Читать книгу A Risk Professional's Survival Guide - Rossi Clifford - Страница 9

At its core, SifiBank, like other commercial banks, engages in profit-maximizing financial intermediation. Profit is defined as:

(1.1)

where r_i represents the rate on earning assets q for the ith product, and i_i is the cost associated with the jth input x, either financial (e.g., deposits) or real (e.g., personnel).

Financial intermediation refers to the process by which banks take in a variety of liabilities such as deposits and debt and transform them into earning assets. Liabilities for banks are inputs into their production process that are used in creating loans, investments and services to bank customers.

Further, the bank is expected to maximize profit subject to technical conditions underlying a production function, P(q₁… qn, x₁, .. x_m) = 0. In developing their strategic plans for the coming year, banks take into consideration a host of other information in setting their asset targets. These include such factors as relative peer profitability and other indicators of performance, and business structural issues such as product concentrations and competitive conditions, among others. Through the production function whereby the bank as a financial intermediary uses its financial inputs – including various forms of deposits including retail and wholesale sources as well as other funding sources – and nonfinancial inputs such as physical premises and personnel, the bank determines its level and combination of assets to produce, taking into account other external factors as described. As a result, the relationship between bank output and inputs could be described by the following first-order condition of the following simple constant elasticity of substitution (CES) production function2:

(1.2)

To illustrate the link between assets and deposits in this construct, assume the bank has a single asset denoted q in the model above that is produced using two types of deposits; x₁ represents retail deposits and x₂ describes brokered deposits.3 The relationship described by the CES production function shows that both inputs as factors of production define the level of assets for the firm. In equilibrium, the bank will select a target level of output q that maximizes the expected utility of profit formally described below. The input combinations of x₁ and x₂ are then optimized by their least cost combination in the profit function subject to any technical production constraint such as funding limitations. External factors driving target output for the bank such as peer performance or other metrics could be subsumed within the constant term C of the production function.

The profit model can be extended to include the production function as well as to introduce uncertainty (risk) into the decision making process.

(1.3)

where is a Lagrange multiplier.4 Introducing output uncertainty into the model, the bank is assumed to maximize expected profit:

(1.4)

where represents the probability of output q_i. The first-order conditions with respect to output and input are as follows:

(1.5)

(1.6)

The term represents the input demand function for the jth input x. In this specification, input demands are a function of input prices i as well as the production function. Taking, for example, brokered deposits as an input variable of interest, the change in expected profit for a unit change in the level of brokered deposits would be dependent upon changes in the costs of its inputs as well as the relationship between bank outputs (assets) and inputs (liabilities and other real inputs) as established by the production function P. In other words, changes in profit arising from changes in brokered deposits are driven by underlying structural economic relationships. Taking these theoretical relationships further, we can postulate the relationship between asset growth and risk-taking that figures prominently in policy discussions of brokered deposits. Adapting the profit model above, assume that the bank maximizes the expected utility of profit as follows:

(1.7)

Setting the derivative of output q equal to zero yields:

(1.8)

Assuming that the bank utility function follows Neumann-Morgenstern expected conditions, a bank that is risk-neutral would exhibit second-order conditions:

(1.9)

In the case that the bank is a risk-taker, it can be shown that the second-order condition must satisfy the following:

(1.10)

which implies that , where q* is the level of bank output that solves the profit maximization problem above. In such situations, q* is greater than the equilibrium level of q that solves. .

The implication from this result is that risk-taking leads to higher output produced by the bank than if the bank were risk-neutral.5 With this result we can establish then that asset growth for the bank must be related to the risk appetite of the firm. With the model establishing input demand as a function of input prices and the production function, the model describes how risk-taking at the bank relates to a target level of output. This framework suggests that deposits certainly are a factor of production, but that asset growth and investment in riskier products is driven more by overall risk-taking of the firm rather than fueled by deposit strategies. In this formulation, output is determined by the least cost combination of inputs subject to various constraints on those inputs. The existence of technical constraints on inputs can influence input allocation. For instance, if banks set a target level of assets for the next year that cannot be funded solely with retail deposits due to capacity constraints, then brokered and other wholesale deposits would be used to fill the gap, subject again to profit maximization conditions. With this framework describing the bank’s conceptual constrained profit maximization problem, it is instructive to dig deeper into some of the structural, market and regulatory aspects of banking that affect the way risk management is performed.