10. Incomplete Contract and Debt Maturity Structure

This chapter talks about studies on the debt maturity structure of non-financial institutions, with the traditional flavor of principal-agent problems under incomplete contract conditions. (Financial institutions are different, but have similar core issues.)

In a frictionless world, i.e. complete information + complete contracts, maturity mismatch doesn't matter — since debt maturity does nothing to tax benefits or management incentives. So the following three scenarios are indifferent to the borrower:

1. borrow debt with the same maturity as the project cash flow (asset maturity);
2. borrow debt with longer maturity than the cash flow (asset maturity), then renegotiate and pay off all debt when receiving cash flow from the project;
3. borrow debt with shorter maturity than the cash flow (asset maturity), then roll over debt until receiving cash flow from the project.

However, in a world with frictions, debt maturity does matter to firms. Below are studies on the trade-offs between long-term debt and short-term debt.

10.1.1 Basic ideas. Diamond (1991) discusses short-term versus long-term debt, with the following key ideas.

• Benefit of short-term debt:
• good firms and bad firms are pooled together;
• good firms have a higher probability of receiving an upgrade credit report to reduce borrowing cost;
• so good firms would like to borrow short-term to roll it over later at a possibly lower rate.
• Cost of short-term debt:
• both good and bad firms have a positive probability of receiving a downgrade credit report;
• if a downgrade happens, firms (even good ones) would face liquidity risk (here liquidity risk means the firm could not roll over its short-term debt and is forced to be physically liquidated).
• Trade-off:
• the model wants liquidation to be efficient for bad firms but inefficient for good firms;
• the model wants the manager (agent) to never want to be liquidated, so it gives the manager a control rent \(C\) independent of firm types, which is the non-pledgeable part of the project cash flow;
• the trade-off is between the lower roll-over interest and the liquidity risk in case of a downgrade report.
• Conclusion: good firms want short-term debt in some cases and long-term debt in other cases.
• Since it's a pooling equilibrium, bad firms always mimic the choice of good firms.

10.1.2 Setup.

• All agents are risk neutral; the discount rate is zero.
• Two types of firms: good \(G\) and bad \(B\). Firms privately know their own type; the creditor only has a probability distribution of the two types. Good firms have positive NPV, bad firms negative NPV.
• Two types of debt:
• Short-term debt: maturity of only one period. The face value of lending 1 at \(t=0\) is \(F_0^S\); the face value of lending \(F_0^S\) at \(t=1\) is \(F_1^S\). The lender could liquidate the firm at \(t=1\) if not repaid by face value and no agreement is reached in renegotiation.
• Long-term debt: maturity of two periods. The face value of lending 1 at \(t=0\) is \(F_0^L\). The lender could not liquidate the firm at \(t=1\).
• Three periods \(t=0,1,2\). Both types need \(I=1\) invested in a project at \(t=0\), which has a cash payoff only at \(t=2\).
• At \(t=0\): the creditor has a prior \(f_0\in(0,1)\) of a good firm and \(1-f_0\) of a bad firm.
• At \(t=2\):
  • type \(G\): cash flow is for sure \(X>1\);
  • type \(B\): if the manager works hard, cash flow is \(X>1\) with probability \(p\) and 0 with probability \(1-p\); if the manager shirks, cash flow is \(X>1\) with probability \(p-\Delta\) and 0 with probability \(1-p+\Delta\) (\(\Delta\in(0,p)\)), and the manager enjoys a private benefit \(b\) from shirking.
  • Managers of both types receive a control rent as compensation at \(t=2\) if cash flow is \(X\).
• At \(t=1\): a credit rating (upgrade or downgrade) is public, telling the Bayesian-updated probability of good/bad; firms with short-term debt could issue new short-term debt to roll over the retiring one; if the lender decides to liquidate, the liquidation value is \(L\ge0\).

10.1.3 Incentive compatibility for the manager's effort. To make the manager always want to exert effort (efficient), the payoff from working hard must be at least the payoff from shirking:

$$p\cdot C\ge(p-\Delta)\cdot C+b$$

$$\Rightarrow \Delta C\ge b$$

$$\Rightarrow C\ge\frac{b}{\Delta} \tag{10.1}$$

(10.1) is the IC constraint for the manager, assumed to hold throughout.

• For bad firms \(B\): probability of upgrade is 0; probability of downgrade is 1.
• For good firms \(G\): probability of upgrade is \(1-e\); probability of downgrade is \(e\).
• Denote \(f_d\) as the Bayesian-updated probability of a type-\(G\) firm after a downgrade at \(t=1\), and \(f_u\) after an upgrade.
• By Bayes' rule:

$$f_d=\frac{f_0\cdot e}{f_0\cdot e+(1-f_0)} \tag{10.2}$$

$$f_u=1$$

• (10.2) implies:

$$f_0 e+(1-f_0)=\frac{f_0}{f_d}e$$

$$\Rightarrow\left(f_0-\frac{f_0}{f_d}\right)e=-(1-f_0)$$

$$\Rightarrow\frac{f_0(f_d-1)}{f_d}e=-(1-f_0)$$

$$\Rightarrow e=\frac{f_d(1-f_0)}{f_0(1-f_d)} \tag{10.3}$$

• (10.3) says the probability of a downgrade report for a type-\(G\) firm is decreasing in the prior belief \(f_0\).
  • If \(f_0\) is high enough (close to 1), then misreport (i.e. \(G\) gets a downgrade) is very unlikely.

The equilibrium is always a pooling equilibrium: good firms \(G\) have positive NPV, bad firms \(B\) negative NPV; in any separating equilibrium bad firms never get financed, so bad firms always mimic the financing choice of good firms.

Long-term debt: its face value \(F_0^L>1\) is set so the creditor breaks even (assuming a competitive credit supply market):

$$F_0^L[f_0+(1-f_0)p]=1\ \Rightarrow\ F_0^L=\frac{1}{f_0+(1-f_0)p} \tag{10.4}$$

Short-term debt: suppose type \(G\) selects short-term debt.

• In the pooling equilibrium, given an upgrade, \(G\) can roll over \(F_0^S\) at zero cost, i.e. \(F_1^S=F_0^S\), which is attractive to a good firm.
• Given a downgrade, the face value \(F_1^S\) of borrowing \(F_0^S\) at \(t=1\) satisfies

$$F_1^S[f_d+(1-f_d)p]=F_0^S\ \Rightarrow\ F_1^S=\frac{F_0^S}{f_d+(1-f_d)p}\ge\frac{1}{f_d+(1-f_d)p} \tag{10.5}$$

• Liquidity risk: if

$$\frac{1}{f_d+(1-f_d)p}>X$$

then by (10.5) it implies \(F_1^S>X\), so refinancing (rolling-over) fails (feasibility requires \(F_1^S\le X\), i.e. you can't borrow more than the maximum you can pay back). - When rolling-over fails, the firm, not wanting to lose the control rent \(C\), would give the whole \(X\) to the creditor at \(t=2\) if not liquidated at \(t=1\). - Then the creditor can liquidate and receive \(L\), or not liquidate and wait until \(t=2\) to receive \([f_d+(1-f_d)p]X\): - if \([f_d+(1-f_d)p]X\ge L\), the creditor will not liquidate and will extend maturity; - if \([f_d+(1-f_d)p]X

Conclusion (varying with the prior \(f_0\)):

• High enough \(f_0\): by (10.3) \(e\) is almost zero, a downgrade for \(G\) is almost impossible, the trade-off becomes trivial (liquidity risk disappears), and good firms prefer short-term debt to long-term — like commercial paper (short-term unsecured debt with very high credit rating and very low default risk).
• Medium \(f_0\): \(e\) is non-trivial, a downgrade for \(G\) is possible, the liquidity risk of short-term debt outweighs its benefit, so \(G\) prefers long-term debt — like corporate bonds.
• Low enough \(f_0\): by (10.4) \(F_0^L=\frac{1}{f_0+(1-f_0)p}\) decreases in \(f_0\), so a low \(f_0\) makes \(F_0^L\) so high that

$$F_0^L>X$$

which is impossible (need \(F_0^L\le X\)), so long-term debt is infeasible and the firm can only issue short-term debt — like bank debt.

Diamond (1991) implicitly assumes a competitive credit supply market, so the face value of both short- and long-term debt always makes the creditor break even. However, with a lending monopoly, the ability to refinance (roll over short-term debt) may not respond to new information (e.g. a credit report). Rajan (1992) discusses public debt versus bank debt and allows for monopoly in the bank-debt credit market.

10.2.1 Setup.

• All agents risk neutral, discount rate zero. Two debt types: short-term (one period; the lender may liquidate at \(t=1\) if not repaid in full and no agreement) and long-term (two periods; no liquidation at \(t=1\)). Three periods \(t=0,1,2\).
• A project needs investment \(I\) at \(t=0\) and only pays \(X\) (success) or 0 (failure) at \(t=2\).
• At \(t=0\): the agent exerts effort \(e\) at cost \(\beta(e)\) (\(\beta'(e)>0\)), which affects the probability of a good signal \(G\) about success at \(t=1\):
• with probability \(q(\beta)\) the signal is \(G\), meaning the project succeeds for sure at \(t=2\);
• with probability \(1-q(\beta)\) the signal is \(B\), meaning success with probability \(p_B\) and failure with \(1-p_B\) at \(t=2\);
• \(q'(\beta)>0\), \(q''(\beta)<0\).
• At \(t=1\), after observing the signal, the creditor may inefficiently liquidate for \(L\ge0\). Assume \(Lp_B X\) (liquidation efficient given \(B\), the risk-neutral creditor always wants to liquidate).
• The manager has zero wealth and must borrow from arm-length investors (no private contact, simply offering public debt) or a bank. Suppose the manager always wants to continue at \(t=1\) (justified by a tiny control rent \(C\)). Suppose the manager has Nash bargaining power \(\mu\in(0,1)\) against the monopoly bank.

10.2.2 First-best effort level. Since liquidation is inefficient under \(G\) and efficient under \(B\), the social-welfare-maximizing planner continues under \(G\) and liquidates under \(B\). The first-best effort cost \(\beta^{FB}\) solves

$$\max_\beta\ q(\beta)X+(1-q(\beta))L-\beta$$

with f.o.c.

$$q'(\beta^{FB})X-q'(\beta^{FB})L=1\ \Rightarrow\ q'(\beta^{FB})=\frac{1}{X-L} \tag{10.6}$$

Note \(q'(\cdot)\) is decreasing in the effort cost \(\beta\) and thus decreasing in effort \(e\).

Arm-length investors offering public debt cannot observe the interim signal \(G\) or \(B\) at \(t=1\), so \(t=1\) is trivial for the public-debt market. Thus suppose the public-debt market only offers long-term debt with face value \(D^P\) (borrowing \(I\) at \(t=0\)). The manager (note: he never voluntarily liquidates unless forced; even though liquidation is efficient under \(B\), the liquidation revenue only goes to the creditor, so he has no way to do so) chooses the optimal effort cost \(\beta^P\) by solving

$$\max_\beta\ q(\beta)(X-D^P)+(1-q(\beta))p_B(X-D^P)-\beta$$

with f.o.c.

$$q'(\beta^P)(X-D^P)-q'(\beta^P)p_B(X-D^P)=1\ \Rightarrow\ q'(\beta^P)=\frac{1}{(X-D^P)(1-p_B)} \tag{10.7}$$

Comparing \(\beta^P\) with \(\beta^{FB}\): for a risk-neutral investor to invest at \(t=0\) we need \(D^P>I\) (strict, from the positive probability of zero payoff); feasibility needs \(X>D^P\). So \(X>D^P>I>L\), which gives

$$p_B X+D^P(1-p_B)>p_B D^P+D^P(1-p_B)=D^P>L$$

$$\Rightarrow(X-D^P)(1-p_B)

$$\Rightarrow\frac{1}{(X-D^P)(1-p_B)}>\frac{1}{X-L}\ \Rightarrow\ q'(\beta^P)>q'(\beta^{FB})\ \Rightarrow\ \beta^P<\beta^{FB}$$

i.e. the manager puts in too low effort (relative to first best) when borrowing long-term public debt.

The manager has zero cash at \(t=1\) and can never repay short-term bank debt at \(t=1\). With a monopolistic bank as the only lender, manager and bank Nash-bargain at \(t=1\).

• Signal \(B\): manager's outside option 0, bank's \(L\); total surplus \(p_B X-L<0\), so no renegotiation and the project is liquidated.
• Signal \(G\): manager's outside option 0, bank's \(L\); total surplus \(X-L>0\), so they renegotiate and the project is not liquidated. The manager gets a \(\mu\) share, \(\mu(X-L)\), and the bank gets \((1-\mu)(X-L)\).

The manager chooses the optimal effort cost \(\beta^{BS}\) by solving

$$\max_\beta\ q(\beta)\mu(X-L)+(1-q(\beta))\cdot 0-\beta$$

with f.o.c.

$$q'(\beta^{BS})\mu(X-L)=1\ \Rightarrow\ q'(\beta^{BS})=\frac{1}{\mu(X-L)} \tag{10.8}$$

Comparing \(\beta^{BS}\) with \(\beta^{FB}\):

$$\mu\in(0,1)\ \Rightarrow\ \mu(X-L)\frac{1}{X-L}\ \Rightarrow\ q'(\beta^{BS})>q'(\beta^{FB})\ \Rightarrow\ \beta^{BS}<\beta^{FB}$$

i.e. the manager puts in too low effort under short-term bank debt.

Effect of the manager's bargaining power: greater power makes \(\frac{1}{\mu(X-L)}\) smaller, so by (10.8) \(\beta^{BS}\) is greater; if \(\mu\to1\) effort approaches the first best \(\beta^{BS}\to\beta^{FB}\). Intuition: higher bargaining power means more rewards for a better outcome, so the manager internalizes more of the benefit of effort, reducing distortion and raising optimal effort.

Let \(D^{BL}\) be the face value of long-term debt (borrowing \(I\) at \(t=0\)). With a monopolistic bank, they Nash-bargain at \(t=1\).

• Signal \(B\): manager's outside option \(p_B(X-D^{BL})\), bank's \(p_B D^{BL}\); total surplus \(L-p_B X>0\), so they renegotiate and the project is liquidated. The manager gets his outside option plus a \(\mu\) share: \(p_B(X-D^{BL})+\mu(L-p_B X)\); the bank gets \(p_B D^{BL}+(1-\mu)(L-p_B X)\).
• Signal \(G\): manager's outside option \(X-D^{BL}>0\), bank's \(D^{BL}>I>L\); the surplus of renegotiation-and-liquidation is \(L-X<0\), so they do not renegotiate.

The manager chooses the optimal effort cost \(\beta^{BL}\) by solving

$$\max_\beta\ q(\beta)(X-D^{BL})+(1-q(\beta))[p_B(X-D^{BL})+\mu(L-p_B X)]-\beta$$

with f.o.c.

$$q'(\beta^{BL})[(X-D^{BL})-p_B(X-D^{BL})-\mu(L-p_B X)]=1$$

$$\Rightarrow q'(\beta^{BL})=\frac{1}{(1-p_B)(X-D^{BL})-\mu(L-p_B X)} \tag{10.9}$$

Comparing \(\beta^{BL}\) with \(\beta^{FB}\):

$$D^{BL}>L\ \Rightarrow\ X-D^{BL}

$$\Rightarrow\frac{1}{(1-p_B)(X-D^{BL})-\mu(L-p_B X)}>\frac{1}{X-L}\ \Rightarrow\ q'(\beta^{BL})>q'(\beta^{FB})\ \Rightarrow\ \beta^{BL}<\beta^{FB}$$

i.e. the manager puts in too low effort under long-term bank debt. Effect of bargaining power: greater power makes \(\frac{1}{(1-p_B)(X-D^{BL})-\mu(L-p_B X)}\) greater, so by (10.9) \(\beta^{BL}\) is lower (the opposite of short-term bank debt).

10.2.6 Comparing public debt and monopolistic bank debt.

• Manager's effort is smaller with public debt: long-term debt without a liquidation threat provides a weaker incentive to exert effort.
• Manager's effort is also smaller with bank debt:
• short-term debt: there is a hold-up problem — the benefits of the manager's effort could be robbed by the bank, so the manager doesn't want to put in too much effort;
• long-term debt: the renegotiation-and-liquidation mechanism reduces the manager's loss from signal \(B\), so he faces a softer penalty for low effort, which makes him reduce effort.
• Short- vs long-term bank debt: \(\beta^{BS}\) increases in bargaining power \(\mu\), while \(\beta^{BL}\) decreases in \(\mu\).
• Very large \(\mu\) → short-term debt is more socially optimal; very small \(\mu\) → long-term debt is better.
• Intuition: large \(\mu\) → the manager internalizes more benefit from not liquidating (less distortion in short-term debt) and enjoys more benefit from the softer penalty (more distortion in long-term debt), so short-term debt is better at providing effort incentives; small \(\mu\) reverses this, so long-term debt is better.

10.2.7 Monopolistic competition of banks. Rajan (1992) further discusses two banks competing, one informed and one uninformed. With competing banks, the manager can raise money from the uninformed bank to repay the short-term debt to the informed bank given signal \(B\), so the project may inefficiently survive even after signal \(B\) — the cost of bank competition.

Rajan (1992) discusses one cost of bank competition (reduced efficient liquidation), i.e. the benefit of monopoly. Petersen and Rajan (1995) discuss another benefit of monopoly:

• initial lending rates have to be high (information asymmetry → high uncertainty about a young firm's quality → banks must lend at a high initial rate to compensate for high risk);
• but young firms cannot borrow at too high a rate (high rates distort incentives, possibly turning the project NPV from positive to negative);
• under monopoly, the bank can lend at a low rate and lose money initially, then raise the rate later to recoup the initial loss;
• the monopoly bank can choose the optimal timing of the rate increase so the project still has positive NPV and its own payoff is highest;
• the bank wants this across-time subsidy because firms may not be able to borrow if the initial rate is too high;
• however, under competition the bank can never make profits and must break even period by period;
• therefore some positive-NPV projects requiring a low initial rate cannot be financed — a cost of bank competition and a benefit of lending monopoly.

10.3.1 Setup. All agents risk neutral, discount rate zero. Start with two periods \(t=0,1\). Two project types both need initial investment \(I\): the safe project pays \(R>I\) next period; the risky project pays \(X>I\) with probability \(p\) and 0 with \(1-p\). Suppose \(X>R>I>pX\). Two firm types: \(G\) with probability \(\theta\), \(B\) with \(1-\theta\); \(G\) can choose safe or risky, \(B\) only risky.

10.3.2 Two-period model. Focus on the standard debt contract (note: not an innocuous assumption). Denote the face value of borrowing \(I\) by \(F\). If the bank lends to \(B\), or to \(G\) and \(G\) does risk-shifting by choosing the risky project, the bank loses money (\(I>pX\ge pF\)). So the bank never wants to lend to \(B\); to lend to \(G\), it must prevent \(G\)'s risk-shifting, i.e. satisfy \(G\)'s IC constraint:

$$R-F\ge p(X-F)$$

$$\Rightarrow(1-p)F\le R-pX$$

$$\Rightarrow F\le\underbrace{\frac{R-pX}{1-p}}_{=F^{\max}} \tag{10.10}$$

i.e. the highest face value is \(F^{\max}=\frac{R-pX}{1-p}\). The ex-ante break-even condition:

$$I=\theta F^{BE}+(1-\theta)pF^{BE}\ \Rightarrow\ [\theta+(1-\theta)p]F^{BE}=I\ \Rightarrow\ F^{BE}=\frac{I}{\theta+(1-\theta)p} \tag{10.11}$$

For the bank to at least break even, the highest face value must make it break even (otherwise the bank never invests), i.e. by (10.10) and (10.11):

$$F^{\max}\ge F^{BE}\ \Rightarrow\ \frac{R-pX}{1-p}\ge\frac{I}{\theta(1-p)+p}$$

$$\Rightarrow[\theta(1-p)+p](R-pX)\ge I(1-p)\ \Rightarrow\ \theta(1-p)\ge\frac{I(1-p)}{R-pX}-p$$

$$\Rightarrow\theta\ge\underbrace{\frac{I(1-p)-(R-pX)p}{(R-pX)(1-p)}}_{=\underline\theta} \tag{10.12}$$

which gives the lowest \(\underline\theta\) that makes lending possible.

From (10.12), if the pool has low quality (\(\theta<\underline\theta\)), lending is impossible in the two-period model.

Now suppose multiple periods:

• in period 0, no bank has information about firms, so every bank uses the prior \(\theta\);
• in period 1, firms reveal themselves as \(B\) or \(G\) through their project choice (\(B\) always invests in risky projects; \(G\) always invests in safe projects given (10.10) holds);
• in period 2, firms raise money again for the same one-period project; this repeats.

Under a competitive bank market: if \(\theta<\underline\theta\), no firm can get financed at the beginning, so the story ends. This is because of incomplete contracts: due to competition, \(G\) can always raise money at \(F=I\) starting from period 2, so the initial bank cannot lock up the lending relationship with this firm. Hence no bank would lend at a loss in period 0.

Under a monopolistic bank market: even with \(\theta<\underline\theta\), the bank can still lend at \(t=0\) with \(FI\), earning profits (not just breaking even) from \(G\) in later rounds to compensate for the first-round loss.

Therefore, in this repeated setup, lending monopoly makes credit more available, and bank competition makes credit less available.