Modeling the bank lending rate

The core theoretical framework that I will use to explain the determinants of credit market equilibrium is based on the Monti-Klein model of a monopolistic bank (Klein 1971)(Monti 1972). The model serves as a theoretical illustration of how factors beyond the term and risk structure of interest rates determine the equilibrium bank lending rates. The Monti-Klein model is the foundation on which more recent theoretical models in the industrial organization literature have extended upon. This framework plays an

important role in motivating the particular econometric approach opted for in this paper as well as most of the other empirical literature, for summaries of papers that follow this approach more or less directly see (G. De Bondt 2002)(G. J. De Bondt 2005) and (ECB 2009).

Following the traditional approach of the industrial organization literature, the role of the bank in the Monti-Klein framework is a profit maximizing firm like any other. The original Monti-Klein framework considers a monopolistic market structure; however, the European banking markets are better described as oligopolistic (Angeloni, Kashyap, and Mojon 2003). After all, there are several competing banks in all countries but the barriers to entry are quite high, so the market structure does not approximate perfect competition. The version of the Monti-Klein framework that I will present is based on that of (Freixas and Rochet 1997) and is modified to accommodate Cournot competition between a finite number of competing banks. Finally both credit risk and minimum reserve capital requirements are included in the model as a further extension following (Wong 1997) and (Corvoisier and Gropp 2002). The optimal lending rate of a representative bank is then derived by imposing equilibrium in the lending market. The presented derivations are directly the work of (Putkuri 2010).

2.4.1 Model assumptions

First I introduce the assumptions on top of which the model is built and then the resulting optimization behavior is characterized.

There is a market where banks and borrowers meet. The banking industry is assumed to be oligopolistic and there are a number of banks N, where N is finite and banks are indexed as 𝑛 ∈ [1, 𝑁]. The banks offer both loans and deposits to their customers. Banks are furthermore forced to hold a certain amount of equity capital as reserve capital.

Furthermore, there is an interbank market where banks amongst themselves can lend and borrow surplus reserve capital.

The balance sheet of a bank is described by the following identity:

𝐿_𝑛+ 𝑀_𝑛= 𝐷_𝑛+ 𝐾_𝑛 [1]

Where 𝐿_𝑛 is the quantity of loans, 𝐷_𝑛 the quantity of deposits, 𝐾_𝑛 the amount of equity capital held by the bank and 𝑀_𝑛 the amount lent or borrowed from the interbank market.

The interest rate in the interbank market, 𝑟_𝑀, is controlled exogenously by the central bank. The minimum amount of equity capital 𝐾_𝑛 that the bank chooses to hold is determined by the reserve requirement set by the central bank:

𝐾_𝑛 ≥ 𝜅𝐿_𝑛 [2]

The cost of holding equity capital, 𝑟_𝐾, is assumed to be greater than the interbank market rate, 𝑟_𝑀 and the rate paid on deposits, 𝑟_𝐷, at all times. This criterion ensures that the reserve requirement is strictly binding and that (1 − 𝜅)𝐿_𝑛 will be financed by either deposits or interbank borrowing.

Individual banks are assumed to have an identical cost function which is strictly increasing and of the form:

𝐶(𝐿, 𝐷) = 𝛾_𝐿𝐿_𝑛+ 𝛾_𝐷𝐷_𝑛 [3]

Where parameters 𝛾_𝐿 and 𝛾_𝐷 are assumed to be constant. The parameters are equal to the marginal costs of providing loans and respectively deposits as can be seen from the separable form of the cost function.

𝛾_𝐿≡𝜕𝐶(𝐿, 𝐷)

𝜕𝐿 𝑎𝑛𝑑 𝛾_𝐷≡𝜕𝐶(𝐿, 𝐷)

𝜕𝐷 𝑤𝑖𝑡ℎ 𝜕𝐶²(𝐿, 𝐷)

𝜕𝐿𝜕𝐷 =𝜕𝐶²(𝐿, 𝐷)

𝜕𝐷𝜕𝐿 = 0 [4]

Demand for bank loans 𝐿(𝑟_𝐿) are decreasing in the loan rate 𝑟_𝐿 and vice-versa the demand for bank deposits 𝐷(𝑟_𝐷) are increasing the deposit rate 𝑟_𝐷.

The credit risk banks face on their loans is represented by the parameter 𝜇 ∈ [0,1] and in the original model is the probability that a given loan will defaulted upon. The parameter is identical for all banks as all banks are assumed to have similar degrees of risk aversion and can be thought of as the share of non-performing loans at the end of the period (Wong 1997) or alternatively the default probability of loans (Corvoisier and Gropp 2002).

2.4.2 Bank profit maximization and equilibrium loan rate

Banks compete in a static Cournot game by setting quantities of supplied loans L and supplied deposits D. Given the total quantities L and D chosen by the banks both lending and deposit interest rates adjust to their market clearing levels 𝑟_𝐿(𝐿) and 𝑟_𝐷(𝐷).

The profit function that a given bank 𝑛 is maximizing, taking into account its budget constraint [1], takes the following form:

𝑤.𝑟.𝑡. 𝐿max_𝑛,𝐷_𝑛𝐸[𝜋_𝑛(𝐿_𝑛, 𝐷_𝑛)] = (1 − 𝜇)𝑟_𝐿(𝐿)𝐿_𝑛+ 𝑟_𝐷(𝐷)𝐷_𝑛− 𝑟_𝐾𝐾_𝑛− 𝐶(𝐿_𝑛, 𝐷_𝑛) [5]

The total expected profit is a simple function of expected interest income less total capital expenditure. The expected profit function [5] can be expressed utilizing the relationships [1], [2] and [3].

𝐸[𝜋_𝑛] = ((1 − 𝜇)𝑟_𝐿(𝐿) − 𝑟_𝑀)𝐿_𝑛− (𝑟_𝐷(𝐷) − 𝑟_𝑀)𝐷_𝑛− (𝑟_𝐾− 𝑟_𝑀)𝜅𝐿_𝑛− 𝛾_𝐿𝐿_𝑛− 𝛾_𝐷𝐷_𝑛 [6]

The optimum quantities of loans and deposits for the individual bank firm to supply are found by taking first order conditions while taking the quantities that the other N-1 banks are supplying as given. However only the first order condition for the optimal number of loans is relevant for the research question in this paper and thus the first order condition for the quantity of deposits is omitted here. The Cournot Nash equilibrium is the following given that the profit function is strictly concave in both 𝐿_𝑛 and 𝐷_𝑛 and twice differentiable. The sum of optimal lending quantities of all banks is denoted 𝐿^∗ and 𝐿^∗_𝑛 indicates the optimal lending quantity of the individual bank 𝑛. Loan quantities are increased until the expected marginal profit is equal to zero.

𝜕𝐸[𝜋_𝑛]

𝜕𝐿_𝑛 = (1 − 𝜇)𝑟_𝐿(𝐿^∗) − (𝑟_𝑀+ (𝑟_𝐾− 𝑟_𝑀)𝜅 + 𝛾_𝐿) + (1 − 𝜇)𝑟_𝐿^′(𝐿^∗)𝐿^∗_𝑛= 0 [7]

The optimal quantity of loans supplied by the individual firm can thus be solved as:

𝐿^∗_𝑛=(𝑟_𝑀+ (𝑟_𝐾− 𝑟_𝑀)𝜅 + 𝛾_𝐿) − (1 − 𝜇)𝑟_𝐿(𝐿^∗) (1 − 𝜇)𝑟_𝐿^′(𝐿^∗) [8]

The independence of 𝑛 implies a symmetric equilibrium where all banks choose the same quantity, i.e. 𝐿^∗_𝑛=^𝐿∗

Solving for the equilibrium interest rate 𝑟_𝐿^∗(𝐿) yields the expression:

𝑟_𝐿^∗(𝐿) = 1

Which can be written as a simple linear relationship between the bank lending rate and

This kind of simplification does imply certain restrictive assumptions on our empirical model. For 𝛽₀ to be constant over time, the variables related to the degree of

competition need to be unchanged over time. In a simple linear model, the constant 𝛽₀ can be augmented with a trend but it still rather restricted.

This simple expression shall serve as a starting point for the empirical investigation of the transmission of the policy rate to bank lending rates that is conducted in this paper.

2.4.3 Implications of the model

From equation [13] one can easily see what restrictions on variables 𝜇, 𝑁, 𝜀_𝐿(𝑟_𝐿), 𝜅, 𝑟_𝐾 𝑎𝑛𝑑 𝛾_𝐿 a linear regression with time invariant 𝛽₀ 𝑎𝑛𝑑 𝛽₁ imply. The comparative statics of the theoretical model are found by differentiating with respect to {𝑟_𝐷, 𝑟_𝑀, 𝑟_𝐾, 𝛾_𝐿, 𝜇, 𝑁, 𝜀_𝐿, 𝜅}. The lending rate is curiously found completely separable from the interest rate on deposits. The markup term 𝛽₀ is found positively dependent on increasing operating costs 𝛾_𝐿, cost of capital 𝑟_𝐾, the required capital to loans ratio 𝜅 and credit risk 𝜇. An increase in the number of competitors 𝑁 and elasticity of substitution 𝜀_𝐿 both have a negative effect on the equilibrium markup. The same parameters except for the cost of capital and the operating costs have an effect on the sensitivity towards the money market rate 𝛽₁, i.e. the pass-through. An increase in credit risk 𝜇 and the capital-to-loans ratio 𝜅 both increase the degree of pass-through. Increasing market power of individual banks, i.e. lower number of competitors 𝑁 and lower elasticity of substitution 𝜀_𝐿, both decrease the interest rate pass-through.