Relationship Lending and Switching Costs under Asymmetric Information about Bank Types

Abstract

This theoretical paper extends the pioneering articles on relationship lending (e.g., Sharpe. J Finance XLV(4): 1069-1087, 1990; Rajan. J Financ 47: 1367–1400, 1992; von Thadden. Financ Res Lett 1(1): 11–23, 2004) by examining relationship lending and hold-up problems in credit markets when borrowers are identical and banks are different. The results show that existing borrowers are informationally captured by good banks and yield profits to them, but new borrowers are unprofitable. In this market, short-term loan contracts and unsecured loans are optimal while loan commitments should not be used. Further, banks and borrowers have long-term relationships. This paper challenges the standard theories on product quality, reputation and experience goods by introducing scenarios in which good and bad banks can retain their existing borrowers. In the standard theories, consumers leave bad producers and search for good ones.

Appendices

Appendix 1

Firms that did not borrow from Bank X in the first period, which are labeled outside firms with respect to Bank X, observe a signal of the bank type in period 1. This signal has a conditional distribution function:

$$ {\displaystyle \begin{array}{c} prob\left(\overset{\sim }{y}=\overset{\sim }{G}|G\right)= prob\left(\overset{\sim }{\gamma }=\overset{\sim }{B}|B\right)=\left(1+\varnothing \right)/2,\\ {} prob\left(\overset{\sim }{y}=\overset{\sim }{G}|B\right)= prob\left(\overset{\sim }{\gamma }=\overset{\sim }{B}|G\right)=\left(1-\varnothing \right)/2,\end{array}}, $$

(26)

0 ≤  ∅  ≤ 1. Here $\overset{\sim }{\gamma }(X)\in \left\{\overset{\sim }{G},\overset{\sim }{B}\right\}$ is a noisy signal regarding Bank X. The signal is either good, $\overset{\sim }{G},$ or bad, $\overset{\sim }{B}.$ All outside firms and Bank X observe the same signal, and they do so without cost. Thus, if ∅ = 0 (∅ = 1), then outside firms learn nothing (have perfect information) about the bank. Consider a bank with a good signal. It represents the good type with probability:

$$ prob\ \left(G|\overset{\sim }{G}\right)=\frac{g\left(1+\varnothing \right)}{g\left(1+\varnothing \right)+\left(1-g\right)\left(1-\varnothing \right)}. $$

(27)

Here $prob\ \left(G|\overset{\sim }{G}\right)$ increases with ∅. If ∅ = 0, we have $prob\ \left(G|\overset{\sim }{G}\right)=g$ . If ∅ = 1, then we get $prob\ \left(G|\overset{\sim }{G}\right)=1$. A loan succeeds with certainty if the bank is good, and with probability p , if it is bad. If a borrower contacts an outside bank with a good signal, their expected return is:

$$ \left[1- prob\ \left(G|\overset{\sim }{G\ }\right)\ \right]\ p\left(Y-R\right)+\kern0.5em prob\ \left(G|\overset{\sim }{G\ }\right)\left(Y-R\right). $$

(28)

The first (second) term shows expected repayments if the new bank proves to be bad (good). Recall from Eq. (3) in subsection 3.1 the firm’s expected return if it retains the initial loan relationship, ${\pi}_2^G=Y-{R}_2^G$. The initial bank can retain the lending relationship if ${\pi}_2^G=Y-{R}_2^G$ is at least (28), that is, the interest rate of the initial bank is at most:

$$ {R}_2^G=R+\left[1- prob\ \left(G|\overset{\sim }{G\ }\right)\ \right]\ \left(1-p\right)\left(Y-R\right). $$

(29)

If the signal has no value, ∅ = 0, then we have $prob\ \left(G|\overset{\sim }{G\ }\right)=g$ and ${R}_2^G$ is the same as in Eq. (4) in subsection 3.1. The initial bank has a great information advantage in period 2, and it can set a high interest rate. If the signal is perfect, ∅ = 1, then we have $prob\ \left(G|\overset{\sim }{G\ }\right)=1$ and ${R}_2^G=R:$ banks operate under perfect competition in period 2, and the information advantage of the initial bank disappears. If ∅ increases, then the transparency of the banking system improves, the information advantage of the initial bank shrinks and ${R}_2^G$ declines. Since a lending relationship yields zero expected profit, the improved transparency rises the loan interest of period 1. Finally, we have implicitly assumed that the noisy signal does not change bank competition. The signal may give market power to banks if the number of banks with a good signal is small compared to the number of firms.

Appendix 2

This appendix shows that there is a proportion of bad banks to good banks where all bad banks make zero profit. To begin, recall the expected profit of a bad bank from Eq. (6)

$$ {\pi}_B(b)=\left(1-b\right)\left(\ p\ {R}_1-r\right)+\delta\ \left(1-b\right)b\left(\ pR-r\ \right)-{C}_B. $$

(30)

First, we show that if the proportion of bad banks is very small, they are profitable. Assume that the proportion of bad banks is b_ε > 0. In period 1, the interest rate is R₁ = R − (1 − p) b_ε(Y − R). Here b_ε can be chosen so that p R₁ > pR − ε with each ε. Therefore, it is possible to have b_ε such that a loan is profitable and covers the cost of bank formation: p R₁ − r − C_B> pR − ε − r − C_B > 0 . The latter inequality is based on Assumption 2, pR − r − C_B > 0 , and the fact that ε is small enough. We know that π_B(b_ε) > 0.

We aim to find out the optimal proportion of bad banks. Now Eq. (30) becomes:

$$ \frac{d{\pi}_B}{d\ b}=-2b\left( pR-r\right)-\left(1-2b\right)p\left(1-p\right)\left(Y-R\right). $$

(31)

Two scenarios occur: Scenario 1: If 2(pR − r) > p(1 − p)(Y − R) in Eq. (31), we have dπ_B/db < 0 with each b and the expected return minimizes if b = 1. If b = 1, then Eq. (30) shows π_B(1) =  − C_B < 0. Given this, π_B(b_ε) > 0 and dπ_B/db < 0, there is b^∗ such that the expected profit is zero, π_B( b^∗)= 0. Scenario 2: If 2(pR − r) < p(1 − p)(Y − R) in Eq. (31), we get dπ_B/db = 0 if b = b_min,

$$ {b}_{min}=\frac{1}{2}\kern0.5em \ast \frac{p\left(1-p\right)\left(Y-R\right)}{p\left(1-p\right)\left(Y-R\right)-\left( pR-r\right)}\kern0.5em . $$

(32)

Here we have 0.5 < b_min < 1. When b = b_min, the bank profit achieves the minimum value. The minimum bank profit without the cost of bank formation, C_B, is:

$$ {\pi}_B\left({b}_{min}\right)=\Phi\ \left[\ \left(1-p\right)p\left(Y-R\right)-\left( pR-r\right)\right]\ \left[\ \left(1-p\right)p\left(Y-R\right)-2\left( pR-r\right)\ \right]. $$

(33)

where $\Phi =\frac{-{b}_{min}\left(1-{b}_{min}\right)}{p\left(1-p\right)\left(Y-R\right)}$.

The terms in both square brackets are positive owing to the definition of Scenario 2. Since Φ < 0, (33) is negative. The bank gets a negative return if b = b_min even without −C_B. When b_min ≤ b < 1, we have dπ_B/db > 0 and π_B(b) < 0 even without −C_B. Therefore, the optimal proportion of bad borrowers is lower than b_min. When 0 ≤ b ≤ b_min, we have π_B(b_ε) > 0, dπ_B/db < 0, and π_B(b_min) < 0 . There is a proportion of bad banks, b^∗∗, 0 < b^∗∗ < b_min, such that a bad bank makes zero profit, π_B(b^∗∗) = 0.

In sum, let b_max = b^∗ if 2(pR − r) > p(1 − p)(Y − R) in Eq. (31) and b_max = b^∗∗ if 2(pR − r) < p(1 − p)(Y − R) in (31). Now, b_max gives the maximal proportion of bad banks such that each of them has zero profit.