Securitization This problem is adapted from Green baum and Thakor (1987). When a firm wants to securitize some of its assets, it typically signs a credit enhancement contract with a bank. In such a contract, the banks promises to insure a fraction θ of the repayment R(θ) promised by the firm to the investors who buy the security, in exchange for a fee Q(θ). This exercise shows how credit enhancement can be used to allow for a self-selection of firms, with better risks buying more credit enhancement. Consider an economy in which risk-neutral firms have an investment project with a return X in case of success, which occurs with probability p, and a zero return in case of failure (probability 1- p). The probability p is known to the firms but not to the investors. Banks or credit insurance contracts characterized by different levels of credit enhancement y, where y is the fraction of the initially promised repayment R(θ) that the investor will receive if the firm's project fails. A credit insurance contract will

1. Write the first- and second-order conditions that are necessary for the contract to be incentive-compatible.

2. Write the individual rationality (IR) constraint of the bank.

3. Assume the IR constraint holds with equality. By differentiating it, show that the mechanism is such that better risks tend to buy more credit enhancement, and the repayment R decreases with the guarantee y.