روشی محاسباتی برای قیمت گذاری خط اعتباری بانک

Using trended Brownian motion to characterize borrower cash needs over time, we are able to derive a probability density function for the time to depletion of a bank credit line as well as the likelihoods for the time to exhausting the sources of liquidity that fund the loan. Armed with these analytic results, we solve for the credit line mark-up rate and the configuration of stored liquidity that maximizes the bank’s intertemporal expected profits from the loan. The optimality conditions produce a system of integral differential equations whose solutions we then simulate over a host of scenarios.

A bank loan commitment is a lending contract that provides a line of credit to a borrower.1 The size of the commitment is the maximum amount that can be borrowed by the loan recipient over the life of the contract. Customer usage of the line of credit is known as the “takedown”; borrowing against the line of credit can also be called “drawing down” the loan account. The contract has a maturity, known as the “duration” of the loan commitment. Commercial banks typically engage in “LIBOR plus” pricing where what is added to LIBOR is the bank’s mark-up rate which is fixed over the life of the loan contract. In addition, banks often impose a variety of fees on a loan commitment.2 These fees include a commitment fee, an annual service fee and a non-usage fee. The commitment fee is an up-front fee imposed at the time the loan commitment is made. The bank may charge an annual service fee on the amount borrowed every year. A non-usage fee is often charged on the amount of credit that is available but not used by the borrower. A bank can charge the customer any combination of these fees; some banks may charge a borrower all three fees, while others may not charge any of these fees.3 The existence of loan commitments can be explained by the benefits it provides to both the borrower and the lender. Commitments allow the borrower to transfer interest rate risk to the creditor. Campbell (1978) provides the first model to document this transfer as he solves for the fee structure of a fixed rate loan commitment that maximizes the bank’s expected utility. Boot et al. (1987) demonstrate that loan commitments provide incentives to the borrower to choose safer projects and put forth greater effort. Both choices result in higher profits for the borrower on average. Thakor (1989) argues that loan commitments allow the borrowing firm to overcome the pre-contract asymmetric information problems that plague debt financing.4 According to Maksimovic (1990), borrowers can gain an advantage over competitors who acquire funds in the spot market by purchasing the option to own a loan commitment. This option allows firms that operate under imperfect competition to determine the method of financing before its production decision is made. Shockley (1995) shows that firms that have more loan commitments than spot loans will benefit from a lower cost of debt, which leads to greater capacity to borrow.5 Loan commitments also provide benefits to the issuing bank. Credit line contracts assist banks in forecasting future loan demand according to Greenbaum et al. (1991). The authors show that the fee structure of the loan commitment gives the customer incentives to disclose information about its anticipated loan takedown. In exchange, the bank offers lower borrowing rates but reaps extra profits gained from the customer’s cooperation.6 In their analysis of financial contracting, Boot et al. (1993) illustrate how loan commitments allow banks to enhance its reputational capital. Banks that honor a financial contract, even if a borrower encounters financial distress, do so at a short-term financial cost, but they enjoy a long run gain in reputation for honoring its lending agreements. Thakor and Udell (1987) demonstrate how banks that offer loan commitments can acquire private information about borrowers that lenders in the spot market cannot obtain. In their discussion of loan commitments as the optimal form of bank financing, the authors show that information about the customer behavior (or type) is revealed by the choice of contract.7Avery and Berger (1991) find empirical evidence suggesting banks with large amounts of credit line loans (relative to traditional lending) have fewer defaults compared to their peers with fewer credit line commitments.8 In order to price a line of credit the bank must consider the implications of borrower needs for the timing and the magnitude of the loan’s takedown. Intertemporal bank revenues and funding costs clearly depend upon the nature of the stochastic drawdown of the loan commitment. If the credit line is entirely depleted within a few days of its inception then usage fees will generate sizable revenues while non-usage fees will be insignificant. The assets used to fund the loan takedown will give rise to very little interest income and asset conversion costs will be fully realized. If a dramatic loan takedown occurs near the maturity of the credit line then the bank’s usage fees will be small while the non-usage revenue will be relatively large. Assets, held to fund borrower takedown, will generate sizable interim income but suffer significant liquidation costs at the end of the loan contract. If the credit line goes nearly unused through the life of the loan commitment then usage fees will be negligible while the bank will enjoy increased income from non-usage fees. Interest earned from loan funding sources will be enhanced and asset liquidation costs will be small. Clearly if expected profits are to be maximized across time then the likelihood of these scenarios, and others, must be taken into consideration by the bank.9

Using trended Brownian motion to characterize borrower cash needs over time, we derive a probability density function for the time to depletion of a bank credit line as well as the likelihoods for the time to exhausting the sources of liquidity that fund the loan. Armed with these analytic results, we solve for the “price” that maximizes the bank’s intertemporal expected profits from the loan. Our results also provide the optimal configuration of the assets that the bank uses to finance the credit line as it is taken down stochastically over time. The optimality conditions produce a system of integral differential equations which refuse to yield reduced form equations for the endogenous variables. Consequently, we simulate the values of the bank’s decision variables which satisfied the FOCs over a host of scenarios. Characterizing View the MathML sourceϕ∗,x1∗, andView the MathML sourcex2∗ numerically is an artifact of a host of assumptions about the statistical characteristics of the borrower’s cash needs, the behavior of LIBOR over time, the rate of return available for overnight lending by banks, the return on treasuries, the cost of liquidating treasuries, the functional form of the loan demand equation, the magnitude of the elasticity of the loan demand equation, and the amount of funding available to finance a loan takedown. In an effort to establish an outcome that may be a little less encumbered, we spoke to the issue of bank coverage of outstanding loan commitments. In particular, we detail in this paper the endogenous behavior of internal bank liquidity if the lender manages the risk of a liquidity shortage while funding a credit line takedown.