**1.2. Outline of the book chapter**

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

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

Our contribution has strong connections with [7], [10], [12] and [13] that deals with subprime

The working paper [7] examines large capital injections by U.S. financial institutions from 2000 to 2009. These infusions include private as well as government cash injections under the Troubled Asset Relief Program (TARP). The sample period covers both business cycle expansions and contractions, and the recent financial crisis. Elyasiani, Mester and Pagano show that more financially constrained institutions were more likely to have raised capital through private market offerings during the period prior to TARP, and firms receiving a TARP injection tended to be riskier and more levered. In the case of TARP recipients, they appeared to finance an increase in lending (as a share of assets) with more stable financing sources such as core deposits, which lowered their liquidity risk. However, in [7], Elyasiani, Mester and Pagano find no evidence that banks' capital adequacy increased after the capital injections. In this book chapter, we regard the LCR as a measure of liquidity risk. We check the tendencies of this measure over the sample period 2002 to 2012 and make conclusions

mortgage funding and liquidity risk and Basel III liquidity regulation, respectively.

(see, for instance, [1] and [13]).

68 Dynamic Programming and Bayesian Inference, Concepts and Applications

instance, [11]).

**1.1. Overview of the literature**

about it (see, also, [13]).

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 does not move below an acceptable level. The control theoretic liquidity problem is to meet LCR targets with as little additional liquidity provisioning (essentially corresponding to cash injections in our chapter) as possible and optimal HQLA allocation. Section 5 provides conclusions and future directions. As was mentioned before, the additional provisioning may arise from an inflow of cash injections. We choose examples to illustrate that Basel III liquidity regulation resulted from both problematic liquidity structures and unexpected cash outflows (see Section 5 for more details).

It is important for banks that *lt* in (2) has to be sufficiently high to ensure high LCRs. In fact, as was mentioned before, Basel III sets the minimum value of the LCR at 1. Obviously, low values of *lt* indicate that the bank has decreased liquidity and is at high risk of causing a

Bank liquidity has a heavy reliance on liquidity provisioning rates. This rate should be reduced for high LCRs and increased beyond the normal rate when bank LCRs are low. In the sequel, the stochastic process *<sup>u</sup>*<sup>1</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> is the *normal rate of liquidity provisioning*

is the normal liquidity provisioning rate per unit of the bank's NCOs over the time period (*t*, *t* + *dt*). A related concept is the *adjustment to the rate of liquidity provisioning per monetary unit of the bank's NCOs for surplus or deficit*, *<sup>u</sup>*<sup>2</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>+, that depends on the LCR. In the case of liquidity deficit, during stress scenarios, this adjustment rate can correspond to a cash injection rate. Here the amount of surplus or deficit is reliant on the excess of HQLAs over NCOs. We denote the sum of *<sup>u</sup>*<sup>1</sup> and *<sup>u</sup>*<sup>2</sup> by the *liquidity provisioning rate u*<sup>3</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>+,

*<sup>t</sup>* + *<sup>u</sup>*<sup>2</sup>

The following assumption is made in order to model the LCR in a stochastic framework

**Assumption 2.2. (Liquidity Provisioning Rate)** *The liquidity provisioning rate, u*3, *is predictable with respect to* {F*t*}*t*≥<sup>0</sup> *and provides us with a means of controlling bank LCR dynamics (see (3) for*

The closed loop system will be defined such that Assumption 2.2 is met, as we shall see in the sequel. In times of deficit, for (3), we should choose the cash injection rate, *u*2, sufficiently large in order to guarantee bank liquidity. In reality, cash injections are subject to more

Before and during the financial crisis, the LCR decreased significantly as extensive cash outflows took place with a consequent rising of NCOs. By contrast, banks predicted continued growth in the financial markets (see, for instance, [14]). The dynamics of the

*<sup>t</sup> dW<sup>e</sup>*

where *et* is the outflows per NCO monetary unit, *<sup>r</sup><sup>e</sup>* : *<sup>T</sup>* <sup>→</sup> R is the rate of outflows per monetary unit of the bank's NCOs, the scalar *<sup>σ</sup><sup>e</sup>* : *<sup>T</sup>* <sup>→</sup> R, is the volatility in the outflows per NCO unit and *<sup>W</sup><sup>e</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> R is standard Brownian motion (compare with [10]). Moreover,

*<sup>t</sup> dW<sup>h</sup>*

*<sup>t</sup>* . In this case, *<sup>u</sup>*<sup>1</sup>

Optimizing Basel III Liquidity Coverage Ratios

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

*<sup>t</sup>* , for all *t*. (3)

*<sup>t</sup>* , *e*(*t*0) = *e*0, (4)

*<sup>t</sup>* , *h*(*t*0) = *h*0, (5)

*<sup>t</sup> dt*

71

*per monetary unit of the bank's NCOs* whose value at time *t* is denoted by *u*<sup>1</sup>

*u*3 *<sup>t</sup>* = *<sup>u</sup>*<sup>1</sup>

*outflows per monetary unit of the bank's NCOs*, *e* : Ω × *T* → R, is given by

*<sup>t</sup> dt* + *<sup>σ</sup><sup>e</sup>*

*<sup>t</sup> dt* + *<sup>σ</sup><sup>h</sup>*

*det* = *r<sup>e</sup>*

*dht* = *r<sup>h</sup>*

credit crunch (see, for instance, [13]).

i.e.,

(compare with [10]).

stringent conditions (see, also, [13]).

*more details).*

we consider
