**1.1. Overview of the literature**

Our contribution has strong connections with [7], [10], [12] and [13] that deals with subprime mortgage funding and liquidity risk and Basel III liquidity regulation, respectively.

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

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

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