Uncertainty, Bank Lending, and Bank-Level Heterogeneity
2.2 Uncertainty and Bank Lending: A Stylized Model
To illustrate the concept of uncertainty which underlies this paper, we present a styl-ized model. Based on the model by Shin (2010), we can assess the impact of higher uncertainty in banking on banks’ loan supply. Assume that at timet, the balance sheet of the bank looks as follows:
Assets Liabilities
Loans lt Deposits dt
Cash and liquid assets ct Equity et
The bank makes loans lt at time t and receives an interest rate on loans at time t+ 1 of ˜rt+1: The loan rate ˜rt+1 is uncertain, as borrowers might not pay back the full loan but larger than zero in expectation (E[˜rt+1]>0). The deposit rate and the return on liquid assets are assumed to be risk-free and equal to zero.
The value of equity at time t+ 1 is then given by:
et+1 =lt(1 + ˜rt+1) +ct−dt=et+ ˜rt+1lt (2.1) The bank defaults in t+ 1 if the value of the equity is negative (et+1 <0), that is, if the return on loans is smaller than the amount of equity per unit of loans available to cover potential losses:
˜
rt+1 <−et
lt (2.2)
The VaR Constraint
We assume that the bank is risk-neutral but operates under a value at risk (VaR) constraint which is given by:
The VaR is defined as the loss not to be exceeded with probability 1−α, that is, V aRα =et/lt. We can think of the VaR constraint as reflecting how the bank manages its risk. A minimum capital requirement imposed by the regulator would have the same effect as a VaR constraint because any violation of the capital requirement would trigger regulatory interventions. The VaR constraint can also be defined as a deviation from the mean measured in terms of its standard deviation σt, which is assumed to be known at time t, that is
P rob(˜rt+1 < µ−φσt)≤1−α (2.4)
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whereφ is some constant.
We assume that the bank maximizes its shareholder value et+1 at timet+ 1. In the absence of the VaR constraint, the risk-neutral bank would give out as many loans as possible. The reason for this is that the expected return is larger than zero. In principle, the size of the balance sheet would thus be indeterminate. However, the VaR constraint under which the bank operates determines the size of the loan portfolio. This can be seen by combining Equations (2.3) and (2.4):
lt= et
φσt−µ (2.5)
Uncertainty in Banking and Loan Rates
We assume that loan rates of bank i follow a stochastic process with time-varying volatility:
˜
rit+1 =µi +σtεit+1 (2.6)
where E[εit] = 0 and εit ∼ N(0,1), which implies that the mean of the loan rate conditional on time t information (Iit) is constant, so E[˜rit+1|Iit] = µi. Although the assumption of µi being constant might appear too restrictive, its interpretation can easily be generalized to the predicted value of the loan rate at time t without making a specific assumption on the underlying prediction model. Therefore, we can simply refer toµi as the predicted portion of the loan rate and to σtεit+1 as the unpredicted portion for each banki.
The volatility of the bank-specific shock εit+1 to the loan rate is time-varying.
Regarding the timing convention, we follow Bloom et al. (2012) and assume that banks know in advance about any potential change in business conditions. This would be reflected in a change in the distribution of shocks and thus the volatilityσt. A higher σt can be interpreted as higher uncertainty because it widens the distribution of ˜rit+1. Hence, it constitutes a measure ofuncertainty in banking. More specifically, we assume that in t, the bank can condition its portfolio decision on the level of uncertainty σt. Although the bank knows that the distribution of shocks has widened, the bank does not learn about the realization of the loan rate ˜rit+1 before t+ 1.
Uncertainty in Banking and Loan Supply
We now show how higher uncertainty in banking affects the structure of banks’ assets and induces the banks to shift from risky to safe assets. Starting from the optimal
Chapter 2: Uncertainty, Bank Lending, and Bank-Level Heterogeneity
periods, we can derive the change in loans from t−1 to t relative to total assets at time t−1 (tat−1 = et−1 +dt−1).3 This will constitute the dependent variable in the
We assume that, at timet, the bank learns about changes in uncertainty in banking and can incorporate this information into its loan supply decision. The effect of higher uncertainty on the change in loans relative to total assets in t−1 is given by the first partial derivative of Equation (2.7) with respect to uncertainty σt:
∂ta∆lt
The bank reduces the volume of loans on its balance sheet, that is, it supplies fewer loans, if uncertainty in banking increases. The inequality holds as long asrt> µ−φσt−1, which implies that the bank is solvent int (see Equation (2.2) combined with Equation (2.5)).
The Role of the Capital Buffer
In the empirical analysis, we will investigate how the response of banks to uncertainty depends on bank-level characteristics. One of these characteristics is the capital buffer that a bank holds. A bank might voluntarily choose to hold capital above the regulatory requirement to shield itself against unexpected losses. The probability to incur such an unexpected loss, in turn, depends on the bank’s capital buffer. A bank with a capital buffer is subject to a more stringent (but voluntarily chosen) VaR constraint at time t−1 which can be relaxed at time t (φt−1 > φt). In the Appendix, we show that a bank holding a capital buffer reduces the loan volume by less as long as it still receives a positive return on its loans (rt>0).
Empirical Implications
The model illustrates a specific mechanism how uncertainty in banking – modeled as an increase in the standard deviation of loan rates – affects banks’ behavior. Under the assumption that loans are the only risky assets of banks, uncertainty in banking directly translates into a higher dispersion of shocks to observable bank-level outcomes such as return on assets or asset growth. We apply this idea to our empirical analysis
3 For notational convenience, we will skip the bank indexiin the following.
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and measure uncertainty as the cross-sectional dispersion of bank-specific shocks. In reality, banks have a more general asset and liability structure and thus might be affected by uncertainty through a range of other channels. In addition to dispersion in shocks to asset growth and return on assets, we will thus also account for shocks to bank productivity and to short-term funding. Heterogeneity in banks’ responses to these shocks, in turn, will be modeled as a function of their capital buffer, the structure of their assets, and their ownership status (domestic vs. foreign).