Abstract
A bank loan commitment is often priced as a European-style put option that is used by a company with a known borrowing need on a known future date to lock in an interest rate. The literature has abstracted some of the important institutional features of a loan commitment contract. First, the timing, number, and size of the loan takedowns under such a contract are often random, rather than fixed. Second, companies often use loan commitment contracts to reduce the transaction costs of frequent borrowing and to serve as a guarantee for large and immediate random liquidity needs. Third, commercial banks maintain liquidity reserves for making random spot loans or random committed loans. Partial loan takedowns raise, rather than lower, the opportunity cost of a committed bank’s holding of excess capacity. This paper introduces a “stochastic needs-based” pricing model that incorporates these features. Simulations are conducted to illustrate the effects of various parameters on the fair price of a loan commitment.
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Notes
Shockley and Thakor (1997) have classified the usage of committed bank loans into seven categories, namely, “commercial paper backup,” “liquidity,” “capital structure,” “general corporate purposes,” “takeover,” “leveraged buyout,” and “debtor-in-possession.”
Recently, Agarwal et al. (2006) argue that a primary advantage of credit lines over term spot loans is that credit lines provide borrowers with financial flexibility. Sufi (2009) has refocused on the role of bank lines of credit as an instrumental component of corporate liquidity management. He suggests that according to Kashyap et al. (2002) and Gatev and Strahan (2006), banks are the most efficient liquidity providers in the economy, which implies that firms should rely on lines of credit over internal cash. However, there is a lack of interaction between the literature on cash and lines of credit for a bank’s liquidity provision.
In practice, many loan commitment contracts have upfront commitment fees and/or usage fees on any undrawn portion of the committed loan that may serve to offset at least partly the costs of holding idle reserves.
THG’s pricing model, on the contrary, takes into account a possible change in the credit risk of the borrower within the loan commitment period. Even though the credit risk of a spot loan is the same as that of a loan commitment contract at contract inception, it may differ significantly from that of a commitment loan when the contract is executed during the commitment period. Assumption (A2) holds only if a borrower’s credit risk does not change significantly during the loan commitment period.
More precisely, if r(0) > 0 and 2 αm ≥ σ 2, then r(t) > 0 for all t ∈ (0,T], where σ is the standard deviation of Brownian motion Z(t). Non-negativity is often regarded as a desirable property for interest rate processes.
Readers may consult Duan et al.’s (1995) article for a more detailed derivation of the non-liquid asset process. Duan et al. and others do not distinguish between liquid and non-liquid assets. In this paper, non-liquid assets involve long-term investments and, therefore, can be evaluated using the standard martingale pricing theory.
A simple discussion of the integration of such a stochastic differential equation is provided by Øksendal (2003).
In practice, a committed bank will hold a certain level of reserve and may rely on the interbank market occasionally. A committed bank that relies on the interbank market for obtaining liquidity when a committed bank loan is executed faces the risk of interbank borrowing rate fluctuation. On the other hand, there is an opportunity cost for holding idle liquidity reserve. The balance between holding a liquidity reserve and relying on the interbank market depends on the risk attitude and other operational characteristics of the committed bank. Taking such optimal behavior into account would introduce too much complication to the model.
Interested readers may obtain the program from the author on request. The Java Development Kit required for running the program can be downloaded from Sun Microsystem’s website at http://www.sun.com.
The random number generator Java program RngStream.java written by L’ecuyer et al. (2002) is available at http://www.iro.umontreal.ca/~lecuyer/myftp/streams00/java/RngStream.java. Accessed July 15, 2009.
The patterns in the loan commitment charges are similar for loan commitments used for general corporate operations.
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The author is Associate Professor of the Department of Finance and Insurance at Lingnan University. Financial support from the University Research Grants and the Academic Programme Research Grants of Lingnan University is highly appreciated. The author would like to thank an anonymous referee for providing valuable suggestions that have led to significant improvements in the paper. All errors belong to the author.
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Hau, A. Pricing of Loan Commitments for Facilitating Stochastic Liquidity Needs. J Financ Serv Res 39, 71–94 (2011). https://doi.org/10.1007/s10693-010-0083-6
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DOI: https://doi.org/10.1007/s10693-010-0083-6