14. Asset Side of Banking

Research on the banking sector was brought back to the center of the stage by the 2008 financial crisis. Broadly speaking, banking research covers financial institutions such as commercial banks, hedge funds and investment banks. The discussion in this part focuses on commercial banks.

Suppose there is no bank. Then every firm raises money directly from investors (direct financing). The key question: why do we need banks to stand in the middle, creating a structure of two layers? Two papers try to answer: Diamond (1984) proposes that banks improve efficiency by delegated monitoring; DeMarzo (2005) argues that banks improve efficiency by pooling and tranching under certain conditions.

The basic idea in Diamond (1984):

• a debt contract is about ex-post penalty;
• ex-ante monitoring could be better than ex-post penalty;
• but direct financing with ex-ante monitoring involves repetitive monitoring by each investor, which is inefficient;
• so a bank can be the delegated monitor to conduct monitoring only once;
• the bank's incentive comes from the debt (penalty) contract with its depositors.

14.1.1 Setup. Three types of agents in a risk-neutral world with information asymmetry:

• \(N\) firms: each needs 1 dollar to invest in its project, with no wealth; each firm \(i\) privately knows its realized cash flow \(\tilde y_i\) (unobservable to others); the investment technology is efficient, every project has positive NPV, \(\mathbb E[\tilde y_i]>1\).
• \(mN\) investors: each has \(\frac1m\) dollar of cash.
• Banks: each monitoring activity costs \(K\); after monitoring, the bank knows exactly the cash flow \(\tilde y_i\) of the firm monitored.

14.1.2 Ex-post penalty. Recall Townsend's (1979) CSV model: it is optimal to have a debt contract with a cutoff state \(j^\star\) that audits when the manager defaults with a payment shortfall; the audit takes away everything from the manager = penalty. In Diamond (1984) the penalty is non-pecuniary \(\xi(\hat y)\) and committed; it is also costly (takes \(\xi(\hat y)\) from the manager) but gives no benefit to anyone (the non-pecuniary condition ensures investors don't benefit from the penalty, so it is costly/inefficient). \(\xi(\hat y)\) is a function of the cash flow report \(\hat y\).

Manager's problem. Denote the true cash flow by \(y\), the report by \(\hat y(y)\), the payment to investors by \(z(\hat y)\). By the revelation principle, focus on the truth-telling equilibrium \(\hat y(y)=y\). The manager solves ex-ante (before observing \(y\)):

$$\max_{z(\cdot),\xi(\cdot)}\ \mathbb{E}[y-z(y)-\xi(y)] \tag{14.1}$$

$$0\le z(y)\le y \quad[\text{Limited Liability (LL)}] \tag{14.2}$$

$$\xi(y)\ge0,\ \forall y \quad[\text{Non-negative Penalty}] \tag{14.3}$$

$$y-z(y)-\xi(y)\ge y-z(y')-\xi(y'),\ \forall y,y' \quad[\text{Incentive Compatibility (IC)}] \tag{14.4}$$

$$\mathbb{E}[z(y)]\ge1 \quad[\text{Investor's participation (IR)}] \tag{14.5}$$

The deadweight loss of the debt contract is the inefficiency penalty \(\mathbb E[\xi(y)]\). Two cases: (1) \(m\) investors sign one contract with a firm; (2) \(m\) investors individually sign contracts with a firm. These are equivalent since \(\xi(y)=h-\min\{y,h\}\) is homogeneous of degree 1 in \((y,h)\). So banks won't play a role in this debt-with-penalty case.

One-investor case \(m=1\). Each investor has 1 dollar to invest in one firm. If monitoring is more efficient than penalty, i.e.

$$K<\mathbb{E}[\xi(y)]$$

each investor would monitor at cost \(K\) — so no bank is needed to reduce monitoring cost. If

$$K>\mathbb{E}[\xi(y)]$$

individual investors won't monitor, but banks are needed to pool projects to improve efficiency.

Multiple-investor case \(m>1\). For large enough \(m\), we have

$$mK>\mathbb{E}[\xi(y)]$$

which comes from the duplication of monitoring activities. So individual monitoring under direct financing is no longer feasible, and we need a bank as the delegated monitor to monitor only once. But:

• the bank cannot make the private cash-flow information public;
• so the bank also has an incentive to collude with the manager to under-report, splitting the benefit;
• so the monitor (bank) also needs to be monitored — but monitoring the bank by monitoring again duplicates the cost;
• instead, depositors sign a debt-with-penalty contract with the bank.

So there are two layers: Layer 1, the bank raises money from depositors through a debt-with-penalty contract; Layer 2, the firm manager raises money from the bank by being monitored.

All agents are risk-neutral, risk-free rate \(R_f=1.05\), monitoring cost \(K=0\). Two states:

• with probability 0.8, the \(G\) state: firm cash flow \(C=1.4\). To make lending possible, the face value \(F
• with probability 0.2, the \(B\) state: firm cash flow 1, defaults and repays 1; assume a deposit default penalty of 1 (e.g. inefficient liquidation destroying value physically), only in the \(B\) state, making the bank's cash flow $1-1=0$.

The face value \(F\) from the bank's break-even:

$$1.05=0.8\cdot F+0.2\cdot 0\ \Rightarrow\ 1.05=0.8F\ \Rightarrow\ F=\frac{1.05}{0.8}=1.3125$$

(in \(B\) the bank's cash flow is $1-1=0$ due to penalty cost). Under perfectly correlated payoffs the bank's solution is strictly dominated (just adds monitoring cost).

Under i.i.d. payoffs, suppose two payoffs are $(1.4,1.4)$ w.p. 0.64; $(1.4,1)$ w.p. 0.16; $(1,1.4)$ w.p. 0.16; $(1,1)$ w.p. 0.04. Without pooling, $(1.4,1)$ and $(1,1.4)$ each incur a deposit default penalty of 1 (inefficient); as long as pooling avoids the penalty in these states, pooling is better, i.e. we need \(C+1>2F\). Depositors receive (by competition) the same expected rate as the alternative \(R_f=1.05\) (only the $(1,1)$ state, w.p. 0.04, triggers a penalty):

$$(1-0.04)F=1.05\ \Rightarrow\ F=\frac{1.05}{0.96}=1.09375$$

i.e. a promised deposit return of 9.375% while the expected deposit return is 5%. To support this, the minimum good cash flow must satisfy

$$C+1>2F\ \Rightarrow\ C_{\min}=2F-1=1.1875$$

Since \(C=1.4>C_{\min}=1.1875\), pooling means the bank is not punished in $(1.4,1.4)$, $(1.4,1)$, $(1,1.4)$, so the penalty (e.g. inefficient liquidation) is less frequent, and pooling i.i.d. projects improves efficiency. By Prop 14.3, as the number of i.i.d. projects goes to infinity, the bank's cash becomes certain with probability 1 by the LLN, so default and inefficient penalty occur with probability zero — optimal for society.

From the numerical Example 14.1 we already see that pooling could be beneficial. DeMarzo (2005) focuses more on the pooling and tranching problem. (Recall DeMarzo and Duffie (1999): tranching is good for the issuer.) DeMarzo (2005) further discusses securitization, pooling and tranching.

14.2.1 Setup. There are \(N\) projects, each with independent payoff \(Y_i=X_i+Z_i\): \(X_i\) is private information to the manager; \(Z_i\) is not known by anyone. Four types of agents:

• informed originator: the firm raising money to finance projects, privately knows \(X_i\);
• informed intermediaries: banks who monitor at cost \(C\), then know \(X_i\) (same information set as the informed originator);
• uninformed originator: the firm raising money, doesn't have private information of \(X_i\);
• uninformed investors: break even in expectation, don't have private information of \(X_i\).

14.2.2 The main ideas. Two competing forces:

• information destruction effect: higher \(N\) means lower information advantage from the private information \(X_i\);
• risk diversification effect: higher \(N\) means lower unobserved errors \(Z_i\) (approaches 0) and thus stronger certainty.

It is shown that the risk diversification effect dominates as \(N\to\infty\). Also, pooling ensures that the bank is not cherry-picking among projects.