Fundamental Structure of Our Model and the Result in the Case with no APR Violations

Abstract

In this chapter, we explain the structure of our basic model and present the benchmark result when the APR is rigorously preserved in the liquidation of the defaulted firm. We consider the model with two periods. In the first period, the firm makes an initial efficient investment and at the end of that period the firm obtains the returns on the investment. Given such a situation, we explore whether the firm makes the inefficient additional investment by fundraising from either outside creditors or the initial creditors. The additional investment yields its returns at the end of the second period. Then, we demonstrate that if the APR is retained, no inefficiency arises in the financial contract, that is, such inefficient additional investment cannot be fundraised, but efficient initial investment can be financed by the investors. The former implies that loan evergreening never occurs, while the latter implies that a credit crunch never arises when the APR is retained. Importantly, the outside creditor cannot finance the additional investment under the APR in this setting. Thus, the initial creditor does not have any incentive to finance inefficient additional borrowing. Hence, there is no inefficiency in the lending behavior of the initial lender.

Appendices

Appendix 3.1: Proof of Lemma 3.0

When $\tilde{\theta } \ge B$, the debt can be repaid at time 1 and there is no remaining senior debt at time 2. We assume that the manager is willing to make the inefficient additional investment, which implies that the following inequality holds.

$$ \int\limits_{\,\Delta D}^{{\,\bar{x}}} {\left\{ {x - \Delta D} \right\}dG(x)} \ge A + (\tilde{\theta } - B). $$

The left-hand side is the residual after debt repayment at time 2 when additional investment is carried out through an outside investor, and the right-hand side is the residual when the firm is liquidated at time 1. (When the existing creditor lends money, it is the same except that the notation ΔD is replaced by the notation ΔB.) The above equation can be rewritten as $\int_{\Delta D}^{{\bar{x}}} {\left\{ {x - \Delta D} \right\}dG(x)} - A \ge \tilde{\theta } - B > 0.$

Furthermore, the left-hand side is

$$ \int\limits_{\Delta D}^{{\bar{x}}} {\left\{ {x - \Delta D} \right\}dG(x)} - A = V_{x} - V_{\Delta D} - A, $$

where $V_{\Delta D} \equiv \int_{0}^{\Delta D} {xdG(x)} - \Delta D\left( {1 - G(\Delta D)} \right)$ expresses the expected repayment value of the additional loan. The participation constraint for the financier requires $V_{\Delta D} = \Delta I$ in the competitive financial market. Thus,

$$ V_{x} - V_{\Delta D} - A = V_{x} - \Delta I - A. $$

However, based on condition (3.2), the right-hand side is strictly negative. These are contradictory.□

Appendix 3.2: Proof of Proposition 3.1

If inefficient additional lending is carried out, then conditions $V_{\Delta D} (0) = \Delta I$ and $V_{B - \theta } (0) \ge A$ must hold simultaneously. Noting that $B - \theta > V_{B - \theta } (0) \ge A$ and using $V_{\Delta D} (0) = \Delta I,$ from inequality (3.5) we obtain the following inequality:

$$ \int\limits_{{B - \tilde{\theta }}}^{{\bar{x}}} {\{ x - (B - \tilde{\theta })\} }dG(x) - \Delta I > 0. $$

This inequality and the condition $V_{B - \theta } (0) \ge A$ give the following inequality.

$$ \int\limits_{{B - \tilde{\theta }}}^{{\bar{x}}} {\left\{ {x - (B - \tilde{\theta })} \right\}dG} - \Delta I + (V_{{B - \tilde{\theta }}} (0) - A) = \int\limits_{0}^{{\,\bar{x}}} {xdG} - \Delta I - A > 0. $$

This contradicts Eq. (3.2).□

Appendix 3.3: Proof of Proposition 3.2

• (i) When $A > V_{{B - \tilde{\theta }}} (0)$, the senior creditor chooses liquidation of the debtor firm. In this case, the debtor firm cannot get any positive gain so that the firm does not fundraise for inefficient additional investment.

• (ii) When $A \le V_{{B - \tilde{\theta }}} (0)$, assume inefficient additional investment can be financed. Then equations $V_{\Delta D} (0) = \Delta I$ and (3.5) must be satisfied simultaneously. This implies following inequality:

$$ \int\limits_{{B - \tilde{\theta }}}^{{\bar{x}}} {\left\{ {x - (B - \tilde{\theta })} \right\}dG(x)} - \Delta I = \int\limits_{0}^{{\bar{x}}} {xdG(x)} - \Delta I - V_{{B - \tilde{\theta }}} (0) \ge \hbox{max} \left\{ {\tilde{\theta } + A - B,0} \right\}. $$

From $A \le V_{{B - \tilde{\theta }}} (0) \le B - \tilde{\theta }$, we know $A + \tilde{\theta } \le B$. Thus, the above inequality can be rewritten as follows:

$$ V_{x} - \Delta I - V_{{B - \tilde{\theta }}} (0) \ge 0. $$

(3.6)

On the other hand, from condition (3.2) and $A \le V_{{B - \tilde{\theta }}} (0)$, the left-hand side is

$$ V_{x} - \Delta I - V_{{B - \tilde{\theta }}} (0) < A - V_{{B - \tilde{\theta }}} (0) \le 0. $$

This is inconsistent with (3.6).

Appendix 3.4: Proof of Proposition 3.3

Without the APR violation, from Propositions 3.1 and 3.2, we know that inefficient additional investment will never be carried out such that the condition under which the firm carries out an initial investment by borrowing is:

$$ \int\limits_{B - A}^{{\bar{\theta }}} {\left\{ {\theta + A - B} \right\}dF(\theta )} \ge A. $$

(3.7)

Substituting this condition into condition (3.1), we obtain

$$ \int\limits_{0}^{B - A} {\left\{ {\theta + A} \right\}dF(\theta )} + B\left( {1 - F(B - A)} \right) \ge I. $$

(3.8)

This indicates the participation condition for the initial creditor to lend the money for initial efficient investment. Thus, under condition (3.1), we can make the debt contract in which both of the participation constraints for the borrowing firm [condition (3.7)] and creditor [condition (3.8)] are satisfied simultaneously.□