Analytical Solutions for Expected Loss and Standard Deviation of Loss with an Additional Loan

Abstract

We evaluate the expected loss and the standard deviation of loss of a bank loan, considering the bank’s strategic control of the expected return on the loan. Assuming that the bank supplies an additional loan to minimize the expected loss of the total loan, we provide analytical formulations for the expected loss and the variance of loss with bivariate normal distribution functions. Using a given expected growth rate and interest rates, two thresholds for the asset/liability ratio can specify three cases. The bank supplies an additional loan to decrease the expected loss in two cases: (i) the asset/liability ratio of the firm is low, and its expected growth rate is high; and (ii) the asset/liability ratio of the firm is high, and the lending interest rate is high. The bank maintains the current loan amount when the asset/liability ratio lies between the two thresholds. Depending on the bank’s strategy, the bank can decrease the initial expected loss of the loan. Conversely, the bank would face a greater risk of the standard deviation of loss.

Appendix: Decision on an Additional Loan at Time \(t\)

1.1 Critical Value of Asset at Time $t$

The first derivative of $f(d)$ given in Eq. (9) is

$$\begin{aligned} \frac{\partial f(d)}{\partial d}&= \frac{\partial \varPhi (d)}{\partial d}-e^{(\mu -r_L )\tau }\frac{\partial \varPhi (d-\sigma \sqrt{\tau })}{\partial d} \nonumber \\&= \phi (d)-\mathrm {e}^{(\mu -r_L )\tau }\phi (d-\sigma \sqrt{\tau }) \nonumber \\&= \left\{ 1-\mathrm {e}^{(\mu -r_L -\sigma ^2/2)\tau }\mathrm {e}^{\sigma \sqrt{\tau }d}\right\} \phi (d), \end{aligned}$$

(41)

where $\phi (\cdot )$ is the standard normal density. The sign of Eq. (41) is given by

$$\begin{aligned} \frac{\partial f(d)}{\partial d} {\left\{ \begin{array}{ll} {>0} &{} \text{ if }\quad {d<\bar{d}}, \\ {<0} &{} \text{ if }\quad {d>\bar{d}}, \end{array}\right. } \end{aligned}$$

(42)

where

$$\begin{aligned} \bar{d}=\frac{r_L -\mu +\sigma ^2/2}{\sigma }\sqrt{\tau }. \end{aligned}$$

(43)

Characteristic values of $f(d)$ are given as

$$\begin{aligned} \mathop {\lim }\limits _{d\rightarrow -\infty } f(d)&= \mathrm {e}^{(r_M -r_L )\tau }-1,\end{aligned}$$

(44)

$$\begin{aligned} \mathop {\lim }\limits _{d\rightarrow \infty } f(d)&= \mathrm {e}^{(r_M -r_L )\tau }-\mathrm {e}^{(\mu -r_L )\tau } =\{\mathrm {e}^{r_M \tau }-\mathrm {e}^{\mu \tau }\}\mathrm {e}^{-r_L \tau },\end{aligned}$$

(45)

$$\begin{aligned} f(\bar{d})&= \mathrm {e}^{(r_M -r_L )\tau }-1+\varPhi (\bar{d}) -\mathrm {e}^{(\mu -r_L )\tau }\varPhi (\bar{d}-\sigma \sqrt{\tau }). \end{aligned}$$

(46)

From Eqs. (44)–(46), on the assumption $f(\bar{d})>0$, there exists $d_{1}^{*}$ such that $f(d_1^{*})=0$ for $d_1^{*}<\bar{d}$ if $r_L >r_M$, and there exists $d_2^{*}$ such that $f(d_2^{*} )=0$ for $\bar{d}<d_2^{*}$ if $\mu >r_M$.

Here, we consider the level of asset $A_t$ at which the bank supplies an additional loan for given values of $t$, $T$, $r_M$, $r_L$, $\mu$, and $\sigma$. We define the function $h(A_t )$ as

$$\begin{aligned} h(A_t )\equiv \lim \limits _{\varDelta \rightarrow 0} \frac{\partial E_t [L_T (\varDelta )]}{\partial \varDelta }=\mathrm {e}^{(r_M -r_L )\tau }-1+\varPhi (\tilde{d}(A_t))-\mathrm {e}^{(\mu -r_L )\tau }\varPhi (\tilde{d}(A_t)-\sigma \sqrt{\tau }), \end{aligned}$$

(47)

where $\tilde{d}(A_t)$ is given as Eq. (14). The bank supplies an additional loan if $h(A_t)<0$. We can confirm that the condition $h(A_t)<0$ is equivalent to the condition $\tilde{d}(A_t)<d_1^{*}$ or $\tilde{d}(A_t)>d_2^{*}$.

If $\tilde{d}(A_t)<d_1^{*}$, then $A_t >D\mathrm {e}^{-d_1^{*} \sigma \sqrt{\tau }-(\mu -\sigma ^2/2)\tau }=D\xi _1^{*}$. If $\tilde{d}(A_t)>d_2^{*}$, then $A_t<D\mathrm {e}^{-d_2^{*} \sigma \sqrt{\tau }-(\mu -\sigma ^2/2)\tau }=D\xi _2^{*}$. $D\xi _1^{*}$ and $D\xi _2^{*}$ are the thresholds of $A_t$ for which the bank supplies an additional loan.

Now, we derive an optimal additional loan. When $A_t >D\xi _1^{*}$, then the relation

$$\begin{aligned} \frac{A_t}{D}>\frac{A_t +\varDelta \mathrm {e}^{-r_L \tau }}{D+\varDelta }\ge \xi _1^{*}, \end{aligned}$$

(48)

holds for an additional loan amount $\varDelta >0$ using $d_1^{*} <\bar{d}$. It implies that the optimal loan amount $\varDelta _1^{*}$ satisfies

$$\begin{aligned} \frac{A_t +\varDelta _1^{*} \mathrm {e}^{-r_L \tau }}{D+\varDelta _1^{*}}=\xi _1^{*}. \end{aligned}$$

(49)

Similarly, when $A_t <D\xi _2^{*}$, the following relation holds.

$$\begin{aligned} \frac{A_t}{D}<\frac{A_t +\varDelta \mathrm {e}^{-r_L \tau }}{D+\varDelta }\le \xi _2^{*}. \end{aligned}$$

(50)

It implies that the optimal loan amount $\varDelta _2^{*}$ satisfies

$$\begin{aligned} \frac{A_t +\varDelta _2^{*} \mathrm {e}^{-r_L \tau }}{D+\varDelta _2^{*}}=\xi _2^{*}. \end{aligned}$$

(51)

From Eqs. (49) and (51), we obtain Eq. (11).

1.2 Parameter Relation for the Optimal Additional Loan

We assume that the firm accepts the additional loan. If the firm maximizes the expected value of the equity after accepting an additional loan $\varDelta$, this assumption is consistent with the firm’s behavior in the case of $\mu >r_L$. The expected value of the equity is evaluated as $(A_t +\varDelta \mathrm {e}^{-r_L \tau })\mathrm {e}^{\mu \tau }\varPhi (d_t (\varDelta )-\sigma \sqrt{\tau })-(D+\varDelta )\varPhi (d_t (\varDelta ))$, and the marginal expected value of the equity is given by $\mathrm {e}^{(\mu -r_L )\tau }\varPhi (d_t (\varDelta )-\sigma \sqrt{\tau })-\varPhi (d_t (\varDelta ))$. If $\mu >r_L$, the marginal expected value at $\varDelta =0$, $\mathrm {e}^{(\mu -r_{L} )\tau }\varPhi (d_t (0)-\sigma \sqrt{\tau })-\varPhi (d_t (0))$, is always nonnegative. It implies that an additional loan increases the expected value of the equity.

The optimal additional loan amount may be infinite if $r_L >r_M$ and $\mu >r_M$. If $r_L \le r_M$ or $\mu \le r_M$, the amount is always finite. The proof is given in Sect. 1 in this “Appendix”.

These relations are summarized as Table 2.

Table 2 Parameter relation for an additional loan at time $t$

1.3 Equivalent Condition that the Optimal Additional Loan is Finite

Under the condition $\partial E_t [L_T (\varDelta )]/\partial \varDelta \vert _{\varDelta =0} <0$, if $\partial E_t [L_T (\varDelta )]/\partial \varDelta \vert _{\varDelta \rightarrow \infty } <0$ then the optimal additional loan amount is infinite. The second derivative of EL $\partial ^2 E_t [L_T (\varDelta )]/\partial \varDelta ^2$ is always non-negative,⁶ and the second derivative converges to 0 as $\varDelta \rightarrow \infty$. It implies that the marginal expected loss converges to a constant as

$$\begin{aligned} \lim \limits _{\varDelta \rightarrow \infty } \frac{\partial E_t [L_T (\varDelta )]}{\partial \varDelta }=\mathrm {e}^{(r_M -r_L )\tau }-1+\varPhi (\bar{d})-\mathrm {e}^{(\mu -r_{L} )\tau }\varPhi (\bar{d}-\sigma \sqrt{\tau }). \end{aligned}$$

(52)

The necessary and sufficient condition for the optimal additional loan amount to be finite is that the right-hand side of Eq. (52) is positive. It is equivalent to

$$\begin{aligned} f(\bar{d})>0. \end{aligned}$$

(53)

1.4 Sufficient Parameter Conditions for the Finite Optimal Additional Loan

In this subsection, we prove that the optimal additional loan is finite if $r_L \le r_M$ or $\mu \le r_M$. For preparation, we show Proposition 1.

Proposition 1

$\varPhi (-\alpha +s/2)-\mathrm {e}^{\alpha s}\varPhi (-\alpha -s/2)>0$ for any $\alpha \in \mathbb {R}$ and $s>0$.

Proof

Let $X$ be a random variable distributed as $\ln X\sim \mathrm {N}(\alpha s-s^2/2,s^2)$. Then,

$$\begin{aligned} \Pr ((1-X)^{+}>0)=\Pr (X<1)=\varPhi (-\alpha +s/2)>0. \end{aligned}$$

(54)

By definition, $(1-X)^{+}\ge 0$ and, from Eq. (54), the probability that $(1-X)^{+}>0$ is positive. Therefore, $0<E[(1-X)^{+}]=\int _{-\infty }^{-\alpha +s/2} {(1-\mathrm {e}^{s y+\alpha s-s^2/2})\phi (y)\mathrm {d}y} =\varPhi (-\alpha +s/2)-\mathrm {e}^{\alpha s}\varPhi (-\alpha -s/2)$. $\square$

If $r_L\le r_M$, we can confirm that $\mathrm {e}^{(r_M -r_L )\tau }-1\ge 0$ and $\varPhi (\bar{d})-\mathrm {e}^{(\mu -r_L )\tau }\varPhi (\bar{d}-\sigma \sqrt{\tau })>0$ by applying Proposition 1 with $\alpha =(\mu -r_L)\sqrt{\tau }/\sigma$ and $s=\sigma \sqrt{\tau }$. This implies that the optimal additional loan is finite if $r_L \le r_M$.

On the other hand, if $\mu \le r_M$, then

$$\begin{aligned} \lim \limits _{\varDelta \rightarrow \infty }\frac{\partial E_t [L_T (\varDelta )]}{\partial \varDelta }&\ge \mathrm {e}^{(\mu -r_L )\tau }-1+\varPhi (\bar{d})-\mathrm {e}^{(\mu -r_L )\tau }\varPhi (\bar{d}-\sigma \sqrt{\tau }) \nonumber \\&= \mathrm {e}^{(\mu -r_L )\tau }\{\varPhi (-\bar{d}+\sigma \sqrt{\tau })-\mathrm {e}^{(r_L -\mu )\tau }\varPhi (-\bar{d})\}. \end{aligned}$$

(55)

Using Proposition 1, $\alpha =(r_L -\mu )\sqrt{\tau }/\sigma$ and $s=\sigma \sqrt{\tau }$, the right-hand side of Eq. (55) becomes positive. It implies that the optimal additional loan is finite if $\mu \le r_M$.