Effects of Cap-and-Trade Mechanism and Financial Gray Rhino Threats on Insurer Performance

Abstract

This paper develops a capped barrier option model to examine how a cap-and-trade mechanism affects an insurer’s guaranteed rate-setting behavior and policyholder protection in a financial gray rhino environment. Toward sustainability, the insurer explicitly captures the credit risk from the borrowing firms, participating in the cap-and-trade scheme to reduce carbon emissions, an essential issue of carbon emission and environmental protection when facing gray rhino threats. In addition, the energy economics and policy analysis are from the fund-providing insurer’s perspective. Green lending policies and life insurance policy loans (i.e., disintermediation related to insurance stability) are crucial to managers and regulators, particularly bridging the borrowing-firm carbon transactions for carbon emission reductions toward sustainability. We show that the shrinking regulatory cap of the cap-and-trade scheme harms policyholder protection, adversely affecting insurance stability. The harm becomes more serious when the gray rhino threat on borrowing firms becomes significant. An increase in policy loans decreases the insurer’s interest margin and policyholder protection. However, increasing the gray rhino threat decreases life insurance policies at a reduced guaranteed rate but increases policyholder protection, contributing to insurance stability. Therefore, the government can use the cap-and-trade scheme to control carbon emissions and improve the environment, but it harms policyholder protection. We suggest that, for example, the government should subsidize the insurer for green lending, affecting insurance stability.

1. Introduction

Green lending has become a common concern worldwide since it can promote and hinder a cleaner environment, affecting health and life [1]. The cap-and-trade mechanism is one of the most effective carbon emission reduction regulations for production [2,3,4]. Flachsland et al. [5] point out several limits to the cap-and-trade mechanism for carbon emission reductions from an international viewpoint. First, international carbon emissions trading may not be welfare-enhancing for all countries due to market distortions or terms-of-trade effects. Second, political benefits come from the reinforced commitment to international climate policy. Still, this reinforcement must accord with the possible incentive to adjust national cap-and-trade transactions in anticipation of linking. Third, regulatory disadvantages may be from the international linked system’s inconsistency with domestic policy targets. However, carbon emissions have a significant link to a life insurance business model since environmental carbon pollution may stimulate the life insurance need for health and life protection [6]. Insurers must invest the life insurance policy premiums they collect from policyholders to pay claims and benefits on their policies and to cover their total liabilities in the asset–liability risk matching management. More specifically, insurers’ primary role is to protect the policyholders. From the standpoint of risk, the emergency fund hypothesis proposes life insurance policy loans used to fund current expenses in economic hardship [7]. Huang [8] also argues that gray rhinos (i.e., highly possible yet ignored threats) are likely to have adverse consequences in the insurance industry. In addition, the International Cooperative and Mutual Insurance Federation reported doubling investment in insurer green finance to USD 84 billion in 2015 and increasing this investment fivefold to USD 420 billion in 2020 (See https://www.icmif.org/blog_articles/insurers-are-significantly-engaged-in-the-climate-debate/, accessed on 27 June 2022). The statistics have indicated that insurers understand the risk associated with climate change and can model their potential impacts on greenness. The green finance provided by insurers is at the very heart of creating resilience toward sustainability. Particularly, this paper aims at a vital issue of life insurance policyholder protection under the cap-and-trade mechanism, considering policy loan activities and financial gray rhino threats. Thus, the essential features of the research emphasize the insurer’s green finance toward sustainability. Toward that end, the insurer explicitly captures the credit risk from the borrowing firms, participating in the cap-and-trade scheme to reduce carbon emissions, an essential issue of carbon emission and environmental protection when facing gray rhino threats. In addition, the analysis is from the fund-providing insurer’s perspective. Green lending policies and life insurance policy loans (i.e., disintermediation related to insurance stability) are crucial to managers and regulators, particularly bridging the borrowing-firm carbon transactions for carbon emission reductions toward sustainability.

It is essential to elaborate on the issue of policyholder protection under the cap-and-trade scheme for environmental protection. Insurers, as fund providers, operate their investment strategies so that expected returns exceed what is guaranteed. In a regulatory carbon emission reduction environment, insurer green/brown lending strategies are most likely related to borrowing-firm productions. The high carbon emitter funded by the insurer’s brown loans must buy a carbon quota in the carbon trading market due to excessive carbon emissions. The low carbon emitter financed by the insurer’s green loans can sell the surplus quotas to earn profits [9]. The insurer’s lending function with borrowing-firm cap-and-trade transactions creates the need to model its equity explicitly considering credit risks from the borrowing firms [10].

On the liability side of the balance sheet, the insurer faces the uncertain withdrawals of life insurance policy loans: the unexpected disintermediation influences the asset-liability risk matching management. Understanding policy loan demand is essential to various stakeholders because policy loans are a form of disintermediation and disrupt insurer cash flow [7]. Huang [8] argues that a potential threat of a gray rhino (i.e., highly possible yet ignored threats, likely to occur with adverse consequences) affects the asset–liability matching management (The top five gray rhino risks in 2019 include US–China tensions, economic and financial fragilities, geopolitical uncertainty (the sum of the first two), cybersecurity/data integrity, and climate change and natural disasters [11]. Besides, EIOPA [12] reports there are more than 51 life insurers who failed in Europe during 1999–2016. NOLHGA [13] also indicates that the following insurance companies recently failed: American Network Insurance Company and Penn Treaty Network America Insurance Company in 2017, North Carolina Mutual Life Insurance Company in 2018, and Northwestern National Insurance Company of Milwaukee Wisconsin and Senior American Company in 2019. Thus, life insurance policyholder protection is crucial under the cap-and-trade system and gray rhino effect). As policyholder protection is crucial to insurer liquidity management, the issues of how it adjusts to uncertainty changes in the insurance environment under the cap-and-trade mechanism in a gray rhino environment deserve closer scrutiny.

Our study contributes to the literature on the following aspects. First, stakeholder theory demonstrates that environmental regulation such as a cap-and-trade scheme may help firms ease the conflict among stakeholders and achieve more healthy and sustainable development [14]. Zhang [15] also argues that environmental pollution’s harmfulness to economic and social development appears, and the public pays more attention to firms’ environmental protection behavior, which causes interest conflicts between them. Our paper complements the literature to study a step earlier regarding the carbon-emission borrowing-firm funding under a cap-and-trade mechanism and focuses on the interest conflicts between their lender’s (i.e., insurer’s) shareholders and policyholders. Thus, exploring the relationship between environmental protection and policyholder protection in our study is interesting, a topic on which the previous literature remains silent. Second, previous works find that green credit help financial intermediaries avoid environmental risks, borrowing-firm transformation, and sustainable economic development [16,17,18,19]. However, only limited research investigates the influence of strategic green credit on insurer performance from the perspective of policyholder protection under the cap-and-trade mechanism. Third, we focus on the gray rhino modeling. The present paper strives to understand better the insurer leading causes of policyholder protection. It is the first paper to enhance supervisory knowledge on policyholder protection against gray rhino threats. A life insurer has unique features and provides an excellent natural laboratory for explaining the causes of failure, an essential issue of policyholder protection and insurance stability. Fourth, Chiaramonte et al. [20] point out that few papers have explored the broader concept of sustainability in the insurance sector. The authors show that sustainability, proxied by environmental, social, and governance scores, enhances insurance stability. Our research models insurer green finance toward sustainability. Thus, our contribution stresses hybrid environment–insurance protection where borrowing firms face the gray rhino situation.

This paper theoretically investigates the impact of the regulatory cap, policy loan rate, gray rhino, and insurer leverage on policyholder protection. Our analytical framework relies on the capped barrier contingent claim to evaluate the insurer’s equity and liabilities. The model considers the capped credit risk from borrowing firms based on Dermine and Lajeri [10] and the barrier to capturing the premature default based on Brockman and Turtle [21]. The model appears appropriate since the carbon-emission borrowing firms operate under the regulatory cap-and-trade scheme in a gray rhino threat environment. The insurer (the lender) faces the uncertainty of demand for life insurance loan policies. The insurer faces an upward-sloping supply curve in an imperfectly competitive life insurance policy market [22]. Thus, determining the optimal guaranteed rate is crucial for the profit-maximization framework. Accordingly, we could explore the effects on policyholder protection evaluated at the optimal guaranteed rate under the cap-and-trade mechanism.

To our best knowledge, this is the first paper to develop a capped barrier option model studying the hybrid environment–insurance protection where the borrowing firms and the insurer face gray rhino threats. The paper investigates the influences of the regulatory cap-and-trade transactions and gray rhino threats on the guaranteed rate-setting behavior and policyholder protection. We present the main results as follows. First, the government attempts to reduce the cap of the cap-and-trade mechanism to protect the natural environment. This reduced cap increases the insurance business at an increased guaranteed rate but harms policyholder protection. In the insurer–borrowing firm situation, the regulatory cap deteriorates the insurer’s performance: reducing the insurer’s optimal interest margin and adversely affecting insurance stability. The negative effect of environmental protection on policyholder protection becomes significant when the gray rhino threats become significant. Second, policy loan is cash disintermediation in the asset-liability matching management. Policyholders are reluctant to conduct loan policies at a higher policy loan rate. The unexpected cash withdrawal bears a penalty since it disrupts the insurer’s liquidity management. Increasing the policy loan rate reduces the insurer’s interest margin and policyholder protection. Third, a gray rhino effect increases the bank’s interest margin and policyholder protection. The gray rhino effect incurs in the borrowing-firm product market, creating carbon pollution in the natural environment. Householders’ health status might worsen, stimulating a high demand for life insurance policies. The insurer benefits from a higher profit (i.e., a higher interest margin at a reduced guaranteed rate) for higher policyholder protection. Overall, this paper contributes to the literature by considering the credit risk from green lending, where the borrowing firms participate in the cap-and-trade transactions in a gray rhino environment. Under the circumstances, the hybrid-protection interaction should interest investors, insurer managers, and regulators.

The paper organizes as follows. Section 2 introduces the literature review as a theoretical background of the article. In Section 3, we develop a research model to address the critical problems we investigate in the paper. Section 4 presents the methodology and data for the numerical analysis. We demonstrate the results in Section 5. Finally, a conclusion section presents the main results and implications and suggests further avenues of research.

2. Literature Review and Theoretical Background

There is some literature on life insurance policyholder protection considering the cap-and-trade scheme; we here primarily review the most related research strands as a theoretical background for our model development. A new substantial literature has emerged that discusses the impact of the borrowing-firm gray rhino on fund-provider policyholder protection considering the optimal guaranteed determination. Next, we focus on the issue of a cap-and-trade mechanism for policyholder protection. The instrument has vital environmental improvement. The third strand is the policy loans on the policyholder protection. The last is the recent related literature to the aim we analyze.

2.1. Gray Rhino

Ferguson [23] argues that either a coronavirus COVID-19 pandemic (which is a natural disaster) or a war (which is artificial) is a rare and sizeable scale–scale disaster that characterizes a black swan, a gray rhino, or a dragon king. Stressing pandemics surprised most people, despite numerous warnings of the likelihood of such a disaster. More specifically, Huang [8] develops a down-and-out call option to examine how capital regulation affects an insurer’s optimal guaranteed rate and survival probability, considering the insurance grey rhino. The investment-oriented grey rhino effect increases the policies at an increased guaranteed rate (and thus, a decreased interest margin) and further decreases the insurer’s survival probability. The policy-oriented grey rhino effect decreases the guaranteed rate and the survival probability. Thus, Huang [8] complements Ferguson [23] by distinguishing the features of the gray rhino in an insurance market. Lin et al. [24] also focus on the insurance gray rhino as an extension of Huang [8]. Li et al. [25] mainly evaluate the relevant study’s objective setting by developing a contingent claim utility model. Their analysis shows that stringent capital regulation enhances policyholder protection but at the expense of the insurer’s equity return, thus adversely affecting insurer survival. While the research discusses the financial gray rhino, considering the cap-and-trade mechanism and policyholder protection makes our study in a different direction.

2.2. Cap-and-Trade Mechanism

There is extensive literature regarding the pricing effect and carbon emission reduction under the cap-and-trade mechanism. For example, Debo et al. [26] studied the pricing and green technology selection problem of remanufacturing products and showed that remanufacturing products are profitable due to green technology choice. Su et al. [27] focus on two alternative green technologies for the market structures of green products and develop a nonlinear programming model to determine optimal and quality level products. Yang et al. [28] discuss the supply chain’s pricing and carbon emission reduction decisions with vertical and horizontal operations. Chen et al. [29] explored a monopolistic firm’s manufacturing, remanufacturing, and collection decisions under the cap-and-trade regulation. Hussain et al. [30] develop a simulation-based model to study the emission reduction subsidy policy on the decision of product pricing and green technology of companies in an imperfectly competitive market. Under the cap-and-trade mechanism, Yang et al. [31] study the effects of allowance allocation rules on green technology investment and product pricing. Xue and Sun [32] consider two competitive supply chains with consumers’ low-carbon preferences under the regulatory cap-and-trade mechanism, each of which consists of one manufacturer and one retailer. Together, the study can model competition or integration in vertical and horizontal directions, four different supply chain structures. Xue et al. [33] investigate firms’ incentive mechanisms for carbon emission reduction in a two-echelon supply chain under the regulatory cap-and-trade scheme, where consumers exhibit low-carbon awareness. Therefore, from the perspective of enterprises (i.e., borrowing firms) under the cap-and-trade mechanism, the present study endeavors to address this research gap by involving the funding characteristic of brown/green loans and studying the optimal life insurance policy pricing to maximize insurer profits. A comparative analysis is on the impact of policyholder protection from the cap-and-trade mechanism.

2.3. Life Insurance Policy Loan

One of the related streams of the literature is the life insurance policy issue under the cap-and-trade mechanism. Carson and Hoyt [34] investigate the impact of redesigned life insurance policy loans on the demand for policy loans. The results suggest that policy loan demand has altered because of the introduction of variable loan rates and the redesign of policies. Specifically, their findings are the need for policy loans driven by reducing arbitrage potential. Liebenberg et al. [7] provide a shred of evidence supporting the hypothesis of the policy loan emergency. Their findings are that the more detailed emergency fund reveals a significant positive relation between loan demand and the expense of income shocks. Srbinoski et al. [35] contribute by providing the most comprehensive systematic review that integrates alternative fields, besides encapsulating studies on demand for life-insurance policy loans. They indicate the most critical drivers of life-insurance policyholder behavior during their lifetime. Li et al. [36] developed a two-stage contingent claim model to determine the optimal guaranteed rate and technology choice. One finding is that increasing life-insurance policy loans decreases insurance businesses at a reduced guaranteed rate. The research also finds that increased advanced technology involvement enhances insurance businesses at an increased guaranteed rate. An increase in the policy loan also increases the policyholder protection. Srbinoski et al. [35] contribute by providing the most comprehensive systematic review that integrates alternative fields, besides encapsulating studies on the demand for life-insurance policy loans. The most critical drivers of life-insurance policyholder behavior appear during the policyholder’s lifetime. The literature remains silent on their funding investment. In particular, the funded borrowing firms get involved in the cap-to-trade carbon reduction transactions. Therefore, it is necessary to make asset-liability matching decisions on policyholder protection and environmental protection to investigate insurer performance.

2.4. Recent Related Literature

Chiaramonte et al. [20] demonstrate that sustainability has received growing attention from alternative stakeholders, but focusing on the insurance sector is little. The existing literature presents evidence of the positive effect of green finance toward sustainability on financial stability but remains silent on insurance stability. Alam [37] investigates the relationships between inflation, short-term interest rate, money supply, and crude oil price in Saudi Arabia. The study shows a negative relationship between short-term interest rate and crude oil price. A study’s central finding contributes to the combined effect of macroeconomic variables and oil price shocks on the Saudi stock market. Our theoretical model could treat oil price shocks in Alam [37] as financial gray rhino threats. Thus, Alam’s [37] study could expand an alternative implication of the cap-and-trade mechanism. Alam et al. [38] explore the relationship of interest margin return on assets, bank investment, and bank lending capacity with gross domestic product in India. A central result indicates a significant relationship between interest margin and asset returns with economic growth. Our research complements Alam et al. [38] by studying the optimal insurer interest margin determination, focusing on the green finance issue. Fazal et al. [39] modify Peter and Clark algorithm to determine the causal nexus between monetary policy and inflation. The results show that energy prices and monetary policy have cost-side effects on inflation; however, the latter becomes counterproductive whenever a high interest rate decreases energy-push inflation. Although our study remains silent on energy-push inflation, we examine the effect of the energy-related cap-and-trade mechanism on insurer performance. We believe that our model can accommodate energy-push inflation as a further avenue of research. Idrees et al. [40] develop a financial liberalization index for Pakistan. The study suggests the liberalization index considering the real-time change in the implementation process. Besides, liberalization reduces the cost of capital and improves corporate governance. One of the vital policies of financial liberalization could be insurer interest margin determination. Corporate governance is one of the essential factors toward sustainability (i.e., the cap-and-trade mechanism in our paper). The conceptual framework of Idrees et al. [40] implicitly paves the way for modeling our research. Overall, the previous literature as a study background enriches our research.

In sum, most literature did not simultaneously consider the gray rhino impact, the regulatory cap-and-trade mechanism, and life insurance policy loans for the insurer’s guaranteed rate determination. Our research fills the gap in the literature to explore policyholder protection with the regulatory cap-and-trade scheme in an imperfect competitive life insurance market.

3. Research Model and Problems

3.1. Assumptions and Framework

The model designs a minimalist fashion with the following characteristics of a life insurer:

• The insurer’s role is to provide funds to its borrowing firms on demand. Particular attention is given to two representative borrowing firms within the pollution and climate transition. The risks and challenges of the environmental change to carbon emission reduction vary from the high-carbon borrowing firm (i.e., the high emitter) to the low-carbon borrowing firm (i.e., the low emitter).
• The cap-and-trade mechanism is an effective carbon emission reduction [3,4]. In our model, the high emitter needs to buy carbon quotas in the carbon trading market; otherwise, it would be fined. The low emitter sells the surplus allowances to earn profits [9].
• The insurer’s total assets are financed partly by profit-sharing life insurance policies, including the policy loan option that allows policyholders to borrow against the policy’s cash value [7]. When policyholders conduct the policy loan option, the insurer will reduce its investment funding in the asset-liability matching management. Besides, the life insurance policy market faced by the insurer is imperfectly competitive [22], where the insurer is a guaranteed rate-setter of the life insurance contract (Hubbard [41] reports that at nearly 13%, MetLife has the largest market share of the life insurance industry for direct premiums written, followed by Equitable Holdings (7.9%) and Prudential (7.8%) in the United States. The statistics indicate an imperfectly competitive life insurance market. See https://www.bankrate.com/insurance/life-insurance/life-insurance-statistics/, accessed on 27 June 2022).
• The insurer and its borrowing firm likely face the financial gray rhino (i.e., highly possible yet ignored threats), occurring with adverse consequences [42]. One possibility is extreme climate change [11]. Thus, green lending is essential in the financial gray rhino environment.
• Overall, the model calls attention to the fact that credit risks from the borrowing firms affect the distribution of the insurer’s asset portfolio, explicitly considering the policy loans in the premature default risk. Thus, the standard Merton [43] methodology such as Briys and de Varenne [44] used to provide a market-based estimation of the insurer’s equity needs to be adapted.

We consider the insurer in a one-period model $(t\in [0,1\left]\right)$ applied to an insurer-two-borrowing-firm situation. It is well-recognized that green loans are the most important financing tools for green projects. The two borrowing firms have the following balance sheets:

$$\mathrm{high} \mathrm{emitter}: V_H=A_H+K_H$$

(1)

$$\mathrm{low} \mathrm{emitter}: V_L=A_L+K_L$$

(2)

In Equation (1), the high emitter is funding an asset ($V_H$) with an insurer’s brown loans (i.e., high carbon emission loans or conventional loans $A_H$) and equity ($K_H$). In Equation (2), the low emitter is funding an asset ($V_L$) with an insurer’s green loans (i.e., low carbon emission loans $A_L$) and equity ($K_L$). In financing to green transition, the annual growth rate of green loans is about 19%, ensuring the share of green loans to total loans is over 10% [45]. Thus, imposing an assumption on the model is $A_L<A_H$.

The insurer’s balance sheet is:

$$A+B=L+K$$

(3)

where

$$L=(1-\alpha )(A+B), K=\alpha (A+B), \mathrm{and} A=A_H+A_L$$

(4)

Equation (3) illustrates that $A$ is the number of risky assets distributed to $A_H$ and $A_L$, $B$ is the number of risk-free assets, $L$ is the amount of life insurance policies, and $K$ is the stock of equity capital. The parameter $0<\alpha <1$ is a leverage variable chosen by the insurer. The policies partly finance the insurer’s investment $(1-\alpha )(A+B)$. Alternatively, the leverage variable could be capital regulation in our model. Thus, the asset loan portfolio distribution explicitly bridges the relationship between the insurer and the borrowing firms.

As mentioned, the insurer’s asset portfolio consists of three assets: brown loans, green loans, and liquid assets. We assume that the insurer is a rate-taker in the investment markets. The expected repayments of brown loans, green loans, and liquid assets are $A_{HH}=(1+R_{AH})A_H$, $A_{LL}=(1+R_{AL})A_L$, and $(1+R_B)B$, respectively. Here, $R_{AH}$ is the expected return rate of brown loans, $R_{AL}$ is the expected return rate of green loans, and $R_B$ is the return rate of liquid assets. Brown loans have been riskier than green loans [45]. Thus, we assume that the expected return rate of brown loans would be higher than that of green loans (i.e., $R_{AH}>R_{AL}$) (The model assumes the brown loan rate is higher than the green loan rate. Liu [45] also indicates that the brown loan is riskier than the green loan. More than 80% of customers consider the greenness of products and services when they purchase them [46], stimulating the growing green lending. Thus, the insurer anticipates the green loans to earn goodwill for asset–liability matching management. We could treat lower risk and greenness goodwill as the expected revenues. However, we remain silent on this aspect in our model, which should open an avenue of research).

We follow Briys and de Varenne [44] to specify the profit-sharing policy where the policyholders are guaranteed an interest rate $R$. Besides this guaranteed rate, the policyholders own a participation level $\delta$ of the net equity of the insurer. We define the insurance liabilities as a form of $(1-\alpha )(A+B)e^R$, i.e., the guaranteed payoff to the policyholders. One situation is that policy loan demand is perhaps due to the emergency fund hypothesis [7]. The policy loan demand is expressed as a part of the life insurance policy. That is the term $\kappa (1-\alpha )(A+B)e^{R_P}$, where $R_P$ is the rate of a policy loan and the condition $0<\kappa <1$. We assume that the state $R_P>R$, since the policy loan rate is the rate with the costs of insurance policy collateral and unexpected demand penalty.

The market values of the underlying assets of the borrowing firms vary over the period according to the stochastic process:

$$\mathrm{high} \mathrm{emitter}: d{E}_H=({\mu }_H-{\varphi }_{G}s)E_{H}dt+({\sigma }_H+{\varphi }_{G}s)E_{H}d{W}_H$$

(5)

$$\mathrm{low} \mathrm{emitter}: d{E}_L=({\mu }_L-{\varphi }_{G}s)E_{L}dt+({\sigma }_L+{\varphi }_{G}s)E_{L}d{W}_L$$

(6)

where

$$E_H=(1+R_H)V_H-(c_H-c_{CAP})V_H$$

$$E_L=(1+R_L)V_L+(c_{CAP}-c_L)V_L$$

In Equation (4), the underlying assets are the high-emitter investment value net of carbon quota costs. In Equation (5), the underlying assets are the low-emitter investment value plus carbon allowance benefits.

In Equation (4), $R_H=$ the return rate on the risky-asset investment of the high emitter $({\mu }_H-{\varphi }_{G}s)=$ the instantaneous expected return net of the gray-rhino effect (${\varphi }_G$) and the structural break ($s$) on the high-emitter investment, $({\sigma }_H-{\varphi }_{G}s)=$ the instantaneous volatility on the high-emitter investment, and $W_H=$ a standard Wiener process. The stochastic process implies that the underlying assets follow a lognormal distribution. Developing a procedure for estimating an appropriate gray rhino measure is critical. We consider a constant case limited to a one-time drop when the gray rhino effect incurs. Under the circumstances, the high emitter will suffer low returns and high risks in its investment activities (i.e., the so-called investment-oriented gray rhino effect) (In our model, it is strong to assume that the reduced return due to the gray rhino effect equals to the increased volatility and both the impacts insurer in the value assessment simultaneously. However, adding this assumption is expected to be not alter the qualitative results derived from our model). For example, a parameter of the gray rhino (${\varphi }_G=$ 0.25) indicates the panic that occurs when the high emitter makes decisions in time. A parameter (${\varphi }_G=$ 1.75) suggests the alarm when the manager does not make decisions in time. These two examples explain a gray rhino risk such as external economic and financial fragilities captured by the term ${\varphi }_{G}s$ in Equation (4). Equation (4) describes the instantaneous returns decreasing less significantly and the instantaneous volatility increasing less effectively when the gray rhino risk is realized and managed in time. Similarly, Equation (5) interpretation follows the symmetric argument as Equation (4).

The two emitters get involved in the regulatory cap-and-trade system. The high emitter needs to buy carbon quota ($CB$, cost with the regulatory rate $c_{CAP}$), and the low emitter sells the surplus allowances ($CS$, revenue with $c_{CAP}$) in the carbon trading market:

$$\mathrm{high} \mathrm{emitter}: CB=(c_H-c_{CAP})V_H>0$$

(7)

$$\mathrm{low} \mathrm{emitter}: CS=(c_{CAP}-c_L)V_L>0$$

(8)

In the carbon trading market, the high emitter’s carbon price per unit of production is $c_H$, which is higher than the regulatory price. We argue why the high emitter needs to buy carbon quota where the price is ($c_H-c_{CAP}$). The low emitter’s carbon price is $c_L$, and they can sell the surplus allowance ($c_{CAP}-c_L$).

3.2. Objective

We apply the option pricing framework integrating Briys and de Varenne [44], with the cap structure in Dermine and Lajeri [10] and the barrier structure in Brockman and Turtle [21] to evaluate the market value of the insurer’s equity and liabilities. As far as the insurer’s equity is concerned, the equity value over the time interval is given by:

$$\begin{array}{ll}S& =Max\{0,Min[(E_H+E_L)-Z,(1-\delta (1-\alpha ))(E_H+E_L)-(1-\delta )Z]\}\\ & =Max[0,(E_H+E_L)-Z]-\delta (1-\alpha )Max[0,(E_H+E_L)-Z/(1-\alpha )]\\ & =C_{SS}((E_H+E_L),Z)-\delta C_{SP}((1-\alpha )(E_H+E_L),Z)\end{array}$$

(9)

where

• $Z=(1-\alpha )(A+B)e^R-\kappa (1-\alpha )(A+B)e^{R_P}-(1+R_B)B$
• $\delta =$ the participation level of the profit-sharing policy

In Equation (8), $C_{SS}$ and $C_{SP}$ are capped down-and-out call options with the strike price $Z$. The strike price is the net liabilities, i.e., life insurance policies net of policy loans and liquid-asset repayments. The capped structure reflects borrowing-firm credit risks, while the barrier structure captures the insurer’s premature default risk. See Appendix A for the specifications of the two options.

As an outgrowth of the model, we also introduce an objective framework application to the insurer’s liabilities (i.e., policyholder protection) as follows:

$$\begin{array}{l}P=Z{e}^{-R_B}-\{Z{e}^{-R_B}N(-d_1)-(E_H+E_L)N(-d_1)\\ \hspace{1em}-(E_H+E_L){(\frac{H}{E_H+E_L})}^{2\eta }N(d_5)+Z{e}^{-R_b}{(\frac{H}{E_H+E_L})}^{2\eta -2}N(d_6)\}\\ \hspace{1em}+\delta \{(1-\alpha )(E_H+E_L)N(c_1)-Z{e}^{-R_B}N(c_2)\\ \hspace{1em}-(1-\alpha )(E_H+E_L){[\frac{H}{(1-\alpha )(E_H+E_L)}]}^{2\eta }N(c_5)\\ \hspace{1em}+Z{e}^{-R_B}{[\frac{H}{(1-\alpha )(E_H+E_L)}]}^{2\eta -2}N(c_6)\}\end{array}$$

(10)

The value of the insurer’s liabilities is similar to the put option, except that the underlying asset is the investments of the borrowing firms, not the loans of the insurer. In Equation (9), the former $\{\cdot \}$ is the standard capped call for equity while the latter $\{\cdot \}$ is the standard capped call for participation.

3.3. Solutions and Comparative Statics

Partially differentiating Equation (8) concerning $R$, the first-order condition is given by:

$$\frac{\partial S}{\partial R}=\frac{\partial C_{SS}((E_H+E_L),Z)}{\partial R}-\delta \frac{\partial C_{SP}((1-\alpha )(E_H+E_L),Z)}{\partial R}=0$$

(11)

Equation (10) reveals that the first-order condition determines the optimal guaranteed rate where the second-order condition (${\partial }^{2}S/\partial R^2<0$) is valid. The first-order condition also determines the optimal insurer interest margin, i.e., the spread between the loan rate ($(R_{AH}+R_{AL})/2$) and the optimal guaranteed rate. The optimal insurer margin conveys vital information for the efficiency of the financial system. We substitute the optimal guaranteed rate to obtain the insurer’s liabilities of Equation (9), staying on the optimization.

Identifying the impacts on the insurer survival from changes in regulatory cap rate, policy loan rate, gray rhino effect, and insurer leverage would help understand changing insurer efficiency and provide policy implications. Toward that end, the differentiation of $P$ evaluated at the optimal $R$ with $i$ ($=c_{CAP}$, $R_P$, ${\varphi }_G$, and $\alpha$) yields as follows:

$$\frac{dP}{di}=\frac{\partial P}{\partial i}+\frac{\partial P}{\partial R}\frac{\partial R}{\partial i}$$

(12)

In Equation (11), the first term on the right-hand side is the direct effect where the optimal $R$ is constant. The second term is the indirect effect, where different optimal states $R$ adjust the effect. We use numerical analysis to demonstrate the intuition about Equation (11) in the following section.

4. Methodology and Data

4.1. Methodology

The critical issues we investigate are Equation (11) in the model. It is obviously very complicated to derive the comparative statics directly. We employ a numerical approach to draw more managerial insights from the theoretical results above. A significant advantage is that we can use the numerical method to solve problems, where an analytical solution is very complicated.

4.2. Data Description

Before proceeding with numerical analysis to provide intuition, we define a reasonable firm-level dataset based on our presented model as follows:

• Supply of life insurance policies: The insurer faces an upward-sloping curve of policies. Holsboer [47] reports the guaranteed rate ranging from 3.25% to 5.70% in Belgium. Thus, we assume the locus from $(R(\%),L)=$ (3.30, 307), (3.50, 323), (3.70, 334), (3.90, 341), (4.10, 345), (4.30, 347), and (4.50, 348). Each bundle’s life insurance policy quantity here is arbitrary since the model deals with a firm-level analysis.
• Green and brown loans and capital stocks: As mentioned previously, the ratio of green loans to total loans is over 10% [45]. Thus, we assume $A_H=$ 289 and $A_L=$ 51. We also follow Dermine and Lajeri [10] and consider the leverages of the high- and low-emitters equaling 30% for simplicity. Hence, we have $K_H=$ 123.86 and $K_L=$ 21.86. We assume the insurer’s leverage ratio is $\alpha =$ 10.00% [44].
• Investment: According to Brockman and Turtle [21], the bond interest rate is $R_B=$ 4.50%. Tan et al. [48] find that the rate of return on investment is 5.10%. Using the return rate as a baseline, we assume the brown loan rate $R_{AH}=$ 5.00% and the green loan rate $R_{AL}=$ 4.60%. The return rate of high-emitter investment is $R_H=$ 7.00%, and that of low-emitter investment is $R_L=$ 5.50%. Thus, we have the condition ($R_{AL}<R_{AH}<R_L<R_H$) held. The reason is that the default rate of green loans is less than that of brown loans [45].
• Policy loans: The interest rate of the policy loans is 4.80% when charged in advance [49]. Thus, we assume $R_P=$ 4.80%. Regarding the ratio of policy loans to total life insurance policies, we arbitrarily take $\kappa =$ 0.40% at a firm-level one for our analysis.
• Cap-and-trade mechanism: Narassimhan et al. [50] report that the cost of compliance with the economic efficiency of the cap-and-trade regime in the European Union in 2016 was USD 72,440 per installation (USD 0.20 per tonne of $C{O}_{2e}$). The administration cost is USD 2750 per installation. The stringency of the cap (% cap reduction/year) is 2.20%. Accordingly, we assume $c_H=$ 2750/72,440 $=$ 3.80%, $c_{CAP}=$ 2.20%, and $c_L=$ 0.60%. The assumptions imply a possible case of carbon neutrality.
• Risks: Markellos and Psychoyios [51] find that the interest rate volatility is approximately 39.43%. Again, Liu [45] implicitly indicates that green loans have been safer than brown loans. Using the finding of Markellos and Psychoyios [51] as a baseline, we assume ${\sigma }_H=$ 49.00% and ${\sigma }_L=$ 29.00%. According to Tan et al. [48], we consider the structural break $s=$ 0.03. We further take ${\varphi }_G=$ 1.00 for the initial state because of the condition $0.25\le {\varphi }_G\le 1.75$, as mentioned previously. The initial barrier is $b=$ 69.20% due to the finding of Brockman and Turtle [21]. The participation rate is $\delta =$ 0.85 [44].

The baseline for the numerical analysis is summarized in

Table 1. Data baseline.

Variable	Approximation and Assumption	Source
$(R(\%),L)$: policy supply	(3.30, 307) (3.50, 323) (3.70, 334) (3.90, 341) (4.10, 345) (4.30, 347) (4.50, 348)	Holsboer [47]
$A_H:$ brown loans	289	Liu [45]
$A_L:$ green loans	51	Liu [45]
$K_H:$ high-emitter capital	123.86	Dermine and Lajeri [10]
$K_L:$ low-emitter capital	21.86	Dermine and Lajeri [10]
$\alpha :$ insurer leverage ratio	0.10	Briys and de Varenne [44]
$R_B$: liquid-asset rate	4.50%	Brockman and Turtle [21]
$R_{AH}$: brown loan rate	5.00%	Tan et al. [48]
$R_{AL}$: green loan rate	4.60%	Tan et al. [48]
$R_H:$ high-emitter investment rate of return	7.00%	Liu [45]
$R_L:$ low-emitter investment rate of return	5.50%	Liu [45]
$R_P:$ policy loan rate	4.80%	Harding [49]
$\kappa :$ policy loans to total policies	0.40%	arbitrary
$c_H:$ high-emitter cap	3.80%	Narassimhan et al. [50]
$c_{CAP}:$ regulatory cap	2.20%	Narassimhan et al. [50]
$c_L:$ low-emitter cap	0.60%	Narassimhan et al. [50]
$\sigma :$ interest rate volatility	39.43%	Markellos and Psychoyios [51]
$s:$ structural break	0.03	Tan et al. [48]
${\varphi }_G:$ gray rhino effect	1.00	arbitrary
$b:$ barrier	69.20%	Brockman and Turtle [21]
$\delta :$ participation rate	0.85	Briys and de Varenne [44]

as follows:

5. Results

5.1. Regulatory Cap Effect

Regulators give producers limited carbon emission quotas, i.e., the regulatory cap [31]. For environmental protection, the regulators reduce the regulatory cap of the cap-and-trade scheme. The study first explores the regulatory cap effect on the optimal guaranteed rate determination at various gray rhino levels.

Table 2. Responsiveness of the optimal guaranteed rate to $c_{CAP}$.

	$(\mathit{R}(\%),\hspace{0.17em}\mathit{L})$
${\mathit{c}}_{\mathit{C}\mathit{A}\mathit{P}}$	(3.30, 307)	(3.50, 323)	(3.70, 334)	(3.90, 341)	(4.10, 345)	(4.30, 347)	(4.50, 348)
	$S$
0.016	28.4975	28.7541	28.8847	28.9176	28.8792	28.7943	28.6865
0.018	28.5397	28.7961	28.9267	28.9595	28.9211	28.8364	28.7285
0.020	28.5817	28.8379	28.9684	29.0012	28.9628	28.8781	28.7704
0.022	28.6233	28.8795	29.0098	29.0426	29.0043	28.9196	28.8119
0.024	28.6648	28.9207	29.0510	29.0838	29.0455	28.9609	28.8533
0.026	28.7060	28.9618	29.0920	29.1247	29.0865	29.0019	28.8943
0.028	28.7469	29.0026	29.1327	29.1654	29.1272	29.0426	28.9352
	$\partial R/\partial c_{CAP}$ (%)
0.016→0.018	-	−12.3184	−8.0694	−2.7831	4.9926	22.2168	-
0.018→0.020	-	−12.3496	−8.0898	−2.7901	5.0053	22.2732	-
0.020→0.022	-	−12.3800	−8.1098	−2.7970	5.0176	22.3281	-
0.022→0.024	-	−12.4096	−8.1292	−2.8037	5.0296	22.3817	-
0.024→0.026	-	−12.4386	−8.1482	−2.8103	5.0414	22.4339	-
0.026→0.028	-	−12.4668	−8.1666	−2.8166	5.0528	22.4848	-

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, $R_P=$ 4.80%, ${\varphi }_G=$ 1.00, and $\alpha =$ 0.10. The second-order condition (${\partial }^{2}S/\partial R^2<0$) consistently holds. The shaded areas illustrate the optimal values evaluated at the optimal guaranteed rate.

reveals that a decrease in the regulatory cap of the cap-and-trade scheme increases the insurer’s guaranteed rate and thus decreases the insurer’s interest margin. The regulator may attempt to tune down the cap of the cap-and-trade system for environmental improvement. As the borrowing firms face a reduced regulatory cap, the high-carbon-emission borrowing firm increases the operational costs by buying more carbon emission quotas. At the same time, the low-carbon-emission borrowing firm reduces the revenues by selling fewer carbon emission quotas. Under the circumstances, the insurer (the lender) must now provide a return to a higher capped risk base. The insurer may attempt to augment its total returns by increasing its investments at an increased guaranteed rate. If the policy supply is relatively rate-elastic, more life insurance policies are possible at an increased guaranteed rate (and thus, at a decreased insurer interest margin). A strict regulatory cap of the cap-and-trade scheme enhances the credit risk from the borrowing firms and the insurance business at a reduced profit for the insurer. Therefore, the regulatory cap has different influences on the stakeholders. Chen et al. [52] conclude that a reduced regulatory cap of the cap-and-trade scheme decreases the bank interest margin. Although focusing on a life insurer, our finding is consistent with the result of Chen et al. [52].

As mentioned previously, reducing the regulatory cap increases the optimal guaranteed rate (and thus, decreases the optimal insurer interest margin). The negative effect on the optimal guaranteed rate becomes less significant when the gray rhino impact becomes more serious (see

Table 3. Responsiveness of the optimal guaranteed rate to $c_{CAP}$ at various gray rhino levels.

	${\mathit{\varphi }}_{\mathit{G}}$
${\mathit{c}}_{\mathit{C}\mathit{A}\mathit{P}}$	0.25	0.50	0.75	1.00	1.25	1.50	1.75
	$\partial R/\partial c_{CAP}$ (%)
0.016→0.018	−3.2953	−3.1113	−2.9408	−2.7831	−2.6375	−2.5031	−2.3792
0.018→0.020	−3.3020	−3.1182	−2.9478	−2.7901	−2.6445	−2.5100	−2.3859
0.020→0.022	−3.3085	−3.1250	−2.9547	−2.7970	−2.6513	−2.5167	−2.3925
0.022→0.024	−3.3147	−3.1315	−2.9613	−2.8037	−2.6580	−2.5233	−2.3989
0.024→0.026	−3.3207	−3.1378	−2.9678	−2.8103	−2.6645	−2.5297	−2.4052
0.026→0.028	−3.3265	−3.1439	−2.9741	−2.8166	−2.6708	−2.5360	−2.4113

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, $R_P=$ 4.80%, and $\alpha =$ 0.10. The second-order condition (${\partial }^{2}S/\partial R^2<0$) consistently holds. All comparative statics are evaluated at the optimal bundle $(R(\%),\hspace{0.17em}L)$ = (3.90, 341).

). This negative effect implies the regulatory cap harms the insurer’s profits by expanding the life insurance businesses at an increased guaranteed rate. The harmfulness becomes less severe in a more severe gray rhino environment because the insurer may use its guaranteed rate-setting behavior to face the potential gray rhino threats. Our argument stresses the importance of the guaranteed rate-setting behavior when the insurer meets an imperfectly competitive insurance market in a gray rhino environment.

In

Table 4. Responsiveness of $P$ to $c_{CAP}$.

	$(\mathit{R}(\%),\hspace{0.17em}\mathit{L})$
${\mathit{c}}_{\mathit{C}\mathit{A}\mathit{P}}$	(3.30, 307)	(3.50, 323)	(3.70, 334)	(3.90, 341)	(4.10, 345)	(4.30, 347)	(4.50, 348)
	$P$
0.016	438.1437	437.8947	437.7678	437.7358	437.7731	437.8556	437.9604
0.018	438.8742	438.6252	438.4983	438.4663	438.5036	438.5861	438.6908
0.020	439.6046	439.3557	439.2288	439.1969	439.2342	439.3166	439.4213
0.022	440.3351	440.0862	439.9594	439.9274	439.9648	440.0471	440.1518
0.024	441.0656	440.8168	440.6900	440.6581	440.6954	440.7777	440.8824
0.026	441.7961	441.5474	441.4206	441.3887	441.4260	441.5083	441.6130
0.028	442.5266	442.2780	442.1513	442.1194	442.1566	442.2389	442.3436
	$dP/d{c}_{CAP}:$ total effect
0.016→0.018	-	365.3939	365.3086	365.2661	365.2660	365.3373	-
0.018→0.020	-	365.4079	365.3227	365.2802	365.2799	365.3512	-
0.020→0.022	-	365.4221	365.3370	365.2945	365.2942	365.3654	-
0.022→0.024	-	365.4366	365.3516	365.3091	365.3086	365.3798	-
0.024→0.026	-	365.4513	365.3664	365.3239	365.3234	365.3945	-
0.026→0.028	-	365.4662	365.3815	365.3390	365.3384	365.4094	-

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, $R_P=$ 4.80%, ${\varphi }_G=$ 1.00, and $\alpha =$ 0.10. The shaded areas illustrate the optimal values evaluated at the optimal guaranteed rate. The direct ($\partial P/\partial c_{CAP}$) and indirect ($(\partial P/\partial R)(\partial R/\partial c_{CAP})$) effects are positive in sign.

, the positive direct effect demonstrates that decreasing the regulatory cap of the cap-and-trade scheme reduces the insurer’s liabilities, holding the optimal guaranteed rate constant. We explain the positive impact because a decrease in the regulatory cap discourages the borrowing-firm productions, ceteris paribus, decreasing their funds needed and further reducing the insurer’s insurance issuing for lending investments. The positive indirect effect shows that lowering the regulatory cap increases the guaranteed rate-setting, as mentioned previously, which further decreases the insurer’s liabilities. The indirect impact reinforces the direct to give an overall positive effect: a reduced regulatory cap harms policyholder protection. Intuitively, the regulatory reduces the cap of the cap-and-trade transactions, discouraging the borrowing-firm production, at least in the short run, and further increasing the cap credit risk. The reduced cap harms the insurer’s interest margin and policyholder protection. Therefore, our result contributes to the literature that the reduced cap of the cap-and-trade scheme improves environmental protection due to production reduction but breaks policyholder protection. Chiaramonte et al. [20] suggest that sustainability, expressed by environmental, social, and governance scores, increases insurance stability. However, the study finds that the regulatory cap of the cap-and-trade mechanism toward sustainability hurts policyholder protection, adversely affecting insurance stability. The controversial results might be from the guaranteed rate-setting behavior conducted by the insurer that Chiaramonte et al. [20] remain silent. Therefore, our study complements the insurance stability literature.

Table 5. Responsiveness of policyholder protection to $c_{CAP}$ at various gray rhino levels.

	${\mathit{\varphi }}_{\mathit{G}}$
${\mathit{c}}_{\mathit{C}\mathit{A}\mathit{P}}$	0.25	0.50	0.75	1.00	1.25	1.50	1.75
	$dP/d{c}_{CAP}:$ total effect
0.016→0.018	364.5029	364.7441	364.9990	365.2661	365.5439	365.8309	366.1258
0.018→0.020	364.5276	364.7652	365.0165	365.2802	365.5547	365.8385	366.1303
0.020→0.022	364.5525	364.7864	365.0343	365.2945	365.5657	365.8463	366.1351
0.022→0.024	364.5777	364.8079	365.0522	365.3091	365.5770	365.8545	366.1402
0.024→0.026	364.6029	364.8296	365.0704	365.3239	365.5885	365.8629	366.1456
0.026→0.028	364.6284	364.8515	365.0889	365.3390	365.6004	365.8716	366.1513

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, $R_P=$ 4.80%, and $\alpha =$ 0.10. All the comparative statics are evaluated at the optimal guaranteed rate. The direct ($\partial P/\partial c_{CAP}$) and indirect ($(\partial P/\partial R)(\partial R/\partial c_{CAP})$) effects are positive in sign.

shows that reducing the regulatory cap jeopardizes policyholder protection. The jeopardy becomes less severe when the potential threats from the gray rhino become obvious: an increasingly threatening effect results in increasing policyholder protection, ceteris paribus. The regulator decreases the cap for environmental protection, stimulating the life insurance activities. The potential threat from the gray rhino brings a need for life insurance policies. Thus, policyholder protection improvement complements environment protection, utilizing the guaranteed rate-setting behavior in a severe gray-rhino threat environment.

5.2. Policy Loan Rate Effect

A policy loan is a form of disintermediation of the asset–liability matching management. The following section will investigate this disintermediation issue.

Policy loans are disintermediation and disrupt insurer cash flow.

Table 6. Responsiveness of the optimal guaranteed rate to the policy loan rate at various levels of $c_{CAP}$.

	${\mathit{c}}_{\mathit{C}\mathit{A}\mathit{P}}$
${\mathit{R}}_{\mathit{P}}$ (%)	0.016	0.018	0.02	0.022	0.024	0.026	0.028
	$\partial R/\partial R_P$ (%)
4.20→4.40	1.438412	1.438422	1.438432	1.438442	1.438453	1.438463	1.438473
4.40→4.60	1.441196	1.441206	1.441216	1.441226	1.441236	1.441247	1.441257
4.60→4.80	1.443985	1.443995	1.444005	1.444015	1.444026	1.444036	1.444046
4.80→5.00	1.446779	1.446789	1.446799	1.446810	1.446820	1.446830	1.446841
5.00→5.20	1.449578	1.449588	1.449599	1.449609	1.449619	1.449630	1.449640
5.20→5.40	1.452383	1.452393	1.452403	1.452414	1.452424	1.452434	1.452445

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, ${\varphi }_G=$ 1.00, and $\alpha =$ 0.10. The second-order condition (${\partial }^{2}S/\partial R^2<0$) consistently holds. All comparative statics are evaluated at the optimal bundle $(R(\%),\hspace{0.17em}L)$ = (3.90, 341).

shows that increasing the policy loan rate increases the guaranteed rate. The intuition is very straightforward. The insurer considers the capped credit risk from its borrowing firms, perhaps resulting in the credit sources being more expensive or difficult to obtain (i.e., the alternative funds hypothesis [7]). In this case, the demand for policy loans is likely higher; thus, the policy loan market rate may increase. The insurer increases its life insurance policy at an increased guaranteed rate to maintain efficient liquidity management. The fund inflow of life insurance policies by increasing the guaranteed rate-setting may replace policy loan fund outflow when the insurer faces an imperfectly competitive life insurance market. Thus, life insurance policies and policy loans are complements, considering the credit risk from borrowing firms explicitly. However, Li et al. [36] argue that life insurance policies and policy loans are substitutes since they consider technology choices but remain silent on the explicit credit risk from borrowing firms. Therefore, whether or not the life insurance policies and policy loans are complements/substitutes depends on the insurer’s margin determinants.

Moreover, the positive effect of the policy loan rate on the guaranteed rate-setting becomes less significant when the regulator conducts the cap reduction of the cap-and-trade scheme. An increasing policy loan rate accompanies fund disintermediation. The insurer may attempt to increase the fund inflowing by increasingly issuing life insurance policies at an increased guaranteed rate. The positive loan rate effect on the guaranteed rate-setting becomes less significant when the regulator conducts the cap reduction policy for environmental improvement. Therefore, the cap reduction policy might hurt insurer fund intermediation due to increasing policy loan rate on less significant increasing guaranteed rate-setting.

In

Table 7. Responsiveness of the policyholder protection to the policy loan rate at various levels of $c_{CAP}$.

	${\mathit{c}}_{\mathit{C}\mathit{A}\mathit{P}}$
${\mathit{R}}_{\mathit{P}}$ (%)	0.016	0.018	0.02	0.022	0.024	0.026	0.028
	$dP/d{R}_P:$ total effect (%)
4.20→4.40	−24.8438	−24.8371	−24.8302	−24.8232	−24.8161	−24.8088	−24.8014
4.40→4.60	−24.8936	−24.8869	−24.8800	−24.8730	−24.8658	−24.8585	−24.8511
4.60→4.80	−24.9435	−24.9367	−24.9298	−24.9228	−24.9156	−24.9083	−24.9009
4.80→5.00	−24.9934	−24.9867	−24.9797	−24.9727	−24.9655	−24.9582	−24.9507
5.00→5.20	−25.0435	−25.0367	−25.0298	−25.0227	−25.0155	−25.0082	−25.0007
5.20→5.40	−25.0937	−25.0869	−25.0799	−25.0728	−25.0656	−25.0583	−25.0508

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, ${\varphi }_G=$ 1.00, and $\alpha =$ 0.10. All comparative statics are evaluated at the optimal bundle $(R(\%),\hspace{0.17em}L)$ = (3.90, 341). The direct ($\partial P/\partial R_P$) and indirect ($(\partial P/\partial R)(\partial R/\partial R_P)$) effects are negative in sign.

, an increase in the policy loan rate decreases the insurer’s liabilities and thus, policyholder protection, holding the optimal guaranteed rate constant (i.e., the direct effect). An increase in the fund outflow at an increased policy loan market rate may harm policyholder protection since the cash flow disruption leads to inefficient liquidity management, ceteris paribus. Besides, increasing the policy loan rate reduces the insurer’s liabilities at every possible state of the optimal guaranteed rate-setting adjustment (i.e., the indirect effect). The indirect impact reinforces the direct impact to give an overall negative impact: increasing the policy loan rate deteriorates policyholder protection, thereby adversely affecting insurance stability. In addition, the negative policy loan effect on policyholder protection increases as the government conducts the cap reduction policy. As a result, the government performs a cap reduction policy for environmental protection, deteriorating policyholder protection when the policy loans increase. The result demonstrates the relationships between hybrid protection and policy loans. Li et al. [36] show that an increase in policy loan rate increases policyholder protection (i.e., insurance stability). However, this study finds that the responsiveness of the policyholder protection to the policy loan rate is negative. We argue that the inconsistent results might be from ignoring the explicit treatment of borrowing-firm credit risk and the cap-and-trade transactions toward sustainability. The discussed points in our study contribute to the insurance stability literature.

5.3. Financial Gray Rhino Effect

As mentioned previously, the financial gray rhino metaphor demonstrates an environment with the highest possible yet ignored threats. In the following section, we will investigate the gray rhino impact on the insurer’s optimal guaranteed rate and policyholder protection.

The metaphor of the financial gray rhino draws attention to the apparent risks, but we neglect them. Understanding the financial gray rhino impact on insurance is crucial to various stakeholders. We find that increasing the significant effect of the financial gray rhino decreases the life insurance businesses at a reduced guaranteed rate-setting by the insurer (see

Table 8. Responsiveness of the optimal guaranteed rate to ${\varphi }_G$.

	$(\mathit{R}(\%),\hspace{0.17em}\mathit{L})$
${\mathit{\varphi }}_{\mathit{G}}$	(3.30, 307)	(3.50, 323)	(3.70, 334)	(3.90, 341)	(4.10, 345)	(4.30, 347)	(4.50, 348)
	$S$
0.25	29.2964	29.5533	29.6841	29.7170	29.6785	29.5936	29.4856
0.50	29.0713	29.3280	29.4586	29.4915	29.4531	29.3682	29.2603
0.75	28.8469	29.1033	29.2338	29.2667	29.2283	29.1435	29.0357
1.00	28.6233	28.8795	29.0098	29.0426	29.0043	28.9196	28.8119
1.25	28.4006	28.6564	28.7866	28.8194	28.7811	28.6965	28.5890
1.50	28.1787	28.4342	28.5642	28.5970	28.5587	28.4743	28.3668
1.75	27.9576	28.2128	28.3427	28.3754	28.3372	28.2528	28.1455
	$\partial R/\partial {\varphi }_G$ (%)
0.25→0.50	-	−0.1478	−0.0946	−0.0323	0.0579	0.2598	-
0.50→0.75	-	−0.1645	−0.1062	−0.0364	0.0653	0.2921	-
0.75→1.00	-	−0.1775	−0.1154	−0.0396	0.0711	0.3174	-
1.00→1.25	-	−0.1872	−0.1223	−0.0421	0.0756	0.3367	-
1.25→1.50	-	−0.1942	−0.1274	−0.0440	0.0789	0.3509	-
1.50→1.75	-	−0.1991	−0.1310	−0.0452	0.0812	0.3608	-

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, $c_{CAP}=$ 2.20%, $R_P=$ 4.80%, and $\alpha =$ 0.10. The second-order condition (${\partial }^{2}S/\partial R^2<0$) consistently holds. The shaded areas illustrate the optimal values evaluated at the optimal guaranteed rate.

). The result is understood because the potential risks reflected by the gray rhino likely yield the insurer a higher interest margin (and thus profits). The neglect encourages the insurer to decrease its liabilities at a reduced guaranteed rate even though the insurer is likely exposed to a potentially high-risk environment. The negligence by the insurer incurs, at least in the short term.

Table 9. Responsiveness of $P$ to ${\varphi }_G$.

	$(\mathit{R}(\%),\hspace{0.17em}\mathit{L})$
${\mathit{\varphi }}_{\mathit{G}}$	(3.30, 307)	(3.50, 323)	(3.70, 334)	(3.90, 341)	(4.10, 345)	(4.30, 347)	(4.50, 348)
	$P$
0.25	440.2360	439.9839	439.8555	439.8231	439.8609	439.9443	440.0504
0.50	440.2746	440.0234	439.8955	439.8633	439.9009	439.9840	440.0897
0.75	440.3076	440.0575	439.9301	439.8981	439.9355	440.0183	440.1235
1.00	440.3351	440.0862	439.9594	439.9274	439.9648	440.0471	440.1518
1.25	440.3570	440.1094	439.9832	439.9514	439.9886	440.0706	440.1747
1.50	440.3735	440.1272	440.0017	439.9701	440.0070	440.0886	440.1922
1.75	440.3846	440.1397	440.0148	439.9834	440.0201	440.1012	440.2043
	$dP/d{\varphi }_G:$ total effect (%)
0.25→0.50	-	15.9964	16.0672	16.0604	16.0091	15.9794	-
0.50→0.75	-	13.8481	13.9247	13.9161	13.8600	13.8297	-
0.75→1.00	-	11.6869	11.7711	11.7618	11.7011	11.6685	-
1.00→1.25	-	9.5220	9.6149	9.6060	9.5410	9.5046	-
1.25→1.50	-	7.3618	7.4644	7.4565	7.3876	7.3464	-
1.50→1.75	-	5.2140	5.3268	5.3206	5.2480	5.2013	-

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, $c_{CAP}=$ 2.20%, $R_P=$ 4.80%, and $\alpha =$ 0.10. The shaded areas illustrate the optimal values evaluated at the optimal guaranteed rate. The direct ($\partial P/\partial {\varphi }_G$) and indirect ($(\partial P/\partial R)(\partial R/\partial {\varphi }_G)$) effects are negative in sign.

demonstrates that increasing the financial gray rhino increases policyholder protection. As mentioned previously, the insurer decreases its guaranteed rate setting when it neglects the potential-risk incurrence of intermediary operations. The more substantial risk-approaching behavior may encourage the insurer to reduce its liabilities and increase its profits. The policyholders benefit from insurer profitability in the gray rhino environment, at least in the short term.

5.4. Leverage Effect

It is necessary to elaborate on the leverage issue. Leverage plays a vital role in asset-liability management.

Table 10. Responsiveness of the guaranteed rate to the insurer’s leverage at various levels of $c_{CAP}$.

	${\mathit{c}}_{\mathit{C}\mathit{A}\mathit{P}}$
$\mathit{\alpha }$	0.016	0.018	0.02	0.022	0.024	0.026	0.028
	$\partial R/\partial \alpha$
0.04→0.06	10.5443	10.5407	10.5372	10.5337	10.5302	10.5267	10.5232
0.06→0.08	6.7210	6.7209	6.7207	6.7206	6.7205	6.7203	6.7202
0.08→0.10	5.8341	5.8322	5.8303	5.8285	5.8267	5.8248	5.8230
0.10→0.12	5.1631	5.1606	5.1580	5.1555	5.1529	5.1504	5.1478
0.12→0.14	3.5196	3.5196	3.5196	3.5196	3.5196	3.5196	3.5196
0.14→0.16	3.2651	3.2644	3.2637	3.2630	3.2624	3.2617	3.2610

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, $R_P=$ 4.80%, and ${\varphi }_G=$ 1.00. The second-order condition (${\partial }^{2}S/\partial R^2<0$) consistently holds. The comparative statics are evaluated at the optimal bundles, which are (3.70, 334) at $\alpha =$ 0.04→0.06, (3.90, 341) at $\alpha =$ 0.06→0.12, and (4.10, 345) at $\alpha =$ 0.12→0.16, respectively.

shows that increasing the insurer’s leverage increases the insurance businesses at an increased guaranteed rate-setting. Intuitively, as the insurer increases its capital relative to its liability level, it must now provide a return to a more extensive equity base. The insurer attempts to augment its total returns by increasing its guaranteed rate-setting to attract more funds from issuing life insurance policies. Under the circumstances, the insurer is more willing to provide funds for borrowing firms. The increased leverage is not from demand-pull investments but supply-push investments. As a result, the increased leverage reduces the insurer’s interest margin. The effect explains that the insurer is reluctant to use its internal capital rather than its external liabilities for providing borrowing-firm funds. Table 10 also shows that the positive impact of insurer leverage on the guaranteed rate (thus, on the insurer interest margin) becomes more significant when the government carries out a cap reduction policy. Therefore, environmental protection costs insurer profitability, considering increases in the insurer’s leverage strategy.

In

Table 11. Responsiveness of the policyholder protection to the insurer’s leverage at various levels of $c_{CAP}$.

	${\mathit{c}}_{\mathit{C}\mathit{A}\mathit{P}}$
$\mathit{\alpha }$	0.016	0.018	0.02	0.022	0.024	0.026	0.028
	$dP/d\alpha :$ total effect
0.04→0.06	−440.6117	−441.7299	−442.8466	−443.9618	−445.0755	−446.1877	−447.2985
0.06→0.08	−449.3021	−450.4014	−451.4993	−452.5956	−453.6904	−454.7837	−455.8755
0.08→0.10	−458.8092	−459.8853	−460.9598	−462.0329	−463.1044	−464.1744	−465.2429
0.10→0.12	−468.8837	−469.9316	−470.9781	−472.0231	−473.0666	−474.1086	−475.1491
0.12→0.14	−478.6010	−479.6176	−480.6328	−481.6465	−482.6587	−483.6695	−484.6789
0.14→0.16	−489.5365	−490.5162	−491.4945	−492.4715	−493.4472	−494.4215	−495.3945

Notes: Unless otherwise indicated, $A_H=$ 289, $A_L=$ 51, $K_H=$ 123.86, $K_L=$ 21.86, $R_B=$ 4.50%, $R_{AH}=$ 5.00%, $R_{AL}=$ 4.60%, $R_H=$ 7.00%, $R_L=$ 5.50%, $\kappa =$ 0.40%, $c_H=$ 3.80%, $c_L=$ 0.60%, $\sigma =$ 39.43%, $s=$ 0.03, $b=$ 69.20%, $\delta =$ 0.85, $R_P=$ 4.80%, and ${\varphi }_G=$ 1.00. The comparative statics are evaluated at the optimal bundles. The direct ($\partial P/\partial \alpha$) and indirect ($(\partial P/\partial R)(\partial R/\partial \alpha )$) effects are negative in sign.

, the direct effect of the insurer’s leverage on the policyholder protection is negative. The negative impact implies that the insurer must hold more capital for liquidity management. Preserving more capital will jeopardize its performance and lead to less lending. Thus, policyholder protection deteriorates. The indirect effect is also harmful. Holding more capital harms policyholder protection through every possible state of the guaranteed rate-setting adjustments. The indirect effect indicates that an increase in leverage increases the guaranteed rate; further, the expanded guaranteed rate decreases policyholder protection. The indirect impact reinforces the direct result to give an overall negative impact: holding more capital harms policyholder protection. The negative effect of insurer leverage on policyholder protection becomes less significant when the government uses the cap reduction policy. The result implies that environmental protection substitutes for policyholder protection when the insurer conducts an increased leverage strategy. Our result supports a finding of Huang et al. [53] when the insurer adopts an aggressive guaranteed rate-setting management.

6. Conclusions and Implications

The paper investigates an insurer-borrowing firm relationship for policyholder protection in a gray rhino environment. Toward that end, the paper models the capped credit risk from the insurer’s green/brown lending with the premature default risk in the equity valuation of the insurer and demonstrates its use of the insurer interest margin determination and policyholder protection.

Several crucial results should interest insurer managers, regulators, and investors. For example, the reduced regulatory cap of the cap-and-trade scheme for environmental protection induces an insurer interest margin reduction and policyholder protection harmfulness in a gray rhino environment. The cap-and-trade mechanism is an efficient carbon emission reduction regulation. However, environmental protection has been at the cost of policyholder protection, considering a case of insurer green/brown lending to production firms. Besides, we also contribute to the literature that the insurer interest margin and policyholder protection are beneficial when the gray rhino impact becomes significant in the production environment. The capped credit risk from borrowing firms yields better insurer performance because carbon emissions demand substantial life-insurance policies. In conclusion, we suggest that the capped barrier option model is intimately relevant to the cap-and-trade scheme and insurer green lending in a gray rhino environment.

Profitability and policyholder protection are key performance issues concerning insurer managers, particularly during a gray rhino environment. Profitability related to insurer interest margin is an essential goal for most financial firms, and insurers often assess their performance relative to each other. Knowing how the regulatory cap, policy loan rate, gray rhino, and leverage affect insurer profitability is paramount for investors and regulators. Notably, a stringent cap of the cap-and-trade mechanism stimulates environmental improvement and enhances insurer profitability. Policyholder protection related to insurer survival is central in strategic decisions made by insurers and in decisions made by regulators concerned about insurance stability. A stringent cap hurts policyholder protection, adversely affecting insurance stability. Overall, we complement the literature that a strict cap of the cap-and-trade mechanism is an effective carbon emission reduction regulation from a profitability but not a stability standpoint.

The model presented here is general and should open at least two further avenues of research. One is to introduce borrowing-firm green technology employment, which could improve the clean environment. Knowing how borrowing-firm green technology employment affects insurer profits and policyholder protection in a gray rhino environment deserves closer scrutiny. The other is introducing reinsurance, since policy loans and gray rhinos are vital factors in our model. When adding reinsurance to our model, we can reconsider the issue of policyholder protection and possible risk-sharing between the insurer and the reinsurer.