# The relationship between regulatory pressure and insurer risk taking.

ABSTRACT

The article examines the risk-taking behavior of property-liability insurers in the presence of risk-based capital regulation. An option pricing model is developed to evaluate the expected regulatory cost and predict a nonlinear relationship between regulatory pressure and insurers' risk taking. We then conduct an empirical test using the simultaneous threshold regression. The result shows that there is a threshold effect of regulatory pressure on insurer risk taking. Poorly capitalized insurers seem to be aware of their proximity to regulatory interventions but do not fully respond to the impending regulatory pressure. This implies either regulatory interventions are not costly enough or they are too late, or both.

INTRODUCTION

Understanding the risk-taking behavior of financial institutions, including banks and insurance firms, is important to not only regulators but also claimholders of these firms because the behavior can substantially affect values of the firms and thus how the values are distributed between claimholders. A number of theories have been proposed to explain the behavior in the banking and insurance literature. However, they often make contradictory predictions regarding the risk-taking behavior of financial institutions. For example, the risk-subsidy hypothesis predicts that banks and insurers will engage in excessive risk taking when deposit insurance for banks or guarantee funds for insurers are nonrisk based. The reason is that the deficit of deposits and insurance policies will be absorbed by government funds when these firms become insolvent, and thus the firms pay little cost for their high risk taking. When banks/insurers attempt to increase their risk taking, they can either increase portfolio risk or lower capital level, or both. Hence, the risk-subsidy hypothesis predicts a negative relationship between capital and risk. Another stem of theories regarding the risk-taking behavior of financial institutions, in contrast, advocates that these firms will operate in a finite-risk paradigm if relatively large costs can result from high risk taking. These costs include transaction costs, (1) agency costs, (2) bankruptcy costs, (3) regulatory costs, (4) and others. (5) This class of theories anticipates that financial institutions will sustain their financial strength by either raising capital when their asset risk is increasing or lowering asset risk when their capitalization is declining, and thus predicts that capital and risk levels will be positively correlated. Despite the fact that these two distinct types of theories have their relative merits, it seems unlikely that the costs and the benefits resulting from risk taking are mutually exclusive. For example, if a firm decides to adjust its risk to a lower level, the rationale behind the change must be that the decrease in the associated costs as a whole is larger than the decrease in the benefits resulting from government subsidy. The logic of this line of reasoning leads us to conclude that banks and insurers choose their optimal mix of capital and risk based upon a trade-off between the associated costs and benefits.

The property-liability (P-L) insurance industry has been characterized by high business risk, intense competition, and relatively stringent regulation from governments. However, in relation to the extensive research on the risk-taking behavior of banks, there are a small number of studies investigating the risk-taking behavior of insurers. Cummins and Sommer (1996) find that insurers are likely to restrain their risk taking due to policyholders' demand for safety when the protection of guarantee funds is incomplete. However, Lee et al. (1997) find that the existence of state guarantee funds generally encourages insurers to increase risk taking. Baranoff and Sager (2002, 2003) find that capital level and asset risk of life insurers are positively correlated. They therefore conclude that life insurers overall were operated in a finite-risk paradigm to curb transaction costs. There have been, nevertheless, relatively few studies devoted to understanding the extent to which capital regulation can influence insurers' risk taking. To fill the gap, the objective of the study is to examine the association between regulatory pressure and the risk-taking behavior for P-L insurers.

Similar to what is observed in the banking industry, the risk-based capital (RBC) (6) standard adopted in 1994 forms the principal basis for the current solvency regulation in the insurance industry. RBC regulation is mainly designed to target problem (i.e., poorly capitalized) insurers and take necessary corrective actions when RBC ratios of these insurers fall below the required standard. These disciplinary interventions will affect insurers' operation and incur high regulatory costs to the insurers. (7) When these regulatory costs exceed the benefits from risk taking, the insurers would stop taking risky investments. This line of reasoning motivates us to contend that there can exist a threshold at which insurers alter their risk taking in response to the cost-benefit trade-off.

Our intention to examine whether risk taking and regulatory pressure are nonlinearly correlated in the insurance industry is also motivated by two related studies in the banking literature. Shrieves and Dahl (1992) document that the extent to which RBC regulation can influence a bank's risk taking depends on how far the bank's capital level is away from the RBC standard. Jacques and Nigro (1997) find that RBC regulation is effective in controlling banks' risk taking and particularly that weak and healthier banks react differently to RBC regulatory pressure, suggesting that the capital level and the magnitude of regulatory pressure need not correlate in a monotonic (i.e., linear) manner. Although Shrieves and Dahl (1992) and Jacques and Nigro (1997) both recognize the nonlinear influence of regulatory pressure on banks' risk taking in their studies, they deal with the issue by simply using a dummy variable to partition banks into two groups depending on whether their capital ratios fall below the explicit RBC standard. These two studies further find that the influence of RBC regulatory pressure on the risk-taking behavior of the two bank groups appears asymmetric. Extending the studies of Shrieves and Dahl (1992) and Jacques and Nigro (1997), we argue that there is an implicit threshold of regulatory pressure at which an insurer will change its risk taking and the threshold, if any exists, does not necessarily equate to the explicit RBC standard.

The study thus proposes a new research scheme to address the nonlinear relationship between regulatory pressure and risk taking for insurers. We first develop an option pricing model to illustrate how regulatory pressure is related to insurers' risk taking. To the best of our knowledge, this makes our study the first to apply the option pricing model to the insurance field in such a context. The theoretical model shows that insurers will alter their risk taking at a regulatory-pressure threshold where savings in the expected regulatory cost are equal to reductions in the put option value expropriated from guarantee funds. The existence of such an implicit threshold in fact predicts a nonlinear relationship between regulatory pressure and risk taking. At the second stage of the study, we use a simultaneous threshold-regression model (8) to examine whether insurers' risk-taking behavior is nonlinear and to evaluate the effectiveness of capital regulation in terms of whether the trigger point of the RBC standard for disciplinary action is adequate and how costly the disciplinary action is to insurers. Our study is also the first to apply simultaneous threshold regression to this area of research. We believe that the threshold-regression model is best suited for the purpose of the study. It can ascertain whether a threshold effect exists and provide an estimate of the implicit threshold when it really exists. Furthermore, it helps divide insurers into groups based upon the estimated thresholds of RBC regulatory pressure and hence allows us to examine if insurers' risk-taking behavior is distinct across these insurer groups. Overall, the framework of our threshold model allows us to assess the effectiveness of RBC regulation from three aspects: (1) the sensitivity of insurers' risk taking to RBC regulatory pressure across the insurer groups, (2) the capital buffer management strategy across the insurer groups, and (3) the speed of insurers' adjustment to capital and risk in response to impending regulatory pressure across the insurer groups.

The regression result shows that there is a threshold effect of regulatory pressure, indicating that insurers' risk-taking behavior is distinct across insurer groups characterized by different degrees of RBC regulatory pressure. The result further shows that the effectiveness of RBC regulation is at best weakly supported. This, coupled with the finding that our lower threshold of 346 percent is much larger than the RBC requirement of 200 percent, suggests that insurers seem to be aware of their proximity to the regulatory intervention but do not fully respond to the impending regulatory pressure. This implies that RBC regulatory interventions are not costly enough to exert due influence on weak insurers and/or that regulatory interventions to regulate poorly capitalized insurers are too late. The result overall hence has a profound policy implication that RBC regulation can become more effective either when the explicit RBC standard is raised to an adequate level to exert timely pressure on insurers or when the regulatory cost is high enough to insurers once receiving regulatory interventions, or both.

The remainder of the article is organized as follows. First, we develop a path-dependent contingent claim model for evaluating the expected regulatory cost, and then use this model to study the nonlinear relationship between regulatory pressure and insurers' risk taking. Second, an empirical study based upon the simultaneous threshold model is conducted to verify this nonlinear risk-taking behavior. Finally, we evaluate and discuss the adequacy of current insurance RBC regulation, and summarize our findings.

OPTION PRICING MODEL WITH REGULATORY COST

Financial option pricing models have long been employed to evaluate the equity and debt values of financial institutions (e.g., Merton, 1977; Furlong and Keeley, 1989; Doherty and Garven, 1986; Cummins, 1988a, 1988b; Cummins et al., 1999; Ibragimov et al., 2010). (9) If policyholders of an insurance company are uninformed about the firm's behavior or if they are informed but protected by nonrisk-based guarantee funds, option pricing models predict that the insurer has an incentive to raise risk to expropriate the value of the put option (Cummins, 1988a; Lee et al., 1997). This prediction is, however, contradictory to many empirical findings concerning insurers' risk taking. A possible explanation is that these option models are oversimplified since they do not consider some important internal and external costs. Cummins and Sommer (1996) use an option model to show that insurance prices are inversely related to the value of the put option, which is determined by the amount of capital and risk an insurer undertakes. The authors therefore predict that insurers will control their risk through balancing their capital ratios against asset risks if the protection from guarantee funds is incomplete. Overall, their empirical results support the finite-risk paradigm regarding insurers' risk taking.

Marcus (1984) was the first to utilize an option model to analyze the risk-taking behavior of banks when charter value is taken into account. Marcus shows that banks will stop taking risk when the expected benefit received from the put option is lower than the expected loss of charter value. If insurers place a cap on their insolvency put option to protect their franchise value, their capital ratios and portfolio/product risks will be positively correlated. (10) The option pricing framework hence suggests that the incentive for an insurer to increase its risk depends on the associated costs and benefits from its capital and asset/product risk decisions. (11)

Aside from the franchise/charter value, regulatory pressure may also substantially affect insurers' risk taking. The incentive to protect the firm's franchise value can be viewed as an internal "self-disciplining" force, while the pressure from solvency regulation can be regarded as an external force. Our option pricing model will take into account effects of the two important forces. It is noteworthy that concerns for regulatory pressure will persist as long as insurers' solvency condition falls below a given level. This suggests that the option pricing model to evaluate the expected regulatory costs should be essentially path dependent. (12)

The option pricing model developed in this study suggests that whether insurers change their risk taking depends on an implicit threshold, measured in terms of the capital ratio, at which the sum of the expected regulatory cost and the expected loss of franchise value equals the gain from the put option expropriated from guarantee funds. To the best of our knowledge, none of the existing option models (e.g., Merton, 1977, 1978; Sharpe, 1978; Cummins, 1988a, 1988b; Furlong and Keeley, 1989) are designed for this purpose.

The Model

Following the common setting of prior research (e.g., Cummins, 1988a, 1988b), we assume that market values of an insurer's aggregated assets ([A.sub.t]) and liabilities ([L.sub.t]) follow geometric Brownian motions under probability measure P:

dA = [[mu].sub.A]Adt + [[sigma].sub.A][AdW.sup.P.sub.A], (1a)

dL = [[mu].sub.L]Ldt + [[sigma].sub.L][LdW.sup.P.sub.L]. (1b)

Note that ([[mu].sub.A],[[sigma].sup.2.sub.A]) and ([[mu].sub.L], [[sigma].sup.2.sub.L]) are pairs of expected instantaneous changes and time-invariant volatilities of assets and liabilities, respectively, and the instantaneous correlation coefficient between assets and liabilities is d [W.sup.P.sub.A]d[W.sup.P.sub.L] = [[rho].sub.AL] d t, where d [W.sup.P] is the associated Wiener stochastic term under probability measure P. (13) Applying Girsanov's theorem, we transform processes (1a) and (1b) into probability measure Q representing a risk-neutral world and a market without arbitrage opportunities. According to Ito's Lemma, the asset-liability stochastic process under probability measure Q can be expressed as follows: (14)

d ln(x) = ([r.sub.h] - [[mu].sub.L] - 1/2 [[sigma].sup.2]) dt + [sigma] d [W.sup.Q.sub.x], (1c)

where x = A/L denotes the asset-liability ratio, [r.sub.h] - [[mu].sub.L] denotes the expected drift of the ratio and is also known as the numeraire of the martingale measure, [r.sub.h] is the expected drift of the hedged security (H), and [sigma] = [square root of [[sigma].sup.2.sub.A] + [[sigma].sup.2.sub.L] - 2[[rho].sub.AL] [[sigma].sub.A] [[sigma].sub.L]] denotes the volatility of the ratio.

Expected Regulatory Cost

RBC regulation in the insurance industry entails a number of potential actions that may incur regulatory cost, whose amount depends on an insurer's capital-ratio level. In particular, the regulatory pressure from RBC will persist as long as the insurer's capital ratio stays below the regulatory requirement ([phi]). (15) The total regulatory cost (RC) is fundamentally proportional to the accumulated sum of the instantaneous asset-liability ratio (x) short of the regulatory boundary ([phi]) during the time horizon. This line of reasoning prompts us to use the average asset-liability ratio minus the regulatory boundary ([phi]) and multiply it by a constant to approximate the aggregate regulatory cost, denoted as RC = -[r.sub.c]([bar.x] - [phi]) where x is the average asset-liability ratio before auditing, and [r.sub.c] can be viewed as a coefficient measuring unit regulatory cost or regulatory pressure. Because the aggregate regulatory cost (RC) is path dependent on the asset-liability ratio process, it can be derived in a way similar to how an Asian (put) option is priced.

To facilitate the derivation of an explicit solution for the expected regulatory cost, let the average asset-liability ratio be represented by its geometric mean, [[bar.x].sub.G](t) = exp(1/t [[integral].sup.t.sub.0] ln(x(u))d u). The analytical form of the expected regulatory cost is then given by (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Expected Equity Value

Many seminal papers (e.g., Merton, 1978; Cummins, 1988a) have shown that the equity value of a bank (or an insurer) can be expressed as a call option, in which the firm's assets constitute the underlying asset and the value of the firm's liabilities represents the (stochastic) exercise price of the option. The contingent-claim model anticipates that insurers can expropriate the debt value of policyholders by increasing risk taking after selling insurance policies. Marcus (1984) contends that a bank's risk taking may behave differently when the effect of franchise (charter) value is considered, because insolvency can lead to a deadweight loss of franchise value. (17) Hence, an insurer's equity value (TV), taking into account both contingent-claim value and franchise value, can be expressed as follows:

TV(A, L, [[sigma].sup.2], T) = max {[A.sub.T] + [K.sub.T] - [L.sub.T], 0},

where [K.sub.T] is the franchise value at terminal time T. We assume that the franchise value of an insurer is closely related to its market power, and thus is roughly proportional to the size of its liabilities, that is, [K.sub.T] = [kappa] [L.sub.T] where [kappa] is the parameter governing the insurer's market power. (18) However, the main objective of this study is to explore how regulatory pressure (or cost), in addition to franchise value, can affect the expected equity value and thus influence the insurer's risk-taking behavior. The payoff to shareholders at audit time T, taking into account regulatory cost (RC), can be written as

TV = max {[A.sub.T] + [K.sub.T] - [L.sub.T] - E [RC], 0}. (3)

Let the contingent equity value TV (A, L, [[sigma].sup.2], t) = LG (x, t) be expressed in terms of unit insurance liability. Under probability measure Q, and subject to the asset-liability process of Equation (1), the expected payoff to equity holders can be shown to be (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Using the equivalent martingale measure, Equation (4) then can be rewritten as (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, the fair equity value in the presence of both franchise value and regulatory cost is given by (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Capital Decisions in the Presence of Regulatory Pressure

The risk-taking behavior of insurers can be studied through the comparative static of the equity value with respect to the asset value and asset risk, respectively. Because the effect of franchise (charter) value on risk taking has been explored by Marcus (1984), this study focuses only on the effect of regulatory pressure (cost). For simplicity, we set the terminal time to be the unit (i.e., T = 1) and subtract one from the partial derivative to represent the net gain from equity when adding one unit of capital: (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

The sensitivity of shareholders' expected payoff to capital comes from changes of three components: put option value expropriated from guarantee funds, franchise value, and regulatory cost. If the effect of franchise value is ignored, then Equation (7) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Solving the first-order condition, we obtain the critical value of regulatory cost ([r.sup.*.sub.c,CAP]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

This critical value is equivalent to the (implicit) threshold at which insurers change their risk-taking behavior. It will be empirically estimated by the threshold-regression method later in this article. The comparative static in Equation (8) indicates that, ceteris paribus, shareholders are willing (reluctant) to increase their capital if unit regulatory cost is greater (smaller) than this threshold (i.e., [r.sub.c] > [<][r.sup.*.sub.c,CAP]), because the reduction of regulatory cost exceeds (falls short) of the loss from contingent claims (i.e., [partial derivative](TV)/[partial derivative]A - 1 > [<] 0). Thus, excessive risk taking can be restrained only when sufficiently great regulatory pressure (i.e., [r.sub.c] > [r.sup.sub.c,CAP]) is imposed. In addition, the marginal change in regulatory cost is a decreasing function of capital (i.e., [MC'.sub.RC] = [[partial derivative].sup.2](RC)/[partial derivative][A.sup.2] < 0), and converges to zero when the insurer's capital ratio is extremely large (i.e., [MC'.sub.RC] = [[partial derivative].sup.2](RC)/[partial derivative][A.sup.2] [right arrow] 0 as A [right arrow] [infinity]). This indicates that capital decisions become relatively inelastic with respect to regulatory pressure if insurers are well capitalized, which can be seen from Figure 1.

[FIGURE 1 OMITTED]

The critical value (i.e., the threshold) of the asset-liability ratio [(A/L).sup.*] can be derived in the same manner:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

It is easy to verify that shareholders will receive negative returns when they inject more capital (i.e., [partial derivative](TV)/[partial derivative]A - 1 < 0) if the firm's asset-liability ratio exceeds the critical value (i.e., A/L > [(A/L).sup.*]). In such a circumstance, a well-capitalized insurer will be reluctant to raise more capital even if regulatory cost is high, because the resulting expected regulatory cost is low. Figure 2 displays how the expected equity value varies according to the extent of regulatory pressure and the level of the asset-liability ratio.

[FIGURE 2 OMITTED]

Risk Decisions in the Presence of Regulatory Pressure

To examine the effect of regulatory pressure on risk decisions, we consider the comparative static of the expected payoff to shareholders with respect to firm risk:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

The fact that the first terms on the right-hand side of Equation (11) are of positive value suggests that the values received from the subsidy of guarantee funds become larger if insurers raise their risk taking. At the same time, the negative second term of Equation (11) indicates that greater risk taking also depletes franchise value. The negative third and fourth terms instead indicate that greater risk taking increases the expected regulatory cost. If the impact of franchise value is ignored, then

Equation (11) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

The critical value of regulatory cost ([r.sup.*.sub.c]) then can be found by setting Equation (12) equal to zero at T = 1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Figure 3 shows how the expected regulatory cost is influenced by the insurer's risk taking. The marginal change in regulatory cost with respect to the change in risk (i.e., [partial derivative](RC)/[partial derivative][sigma]) approaches zero when the asset-liability ratio is either extremely high or extremely low. It implies that capital regulation becomes relatively ineffective when an insurer's capitalization is either very weak or strong. However, the difference in effectiveness of capital regulation between various levels of capitalization becomes trivial when the expected regulatory cost is relatively low. In addition, Figure 3 reveals that [partial derivative](RC)/[partial derivative][sigma] becomes more sensitive to the change in the asset-liability ratio when this ratio is getting closer to the critical value (i.e., [(A/L).sup.*]).

Figure 4 illustrates how equity value varies according to the change in risk under various extents of regulatory pressure and levels of the asset-liability ratio. We find that additional risk taking reduces equity value (i.e., [partial derivative](TV)/[partial derivative][sigma] < 0) only when the regulatory cost is greater than the critical value ([r.sub.c] > [r.sup.*.sub.c,RISK]). (23)

Effects of Stochastic Volatility on Insurers' Options

Compelling empirical evidence has revealed that returns of many market assets are not normally distributed. In fact, these distributions are either characterized by fat tails or high skewness, or both. This conflicts with underlying assumptions of the standard Black-Scholes (BS) model and may lead to significant pricing biases if the basic BS formula is applied. Although nonnormal asset returns may also be caused by price jumps, one major reason for the nonnormal returns is that the volatility of the asset price process is stochastic, conflicting with the constant-volatility assumption of the standard BS model. On the other hand, empirical evidence has shown that

implied volatilities calculated from away-from-the-money European call and put options are greater than those derived from near-the-money options. This so-called "implied volatility smile" in fact casts doubt on the plausibility of the constant-volatility assumption. As shown in Heston (1993), the extent to which the BS model misprices European options when volatility is stochastic depends on a number of factors including the moneyness of asset prices (i.e., they are in the money or out of the money), the correlation between asset returns and volatility innovations, and the volatility of volatility. (24) The pricing bias due to stochastic volatility (SV) may be summarized as follows. First, when asset returns and volatility innovations are uncorrelated, the constant-volatility BS model tends to underprice away-from-the-money options, including both out-of-the-money options and in-the-money options. Second, if asset returns and volatility innovations are negatively correlated, (25) the return distribution can be left-skewed because extremely negative returns are more probable than extremely positive returns. Hence, the BS model tends to underprice in-the-money call options but overprice out-of-the-money options. This suggests that the incentive of solvent insurers (i.e., those with in-the-money options) to exploit the put option value of the guarantee fund would be actually stronger than that predicted by the standard BS model. The opposite is true when the correlation is positive.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

EMPIRICAL MODELS

The option pricing model developed in previous sections of the study predicts that insurers with various levels of capital and risk will react to regulatory pressure differently. In particular, an insurer changes risk taking only when its expected regulatory cost exceeds a critical value (i.e., the implicit threshold). If RBC regulation is effective, insurers will change their risk taking when they come under mounting regulatory pressure. The threshold at which insurers actually alter their risk taking is, however, unobservable. In particular, it is not necessarily equal to the explicit RBC standard (200 percent). This implies that using the RBC standard as the threshold to depict insurers' risk-taking behavior could be misleading. In this section, we conduct an empirical study to test whether an insurer's risk taking is nonlinearly related to regulatory pressure it faces. Because the actual regulatory cost imposed on insurers is unobservable in the real world, the threshold-regression approach enables us to find the implicit regulatory threshold. To the best of our knowledge, the present article is the first to apply such a research scheme to study the nonlinear risk-taking behavior of financial institutions.

The threshold-regression approach, originally developed by Hansen (1999), divides samples optimally into a number of regimes according to the estimated threshold values. Given that most previous studies (e.g., Shrieves and Dahl, 1992; Cummins and Sommer, 1996; Baranoff and Sager, 2002, 2003; Baranoff et. al., 2007) treat capital and risk decisions of financial institutions as simultaneous, we extend our threshold regression to be a simultaneous model, which was recently developed by Caner and Hansen (2004). (26) Using threshold regression not only allows us to depict nonlinear risk-taking behavior but, more importantly, enables us to evaluate when and to what extent RBC regulation can influence insurers' risk taking. For example, consider a case where the implicit threshold estimated by the regression model is higher than the explicit regulatory threshold (i.e., the 200 percent RBC standard). Under such a circumstance, if insurers with RBC ratios lower than the implicit threshold are found to reduce risk taking, it can be argued that regulatory intervention is relatively costly to the insurers. In contrast, if the insurers increase risk taking, then this seems to indicate that regulatory intervention appears to be nonbinding and/or late.

Just as Shrieves and Dahl (1992), we test our hypotheses by investigating the relationship between changes in risk and changes in capital rather than the relationship between levels of capital and of risk. Following Baranoff and Sager (2002, 2003), we consider both asset risk and product risk born by a P-L insurer. Consequently, there are three alternative measures of insurers' risk change (i.e., [DELTA][CAPR.sub.it], [DELTA][ARISK.sub.it], and [DELTA][[PRISK.sub.it]) and they are assumed to be endogenously determined. Because our primary goal is to examine the impact of capital regulation on insurers' risk taking, the RBC ratio is thus used as not only the proxy of regulatory pressure but also the threshold variable (i.e., [q.sub.it]) in the model. Our empirical simultaneous threshold models (27) are given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

where [X.sub.it], [Y.sub.it], and [Z.sub.it] denote vectors of instrumental variables that can affect insurers' risk taking and have been commonly considered in previous studies, (28) [.sub.[*]] is an indicator function, and y is the threshold value to be estimated.

Hypotheses

As previously noted, the objectives of this study are twofold: (1) to test whether insurers' risk-taking behavior is nonlinear, which is implied by our option pricing model, and (2) to evaluate the effectiveness of capital regulation in terms of whether the trigger point of the RBC standard for disciplinary action is adequate and how costly the disciplinary action is to insurers. Our first goal can be reached through testing whether RBC regulatory pressure has a threshold effect on insurers' risk taking. The existence of a threshold effect indicates that insurers' risk-taking behavior is distinct across insurer regimes characterized by different degrees of RBC regulatory pressure. On the other hand, the effectiveness of current capital regulation can be evaluated by comparing the estimate of the implicit threshold with the explicit RBC standard and examining the risk-taking behavior of insurers particularly in the high-regulatory-pressure regime.

Our theoretical option model predicts that insurers will reduce risk taking only if their RBC ratios fall below the implicit threshold and the expected regulatory cost is sufficiently large. Fortunately, the simultaneous threshold model employed in the study can help estimate the implicit threshold and discern the risk-taking behavior of an insurer. The following discussions illustrate the complementary roles of our theoretical option pricing and empirical threshold regression models in providing new insights into the literature.

Prior insurance studies often implicitly assume that their findings of insurers' risk-taking behavior can be a common rule to all insurers. For example, if RBC regulation were effective, then RBC-ratio changes would be negatively correlated with capital-ratio changes but positively correlated with asset/product-risk changes. Unlike previous studies, the theoretical model proposed in this study predicts that the capital-risk relationship should vary across regulatory-pressure regimes. If RBC regulation is effective, only poorly capitalized insurers (i.e., insurers with relatively low RBC ratios) will reduce their risk taking. On the other hand, our theoretical model also predicts that the risk-taking behavior of well-capitalized insurers (i.e., those with relatively high RBC ratios) will be relatively insensitive to regulatory pressure. This is because the probability of those insurers to provoke disciplinary action from regulators is quite low. Therefore, if empirical results show that the implicit threshold is higher than the explicit RBC threshold and that insurers in the high-regulatory-pressure (i.e., low-RBC) regime reduce their risk taking, our theoretical option pricing model can help conclude that RBC regulation is considered to be highly costly to insurers so that they take early action to control their risk even when their RBC ratios are still above the explicit regulatory threshold. In contrast, if insurers in the high-regulatory-pressure regime are found to increase their risk taking, our theoretical option pricing model suggests that either the associated regulatory cost is not large enough to prevent insurers facing impending regulatory interventions from taking more risk or regulatory interventions come too late to constrain these insurers from taking more risk, or both. Under such a scenario, insurers may reduce their capital levels and/or increase their asset/product risks to expropriate the put option value from guarantee funds. Finally, suppose that the estimated RBC threshold is lower than the explicit RBC threshold. The result that insurers in the high-regulatory-pressure regime are found to reduce risk taking may indicate effective but too early RBC interventions. The result that insurers in the high-regulatory-pressure regime increase risk taking, in contrast, may indicate early but ineffective RBC regulation.

Recall that insurers can change their capital buffers (29) (i.e., safety levels) by adjusting either capital ratios or asset/product risks, or both. If an insurer chooses to maintain its capital buffer (i.e., to cap its insolvency put option), as predicted in Cummins and Sommers (1996), it can be achieved by increasing (or decrease) capital ratios and asset/product risks at the same time. Under such a circumstance, capital-ratio changes and asset/product-risk changes are likely to be positively correlated. A negative relationship between capital-ratio changes and asset/product-risk changes, however, provides no clue about the dynamics of an insurer's capital buffer. It may indicate that an insurer chooses to rebuild its capital buffer (i.e., to decrease the value of the insolvency put) by raising capital ratios and/or reducing asset/product risks, which would be most likely when the insurer senses impending regulatory pressure and high insolvency cost. It, on the other hand, may suggest that an insurer downsizes its capital buffer (i.e., increases the value of the insolvency put) through lowering its capital ratios and/or taking more asset/product risks especially when the insurer considers the benefit of expropriating the option value from guarantee funds is greater than the sum

of the regulatory cost and the loss of the franchise value. This line of reasoning makes us anticipate that under an effective RBC regulatory system, poorly capitalized insurers are prone to adopt a capital-buffer rebuilding strategy. Well-capitalized insurers, in contrast, have more leeway in developing their capital-buffer strategy regardless of the effectiveness of RBC regulation, since their RBC ratios are well above the required standard.

Last but not least, because our empirical model is carried out under a partial-adjustment framework, the rate of risk adjustment thus can be measured by the lagged term of capital and risk variables. If RBC regulation is effective, we anticipate that weak insurers should adjust their capital ratios and risks faster than healthy insurers.

EMPIRICAL RESULTS

DATA

Data sources of the study include the NAIC P-L insurance annual report, the NAIC newsletter, and the A.M. Best Key Rating Guide. The sample period spans 2000 through 2009. We exclude observations with incorrect data values. (30) After excluding these observations, the final sample size totals 15,470 firm-year observations.

Table 1 reports descriptive statistics for important characteristics of our sample insurers. Panel A shows that: (1) the principal risk faced by P-L insurers came from insurance policies sold (i.e., product risks); (2) on average, capital ratios and product risks experienced a decreasing trend whereas asset risks showed an increasing pattern over the sample period; and (3) most P-L insurers were well capitalized (31) but their RBC ratios exhibit considerable variation.

Panel A of Table 2 provides summary statistics for firm characteristics of insurers. Of the sample insurers, 71 percent are stock firms and 67 percent are affiliated organizations. Nearly 73 percent of insurers use the independent agency as their major distribution channel and 35 percent of insurers operate businesses nationwide. Finally, Panel B of Table 2 shows the frequency distribution of RBC ratios. Overall, the distribution of the RBC ratio is found to be highly skewed to the right, with most insurers (i.e., about 76 percent) having RBC ratios within the range from 200 percent to 1,500 percent.

Simultaneous Threshold Regression

To implement the threshold regression with instrumental variables, it is necessary first to identify the thresholds of the reduced-form equations. (32) Panel A of Table 3 shows that two thresholds might exist. To test the significance of the threshold effect, Hansen (1999) proposes a powerful F-statistic, which can be calculated using a bootstrapping procedure. (33) The test results indicate that threshold effects of one threshold and two thresholds are both significant at the 1 percent level, and the best goodness of fit occurs when two thresholds are considered. The two threshold estimates are used to obtain the expected values of the endogenous variables (through instrumental-variable estimation), which are then plugged into structural equations to derive the threshold values in the second stage. Panel B of Table 3 reports the estimated thresholds in the second stage. A bootstrapped F-test is further used to examine whether the threshold effect exists in the structural equation model, showing that the two structural threshold estimates are statistically significant at the 1 percent level. The result confirms what is predicted by our theoretical model: that the risk-taking behavior of P-L insurers is nonlinearly related to RBC regulatory pressure.

To verify the effectiveness of the threshold regression, we would like to know if our threshold model outperforms other models such as those splitting the sample based upon some salient cutoff points including the median and mean. We use the error sum of squares to measure the performance of a model. Table 4 reports the result. As expected, the model with no threshold has the largest error sum of squares. The two threshold regressions outperform the two OLS regressions with mean/median as the threshold. Finally, we find that the error sum of squares declines substantially when the second threshold is added into the threshold regression. Drawing on the finding that it is optimal to segregate insurers into three regimes, we refer to those insurers with RBC ratios below the lower threshold (i.e., 3.46) as the regime with "high regulatory pressure" and those with RBC ratios exceeding the upper threshold (i.e., 15.09) as the regime with "low regulatory pressure." (34)

Table 5 presents descriptive statistics for insurers of various groups: the whole sample and the three regime samples stratified by the two RBC thresholds. For the group of all insurers, capital ratios and product risks on average decline during the sample period, whereas asset risks increase merely by a relatively small amount. These observations suggest that insurers as a whole did not inflate their risks during our sample period.

Like what is observed in the whole sample case, the by-regime statistics indicate that capital ratios and product risks exhibit a relatively clear upward/downward trend while asset risks show no obvious trend during the sample period. A comparison of changes in capital ratios, changes in product risks, and changes in asset risks across the three regime groups, nevertheless, reveals insightful results. Although changes in asset risks are uniformly trivial across the three insurer groups, changes in capital ratios and product risks show stimulating patterns. Insurers in the low-RBC regime (i.e., those facing high regulatory pressure) experience the largest decline in capital ratios and the largest increase in product risks. In contrast, insurers in the high-RBC regime (i.e., those facing low regulatory pressure) experience the largest increase in capital ratios and the largest decline in product risks. This suggests that insurers facing high regulatory pressure are more likely to increase risk taking in comparison to other insurers. The Scheffe test further confirms that with the exception of the change in asset risks, all the variables reported in Table 5 have unequal means across the three regime groups. This implies that the threshold regression has successfully partitioned our sample insurers into three regimes characterized by distinct risk-taking behavior. We find that the average RBC ratio for insurers of low RBC ratios is about 259 percent, slightly above the RBC regulatory standard. Besides, insurers in the low-RBC regime are found to have the largest loss ratio, the largest leverage ratio, and the lowest ROA, whereas insurers in the high-RBC regime are characterized by the lowest loss ratio, the largest reinsurance ratio, the lowest leverage, and the largest ROA.

Joint Tests of Coefficient Equality Across Regimes

To verify the nonlinearity of insurers' risk-taking behavior, we first carry out coefficient-equality tests for each endogenous/exogenous variable across the three RBC regimes. This test is performed based upon Equations (14)-(16) one at a time and three at a time, respectively. Table 6 shows that the hypothesis of equal coefficients across the three regimes cannot be rejected for all variables in the asset-risk regression model. However, the number of variables with unequal coefficients across the three regimes is 3 when the capital regression is considered alone, 9 when the product-risk regression is considered alone, and 14 when the three regressions are considered jointly. The result that the joint test based upon the three equations is highly significant suggests that the influence of RBC regulation on insurers' risk taking is significantly different across the three insurer groups. With respect to the coefficient equality test of the major variables in the study, a similar pattern prevails. Regarding results based upon an individual equation, the coefficient of the capital-ratio change is significantly different across regimes only in the product-risk equation, the coefficient of the product-risk change is significantly different only in the capital equation, but the coefficient of the asset-risk change is not significantly different. The coefficient equality hypothesis is, however, rejected for each of the three variables when the three regression models are considered jointly. Overall, the result reported in Table 6 supports the existence of a threshold effect of regulatory pressure on insurers' risk taking, as predicted by our theoretical option pricing model.

Sensitivity of Risk Taking to RBC Regulatory Pressure

We now consider the relationship between insurers' risk taking and their pressure from RBC regulation. If RBC regulation is effective, then the coefficient of the RBC ratio in the capital equation will be significantly negative and will be significantly positive in the asset- and product-risk equations. To compare with previous studies, we first run the three regressions simultaneously with no thresholds. The result reported in the "whole sample" column across the three panels of Table 7 shows that insurers to a lesser extent control their product risks but concurrently lower capital ratios and raise asset risks to a greater extent in response to the impending regulatory pressure. This suggests that RBC regulation is, on average, weakly effective in controlling insurers' risk taking.

The aforementioned prediction about the sign of the RBC ratio in the three equations is most likely to be confirmed for poorly capitalized (i.e., the low-RBC regime) insurers. We therefore estimate the three equations simultaneously based upon the threshold-regression approach. The results are also shown in the regime columns of Table 7. We find that, in general, insurers' risk taking is not quite sensitive to RBC regulatory pressure across the three insurer groups. The RBC regulatory pressure is evidently nonbinding with respect to risk taking for the middle-RBC and the highRBC regime insurers. However, the relationship between risk taking and regulatory pressure becomes relatively complicated for the low-RBC regime insurers. In particular, the coefficient of the RBC ratio is insignificant in the capital equation, but it turns significantly negative in the asset-risk equation and significantly positive in the product-risk equation. This suggests that RBC regulation seems to influence weak insurers to raise asset risks and to lower their product risks while it plays no role in their capital decision. Therefore, there is at best weak, if any exists, evidence that RBC regulation can effectively curb insurers' risk taking. Overall, our findings support the prediction of our theoretical model that insurer's risk reduction is not quite sensitive to changes of regulatory pressure if regulatory pressure (cost) is relatively low (see, e.g., Figure 3). Furthermore, the result, coupled with the finding that our lower threshold of 346 percent, is much larger than the RBC requirement of 200 percent, (35) suggests that insurers appear to be aware of their proximity to the regulatory threshold but do not fully respond to the impending regulatory pressure. The result also implies that RBC regulation will become more effective either when the explicit RBC standard can be raised to an adequate level to exert timely pressure on insurers or when the regulatory cost can be high enough to insurers once receiving regulatory interventions, or both.

Capital Buffer Strategy

With regard to insurers' capital buffer strategy, the result of Table 7 reveals that capital changes and asset-risk changes are positively correlated whereas capital changes and product-risk changes are negatively correlated for the whole sample. This suggests that insurers, on average, adopt the capital-buffer-maintaining strategy to manage their capital and asset risk. The negative relationship between capital changes and product-risk changes, on the other hand, can have two entirely distinct implications, as argued in previous sections of the study. We are, however, inclined to interpret it as evidence supporting that insurers tend to adopt the capital-buffer-rebuilding strategy rather than the capital-buffer-downsizing strategy to deal with their capital and product risk. (36) These results to some extent support the finding of Cummins and Sommer (1996) and Baranoff and Sager (2002,2003) that insurers choose to operate in a finite-risk framework. Based upon a sample of life insurers, Baranoff and Sager (2002) contended that the negative relationship between capital and product risk might be associated with the effect of guarantee fund protection.

As argued in previous sections, regulatory pressure can have an impact on insurers' capital buffer strategy. In particular, if RBC regulation is effective, insurers subject to the greatest regulatory pressure (i.e., insurers in the low-RBC regime) should adopt the capital-buffer-rebuilding strategy to reduce their insolvency put option, thus yielding a negative relationship between capital changes and risk changes. Comparing capital buffer strategy of insurers under various degrees of regulatory pressure would, therefore, provide meaningful insights in that regard. Our simultaneous threshold-regression model can do exactly that. As shown in Table 7, within the insurer group of the low-RBC regime, the relationship between capital changes and asset-risk changes is significantly positive while the relationship between capital changes and product-risk changes is significantly negative. This suggests that poorly capitalized insurers tend to maintain their capital buffers when managing their capital and assets. In contrast, they adopt the capital-buffer-downsizing strategy to manage their capital and products. The reason behind this argument is that insurers in the low-RBC regime are found to decrease their capital but increase their product risks over the sample period, as reported in Table 5. The result in general casts doubt on the effectiveness of RBC regulation.

With regard to the result for well-capitalized insurers, Table 7 shows a positive association between capital changes and asset-risk changes but a negative association between capital changes and product-risk changes. (37) In relation to the result for insurers in the low-RBC regime, relationships between capital changes and asset-/ product-risk changes for well-capitalized insurers are, however, weak. For insurers in the low-RBC regime, the positive relationship between capital changes and asset-risk changes prevails in both the capital and the asset-risk equations, and the negative relationship between capital changes and product-risk changes holds in both the capital and the product-risk equations. However, the positive relationship exists only in the asset-risk equation and the negative relationship holds only in the capital equation for insurers in the high-RBC regime. The relatively weak result for well-capitalized insurers is consistent with our contention that these insurers have more discretion as to their capital buffer strategy, further confirmed by the observation that the result of the middle-RBC insurer group bears a closer resemblance to that of the low-RBC insurer group than that of the high-RBC insurer group.

In sum, the negative relationship between capital changes and product-risk changes along with the positive relationship between capital changes and asset-risk changes suggests that insurers, on average, manage to increase product risks rather than asset risks when they seek to assume more risk. This also suggests that insurers tend to develop a capital-buffer-maintaining strategy to manage their capital and asset-risk, and a capital-buffer-rebuilding strategy rather than a capital-buffer-downsizing strategy to deal with their capital and product-risk. This finding is consistent with those of Cummins and Sommer (1996) and Baranoff and Sager (2002, 2003). The result based upon the threshold-regression model, in contrast, reveals that

poorly capitalized insurers adopt a capital-buffer-maintaining strategy to manage capital and assets but a capital-buffer-downsizing strategy to manage capital and product-risk, indicating that RBC regulation cannot effectively prevent weak insurers from taking excessive risk.

Speed of Insurers' Adjustment to Risk Taking

The third perspective from which we assess the effectiveness of RBC regulation is the speed of insurers' adjustment to their capital and risk in response to impending regulatory pressure, made possible through our empirical design. Since insurers' risk-taking behavior is modeled in a partial adjustment framework, the lagged value of capital-ratio, asset-risk, and product-risk variables can be used to reflect the speed of firms' adjustment to capital, asset-risk, and product-risk, respectively, similar to what was done in Shrieves and Dahl (1992). If RBC regulation is effective, the speed of adjustment to capital and risk for insurers in the low-RBC regime is expected to be much faster than that for insurers in the high-RBC regime. As shown in Table 7, we find that the adjustment rate of capital level for poorly capitalized insurers is only slightly faster than that for well-capitalized insurers (i.e., 8.83 percent vs. 6.56 percent). Similarly, we find that insurers in the low-RBC regime also adjust their asset risks and product risks at a speed not much faster than insurers in the high-RBC regimes (i.e., 23.36 percent vs. 15.22 percent for asset risks and 23.87 percent vs. 18.59 percent for product risks). In sum, the result at best lends only weak support to the effectiveness of RBC regulation.

CONCLUSIONS

The study examines effects of RBC regulatory pressure on the risk-taking behavior of P-L insurers. It is the first study that develops a path-dependent option pricing model to evaluate the expected regulatory cost, which provides a theoretical foundation for insurers' nonlinear risk-taking behavior with respect to regulatory pressure. The model suggests that insurers will change their risk taking at an implicit RBC threshold where the expected regulatory cost equals the put option value expropriated from guarantee funds. To verify the prediction of our theoretical model, an empirical study is carried out to assess the nonlinear risk-taking behavior and test the effectiveness of RBC regulation. We find that the simultaneous threshold-regression model is best suited for our purposes. It is therefore used to demonstrate a threshold effect and obtain two RBC-ratio thresholds, consistent with the prediction of our theoretical model that insurers' risk-taking behavior is nonlinearly related to RBC regulatory pressure.

The threshold model then divides insurers into three regimes based upon the two estimated thresholds and examines insurers' risk-taking behavior for each insurer group. Through investigating the risk-taking behavior of insurers in the three regimes, particularly the low-RBC regime, the framework of our threshold model allows to assess the effectiveness of RBC regulation from three perspectives. The result based upon the sensitivity of insurers' risk taking to RBC regulatory pressure shows that RBC regulation to some extent influences poorly capitalized insurers to raise asset risks and to lower product risks while it plays no role in the capital decision. There is hence at best weak evidence that RBC regulation can effectively curb insurers' risk taking. With regard to the capital buffer management strategy, the threshold-regression result shows that poorly capitalized insurers tend to adopt a capital-buffer-maintaining strategy to manage capital and assets but a capital-buffer-downsizing strategy to manage capital and products, indicating that RBC regulation cannot prevent weak insurers from taking excessive risk. The third perspective, provided by our threshold model, from which we can assess the effectiveness of RBC regulation is the speed of insurers' adjustment to capital and risk in response to impending regulatory pressure. The result indicates that the speeds at which insurers facing high regulatory pressure adjust their capital, asset risks, and product risks is only slightly higher than those for well-capitalized insurers, providing only weak evidence for the effectiveness of RBC regulation.

In sum, the empirical evidence shows that the effectiveness of RBC regulation is at best weakly supported. This result, coupled with the finding that our lower threshold of 346 percent is much larger than the RBC requirement of 200 percent, suggests that insurers seem to be aware of their proximity to the regulatory threshold but do not fully respond to the impending regulatory pressure. We are therefore inclined to conclude that current RBC regulatory cost is not high enough to exert due influence on weak insurers and/or that regulatory interventions to regulate poorly capitalized insurers appear to be late. This result hence has a profound policy implication that RBC regulation will become more effective either when the explicit RBC standard can be raised to an adequate level to exert timely pressure on insurers or when the regulatory cost can be high enough to insurers once receiving regulatory interventions, or both.

Overall, the study offers an innovative approach to investigating insurers' risk-taking behavior and testing the adequacy of capital regulation. We believe that the methodologies developed in this article may also be used to investigate the risk-taking behavior of other financial institutions such as banks and life insurance companies with only minor modifications.

DOI: 10.1111/j.1539-6975.2012.01505.x

REFERENCES

Aggarwal, R., and K. T. Jacques, 2001, The Impact of FDICIA and Prompt Corrective Action on Bank Capital and Risk: Estimates Using a Simultaneous Equations Model, Journal of Banking and Finance, 25(6): 1139-1160.

Bai, 1997, Estimating Multiple Breaks One at a Time, Econometric Theory, 13:315-352.

Baranoff, E. G., S. Papadopoulos, and T. W. Sager, 2007, Capital and Risk Revisited: A Structure Equation Model Approach for Life Insurers, Journal of Risk and Insurance, 74(3): 653-681.

Baranoff, E. G., and T. W. Sager, 2002, The Relations Among Asset Risk, Product Risk, and Capital in the Life Insurance Industry, Journal of Banking and Finance, 26(6): 1181-1197.

Baranoff, E. G., and T. W. Sager, 2003, The Relations Among Organizational and Distribution Forms and Capital and Asset Risk Structures in the Life Insurance Industry, Journal of Risk and Insurance, 70(3): 375-400.

Caner, M., and B. E. Flansen, 2004, Instrumental Variable Estimation of a Threshold Model, Econometric Theory, 20: 813-843.

Chen, S. K., B. Lin, H. R. Oppenheimer, and T. Yu, 2007, The Impact of Firm Franchise Value on Asset Allocation: An Analysis of Property-Liability Insurance Firms, Risk Management and Insurance Review, 11(1): 159-180.

Cheng, J., and M. A. Weiss, 2012, The Role of RBC, Hurricane Exposure, Bond Portfolio Duration and Macroeconomic and Industry Wide Factors in Property-Liability Insolvency Prediction, Journal of Risk and Insurance, 79(3): 723-750.

Cummins, J. D., 1988a, Risk-Based Premium for Insurance Guaranty Fund, Journal of Finance, 43(4): 823-839.

Cummins, J. D., 1988b, Capital Structure and Fair Profits in Property-Liability Insurance, in: J. D. Cummins and R. A. Derrig, eds., Financial Insurance Solvency Theory (Boston: Kluwer Academic Publishers), pp. 295-307.

Cummins, J. D., M. F. Grace, and R. D. Phillips, 1999, Regulatory Solvency Prediction in Property-Liability Insurance: Risk-Based Capital, Audit Ratios, and Cash Flow Simulation, Journal of Risk and Insurance, 66(3): 417-458.

Cummins, J. D., and D. W. Sommer, 1996, Capital and Risk in Property-Liability Insurance Markets, Journal of Banking and Finance, 20(6): 1069-1092.

Doherty, N. A., and J. R. Garven, 1986, Price Regulation in Property-Liability Insurance: A Contingent-Claims Approach, Journal of Finance, 41,1031-1050.

Ediz, T., I. Michael, and W. Perraudin, 1998, The Impact of Capital Requirements on U.K. Bank Behaviour, Federal Reserve Bank of New York Economic Policy Review, 4(3): 15-22.

Fischer, S., 1978, Call Option Pricing When the Exercise Price Is Uncertain and the Valuation of Index Bonds, Journal of Finance, 33(1): 169-176.

Furlong, F. T., and M. C. Keeley, 1989, Capital Regulation and Bank Risk Taking: A Note, Journal of Banking and Finance, 13(4): 883-891.

Hansen, B. E., 1999, Threshold Effects in Non-Dynamic Panels: Estimation, Testing and Inference, Journal of Econometrics, 93(2): 345-368.

Heston, S., 1993, A Closed-form Solution for Options with Stochastic Volatility, With Applications to Bond and Currency Options, Review of Financial Studies, 6: 327-343.

Ibragimov, R., D. Jaffee, and J. Walden, 2010, Pricing and Capital Allocation for Multiline Insurance firms, Journal of Risk and Insurance, 77(3): 551-578.

Jacques, K. and P. Nigro, 1997, Risk-Based Capital, Portfolio Risk, and Bank Capital: A Simultaneous Equation Approach, Journal of Economics and Business, 49(6): 533-547.

Lai, Y.-H., 2005, An Integrated Analysis of Regulatory Pressure and Franchise Value on the Risk-Taking Behavior of a P/L Insurance Firm, Unpublished Dissertation, National Chung Cheng University, Taiwan.

Lee, S.-J., D. Mayers, and C. Smith, Jr., 1997, Guarantee Funds and Risk-taking: Evidence from the Insurance Industry, Journal of Financial Economics, 44: 3-24.

Leverty, J. T., and M. F. Grace, 2012, Dupes or Incompetents? An Examination of Management's Impact on Firm Distress, Journal of Risk and Insurance, 79(3): 751-783.

Marcus, A. J., 1984, Deregulation and Bank Financial Policy, Journal of Banking and Finance, 8(4): 557-565.

Merton, R. C., 1977, An Analytic Derivation of the Cost of Deposit Insurance and Loan Guarantees, Journal of Banking and Finance, 1(1): 3-11.

Merton, R. C., 1978, On the Cost of Deposit Insurance When There Are Surveillance Costs, Journal of Business, 51(3): 439-453.

Park, S., 1997, Risk-Taking Behavior of Banks Under Regulation, Journal of Banking and Finance, 21: 491-507.

Repullo, R., 2004, Capital Requirements, Market Power, and Risk Taking in Banking, Journal of Financial Intermediation, 13(2): 156-182.

Rime, B., 2001, Capital Requirements and Bank Behaviour: Empirical Evidence for Switzerland, Journal of Banking and Finance, 25(4): 798-805.

Sharpe, W. F., 1978, Bank Capital Adequacy, Deposit Insurance and Security Values, Journal of Financial and Quantitative Analysis, 13(4): 701-718.

Shrieves, R. E., and D. Dahl, 1992, The Relationship Between Risk and Capital in Commercial Banks, Journal of Banking and Finance, 16(4): 439-457.

So, J., and J. Z. Wei, 2004, Deposit Insurance and Forbearance Under Moral Hazard, Journal of Risk and Insurance, 71(4): 707-735.

Staking, K. B., and D. F. Babbel, 1995, The Relation Between Capital Structure, Interest Rate Sensitivity and Market Value in the Property-Liability Insurance Industry, Journal of Risk and Insurance, 62(4): 690-718.

Williamson, O. E., 1988, Corporate Finance and Corporate Governance, Journal of Finance, 18(3): 567-591.

Wen-Chang Lin is an associate professor in the Department of Finance, National Chung Cheng University, Taiwan. Yi-Hsun Lai is an assistant professor in the Department of Finance, National Yunlin University of Science and Technology, Taiwan. Michael R. Powers is a professor in the Department of Finance, Tsinghua University, and on leave from the Department of Risk Management and Insurance, Temple University. Wen-Chang Lin can be contacted via e-mail: finwcl@ccu.edu.tw.

(1) Transaction-cost economics (see, e.g., Williamson, 1988) posits that financial institutions will limit their product risks to forestall paying high financing costs, as the cost of debt financing will increase if they sell risky products.

(2) The agency theory contends that the separation of management and ownership can lead to lower risk taking by managers as they do not share residual profits with owners.

(3) Bankruptcy costs are those such as the cost of liquidating the firm and the loss of franchise value.

(4) Regulatory costs have received relatively considerable research attention in the banking literature (e.g., Shrieves and Dahl, 1992; Jacques and Nigro, 1997; Park, 1997; Ediz et al., 1998; Aggarwal and Jacques, 2001; Rime, 2001).

(5) For example, Repullo (2004) contends that risk taking of banks may be associated with the degree of market competition. When markets are overcompetitive or appear to be monopolistic, firms may increase their risk taking. In contrast, under a moderate competitive market, firms may lower their risk taking to protect their franchise value.

(6) RBC regulation aims to prevent costly insolvencies by mandating insurers to hold a satisfactory level of capital. The required capital is determined by a formula-based risk measure encompassing asset risks, underwriting risks, credit risks, and miscellaneous risks.

(7) Recently, Cheng and Weiss (2012) find that the ability of the RBC ratio in predicting insurer insolvencies is not consistent over time.

(8) Threshold regression was originally developed by Hansen (1999) and simultaneous threshold regression was recently developed by Caner and Hansen (2004).

(9) Merton (1977) shows that banks with limited liability can inflate shareholder value by reducing capital and/or raising risk in the contingent claim framework. Furlong and Keely (1989) suggest that flat capital requirements can reduce, but not eliminate, moral-hazard incentives. This is because a bank can set its amount of capital against credit risk, which does not depend on the bank's asset quality.

(10) Staking and Babbel (1995) and Chen et al. (2007) conduct empirical studies to test whether the "self-discipline effect" dominates the "moral-hazard effect" for insurers with considerable franchise values.

(11) Recently, Leverty and Grace (2012) find that managerial ability can also affect the cost of insurer failure.

(12) The path-dependent option is an option whose payoff depends on the whole path of the underlying asset. If the exercise price of a path dependent option is set to be the sum or the mean of past prices of the underlying asset, it is known as an Asian option. The option models developed in this article are modified from Lai (2005).

(13) For detailed assumptions underlying the asset and liability processes, see Cummins (1988a).

(14) Because insurance liabilities are typically considered to be nontradable in the market, it is necessary to construct a tradable hedging security (H) that is highly correlated with these liabilities. See Fisher (1978) and Cummins (1988a, 1988b) for more details. The proof is similar to that of Cummins (1988a, 1988b), and is available from the authors.

(15) Current RBC regulation requires that insurers' capital ratios should exceed the RBC ratio of 200 percent. Note that the boundary ([phi]) can be determined based upon the RBC formula.

(16) The closed-form solution exists only when [[bar.x].sub.G] is a geometric mean; otherwise, numerical methods are needed. The proof is similar to the derivation of the price of an Asian option and is available from the authors.

(17) The franchise value of an insurer is closely related to its market power and/or economic scale. The value may also be generated from policy renewals, underwriting experience, etc.

(18) Many previous studies posit that franchise (charter) values of insurers (banks) are closely related to their economic scales. For example, So and Wei (2004) derive the charter values of banks in terms of savings and monopoly rents.

(19) The proof is available from the authors.

(20) The proof is available from the authors.

(21) The last two terms of Equation (6) are associated with the expected discounted regulatory cost.

(22) This suggests that if the sign is positive, then, ceteris paribus, the equity value increases by more than one dollar when shareholders add one dollar to the firm's assets.

(23) In the numerical example (Figure 4), a value of [r.sub.c] greater than 0.25 may result in a decrease in equity value when the insurer increases risk taking and its asset-liability ratio is around 2.

(24) Although the option models (with stochastic exercise prices) in our study are to some extent different from the standard BS model, the influence of SV on these models is similar to that discussed in Heston (1993) because they are essentially European-type options.

(25) Return distributions of many market indexes, such as S&P 500 and FTSE 100, are found to be negatively skewed.

(26) This may involve several instrumental (exogenous) variables and lead to more complicated computations than the simple threshold model. For more details about threshold models with endogenous variables, see Caner and Hansen (2004).

(27) As pointed out by Hansen (1999), multiple thresholds can exist in a model.

(28) For detailed descriptions of exogenous variables, see the notes of Table 1.

(29) The capital buffer is conceptually analogous to the insolvency put option discussed in Merton (1977) and Cummins (1988a, 1988b).

(30) We exclude observations whose capital ratios or total admitted assets are less than or equal to zero. In addition, we delete observations with unreasonable values. For example, those with capital ratios greater than one, or with product risks or asset risks equal to zero are excluded.

(31) The average RBC ratio for all insurers is 1,075 percent.

(32) We calculate the likelihood ratio to test the existence of multiple thresholds (see Caner and Hansen, 2004).

(33) For the details of how the F-statistic is derived, see Hansen (1999) and Bai (1997).

(34) The low regime includes those insurers with RBC ratios smaller than 3.46, insurers belonging to the middle regime with RBC ratios ranging from 3.46 to 15.09, and those in the high regime with RBC ratios larger than 15.09.

(35) The associated test statistic for [H.sub.0]: [gamma] = [[gamma].sub.0] is L[R.sub.1]([gamma]) = [S([gamma]) - S ([[gamma].sub.0])]/[[delta].sup.2], where S([gamma]) is the error sum of squares under threshold [gamma], and [[delta].sup.2] is the residual variance.

(36) This is because product risk is on average decreasing for all insurers during the sample period.

(37) In contrast to the negative relationship between capital changes and product-risk changes for poorly capitalized insurers, the negative relationship here implies that well-capitalized insurers tend to adopt a capital-buffer-rebuilding strategy to manage capital and products, which is supported by the finding of Table 5 that insurers in the high-RBC regime experience an increase in capital ratios and a decline in product risks.

The article examines the risk-taking behavior of property-liability insurers in the presence of risk-based capital regulation. An option pricing model is developed to evaluate the expected regulatory cost and predict a nonlinear relationship between regulatory pressure and insurers' risk taking. We then conduct an empirical test using the simultaneous threshold regression. The result shows that there is a threshold effect of regulatory pressure on insurer risk taking. Poorly capitalized insurers seem to be aware of their proximity to regulatory interventions but do not fully respond to the impending regulatory pressure. This implies either regulatory interventions are not costly enough or they are too late, or both.

INTRODUCTION

Understanding the risk-taking behavior of financial institutions, including banks and insurance firms, is important to not only regulators but also claimholders of these firms because the behavior can substantially affect values of the firms and thus how the values are distributed between claimholders. A number of theories have been proposed to explain the behavior in the banking and insurance literature. However, they often make contradictory predictions regarding the risk-taking behavior of financial institutions. For example, the risk-subsidy hypothesis predicts that banks and insurers will engage in excessive risk taking when deposit insurance for banks or guarantee funds for insurers are nonrisk based. The reason is that the deficit of deposits and insurance policies will be absorbed by government funds when these firms become insolvent, and thus the firms pay little cost for their high risk taking. When banks/insurers attempt to increase their risk taking, they can either increase portfolio risk or lower capital level, or both. Hence, the risk-subsidy hypothesis predicts a negative relationship between capital and risk. Another stem of theories regarding the risk-taking behavior of financial institutions, in contrast, advocates that these firms will operate in a finite-risk paradigm if relatively large costs can result from high risk taking. These costs include transaction costs, (1) agency costs, (2) bankruptcy costs, (3) regulatory costs, (4) and others. (5) This class of theories anticipates that financial institutions will sustain their financial strength by either raising capital when their asset risk is increasing or lowering asset risk when their capitalization is declining, and thus predicts that capital and risk levels will be positively correlated. Despite the fact that these two distinct types of theories have their relative merits, it seems unlikely that the costs and the benefits resulting from risk taking are mutually exclusive. For example, if a firm decides to adjust its risk to a lower level, the rationale behind the change must be that the decrease in the associated costs as a whole is larger than the decrease in the benefits resulting from government subsidy. The logic of this line of reasoning leads us to conclude that banks and insurers choose their optimal mix of capital and risk based upon a trade-off between the associated costs and benefits.

The property-liability (P-L) insurance industry has been characterized by high business risk, intense competition, and relatively stringent regulation from governments. However, in relation to the extensive research on the risk-taking behavior of banks, there are a small number of studies investigating the risk-taking behavior of insurers. Cummins and Sommer (1996) find that insurers are likely to restrain their risk taking due to policyholders' demand for safety when the protection of guarantee funds is incomplete. However, Lee et al. (1997) find that the existence of state guarantee funds generally encourages insurers to increase risk taking. Baranoff and Sager (2002, 2003) find that capital level and asset risk of life insurers are positively correlated. They therefore conclude that life insurers overall were operated in a finite-risk paradigm to curb transaction costs. There have been, nevertheless, relatively few studies devoted to understanding the extent to which capital regulation can influence insurers' risk taking. To fill the gap, the objective of the study is to examine the association between regulatory pressure and the risk-taking behavior for P-L insurers.

Similar to what is observed in the banking industry, the risk-based capital (RBC) (6) standard adopted in 1994 forms the principal basis for the current solvency regulation in the insurance industry. RBC regulation is mainly designed to target problem (i.e., poorly capitalized) insurers and take necessary corrective actions when RBC ratios of these insurers fall below the required standard. These disciplinary interventions will affect insurers' operation and incur high regulatory costs to the insurers. (7) When these regulatory costs exceed the benefits from risk taking, the insurers would stop taking risky investments. This line of reasoning motivates us to contend that there can exist a threshold at which insurers alter their risk taking in response to the cost-benefit trade-off.

Our intention to examine whether risk taking and regulatory pressure are nonlinearly correlated in the insurance industry is also motivated by two related studies in the banking literature. Shrieves and Dahl (1992) document that the extent to which RBC regulation can influence a bank's risk taking depends on how far the bank's capital level is away from the RBC standard. Jacques and Nigro (1997) find that RBC regulation is effective in controlling banks' risk taking and particularly that weak and healthier banks react differently to RBC regulatory pressure, suggesting that the capital level and the magnitude of regulatory pressure need not correlate in a monotonic (i.e., linear) manner. Although Shrieves and Dahl (1992) and Jacques and Nigro (1997) both recognize the nonlinear influence of regulatory pressure on banks' risk taking in their studies, they deal with the issue by simply using a dummy variable to partition banks into two groups depending on whether their capital ratios fall below the explicit RBC standard. These two studies further find that the influence of RBC regulatory pressure on the risk-taking behavior of the two bank groups appears asymmetric. Extending the studies of Shrieves and Dahl (1992) and Jacques and Nigro (1997), we argue that there is an implicit threshold of regulatory pressure at which an insurer will change its risk taking and the threshold, if any exists, does not necessarily equate to the explicit RBC standard.

The study thus proposes a new research scheme to address the nonlinear relationship between regulatory pressure and risk taking for insurers. We first develop an option pricing model to illustrate how regulatory pressure is related to insurers' risk taking. To the best of our knowledge, this makes our study the first to apply the option pricing model to the insurance field in such a context. The theoretical model shows that insurers will alter their risk taking at a regulatory-pressure threshold where savings in the expected regulatory cost are equal to reductions in the put option value expropriated from guarantee funds. The existence of such an implicit threshold in fact predicts a nonlinear relationship between regulatory pressure and risk taking. At the second stage of the study, we use a simultaneous threshold-regression model (8) to examine whether insurers' risk-taking behavior is nonlinear and to evaluate the effectiveness of capital regulation in terms of whether the trigger point of the RBC standard for disciplinary action is adequate and how costly the disciplinary action is to insurers. Our study is also the first to apply simultaneous threshold regression to this area of research. We believe that the threshold-regression model is best suited for the purpose of the study. It can ascertain whether a threshold effect exists and provide an estimate of the implicit threshold when it really exists. Furthermore, it helps divide insurers into groups based upon the estimated thresholds of RBC regulatory pressure and hence allows us to examine if insurers' risk-taking behavior is distinct across these insurer groups. Overall, the framework of our threshold model allows us to assess the effectiveness of RBC regulation from three aspects: (1) the sensitivity of insurers' risk taking to RBC regulatory pressure across the insurer groups, (2) the capital buffer management strategy across the insurer groups, and (3) the speed of insurers' adjustment to capital and risk in response to impending regulatory pressure across the insurer groups.

The regression result shows that there is a threshold effect of regulatory pressure, indicating that insurers' risk-taking behavior is distinct across insurer groups characterized by different degrees of RBC regulatory pressure. The result further shows that the effectiveness of RBC regulation is at best weakly supported. This, coupled with the finding that our lower threshold of 346 percent is much larger than the RBC requirement of 200 percent, suggests that insurers seem to be aware of their proximity to the regulatory intervention but do not fully respond to the impending regulatory pressure. This implies that RBC regulatory interventions are not costly enough to exert due influence on weak insurers and/or that regulatory interventions to regulate poorly capitalized insurers are too late. The result overall hence has a profound policy implication that RBC regulation can become more effective either when the explicit RBC standard is raised to an adequate level to exert timely pressure on insurers or when the regulatory cost is high enough to insurers once receiving regulatory interventions, or both.

The remainder of the article is organized as follows. First, we develop a path-dependent contingent claim model for evaluating the expected regulatory cost, and then use this model to study the nonlinear relationship between regulatory pressure and insurers' risk taking. Second, an empirical study based upon the simultaneous threshold model is conducted to verify this nonlinear risk-taking behavior. Finally, we evaluate and discuss the adequacy of current insurance RBC regulation, and summarize our findings.

OPTION PRICING MODEL WITH REGULATORY COST

Financial option pricing models have long been employed to evaluate the equity and debt values of financial institutions (e.g., Merton, 1977; Furlong and Keeley, 1989; Doherty and Garven, 1986; Cummins, 1988a, 1988b; Cummins et al., 1999; Ibragimov et al., 2010). (9) If policyholders of an insurance company are uninformed about the firm's behavior or if they are informed but protected by nonrisk-based guarantee funds, option pricing models predict that the insurer has an incentive to raise risk to expropriate the value of the put option (Cummins, 1988a; Lee et al., 1997). This prediction is, however, contradictory to many empirical findings concerning insurers' risk taking. A possible explanation is that these option models are oversimplified since they do not consider some important internal and external costs. Cummins and Sommer (1996) use an option model to show that insurance prices are inversely related to the value of the put option, which is determined by the amount of capital and risk an insurer undertakes. The authors therefore predict that insurers will control their risk through balancing their capital ratios against asset risks if the protection from guarantee funds is incomplete. Overall, their empirical results support the finite-risk paradigm regarding insurers' risk taking.

Marcus (1984) was the first to utilize an option model to analyze the risk-taking behavior of banks when charter value is taken into account. Marcus shows that banks will stop taking risk when the expected benefit received from the put option is lower than the expected loss of charter value. If insurers place a cap on their insolvency put option to protect their franchise value, their capital ratios and portfolio/product risks will be positively correlated. (10) The option pricing framework hence suggests that the incentive for an insurer to increase its risk depends on the associated costs and benefits from its capital and asset/product risk decisions. (11)

Aside from the franchise/charter value, regulatory pressure may also substantially affect insurers' risk taking. The incentive to protect the firm's franchise value can be viewed as an internal "self-disciplining" force, while the pressure from solvency regulation can be regarded as an external force. Our option pricing model will take into account effects of the two important forces. It is noteworthy that concerns for regulatory pressure will persist as long as insurers' solvency condition falls below a given level. This suggests that the option pricing model to evaluate the expected regulatory costs should be essentially path dependent. (12)

The option pricing model developed in this study suggests that whether insurers change their risk taking depends on an implicit threshold, measured in terms of the capital ratio, at which the sum of the expected regulatory cost and the expected loss of franchise value equals the gain from the put option expropriated from guarantee funds. To the best of our knowledge, none of the existing option models (e.g., Merton, 1977, 1978; Sharpe, 1978; Cummins, 1988a, 1988b; Furlong and Keeley, 1989) are designed for this purpose.

The Model

Following the common setting of prior research (e.g., Cummins, 1988a, 1988b), we assume that market values of an insurer's aggregated assets ([A.sub.t]) and liabilities ([L.sub.t]) follow geometric Brownian motions under probability measure P:

dA = [[mu].sub.A]Adt + [[sigma].sub.A][AdW.sup.P.sub.A], (1a)

dL = [[mu].sub.L]Ldt + [[sigma].sub.L][LdW.sup.P.sub.L]. (1b)

Note that ([[mu].sub.A],[[sigma].sup.2.sub.A]) and ([[mu].sub.L], [[sigma].sup.2.sub.L]) are pairs of expected instantaneous changes and time-invariant volatilities of assets and liabilities, respectively, and the instantaneous correlation coefficient between assets and liabilities is d [W.sup.P.sub.A]d[W.sup.P.sub.L] = [[rho].sub.AL] d t, where d [W.sup.P] is the associated Wiener stochastic term under probability measure P. (13) Applying Girsanov's theorem, we transform processes (1a) and (1b) into probability measure Q representing a risk-neutral world and a market without arbitrage opportunities. According to Ito's Lemma, the asset-liability stochastic process under probability measure Q can be expressed as follows: (14)

d ln(x) = ([r.sub.h] - [[mu].sub.L] - 1/2 [[sigma].sup.2]) dt + [sigma] d [W.sup.Q.sub.x], (1c)

where x = A/L denotes the asset-liability ratio, [r.sub.h] - [[mu].sub.L] denotes the expected drift of the ratio and is also known as the numeraire of the martingale measure, [r.sub.h] is the expected drift of the hedged security (H), and [sigma] = [square root of [[sigma].sup.2.sub.A] + [[sigma].sup.2.sub.L] - 2[[rho].sub.AL] [[sigma].sub.A] [[sigma].sub.L]] denotes the volatility of the ratio.

Expected Regulatory Cost

RBC regulation in the insurance industry entails a number of potential actions that may incur regulatory cost, whose amount depends on an insurer's capital-ratio level. In particular, the regulatory pressure from RBC will persist as long as the insurer's capital ratio stays below the regulatory requirement ([phi]). (15) The total regulatory cost (RC) is fundamentally proportional to the accumulated sum of the instantaneous asset-liability ratio (x) short of the regulatory boundary ([phi]) during the time horizon. This line of reasoning prompts us to use the average asset-liability ratio minus the regulatory boundary ([phi]) and multiply it by a constant to approximate the aggregate regulatory cost, denoted as RC = -[r.sub.c]([bar.x] - [phi]) where x is the average asset-liability ratio before auditing, and [r.sub.c] can be viewed as a coefficient measuring unit regulatory cost or regulatory pressure. Because the aggregate regulatory cost (RC) is path dependent on the asset-liability ratio process, it can be derived in a way similar to how an Asian (put) option is priced.

To facilitate the derivation of an explicit solution for the expected regulatory cost, let the average asset-liability ratio be represented by its geometric mean, [[bar.x].sub.G](t) = exp(1/t [[integral].sup.t.sub.0] ln(x(u))d u). The analytical form of the expected regulatory cost is then given by (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Expected Equity Value

Many seminal papers (e.g., Merton, 1978; Cummins, 1988a) have shown that the equity value of a bank (or an insurer) can be expressed as a call option, in which the firm's assets constitute the underlying asset and the value of the firm's liabilities represents the (stochastic) exercise price of the option. The contingent-claim model anticipates that insurers can expropriate the debt value of policyholders by increasing risk taking after selling insurance policies. Marcus (1984) contends that a bank's risk taking may behave differently when the effect of franchise (charter) value is considered, because insolvency can lead to a deadweight loss of franchise value. (17) Hence, an insurer's equity value (TV), taking into account both contingent-claim value and franchise value, can be expressed as follows:

TV(A, L, [[sigma].sup.2], T) = max {[A.sub.T] + [K.sub.T] - [L.sub.T], 0},

where [K.sub.T] is the franchise value at terminal time T. We assume that the franchise value of an insurer is closely related to its market power, and thus is roughly proportional to the size of its liabilities, that is, [K.sub.T] = [kappa] [L.sub.T] where [kappa] is the parameter governing the insurer's market power. (18) However, the main objective of this study is to explore how regulatory pressure (or cost), in addition to franchise value, can affect the expected equity value and thus influence the insurer's risk-taking behavior. The payoff to shareholders at audit time T, taking into account regulatory cost (RC), can be written as

TV = max {[A.sub.T] + [K.sub.T] - [L.sub.T] - E [RC], 0}. (3)

Let the contingent equity value TV (A, L, [[sigma].sup.2], t) = LG (x, t) be expressed in terms of unit insurance liability. Under probability measure Q, and subject to the asset-liability process of Equation (1), the expected payoff to equity holders can be shown to be (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Using the equivalent martingale measure, Equation (4) then can be rewritten as (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, the fair equity value in the presence of both franchise value and regulatory cost is given by (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Capital Decisions in the Presence of Regulatory Pressure

The risk-taking behavior of insurers can be studied through the comparative static of the equity value with respect to the asset value and asset risk, respectively. Because the effect of franchise (charter) value on risk taking has been explored by Marcus (1984), this study focuses only on the effect of regulatory pressure (cost). For simplicity, we set the terminal time to be the unit (i.e., T = 1) and subtract one from the partial derivative to represent the net gain from equity when adding one unit of capital: (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

The sensitivity of shareholders' expected payoff to capital comes from changes of three components: put option value expropriated from guarantee funds, franchise value, and regulatory cost. If the effect of franchise value is ignored, then Equation (7) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Solving the first-order condition, we obtain the critical value of regulatory cost ([r.sup.*.sub.c,CAP]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

This critical value is equivalent to the (implicit) threshold at which insurers change their risk-taking behavior. It will be empirically estimated by the threshold-regression method later in this article. The comparative static in Equation (8) indicates that, ceteris paribus, shareholders are willing (reluctant) to increase their capital if unit regulatory cost is greater (smaller) than this threshold (i.e., [r.sub.c] > [<][r.sup.*.sub.c,CAP]), because the reduction of regulatory cost exceeds (falls short) of the loss from contingent claims (i.e., [partial derivative](TV)/[partial derivative]A - 1 > [<] 0). Thus, excessive risk taking can be restrained only when sufficiently great regulatory pressure (i.e., [r.sub.c] > [r.sup.sub.c,CAP]) is imposed. In addition, the marginal change in regulatory cost is a decreasing function of capital (i.e., [MC'.sub.RC] = [[partial derivative].sup.2](RC)/[partial derivative][A.sup.2] < 0), and converges to zero when the insurer's capital ratio is extremely large (i.e., [MC'.sub.RC] = [[partial derivative].sup.2](RC)/[partial derivative][A.sup.2] [right arrow] 0 as A [right arrow] [infinity]). This indicates that capital decisions become relatively inelastic with respect to regulatory pressure if insurers are well capitalized, which can be seen from Figure 1.

[FIGURE 1 OMITTED]

The critical value (i.e., the threshold) of the asset-liability ratio [(A/L).sup.*] can be derived in the same manner:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

It is easy to verify that shareholders will receive negative returns when they inject more capital (i.e., [partial derivative](TV)/[partial derivative]A - 1 < 0) if the firm's asset-liability ratio exceeds the critical value (i.e., A/L > [(A/L).sup.*]). In such a circumstance, a well-capitalized insurer will be reluctant to raise more capital even if regulatory cost is high, because the resulting expected regulatory cost is low. Figure 2 displays how the expected equity value varies according to the extent of regulatory pressure and the level of the asset-liability ratio.

[FIGURE 2 OMITTED]

Risk Decisions in the Presence of Regulatory Pressure

To examine the effect of regulatory pressure on risk decisions, we consider the comparative static of the expected payoff to shareholders with respect to firm risk:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

The fact that the first terms on the right-hand side of Equation (11) are of positive value suggests that the values received from the subsidy of guarantee funds become larger if insurers raise their risk taking. At the same time, the negative second term of Equation (11) indicates that greater risk taking also depletes franchise value. The negative third and fourth terms instead indicate that greater risk taking increases the expected regulatory cost. If the impact of franchise value is ignored, then

Equation (11) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

The critical value of regulatory cost ([r.sup.*.sub.c]) then can be found by setting Equation (12) equal to zero at T = 1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Figure 3 shows how the expected regulatory cost is influenced by the insurer's risk taking. The marginal change in regulatory cost with respect to the change in risk (i.e., [partial derivative](RC)/[partial derivative][sigma]) approaches zero when the asset-liability ratio is either extremely high or extremely low. It implies that capital regulation becomes relatively ineffective when an insurer's capitalization is either very weak or strong. However, the difference in effectiveness of capital regulation between various levels of capitalization becomes trivial when the expected regulatory cost is relatively low. In addition, Figure 3 reveals that [partial derivative](RC)/[partial derivative][sigma] becomes more sensitive to the change in the asset-liability ratio when this ratio is getting closer to the critical value (i.e., [(A/L).sup.*]).

Figure 4 illustrates how equity value varies according to the change in risk under various extents of regulatory pressure and levels of the asset-liability ratio. We find that additional risk taking reduces equity value (i.e., [partial derivative](TV)/[partial derivative][sigma] < 0) only when the regulatory cost is greater than the critical value ([r.sub.c] > [r.sup.*.sub.c,RISK]). (23)

Effects of Stochastic Volatility on Insurers' Options

Compelling empirical evidence has revealed that returns of many market assets are not normally distributed. In fact, these distributions are either characterized by fat tails or high skewness, or both. This conflicts with underlying assumptions of the standard Black-Scholes (BS) model and may lead to significant pricing biases if the basic BS formula is applied. Although nonnormal asset returns may also be caused by price jumps, one major reason for the nonnormal returns is that the volatility of the asset price process is stochastic, conflicting with the constant-volatility assumption of the standard BS model. On the other hand, empirical evidence has shown that

implied volatilities calculated from away-from-the-money European call and put options are greater than those derived from near-the-money options. This so-called "implied volatility smile" in fact casts doubt on the plausibility of the constant-volatility assumption. As shown in Heston (1993), the extent to which the BS model misprices European options when volatility is stochastic depends on a number of factors including the moneyness of asset prices (i.e., they are in the money or out of the money), the correlation between asset returns and volatility innovations, and the volatility of volatility. (24) The pricing bias due to stochastic volatility (SV) may be summarized as follows. First, when asset returns and volatility innovations are uncorrelated, the constant-volatility BS model tends to underprice away-from-the-money options, including both out-of-the-money options and in-the-money options. Second, if asset returns and volatility innovations are negatively correlated, (25) the return distribution can be left-skewed because extremely negative returns are more probable than extremely positive returns. Hence, the BS model tends to underprice in-the-money call options but overprice out-of-the-money options. This suggests that the incentive of solvent insurers (i.e., those with in-the-money options) to exploit the put option value of the guarantee fund would be actually stronger than that predicted by the standard BS model. The opposite is true when the correlation is positive.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

EMPIRICAL MODELS

The option pricing model developed in previous sections of the study predicts that insurers with various levels of capital and risk will react to regulatory pressure differently. In particular, an insurer changes risk taking only when its expected regulatory cost exceeds a critical value (i.e., the implicit threshold). If RBC regulation is effective, insurers will change their risk taking when they come under mounting regulatory pressure. The threshold at which insurers actually alter their risk taking is, however, unobservable. In particular, it is not necessarily equal to the explicit RBC standard (200 percent). This implies that using the RBC standard as the threshold to depict insurers' risk-taking behavior could be misleading. In this section, we conduct an empirical study to test whether an insurer's risk taking is nonlinearly related to regulatory pressure it faces. Because the actual regulatory cost imposed on insurers is unobservable in the real world, the threshold-regression approach enables us to find the implicit regulatory threshold. To the best of our knowledge, the present article is the first to apply such a research scheme to study the nonlinear risk-taking behavior of financial institutions.

The threshold-regression approach, originally developed by Hansen (1999), divides samples optimally into a number of regimes according to the estimated threshold values. Given that most previous studies (e.g., Shrieves and Dahl, 1992; Cummins and Sommer, 1996; Baranoff and Sager, 2002, 2003; Baranoff et. al., 2007) treat capital and risk decisions of financial institutions as simultaneous, we extend our threshold regression to be a simultaneous model, which was recently developed by Caner and Hansen (2004). (26) Using threshold regression not only allows us to depict nonlinear risk-taking behavior but, more importantly, enables us to evaluate when and to what extent RBC regulation can influence insurers' risk taking. For example, consider a case where the implicit threshold estimated by the regression model is higher than the explicit regulatory threshold (i.e., the 200 percent RBC standard). Under such a circumstance, if insurers with RBC ratios lower than the implicit threshold are found to reduce risk taking, it can be argued that regulatory intervention is relatively costly to the insurers. In contrast, if the insurers increase risk taking, then this seems to indicate that regulatory intervention appears to be nonbinding and/or late.

Just as Shrieves and Dahl (1992), we test our hypotheses by investigating the relationship between changes in risk and changes in capital rather than the relationship between levels of capital and of risk. Following Baranoff and Sager (2002, 2003), we consider both asset risk and product risk born by a P-L insurer. Consequently, there are three alternative measures of insurers' risk change (i.e., [DELTA][CAPR.sub.it], [DELTA][ARISK.sub.it], and [DELTA][[PRISK.sub.it]) and they are assumed to be endogenously determined. Because our primary goal is to examine the impact of capital regulation on insurers' risk taking, the RBC ratio is thus used as not only the proxy of regulatory pressure but also the threshold variable (i.e., [q.sub.it]) in the model. Our empirical simultaneous threshold models (27) are given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

where [X.sub.it], [Y.sub.it], and [Z.sub.it] denote vectors of instrumental variables that can affect insurers' risk taking and have been commonly considered in previous studies, (28) [.sub.[*]] is an indicator function, and y is the threshold value to be estimated.

Hypotheses

As previously noted, the objectives of this study are twofold: (1) to test whether insurers' risk-taking behavior is nonlinear, which is implied by our option pricing model, and (2) to evaluate the effectiveness of capital regulation in terms of whether the trigger point of the RBC standard for disciplinary action is adequate and how costly the disciplinary action is to insurers. Our first goal can be reached through testing whether RBC regulatory pressure has a threshold effect on insurers' risk taking. The existence of a threshold effect indicates that insurers' risk-taking behavior is distinct across insurer regimes characterized by different degrees of RBC regulatory pressure. On the other hand, the effectiveness of current capital regulation can be evaluated by comparing the estimate of the implicit threshold with the explicit RBC standard and examining the risk-taking behavior of insurers particularly in the high-regulatory-pressure regime.

Our theoretical option model predicts that insurers will reduce risk taking only if their RBC ratios fall below the implicit threshold and the expected regulatory cost is sufficiently large. Fortunately, the simultaneous threshold model employed in the study can help estimate the implicit threshold and discern the risk-taking behavior of an insurer. The following discussions illustrate the complementary roles of our theoretical option pricing and empirical threshold regression models in providing new insights into the literature.

Prior insurance studies often implicitly assume that their findings of insurers' risk-taking behavior can be a common rule to all insurers. For example, if RBC regulation were effective, then RBC-ratio changes would be negatively correlated with capital-ratio changes but positively correlated with asset/product-risk changes. Unlike previous studies, the theoretical model proposed in this study predicts that the capital-risk relationship should vary across regulatory-pressure regimes. If RBC regulation is effective, only poorly capitalized insurers (i.e., insurers with relatively low RBC ratios) will reduce their risk taking. On the other hand, our theoretical model also predicts that the risk-taking behavior of well-capitalized insurers (i.e., those with relatively high RBC ratios) will be relatively insensitive to regulatory pressure. This is because the probability of those insurers to provoke disciplinary action from regulators is quite low. Therefore, if empirical results show that the implicit threshold is higher than the explicit RBC threshold and that insurers in the high-regulatory-pressure (i.e., low-RBC) regime reduce their risk taking, our theoretical option pricing model can help conclude that RBC regulation is considered to be highly costly to insurers so that they take early action to control their risk even when their RBC ratios are still above the explicit regulatory threshold. In contrast, if insurers in the high-regulatory-pressure regime are found to increase their risk taking, our theoretical option pricing model suggests that either the associated regulatory cost is not large enough to prevent insurers facing impending regulatory interventions from taking more risk or regulatory interventions come too late to constrain these insurers from taking more risk, or both. Under such a scenario, insurers may reduce their capital levels and/or increase their asset/product risks to expropriate the put option value from guarantee funds. Finally, suppose that the estimated RBC threshold is lower than the explicit RBC threshold. The result that insurers in the high-regulatory-pressure regime are found to reduce risk taking may indicate effective but too early RBC interventions. The result that insurers in the high-regulatory-pressure regime increase risk taking, in contrast, may indicate early but ineffective RBC regulation.

Recall that insurers can change their capital buffers (29) (i.e., safety levels) by adjusting either capital ratios or asset/product risks, or both. If an insurer chooses to maintain its capital buffer (i.e., to cap its insolvency put option), as predicted in Cummins and Sommers (1996), it can be achieved by increasing (or decrease) capital ratios and asset/product risks at the same time. Under such a circumstance, capital-ratio changes and asset/product-risk changes are likely to be positively correlated. A negative relationship between capital-ratio changes and asset/product-risk changes, however, provides no clue about the dynamics of an insurer's capital buffer. It may indicate that an insurer chooses to rebuild its capital buffer (i.e., to decrease the value of the insolvency put) by raising capital ratios and/or reducing asset/product risks, which would be most likely when the insurer senses impending regulatory pressure and high insolvency cost. It, on the other hand, may suggest that an insurer downsizes its capital buffer (i.e., increases the value of the insolvency put) through lowering its capital ratios and/or taking more asset/product risks especially when the insurer considers the benefit of expropriating the option value from guarantee funds is greater than the sum

of the regulatory cost and the loss of the franchise value. This line of reasoning makes us anticipate that under an effective RBC regulatory system, poorly capitalized insurers are prone to adopt a capital-buffer rebuilding strategy. Well-capitalized insurers, in contrast, have more leeway in developing their capital-buffer strategy regardless of the effectiveness of RBC regulation, since their RBC ratios are well above the required standard.

Last but not least, because our empirical model is carried out under a partial-adjustment framework, the rate of risk adjustment thus can be measured by the lagged term of capital and risk variables. If RBC regulation is effective, we anticipate that weak insurers should adjust their capital ratios and risks faster than healthy insurers.

EMPIRICAL RESULTS

DATA

Data sources of the study include the NAIC P-L insurance annual report, the NAIC newsletter, and the A.M. Best Key Rating Guide. The sample period spans 2000 through 2009. We exclude observations with incorrect data values. (30) After excluding these observations, the final sample size totals 15,470 firm-year observations.

Table 1 reports descriptive statistics for important characteristics of our sample insurers. Panel A shows that: (1) the principal risk faced by P-L insurers came from insurance policies sold (i.e., product risks); (2) on average, capital ratios and product risks experienced a decreasing trend whereas asset risks showed an increasing pattern over the sample period; and (3) most P-L insurers were well capitalized (31) but their RBC ratios exhibit considerable variation.

Panel A of Table 2 provides summary statistics for firm characteristics of insurers. Of the sample insurers, 71 percent are stock firms and 67 percent are affiliated organizations. Nearly 73 percent of insurers use the independent agency as their major distribution channel and 35 percent of insurers operate businesses nationwide. Finally, Panel B of Table 2 shows the frequency distribution of RBC ratios. Overall, the distribution of the RBC ratio is found to be highly skewed to the right, with most insurers (i.e., about 76 percent) having RBC ratios within the range from 200 percent to 1,500 percent.

Simultaneous Threshold Regression

To implement the threshold regression with instrumental variables, it is necessary first to identify the thresholds of the reduced-form equations. (32) Panel A of Table 3 shows that two thresholds might exist. To test the significance of the threshold effect, Hansen (1999) proposes a powerful F-statistic, which can be calculated using a bootstrapping procedure. (33) The test results indicate that threshold effects of one threshold and two thresholds are both significant at the 1 percent level, and the best goodness of fit occurs when two thresholds are considered. The two threshold estimates are used to obtain the expected values of the endogenous variables (through instrumental-variable estimation), which are then plugged into structural equations to derive the threshold values in the second stage. Panel B of Table 3 reports the estimated thresholds in the second stage. A bootstrapped F-test is further used to examine whether the threshold effect exists in the structural equation model, showing that the two structural threshold estimates are statistically significant at the 1 percent level. The result confirms what is predicted by our theoretical model: that the risk-taking behavior of P-L insurers is nonlinearly related to RBC regulatory pressure.

To verify the effectiveness of the threshold regression, we would like to know if our threshold model outperforms other models such as those splitting the sample based upon some salient cutoff points including the median and mean. We use the error sum of squares to measure the performance of a model. Table 4 reports the result. As expected, the model with no threshold has the largest error sum of squares. The two threshold regressions outperform the two OLS regressions with mean/median as the threshold. Finally, we find that the error sum of squares declines substantially when the second threshold is added into the threshold regression. Drawing on the finding that it is optimal to segregate insurers into three regimes, we refer to those insurers with RBC ratios below the lower threshold (i.e., 3.46) as the regime with "high regulatory pressure" and those with RBC ratios exceeding the upper threshold (i.e., 15.09) as the regime with "low regulatory pressure." (34)

Table 5 presents descriptive statistics for insurers of various groups: the whole sample and the three regime samples stratified by the two RBC thresholds. For the group of all insurers, capital ratios and product risks on average decline during the sample period, whereas asset risks increase merely by a relatively small amount. These observations suggest that insurers as a whole did not inflate their risks during our sample period.

Like what is observed in the whole sample case, the by-regime statistics indicate that capital ratios and product risks exhibit a relatively clear upward/downward trend while asset risks show no obvious trend during the sample period. A comparison of changes in capital ratios, changes in product risks, and changes in asset risks across the three regime groups, nevertheless, reveals insightful results. Although changes in asset risks are uniformly trivial across the three insurer groups, changes in capital ratios and product risks show stimulating patterns. Insurers in the low-RBC regime (i.e., those facing high regulatory pressure) experience the largest decline in capital ratios and the largest increase in product risks. In contrast, insurers in the high-RBC regime (i.e., those facing low regulatory pressure) experience the largest increase in capital ratios and the largest decline in product risks. This suggests that insurers facing high regulatory pressure are more likely to increase risk taking in comparison to other insurers. The Scheffe test further confirms that with the exception of the change in asset risks, all the variables reported in Table 5 have unequal means across the three regime groups. This implies that the threshold regression has successfully partitioned our sample insurers into three regimes characterized by distinct risk-taking behavior. We find that the average RBC ratio for insurers of low RBC ratios is about 259 percent, slightly above the RBC regulatory standard. Besides, insurers in the low-RBC regime are found to have the largest loss ratio, the largest leverage ratio, and the lowest ROA, whereas insurers in the high-RBC regime are characterized by the lowest loss ratio, the largest reinsurance ratio, the lowest leverage, and the largest ROA.

Joint Tests of Coefficient Equality Across Regimes

To verify the nonlinearity of insurers' risk-taking behavior, we first carry out coefficient-equality tests for each endogenous/exogenous variable across the three RBC regimes. This test is performed based upon Equations (14)-(16) one at a time and three at a time, respectively. Table 6 shows that the hypothesis of equal coefficients across the three regimes cannot be rejected for all variables in the asset-risk regression model. However, the number of variables with unequal coefficients across the three regimes is 3 when the capital regression is considered alone, 9 when the product-risk regression is considered alone, and 14 when the three regressions are considered jointly. The result that the joint test based upon the three equations is highly significant suggests that the influence of RBC regulation on insurers' risk taking is significantly different across the three insurer groups. With respect to the coefficient equality test of the major variables in the study, a similar pattern prevails. Regarding results based upon an individual equation, the coefficient of the capital-ratio change is significantly different across regimes only in the product-risk equation, the coefficient of the product-risk change is significantly different only in the capital equation, but the coefficient of the asset-risk change is not significantly different. The coefficient equality hypothesis is, however, rejected for each of the three variables when the three regression models are considered jointly. Overall, the result reported in Table 6 supports the existence of a threshold effect of regulatory pressure on insurers' risk taking, as predicted by our theoretical option pricing model.

Sensitivity of Risk Taking to RBC Regulatory Pressure

We now consider the relationship between insurers' risk taking and their pressure from RBC regulation. If RBC regulation is effective, then the coefficient of the RBC ratio in the capital equation will be significantly negative and will be significantly positive in the asset- and product-risk equations. To compare with previous studies, we first run the three regressions simultaneously with no thresholds. The result reported in the "whole sample" column across the three panels of Table 7 shows that insurers to a lesser extent control their product risks but concurrently lower capital ratios and raise asset risks to a greater extent in response to the impending regulatory pressure. This suggests that RBC regulation is, on average, weakly effective in controlling insurers' risk taking.

The aforementioned prediction about the sign of the RBC ratio in the three equations is most likely to be confirmed for poorly capitalized (i.e., the low-RBC regime) insurers. We therefore estimate the three equations simultaneously based upon the threshold-regression approach. The results are also shown in the regime columns of Table 7. We find that, in general, insurers' risk taking is not quite sensitive to RBC regulatory pressure across the three insurer groups. The RBC regulatory pressure is evidently nonbinding with respect to risk taking for the middle-RBC and the highRBC regime insurers. However, the relationship between risk taking and regulatory pressure becomes relatively complicated for the low-RBC regime insurers. In particular, the coefficient of the RBC ratio is insignificant in the capital equation, but it turns significantly negative in the asset-risk equation and significantly positive in the product-risk equation. This suggests that RBC regulation seems to influence weak insurers to raise asset risks and to lower their product risks while it plays no role in their capital decision. Therefore, there is at best weak, if any exists, evidence that RBC regulation can effectively curb insurers' risk taking. Overall, our findings support the prediction of our theoretical model that insurer's risk reduction is not quite sensitive to changes of regulatory pressure if regulatory pressure (cost) is relatively low (see, e.g., Figure 3). Furthermore, the result, coupled with the finding that our lower threshold of 346 percent, is much larger than the RBC requirement of 200 percent, (35) suggests that insurers appear to be aware of their proximity to the regulatory threshold but do not fully respond to the impending regulatory pressure. The result also implies that RBC regulation will become more effective either when the explicit RBC standard can be raised to an adequate level to exert timely pressure on insurers or when the regulatory cost can be high enough to insurers once receiving regulatory interventions, or both.

Capital Buffer Strategy

With regard to insurers' capital buffer strategy, the result of Table 7 reveals that capital changes and asset-risk changes are positively correlated whereas capital changes and product-risk changes are negatively correlated for the whole sample. This suggests that insurers, on average, adopt the capital-buffer-maintaining strategy to manage their capital and asset risk. The negative relationship between capital changes and product-risk changes, on the other hand, can have two entirely distinct implications, as argued in previous sections of the study. We are, however, inclined to interpret it as evidence supporting that insurers tend to adopt the capital-buffer-rebuilding strategy rather than the capital-buffer-downsizing strategy to deal with their capital and product risk. (36) These results to some extent support the finding of Cummins and Sommer (1996) and Baranoff and Sager (2002,2003) that insurers choose to operate in a finite-risk framework. Based upon a sample of life insurers, Baranoff and Sager (2002) contended that the negative relationship between capital and product risk might be associated with the effect of guarantee fund protection.

As argued in previous sections, regulatory pressure can have an impact on insurers' capital buffer strategy. In particular, if RBC regulation is effective, insurers subject to the greatest regulatory pressure (i.e., insurers in the low-RBC regime) should adopt the capital-buffer-rebuilding strategy to reduce their insolvency put option, thus yielding a negative relationship between capital changes and risk changes. Comparing capital buffer strategy of insurers under various degrees of regulatory pressure would, therefore, provide meaningful insights in that regard. Our simultaneous threshold-regression model can do exactly that. As shown in Table 7, within the insurer group of the low-RBC regime, the relationship between capital changes and asset-risk changes is significantly positive while the relationship between capital changes and product-risk changes is significantly negative. This suggests that poorly capitalized insurers tend to maintain their capital buffers when managing their capital and assets. In contrast, they adopt the capital-buffer-downsizing strategy to manage their capital and products. The reason behind this argument is that insurers in the low-RBC regime are found to decrease their capital but increase their product risks over the sample period, as reported in Table 5. The result in general casts doubt on the effectiveness of RBC regulation.

With regard to the result for well-capitalized insurers, Table 7 shows a positive association between capital changes and asset-risk changes but a negative association between capital changes and product-risk changes. (37) In relation to the result for insurers in the low-RBC regime, relationships between capital changes and asset-/ product-risk changes for well-capitalized insurers are, however, weak. For insurers in the low-RBC regime, the positive relationship between capital changes and asset-risk changes prevails in both the capital and the asset-risk equations, and the negative relationship between capital changes and product-risk changes holds in both the capital and the product-risk equations. However, the positive relationship exists only in the asset-risk equation and the negative relationship holds only in the capital equation for insurers in the high-RBC regime. The relatively weak result for well-capitalized insurers is consistent with our contention that these insurers have more discretion as to their capital buffer strategy, further confirmed by the observation that the result of the middle-RBC insurer group bears a closer resemblance to that of the low-RBC insurer group than that of the high-RBC insurer group.

In sum, the negative relationship between capital changes and product-risk changes along with the positive relationship between capital changes and asset-risk changes suggests that insurers, on average, manage to increase product risks rather than asset risks when they seek to assume more risk. This also suggests that insurers tend to develop a capital-buffer-maintaining strategy to manage their capital and asset-risk, and a capital-buffer-rebuilding strategy rather than a capital-buffer-downsizing strategy to deal with their capital and product-risk. This finding is consistent with those of Cummins and Sommer (1996) and Baranoff and Sager (2002, 2003). The result based upon the threshold-regression model, in contrast, reveals that

poorly capitalized insurers adopt a capital-buffer-maintaining strategy to manage capital and assets but a capital-buffer-downsizing strategy to manage capital and product-risk, indicating that RBC regulation cannot effectively prevent weak insurers from taking excessive risk.

Speed of Insurers' Adjustment to Risk Taking

The third perspective from which we assess the effectiveness of RBC regulation is the speed of insurers' adjustment to their capital and risk in response to impending regulatory pressure, made possible through our empirical design. Since insurers' risk-taking behavior is modeled in a partial adjustment framework, the lagged value of capital-ratio, asset-risk, and product-risk variables can be used to reflect the speed of firms' adjustment to capital, asset-risk, and product-risk, respectively, similar to what was done in Shrieves and Dahl (1992). If RBC regulation is effective, the speed of adjustment to capital and risk for insurers in the low-RBC regime is expected to be much faster than that for insurers in the high-RBC regime. As shown in Table 7, we find that the adjustment rate of capital level for poorly capitalized insurers is only slightly faster than that for well-capitalized insurers (i.e., 8.83 percent vs. 6.56 percent). Similarly, we find that insurers in the low-RBC regime also adjust their asset risks and product risks at a speed not much faster than insurers in the high-RBC regimes (i.e., 23.36 percent vs. 15.22 percent for asset risks and 23.87 percent vs. 18.59 percent for product risks). In sum, the result at best lends only weak support to the effectiveness of RBC regulation.

CONCLUSIONS

The study examines effects of RBC regulatory pressure on the risk-taking behavior of P-L insurers. It is the first study that develops a path-dependent option pricing model to evaluate the expected regulatory cost, which provides a theoretical foundation for insurers' nonlinear risk-taking behavior with respect to regulatory pressure. The model suggests that insurers will change their risk taking at an implicit RBC threshold where the expected regulatory cost equals the put option value expropriated from guarantee funds. To verify the prediction of our theoretical model, an empirical study is carried out to assess the nonlinear risk-taking behavior and test the effectiveness of RBC regulation. We find that the simultaneous threshold-regression model is best suited for our purposes. It is therefore used to demonstrate a threshold effect and obtain two RBC-ratio thresholds, consistent with the prediction of our theoretical model that insurers' risk-taking behavior is nonlinearly related to RBC regulatory pressure.

The threshold model then divides insurers into three regimes based upon the two estimated thresholds and examines insurers' risk-taking behavior for each insurer group. Through investigating the risk-taking behavior of insurers in the three regimes, particularly the low-RBC regime, the framework of our threshold model allows to assess the effectiveness of RBC regulation from three perspectives. The result based upon the sensitivity of insurers' risk taking to RBC regulatory pressure shows that RBC regulation to some extent influences poorly capitalized insurers to raise asset risks and to lower product risks while it plays no role in the capital decision. There is hence at best weak evidence that RBC regulation can effectively curb insurers' risk taking. With regard to the capital buffer management strategy, the threshold-regression result shows that poorly capitalized insurers tend to adopt a capital-buffer-maintaining strategy to manage capital and assets but a capital-buffer-downsizing strategy to manage capital and products, indicating that RBC regulation cannot prevent weak insurers from taking excessive risk. The third perspective, provided by our threshold model, from which we can assess the effectiveness of RBC regulation is the speed of insurers' adjustment to capital and risk in response to impending regulatory pressure. The result indicates that the speeds at which insurers facing high regulatory pressure adjust their capital, asset risks, and product risks is only slightly higher than those for well-capitalized insurers, providing only weak evidence for the effectiveness of RBC regulation.

In sum, the empirical evidence shows that the effectiveness of RBC regulation is at best weakly supported. This result, coupled with the finding that our lower threshold of 346 percent is much larger than the RBC requirement of 200 percent, suggests that insurers seem to be aware of their proximity to the regulatory threshold but do not fully respond to the impending regulatory pressure. We are therefore inclined to conclude that current RBC regulatory cost is not high enough to exert due influence on weak insurers and/or that regulatory interventions to regulate poorly capitalized insurers appear to be late. This result hence has a profound policy implication that RBC regulation will become more effective either when the explicit RBC standard can be raised to an adequate level to exert timely pressure on insurers or when the regulatory cost can be high enough to insurers once receiving regulatory interventions, or both.

Overall, the study offers an innovative approach to investigating insurers' risk-taking behavior and testing the adequacy of capital regulation. We believe that the methodologies developed in this article may also be used to investigate the risk-taking behavior of other financial institutions such as banks and life insurance companies with only minor modifications.

DOI: 10.1111/j.1539-6975.2012.01505.x

REFERENCES

Aggarwal, R., and K. T. Jacques, 2001, The Impact of FDICIA and Prompt Corrective Action on Bank Capital and Risk: Estimates Using a Simultaneous Equations Model, Journal of Banking and Finance, 25(6): 1139-1160.

Bai, 1997, Estimating Multiple Breaks One at a Time, Econometric Theory, 13:315-352.

Baranoff, E. G., S. Papadopoulos, and T. W. Sager, 2007, Capital and Risk Revisited: A Structure Equation Model Approach for Life Insurers, Journal of Risk and Insurance, 74(3): 653-681.

Baranoff, E. G., and T. W. Sager, 2002, The Relations Among Asset Risk, Product Risk, and Capital in the Life Insurance Industry, Journal of Banking and Finance, 26(6): 1181-1197.

Baranoff, E. G., and T. W. Sager, 2003, The Relations Among Organizational and Distribution Forms and Capital and Asset Risk Structures in the Life Insurance Industry, Journal of Risk and Insurance, 70(3): 375-400.

Caner, M., and B. E. Flansen, 2004, Instrumental Variable Estimation of a Threshold Model, Econometric Theory, 20: 813-843.

Chen, S. K., B. Lin, H. R. Oppenheimer, and T. Yu, 2007, The Impact of Firm Franchise Value on Asset Allocation: An Analysis of Property-Liability Insurance Firms, Risk Management and Insurance Review, 11(1): 159-180.

Cheng, J., and M. A. Weiss, 2012, The Role of RBC, Hurricane Exposure, Bond Portfolio Duration and Macroeconomic and Industry Wide Factors in Property-Liability Insolvency Prediction, Journal of Risk and Insurance, 79(3): 723-750.

Cummins, J. D., 1988a, Risk-Based Premium for Insurance Guaranty Fund, Journal of Finance, 43(4): 823-839.

Cummins, J. D., 1988b, Capital Structure and Fair Profits in Property-Liability Insurance, in: J. D. Cummins and R. A. Derrig, eds., Financial Insurance Solvency Theory (Boston: Kluwer Academic Publishers), pp. 295-307.

Cummins, J. D., M. F. Grace, and R. D. Phillips, 1999, Regulatory Solvency Prediction in Property-Liability Insurance: Risk-Based Capital, Audit Ratios, and Cash Flow Simulation, Journal of Risk and Insurance, 66(3): 417-458.

Cummins, J. D., and D. W. Sommer, 1996, Capital and Risk in Property-Liability Insurance Markets, Journal of Banking and Finance, 20(6): 1069-1092.

Doherty, N. A., and J. R. Garven, 1986, Price Regulation in Property-Liability Insurance: A Contingent-Claims Approach, Journal of Finance, 41,1031-1050.

Ediz, T., I. Michael, and W. Perraudin, 1998, The Impact of Capital Requirements on U.K. Bank Behaviour, Federal Reserve Bank of New York Economic Policy Review, 4(3): 15-22.

Fischer, S., 1978, Call Option Pricing When the Exercise Price Is Uncertain and the Valuation of Index Bonds, Journal of Finance, 33(1): 169-176.

Furlong, F. T., and M. C. Keeley, 1989, Capital Regulation and Bank Risk Taking: A Note, Journal of Banking and Finance, 13(4): 883-891.

Hansen, B. E., 1999, Threshold Effects in Non-Dynamic Panels: Estimation, Testing and Inference, Journal of Econometrics, 93(2): 345-368.

Heston, S., 1993, A Closed-form Solution for Options with Stochastic Volatility, With Applications to Bond and Currency Options, Review of Financial Studies, 6: 327-343.

Ibragimov, R., D. Jaffee, and J. Walden, 2010, Pricing and Capital Allocation for Multiline Insurance firms, Journal of Risk and Insurance, 77(3): 551-578.

Jacques, K. and P. Nigro, 1997, Risk-Based Capital, Portfolio Risk, and Bank Capital: A Simultaneous Equation Approach, Journal of Economics and Business, 49(6): 533-547.

Lai, Y.-H., 2005, An Integrated Analysis of Regulatory Pressure and Franchise Value on the Risk-Taking Behavior of a P/L Insurance Firm, Unpublished Dissertation, National Chung Cheng University, Taiwan.

Lee, S.-J., D. Mayers, and C. Smith, Jr., 1997, Guarantee Funds and Risk-taking: Evidence from the Insurance Industry, Journal of Financial Economics, 44: 3-24.

Leverty, J. T., and M. F. Grace, 2012, Dupes or Incompetents? An Examination of Management's Impact on Firm Distress, Journal of Risk and Insurance, 79(3): 751-783.

Marcus, A. J., 1984, Deregulation and Bank Financial Policy, Journal of Banking and Finance, 8(4): 557-565.

Merton, R. C., 1977, An Analytic Derivation of the Cost of Deposit Insurance and Loan Guarantees, Journal of Banking and Finance, 1(1): 3-11.

Merton, R. C., 1978, On the Cost of Deposit Insurance When There Are Surveillance Costs, Journal of Business, 51(3): 439-453.

Park, S., 1997, Risk-Taking Behavior of Banks Under Regulation, Journal of Banking and Finance, 21: 491-507.

Repullo, R., 2004, Capital Requirements, Market Power, and Risk Taking in Banking, Journal of Financial Intermediation, 13(2): 156-182.

Rime, B., 2001, Capital Requirements and Bank Behaviour: Empirical Evidence for Switzerland, Journal of Banking and Finance, 25(4): 798-805.

Sharpe, W. F., 1978, Bank Capital Adequacy, Deposit Insurance and Security Values, Journal of Financial and Quantitative Analysis, 13(4): 701-718.

Shrieves, R. E., and D. Dahl, 1992, The Relationship Between Risk and Capital in Commercial Banks, Journal of Banking and Finance, 16(4): 439-457.

So, J., and J. Z. Wei, 2004, Deposit Insurance and Forbearance Under Moral Hazard, Journal of Risk and Insurance, 71(4): 707-735.

Staking, K. B., and D. F. Babbel, 1995, The Relation Between Capital Structure, Interest Rate Sensitivity and Market Value in the Property-Liability Insurance Industry, Journal of Risk and Insurance, 62(4): 690-718.

Williamson, O. E., 1988, Corporate Finance and Corporate Governance, Journal of Finance, 18(3): 567-591.

Wen-Chang Lin is an associate professor in the Department of Finance, National Chung Cheng University, Taiwan. Yi-Hsun Lai is an assistant professor in the Department of Finance, National Yunlin University of Science and Technology, Taiwan. Michael R. Powers is a professor in the Department of Finance, Tsinghua University, and on leave from the Department of Risk Management and Insurance, Temple University. Wen-Chang Lin can be contacted via e-mail: finwcl@ccu.edu.tw.

(1) Transaction-cost economics (see, e.g., Williamson, 1988) posits that financial institutions will limit their product risks to forestall paying high financing costs, as the cost of debt financing will increase if they sell risky products.

(2) The agency theory contends that the separation of management and ownership can lead to lower risk taking by managers as they do not share residual profits with owners.

(3) Bankruptcy costs are those such as the cost of liquidating the firm and the loss of franchise value.

(4) Regulatory costs have received relatively considerable research attention in the banking literature (e.g., Shrieves and Dahl, 1992; Jacques and Nigro, 1997; Park, 1997; Ediz et al., 1998; Aggarwal and Jacques, 2001; Rime, 2001).

(5) For example, Repullo (2004) contends that risk taking of banks may be associated with the degree of market competition. When markets are overcompetitive or appear to be monopolistic, firms may increase their risk taking. In contrast, under a moderate competitive market, firms may lower their risk taking to protect their franchise value.

(6) RBC regulation aims to prevent costly insolvencies by mandating insurers to hold a satisfactory level of capital. The required capital is determined by a formula-based risk measure encompassing asset risks, underwriting risks, credit risks, and miscellaneous risks.

(7) Recently, Cheng and Weiss (2012) find that the ability of the RBC ratio in predicting insurer insolvencies is not consistent over time.

(8) Threshold regression was originally developed by Hansen (1999) and simultaneous threshold regression was recently developed by Caner and Hansen (2004).

(9) Merton (1977) shows that banks with limited liability can inflate shareholder value by reducing capital and/or raising risk in the contingent claim framework. Furlong and Keely (1989) suggest that flat capital requirements can reduce, but not eliminate, moral-hazard incentives. This is because a bank can set its amount of capital against credit risk, which does not depend on the bank's asset quality.

(10) Staking and Babbel (1995) and Chen et al. (2007) conduct empirical studies to test whether the "self-discipline effect" dominates the "moral-hazard effect" for insurers with considerable franchise values.

(11) Recently, Leverty and Grace (2012) find that managerial ability can also affect the cost of insurer failure.

(12) The path-dependent option is an option whose payoff depends on the whole path of the underlying asset. If the exercise price of a path dependent option is set to be the sum or the mean of past prices of the underlying asset, it is known as an Asian option. The option models developed in this article are modified from Lai (2005).

(13) For detailed assumptions underlying the asset and liability processes, see Cummins (1988a).

(14) Because insurance liabilities are typically considered to be nontradable in the market, it is necessary to construct a tradable hedging security (H) that is highly correlated with these liabilities. See Fisher (1978) and Cummins (1988a, 1988b) for more details. The proof is similar to that of Cummins (1988a, 1988b), and is available from the authors.

(15) Current RBC regulation requires that insurers' capital ratios should exceed the RBC ratio of 200 percent. Note that the boundary ([phi]) can be determined based upon the RBC formula.

(16) The closed-form solution exists only when [[bar.x].sub.G] is a geometric mean; otherwise, numerical methods are needed. The proof is similar to the derivation of the price of an Asian option and is available from the authors.

(17) The franchise value of an insurer is closely related to its market power and/or economic scale. The value may also be generated from policy renewals, underwriting experience, etc.

(18) Many previous studies posit that franchise (charter) values of insurers (banks) are closely related to their economic scales. For example, So and Wei (2004) derive the charter values of banks in terms of savings and monopoly rents.

(19) The proof is available from the authors.

(20) The proof is available from the authors.

(21) The last two terms of Equation (6) are associated with the expected discounted regulatory cost.

(22) This suggests that if the sign is positive, then, ceteris paribus, the equity value increases by more than one dollar when shareholders add one dollar to the firm's assets.

(23) In the numerical example (Figure 4), a value of [r.sub.c] greater than 0.25 may result in a decrease in equity value when the insurer increases risk taking and its asset-liability ratio is around 2.

(24) Although the option models (with stochastic exercise prices) in our study are to some extent different from the standard BS model, the influence of SV on these models is similar to that discussed in Heston (1993) because they are essentially European-type options.

(25) Return distributions of many market indexes, such as S&P 500 and FTSE 100, are found to be negatively skewed.

(26) This may involve several instrumental (exogenous) variables and lead to more complicated computations than the simple threshold model. For more details about threshold models with endogenous variables, see Caner and Hansen (2004).

(27) As pointed out by Hansen (1999), multiple thresholds can exist in a model.

(28) For detailed descriptions of exogenous variables, see the notes of Table 1.

(29) The capital buffer is conceptually analogous to the insolvency put option discussed in Merton (1977) and Cummins (1988a, 1988b).

(30) We exclude observations whose capital ratios or total admitted assets are less than or equal to zero. In addition, we delete observations with unreasonable values. For example, those with capital ratios greater than one, or with product risks or asset risks equal to zero are excluded.

(31) The average RBC ratio for all insurers is 1,075 percent.

(32) We calculate the likelihood ratio to test the existence of multiple thresholds (see Caner and Hansen, 2004).

(33) For the details of how the F-statistic is derived, see Hansen (1999) and Bai (1997).

(34) The low regime includes those insurers with RBC ratios smaller than 3.46, insurers belonging to the middle regime with RBC ratios ranging from 3.46 to 15.09, and those in the high regime with RBC ratios larger than 15.09.

(35) The associated test statistic for [H.sub.0]: [gamma] = [[gamma].sub.0] is L[R.sub.1]([gamma]) = [S([gamma]) - S ([[gamma].sub.0])]/[[delta].sup.2], where S([gamma]) is the error sum of squares under threshold [gamma], and [[delta].sup.2] is the residual variance.

(36) This is because product risk is on average decreasing for all insurers during the sample period.

(37) In contrast to the negative relationship between capital changes and product-risk changes for poorly capitalized insurers, the negative relationship here implies that well-capitalized insurers tend to adopt a capital-buffer-rebuilding strategy to manage capital and products, which is supported by the finding of Table 5 that insurers in the high-RBC regime experience an increase in capital ratios and a decline in product risks.

TABLE 1 Summary Statistics of Risk-Taking Variables and Instrumental Variables Variables Mean S.D. Max Min Panel A: Summary Statistics of Risk-Taking Variables [DELTA]CAPR -0.0020 0.0684 0.7847 -0.5608 [DELTA]ARISK 0.0001 0.0071 0.0805 -0.0614 [DELTA]PRISK -0.0067 0.1223 3.3908 -2.1372 [CAPR.sub.t-1] 0.4295 0.1739 0.9996 0.0349 [ARISK.sub.t-1] 0.0591 0.0139 0.1394 0.0001 [PRISK.sub.t-1] 0.4715 0.2532 3.9528 0.0000 Panel B: Summary Statistics of Instrumental Variables RBC 10.7539 11.4678 99.9362 0.0000 In (TA) 18.3537 1.9047 25.3757 12.6640 ROA 0.0253 0.0565 0.8805 -0.7153 ULOSS 0.6741 0.3627 8.6522 -7.6866 LEVERAGE 0.5725 0.1725 0.9571 0.0000 REINSURANCE 0.3580 0.2921 2.6451 -2.9367 TABLE 2 Summary Statistics of Discrete Instrumental Variables and the Distribution of RBC Ratios Percentage Variables No. Obs. (%) Panel A: Summary Statistics of Discrete Instrumental Variables Organizational form Stock firm 11,024 71.26 Mutual firm 3,059 19.77 Affiliated or unaffiliated firm Single firm 5,133 33.18 Grouped firm 10,337 66.82 National or regional firm National firm 5,463 35.31 Regional firm 10,007 64.69 Distribution system Independent agent 11,238 72.64 Nonindependent agent 4,232 27.36 Licensed in New York NY 1,178 7.61 Non-NY 14,292 92.39 Panel B: The Distribution of the RBC Ratio RBC < 2 255 1.65 2 [less than or equal to] RBC < 5 3,721 24.05 5 [less than or equal to] RBC < 10 644 36.88 10 [less than or equal to] RBC < 15 2,539 16.41 15 [less than or equal to] RBC < 20 1,012 6.54 20 [less than or equal to] RBC < 25 48 2.75 25 [less than or equal to] RBC < 30 276 1.78 30 [less than or equal to] RBC < 35 183 1.18 35 [less than or equal to] RBC < 40 16 0.92 40 [less than or equal to] RBC < 45 95 0.61 45 [less than or equal to] RBC 389 2.51 Notes: Major organizational forms of P-L insurers are classified as stock, mutual, and others, respectively. Affiliation indicates whether insurers belong to insurance groups. National is used to identify whether an insurer is considered to operate nationwide (i.e., if it is licensed in more than 16 states) or regionally. The distribution system of an insurer is used to identify whether the insurer uses the independent agency as its primarily distribution channel. NY denotes that the insurer is licensed in New York State. The sample period is 2000-2009 and the sample size totals 15,470 firm-year observations TABLE 3 Results of Threshold-Effect Tests Multiple Threshold Threshold- p-Value Thresholds Value Effect ([??]) F-Tests Panel A: Reduced-Form Equation First threshold 3.462 568.80 *** 0.000 Second threshold 8.424 184.11 *** 0.000 Panel B: Structural-Form Equation First threshold 3.462 1224.990 *** 0.000 Second threshold 15.087 167.549 *** 0.000 Notes: The table shows the estimated threshold values and the results of threshold/effect tests for the reduced/form and structural/form equations, respectively. The threshold/effect F/test statistic used to test whether the regression coefficients in regimes i and j are equal (i.e., [H.sub.0]: [[beta].sub.i] = [[beta].sub.j]) is defined as F = [[S.sub.0] -- S([??])]/ [[??].sup.2], where [S.sub.0] and S([gamma]) are the error sums of squares without threshold and with threshold [gamma], respectively, and [[??].sup.2] is the residual variance. TABLE 4 Model Performance Under Various Threshold Values Model Error Sum of Squares OLS regression without threshold 239.6092 OLS regression with mean as the threshold 235.9405 OLS regression with median as the threshold 235.8419 Threshold regression with one threshold 217.3710 Threshold regression with two thresholds 130.8331 Notes: The table shows the effectiveness of threshold regression in terms of model performance, as measured by the error sum of squares. Thresholds of the competing models are determined according to a variety of criteria. TABLE 5 Summary Statistics of Variables for Various RBC Regimes Statistics by RBC Regimes (Two Thresholds) Whole Sample Low RBC Variables Mean S.D. Mean S.D. [DELTA]CAPR -0.0020 0.0684 -0.0224 0.0847 [DELTA]ARISK 0.0001 0.0071 0.0000 0.0086 [DELTA]PRISK -0.0067 0.1223 0.0012 0.1941 RBC 10.7539 11.4678 2.5887 0.6568 REINSURANCE 0.3580 0.2921 0.3658 0.3072 ULOSS 0.6741 0.3627 0.7730 0.4978 NATIONAL 0.3531 0.4780 0.3187 0.4661 ROA 0.0253 0.0565 -0.0086 0.0797 GROUP 0.6682 0.4709 0.5324 0.4991 ln(TA) 18.3537 1.9047 18.2432 2.1772 LEVERAGE 0.5725 0.1725 0.7130 0.1414 LA 0.7325 0.4426 0.7310 0.4436 STOCK 0.7126 0.4526 0.7329 0.4426 MUTUAL 0.1977 0.3983 0.1020 0.3028 NY 0.0761 0.2652 0.1128 0.3165 [CAPR.sub.t-1] 0.4295 0.1739 0.3094 0.1403 [DELTA][RISK.sub.t-1] 0.0591 0.0139 0.0529 0.0161 [PRISK.sub.t-1] 0.4715 0.2532 0.5452 0.3127 Statistics by RBC Regimes (Two Thresholds) Middle RBC High RBC Variables Mean S.D. Mean S.D. [DELTA]CAPR -0.0017 0.0627 0.0099 0.0771 [DELTA]ARISK 0.0001 0.0069 -0.0001 0.0073 [DELTA]PRISK -0.0060 0.1127 -0.0148 0.0996 RBC 7.7051 2.9267 29.6733 17.9219 REINSURANCE 0.3458 0.2727 0.4073 0.3537 ULOSS 0.6743 0.3038 0.6082 0.4682 NATIONAL 0.3816 0.4858 0.2493 0.4327 ROA 0.0262 0.0494 0.0434 0.0585 GROUP 0.6890 0.4629 0.6651 0.4721 ln(TA) 18.5665 1.8621 17.4806 1.6253 LEVERAGE 0.5976 0.1355 0.3687 0.1715 LA 0.7427 0.4372 0.6880 0.4634 STOCK 0.7052 0.4560 0.7321 0.4429 MUTUAL 0.2132 0.4096 0.1921 0.3940 NY 0.0708 0.2565 0.0757 0.2646 [CAPR.sub.t-1] 0.4041 0.1425 0.6214 0.1794 [DELTA][RISK.sub.t-1] 0.0598 0.0131 0.0603 0.0149 [PRISK.sub.t-1] 0.4956 0.2371 0.3160 0.2161 Statistics by RBC Regimes (Two Thresholds) Scheffe Test Variables F-Statistic [DELTA]CAPR 113.9152 *** [DELTA]ARISK 1.019293 [DELTA]PRISK 9.154971 *** RBC 9533.492 *** REINSURANCE 46.84996 *** ULOSS 105.1285 *** NATIONAL 84.99311 *** ROA 455.9699 *** GROUP 81.1199 *** ln(TA) 354.9913 *** LEVERAGE 3566.669 *** LA 15.69327 *** STOCK 5.546249 *** MUTUAL 57.18776 *** NY 18.26753 *** [CAPR.sub.t-1] 2810.504 *** [DELTA][RISK.sub.t-1] 194.0712 *** [PRISK.sub.t-1] 649.9563 *** Notes: The table shows the averages and standard deviations of the endogenous variables and instrumental variables for the whole sample and the three insurer regimes. Sample insurers are partitioned into three regimes based upon the two distinct thresholds. The low regime includes insurers with RBC ratios less than 3.462, insurers of the middle regime have RBC ratios lying between 3.462 and 15.087, and insurers of the high regime have RBC ratios over 15.087. NATIONAL equals 1 if an insurer is licensed in more than 16 states, and 0 otherwise. GROUP equals 1 if an insurer is affiliated with or belongs to an insurer group, and 0 otherwise. IA equals 1 if independent agency is the primary distribution channel, and 0 otherwise. STOCK equals 1 if an insurer is a stock firm, and 0 otherwise. MUTUAL equals 1 if an insurer is a mutual firm, and 0 otherwise. NY equals 1 if an insurer is licensed in New York State, and 0 otherwise. The sample period is 2000-2009 and the sample size totals 15,470 firm-year observations. TABLE 6 Coefficient-Equality Tests Under Various Regimes Chi-Square Capital Asset-Risk Product-Risk All Statistic Equation Equation Equation Equations [DELTA]CAPR 0.117 101.271 *** 1796.365 *** [DELTA]ARISK 0.548 5.176 41.471 *** [DELTA]PRISK 17.012 *** 0.026 44.219 *** REINSURANCE 1.979 0.035 1.181 3.196 NATIONAL 0.080 0.022 0.047 0.149 GROUP 2.071 0.003 8.789 ** 10.865 In (TA) 9.426 ** 0.023 28.850 *** 38.335 *** IA 1.094 0.020 35.685 *** 36.805 *** STOCK 1.336 0.035 40.116 *** 41.495 *** MUTUAL 3.918 0.004 4.506 8.430 NY 2.035 0.004 0.762 2.802 RBC 2.760 0.014 29.634 *** 32.419 *** ROA 37.686 *** 122.910 *** ULOSS 0.001 1.791 17.689 ** LEVERAGE 0.054 15.902 *** 288.386 *** LEVERAGE2 0.077 12.616 *** 472.418 *** [CAPR.sub.t-1] 1.512 107.798 *** [ARISK.sub.t-1] 0.011 126.016 *** [PRISK.sub.t-1] 33.754 *** 471.837 *** Notes: The table shows the results of coefficient/equality tests for each endogenous/exogenous variable across regimes. The null hypothesis is [H.sub.0]: [[beta].sub.ij1] = [[beta].sub.ij2] = [[beta].sub.ij3], where denotes the coefficient of variable i in equation j under regime k (k = 1, 2, 3). TABLE 7 Risk-Taking Behavior Based Upon Simultaneous Threshold Regression Whole Sample Low RBC Variables Coeff. t-Value Coeff. t-Value Panel A: Capital Equation Intercept -1.7442 -1.43 0.2508 0.08 [DELTA]ARISK 8.9451 27.15 *** 5.1900 7.43 *** [DELTA]PRISK -0.7611 -37.61 *** -0.1355 -3.29 *** REINSURANCE -1.7920 -5.55 *** 1.1899 1.45 NATIONAL 0.3788 1.78 * 0.7045 1.20 GROUP 0.4663 2.04 ** -0.4751 -0.89 ln(TA) -0.0269 -0.44 -0.0257 -0.17 IA 1.2828 6.21 *** -0.1164 -0.23 STOCK 1.1280 3.39 *** 1.5758 2.50 ** MUTUAL 1.7802 4.90 *** 2.9593 4.19 *** NY 0.0290 0.09 0.3625 0.64 RBC 0.0619 6.43 *** -0.3599 -1.05 ROA 3.6344 4.83 *** 71.9417 11.13 *** [CAPR.sub.t-1] -2.7320 -3.92 *** -8.8286 -4.03' ** Panel B: Asset-Risk Equation Intercept 0.5082 7.21 *** -0.2116 -0.83 [DELTA]CAPR 0.0493 53.55 *** 0.0123 3.84 *** [DELTA]PRISK 0.0312 21.82 *** 0.0028 0.86 REINSURANCE -0.1138 -5.60 *** -0.5073 -6.08 *** NATIONAL -0.0376 -2.88 *** -0.1655 -2.97 *** GROUP 0.0053 0.38 0.1136 2.14 ** ln(TA) 0.0297 7.54 *** 0.0820 5.73 *** IA -0.0644 -5.11 *** -0.0539 -1.05 STOCK -0.0605 -2.98 *** -0.1252 -1.97 ** MUTUAL -0.0794 -3.56 *** -0.0781 -1.06 NY 0.0192 0.95 0.0562 0.95 RBC -0.0022 -3.76 *** -0.0658 -1.69 * ULOSS -0.0006 -0.05 -0.0083 -0.20 LEVERAGE 0.1835 1.46 2.6347 4.00 *** LEVERAGE (2) -0.4561 -3.92 *** -2.6780 -4.64 *** [ARISK.sub.t-1] -13.1565 -31.00 *** -23.3639 -13.74 *** Panel C: Product-Risk Equation Intercept 0.8093 0.68 50.6056 5.62 *** A CAPR -0.9585 -77.05 *** -0.4569 -4.45 *** AARISK 8.0440 22.02 *** 4.1529 2.75 *** REINSURANCE -3.6729 -10.75 *** -10.3079 -6.09 *** NATIONAL 0.3161 1.41 -1.4685 -1.27 GROUP 0.6980 2.91 *** 1.7677 1.62 In (TA) -0.1880 -2.86 *** -1.0699 -2.61 *** IA 1.2696 5.87 *** 1.6267 1.50 Panel C: Product-Risk Equation STOCK 1.6830 4.80 ** 4.6080 3.41 *** MUTUAL 2.4994 6.45 *** 5.2853 2.87 *** NY -0.1744 -0.50 0.0239 0.02 RBC 0.0179 1.80 * 1.8427 2.39 ** ULOSS -0.6305 -3.84 *** -3.6354 -3.45 *** LEVERAGE 8.5581 4.76 *** -43.7487 -1.68 * LEVERAGE2 -5.8282 -3.77 *** 15.7617 0.80 PRISKt-x -5.7585 -15.49 *** -23.8658 -6.21 *** System 0.4633 0.3161 weighted [R.sup.2] Middle RBC High RBC Variables Coeff. t-Value Coeff. f-Value Panel A: Capital Equation Intercept -3.3552 -2.06 ** 14.8344 1.89 * [DELTA]ARISK 4.0489 7.08 *** 0.9338 0.38 [DELTA]PRISK -0.7758 -14.98 *** -0.7470 -3.60 *** REINSURANCE -1.0490 -2.22 ** -0.2542 -0.15 NATIONAL 0.3656 1.56 2.6832 2.03 ** GROUP -0.2708 -0.92 1.6556 1.32 ln(TA) 0.0759 1.01 -0.5896 -1.73 * IA 0.2717 1.05 1.2882 1.20 STOCK 0.8657 1.88 * -2.6830 -1.27 MUTUAL 1.2092 2.62" -1.1590 -0.54 NY 0.2026 0.59 0.9622 0.64 RBC -0.0853 -1.09 -0.0311 -0.83 ROA 24.0765 6.91 *** 28.2769 2.42 ** [CAPR.sub.t-1] 1.5941 1.23 -6.5609 -1.64 * Panel B: Asset-Risk Equation Intercept 0.3517 1.92 * 0.4701 1.20 [DELTA]CAPR 0.0255 9.99 *** 0.0163 2.58 * [DELTA]PRISK 0.0073 2.72 *** 0.0024 0.26 REINSURANCE -0.2656 --7.41 *** -0.1112 -1.21 NATIONAL -0.0362 -1.93 * -0.0297 -0.40 GROUP 0.0288 1.33 -0.0036 -0.05 ln(TA) 0.0468 7.00 *** 0.0207 1.06 IA -0.0209 -1.10 -0.0693 -1.22 STOCK -0.0485 -1.52 0.2108 1.85 * MUTUAL -0.0480 -1.48 0.1647 1.46 NY -0.0354 -1.29 0.1888 2.00 * RBC 0.0073 1.17 -0.0007 -0.23 ULOSS -0.0477 -1.20 0.0002 0.00 LEVERAGE 0.9094 1.58 1.0205 1.46 LEVERAGE (2) -0.9062 -1.89 * -2.5018 -2.72 *** [ARISK.sub.t-1] -21.2656 -20.33 *** 15.2240 -5.81 *** Panel C: Product-Risk Equation Intercept 9.9287 3.17 *** -7.8339 -1.40 A CAPR -0.5542 -13.17 *** -0.1018 -0.96 AARISK 2.2834 3.34 *** -5.4365 -2.50 ** REINSURANCE -5.8277 -8.69 *** -5.2755 -3.20 *** NATIONAL 0.1917 0.67 -0.7100 -0.59 GROUP 1.6077 3.97 *** -0.2064 -0.18 In (TA) -0.7136 -6.04 *** 0.8348 2.82 *** IA 0.3207 1.02 0.5890 0.62 Panel C: Product-Risk Equation STOCK 2.5056 4.47 *** -3.3437 -1.78 * MUTUAL 3.8398 6.41 *** -2.1604 -1.14 NY -1.3972 -3.39 *** 3.3060 2.53 ** RBC -0.3489 -3.50 *** -0.0321 -0.70 ULOSS -1.7252 -3.09 *** 0.9332 1.21 LEVERAGE 48.9696 3.56 *** 16.3759 1.12 LEVERAGE2 -42.9380 -3.81 *** -18.4815 -1.11 PRISKt-x -17.4377 -10.48 *** -18.5891 -3.71 *** System 0.5236 0.4773 weighted [R.sup.2] Notes: The table reports results based upon simultaneous regression with threshold and without threshold, where capital changes, asset-risk changes, and product-risk changes are treated as endogenous variables and other determinants are considered exogenous. The sample insurers are divided optimally into three regimes. The low regime includes the most poorly capitalized insurers (i.e., those subject to the greatest regulatory pressure), and the high regime includes the best-capitalized insurers.

Printer friendly Cite/link Email Feedback | |

Author: | Lin, Wen-Chang; Lai, Yi-Hsun; Powers, Michael R. |
---|---|

Publication: | Journal of Risk and Insurance |

Geographic Code: | 1USA |

Date: | Jun 1, 2014 |

Words: | 13426 |

Previous Article: | What policy features determine life insurance lapse? An analysis of the German market. |

Next Article: | Scenario analysis in the measurement of operational risk capital: a change of measure approach. |

Topics: |