Borrowing-Firm Emission Trading, Bank Rate-Setting Behavior, and Carbon-Linked Lending under Capital Regulation

Abstract

The article develops a capped barrier option model to evaluate a bank’s equity. We explore the effects of borrowing-firm carbon emission trading on bank carbon-linked lending, explicitly considering borrowing-firm credit risks under capital regulation. We also integrate the regulatory compensation for bank low-carbon lending with borrowing-firm carbon allowance transactions in the emission trade scheme. Results show that an increase in the regulatory low-emitter lending compensation decreases loans at an increased interest margin, contributing to bank profitability and stability. The stringent regulatory cap for carbon emission allowances hurts profitability and stability. Strict capital regulation would jeopardize bank performance.

1. Introduction

The recent coronavirus pandemic is a wake-up call to the existential challenges of climate, pollution, and waste crises we face. It has reinforced the importance of accelerating the economy’s transition toward a sustainable and green recovery. A green banking intermediation development is imperative to achieve zero-carbon economies under the cap-and-trade mechanism. Studies have assessed the impact of carbon-linked lending on credit risk [1,2,3,4]. Besides, various studies demonstrate that emission trade schemes via carbon allowance transactions reduce carbon emissions [5,6,7,8,9]. The paper explores carbon-linked lending under the cap-and-trade mechanism, explicitly considering the capped borrowing-firm credit risk the previous studies ignored. Thus, this paper calls attention to the fact that credit risk affects the distribution of bank asset returns. The standard Merton [10] methodology used to provide a market-based estimation of bank equity needs to be adapted to consider the optimal loan rate-setting behavior in asset–liability matching management.

Green banking finance is necessary because it can have a positive impact by controlling environmental pollution. A bank can offer a market-oriented financing solution. The market needs to address the substantial gap in green finance between the supply (i.e., banks) and demand (i.e., borrowing firms). The funds supplied by banks are used for asset–liability matching management. The funds demanded by borrowing firms may be used for green/non-green investments and to cover expected losses from these investments. A cap-and-trade mechanism in the market might be an effective carbon emission reduction regulation [11,12]. Zhao et al. [13] further argue that the efficiency of the cap-and-trade mechanism could improve with bank green lending compensated by the government. Thus, it is critical to integrate bank green lending and the definitive borrowing-firm risk treatment with government green compensation under the cap-and-trade mechanism.

Asset–liability matching management under the cap-and-trade mechanism is an important issue that concerns bank managers. Our work complements the banking management literature by determining the optimal bank interest margin (i.e., the spread between optimal loan and market deposit rates), considering the cap-and-trade mechanism. The margin significantly accounts for bank profits [14], conveying vital information for banking efficiency [15]. Second, we also consider the definitive treatment of borrowing-firm credit risk [16]. This consideration is crucial because the bank can capture the credit risk in the green lending of the equity valuation. Thus, our paper focuses on the critical role of the structure–conduct–performance paradigm with portfolio-theoretic performance in asset–liability management under the cap-and-trade mechanism.

The paper aims to develop a capped barrier option model to investigate the effects of government compensation for green lending, cap-and-trade allowance, and capital regulation on bank spread behavior. The capped option explicitly considers the borrower-firm’s credit risk [16], while the barrier option relies on down-and-out valuation with the premature default structure [17] (The original contingent claims approach to the valuation of corporate securities treated equity as a call option on a firm’s assets [18] and debt as a cash portfolio and a short put option on the firm’s value [19]. Dermine and Lajeri [16] developed a “capped” call option model to evaluate the equity and deposit insurance premia of a bank. Brockman and Turtle [20] evaluated the equity and debt claims via barrier options, a particular type of path-dependent options. Episcopos [17] applied the barrier options theory of corporate security valuation to the contingent claims of a regulated bank. These studies show that broader contingent claim approaches are flexible in theoretical studies on corporate securities valuation and can be strongly supported by empirical evidence. The contingent claims approach has also found its natural application, especially in evaluating financial intermediation utility-free and market-value based, and empirical soundness in bank regulation [21,22]). Since changes in regulation parameters can affect bank spread behavior and, thus, bank interest margins, our research mainly addresses the following questions: (i) What are the most likely effects of the green lending compensation on bank interest margin? (ii) How does the shrinking regulatory cap affect the margin? (iii) How does the swap hedging conducted by the bank affect the margin? Moreover, (iv) what are the impacts of the risk-based capital guidelines on bank interest margin? In addition, we also investigate the first three questions with different scenarios, which can provide for alternative regulatory considerations. The research aims to provide solutions for the four questions based on our developed theoretical model. To the best of the authors’ knowledge, this is the first paper to explicitly introduce borrowing-firm green lending risk in the literature on the market-based estimation of bank equity under the cap-and-trade mechanism.

The main results are as follows. First, an increase in the government’s compensation for the low-emitter borrowing-firm lending (i.e., green lending compensation) decreases the green loans at an increased loan rate-setting. The green lending compensation enhances the bank’s interest margin and banking stability. However, the reduction in bank lending to high emitters is less than the increase in lending to low emitters when increasing the green lending compensation. Overall, we could argue that carbon emissions might not meet the expected reduction under the increased green lending compensation. Second, a decrease in the regulatory cap rate for allowance transactions in the cap-and-trade scheme decreases the bank’s interest margin and the banking stability. The decreasing cap rate for allowance transactions is efficient since reducing carbon emissions by increasing high- vs. low-emitter loans meets the expectation. Third, we show that the effect of swap hedging conducted by the bank on the bank’s interest margin is negative except in the capped barrier case. The result implies the importance of considering alternative equity valuation approaches. Finally, stringent capital regulation decreases the bank’s interest margin and deteriorates banking stability under the cap-and-trade mechanism. Our findings help better appreciate the carbon emission trading policy through which borrowing-firm carbon trading affects bank lending strategies under capital regulation during a worsening environment.

We present the literature review as a paper background in Section 2 in the paper structure. Section 3 models a theoretical framework. We further derive the optimal solution and comparative statics. Section 4 conducts a numerical analysis based on the comparative static results to interpret the economic intuition. Section 5 comprises the conclusion and presents our policy suggestions.

2. Background

Our theory of carbon-linked lending management is related to three strands of the literature. The first is the recent literature on carbon pricing effectiveness. Gloaguen and Alberola [23] show that carbon price is statistically insignificant to carbon emissions. Jaraite-Kažukauske and Di Maria [24] support the finding of Gloaguen and Alberola [23]. In addition, Abrell et al. [25] find that carbon price support reduces carbon emissions. A finding of Leroutier [26] is consistent with Abrell et al. [25]. While we also examine carbon price, our focus on the regulatory carbon emission aspects of bank loan support to low-emitter borrowing firms takes our analysis in a different direction.

The second strand is the emission trade system literature. There is a voluminous literature on the topic [5,6,7,8,9]. Haites [27] reviews the performance of carbon pricing policies based on emissions reductions and cost effectiveness. The review indicates that emissions trade systems (ETSs) have limited impacts since emissions fall faster than the cap in every jurisdiction, undermining carbon-policy effectiveness. Green [28] reviews the issue of carbon pricing and emission reduction. The author indicates that carbon taxes are generally more efficient than emission trade systems. Given the views in the literature, in this paper, we confront the issue of the effect that the regulatory cap rate of the allowance transactions in the emission trade system has on bank fund allocation. In particular, the research models the explicitly capped credit risk of the high-/low-emitter borrowing firms and the bank’s loan rate-setting behavior incorporated into the trading system. The features of the borrowing firm–bank relationships document the difference between our model and those studies.

The third strand is the literature on carbon finance risks. The correlation between green lending, credit risks, and institutional pressure on environmental and financial stakes is studied [1]. Zhang and Li [29] analyze firm carbon-linked borrowing with the credit and market risk. Lee and Choi [30] examine how the capital market reacts to firms’ carbon risk management and present the effects of carbon risk management with government carbon policies on the cost of debt capital. The findings indicate some strategic choices to improve carbon risk management and significant impacts of regulatory carbon emission reduction policies nationwide. Vigorously developing green finance will help promote sustainable energy development [2]. Si et al. [3] identify the causality between bank lending rate liberalization and borrowing-firm operational risks. Bank lending rate deregulation could lower energy borrowing-firm operational risks. In particular, the banking deregulation through easing financing constraints and reversing borrowing-firm financialization affects energy borrowing-firm operational risks. Umar et al. [4] explore the impact of carbon-neutral lending on credit risk. Their findings are that exposure to carbon-neutral lending is negatively related to the default risk. Thus, financial institutions can benefit from more inadequate loan loss provisions and economic capital requirements. Our model complements the literature by exploring the credit risk from bank lending to high-/low-emitter borrowing firms subject to non-performance. Besides, modeling premature default risk reflected by barrier options in bank equity valuation is essential for carbon lending. Overall, what distinguishes our work from the carbon finance literature is our focus on commingling the assessment of borrowing-firm credit risk with bank carbon-loan rate-setting behavior in borrowing-firm emission allowance transactions.

3. Model Setup and Solution

3.1. Conceptual Framework

We model a bank’s carbon-linked lending framework. For simplicity, (i) the bank provides funds to two representative borrowing firms. (ii) The two borrowing firms participate in the cap-and-trade scheme ETS. One borrowing firm is a lower carbon emitter; the other is a higher one. The lower emitter can sell its extra allowances to the trading market. At the same time, the higher one can buy its insufficient allowances due to the cap from the trading market where the demand for the allowances equals the supply of the allowances. (iii) The low emitter can borrow funds from the bank at a lower loan rate than the high emitter [31]. The government subsidizes the difference between the two loan rates for the bank due to the International Accounting Standard 20 (IAS20). (iv) The bank may conduct a swap hedging for the high-emitter loans because the borrowing firm has extra cost burdens of eliminating emissions and buying allowances for its investment. Accordingly, our focus is on a bank’s high- vs. low-emitter lending, considering hedging and ETS.

3.2. Balance-Sheet Activities

At the beginning of the period, the high-emitter borrowing firm funds an asset ($A_H$) with the bank loans ($L_H$) and its equity ($E_H$). The low-emitter one supports an asset ($A_L$) with the bank loans ($L_L$) and with its equity ($E_L$). The bank funds the loans to the two borrowing firms ($L_H+L_L$) and liquid-asset investment ($B$) with deposits ($D$) and equity ($K$). The balance sheets of the high emitter, the low emitter, and the bank are, respectively,

$$A_H=L_H+E_H,$$

(1)

$$A_L=L_L+E_L,$$

(2)

$$L_H+L_L+B=D+K=K(1/q+1).$$

(3)

The capital-to-deposits ratio ($q=K/D$) designed by the regulatory authority forces the bank’s capital position to reflect its asset portfolio risk (i.e., a risk-based system of capital standards).

Differentiated lendings and borrowing-firm carbon-linked allowance trades are two key issues that concern bank managers for lending determination. The bank faces two loan demands: $L_H(R_H)$ and $L_L(R_H-R_G)$. The loan rate ($R_H$) is the interest rate for high-emitter lending. The loan rate for the low emitter is the interest rate ($R_L$), where $R_L$ equals the difference between $R_H$ and $R_G$, and the interest rate ($R_G$) is a constant interest rate compensated by the government. It is reasonable to have the condition $R_H<R_L(=R_H-R_G)<R$ (where $R$ is the liquid-asset rate) held for our model because the low-emitting loans are also subject to non-performance. Accordingly, we can model the two-loan demand function ($L(R_H)=L_H(R_H)+L_L(R_H-R_G)$) with the conditions ($\partial L/\partial R_H<0$, $\partial L_H/\partial R_H<0$, and $\partial L_L/\partial R_H<0$) since the bank is the loan rate setter, capturing the loan rate-setting behavior in the imperfectly competitive market [32].

Next, we focus on borrowing-firm allowance transactions in ETS. The investment returns of the high emitter at the end of the period are ($1+R_A$)$A_H$ net of the purchased extra carbon allowance costs ($(c_H-c_G)A_H$). $R_A$ is the market rate of investment return. The rate $c_H$ is the marginal cost of the level of carbon emissions, and the rate $c_G$ is the regulatory cap for the carbon emission allowances. We assume the condition ($c_H>c_G$) held for the high emitter, a buyer in the allowance market. The high-emitter’s investment is financed partly by $(1+R_H)L_H$. The low-emitter’s total returns include the investment returns ($(1+R_A)A_L$) and the revenues from the selling of their extra carbon allowance ($(c_G-c_L)A_L$). The low emitter also has the same rate of return ($R_A$), implying that both the borrowing firms face the same investment market. The rate $c_L$ is the marginal cost of the carbon allowances. Here, the condition $(c_G-c_L)>0$ holds for the low emitter. The fund cost for the low emitter is ($(1+(R_H-R_G))L_L$). In addition to ETS, the model also considers swap hedging conducted by the bank, since the bank looks at the situations with multiple uncertainty sources, i.e., asset risk, credit risk, green environment risk, and early-default risk.

3.3. Capped Barrier Call and Alternative Objectives

This section integrates Dermine and Lajeri [16] with Episcopos [17] as our developed model, and more importantly, we introduce loan rate-setting features for the analysis. The bank’s equity can be in the form of a capped barrier call option on the two borrowing-firms’ assets, considering the problem of early bank closure. Conceptually, the capped barrier call option consists of two parts. The first part is the value of a barrier call on the assets of the borrowing firms at a strike price of the bank’s net obligations (i.e., $DO{C}_{AZ}$, the down-and-out call option). The second one is the same value at a strike price of the bank’s loan repayments (i.e., $DO{C}_{AV}$, the capped down-and-out call option). The latter represents the loss of value resulting from the capped barrier. We formulate the two options as follows.

The total market value of the two borrowing-firm asset portfolios ($A$) varies continuously over the period according to the stochastic process:

$$dA=\mu Adt+{\sigma }_{A}AdW,$$

(4)

where

$A=(1+R_A)(A_H+A_L)-(c_H-c_G)A_H+(c_G-c_L)A_L$;

$\mu =$ the instantaneous expected rate of return on the earning-asset portfolio;

${\sigma }_A=$ the instantaneous standard deviation of the return;

$W=$ a Wiener process.

As is well known from option theory [19], the market value of the bank’s equity ($DO{C}_{AZ}$) is a barrier call option on the borrowing-firm’s assets, that is,

$$DO{C}_{AZ}=S{C}_{AZ}(A,Z)-DI{C}_{AZ}(A,Z),$$

(5)

where

$$Z=(1+R_D)(K/q)-(1+R)[K(1/q+1)-L_H-L_L]-\alpha [(1+R)L_H+(1+R)L_L+R_G{L}_L],$$

$$S{C}_{AZ}=AN(d_1)-Z{e}^{-R}N(d_1-{\sigma }_A);$$

$$DI{C}_{AZ}=A{(H/A)}^{2\eta }N(d_2)-Z{e}^{-R}{(H/A)}^{2\eta -2}N(d_2-{\sigma }_A);$$

$$0\le \alpha <1=\mathrm{swap}\mathrm{hedging}\mathrm{ratio};$$

$$H=\theta Z,0\le \theta <1=\mathrm{barrier}\mathrm{ratio};$$

$$\eta =R/{\sigma }_A^2+1/2;$$

$$d_1=[\ln (A/Z)+R+{\sigma }_A^2/2]/{\sigma }_A,d_2=[\ln (H^2/AZ)+R+{\sigma }_A^2/2]/{\sigma }_A;$$

$N(\cdot )=$ the cumulative density function of the standard normal distribution.

Equation (5) demonstrates the bank’s equity value without considering the capped credit risks from the borrowing firms. The first term is the standard call option with the underlying assets ($A$) at the strike price ($Z$). The strike price consists of depositor payments, repayments from the bank’s liquid-asset investment, and hedging funds. The value $S{C}_{AZ}$ is a path-independent call option that reaches maturity at the end of the period. The second term is the down-and-in call option: the value $DI{C}_{AZ}$ is a path-dependent option in which the bank’s default occurs before maturity. The parameter $\theta$ is the barrier ($H$)-to-liabilities ($Z$) ratio, capturing the premature default. Banks with high asset volatility or low capital reserve are more likely to hit the barrier before the maturity date than banks without such characteristics, as pointed out by Brockman and Turtle [20]. An appropriate implication is that a high barrier ratio resulting from the COVID-19 pandemic may reflect banks with high asset volatility.

Next, we write the capped credit risk from the borrowing firms as follows:

$$DO{C}_{AV}=S{C}_{AV}(A,V)-DI{C}_{AV}(A,V),$$

(6)

where

$$V=(1-\alpha )[(1+R_H)L_H+(1+(R_H-R_G))L_L+R_G{L}_L];$$

$$S{C}_{AV}(V,A)=AN(b_1)-V{e}^{-R}N(b_1-{\sigma }_A);$$

$$DI{C}_{AV}(V,A)=A{(H/V)}^{2\eta }N(b_2)-V{e}^{-R}{(H/V)}^{2\eta -2}N(b_2-{\sigma }_A);$$

$$b_1=[\ln (A/V)+R+{\sigma }_A^2/2]/{\sigma }_A,b_2=[\ln (H^2/AV)+R+{\sigma }_A^2/2]/{\sigma }_A.$$

Similarly, the first term on the right-hand side of Equation (6) identifies the standard call as the underlying assets (i.e., the borrowing-firm investment returns) at the strike price of the bank’s loan repayments. This value demonstrates the capped credit risk from the borrowing firms where the default occurs at maturity. The second term identifies the down-and-in barrier feature of the capped credit risk. It is worthwhile to mention that the risk asset $V$ indicates the assets without hedging consideration. Thus, the value of the difference between the standard call and the down-and-in call expresses the capped credit risk in the path-dependent option valuation.

Combining Equations (5) and (6), we express the bank’s equity value as follows:

$$S=[S{C}_{AZ}(A,Z)-DI{C}_{AZ}(A,Z)]-[S{C}_{AV}(A,V)-DI{C}_{AV}(A,V)].$$

(7)

The two relevant distinctions for the argument are whether to consider the barrier and whether to consider the cap. Together they lead to the following four scenarios:

Scenario I (the capped barrier case): Considering the barrier and cap reveals Equation (7). The method makes the bank determine the optimal interest margin, explicitly considering the credit risk from its borrowing firms and the premature default environment. Our Equation (7) setting contributes to the literature on bank equity valuation by admitting a highly volatile state such as the COVID-19 pandemic.

Scenario II (the cap call case): We follow method I, except we ignore the barrier terms. The issue is in the spirit of the risk-neutral valuation adopted by Dermine and Lajeri [16]. The case focuses on the explicit treatment of borrowing-firm credit risk evaluated by the bank. The lending environment faced by the bank might be significantly uncertain, e.g., the shocks to the petroleum industry.

Scenario III (the barrier case): We ignore the cap terms and rewrite the bank’s equity in the spirit of Episcopos [17]. That is, $S{C}_{VZ}(V,Z)-DI{C}_{VZ}(V,Z)$ (See Appendix A). The same pattern applies as previously. The case may apply to an economic environment in which the bank anticipates loan repayments with uncertainty and liquidity problems, resulting in possible defaults before auditing maturity.

Scenario IV (the standard call case): Following scenario III, we ignore both the cap and the barrier. The Black and Scholes [18] formula for the call option states the market value of the bank’s equity.

3.4. Solutions

The following section will derive the optimal solutions and comparative static results. Then, we will compare the four cases in the numerical analysis section.

Partially differentiating Equation (7) to the loan rate ($R_H$) yields the first-order condition as $\partial S/\partial R_H=0$. A sufficient condition for an optimum is that the bank’s equity value function is strictly concave (${\partial }^{2}S/\partial R_H^2<0$), at least in the short run. Now, we consider the impact on the optimal loan rate ($R_H$) of changes in some parameters related to carbon banking issues. The parameters include the compensated rate ($R_G$), the regulatory cap rate for the carbon emission allowance ($c_G$), hedging ($\alpha$), and the capital-to-deposits ratio ($q$). Differentiation of the first-order condition with $i=$$R_G$, $c_G$, $\alpha$, and $q$ yields:

$$\frac{\partial R_H}{\partial i}=-\frac{{\partial }^{2}S}{\partial R_H\partial i}/\frac{{\partial }^{2}S}{\partial R_H^2}.$$

(8)

The four comparative statics in Equation (8) pave the way for the numerical analysis in the following section.

4. Numerical Analysis

The following section adopts a numerical-analysis approach with empirical data to explain the four scenarios’ central results based on Equation (8). The alternative scenarios provide the cases for parameter changes on bank interest margin’s effects; further, they offer soundness tests for the comparative statics.

4.1. Data Description

We present the data based on some evidence in our carbon-linked lending environment model for numerical analysis.

Table 1. Data description for the numerical analysis ¹.

Variables	Approximation and Assumption
$(R_H(\%),L_H)$: high-emitter loan bundle	(4.25, 219) (4.50, 218) (4.75, 216) (5.00, 212) (5.25, 204) (5.50, 188) (5.75, 156)
$((R_H-R_G)(\%)),L_L)$: low-emitter loan bundle	(3.75, 22.0) (4.00, 22.1) (4.25, 22.3) (4.50, 22.6) (4.75, 23.0) (5.00, 23.5) (5.25, 24.1)
$R$: liquid-asset rate	2.86%
$R_A$: investment return rate	6.50%
$E_H$: high-emitter equity capital	10
$E_L$: low-emitter equity capital	1
$R_G$: compensated rate	0.50%
$c_G$: cap rate	2.20%
$c_H$: the high-emitter marginal cost of the carbon allowances	3.80%
$c_L$: the low-emitter marginal cost of the carbon allowances	0.60%
$R_D$: deposit interest rate	1.75%
$\theta$: barrier ratio	69.66%
$q$: capital-to-deposits ratio	3.95%
$B$: liquid-asset	5
${\sigma }_A$: instantaneous volatility	0.50
$\alpha$: swap hedging ratio	0.25

¹ Sources: see Appendix B.

summarizes the data as follows.

4.2. Effect of Compensated Rate

In the subsection, we discuss the effect of the government’s support for bank lending to the low emitter on bank loan rate-setting behavior. The discussion also includes the impact of the regulatory compensated rate on bank interest margin determination of the alternative equity valuation scenarios.

Table 2. Effects of compensated rate $R_G$ on the optimal loan rate at various scenarios ¹.

$$\mathbf{\Delta }{\mathit{R}}_{\mathit{G}} (\%)$$	Scenario I	Scenario II	Scenario III	Scenario IV
	$\partial R_H/\partial R_G \left({10}^{-6}\right)$
0.20 → 0.30	3128.6983	3312.6703	2345.4098	2168.5318
0.30 → 0.40	3127.9940	3311.1054	2345.4616	2168.7928
0.40 → 0.50	3127.2852	3309.5400	2345.5140	2169.0525
0.50 → 0.60	3126.5792	3307.9739	2345.5664	2169.3134
0.60 → 0.70	3125.8710	3306.4098	2345.6187	2169.5730
0.70 → 0.80	3125.1620	3304.8476	2345.6705	2169.8332

¹ Parameter values, unless stated otherwise, $R=$ 2.86%, $R_A=$ 6.50%, $E_H=$ 10, $E_L=$ 1, $c_H=$ 3.80%, $c_L=$ 0.60%, $R_D=$ 1.75%, $\theta =$ 69.66%, $B=$ 5, ${\sigma }_A=$ 0.50, $c_G=$ 2.20%, $\alpha =$ 0.25, and $q=$ 3.95%. The optimal loan bundle $(R_H(\%),L_H)$ is at (5.00, 212) for Scenario I and II and (4.75, 208) for Scenario III and IV.

demonstrates three main results. First, increasing the government’s compensation for bank low-carbon-emission lending increases the loan rates for both the high-carbon-emission borrowing firms and low-carbon-emission borrowing firms (In the model, we assume that the loan rate ($R_H$) is the interest rate for the high-emitter loans. The interest rate for the high-emitter loan net of the compensated rate ($R_H-R_G$) is defined as the low-emitter loan rate. Thus, increasing the high-emitter loan rate implies an increase in the low-emitter loan rate, holding the compensated rate constant. Table 2 presents that a rise of 0.10% of the compensated rate increases about 0.30% of the high-emitter loan rate (see the result of 3128.6983 in Table 2) and about 0.20% of the low-emitter loan rate. The same pattern as in Table 2 applies to

Table 3. Effects of regulatory cap rate $c_G$ on the optimal loan rate at various scenarios ¹.

$${\mathit{c}}_{\mathit{G}}(\%)$$	Scenario I	Scenario II
	$\partial R_H/\partial c_G\left({10}^{-6}\right)$
1.90 → 2.00	2564.9932	1784.3066
2.00 → 2.10	2561.2297	1779.1733
2.10 → 2.20	2557.4436	1774.0563
2.20 → 2.30	2553.6386	1768.9528
2.30 → 2.40	2549.8110	1763.8706
2.40 → 2.50	2545.9644	1758.8005

¹ Parameter values, unless stated otherwise, $R=$ 2.86%, $R_A=$ 6.50%, $E_H=$ 10, $E_L=$ 1, $c_H=$ 3.80%, $c_L=$ 0.60%, $R_D=$ 1.75%, $\theta =$ 69.66%, $B=$ 5, ${\sigma }_A=$ 0.50, $R_G=$ 0.50%, $\alpha =$ 0.25, and $q=$ 3.95%. The optimal loan bundle $(R_H(\%),L_H)$ is at (5.00, 212) for Scenarios I and II.

,

Table 4. Effects of swap hedging ratio $\alpha$ on the optimal loan rate at various scenarios ¹.

$$\mathit{\alpha }$$	Scenario I	Scenario II	Scenario III	Scenario IV
	$\partial R_H/\partial \alpha \left({10}^{-3}\right)$
0.10 → 0.15	85.6912	−82.4970	−55.3069	−21.3134
0.15 → 0.20	103.6889	−134.7135	−58.1734	−22.5496
0.20 → 0.25	77.9865	−186.7910	−61.3570	−23.9485
0.25 → 0.30	20.1971	−238.1676	−64.9145	−25.5467
0.30 → 0.35	−59.7193	−288.1573	−68.9182	−27.3938
0.35 → 0.40	−151.6846	−335.9474	−73.4616	−29.5584

¹ Parameter values, unless stated otherwise, $R=$ 2.86%, $R_A=$ 6.50%, $E_H=$ 10, $E_L=$ 1, $c_H=$ 3.80%, $c_L=$ 0.60%, $R_D=$ 1.75%, $\theta =$ 69.66%, $B=$ 5, ${\sigma }_A=$ 0.50, $R_G=$ 0.50%, $c_G=$ 2.20%, and $q=$ 3.95%. The optimal loan bundle $(R_H(\%),L_H)$ is at (5.00, 212) for Scenario I and II and (4.75, 208) for Scenario III and IV.

and

Table 5. Effects of capital-to-deposits ratio $q$ on the optimal loan rate at various scenarios ¹.

$\mathbf{\Delta }\mathit{q} (\%)$	Scenario I	Scenario II	Scenario III	Scenario IV
	$\partial R_H/\partial q$ (10−3)
3.65 → 3.75	−21.2108	−31.6794	−2.9865	−1.4868
3.75 → 3.85	−21.0481	−31.2066	−2.9753	−1.4891
3.85 → 3.95	−20.8876	−30.7473	−2.9643	−1.4913
3.95 → 4.05	−20.7293	−30.3010	−2.9533	−1.4935
4.05 → 4.15	−20.5732	−29.8671	−2.9425	−1.4957
4.15 → 4.25	−20.4191	−29.4451	−2.9318	−1.4979

¹ Parameter values, unless stated otherwise, $R=$ 2.86%, $R_A=$ 6.50%, $E_H=$ 10, $E_L=$ 1, $c_H=$ 3.80%, $c_L=$ 0.60%, $R_D=$ 1.75%, $\theta =$ 69.66%, $B=$ 5, ${\sigma }_A=$ 0.50, $R_G=$ 0.50%, $c_G=$ 2.20%, and $\alpha =$ 0.25. The optimal loan bundle $(R_H(\%),L_H)$ is at (5.00, 212) for Scenario I and II and (4.75, 208) for Scenario III and IV.

). Intuitively, as the government compensates for low-carbon-emission borrowing, the bank returns to a smaller-scale loan base. The bank may attempt to augment its total returns by shifting its investments to liquid assets and away from its loan portfolio. If loan portfolio demand is relatively rate-elastic, a smaller-scale loan portfolio is possible at an increased margin. Hence, the increased compensated rate reduces bank lending at increased loan rates (and, thus, increases bank interest margins), thereby contributing to bank profitability and stability. Under the circumstances, the reduced banking funding decreases borrowing-firm carbon emissions by reducing production activities. Overall, we could show that the government’s compensation is efficient for bank profitability, banking stability, and carbon emissions. Our findings are consistent with those of Cui et al. [1]: green credit helps banks avoid risks, firms’ green transformation, and sustainable economic development.

Second, we show that the positive effect of the government’s support on the loan rate for the high-emitter borrowing firm is much less significant than that for the low-emitter borrowing firm. The reduction in the high emitter’s loan funding for production is less than the increase in the low emitter’s funding for production. Thus, the high-carbon emitter reduces production, and the low-carbon emitter reduces production. Carbon emissions might not meet the expected reduction when the government proposes the support scheme under the cap-and-trade mechanism due to the optimal interest margin determination for the bank’s liquidity management. Our finding is consistent with that of Abrell et al. [25]: carbon price support decreases carbon emissions, but the reduction is below the expected level.

Third, comparing the four scenarios, scenario II (i.e., the cap call case) is the most impactful, and scenario IV (i.e., the standard call case) is the least impactful. From the standpoint of the government’s intervention in carbon emission lending, the traditional call is efficient. However, this cap call is efficient from the perspectives of bank profitability and banking stability. Therefore, the effect of the carbon emission lending compensation from the government depends on loan rate-setting behavior and bank liquidity management scenarios. The comparative static results also demonstrate the soundness of our finding: an increase in the government’s support for bank lending to reduce carbon emissions decreases bank loans at an enhanced margin. This carbon-reduced support is efficient from the standpoint of bank profitability.

4.3. Effect of Regulatory Cap Rate

It is interesting to investigate the effect of the regulatory cap rate on the bank’s interest margin under the cap-and-trade mechanism.

Scenarios III and IV are the excluded cases since both the equity valuations focus on liquidity management, not explicitly considering the cap role of borrowing firms. Table 3 shows that an increase in the regulatory cap rate for the carbon emission allowances increases the bank’s loan rates for high- vs. low-emitter lendings. The interpretation of this result follows a similar argument as in the case of a change in $R_G$. Decreasing the cap rate encourages the bank to shift investments from its liquid assets to the loans. In the imperfect loan markets, the bank must reduce the size of the margin to increase the number of loans. Decreasing loan rates enhance bank lending, contributing to banking stability. The effect of carbon emission through allowance transactions regulated by the reduced cap rate is harmful. Both the emitters minimize carbon emissions with more production due to increased bank funding. Again, the effect of the regulatory cap rate on the loan rate-setting behavior is less significant in the first scenario than in the second scenario. Thus, the cap barrier case is more efficient for liquidity management from the profitability and stability viewpoints. Explicitly considering the features of borrowing-firm credit risks and the premature barriers is crucial to bank interest margin determination when determining the regulatory cap for the allowance carbon emission transactions. Wakabayashi and Kimura [7] argue that the emission trade system may not significantly affect carbon emissions. Our model, therefore, encompasses their argument.

4.4. Effect of Hedging

Hedging is an essential operation in bank liquidity management that concerns bank managers. Hedging is central not only to banks’ strategic decisions but also to regulators concerned about banking stability. Table 4 shows that hedging ambiguity increases bank loan rate-setting in the capped barrier case. The cap explicitly captures borrowing-firm credit risks, and the barrier captures the premature default performance. Therefore, the interaction between cap and barrier yields ambiguity. Except for scenario I, hedging increases the bank’s loans at a reduced loan rate setting. The result implies that bank hedging increases lending activities, decreases profits, and deteriorates banking stability. Hedging helps high-/low-emitter funding for their production. As Liu [33] pointed out, the carbon-linked loans relative to total loans in China are about 10%. Accordingly, we argue that bank hedging might not help carbon emission reduction, at least at a relatively low level of bank carbon-linked lending.

Further, comparing the three scenarios (II, III, and IV), the negative hedging impact on bank interest margin is the most significant in cap call valuation. The least one is in the standard call valuation. From the standpoints of bank profits and banking stability, it is redundant to conduct hedging since it has explicitly considered the borrowing-firm credit risk, increasing loans at a reduced margin. Thus, explicitly considering borrowing-firm credit risk and active hedging relative to achieving hedging alone are inappropriate from the profitability viewpoint.

4.5. Effect of Capital-to-Deposits Ratio

Table 5 depicts that stringent capital regulation increases bank lending at a reduced loan rate. This results in a decreasing bank interest margin but adversely affects banking stability. The harmfulness of this result applies to the four scenarios. Furthermore, the negative effect of capital regulation on bank interest margin is the most significant in the second scenario, and the least significant one is the fourth scenario. Thus, we suggest that a stringent capital regulation should consider the bank’s liquidity structure not to deteriorate banking stability, mainly when the bank conducts carbon-linked lending.

In the model, we validate the simulation results by conducting different equity objective valuations. Overall, the effects of the compensation to green lending, the regulatory cap, and capital regulation on the optimal bank interest margin are consistent in various objective evaluations (Table 2, Table 3 and Table 5). However, the soundness does not hold for Table 4. We suggest that the swap hedging effect on the optimal bank interest margin needs to be more nuanced, considering alternative objective valuations. Results derived from our model may not extend to the cases where other parameters, such as various barriers, in the model are used for soundness tests.

Further, Santomero [34] argues that choosing an appropriate goal in modeling the bank’s objective remains a controversial issue. As mentioned previously, the capped barrier option to evaluate the bank’s equity is an option focusing on borrowing-firm credit risk and premature environment. The option allows the inclusion of a more realistic environment along with the more appropriate behavioral mode of loan rate-setting and, thus, bank interest margin determinations. However, we remain silent on an empirical verification based on the four alternative objectives, which can be recognized as a future research direction.

5. Conclusions

The paper proposes a contingent claim approach for bank carbon-linked lending and borrowing-firm emission allowance trading. The article confirms the contingent claim models with numerical analyses from one bank and two borrowing firms. It demonstrates their usefulness in borrowing-firm emission allowance trading when considering regulatory low-emitter lending compensation. Several results should be of interest to investors, analysts, and policymakers. For example, an increase in the regulatory low-emitter lending support increases bank interest margin and banking stability, but carbon emission might not meet the expected reduction. We also show that increasing borrowing-firm carbon allowance transactions enhances bank profitability and stability. Carbon emission, as expected, reduces due to decreased borrowing-firm production with reduced bank funding. Stringent capital regulation deteriorates bank interest margin and banking stability. Overall, we suggest that the carbon emission allowances for carbon emission reduction enhance fund-lender profits and banking stability, but the regulatory cap harms banking stability.

One implication of the research is to evaluate the plethora of green arrangements proposed as alternatives for future regulations. The carbon emission allowance transactions stimulate borrowing-firm carbon emission trading. Indeed, the transactions improve the green environment and further increase the bank’s (the fund provider’s) interest margin and banking stability. We suggest that the carbon emission allowance transaction is a win–win for green improvement and banking stability. Alternatively, the regulatory cap also encourages borrowing firms to reduce carbon emissions but brings banking instability. Under the circumstances, we suggest that the regulators reconsider the regulatory cap policy from a welfare analysis standpoint under the circumstances. In addition, stringent capital regulation harms the bank interest margin (i.e., increasing bank lending at a reduced loan rate setting) and the banking stability. The strict rule may encourage bank lending with a higher risk to the borrowing firms. The spillover may harm green environment improvement. Accordingly, we suggest that a stringent capital regulation might need to be more nuanced.

The framework opens a further avenue of research: an outgrowth of the model is to introduce so-called swaption valuations. The swaption focuses on a synergy-banking equity valuation rather than on a narrow-banking valuation used in the current study. The advantage of synergy banking considers the equity valuation with simultaneous asset–liability matching management. The case might alter the effects of bank spread behavior from changes in carbon-linked lending and allowance transactions. Then, the alterations act as regulators for environmental policy reconsideration.