**1. Introduction**

The recent financial crises, viz., the subprime mortgage and 2007-2009 financial crises as well as the sovereign debt crisis, are characterized by too-big-to fail banks that suffered from a lack of liquidity (see, for instance, [13] and [14]). The actualization of such liquidity risk led to credit crunches and had negative effects on global financial markets. In response to this, among other things, the Basel Committee on Banking Supervision (BCBS) is proposing that banks should always have a 30-day liquidity cover for stress scenarios (see, for instance, [2], [3], [4] and [5]).

In this regard, the level of high-quality liquid assets (HQLAs) is important in order for banks to function optimally (see, for instance, [6] and [8]). As far as Basel III liquidity proposals are concerned, the BCBS is suggesting a liquidity coverage ratio (LCR) defined as follows.

The LCR has two components:

(a) total stock of HQLAs; and

(b) total nett cash outflows,

and is expressed as

LCR <sup>=</sup> Total Stock of High-Quality Liquid Assets (HQLAs) Total Nett Cash Outflows (NCOs) Over the Next 30 Calendar Days <sup>≥</sup> 1. (1)

The numerator of the LCR is the stock of HQLA. Under the standard, banks must hold a stock of unencumbered HQLA to cover the total net cash outflows over a 30-day period under the

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prescribed stress scenario. In order to qualify as HQLA, assets should be liquid in markets during a time of stress and, in most cases, be eligible for use in central bank operations. Certain types of assets within HQLA are subject to a range of haircuts. HQLA are comprised of Level 1 assets (L1As) and Level 2 assets (L2As). L1As generally include cash, central bank reserves, and certain marketable securities backed by sovereigns and central banks, among others. These assets are typically of the highest quality and the most liquid, and there is no limit on the extent to which a bank can hold these assets to meet the LCR. L2As are comprised of Level 2A assets (L2AAs) and Level 2B assets (L2BAs). L2AAs include, for example, certain government securities, covered bonds and corporate debt securities. L2BAs include lower rated corporate bonds, residential mortgage backed securities and equities that meet certain conditions. L2As may not in aggregate account for more than 40 % of a bank's stock of HQLA. L2BAs may not account for more than 15 % of a bank's total stock of HQLA (see, for instance, [1] and [13]).

The technique employed in this book chapter is heavily reliant on the one used in [10]. In that paper, Mukuddem-Petersen and Petersen consider the application of stochastic optimization theory to asset and capital adequacy management in banking. The study is motivated by new banking regulation that emphasizes risk minimization practices associated with assets and regulatory capital. The analysis in [10] depends on the dynamics of the capital adequacy ratio (CAR), that we compute by dividing regulatory bank capital (RBC) by risk weighted assets (RWAs). Furthermore, Mukuddem-Petersen and Petersen demonstrate how the CAR can be optimized in terms of bank equity allocation and the rate at which additional debt and equity is raised. In either case, the dynamic programming algorithm for stochastic optimization is employed to verify the results. Also, in [10], Mukuddem-Petersen and Petersen provide an illustration of aspects of bank management practice in relation to this regulation. In the

Optimizing Basel III Liquidity Coverage Ratios

http://dx.doi.org/10.5772/58395

69

In [12], we use actuarial methods to solve a nonlinear stochastic optimal liquidity risk management problem with deposit inflow rates and marketable securities allocation as controls. The main objective in [12] 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 mortgage and deposit reference processes. However, the current chapter is an improvement on [12] in that bank balance sheet features play a more

In order to construct our LCR model, we take into account results obtained in [13] in a discrete-time framework (see, also, [8]). In the aforementioned book, we estimate the LCR and NSFR by applying approximation techniques to banking data from a cross section of countries. We find that these Basel III risk measures have low information values and are relatively poor indicators of liquidity risk. Our results, in [13], show that as the LCR increases (decreases) the probability of failure decreases (increases) for both Class I (internationally active banks with Tier 1 capital in excess of US \$ 4 billion) and II (the rest) banks. Our contribution is distinct from the aforementioned in the following respects. Firstly, our analysis has a heavy reliance on the derivation of a stochastic model for LCR dynamics that depends mainly on the liquidity provisioning rate, HQLA returns and NCO outflows. Secondly, we obtain an analytic solution of a particular type to our stochastic bank LCR problem (with a quadratic objective function) that we pose. Finally, the optimal choices for the cash injection and asset allocation are both expressed in terms of a LCR reference process. To our knowledge such processes have not been considered for LCRs before. The study is particularly significant because the Basel III LCR will be implemented on Thursday, 1 January 2015 on a global scale (see, for instance, [2], [3], [4] and [5]). In this book chapter,

In short, this book chapter advances our knowledge of Basel III liquidity by investigating the LCR global liquidity standard (see, for instance, [6] and [14]) in an optimization context. In particular, in Section 2, a theoretical-quantitative model is constructed by considering the dynamics of the HQLAs and NCOs. Section 3 produces two parameters that are able to be controlled, viz., the liquidity provisioning rate and HQLA allocation. The main motivation for studying LCR dynamics is to show that, in principle, banks are able to control their liquidity via an appropriate provisioning strategy. This should ensure that the said ratio

current chapter, the same technique is employed (see, also, [13]).

prominent role (see Sections 2, 3 and 5 for more details).

we extend the analysis in [13] to continuous time.

**1.2. Outline of the book chapter**

The denominator of the LCR is the total net cash outflows. It is defined as total expected cash outflows, minus total expected cash inflows, in the specified stress scenario for the subsequent 30 calendar days. Total expected cash outflows are calculated by multiplying the outstanding balances of various categories or types of liabilities and off-balance sheet commitments by the rates at which they are expected to run off or be drawn down. Total expected cash inflows are calculated by multiplying the outstanding balances of various categories of contractual receivables by the rates at which they are expected to flow in. Total cash inflows are subject to an aggregate cap of 75 % of total expected cash outflows, thereby ensuring a minimum level of HQLA holdings at all times (see, for instance, [11] and [13]).

The standard requires that, in the absence of financial stress, the value of the ratio be no lower than 100 % (i.e., the stock of HQLA should at least equal total net cash outflows). Banks are expected to meet this requirement on an ongoing basis and hold a stock of unencumbered HQLA as a defence against the potential onset of liquidity stress. During a period of financial stress, however, banks may use their stock of HQLA, thereby falling below 100 % (see, for instance, [11]).
