Abstract
This paper investigates the implications of the uncertain timing and usage of loan commitments for the optimal level of bank capital. We use trended Brownian motion to proxy the stochastic takedown of credit lines. Relying on “time to first passage” mathematics, we derive a probability density function for the time to depletion of the bank credit line as well as the likelihood for the time to exhausting the sources of liquidity that fund the loan takedown. Armed with these analytical results, we solve for the optimal level of bank capital within a simultaneous equation framework in order to capture the interrelationships of the endogenous variables. The optimality conditions produce a system of integral differential equations which refuse to yield reduced form solutions and provide no immediate intuition. Therefore, the maximizing values of the bank’s decision variables were simulated over a host of realistic scenarios. We document the comparative static behavior of the bank’s decision variables when equity is unencumbered by capital requirements and, also, examine the impact of the same parametric changes on bank behavior when equity is a fixed proportion of lending. Further simulations produce the expected time to liquidity depletion under different capital requirement schemes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Wilson (2013) analyzes the performance of banks accepting TARP funds and finds that banks with weak capital ratios, a large amount of charged off loans, and more non-performing loans were more likely to miss payment on dividends of TARP preferred stock. Additionally, banks that issued non-cumulative preferred stock were more likely to miss dividends.
We model the cash needs of the borrower as trended Brownian motion,
\(p(c_{o} ,c;t) = \frac{1}{{\sigma \sqrt {2\pi t} }}e^{{ - (c - c_{o} - \mu t)^{2} /2\sigma^{2} t}} ,\) so that the loan applicant’s cash demands at any point in time “t”, are normally distributed. If borrower demands are expected to grow rapidly then the parameter µ will be relatively large. If there is a great deal of uncertainty regarding the loan recipient’s financial needs then σ will be large. Using p(co, c; t), we are able to characterize the borrowing from the bank over time. In addition, the statistical properties of trended Brownian motion allow us to derive a probability density function for the time to depletion of the loan account, as well as likelihoods for the time to exhausting the respective bank assets that fund the loan takedown.
In our model, the credit line limit W* parameterizes the mean of the Brownian motion that describes credit line take down over time. Consider a simple deterministic example to illustrate our point. Let us say the optimal credit line limit, W*, for a seven day commitment is $14. If we presume the credit line is taken down smoothly over time and is exhausted at t = 7, then the takedown rate is $2 per day. Alternatively, if W* is 21, then the take down rate for a credit line that is drawn down evenly over time (and is completely exhausted at t = 7) would be $3 per day. We allow the W* to dictate the “mean rate”, μ, of the stochastic take down in this same fashion. In effect, we presume that, on average, the draw down is smooth and the borrower takes down the last dollar of a one year credit commitment on day 365.
Confirmation that \(\int_{0}^{\infty } {g(t)dt = 1}\) and that \(\int_{0}^{\infty } {t\,g(t)d\,t = \frac{{W - c_{0} }}{\mu }}\) is available from the authors upon request.
In our attempt to model the bank’s intertemporal expected profit, we include rates and fees that are common in commercial lending and consistent with current academic research. Banks often combine quarterly-adjusted market rates (for instance, LIBORj where j = 1, 2, 3, 4) with a fixed mark up rate ϕ, to determine the rate they charge for that part of the credit line that has been taken down by the borrower (LIBORj + ϕ).
While \(r_{N}\) (non-usage fee) is included in our intertemporal profit function, it is not clear that the non-usage fee should be a decision variable in this analysis. In particular, most of the academic literature assigns an informational role to \(r_{N}\), not one that is related to credit line pricing per se. Consequently, in our analysis, the value of \(r_{N}\) is given exogenously.
Clearly the bank’s expected profits depend upon the time to when R, G and the credit line are each depleted. Using the statistical properties of the borrower’s cash needs, we find that the time to depletion to each of these three accounts satisfy Kolmogorov’s diffusion equation. The solution to each of the three differential equations allows us to determine the probability density function of the time to depletion of each account; the loan account, R and G. Since solving for the time to exhaustion of any one of the accounts is much like solving for either of the other two, our analysis here will emphasize the time to depletion of the bank’s credit lines and the time to depletion of the remaining accounts will be utilized in this paper without explanation.
Over the last 30 years, due to advances in banking technology, banking competition, and the globalization of financial markets, bank asset conversion costs have certainly fallen. However, in the commercial banking industry asset liquidation costs still exist and they continue to be an aspect of academic models used in banking research.
In order to convert Treasuries into funds for financing a loan takedown, a bank must stand ready to make transactions in the capital markets on a daily basis. The decisions involved in effecting these trades are likely made by the bank’s own financial managers located on a trading desk at corporate headquarters. Whether a bank actually makes daily trades for billions of dollars, or a monthly trade for as little as a million dollars, the bank must maintain an operation capable of gathering market information and executing trades to fulfill the needs of loan commitment customers.
\(FC_{G}\) and \(FC_{EF}\) are set equal to zero in the simulations due to the difficulty of providing numeric characterizations for these fixed costs.
Palia and Porter (2004) find that capital levels have a positive impact on the total market value of the bank relative to total asset value.
Chang and Li (2011) model the equity of banks as a swap option on coupon bearing bonds.
An exponential functional form was chosen to represent the demand for bank capital, k, where \(k = e^{{ - \gamma r_{k} }}\). The partial derivative of the demand for equity by the bank with respect to rk is simply
\(\frac{\partial k}{\partial rk} = - \gamma e^{{ - \gamma r_{k} }} = - \gamma k\)
which details an inverse relationship between the magnitude of the demand for capital and the cost of equity. We picked a value for γ that allowed the elasticity of the demand to be in the neighborhood of −1. That is,
\(\frac{\partial k}{\partial rk}\frac{rk}{k} = ( - \gamma k)\frac{rk}{k} = - \gamma rk\)
and, for instance, if rk was 5 % then γ was given the value of 20.
The simulated values of the bank’s decision variables were based on discounted cash flows. However, for the sake of transparency and expository convenience, dozens of entries of a continuous time discount factor were suppressed in the presentation of our, already lengthy, four page objective function.
Chakravarty and Sarkar (1998) report a range of U.S. Treasury transactions cost where the mean is .1 % and standard deviation is 1.7 %. It is thus easy to imagine transactions cost during times of stress being .5 % or more.
Newton’s approach to the numeric solution was chosen for its simplicity and its convergence properties. A tolerance level of 10−7 was used in the simulations.
Barth et al. (1998) find that the relation between the capital/asset ratio and return on equity can be either positive or negative depending on the country of bank location.
This suggestion is from the Basel Committee consultative document entitled “International Framework for Liquidity Risk Measurement, Standards and Monitoring.” December 2009.
References
Asquith P, Mullins DW (1986) Equity issues and offering dilution. J Financ Econ 15:61–89
Barth JR, Gropper DM, Jahera JS (1998) A multi-country analysis of bank capital and Earnings. Rev Pac Basin Financ Markets Polic 1:123–155
Basel Committee Consultative Document (2009) International framework for liquidity risk measurement standards and monitoring
Chakravarty S, Sarkar A (1998) Liquidity in U.S. fixed income markets: a comparison of the bid-ask spread in Corporate, Government and Municipal Bond Markets. Working paper, New York Federal Reserve Bank
Chang CP, Li JH (2011) Optimal bank interest margin with synergy banking under capital regulation and deposit insurance: a swaption approach. Rev Pac Basin Financ Markets Polic 14:327–346
Deshnukh S, Greenbaum S, Kanatas G (1982) Bank forward lending in alternative funding environments. J Financ 37:925–940
Diamond D, Rajan R (2005) Liquidity shortages and banking crises. J Financ 60:615–647
Gao X, Ritter JR (2010) The marketing of seasoned equity offerings. J Financ Econ 97:33–52
Gatev E, Strahan P (2006) Banks advantage in hedging liquidity risk: theory and evidence from the commercial paper market. J Financ 59:867–892
Gatev E, Schuermann T, Strahan P (2009) Managing bank liquidity risk: how deposit-loan synergies vary with market conditions. Rev Financ Stud 22:955–1020
Kashyap A, Raghuram R, Stein J (2002) Banks as liquidity providers: an explanation for the coexistence of lending and deposit-taking. J Financ 57:33–73
Klein M (1971) A theory of the banking firm. J Money Credit Bank 3:205–218
Kopecky KJ, VanHoose D (2004) A model of the monetary sector with and without binding capital requirements. J Bank Financ 28:633–645
Landskroner Y, Paroush J (2008) Liquidity risk and competition in banking. Working paper
Lins K, Henri S, Tufano P (2008) What drives corporate liquidity? An international survey of strategic cash and lines of credit. American Finance Association
Monti M (1972) Deposit, credit, and interest rate determination under alternative bank objectives. In: Szego GP, Shell K (eds) Mathematical methods in investment and finance. North-Holland, Amsterdam
Palia D, Porter R (2004) The impact of capital requirements and managerial compensation on bank charter value. Rev Quant Financ Acc 23:191–206
Saunders A, Cornett MM (2014) Financial institution management: a risk management approach, chapter 16. McGraw-Hill, New York
Stein JC (1998) An adverse-selection model of bank asset and liability management with implications for the transmission of monetary policy. Rand J Econ 29:466–486
Sufi A (2009) Bank lines of credit in corporate finance: an empirical analysis. Rev Financ Stud 22:1057–1088
Wilson L (2013) TARP’s deadbeat banks. Rev Quant Financ Acc 41:651–674
Acknowledgments
The authors would like to express their appreciation to Fang Lin for his computational assistance.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Stanhouse, B., Stock, D. Credit line takedown and endogenous bank capital. Rev Quant Finan Acc 46, 691–723 (2016). https://doi.org/10.1007/s11156-014-0483-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11156-014-0483-z