# Incomplete contract

An incomplete contract , also a relational contract , is a contract between market participants in which not all eventualities can be contractually defined or taken into account ex ante . Mostly these are agreements that are aimed at a longer period of time and contain gaps for future contingencies in order to counteract incomplete foresight.

When the contract is concluded, only the shell or framework contract is agreed, while the details will be concretized over time.

The point of such an agreement is that the definition of all possible contingencies in a complete contract can involve high transaction costs .

Employment contracts are an example of a common form of relational contract . The employee undertakes to follow the employer's instructions within a certain framework. However, it is usually not specified in employment contracts which concrete results the employee has to provide for the employer over the course of the contract, because the desired results are usually not foreseeable at the time of the contract.

## Properties and measures

Characteristics of incomplete contracts:

• they are implicit
• they are largely informal
• they are only legally binding to a limited extent
• they are very flexible
• Contract loyalty can hardly be enforced

The problem arises from the last point that the actors, especially in the case of asymmetrical information , have an incentive to behave opportunistically . The contracts are therefore time inconsistent.

Mechanisms to rule out opportunistic behavior are:

The principal-agent theory examines ways to overcome time inconsistencies in contracts .

## Subsequent changes to the contract

Changes resulting from renegotiations are agreed during the contract period. This contradicts the Pacta sunt servanda principle - contracts must be adhered to.

The subject of investigation are the effects of such a measure.

In the case of credit relationships, renegotiations typically proceed as follows:

• The market value of the remaining customer business is determined.
• A transfer fee is determined
• The present value of the condition contribution is determined.

If necessary, compensation is then set, for example in the case of repayment before the due date.

## Unsafe follow-up business

Unsafe repeat business includes:

• Rollover
• Roll over credit
• embedded options (with bank products options are embedded, whereby the payment flow is partially or not at all fixed)

## Theoretical Analysis of Incomplete Contracts: The Gorton and Kahn Model

Gorton and Kahn developed a model for determining the credit price, taking special account of renegotiation options. Gorton and Kahn look at the loan agreement over time.

### target

The aim is to show the advantages of renegotiations and to determine optimal conditions for bank loans.

### Basic model

#### Contract situation

At Gorton and Kahn, loans are incomplete contracts. There is a blanket termination clause for the lender. This enables the lender to renegotiate the contract at any time.

The aim of the lender is to force the borrower to behave in a manner that is compliant with the lender . Note: The term lender is not to be equated with a bank here.

#### Information distribution

The structure of the model is based on risk-neutral and equally informed lenders and borrowers . Information is distributed symmetrically at all times.

They conclude a loan agreement with repayment in t = 2 for a period of two periods. The capital market interest rate is zero.

In the initial time t = 0, there is uncertainty about the expected return of the project: , , distribution of income by using probability 0.5. ${\ displaystyle y_ {h}}$${\ displaystyle y_ {L}}$${\ displaystyle \ pm \ sigma}$

After a period t = 1, the probability of the project's success becoming known. However, this cannot be verified by third parties. Therefore, no agreements on state-dependent actions are made ex ante. ${\ displaystyle y_ {h}}$

After two periods, the project yields itself.

#### Incomplete Contracts Concept

When concluding the contract, the lender refrains from defining possible courses of action that depend on the information that occurs after a period.

In t = 1, the lender has the right to terminate, which gives him the opportunity to renegotiate

The lender agrees on a termination option in order to be able to renegotiate the contract flexibly in his favor if there is new information.

### Increase in project risk

Borrowers tend to increase project risk over the life of the loan .

In t = 1, the borrower can increase the project risk by the parameter S at cost c

This means that the project outcome can assume both higher and much lower values.

#### Probability tree

After a period it becomes known how high the probabilities of a good (p) and a bad project progress (1-p) are:

• With probability p: ${\ displaystyle y_ {h} \}$
• Wkt. 0.5:${\ displaystyle y_ {h} + \ sigma \}$
• Wkt. 0.5:${\ displaystyle y_ {h} - \ sigma \}$
• With probability (1-p):${\ displaystyle y_ {l} \}$
• Wkt. 0.5:${\ displaystyle y_ {l} + \ sigma \}$
• Wkt. 0.5:${\ displaystyle y_ {l} - \ sigma \}$

The agreed repayment amount is above the worst project outcome: ${\ displaystyle R> y_ {l} - \ sigma}$

##### Increased risk

Gorton and Kahn show that the borrower will increase the project risk as soon as the probability of project success falls below a critical value. This results in the following probability tree:

• With probability p: ${\ displaystyle y_ {h}}$
• Wkt. 0.5:${\ displaystyle y_ {h} + Sc \}$
• Wkt. 0.5:${\ displaystyle y_ {h} + S-c + \ sigma \}$
• Wkt. 0.5:${\ displaystyle y_ {h} + Sc- \ sigma \}$
• Wkt. 0.5:${\ displaystyle y_ {h} - \ sigma \}$
• Wkt. 0.5:${\ displaystyle y_ {h} -S-c + \ sigma \}$
• Wkt. 0.5:${\ displaystyle y_ {h} -Sc- \ sigma \}$
• With probability (1-p): ${\ displaystyle y_ {l}}$
• Wkt. 0.5:${\ displaystyle y_ {l} + S \}$
• Wkt. 0.5:${\ displaystyle y_ {l} + S-c + \ sigma \}$
• Wkt. 0.5:${\ displaystyle y_ {l} + Sc- \ sigma \}$
• Wkt. 0.5:${\ displaystyle y_ {l} -S \}$
• Wkt. 0.5:${\ displaystyle y_ {l} -S-c + \ sigma \}$
• Wkt. 0.5:${\ displaystyle y_ {l} -Sc- \ sigma \}$

The agreed repayment amount is above the third worst project outcome: ${\ displaystyle R> y_ {l} + S- \ sigma -c}$

#### Repayment amount

In the initial situation, the following is assumed with regard to the repayment amount: Otherwise there will be liquidation in t = 2 with the liquidation proceeds of Bei , liquidation in t = 1:${\ displaystyle R> y_ {l} - \ sigma}$${\ displaystyle L_ {2}}$${\ displaystyle R> L_ {1}> L_ {2}}$${\ displaystyle L_ {1}}$

Acceptance of the repayment amount if the risk increases . If R is smaller, the liquidation takes place in t = 2 with the costs . ${\ displaystyle R> y_ {L} + S- \ sigma -c}$${\ displaystyle L_ {2} -c}$

Because of the costs c incurred, an increase in risk is not always advantageous. The borrower will increase the risk at t = 1 if this increases his expected profit. The increase in risk has a negative impact on the expected profit of the investor. This can cause him to terminate or renegotiate.

#### Possibility of renegotiations

The lender, in turn, will modify the repayment of the loan so as to maximize its expected return. An increase in the required repayment is also accompanied by the effect that the probability that the debtor will become insolvent increases. On the other hand, the lender cannot arbitrarily reduce the repayment amount in order to prevent the borrower from increasing the risk.

There are opportunities to renegotiate, particularly in the case of bank loans, with the project income being redistributed. However, there is also a free rider problem.

If the borrower does not want to increase the risk anyway, there is no need for action.

Possible actions of the lender in t = 1 on problematic debtors that increase the project risk are:

• The riskier project has an expected value that is less than the liquidation value . A credit institution can thus threaten liquidation .
• Liquidate : The expected profit on continuation of the project is lower than on liquidation. The expected value of the project is lower than the liquidation value and an interest rate hike cannot reach the liquidation value either.
• Interest rate increase : the project's expected value is lower than the liquidation value, but a credible interest rate increase brings the expected profit above the liquidation value. The disadvantages from the increase in risk can be partially offset by an interest rate increase.
• The threat of liquidation is not credible as the bank would harm itself:
• Doing nothing : The investor cannot do better by doing other things (liquidating, raising or lowering interest rates). Ineffective liquidation and debt relief does more harm than risk increase
• Lower interest rates : The advantages of the risk increase avoided by a limited interest rate cut outweigh the disadvantages of the low interest income. Debt relief does not increase project risk, which increases profit.

From a threshold p, borrowers will not increase the risk.

### Results

The optimal actions of the borrower and the investor depend on the probability of success p. There is no monotonous relationship between borrower quality and the amount of the required repayment amount.

Loan financing through a financial intermediary can be advantageous over a market solution because of the lower coordination costs.

Possible renegotiation results and their probabilities should be taken into account when determining the loan conditions at the beginning.