𝑡+𝑛𝑚=1𝑥1𝑡̃𝑟𝑡𝑦,𝑚0𝜋𝑚𝑡=𝐵0𝑥𝑡𝑢0𝑡+𝑛𝑚=1̃𝑟𝑡𝑦,𝑚0𝑥00𝑡𝜋𝑚𝑡=𝑛𝑚=0𝐵𝑚𝑥𝑡𝑢𝑚𝑡,𝑆𝑥𝑡,𝑢𝑡𝑑𝑊𝑡=𝜋𝑇𝑡𝐶𝑡𝜋𝑡1/20𝑥00𝑡𝑑𝑊1𝑡+0𝜎𝑒0𝜎𝑒𝑥𝑡𝑑𝑊2𝑡+000𝜎𝑖𝑥𝑡𝑑𝑊3𝑡=3𝑗=1𝑀𝑗𝑗𝑢𝑡𝑥𝑡𝑑𝑊𝑡𝑗𝑗,(2.16) where 𝐵 and 𝑀 are only used for notational purposes to simplify the equations. From the stochastic system given by (2.14) it is clear that 𝑢=(𝑢2,𝜋) affects only the stochastic differential equation of 𝑥1𝑡 but not that of 𝑥2𝑡. In particular, for (2.14) we have that 𝜋 affects the variance of 𝑥1𝑡 and the drift of 𝑥1𝑡 via the term 𝑥1𝑡𝑟𝑦𝑡𝑇𝜋𝑡. On the other hand, 𝑢2 affects only the drift of 𝑥1𝑡. Then (2.14) becomes 𝑑𝑥𝑡=𝐴𝑡𝑥𝑡𝑑𝑡+𝑛𝑗=0𝐵𝑗𝑥𝑡𝑢𝑗𝑡𝑑𝑡+𝑎𝑡𝑑𝑡+3𝑗=1𝑀𝑗𝑢𝑡𝑥𝑡𝑑𝑊𝑡𝑗𝑗.(2.17) The model can be simplified if attention is restricted to the system with the LCR, as stated earlier, denoted in this section by 𝑥𝑡=𝑥1𝑡/𝑥2𝑡.

Definition 2.5 (stochastic model for a simplified LCR). Define the simplified LCR system by the SDE 𝑑𝑥𝑡=𝑥𝑡𝑟𝐶(𝑡)+𝑟𝑒𝑡𝑟𝑖𝑡+(𝜎𝑒)2+𝜎𝑖2+𝑟𝑦𝑡𝑇𝜋𝑡+𝑢𝑑𝑡1𝑡+𝑢2𝑡𝑟𝑒𝑡(𝜎𝑒)2+𝑑𝑡(𝜎𝑒)21𝑥𝑡2+𝜎𝑖2𝑥2𝑡+𝑥2𝑡𝜋𝑇𝑡𝐶𝑡𝜋𝑡1/2𝑑𝑊𝑡𝑡,𝑥0=𝑥0.(2.18) Note that in the drift of the SDE (2.18), the term 𝑟𝑒𝑡+𝑥𝑡𝑟𝑒𝑡=𝑟𝑒𝑡𝑥𝑡1,(2.19) appears because it models the effect of the decline of both liquid assets and nett cash outflows. Similarly the term (𝜎𝑒)2+𝑥𝑡(𝜎𝑒)2=(𝜎𝑒)2(𝑥𝑡1) appears.

3. Bank Liquidity in a Numerical Quantitative Framework

In this section, we discuss bank liquidity in a numerical quantitative framework. Recently the finance literature has devoted more attention to modeling and assessing liquidity risk in a numerical quantitative framework (see, e.g., [5, 8, 9]).

3.1. Bank Liquidity: Numerical Example  1

In this subsection, we use the data supplied in [6] (see, also, Appendices A.1 and A.2) to assess the liquidity of banks. The dataset originates from a supervisory liquidity report for Dutch banks. It covers a detailed breakdown of liquid assets and liabilities including cash in- and outflows of banks (see, also, [5]).

3.1.1. Data Description: Numerical Example  1

The aforementioned supervisory liquidity report includes on- and off-balance sheet items for about 85 Dutch banks (foreign bank subsidiaries included) with a breakdown per item (average granularity of about 7 items per bank). The report presents month end data available for the period October 2003 to March 2009. In this case, supervisory requirements dictate that actual bank liquidity must exceed required liquidity, at both a one-week and a one-month horizon. Actual liquidity is defined as the stock of liquid assets (weighted for haircuts) and recognized cash inflows (weighted for their liquidity value) during the test period. Required liquidity is defined as the assumed calls on contingent liquidity lines, assumed withdrawals of deposits, drying up of wholesale funding, and liabilities due to derivatives. In this way, the liquidity report comprises a combined stock and cash flow approach, in which respect it is a forward looking concept. The weights, 𝑤𝑖, of the assumed haircuts on liquid assets and run-off rates of liabilities are presented in last two columns of Tables 1 and 2 below. In this regard, the pecking order hypothesis is tested empirically in [5] by classifying the assets and liabilities of the banks in our sample according to the month weights in the liquidity report (as presented in the last column of Tables 1 and 2). In the report, the 𝑤𝑖 values are fixed (see, e.g., [6]) and reflect the bank-specific and market-wide situation. The 𝑤𝑖 values are based on best practices of values of haircuts on liquid assets and run-off rates of liabilities of the banking industry and credit rating agencies.

Table 1: Assets for liquidity testing.
Table 2: Liabilities for liquidity testing.

The various balance sheet and cash flow items in the prudential report [6] are assumed to reflect the instruments which banks use in liquidity risk management by way of responding to shocks. The instruments are expressed in gross amounts. To enhance the economic interpretation we define coherent groups of instruments and the sum of item amounts per group. The first column of Tables 1 and 2 below provides the group classification. Here, the second columns in these tables describe the particular class of assets and liabilities. For the liquidity test for the full month, a distinction is made between non-scheduled items and scheduled items. By contrast to non-scheduled items, scheduled items are included on the basis of their possible or probable due dates. For the liquidity test for the first week, scheduled items are only included if they are explicitly taken into account in day-to-day liquidity management Treasury operations. In Tables 1 and 2 below, scheduled items are indicated by the letter S.

3.1.2. Data Presentation: Numerical Example  1

In this section, we firstly represent data related to assets and then data related to liabilities.

3.1.3. Data Analysis: Numerical Example  1

From Tables 1 and 2, we have seen that the behavior of banks can be described by rather simple indicators constructed from firm-specific balance sheet data. Although they are descriptive in nature, the measures identify trends in banks behavior that convey forward looking information on market-wide developments. A key insight from the analysis is that while banks usually follow a pecking order in their balance sheet adjustments (by making larger adjustments to the most liquid balance sheet items compared to less liquid items), during the crisis banks were more inclined to a static response. This suggests that they have less room to follow a pecking order in their liquidity risk management in stressed circumstances. It implies that banks responses in crises may have more material effects on the economy, since a static response rule means that banks are more likely to adjust their (less liquid) retail lending and deposits than under normal market conditions. A sufficient stock of liquid buffers could prevent that banks are forced to such detrimental static responses, which lends support to the initiatives of the Basel Committee to tighten liquidity regulation for banks (see, e.g., [1]).

The measures for size and the number of extreme balance sheet adjustments gauge the time dimension of macroprudential risk, and indicators of the dependency and concentration of reactions capture the cross-sectoral dimension. The measures are robust to different specifications and distributions of the data. Applied to Dutch banks, the measures show that the number, size, and similarity of responses substantially changed during the crisis, in particular on certain market segments. They also indicate that the nature of banks behavior is asymmetric, being more intense in busts than in booms. Furthermore, during the crisis the deleveraging of large banks started earlier was more intense and more advanced than the deleveraging of smaller banks.

Given these findings, the indicators are useful for macroprudential analysis, for instance with regard to monitoring frameworks. Our analysis underscores the relevance of using several indicators of liquidity risk at the same time, given the different leads and lags of the measures with systemic risk. The empirical results also provide useful information for financial stability models. A better understanding of banks behavior helps to improve the microfoundations of such models, especially with regard to the behavioral assumptions of heterogeneous institutions. Finally, the empirical findings in our study are relevant to understand the role of banks in monetary transmission and to assess the potential demand for CB finance in stress situations. The measures explain developments of financial intermediation channels (wholesale and retail, unsecured, secured, etc.) along the cross-sectional and time dimensions. They also shed more light on the size and number of banks that rely on CB financing.

3.2. Bank Liquidity: Numerical Example  2

In this section, we provide a simulation of the LCR dynamics given in (2.18).

3.2.1. Simulation: Numerical Example  2

In this subsection, we provide parameters and values for a numerical simulation. The parameters and their corresponding values for the simulation are shown in Table 3.

Table 3: Choices of liquidity coverage ratio parameters.
3.2.2. LCR Dynamics: Numerical Example  2

In Figure 1, we provide the LCR dynamics in the form of a trajectory derived from (2.18).

Figure 1: Trajectory of the LCR for low liquidity assets.
3.2.3. Properties of the LCR Trajectory: Numerical Example  2

Figure 1 shows the simulated trajectory for the LCR of low liquidity assets. Here different values of banking parameters are collected in Table 3. The number of jumps of the trajectory was limited to 1001, with the initial values for 𝑙 fixed at 20.

As we know, banks manage their liquidity by offsetting liabilities via assets. It is actually the diversification of the bank’s assets and liabilities that expose them to liquidity shocks. Here, we use ratio analysis (in the form of the LCR) to manage liquidity risk relating various components in the bank’s balance sheets. In Figure 1, we observe that between 𝑡=2000 and 𝑡=2005, there was a significant decrease in the trajectory which shows that either liquid assets declined or nett cash outflows increased.

There was also an increase between 𝑡=2005 and 𝑡=2007 which suggests that either liquid assets increased or nett cash outflows decreased. There was an even sharper increase subsequent to 𝑡=2007 which comes as somewhat of a surprise. In order to mitigate the aforementioned increase in liquidity risk, banks can use several facilities such as repurchase agreements to secure more funding. However, a significant increase was recorded between 𝑡=2005 and 𝑡=2008, with the trend showing that banks have more liquid assets on their books. If 𝑙>0, it demonstrated that the banks has kept a high volume of liquid assets which might be stemming from quality liquidity risk management. In order for banks to improve liquidity they may use debt securities that allow savings from nonfinancial private sectors, a good network of branches and other competitive strategies.

The LCR has some limitations regarding the characterization of the banks liquidity position. Therefore, other ratios could be used for a more complete analysis taking into account the structure of the short-term assets and liabilities of residual maturities.

4. Bank Liquidity in a Theoretical Quantitative Framework

In this section, we investigate bank liquidity in a theoretical quantitative framework. In particular, we characterize a liquidity provisioning strategy and discuss residual aggregate risk in order to eventually determine the appropriate value of the price process. In order to model uncertainty, in the sequel, we consider the filter probability space (Ω,𝔽,(𝑡)0𝑡𝑇,),𝑇 described in Assumption 2.1.

4.1. Preliminaries about the Liquidity Provisioning Strategy

Firstly, we consider a dynamic liquidity provisioning strategy for a risky underlying illiquid asset process, (Λ𝑡)0𝑡𝑇. For purposes of relating the discussion below to the GFC, we choose Λ to be residential mortgage loans hereafter known simply as mortgages. Mortgages were very illiquid (nonmarketable) before and during the GFC. In this case, for liquidity provisioning purposes, the more liquid marketable securities, 𝑆—judging by their credit rating before and during the GFC—are used as a substitute for mortgages. This was true during the period before and during the GFC, with mortgage-backed securities being traded more easily than the underlying mortgages. Furthermore, we assume that the bank mainly holds illiquid mortgages and marketable securities (compare with the assets presented in Tables 1 and 2) with cash for investment being injected by outside investors. The liquid marketable securities, 𝑆, are not completely correlated with the illiquid mortgages, Λ, creating market incompleteness. Under the probability measure, , the price of the traded substitute securities and the illiquid underlying mortgages are given by 𝑑𝑆𝑡=𝑆𝑡𝜇𝑠𝑑𝑡+𝜎𝑠𝑑𝑊𝑆𝑡,𝑑Λ𝑡=Λ𝑡𝜇Λ𝑑𝑡+𝜎Λ𝑑𝑊Λ𝑡,(4.1) respectively, where 𝜇 and 𝜎 are constants. We define the constant market price of risk for securities as 𝜆𝑠=𝜇𝑠𝑟𝜎.(4.2) We note that if the market correlation |𝜌| between 𝑊𝑆 and 𝑊Λ is equal to one, then the securities and mortgages are completely correlated and the market is complete.

Let Θ be a liquidity provisioning strategy for the bank’s asset portfolio. The dynamics of its wealth process is given by 𝑑Π𝑡=𝑛𝑆𝑡𝑑𝑆𝑡+Π𝑡𝑛𝑆𝑡𝑆𝑡𝑟𝑑𝑡+𝑑𝐶𝑡,(4.3) where 𝑑𝐶𝑡 is an amount of cash infused into the portfolio, 𝑛𝑆𝑡 is the number of shares of securities held in the portfolio at time 𝑡,  Π𝑡 is the value of the wealth process, and 𝑟 is the riskless interest rate. The cumulative cost process 𝐶(Θ) associated with the strategy, Θ, is 𝐶𝑡(̀ΠΘ)=𝑡(Θ)𝑡0𝑛𝑆𝑢𝑑̀𝑆𝑢,0𝑡𝑇.(4.4) The cost process is the total amount of cash that has been injected from date 0 to date 𝑡. We determine a provisioning strategy that generate a payoff (Λ𝑇𝐾)+ at the maturity 𝑇. The quantity 𝑇𝑡exp{𝑟(𝑠𝑡)}𝑑𝐶𝑠 is the discounted cash amount that needs to be injected into the portfolio between dates 𝑡 and 𝑇. Since 𝑇𝑡exp{𝑟(𝑠𝑡)}𝑑𝐶𝑠 is uncertain, the risk-averse agent will focus on minimizing the associated ex-ante aggregate liquidity risk 𝚁𝑡(Θ)=𝔼𝑇𝑡exp{𝑟(𝑠𝑡)}𝑑𝐶𝑢2,0𝑡𝑇.(4.5) It is clear that this concept is related to the conditioned expected square value of future costs. The strategy Θ, 0𝑡𝑇 is mean self-financing if its corresponding cost process 𝐶=(𝐶𝑡)0𝑡=𝑇 is a martingale. Furthermore, the strategy Θ is self-financing if and only if ̀Π𝑡(̀ΠΘ)=0(Θ)+𝑡0𝑛𝑆𝑢𝑑̀𝑆𝑢,0𝑡𝑇.(4.6) A strategy Θ is called an admissible continuation of Θ if Θ coincides with Θ at all times before 𝑡 and Π𝑡(Θ)=𝐿, a.s. Moreover, a provisioning strategy is called liquidity risk minimizing if for any 𝑡[0,𝑇], Θ minimizes the remaining liquidity risk. In other words, for any admissible continuous Θ of Θ at 𝑡 we have 𝚁𝑡(Θ)𝚁𝑡Θ.(4.7) Criterion given in (4.5) can be formally rewritten as 𝑡min(𝑛𝑆,Π)𝚁𝑡,subjecttoΠ𝑡=Λ𝑇𝐾+.(4.8) We define the expected squared error of the cost over the next period as 𝔼Δ𝐶𝑡2=𝔼𝑡Π𝑡+Δ𝑡Π𝑡𝑛𝑆𝑡𝑆𝑡+Δ𝑡𝑆𝑡Π𝑡𝑛𝑆𝑡𝑆𝑡(exp{𝑟(𝑡+Δ𝑡)}exp{𝑟𝑡})2.(4.9) In the next section, we minimize the above quantity at each date, with respect to (𝑛𝑆0,𝑛𝑆Δ,,𝑛𝑆𝑡+Δ𝑡) and also discuss the notion of a liquidity provisioning strategy.

4.2. Characterization of the Liquidity Provisioning Strategy

During the GFC, liquidity provisioning strategies involved several interesting elements. Firstly, private provisioning of liquidity was provided via the financial system. Secondly, there was a strong connection between financial fragility and cash-in-the-market pricing. Also, contagion and asymmetric information played a major role in the GFC. Finally, much of the debate on liquidity provisioning has been concerned with the provisioning of liquidity to financial institutions and resulting spillovers to the real economy. The next result characterizes the liquidity provisioning strategy that we study.

Theorem 4.1 (characterization of the provisioning strategy). The locally liquidity risk minimizing strategy is described by the following.(1)The investment in mortgages is ̀𝑛𝑆𝑡=𝜎ΛΛ𝑡𝜎𝑆𝑆𝑡𝜌𝐶Λ𝑡,Λ𝑡=𝜎ΛΛ𝑡𝜎𝑆𝑆𝑡𝜇𝜌expΛ𝑟𝜌𝜎Λ𝜆𝑆𝙽𝑑(𝑇𝑡)1,,𝑡(4.10) where 𝜆𝑠 is the Sharpe ratio and 𝐶(𝑡,Λ𝑡) is the minimal entropy price 𝐶𝑡,Λ𝑡=exp{𝑟(𝑇𝑡)}𝔼Λ𝑇𝐾+𝜇=expΛ𝑟𝜌𝜎Λ𝜆𝑆Λ(𝑇𝑡)𝑡𝙽𝑑1𝑑,𝑡𝐾exp{𝑟(𝑇𝑡)}𝙽2,,𝑡(4.11) where is the minimal martingale measure defined as 𝑑|||𝑑𝑡1=exp2Λ𝑆2(𝑇𝑡)𝜆𝑆𝑊𝑆𝑇𝑊𝑆𝑡,𝑑1,𝑡=1𝜎ΛΛ𝑇𝑡ln𝑡𝐾+𝜇Λ𝜌𝜎Λ𝜆𝑆+𝜎Λ22(𝑇𝑡),𝑑2,𝑡=𝑑1,𝑡𝜎Λ𝑇𝑡.(4.12)(2) The cash investment is 𝐶𝑡,Λ𝑡̀𝑛𝑆𝑡𝑆𝑡.(4.13) If the Sharpe ratio, 𝜆𝑠, of the traded substitute securities is equal to zero, the minimal martingale measure coincides with the original measure , and the above strategy is globally liquidity risk minimizing.

Proof. Let ̀𝑆𝑡exp{𝑟𝑡}𝑆𝑡 be the discounted value of the traded securities at time 𝑡. This process follows a martingale under the martingale measure, , since we have 𝑑̀𝑆𝑡=̀𝑆𝑡𝜎𝑆𝑑𝑊𝑡𝑆,,(4.14) where 𝑑𝑊𝑡𝑆,𝑑𝑊𝑆𝑡+𝜆𝑆𝑑𝑡 is the increment to a -Brownian motion. Hence, we can write the Kunita-Watanabe decomposition of the discounted option payoff under : Λexp{𝑟𝑡}𝑇𝐾+=𝐻0+𝑇0𝜁𝑡𝑑̀𝑆𝑡+𝑉𝐻𝑇,(4.15) where 𝑉𝐻 is a -martingale orthogonal to ̀𝑆 under . Lévy’s Theorem shows that the process 𝙰 defined by 𝑑𝙰𝑡=𝑑𝑊Λ𝑡𝜌𝑑𝑊𝑆𝑡1𝜌2(4.16) is a -Brownian motion and that it is independent of 𝑊𝑆. Then, by Girsanov’s theorem, (𝑊𝑆,,𝙰) also follows two-dimensional -Brownian motion. Since 𝑉𝐻 is a martingale under and is orthogonal to 𝐹, the martingale representation theorem shows that we have 𝑉𝐻𝑡=𝜓𝑑𝙰𝑡 for some process 𝜓. In particular, 𝑉𝐻 is orthogonal under to the martingale part of ̀𝑆, where the martingale part of ̀𝑆 under is defined as 𝐺𝑡=𝑡0𝜎𝑆𝑆𝑠𝑑𝑊𝑆𝑠.(4.17) Next, we suppose that 𝑃𝑡=exp{𝑟(𝑇𝑡)}𝔼Λ𝑇𝐾+.(4.18) Using (4.15) we obtain 𝑃𝑡𝐻=exp{𝑟𝑡}0+𝑡0𝜁𝑠𝑑̀𝑆𝑠+V𝐻𝑡.(4.19) Consider now the non-self-financing strategy with value ̀Π𝑡=𝑃𝑡 and the number of securities given by ̀𝑛𝑆𝑡=𝜁𝑡. Given (4.1) and (4.19), we obtain that 𝑑̀𝐶𝑡=exp{𝑟𝑡}𝑑𝑉𝐻𝑡. This shows that, 𝑉𝐻,̀𝐶 is a -martingale orthogonal to 𝐺. We recall that a strategy (Π,𝑛𝑆) is locally risk minimizing if and only if the associated cost process follows a -martingale orthogonal to 𝐺. Hence the strategy (̀Π,̀𝑛𝑆) is locally risk minimizing.
We now prove an explicitly expression for the random variable 𝑃𝑡, which is called the minimum entropy price. The Black-Scholes formula implies that 𝑃𝑡=𝐶𝑡,Λ𝑡𝜇=expΛ𝑟𝜎Λ𝜌𝜆𝑆Λ(𝑇𝑡)𝑡𝙽𝑑1𝜇,𝑡𝐾expΛ𝜌𝜎Λ𝜆𝑆𝙽𝑑(𝑇𝑡)2,,𝑡(4.20) which can be written as a function 𝐶(𝑡,Λ𝑡) of 𝑡 and Λ𝑡. Using (4.19), we obtain that 𝜁𝑡=𝜎ΛΛ𝑡𝜎𝑆𝑆𝑡𝜌𝐶Λ𝑡,Λ𝑡.(4.21) The required expression for ̀𝑛𝐹𝑡 follows immediately.

Our paper addresses the problem of dynamic bank provisioning for (illiquid) nonmarketable mortgages, Λ, for which substitute (liquid) marketable securities, 𝑆, is part of the liquidity provisioning strategy. Due to the presence of cross-hedge liquidity risk we operate in an incomplete market setting. In this regard, we employ a non-self-financing strategy to ensure that uncertainty is reduced and trading is conducted in the Treasuries market. Moreover, the strategy is designed to influence a perfect replication at the cost of continuous cash infusion into the replicating bank portfolio. Since the cash infusion is random, the risk-averse agent would require that the total uncertainty involved over the remaining life of the mortgage be minimized. As a consequence, for 0𝑡𝑇, the associated ex-ante aggregated liquidity risk is given by 𝚁𝑡(Θ)=𝔼𝑇𝑡exp{𝑟(𝑠𝑡)}𝑑𝐶𝑢2,0𝑡𝑇.(4.22) Now 𝑇𝑡exp{𝑟(𝑠𝑡)}𝑑𝐶𝑢 is stochastic, so we will focus on minimizing the risk in (4.5). We apply a local risk minimization criterion which entails that instead of minimizing the uncertainty with respect to the cash infusion, 𝐶𝑡, over the process, the strategy attempts to minimize, at each date, the uncertainty over the next infinitesimal period. Also, the incompleteness entails the existence of infinitely many equivalent martingale measures. In order to determine the appropriate price of the asset value one should choose an appropriate equivalent martingale measure. In this case, the process is -Brownian motion so that the discounted price process exp{𝑟𝑡}𝑆𝑡 follows martingale pricing. The equivalent martingale measure will be determine according to the risk minimization criterion in Theorem 4.1. Let us consider the discounted price to be Λexp{𝑟𝑡}𝑇𝐾+.(4.23) Applying the Kunita-Watanabe decomposition for the discounted price under a measure , we get 𝐾Π=𝐻0+𝑇0𝜁𝑡𝑑̀𝑆𝑢+𝑉𝐻𝑇,(4.24) where 𝑉𝐻 is a -martingale orthogonal to ̀𝑆 under the measure . Let 𝑃𝑡=exp{𝑟(𝑇𝑡)}𝔼𝑡Λ𝑇𝐾+(4.25) which can be rewritten as 𝑃𝑡𝐻=exp{𝑟𝑡}0+𝑡0𝜁𝑢𝑑̀𝑆𝑢+𝐿𝐻𝑡.(4.26)

4.3. Residual Aggregate Liquidity Risk

During the GFC, two types of uncertainty concerning liquidity requirements arose. Firstly, each individual bank was faced with idiosyncratic liquidity risk. At any given time its depositors may have more or less liquidity needs. Uncertainty also arose from the fact that banks face aggregate liquidity risk. In some periods aggregate liquidity demand is high while in others it is low. Thus, aggregate risk exposes all banks to the same shock, by increasing or decreasing the demand for liquidity that they face simultaneously. The ability of banks to hedge themselves against these liquidity risks crucially depend on the functioning, or, more precisely, the completeness of financial markets. The next theorem provides an explicit expression for the aggregate liquidity risk when a locally risk minimizing strategy is utilized in an incomplete market.

Theorem 4.2 (residual aggregate liquidity risk). The aggregate liquidity risk when a locally risk minimizing strategy at time 𝑡 is implemented is equal to 𝚁𝑡𝑟𝑚=𝜎Λ21𝜌2𝑇𝑡exp{2𝑟(𝑠𝑡)}𝔼Λ𝑆2𝐶Λ𝑠,Λ𝑠2𝑑𝑠.(4.27) This can be approximated by 𝚁𝑡𝑟𝑚𝜎Λ2(1𝜌)2𝐶Λ0,Λ02Λ201exp{2𝑟(𝑇𝑡)}.2𝑟(4.28)

Proof. Let us now assume Φ=Φ and Π𝑡=𝐶(𝑡,Λ𝑡). Under , the wealth process, Π, evolves as 𝑑Π𝑡=𝑟Π𝑡𝑑𝑡+𝜌𝜎ΛΛ𝑡𝐶Λ𝑡,Λ𝑡𝑑𝑊𝑡𝑆,̀𝐶+𝑑𝑡.(4.29) In addition, (exp{𝑟𝑡}𝐶(𝑡,Λ𝑡))𝑡 follows a -martingale, where 𝑑𝐶𝑡,Λ𝑡=𝑟𝐶𝑡,Λ𝑡𝑑𝑡+𝐶Λ𝑡,Λ𝑡𝜎ΛΛ𝑡𝑑𝑊𝑡Λ,,(4.30)𝑑𝑊𝑡Λ,=𝑑𝑊Λ𝑡+𝜌𝜆𝑆𝑑𝑡(4.31) defines a -Brownian motion. One can write it as 𝑑𝑊𝑡Λ,=𝑑𝑊Λ𝑡𝜌𝑑𝑊𝑆𝑡+𝜌𝑑𝑊𝑡𝑆,=1𝜌2𝑑𝑊2𝑡+𝜌𝑑𝑊𝑡Λ,.(4.32) Comparing (4.29) and (4.30) we obtain that exp{𝑟𝑡}𝑑𝐶𝑡=exp{𝑟𝑡}𝐶Λ𝑡,Λ𝑡𝜎ΛΛ𝑡1𝜌2𝑑𝑊2𝑡,(4.33) hence (4.27).
In what follows, we let 𝛿𝑡 be the delta of the mortgage process at time 𝑡 that is computed from the minimal entropy price so that 𝛿𝑡=𝐶Λ(𝑡,Λ𝑡). We must now compute 𝔼[𝛿2𝑡Λ2𝑡] for all 𝑡 in [0,𝑇]. If (𝛿2𝑡,Λ2𝑡)𝑡>0 were a martingale, the task would be easy since we would have 𝔼[𝛿2𝑡Λ2𝑡]=𝛿20Λ20. But (𝛿2𝑡Λ2𝑡)𝑡0 is not a martingale. However, it can be shown that for small 𝜎Λ2𝑇, the expectation 𝔼[𝛿2𝑡Λ2𝑡] is approximated by the constant 𝛿20Λ20. The formal proof follows from the fact that 𝔼[𝛾𝑡Λ2𝑡]𝛾0Λ20,𝛾𝑡=𝐶ΛΛ(𝑡,Λ𝑡), denoting the gamma of the value of the asset. Therefore, we finally have that 𝜎Λ21𝜌2𝑇𝑡exp{2𝑟𝑠}𝔼𝛿2𝑠Λ𝑆2𝑑𝑠𝜎Λ21𝜌2𝛿20Λ201exp{2𝑟(𝑇𝑡)}.2𝑟(4.34)

Applying a non-self-financing strategy and considering ̀Π𝑡=𝑃𝑡 and ̀𝑛𝑆𝑡=𝜁𝑡, we obtain that 𝑑̀𝐶𝑡=exp{𝑟𝑡}𝑑𝑉𝐻𝑡.(4.35) This implies that ̀𝐶 is -martingale orthogonal to 𝐺. In this regard, the strategy (Π,𝑛𝑆) is locally risk minimizing if and only if the associated cost process 𝐶(Θ) follows a -martingale orthogonal to 𝐺. This means the strategy minimizes at each date the uncertainty over the next infinitesimal period. In applying the risk-minimization strategy there remains some “residual” aggregate liquidity risk stemming from the imperfection of the Brownian motion processes 𝑊𝑆 and 𝑊Λ. After the bank has implemented the locally risk minimizing strategy at time 𝑡, the aggregate liquidity risk is 𝚁𝑡𝑟𝑚=𝜎Λ21𝜌2𝑇𝑡exp{2𝑟(𝑠𝑡)}𝔼Λ2𝐶Λ𝑆,Λ𝑢2.(4.36) For 𝛿𝑡 associated with the value process at time 𝑡 computed via the minimized entropy price, we now need to compute 𝔼[𝛿2𝑡,Λ2𝑡] for all 𝑡 in [0,𝑇]. Since 𝔼[𝛿2𝑡,Λ2𝑡] is approximated by the constant 𝛿20Λ20, then 𝔼[𝛾𝑡Λ2𝑡]𝛾0Λ2𝑡, 𝛾𝑡=𝐶Λ(𝑡,Λ𝑡) which is the gamma of the mortgage value. Therefore, the residual liquidity risk at time 𝑡 is 𝜎Λ2(1𝜌)2𝑇𝑡exp{2𝑟𝑠}𝔼𝛿2𝑢Λ𝑆𝑢𝑑𝑢𝜎Λ2(1𝜌)2𝛿20Λ20exp{2𝑟(𝑇𝑡)}.2𝑟(4.37)

5. Conclusions and Future Directions

In this paper, we discuss liquidity risk management for banks. We investigate the stochastic dynamics of bank items such as loans, reserves, securities, deposits, borrowing and bank capital (compare with Question 1). In accordance with Basel III, our paper proposes that overall liquidity risk is best analyzed using ratio analysis approaches. Here, liquidity risk is measured via the LCR. In this case, we provide numerical results to highlight some important issues. Our numerical quantitative model shows that a low LCR stems from a low level of liquid assets or high nett cash outflows (compare with Question 2). Moreover, we provide a characterization of liquidity risk provisioning by considering an (illiquid) nonmarketable mortgage as an underlying asset and using (liquid) marketable securities for provisioning. In this case, we use non-self-financing strategy that considers market incompleteness to provision for liquidity risk. Then, we provide a quantitative framework for assessing residual risk stemming from the above strategy (compare with Question 3).

Future research should focus on other features of the GFC that are related to liquidity provisioning. The first involves the decrease in prices of AAA-rated tranches of structured financial products below fundamental values. The second is the effect of the GFC on interbank markets for term funding and collateralized money markets. Thirdly, further investigations should address the fear of contagion should a major institution fail. Finally, the effects on the real economy should be considered. In addition, the stochastic dynamic model we have consider in this paper does not take assets and liabilities with residual maturities into account. Such a model should be developed.


A. More about Liquidity Risk

In this section, we provide more information about measures by cash flow, liquidity monitoring approaches, liquidity risk ratings and national approaches to liquidity risk.

A.1. Measures by Cash Flow

Banks use the intensity of the cash flow to predict the level of stress events. In this case, we determine the level of both cash in flows and cash out flows depending on both supply and demand for liquidity in the normal market performance. In this regard, the bank cash flow predicts the level of stress event 𝑠. Moreover, the use of proforma is an acceptable standard which determine the uses and sources of funds in the bank. It identifies where the bank funding short fall and liquidity gap lies.

A.2. Liquidity Monitoring Approaches

The BCBS has set international standards for sound management of liquidity risk. In this regard, the monitoring and evaluation of the banks operational activities is an internal control measure. However, the monitoring approach is divided into three levels, that is, the liquid assets approach, the cash flow approach, and a mixture of both. Liquid asset approach is mostly appropriate used in the Treasury bond market. In this regard, banks are required to maintain some liquid asset in their balance sheet that could be used during the hard period. Assets such as government securities are appropriate to maintain in the balance sheet because they can easily enable the bank to secure funding through securitization. While Cash flow matching approach enable banks to match the cash in flows with the cash out flows from the balance sheet activities.

The monitoring approaches for assessing liquidity risk is divided into three classes, that is, liquid asset approach, the cash flow approach and the combination of both. In the liquid asset approach a bank prescribed to a minimum level of cash or high-quality liquid or marketable assets in relation to the deposits and other sources of funds. While maturity mismatch classify the expected inflows and outflows of funds into time bands of their residual maturity.

A.3. Liquidity Risk Rating

The rating of liquidity risk is categorized into two sets of indicators, that is, the quantitative and qualitative liquidity risk indicators.

Table 4 shows the quantitative and qualitative liquidity risk indicators. In light of the above, the rating for quantitative liquidity risk management is classified into three levels, that is, low, moderate level, and high level of liquidity risk. Therefore, a bank with a full set of all the indicated quantitative indicators has a low level of liquidity risk. Moreover, the rating for qualitative liquidity risk is divided into three levels, that is, strong, satisfactory, and weak quality of management of liquidity risk. In the above, we indicated that rating of liquidity risk is divided into two sets of indicators, namely, the quantitative liquidity risk indicators and qualitative liquidity risk indicators. According to Table 4, a bank with a full set of all the indicated qualitative indicators has a low level of liquidity risk, while a bank with a full set of all indicated qualitative indicators has a higher level of liquidity risk management.

Table 4
A.4. National Approaches to Liquidity Risk

In this section, we discuss a useful principle which needs to be developed by individual countries to ensure sound management of liquidity risk and appropriate level of liquidity insurance by banks. This principle could be enforced via policies that assess liquidity as an internal measure; stress testing and other scenario analysis which determine the probability of a bank culminating into liquidity crisis; contingency funding to provide reliable sources of funds to cover the short fall; setting limitations such as holding of liquid assets, minimum liquid assets, limits on maturity mismatches, and limits on a particular funding sources; reporting about liquidity risks and sources of liquidity as well as through public disclosure to enable investors to access bank information.


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